\(\int \frac {1}{(d+e x) (a+b x+c x^2)^{3/4}} \, dx\) [751]

Optimal result

Integrand size = 22, antiderivative size = 709 \[ \int \frac {1}{(d+e x) \left (a+b x+c x^2\right )^{3/4}} \, dx=-\frac {\left (-b^2+4 a c\right )^{3/4} \sqrt {e} \left (-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4} \arctan \left (\frac {\sqrt [4]{-b^2+4 a c} \sqrt {e} \sqrt [4]{1-\frac {(b+2 c x)^2}{b^2-4 a c}}}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{c d^2-b d e+a e^2}}\right )}{c^{3/4} \left (c d^2-b d e+a e^2\right )^{3/4} \left (a+b x+c x^2\right )^{3/4}}-\frac {\left (-b^2+4 a c\right )^{3/4} \sqrt {e} \left (-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{-b^2+4 a c} \sqrt {e} \sqrt [4]{1-\frac {(b+2 c x)^2}{b^2-4 a c}}}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{c d^2-b d e+a e^2}}\right )}{c^{3/4} \left (c d^2-b d e+a e^2\right )^{3/4} \left (a+b x+c x^2\right )^{3/4}}-\frac {\left (b^2-4 a c\right ) (2 c d-b e) \sqrt {\frac {(b+2 c x)^2}{b^2-4 a c}} \left (-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4} \operatorname {EllipticPi}\left (-\frac {\sqrt {-b^2+4 a c} e}{2 \sqrt {c} \sqrt {c d^2-b d e+a e^2}},\arcsin \left (\sqrt [4]{1-\frac {(b+2 c x)^2}{b^2-4 a c}}\right ),-1\right )}{\sqrt {2} c \left (c d^2-b d e+a e^2\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/4}}-\frac {\left (b^2-4 a c\right ) (2 c d-b e) \sqrt {\frac {(b+2 c x)^2}{b^2-4 a c}} \left (-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4} \operatorname {EllipticPi}\left (\frac {\sqrt {-b^2+4 a c} e}{2 \sqrt {c} \sqrt {c d^2-b d e+a e^2}},\arcsin \left (\sqrt [4]{1-\frac {(b+2 c x)^2}{b^2-4 a c}}\right ),-1\right )}{\sqrt {2} c \left (c d^2-b d e+a e^2\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/4}} \] Output:

-(4*a*c-b^2)^(3/4)*e^(1/2)*(-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)/(a*e^2-b*d*e+c*d^2)^(3/4)/(c*x^2+b 
*x+a)^(3/4)-(4*a*c-b^2)^(3/4)*e^(1/2)*(-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)/(a*e^2-b*d*e+c*d^2)^(3 
/4)/(c*x^2+b*x+a)^(3/4)-1/2*(-4*a*c+b^2)*(-b*e+2*c*d)*((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/(a*e^2-b*d*e+c*d^2)/(2*c*x+b)/(c*x^2+b*x+a)^(3/4)-1/2 
*(-4*a*c+b^2)*(-b*e+2*c*d)*((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/(a*e^2- 
b*d*e+c*d^2)/(2*c*x+b)/(c*x^2+b*x+a)^(3/4)
 

Mathematica [A] (verified)

Time = 11.19 (sec) , antiderivative size = 529, normalized size of antiderivative = 0.75 \[ \int \frac {1}{(d+e x) \left (a+b x+c x^2\right )^{3/4}} \, dx=-\frac {\sqrt [4]{a+x (b+c x)} \left (\sqrt {2} \sqrt [4]{c} \sqrt {e} \sqrt [4]{c d^2+e (-b d+a e)} (b+2 c x) \left (\arctan \left (\frac {\sqrt [4]{-b^2+4 a c} \sqrt {e} \sqrt [4]{\frac {c (a+x (b+c x))}{-b^2+4 a c}}}{\sqrt [4]{c} \sqrt [4]{c d^2+e (-b d+a e)}}\right )+\text {arctanh}\left (\frac {\sqrt [4]{-b^2+4 a c} \sqrt {e} \sqrt [4]{\frac {c (a+x (b+c x))}{-b^2+4 a c}}}{\sqrt [4]{c} \sqrt [4]{c d^2+e (-b d+a e)}}\right )\right )+\sqrt [4]{-b^2+4 a c} (-2 c d+b e) \sqrt {\frac {(b+2 c x)^2}{b^2-4 a c}} \operatorname {EllipticPi}\left (-\frac {\sqrt {-b^2+4 a c} e}{2 \sqrt {c} \sqrt {c d^2+e (-b d+a e)}},\arcsin \left (\sqrt {2} \sqrt [4]{\frac {c (a+x (b+c x))}{-b^2+4 a c}}\right ),-1\right )+\sqrt [4]{-b^2+4 a c} (-2 c d+b e) \sqrt {\frac {(b+2 c x)^2}{b^2-4 a c}} \operatorname {EllipticPi}\left (\frac {\sqrt {-b^2+4 a c} e}{2 \sqrt {c} \sqrt {c d^2+e (-b d+a e)}},\arcsin \left (\sqrt {2} \sqrt [4]{\frac {c (a+x (b+c x))}{-b^2+4 a c}}\right ),-1\right )\right )}{\sqrt {2} \sqrt [4]{-b^2+4 a c} \left (c d^2+e (-b d+a e)\right ) (b+2 c x) \sqrt [4]{\frac {c (a+x (b+c x))}{-b^2+4 a c}}} \] Input:

Integrate[1/((d + e*x)*(a + b*x + c*x^2)^(3/4)),x]
 

Output:

-(((a + x*(b + c*x))^(1/4)*(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[Sq 
rt[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]))/(Sqrt[2]*(-b^2 + 4*a* 
c)^(1/4)*(c*d^2 + e*(-(b*d) + a*e))*(b + 2*c*x)*((c*(a + x*(b + c*x)))/(-b 
^2 + 4*a*c))^(1/4)))
 

Rubi [A] (warning: unable to verify)

Time = 1.41 (sec) , antiderivative size = 723, normalized size of antiderivative = 1.02, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.682, Rules used = {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.

\(\displaystyle \int \frac {1}{(d+e x) \left (a+b x+c x^2\right )^{3/4}} \, dx\)

\(\Big \downarrow \) 1174

\(\displaystyle \frac {\left (-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4} \int \frac {1}{(d+e x) \left (-\frac {c^2 x^2}{b^2-4 a c}-\frac {b c x}{b^2-4 a c}-\frac {a c}{b^2-4 a c}\right )^{3/4}}dx}{\left (a+b x+c x^2\right )^{3/4}}\)

\(\Big \downarrow \) 1173

\(\displaystyle \frac {2 \sqrt {2} \left (-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4} \int -\frac {1}{\left (\frac {c (2 c d-b e)}{b^2-4 a c}-e \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )\right ) \left (1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}\right )^{3/4}}d\left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )}{\left (a+b x+c x^2\right )^{3/4}}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {2 \sqrt {2} \left (-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4} \int \frac {1}{\left (\frac {c (2 c d-b e)}{b^2-4 a c}-e \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )\right ) \left (1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}\right )^{3/4}}d\left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )}{\left (a+b x+c x^2\right )^{3/4}}\)

\(\Big \downarrow \) 504

\(\displaystyle \frac {2 \sqrt {2} \left (-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4} \left (-\frac {c (2 c d-b e) \int \frac {1}{\left (1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}\right )^{3/4} \left (\frac {c^2 (2 c d-b e)^2}{\left (b^2-4 a c\right )^2}-e^2 \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2\right )}d\left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )}{b^2-4 a c}-e \int \frac {-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}}{\left (1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}\right )^{3/4} \left (\frac {c^2 (2 c d-b e)^2}{\left (b^2-4 a c\right )^2}-e^2 \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2\right )}d\left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )\right )}{\left (a+b x+c x^2\right )^{3/4}}\)

\(\Big \downarrow \) 312

\(\displaystyle \frac {2 \sqrt {2} \left (-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4} \left (-e \int \frac {-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}}{\left (1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}\right )^{3/4} \left (\frac {c^2 (2 c d-b e)^2}{\left (b^2-4 a c\right )^2}-e^2 \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2\right )}d\left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )-\frac {c \sqrt {\frac {\left (b^2-4 a c\right ) \left (-\frac {2 c^2 x}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}} (2 c d-b e) \int \frac {1}{\sqrt {\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}} \left (1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}\right )^{3/4} \left (\frac {c^2 (2 c d-b e)^2}{\left (b^2-4 a c\right )^2}-e^2 \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2\right )}d\left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{2 \left (b^2-4 a c\right ) \left (-\frac {2 c^2 x}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )}\right )}{\left (a+b x+c x^2\right )^{3/4}}\)

\(\Big \downarrow \) 118

\(\displaystyle \frac {2 \sqrt {2} \left (-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4} \left (\frac {2 c \sqrt {\frac {\left (b^2-4 a c\right ) \left (-\frac {2 c^2 x}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}} (2 c d-b e) \int \frac {1}{\sqrt {1-\left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^8} \left (e^2 \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^8+\frac {4 c \left (c d^2-b e d+a e^2\right )}{b^2-4 a c}\right )}d\sqrt [4]{1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}}}{\left (b^2-4 a c\right ) \left (-\frac {2 c^2 x}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )}-e \int \frac {-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}}{\left (1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}\right )^{3/4} \left (\frac {c^2 (2 c d-b e)^2}{\left (b^2-4 a c\right )^2}-e^2 \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2\right )}d\left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )\right )}{\left (a+b x+c x^2\right )^{3/4}}\)

\(\Big \downarrow \) 353

\(\displaystyle \frac {2 \sqrt {2} \left (-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4} \left (\frac {2 c \sqrt {\frac {\left (b^2-4 a c\right ) \left (-\frac {2 c^2 x}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}} (2 c d-b e) \int \frac {1}{\sqrt {1-\left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^8} \left (e^2 \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^8+\frac {4 c \left (c d^2-b e d+a e^2\right )}{b^2-4 a c}\right )}d\sqrt [4]{1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}}}{\left (b^2-4 a c\right ) \left (-\frac {2 c^2 x}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )}-\frac {1}{2} e \int \frac {1}{\left (1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}\right )^{3/4} \left (\frac {c^2 (2 c d-b e)^2}{\left (b^2-4 a c\right )^2}-e^2 \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2\right )}d\left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2\right )}{\left (a+b x+c x^2\right )^{3/4}}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {2 \sqrt {2} \left (-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4} \left (\frac {2 c \sqrt {\frac {\left (b^2-4 a c\right ) \left (-\frac {2 c^2 x}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}} (2 c d-b e) \int \frac {1}{\sqrt {1-\left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^8} \left (e^2 \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^8+\frac {4 c \left (c d^2-b e d+a e^2\right )}{b^2-4 a c}\right )}d\sqrt [4]{1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}}}{\left (b^2-4 a c\right ) \left (-\frac {2 c^2 x}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )}+\frac {2 c^2 e \int \frac {1}{\frac {c^2 e^2 \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^8}{b^2-4 a c}+\frac {4 c^3 \left (c d^2-b e d+a e^2\right )}{\left (b^2-4 a c\right )^2}}d\sqrt [4]{1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}}}{b^2-4 a c}\right )}{\left (a+b x+c x^2\right )^{3/4}}\)

\(\Big \downarrow \) 756

\(\displaystyle \frac {2 \sqrt {2} \left (-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4} \left (\frac {2 c \sqrt {\frac {\left (b^2-4 a c\right ) \left (-\frac {2 c^2 x}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}} (2 c d-b e) \int \frac {1}{\sqrt {1-\left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^8} \left (e^2 \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^8+\frac {4 c \left (c d^2-b e d+a e^2\right )}{b^2-4 a c}\right )}d\sqrt [4]{1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}}}{\left (b^2-4 a c\right ) \left (-\frac {2 c^2 x}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )}+\frac {2 c^2 e \left (\frac {\left (b^2-4 a c\right )^2 \int \frac {1}{2 \sqrt {c} \sqrt {c d^2-b e d+a e^2}-\sqrt {4 a c-b^2} e \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^4}d\sqrt [4]{1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}}}{4 c^{5/2} \sqrt {a e^2-b d e+c d^2}}+\frac {\left (b^2-4 a c\right )^2 \int \frac {1}{\sqrt {4 a c-b^2} e \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^4+2 \sqrt {c} \sqrt {c d^2-b e d+a e^2}}d\sqrt [4]{1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}}}{4 c^{5/2} \sqrt {a e^2-b d e+c d^2}}\right )}{b^2-4 a c}\right )}{\left (a+b x+c x^2\right )^{3/4}}\)

\(\Big \downarrow \) 218

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

\(\Big \downarrow \) 221

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

\(\Big \downarrow \) 925

\(\displaystyle \frac {2 \sqrt {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 )}{\left (c x^2+b x+a\right )^{3/4}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 \sqrt {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 )}{\left (c x^2+b x+a\right )^{3/4}}\)

\(\Big \downarrow \) 1537

\(\displaystyle \frac {2 \sqrt {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 )}{\left (c x^2+b x+a\right )^{3/4}}\)

\(\Big \downarrow \) 412

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

Input:

Int[1/((d + e*x)*(a + b*x + c*x^2)^(3/4)),x]
 

Output:

(2*Sqrt[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*c) + (2*c^2*x)/(b^2 - 4*a*c)], -1])/( 
c*(c*d^2 - b*d*e + a*e^2)) - ((b^2 - 4*a*c)*EllipticPi[(Sqrt[-b^2 + 4*a*c] 
*e)/(2*Sqrt[c]*Sqrt[c*d^2 - b*d*e + a*e^2]), ArcSin[(b*c)/(b^2 - 4*a*c) + 
(2*c^2*x)/(b^2 - 4*a*c)], -1])/(8*c*(c*d^2 - b*d*e + a*e^2))))/((b^2 - 4*a 
*c)*(-((b*c)/(b^2 - 4*a*c)) - (2*c^2*x)/(b^2 - 4*a*c)))))/(a + b*x + c*x^2 
)^(3/4)
 

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 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 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 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]

\[\int \frac {1}{\left (e x +d \right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{4}}}d x\]

Input:

int(1/(e*x+d)/(c*x^2+b*x+a)^(3/4),x)
 

Output:

int(1/(e*x+d)/(c*x^2+b*x+a)^(3/4),x)
 

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{(d+e x) \left (a+b x+c x^2\right )^{3/4}} \, dx=\text {Timed out} \] Input:

integrate(1/(e*x+d)/(c*x^2+b*x+a)^(3/4),x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F]

\[ \int \frac {1}{(d+e x) \left (a+b x+c x^2\right )^{3/4}} \, dx=\int \frac {1}{\left (d + e x\right ) \left (a + b x + c x^{2}\right )^{\frac {3}{4}}}\, dx \] Input:

integrate(1/(e*x+d)/(c*x**2+b*x+a)**(3/4),x)
 

Output:

Integral(1/((d + e*x)*(a + b*x + c*x**2)**(3/4)), x)
 

Maxima [F]

\[ \int \frac {1}{(d+e x) \left (a+b x+c x^2\right )^{3/4}} \, dx=\int { \frac {1}{{\left (c x^{2} + b x + a\right )}^{\frac {3}{4}} {\left (e x + d\right )}} \,d x } \] Input:

integrate(1/(e*x+d)/(c*x^2+b*x+a)^(3/4),x, algorithm="maxima")
 

Output:

integrate(1/((c*x^2 + b*x + a)^(3/4)*(e*x + d)), x)
 

Giac [F]

\[ \int \frac {1}{(d+e x) \left (a+b x+c x^2\right )^{3/4}} \, dx=\int { \frac {1}{{\left (c x^{2} + b x + a\right )}^{\frac {3}{4}} {\left (e x + d\right )}} \,d x } \] Input:

integrate(1/(e*x+d)/(c*x^2+b*x+a)^(3/4),x, algorithm="giac")
 

Output:

integrate(1/((c*x^2 + b*x + a)^(3/4)*(e*x + d)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(d+e x) \left (a+b x+c x^2\right )^{3/4}} \, dx=\int \frac {1}{\left (d+e\,x\right )\,{\left (c\,x^2+b\,x+a\right )}^{3/4}} \,d x \] Input:

int(1/((d + e*x)*(a + b*x + c*x^2)^(3/4)),x)
 

Output:

int(1/((d + e*x)*(a + b*x + c*x^2)^(3/4)), x)
 

Reduce [F]

\[ \int \frac {1}{(d+e x) \left (a+b x+c x^2\right )^{3/4}} \, dx=\int \frac {1}{\left (c \,x^{2}+b x +a \right )^{\frac {3}{4}} d +\left (c \,x^{2}+b x +a \right )^{\frac {3}{4}} e x}d x \] Input:

int(1/(e*x+d)/(c*x^2+b*x+a)^(3/4),x)
                                                                                    
                                                                                    
 

Output:

int(1/((a + b*x + c*x**2)**(3/4)*d + (a + b*x + c*x**2)**(3/4)*e*x),x)