\(\int \frac {(a+b x+c x^2)^{5/4}}{d+e x} \, dx\) [738]

Optimal result

Integrand size = 22, antiderivative size = 936 \[ \int \frac {\left (a+b x+c x^2\right )^{5/4}}{d+e x} \, dx=\frac {\left (12 c^2 d^2+b^2 e^2-2 c e (7 b d-6 a e)-2 c e (2 c d-b e) x\right ) \sqrt [4]{a+b x+c x^2}}{6 c e^3}+\frac {2 \left (a+b x+c x^2\right )^{5/4}}{5 e}-\frac {\left (-b^2+4 a c\right )^{3/4} \left (c d^2-b d e+a e^2\right )^{5/4} \left (-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4} \arctan \left (\frac {\sqrt [4]{-b^2+4 a c} \sqrt {e} \sqrt [4]{1-\frac {(b+2 c x)^2}{b^2-4 a c}}}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{c d^2-b d e+a e^2}}\right )}{c^{3/4} e^{7/2} \left (a+b x+c x^2\right )^{3/4}}-\frac {\left (-b^2+4 a c\right )^{3/4} \left (c d^2-b d e+a e^2\right )^{5/4} \left (-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{-b^2+4 a c} \sqrt {e} \sqrt [4]{1-\frac {(b+2 c x)^2}{b^2-4 a c}}}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{c d^2-b d e+a e^2}}\right )}{c^{3/4} e^{7/2} \left (a+b x+c x^2\right )^{3/4}}-\frac {\sqrt {-b^2+4 a c} (2 c d-b e) \left (12 c^2 d^2-b^2 e^2-4 c e (3 b d-4 a e)\right ) \left (-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right ),2\right )}{6 \sqrt {2} c^2 e^4 \left (a+b x+c x^2\right )^{3/4}}-\frac {\left (b^2-4 a c\right ) (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \sqrt {\frac {(b+2 c x)^2}{b^2-4 a c}} \left (-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4} \operatorname {EllipticPi}\left (-\frac {\sqrt {-b^2+4 a c} e}{2 \sqrt {c} \sqrt {c d^2-b d e+a e^2}},\arcsin \left (\sqrt [4]{1-\frac {(b+2 c x)^2}{b^2-4 a c}}\right ),-1\right )}{\sqrt {2} c e^4 (b+2 c x) \left (a+b x+c x^2\right )^{3/4}}-\frac {\left (b^2-4 a c\right ) (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \sqrt {\frac {(b+2 c x)^2}{b^2-4 a c}} \left (-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4} \operatorname {EllipticPi}\left (\frac {\sqrt {-b^2+4 a c} e}{2 \sqrt {c} \sqrt {c d^2-b d e+a e^2}},\arcsin \left (\sqrt [4]{1-\frac {(b+2 c x)^2}{b^2-4 a c}}\right ),-1\right )}{\sqrt {2} c e^4 (b+2 c x) \left (a+b x+c x^2\right )^{3/4}} \] Output:

1/6*(12*c^2*d^2+b^2*e^2-2*c*e*(-6*a*e+7*b*d)-2*c*e*(-b*e+2*c*d)*x)*(c*x^2+ 
b*x+a)^(1/4)/c/e^3+2/5*(c*x^2+b*x+a)^(5/4)/e-(4*a*c-b^2)^(3/4)*(a*e^2-b*d* 
e+c*d^2)^(5/4)*(-c*(c*x^2+b*x+a)/(-4*a*c+b^2))^(3/4)*arctan(1/2*(4*a*c-b^2 
)^(1/4)*e^(1/2)*(1-(2*c*x+b)^2/(-4*a*c+b^2))^(1/4)*2^(1/2)/c^(1/4)/(a*e^2- 
b*d*e+c*d^2)^(1/4))/c^(3/4)/e^(7/2)/(c*x^2+b*x+a)^(3/4)-(4*a*c-b^2)^(3/4)* 
(a*e^2-b*d*e+c*d^2)^(5/4)*(-c*(c*x^2+b*x+a)/(-4*a*c+b^2))^(3/4)*arctanh(1/ 
2*(4*a*c-b^2)^(1/4)*e^(1/2)*(1-(2*c*x+b)^2/(-4*a*c+b^2))^(1/4)*2^(1/2)/c^( 
1/4)/(a*e^2-b*d*e+c*d^2)^(1/4))/c^(3/4)/e^(7/2)/(c*x^2+b*x+a)^(3/4)-1/12*( 
4*a*c-b^2)^(1/2)*(-b*e+2*c*d)*(12*c^2*d^2-b^2*e^2-4*c*e*(-4*a*e+3*b*d))*(- 
c*(c*x^2+b*x+a)/(-4*a*c+b^2))^(3/4)*InverseJacobiAM(1/2*arctan((2*c*x+b)/( 
4*a*c-b^2)^(1/2)),2^(1/2))*2^(1/2)/c^2/e^4/(c*x^2+b*x+a)^(3/4)-1/2*(-4*a*c 
+b^2)*(-b*e+2*c*d)*(a*e^2-b*d*e+c*d^2)*((2*c*x+b)^2/(-4*a*c+b^2))^(1/2)*(- 
c*(c*x^2+b*x+a)/(-4*a*c+b^2))^(3/4)*EllipticPi((1-(2*c*x+b)^2/(-4*a*c+b^2) 
)^(1/4),-1/2*(4*a*c-b^2)^(1/2)*e/c^(1/2)/(a*e^2-b*d*e+c*d^2)^(1/2),I)*2^(1 
/2)/c/e^4/(2*c*x+b)/(c*x^2+b*x+a)^(3/4)-1/2*(-4*a*c+b^2)*(-b*e+2*c*d)*(a*e 
^2-b*d*e+c*d^2)*((2*c*x+b)^2/(-4*a*c+b^2))^(1/2)*(-c*(c*x^2+b*x+a)/(-4*a*c 
+b^2))^(3/4)*EllipticPi((1-(2*c*x+b)^2/(-4*a*c+b^2))^(1/4),1/2*(4*a*c-b^2) 
^(1/2)*e/c^(1/2)/(a*e^2-b*d*e+c*d^2)^(1/2),I)*2^(1/2)/c/e^4/(2*c*x+b)/(c*x 
^2+b*x+a)^(3/4)
 

Mathematica [A] (warning: unable to verify)

Time = 14.38 (sec) , antiderivative size = 699, normalized size of antiderivative = 0.75 \[ \int \frac {\left (a+b x+c x^2\right )^{5/4}}{d+e x} \, dx=\frac {2 (a+x (b+c x))^{5/4}}{5 e}+\frac {\sqrt [4]{a+x (b+c x)} \left (b^2 e^2+4 c^2 d (3 d-e x)+2 c e (-7 b d+6 a e+b e x)\right )}{6 c e^3}+\frac {\left (\frac {c (a+x (b+c x))}{-b^2+4 a c}\right )^{3/4} \left (-\sqrt {b^2-4 a c} (-2 c d+b e) \left (-12 c^2 d^2+b^2 e^2+4 c e (3 b d-4 a e)\right ) \operatorname {EllipticF}\left (\frac {1}{2} \arcsin \left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right ),2\right )-\frac {6 c \left (-b^2+4 a c\right )^{3/4} \left (c d^2+e (-b d+a e)\right ) \left (\sqrt {2} \sqrt [4]{c} \sqrt {e} \sqrt [4]{c d^2+e (-b d+a e)} (b+2 c x) \left (\arctan \left (\frac {\sqrt [4]{-b^2+4 a c} \sqrt {e} \sqrt [4]{\frac {c (a+x (b+c x))}{-b^2+4 a c}}}{\sqrt [4]{c} \sqrt [4]{c d^2+e (-b d+a e)}}\right )+\text {arctanh}\left (\frac {\sqrt [4]{-b^2+4 a c} \sqrt {e} \sqrt [4]{\frac {c (a+x (b+c x))}{-b^2+4 a c}}}{\sqrt [4]{c} \sqrt [4]{c d^2+e (-b d+a e)}}\right )\right )+\sqrt [4]{-b^2+4 a c} (-2 c d+b e) \sqrt {\frac {(b+2 c x)^2}{b^2-4 a c}} \operatorname {EllipticPi}\left (-\frac {\sqrt {-b^2+4 a c} e}{2 \sqrt {c} \sqrt {c d^2+e (-b d+a e)}},\arcsin \left (\sqrt {2} \sqrt [4]{\frac {c (a+x (b+c x))}{-b^2+4 a c}}\right ),-1\right )+\sqrt [4]{-b^2+4 a c} (-2 c d+b e) \sqrt {\frac {(b+2 c x)^2}{b^2-4 a c}} \operatorname {EllipticPi}\left (\frac {\sqrt {-b^2+4 a c} e}{2 \sqrt {c} \sqrt {c d^2+e (-b d+a e)}},\arcsin \left (\sqrt {2} \sqrt [4]{\frac {c (a+x (b+c x))}{-b^2+4 a c}}\right ),-1\right )\right )}{b+2 c x}\right )}{6 \sqrt {2} c^2 e^4 (a+x (b+c x))^{3/4}} \] Input:

Integrate[(a + b*x + c*x^2)^(5/4)/(d + e*x),x]
 

Output:

(2*(a + x*(b + c*x))^(5/4))/(5*e) + ((a + x*(b + c*x))^(1/4)*(b^2*e^2 + 4* 
c^2*d*(3*d - e*x) + 2*c*e*(-7*b*d + 6*a*e + b*e*x)))/(6*c*e^3) + (((c*(a + 
 x*(b + c*x)))/(-b^2 + 4*a*c))^(3/4)*(-(Sqrt[b^2 - 4*a*c]*(-2*c*d + b*e)*( 
-12*c^2*d^2 + b^2*e^2 + 4*c*e*(3*b*d - 4*a*e))*EllipticF[ArcSin[(b + 2*c*x 
)/Sqrt[b^2 - 4*a*c]]/2, 2]) - (6*c*(-b^2 + 4*a*c)^(3/4)*(c*d^2 + e*(-(b*d) 
 + a*e))*(Sqrt[2]*c^(1/4)*Sqrt[e]*(c*d^2 + e*(-(b*d) + a*e))^(1/4)*(b + 2* 
c*x)*(ArcTan[((-b^2 + 4*a*c)^(1/4)*Sqrt[e]*((c*(a + x*(b + c*x)))/(-b^2 + 
4*a*c))^(1/4))/(c^(1/4)*(c*d^2 + e*(-(b*d) + a*e))^(1/4))] + ArcTanh[((-b^ 
2 + 4*a*c)^(1/4)*Sqrt[e]*((c*(a + x*(b + c*x)))/(-b^2 + 4*a*c))^(1/4))/(c^ 
(1/4)*(c*d^2 + e*(-(b*d) + a*e))^(1/4))]) + (-b^2 + 4*a*c)^(1/4)*(-2*c*d + 
 b*e)*Sqrt[(b + 2*c*x)^2/(b^2 - 4*a*c)]*EllipticPi[-1/2*(Sqrt[-b^2 + 4*a*c 
]*e)/(Sqrt[c]*Sqrt[c*d^2 + e*(-(b*d) + a*e)]), ArcSin[Sqrt[2]*((c*(a + x*( 
b + c*x)))/(-b^2 + 4*a*c))^(1/4)], -1] + (-b^2 + 4*a*c)^(1/4)*(-2*c*d + b* 
e)*Sqrt[(b + 2*c*x)^2/(b^2 - 4*a*c)]*EllipticPi[(Sqrt[-b^2 + 4*a*c]*e)/(2* 
Sqrt[c]*Sqrt[c*d^2 + e*(-(b*d) + a*e)]), ArcSin[Sqrt[2]*((c*(a + x*(b + c* 
x)))/(-b^2 + 4*a*c))^(1/4)], -1]))/(b + 2*c*x)))/(6*Sqrt[2]*c^2*e^4*(a + x 
*(b + c*x))^(3/4))
 

Rubi [A] (warning: unable to verify)

Time = 2.16 (sec) , antiderivative size = 1116, normalized size of antiderivative = 1.19, number of steps used = 22, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.955, Rules used = {1162, 1231, 27, 1269, 1094, 761, 1174, 1173, 25, 504, 312, 118, 353, 73, 756, 218, 221, 925, 27, 1537, 412}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\left (a+b x+c x^2\right )^{5/4}}{d+e x} \, dx\)

\(\Big \downarrow \) 1162

\(\displaystyle \frac {2 \left (a+b x+c x^2\right )^{5/4}}{5 e}-\frac {\int \frac {(b d-2 a e+(2 c d-b e) x) \sqrt [4]{c x^2+b x+a}}{d+e x}dx}{2 e}\)

\(\Big \downarrow \) 1231

\(\displaystyle \frac {2 \left (a+b x+c x^2\right )^{5/4}}{5 e}-\frac {-\frac {\int \frac {6 c e (b d-2 a e)^2-d (2 c d-b e) \left (-e b^2+6 c d b-8 a c e\right )-(2 c d-b e) \left (12 c^2 d^2-b^2 e^2-4 c e (3 b d-4 a e)\right ) x}{4 (d+e x) \left (c x^2+b x+a\right )^{3/4}}dx}{3 c e^2}-\frac {\sqrt [4]{a+b x+c x^2} \left (-2 c e (7 b d-6 a e)+b^2 e^2-2 c e x (2 c d-b e)+12 c^2 d^2\right )}{3 c e^2}}{2 e}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 \left (a+b x+c x^2\right )^{5/4}}{5 e}-\frac {-\frac {\int \frac {6 c e (b d-2 a e)^2-d (2 c d-b e) \left (-e b^2+6 c d b-8 a c e\right )-(2 c d-b e) \left (12 c^2 d^2-b^2 e^2-4 c e (3 b d-4 a e)\right ) x}{(d+e x) \left (c x^2+b x+a\right )^{3/4}}dx}{12 c e^2}-\frac {\sqrt [4]{a+b x+c x^2} \left (-2 c e (7 b d-6 a e)+b^2 e^2-2 c e x (2 c d-b e)+12 c^2 d^2\right )}{3 c e^2}}{2 e}\)

\(\Big \downarrow \) 1269

\(\displaystyle \frac {2 \left (a+b x+c x^2\right )^{5/4}}{5 e}-\frac {-\frac {\frac {24 c \left (a e^2-b d e+c d^2\right )^2 \int \frac {1}{(d+e x) \left (c x^2+b x+a\right )^{3/4}}dx}{e}-\frac {(2 c d-b e) \left (-4 c e (3 b d-4 a e)-b^2 e^2+12 c^2 d^2\right ) \int \frac {1}{\left (c x^2+b x+a\right )^{3/4}}dx}{e}}{12 c e^2}-\frac {\sqrt [4]{a+b x+c x^2} \left (-2 c e (7 b d-6 a e)+b^2 e^2-2 c e x (2 c d-b e)+12 c^2 d^2\right )}{3 c e^2}}{2 e}\)

\(\Big \downarrow \) 1094

\(\displaystyle \frac {2 \left (a+b x+c x^2\right )^{5/4}}{5 e}-\frac {-\frac {\frac {24 c \left (a e^2-b d e+c d^2\right )^2 \int \frac {1}{(d+e x) \left (c x^2+b x+a\right )^{3/4}}dx}{e}-\frac {4 \sqrt {(b+2 c x)^2} (2 c d-b e) \left (-4 c e (3 b d-4 a e)-b^2 e^2+12 c^2 d^2\right ) \int \frac {1}{\sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}d\sqrt [4]{c x^2+b x+a}}{e (b+2 c x)}}{12 c e^2}-\frac {\sqrt [4]{a+b x+c x^2} \left (-2 c e (7 b d-6 a e)+b^2 e^2-2 c e x (2 c d-b e)+12 c^2 d^2\right )}{3 c e^2}}{2 e}\)

\(\Big \downarrow \) 761

\(\displaystyle \frac {2 \left (a+b x+c x^2\right )^{5/4}}{5 e}-\frac {-\frac {\frac {24 c \left (a e^2-b d e+c d^2\right )^2 \int \frac {1}{(d+e x) \left (c x^2+b x+a\right )^{3/4}}dx}{e}-\frac {\sqrt {2} \sqrt [4]{b^2-4 a c} \sqrt {(b+2 c x)^2} \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right ) \sqrt {\frac {4 c \left (a+b x+c x^2\right )-4 a c+b^2}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right )^2}} (2 c d-b e) \left (-4 c e (3 b d-4 a e)-b^2 e^2+12 c^2 d^2\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right ),\frac {1}{2}\right )}{\sqrt [4]{c} e (b+2 c x) \sqrt {4 c \left (a+b x+c x^2\right )-4 a c+b^2}}}{12 c e^2}-\frac {\sqrt [4]{a+b x+c x^2} \left (-2 c e (7 b d-6 a e)+b^2 e^2-2 c e x (2 c d-b e)+12 c^2 d^2\right )}{3 c e^2}}{2 e}\)

\(\Big \downarrow \) 1174

\(\displaystyle \frac {2 \left (a+b x+c x^2\right )^{5/4}}{5 e}-\frac {-\frac {\frac {24 c \left (-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4} \left (a e^2-b d e+c d^2\right )^2 \int \frac {1}{(d+e x) \left (-\frac {c^2 x^2}{b^2-4 a c}-\frac {b c x}{b^2-4 a c}-\frac {a c}{b^2-4 a c}\right )^{3/4}}dx}{e \left (a+b x+c x^2\right )^{3/4}}-\frac {\sqrt {2} \sqrt [4]{b^2-4 a c} \sqrt {(b+2 c x)^2} \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right ) \sqrt {\frac {4 c \left (a+b x+c x^2\right )-4 a c+b^2}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right )^2}} (2 c d-b e) \left (-4 c e (3 b d-4 a e)-b^2 e^2+12 c^2 d^2\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right ),\frac {1}{2}\right )}{\sqrt [4]{c} e (b+2 c x) \sqrt {4 c \left (a+b x+c x^2\right )-4 a c+b^2}}}{12 c e^2}-\frac {\sqrt [4]{a+b x+c x^2} \left (-2 c e (7 b d-6 a e)+b^2 e^2-2 c e x (2 c d-b e)+12 c^2 d^2\right )}{3 c e^2}}{2 e}\)

\(\Big \downarrow \) 1173

\(\displaystyle \frac {2 \left (a+b x+c x^2\right )^{5/4}}{5 e}-\frac {-\frac {\frac {48 \sqrt {2} c \left (-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4} \left (a e^2-b d e+c d^2\right )^2 \int -\frac {1}{\left (\frac {c (2 c d-b e)}{b^2-4 a c}-e \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )\right ) \left (1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}\right )^{3/4}}d\left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )}{e \left (a+b x+c x^2\right )^{3/4}}-\frac {\sqrt {2} \sqrt [4]{b^2-4 a c} \sqrt {(b+2 c x)^2} \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right ) \sqrt {\frac {4 c \left (a+b x+c x^2\right )-4 a c+b^2}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right )^2}} (2 c d-b e) \left (-4 c e (3 b d-4 a e)-b^2 e^2+12 c^2 d^2\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right ),\frac {1}{2}\right )}{\sqrt [4]{c} e (b+2 c x) \sqrt {4 c \left (a+b x+c x^2\right )-4 a c+b^2}}}{12 c e^2}-\frac {\sqrt [4]{a+b x+c x^2} \left (-2 c e (7 b d-6 a e)+b^2 e^2-2 c e x (2 c d-b e)+12 c^2 d^2\right )}{3 c e^2}}{2 e}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {2 \left (a+b x+c x^2\right )^{5/4}}{5 e}-\frac {-\frac {-\frac {48 \sqrt {2} c \left (-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4} \left (a e^2-b d e+c d^2\right )^2 \int \frac {1}{\left (\frac {c (2 c d-b e)}{b^2-4 a c}-e \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )\right ) \left (1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}\right )^{3/4}}d\left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )}{e \left (a+b x+c x^2\right )^{3/4}}-\frac {\sqrt {2} \sqrt [4]{b^2-4 a c} \sqrt {(b+2 c x)^2} \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right ) \sqrt {\frac {4 c \left (a+b x+c x^2\right )-4 a c+b^2}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right )^2}} (2 c d-b e) \left (-4 c e (3 b d-4 a e)-b^2 e^2+12 c^2 d^2\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right ),\frac {1}{2}\right )}{\sqrt [4]{c} e (b+2 c x) \sqrt {4 c \left (a+b x+c x^2\right )-4 a c+b^2}}}{12 c e^2}-\frac {\sqrt [4]{a+b x+c x^2} \left (-2 c e (7 b d-6 a e)+b^2 e^2-2 c e x (2 c d-b e)+12 c^2 d^2\right )}{3 c e^2}}{2 e}\)

\(\Big \downarrow \) 504

\(\displaystyle \frac {2 \left (c x^2+b x+a\right )^{5/4}}{5 e}-\frac {-\frac {\sqrt [4]{c x^2+b x+a} \left (12 c^2 d^2+b^2 e^2-2 c e (7 b d-6 a e)-2 c e (2 c d-b e) x\right )}{3 c e^2}-\frac {\frac {48 \sqrt {2} c \left (c d^2-b e d+a e^2\right )^2 \left (-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}\right )^{3/4} \left (-\frac {c (2 c d-b e) \int \frac {1}{\left (1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}\right )^{3/4} \left (\frac {c^2 (2 c d-b e)^2}{\left (b^2-4 a c\right )^2}-e^2 \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2\right )}d\left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )}{b^2-4 a c}-e \int \frac {-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}}{\left (1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}\right )^{3/4} \left (\frac {c^2 (2 c d-b e)^2}{\left (b^2-4 a c\right )^2}-e^2 \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2\right )}d\left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )\right )}{e \left (c x^2+b x+a\right )^{3/4}}-\frac {\sqrt {2} \sqrt [4]{b^2-4 a c} (2 c d-b e) \left (12 c^2 d^2-b^2 e^2-4 c e (3 b d-4 a e)\right ) \sqrt {(b+2 c x)^2} \left (\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}+1\right ) \sqrt {\frac {b^2-4 a c+4 c \left (c x^2+b x+a\right )}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right ),\frac {1}{2}\right )}{\sqrt [4]{c} e (b+2 c x) \sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}}{12 c e^2}}{2 e}\)

\(\Big \downarrow \) 312

\(\displaystyle \frac {2 \left (c x^2+b x+a\right )^{5/4}}{5 e}-\frac {-\frac {\sqrt [4]{c x^2+b x+a} \left (12 c^2 d^2+b^2 e^2-2 c e (7 b d-6 a e)-2 c e (2 c d-b e) x\right )}{3 c e^2}-\frac {\frac {48 \sqrt {2} c \left (c d^2-b e d+a e^2\right )^2 \left (-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}\right )^{3/4} \left (-e \int \frac {-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}}{\left (1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}\right )^{3/4} \left (\frac {c^2 (2 c d-b e)^2}{\left (b^2-4 a c\right )^2}-e^2 \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2\right )}d\left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )-\frac {c (2 c d-b e) \sqrt {\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}} \int \frac {1}{\sqrt {\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}} \left (1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}\right )^{3/4} \left (\frac {c^2 (2 c d-b e)^2}{\left (b^2-4 a c\right )^2}-e^2 \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2\right )}d\left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{2 \left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )}\right )}{e \left (c x^2+b x+a\right )^{3/4}}-\frac {\sqrt {2} \sqrt [4]{b^2-4 a c} (2 c d-b e) \left (12 c^2 d^2-b^2 e^2-4 c e (3 b d-4 a e)\right ) \sqrt {(b+2 c x)^2} \left (\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}+1\right ) \sqrt {\frac {b^2-4 a c+4 c \left (c x^2+b x+a\right )}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right ),\frac {1}{2}\right )}{\sqrt [4]{c} e (b+2 c x) \sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}}{12 c e^2}}{2 e}\)

\(\Big \downarrow \) 118

\(\displaystyle \frac {2 \left (c x^2+b x+a\right )^{5/4}}{5 e}-\frac {-\frac {\sqrt [4]{c x^2+b x+a} \left (12 c^2 d^2+b^2 e^2-2 c e (7 b d-6 a e)-2 c e (2 c d-b e) x\right )}{3 c e^2}-\frac {\frac {48 \sqrt {2} c \left (c d^2-b e d+a e^2\right )^2 \left (-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}\right )^{3/4} \left (\frac {2 c (2 c d-b e) \sqrt {\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}} \int \frac {1}{\sqrt {1-\left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^8} \left (e^2 \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^8+\frac {4 c \left (c d^2-b e d+a e^2\right )}{b^2-4 a c}\right )}d\sqrt [4]{1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}}}{\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )}-e \int \frac {-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}}{\left (1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}\right )^{3/4} \left (\frac {c^2 (2 c d-b e)^2}{\left (b^2-4 a c\right )^2}-e^2 \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2\right )}d\left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )\right )}{e \left (c x^2+b x+a\right )^{3/4}}-\frac {\sqrt {2} \sqrt [4]{b^2-4 a c} (2 c d-b e) \left (12 c^2 d^2-b^2 e^2-4 c e (3 b d-4 a e)\right ) \sqrt {(b+2 c x)^2} \left (\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}+1\right ) \sqrt {\frac {b^2-4 a c+4 c \left (c x^2+b x+a\right )}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right ),\frac {1}{2}\right )}{\sqrt [4]{c} e (b+2 c x) \sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}}{12 c e^2}}{2 e}\)

\(\Big \downarrow \) 353

\(\displaystyle \frac {2 \left (c x^2+b x+a\right )^{5/4}}{5 e}-\frac {-\frac {\sqrt [4]{c x^2+b x+a} \left (12 c^2 d^2+b^2 e^2-2 c e (7 b d-6 a e)-2 c e (2 c d-b e) x\right )}{3 c e^2}-\frac {\frac {48 \sqrt {2} c \left (c d^2-b e d+a e^2\right )^2 \left (-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}\right )^{3/4} \left (\frac {2 c (2 c d-b e) \sqrt {\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}} \int \frac {1}{\sqrt {1-\left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^8} \left (e^2 \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^8+\frac {4 c \left (c d^2-b e d+a e^2\right )}{b^2-4 a c}\right )}d\sqrt [4]{1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}}}{\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )}-\frac {1}{2} e \int \frac {1}{\left (1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}\right )^{3/4} \left (\frac {c^2 (2 c d-b e)^2}{\left (b^2-4 a c\right )^2}-e^2 \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2\right )}d\left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2\right )}{e \left (c x^2+b x+a\right )^{3/4}}-\frac {\sqrt {2} \sqrt [4]{b^2-4 a c} (2 c d-b e) \left (12 c^2 d^2-b^2 e^2-4 c e (3 b d-4 a e)\right ) \sqrt {(b+2 c x)^2} \left (\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}+1\right ) \sqrt {\frac {b^2-4 a c+4 c \left (c x^2+b x+a\right )}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right ),\frac {1}{2}\right )}{\sqrt [4]{c} e (b+2 c x) \sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}}{12 c e^2}}{2 e}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {2 \left (c x^2+b x+a\right )^{5/4}}{5 e}-\frac {-\frac {\sqrt [4]{c x^2+b x+a} \left (12 c^2 d^2+b^2 e^2-2 c e (7 b d-6 a e)-2 c e (2 c d-b e) x\right )}{3 c e^2}-\frac {\frac {48 \sqrt {2} c \left (c d^2-b e d+a e^2\right )^2 \left (-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}\right )^{3/4} \left (\frac {2 e \int \frac {1}{\frac {c^2 e^2 \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^8}{b^2-4 a c}+\frac {4 c^3 \left (c d^2-b e d+a e^2\right )}{\left (b^2-4 a c\right )^2}}d\sqrt [4]{1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}} c^2}{b^2-4 a c}+\frac {2 (2 c d-b e) \sqrt {\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}} \int \frac {1}{\sqrt {1-\left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^8} \left (e^2 \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^8+\frac {4 c \left (c d^2-b e d+a e^2\right )}{b^2-4 a c}\right )}d\sqrt [4]{1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}} c}{\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )}\right )}{e \left (c x^2+b x+a\right )^{3/4}}-\frac {\sqrt {2} \sqrt [4]{b^2-4 a c} (2 c d-b e) \left (12 c^2 d^2-b^2 e^2-4 c e (3 b d-4 a e)\right ) \sqrt {(b+2 c x)^2} \left (\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}+1\right ) \sqrt {\frac {b^2-4 a c+4 c \left (c x^2+b x+a\right )}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right ),\frac {1}{2}\right )}{\sqrt [4]{c} e (b+2 c x) \sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}}{12 c e^2}}{2 e}\)

\(\Big \downarrow \) 756

\(\displaystyle \frac {2 \left (c x^2+b x+a\right )^{5/4}}{5 e}-\frac {-\frac {\sqrt [4]{c x^2+b x+a} \left (12 c^2 d^2+b^2 e^2-2 c e (7 b d-6 a e)-2 c e (2 c d-b e) x\right )}{3 c e^2}-\frac {\frac {48 \sqrt {2} c \left (c d^2-b e d+a e^2\right )^2 \left (-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}\right )^{3/4} \left (\frac {2 e \left (\frac {\int \frac {1}{2 \sqrt {c} \sqrt {c d^2-b e d+a e^2}-\sqrt {4 a c-b^2} e \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^4}d\sqrt [4]{1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}} \left (b^2-4 a c\right )^2}{4 c^{5/2} \sqrt {c d^2-b e d+a e^2}}+\frac {\int \frac {1}{\sqrt {4 a c-b^2} e \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^4+2 \sqrt {c} \sqrt {c d^2-b e d+a e^2}}d\sqrt [4]{1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}} \left (b^2-4 a c\right )^2}{4 c^{5/2} \sqrt {c d^2-b e d+a e^2}}\right ) c^2}{b^2-4 a c}+\frac {2 (2 c d-b e) \sqrt {\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}} \int \frac {1}{\sqrt {1-\left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^8} \left (e^2 \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^8+\frac {4 c \left (c d^2-b e d+a e^2\right )}{b^2-4 a c}\right )}d\sqrt [4]{1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}} c}{\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )}\right )}{e \left (c x^2+b x+a\right )^{3/4}}-\frac {\sqrt {2} \sqrt [4]{b^2-4 a c} (2 c d-b e) \left (12 c^2 d^2-b^2 e^2-4 c e (3 b d-4 a e)\right ) \sqrt {(b+2 c x)^2} \left (\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}+1\right ) \sqrt {\frac {b^2-4 a c+4 c \left (c x^2+b x+a\right )}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right ),\frac {1}{2}\right )}{\sqrt [4]{c} e (b+2 c x) \sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}}{12 c e^2}}{2 e}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {2 \left (c x^2+b x+a\right )^{5/4}}{5 e}-\frac {-\frac {\sqrt [4]{c x^2+b x+a} \left (12 c^2 d^2+b^2 e^2-2 c e (7 b d-6 a e)-2 c e (2 c d-b e) x\right )}{3 c e^2}-\frac {\frac {48 \sqrt {2} c \left (c d^2-b e d+a e^2\right )^2 \left (-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}\right )^{3/4} \left (\frac {2 e \left (\frac {\arctan \left (\frac {\sqrt [4]{4 a c-b^2} \sqrt {e} \sqrt [4]{1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}}}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{c d^2-b e d+a e^2}}\right ) \left (b^2-4 a c\right )^2}{4 \sqrt {2} c^{11/4} \sqrt [4]{4 a c-b^2} \sqrt {e} \left (c d^2-b e d+a e^2\right )^{3/4}}+\frac {\int \frac {1}{2 \sqrt {c} \sqrt {c d^2-b e d+a e^2}-\sqrt {4 a c-b^2} e \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^4}d\sqrt [4]{1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}} \left (b^2-4 a c\right )^2}{4 c^{5/2} \sqrt {c d^2-b e d+a e^2}}\right ) c^2}{b^2-4 a c}+\frac {2 (2 c d-b e) \sqrt {\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}} \int \frac {1}{\sqrt {1-\left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^8} \left (e^2 \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^8+\frac {4 c \left (c d^2-b e d+a e^2\right )}{b^2-4 a c}\right )}d\sqrt [4]{1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}} c}{\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )}\right )}{e \left (c x^2+b x+a\right )^{3/4}}-\frac {\sqrt {2} \sqrt [4]{b^2-4 a c} (2 c d-b e) \left (12 c^2 d^2-b^2 e^2-4 c e (3 b d-4 a e)\right ) \sqrt {(b+2 c x)^2} \left (\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}+1\right ) \sqrt {\frac {b^2-4 a c+4 c \left (c x^2+b x+a\right )}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right ),\frac {1}{2}\right )}{\sqrt [4]{c} e (b+2 c x) \sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}}{12 c e^2}}{2 e}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {2 \left (c x^2+b x+a\right )^{5/4}}{5 e}-\frac {-\frac {\sqrt [4]{c x^2+b x+a} \left (12 c^2 d^2+b^2 e^2-2 c e (7 b d-6 a e)-2 c e (2 c d-b e) x\right )}{3 c e^2}-\frac {\frac {48 \sqrt {2} c \left (c d^2-b e d+a e^2\right )^2 \left (-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}\right )^{3/4} \left (\frac {2 e \left (\frac {\arctan \left (\frac {\sqrt [4]{4 a c-b^2} \sqrt {e} \sqrt [4]{1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}}}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{c d^2-b e d+a e^2}}\right ) \left (b^2-4 a c\right )^2}{4 \sqrt {2} c^{11/4} \sqrt [4]{4 a c-b^2} \sqrt {e} \left (c d^2-b e d+a e^2\right )^{3/4}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{4 a c-b^2} \sqrt {e} \sqrt [4]{1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}}}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{c d^2-b e d+a e^2}}\right ) \left (b^2-4 a c\right )^2}{4 \sqrt {2} c^{11/4} \sqrt [4]{4 a c-b^2} \sqrt {e} \left (c d^2-b e d+a e^2\right )^{3/4}}\right ) c^2}{b^2-4 a c}+\frac {2 (2 c d-b e) \sqrt {\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}} \int \frac {1}{\sqrt {1-\left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^8} \left (e^2 \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^8+\frac {4 c \left (c d^2-b e d+a e^2\right )}{b^2-4 a c}\right )}d\sqrt [4]{1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}} c}{\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )}\right )}{e \left (c x^2+b x+a\right )^{3/4}}-\frac {\sqrt {2} \sqrt [4]{b^2-4 a c} (2 c d-b e) \left (12 c^2 d^2-b^2 e^2-4 c e (3 b d-4 a e)\right ) \sqrt {(b+2 c x)^2} \left (\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}+1\right ) \sqrt {\frac {b^2-4 a c+4 c \left (c x^2+b x+a\right )}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right ),\frac {1}{2}\right )}{\sqrt [4]{c} e (b+2 c x) \sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}}{12 c e^2}}{2 e}\)

\(\Big \downarrow \) 925

\(\displaystyle \frac {2 \left (c x^2+b x+a\right )^{5/4}}{5 e}-\frac {-\frac {\sqrt [4]{c x^2+b x+a} \left (12 c^2 d^2+b^2 e^2-2 c e (7 b d-6 a e)-2 c e (2 c d-b e) x\right )}{3 c e^2}-\frac {\frac {48 \sqrt {2} c \left (c d^2-b e d+a e^2\right )^2 \left (-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}\right )^{3/4} \left (\frac {2 e \left (\frac {\arctan \left (\frac {\sqrt [4]{4 a c-b^2} \sqrt {e} \sqrt [4]{1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}}}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{c d^2-b e d+a e^2}}\right ) \left (b^2-4 a c\right )^2}{4 \sqrt {2} c^{11/4} \sqrt [4]{4 a c-b^2} \sqrt {e} \left (c d^2-b e d+a e^2\right )^{3/4}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{4 a c-b^2} \sqrt {e} \sqrt [4]{1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}}}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{c d^2-b e d+a e^2}}\right ) \left (b^2-4 a c\right )^2}{4 \sqrt {2} c^{11/4} \sqrt [4]{4 a c-b^2} \sqrt {e} \left (c d^2-b e d+a e^2\right )^{3/4}}\right ) c^2}{b^2-4 a c}+\frac {2 (2 c d-b e) \sqrt {\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}} \left (\frac {\left (b^2-4 a c\right ) \int \frac {2 \sqrt {c}}{\left (2 \sqrt {c}-\frac {\sqrt {4 a c-b^2} e \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^4}{\sqrt {c d^2-b e d+a e^2}}\right ) \sqrt {1-\left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^8}}d\sqrt [4]{1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}}}{8 c \left (c d^2-b e d+a e^2\right )}+\frac {\left (b^2-4 a c\right ) \int \frac {2 \sqrt {c}}{\left (\frac {\sqrt {4 a c-b^2} e \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^4}{\sqrt {c d^2-b e d+a e^2}}+2 \sqrt {c}\right ) \sqrt {1-\left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^8}}d\sqrt [4]{1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}}}{8 c \left (c d^2-b e d+a e^2\right )}\right ) c}{\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )}\right )}{e \left (c x^2+b x+a\right )^{3/4}}-\frac {\sqrt {2} \sqrt [4]{b^2-4 a c} (2 c d-b e) \left (12 c^2 d^2-b^2 e^2-4 c e (3 b d-4 a e)\right ) \sqrt {(b+2 c x)^2} \left (\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}+1\right ) \sqrt {\frac {b^2-4 a c+4 c \left (c x^2+b x+a\right )}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right ),\frac {1}{2}\right )}{\sqrt [4]{c} e (b+2 c x) \sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}}{12 c e^2}}{2 e}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 \left (c x^2+b x+a\right )^{5/4}}{5 e}-\frac {-\frac {\sqrt [4]{c x^2+b x+a} \left (12 c^2 d^2+b^2 e^2-2 c e (7 b d-6 a e)-2 c e (2 c d-b e) x\right )}{3 c e^2}-\frac {\frac {48 \sqrt {2} c \left (c d^2-b e d+a e^2\right )^2 \left (-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}\right )^{3/4} \left (\frac {2 e \left (\frac {\arctan \left (\frac {\sqrt [4]{4 a c-b^2} \sqrt {e} \sqrt [4]{1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}}}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{c d^2-b e d+a e^2}}\right ) \left (b^2-4 a c\right )^2}{4 \sqrt {2} c^{11/4} \sqrt [4]{4 a c-b^2} \sqrt {e} \left (c d^2-b e d+a e^2\right )^{3/4}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{4 a c-b^2} \sqrt {e} \sqrt [4]{1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}}}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{c d^2-b e d+a e^2}}\right ) \left (b^2-4 a c\right )^2}{4 \sqrt {2} c^{11/4} \sqrt [4]{4 a c-b^2} \sqrt {e} \left (c d^2-b e d+a e^2\right )^{3/4}}\right ) c^2}{b^2-4 a c}+\frac {2 (2 c d-b e) \sqrt {\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}} \left (\frac {\left (b^2-4 a c\right ) \int \frac {1}{\left (2 \sqrt {c}-\frac {\sqrt {4 a c-b^2} e \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^4}{\sqrt {c d^2-b e d+a e^2}}\right ) \sqrt {1-\left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^8}}d\sqrt [4]{1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}}}{4 \sqrt {c} \left (c d^2-b e d+a e^2\right )}+\frac {\left (b^2-4 a c\right ) \int \frac {1}{\left (\frac {\sqrt {4 a c-b^2} e \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^4}{\sqrt {c d^2-b e d+a e^2}}+2 \sqrt {c}\right ) \sqrt {1-\left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^8}}d\sqrt [4]{1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}}}{4 \sqrt {c} \left (c d^2-b e d+a e^2\right )}\right ) c}{\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )}\right )}{e \left (c x^2+b x+a\right )^{3/4}}-\frac {\sqrt {2} \sqrt [4]{b^2-4 a c} (2 c d-b e) \left (12 c^2 d^2-b^2 e^2-4 c e (3 b d-4 a e)\right ) \sqrt {(b+2 c x)^2} \left (\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}+1\right ) \sqrt {\frac {b^2-4 a c+4 c \left (c x^2+b x+a\right )}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right ),\frac {1}{2}\right )}{\sqrt [4]{c} e (b+2 c x) \sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}}{12 c e^2}}{2 e}\)

\(\Big \downarrow \) 1537

\(\displaystyle \frac {2 \left (c x^2+b x+a\right )^{5/4}}{5 e}-\frac {-\frac {\sqrt [4]{c x^2+b x+a} \left (12 c^2 d^2+b^2 e^2-2 c e (7 b d-6 a e)-2 c e (2 c d-b e) x\right )}{3 c e^2}-\frac {\frac {48 \sqrt {2} c \left (c d^2-b e d+a e^2\right )^2 \left (-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}\right )^{3/4} \left (\frac {2 e \left (\frac {\arctan \left (\frac {\sqrt [4]{4 a c-b^2} \sqrt {e} \sqrt [4]{1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}}}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{c d^2-b e d+a e^2}}\right ) \left (b^2-4 a c\right )^2}{4 \sqrt {2} c^{11/4} \sqrt [4]{4 a c-b^2} \sqrt {e} \left (c d^2-b e d+a e^2\right )^{3/4}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{4 a c-b^2} \sqrt {e} \sqrt [4]{1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}}}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{c d^2-b e d+a e^2}}\right ) \left (b^2-4 a c\right )^2}{4 \sqrt {2} c^{11/4} \sqrt [4]{4 a c-b^2} \sqrt {e} \left (c d^2-b e d+a e^2\right )^{3/4}}\right ) c^2}{b^2-4 a c}+\frac {2 (2 c d-b e) \sqrt {\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}} \left (\frac {\left (b^2-4 a c\right ) \int \frac {1}{\sqrt {1-\left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^4} \sqrt {\left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^4+1} \left (2 \sqrt {c}-\frac {\sqrt {4 a c-b^2} e \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^4}{\sqrt {c d^2-b e d+a e^2}}\right )}d\sqrt [4]{1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}}}{4 \sqrt {c} \left (c d^2-b e d+a e^2\right )}+\frac {\left (b^2-4 a c\right ) \int \frac {1}{\sqrt {1-\left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^4} \sqrt {\left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^4+1} \left (\frac {\sqrt {4 a c-b^2} e \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^4}{\sqrt {c d^2-b e d+a e^2}}+2 \sqrt {c}\right )}d\sqrt [4]{1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}}}{4 \sqrt {c} \left (c d^2-b e d+a e^2\right )}\right ) c}{\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )}\right )}{e \left (c x^2+b x+a\right )^{3/4}}-\frac {\sqrt {2} \sqrt [4]{b^2-4 a c} (2 c d-b e) \left (12 c^2 d^2-b^2 e^2-4 c e (3 b d-4 a e)\right ) \sqrt {(b+2 c x)^2} \left (\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}+1\right ) \sqrt {\frac {b^2-4 a c+4 c \left (c x^2+b x+a\right )}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right ),\frac {1}{2}\right )}{\sqrt [4]{c} e (b+2 c x) \sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}}{12 c e^2}}{2 e}\)

\(\Big \downarrow \) 412

\(\displaystyle \frac {2 \left (c x^2+b x+a\right )^{5/4}}{5 e}-\frac {-\frac {\sqrt [4]{c x^2+b x+a} \left (12 c^2 d^2+b^2 e^2-2 c e (7 b d-6 a e)-2 c e (2 c d-b e) x\right )}{3 c e^2}-\frac {\frac {48 \sqrt {2} c \left (c d^2-b e d+a e^2\right )^2 \left (-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}\right )^{3/4} \left (\frac {2 e \left (\frac {\arctan \left (\frac {\sqrt [4]{4 a c-b^2} \sqrt {e} \sqrt [4]{1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}}}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{c d^2-b e d+a e^2}}\right ) \left (b^2-4 a c\right )^2}{4 \sqrt {2} c^{11/4} \sqrt [4]{4 a c-b^2} \sqrt {e} \left (c d^2-b e d+a e^2\right )^{3/4}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{4 a c-b^2} \sqrt {e} \sqrt [4]{1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}}}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{c d^2-b e d+a e^2}}\right ) \left (b^2-4 a c\right )^2}{4 \sqrt {2} c^{11/4} \sqrt [4]{4 a c-b^2} \sqrt {e} \left (c d^2-b e d+a e^2\right )^{3/4}}\right ) c^2}{b^2-4 a c}+\frac {2 (2 c d-b e) \sqrt {\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}} \left (-\frac {\left (b^2-4 a c\right ) \operatorname {EllipticPi}\left (-\frac {\sqrt {4 a c-b^2} e}{2 \sqrt {c} \sqrt {c d^2-b e d+a e^2}},\arcsin \left (\frac {2 x c^2}{b^2-4 a c}+\frac {b c}{b^2-4 a c}\right ),-1\right )}{8 c \left (c d^2-b e d+a e^2\right )}-\frac {\left (b^2-4 a c\right ) \operatorname {EllipticPi}\left (\frac {\sqrt {4 a c-b^2} e}{2 \sqrt {c} \sqrt {c d^2-b e d+a e^2}},\arcsin \left (\frac {2 x c^2}{b^2-4 a c}+\frac {b c}{b^2-4 a c}\right ),-1\right )}{8 c \left (c d^2-b e d+a e^2\right )}\right ) c}{\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )}\right )}{e \left (c x^2+b x+a\right )^{3/4}}-\frac {\sqrt {2} \sqrt [4]{b^2-4 a c} (2 c d-b e) \left (12 c^2 d^2-b^2 e^2-4 c e (3 b d-4 a e)\right ) \sqrt {(b+2 c x)^2} \left (\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}+1\right ) \sqrt {\frac {b^2-4 a c+4 c \left (c x^2+b x+a\right )}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right ),\frac {1}{2}\right )}{\sqrt [4]{c} e (b+2 c x) \sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}}{12 c e^2}}{2 e}\)

Input:

Int[(a + b*x + c*x^2)^(5/4)/(d + e*x),x]
 

Output:

(2*(a + b*x + c*x^2)^(5/4))/(5*e) - (-1/3*((12*c^2*d^2 + b^2*e^2 - 2*c*e*( 
7*b*d - 6*a*e) - 2*c*e*(2*c*d - b*e)*x)*(a + b*x + c*x^2)^(1/4))/(c*e^2) - 
 (-((Sqrt[2]*(b^2 - 4*a*c)^(1/4)*(2*c*d - b*e)*(12*c^2*d^2 - b^2*e^2 - 4*c 
*e*(3*b*d - 4*a*e))*Sqrt[(b + 2*c*x)^2]*(1 + (2*Sqrt[c]*Sqrt[a + b*x + c*x 
^2])/Sqrt[b^2 - 4*a*c])*Sqrt[(b^2 - 4*a*c + 4*c*(a + b*x + c*x^2))/((b^2 - 
 4*a*c)*(1 + (2*Sqrt[c]*Sqrt[a + b*x + c*x^2])/Sqrt[b^2 - 4*a*c])^2)]*Elli 
pticF[2*ArcTan[(Sqrt[2]*c^(1/4)*(a + b*x + c*x^2)^(1/4))/(b^2 - 4*a*c)^(1/ 
4)], 1/2])/(c^(1/4)*e*(b + 2*c*x)*Sqrt[b^2 - 4*a*c + 4*c*(a + b*x + c*x^2) 
])) + (48*Sqrt[2]*c*(c*d^2 - b*d*e + a*e^2)^2*(-((c*(a + b*x + c*x^2))/(b^ 
2 - 4*a*c)))^(3/4)*((2*c^2*e*(((b^2 - 4*a*c)^2*ArcTan[((-b^2 + 4*a*c)^(1/4 
)*Sqrt[e]*(1 - ((b^2 - 4*a*c)*(-((b*c)/(b^2 - 4*a*c)) - (2*c^2*x)/(b^2 - 4 
*a*c))^2)/c^2)^(1/4))/(Sqrt[2]*c^(1/4)*(c*d^2 - b*d*e + a*e^2)^(1/4))])/(4 
*Sqrt[2]*c^(11/4)*(-b^2 + 4*a*c)^(1/4)*Sqrt[e]*(c*d^2 - b*d*e + a*e^2)^(3/ 
4)) + ((b^2 - 4*a*c)^2*ArcTanh[((-b^2 + 4*a*c)^(1/4)*Sqrt[e]*(1 - ((b^2 - 
4*a*c)*(-((b*c)/(b^2 - 4*a*c)) - (2*c^2*x)/(b^2 - 4*a*c))^2)/c^2)^(1/4))/( 
Sqrt[2]*c^(1/4)*(c*d^2 - b*d*e + a*e^2)^(1/4))])/(4*Sqrt[2]*c^(11/4)*(-b^2 
 + 4*a*c)^(1/4)*Sqrt[e]*(c*d^2 - b*d*e + a*e^2)^(3/4))))/(b^2 - 4*a*c) + ( 
2*c*(2*c*d - b*e)*Sqrt[((b^2 - 4*a*c)*(-((b*c)/(b^2 - 4*a*c)) - (2*c^2*x)/ 
(b^2 - 4*a*c))^2)/c^2]*(-1/8*((b^2 - 4*a*c)*EllipticPi[-1/2*(Sqrt[-b^2 + 4 
*a*c]*e)/(Sqrt[c]*Sqrt[c*d^2 - b*d*e + a*e^2]), ArcSin[(b*c)/(b^2 - 4*a...
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 118

Int[1/(((a_.) + (b_.)*(x_))*Sqrt[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^( 
3/4)), x_] :> Simp[-4   Subst[Int[1/((b*e - a*f - b*x^4)*Sqrt[c - d*(e/f) + 
 d*(x^4/f)]), x], x, (e + f*x)^(1/4)], x] /; FreeQ[{a, b, c, d, e, f}, x] & 
& GtQ[-f/(d*e - c*f), 0]
 

rule 218

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R 
t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
 

rule 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

rule 312

Int[1/(((a_) + (b_.)*(x_)^2)^(3/4)*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Sim 
p[Sqrt[(-b)*(x^2/a)]/(2*x)   Subst[Int[1/(Sqrt[(-b)*(x/a)]*(a + b*x)^(3/4)* 
(c + d*x)), x], x, x^2], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]
 

rule 353

Int[(x_)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_Symbol] 
 :> Simp[1/2   Subst[Int[(a + b*x)^p*(c + d*x)^q, x], x, x^2], x] /; FreeQ[ 
{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0]
 

rule 412

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x 
_)^2]), x_Symbol] :> Simp[(1/(a*Sqrt[c]*Sqrt[e]*Rt[-d/c, 2]))*EllipticPi[b* 
(c/(a*d)), ArcSin[Rt[-d/c, 2]*x], c*(f/(d*e))], x] /; FreeQ[{a, b, c, d, e, 
 f}, x] &&  !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] &&  !( !GtQ[f/e, 0] && S 
implerSqrtQ[-f/e, -d/c])
 

rule 504

Int[((a_) + (b_.)*(x_)^2)^(p_)/((c_) + (d_.)*(x_)), x_Symbol] :> Simp[c   I 
nt[(a + b*x^2)^p/(c^2 - d^2*x^2), x], x] - Simp[d   Int[x*((a + b*x^2)^p/(c 
^2 - d^2*x^2)), x], x] /; FreeQ[{a, b, c, d, p}, x]
 

rule 756

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2 
]], s = Denominator[Rt[-a/b, 2]]}, Simp[r/(2*a)   Int[1/(r - s*x^2), x], x] 
 + Simp[r/(2*a)   Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !GtQ[a 
/b, 0]
 

rule 761

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[( 
1 + q^2*x^2)*(Sqrt[(a + b*x^4)/(a*(1 + q^2*x^2)^2)]/(2*q*Sqrt[a + b*x^4]))* 
EllipticF[2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]
 

rule 925

Int[1/(Sqrt[(a_) + (b_.)*(x_)^4]*((c_) + (d_.)*(x_)^4)), x_Symbol] :> Simp[ 
1/(2*c)   Int[1/(Sqrt[a + b*x^4]*(1 - Rt[-d/c, 2]*x^2)), x], x] + Simp[1/(2 
*c)   Int[1/(Sqrt[a + b*x^4]*(1 + Rt[-d/c, 2]*x^2)), x], x] /; FreeQ[{a, b, 
 c, d}, x] && NeQ[b*c - a*d, 0]
 

rule 1094

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[4*(Sqrt[(b 
+ 2*c*x)^2]/(b + 2*c*x))   Subst[Int[x^(4*(p + 1) - 1)/Sqrt[b^2 - 4*a*c + 4 
*c*x^4], x], x, (a + b*x + c*x^2)^(1/4)], x] /; FreeQ[{a, b, c}, x] && Inte 
gerQ[4*p]
 

rule 1162

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[(d + e*x)^(m + 1)*((a + b*x + c*x^2)^p/(e*(m + 2*p + 1))), x 
] - Simp[p/(e*(m + 2*p + 1))   Int[(d + e*x)^m*Simp[b*d - 2*a*e + (2*c*d - 
b*e)*x, x]*(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x 
] && GtQ[p, 0] && NeQ[m + 2*p + 1, 0] && ( !RationalQ[m] || LtQ[m, 1]) && 
!ILtQ[m + 2*p, 0] && IntQuadraticQ[a, b, c, d, e, m, p, x]
 

rule 1173

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)/((d_.) + (e_.)*(x_)), x_Symbol 
] :> Simp[1/(-4*(c/(b^2 - 4*a*c)))^p   Subst[Int[Simp[1 - x^2/(b^2 - 4*a*c) 
, x]^p/Simp[2*c*d - b*e + e*x, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c, 
 d, e, p}, x] && GtQ[4*a - b^2/c, 0] && IntegerQ[4*p]
 

rule 1174

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)/((d_.) + (e_.)*(x_)), x_Symbol 
] :> Simp[(a + b*x + c*x^2)^p/((-c)*((a + b*x + c*x^2)/(b^2 - 4*a*c)))^p 
Int[((-a)*(c/(b^2 - 4*a*c)) - b*c*(x/(b^2 - 4*a*c)) - c^2*(x^2/(b^2 - 4*a*c 
)))^p/(d + e*x), x], x] /; FreeQ[{a, b, c, d, e, p}, x] &&  !GtQ[4*a - b^2/ 
c, 0] && IntegerQ[4*p]
 

rule 1231

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) 
 - g*(c*d + 2*c*d*p - b*e*p) + g*c*e*(m + 2*p + 1)*x)*((a + b*x + c*x^2)^p/ 
(c*e^2*(m + 2*p + 1)*(m + 2*p + 2))), x] - Simp[p/(c*e^2*(m + 2*p + 1)*(m + 
 2*p + 2))   Int[(d + e*x)^m*(a + b*x + c*x^2)^(p - 1)*Simp[c*e*f*(b*d - 2* 
a*e)*(m + 2*p + 2) + g*(a*e*(b*e - 2*c*d*m + b*e*m) + b*d*(b*e*p - c*d - 2* 
c*d*p)) + (c*e*f*(2*c*d - b*e)*(m + 2*p + 2) + g*(b^2*e^2*(p + m + 1) - 2*c 
^2*d^2*(1 + 2*p) - c*e*(b*d*(m - 2*p) + 2*a*e*(m + 2*p + 1))))*x, x], x], x 
] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && GtQ[p, 0] && (IntegerQ[p] ||  !R 
ationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) &&  !ILtQ[m + 2*p, 0] && (Integer 
Q[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
 

rule 1269

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g/e   Int[(d + e*x)^(m + 1)*(a + b*x + 
 c*x^2)^p, x], x] + Simp[(e*f - d*g)/e   Int[(d + e*x)^m*(a + b*x + c*x^2)^ 
p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] &&  !IGtQ[m, 0]
 

rule 1537

Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> With[ 
{q = Rt[(-a)*c, 2]}, Simp[Sqrt[-c]   Int[1/((d + e*x^2)*Sqrt[q + c*x^2]*Sqr 
t[q - c*x^2]), x], x]] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] & 
& GtQ[a, 0] && LtQ[c, 0]
 

Maple [F]

\[\int \frac {\left (c \,x^{2}+b x +a \right )^{\frac {5}{4}}}{e x +d}d x\]

Input:

int((c*x^2+b*x+a)^(5/4)/(e*x+d),x)
 

Output:

int((c*x^2+b*x+a)^(5/4)/(e*x+d),x)
 

Fricas [F(-1)]

Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^{5/4}}{d+e x} \, dx=\text {Timed out} \] Input:

integrate((c*x^2+b*x+a)^(5/4)/(e*x+d),x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F]

\[ \int \frac {\left (a+b x+c x^2\right )^{5/4}}{d+e x} \, dx=\int \frac {\left (a + b x + c x^{2}\right )^{\frac {5}{4}}}{d + e x}\, dx \] Input:

integrate((c*x**2+b*x+a)**(5/4)/(e*x+d),x)
 

Output:

Integral((a + b*x + c*x**2)**(5/4)/(d + e*x), x)
 

Maxima [F]

\[ \int \frac {\left (a+b x+c x^2\right )^{5/4}}{d+e x} \, dx=\int { \frac {{\left (c x^{2} + b x + a\right )}^{\frac {5}{4}}}{e x + d} \,d x } \] Input:

integrate((c*x^2+b*x+a)^(5/4)/(e*x+d),x, algorithm="maxima")
 

Output:

integrate((c*x^2 + b*x + a)^(5/4)/(e*x + d), x)
 

Giac [F]

\[ \int \frac {\left (a+b x+c x^2\right )^{5/4}}{d+e x} \, dx=\int { \frac {{\left (c x^{2} + b x + a\right )}^{\frac {5}{4}}}{e x + d} \,d x } \] Input:

integrate((c*x^2+b*x+a)^(5/4)/(e*x+d),x, algorithm="giac")
 

Output:

integrate((c*x^2 + b*x + a)^(5/4)/(e*x + d), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^{5/4}}{d+e x} \, dx=\int \frac {{\left (c\,x^2+b\,x+a\right )}^{5/4}}{d+e\,x} \,d x \] Input:

int((a + b*x + c*x^2)^(5/4)/(d + e*x),x)
 

Output:

int((a + b*x + c*x^2)^(5/4)/(d + e*x), x)
 

Reduce [F]

\[ \int \frac {\left (a+b x+c x^2\right )^{5/4}}{d+e x} \, dx=\int \frac {\left (c \,x^{2}+b x +a \right )^{\frac {5}{4}}}{e x +d}d x \] Input:

int((c*x^2+b*x+a)^(5/4)/(e*x+d),x)
 

Output:

int((c*x^2+b*x+a)^(5/4)/(e*x+d),x)