Integrand size = 22, antiderivative size = 897 \[ \int \frac {\left (a+b x+c x^2\right )^{5/4}}{(d+e x)^2} \, dx=-\frac {5 (3 c d-2 b e-c e x) \sqrt [4]{a+b x+c x^2}}{3 e^3}-\frac {\left (a+b x+c x^2\right )^{5/4}}{e (d+e x)}+\frac {5 \left (-b^2+4 a c\right )^{3/4} (2 c d-b e) \sqrt [4]{c d^2-b d e+a e^2} \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 )}{4 c^{3/4} e^{7/2} \left (a+b x+c x^2\right )^{3/4}}+\frac {5 \left (-b^2+4 a c\right )^{3/4} (2 c d-b e) \sqrt [4]{c d^2-b d e+a e^2} \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 )}{4 c^{3/4} e^{7/2} \left (a+b x+c x^2\right )^{3/4}}+\frac {5 \sqrt {-b^2+4 a c} \left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-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 )}{3 \sqrt {2} c e^4 \left (a+b x+c x^2\right )^{3/4}}+\frac {5 \left (b^2-4 a c\right ) (2 c d-b e)^2 \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 )}{4 \sqrt {2} c e^4 (b+2 c x) \left (a+b x+c x^2\right )^{3/4}}+\frac {5 \left (b^2-4 a c\right ) (2 c d-b e)^2 \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 )}{4 \sqrt {2} c e^4 (b+2 c x) \left (a+b x+c x^2\right )^{3/4}} \] Output:
-5/3*(-c*e*x-2*b*e+3*c*d)*(c*x^2+b*x+a)^(1/4)/e^3-(c*x^2+b*x+a)^(5/4)/e/(e *x+d)+5/4*(4*a*c-b^2)^(3/4)*(-b*e+2*c*d)*(a*e^2-b*d*e+c*d^2)^(1/4)*(-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)+5/4*(4*a*c-b^2)^(3/4)*(-b*e+2*c*d)*(a*e^ 2-b*d*e+c*d^2)^(1/4)*(-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)+5/6*(4*a*c- b^2)^(1/2)*(6*c^2*d^2+b^2*e^2-2*c*e*(-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/e^4/(c*x^2+b*x+a)^(3/4)+5/8*(-4*a*c+b^2)*(-b*e+2*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)*Ellipt icPi((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)+5/ 8*(-4*a*c+b^2)*(-b*e+2*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)
Time = 12.93 (sec) , antiderivative size = 658, normalized size of antiderivative = 0.73 \[ \int \frac {\left (a+b x+c x^2\right )^{5/4}}{(d+e x)^2} \, dx=\frac {5 (-3 c d+2 b e+c e x) \sqrt [4]{a+x (b+c x)}}{3 e^3}-\frac {(a+x (b+c x))^{5/4}}{e (d+e x)}+\frac {5 \left (\frac {c (a+x (b+c x))}{-b^2+4 a c}\right )^{3/4} \left (\sqrt {b^2-4 a c} \left (6 c^2 d^2+b^2 e^2+2 c e (-3 b d+a e)\right ) \operatorname {EllipticF}\left (\frac {1}{2} \arcsin \left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right ),2\right )-\frac {3 \left (-b^2+4 a c\right )^{3/4} (-2 c d+b e) \left (\sqrt {2} \sqrt [4]{c} \sqrt {e} \sqrt [4]{c d^2+e (-b d+a e)} (b+2 c x) \left (\arctan \left (\frac {\sqrt [4]{-b^2+4 a c} \sqrt {e} \sqrt [4]{\frac {c (a+x (b+c x))}{-b^2+4 a c}}}{\sqrt [4]{c} \sqrt [4]{c d^2+e (-b d+a e)}}\right )+\text {arctanh}\left (\frac {\sqrt [4]{-b^2+4 a c} \sqrt {e} \sqrt [4]{\frac {c (a+x (b+c x))}{-b^2+4 a c}}}{\sqrt [4]{c} \sqrt [4]{c d^2+e (-b d+a e)}}\right )\right )+\sqrt [4]{-b^2+4 a c} (-2 c d+b e) \sqrt {\frac {(b+2 c x)^2}{b^2-4 a c}} \operatorname {EllipticPi}\left (-\frac {\sqrt {-b^2+4 a c} e}{2 \sqrt {c} \sqrt {c d^2+e (-b d+a e)}},\arcsin \left (\sqrt {2} \sqrt [4]{\frac {c (a+x (b+c x))}{-b^2+4 a c}}\right ),-1\right )+\sqrt [4]{-b^2+4 a c} (-2 c d+b e) \sqrt {\frac {(b+2 c x)^2}{b^2-4 a c}} \operatorname {EllipticPi}\left (\frac {\sqrt {-b^2+4 a c} e}{2 \sqrt {c} \sqrt {c d^2+e (-b d+a e)}},\arcsin \left (\sqrt {2} \sqrt [4]{\frac {c (a+x (b+c x))}{-b^2+4 a c}}\right ),-1\right )\right )}{4 (b+2 c x)}\right )}{3 \sqrt {2} c e^4 (a+x (b+c x))^{3/4}} \] Input:
Integrate[(a + b*x + c*x^2)^(5/4)/(d + e*x)^2,x]
Output:
(5*(-3*c*d + 2*b*e + c*e*x)*(a + x*(b + c*x))^(1/4))/(3*e^3) - (a + x*(b + c*x))^(5/4)/(e*(d + e*x)) + (5*((c*(a + x*(b + c*x)))/(-b^2 + 4*a*c))^(3/ 4)*(Sqrt[b^2 - 4*a*c]*(6*c^2*d^2 + b^2*e^2 + 2*c*e*(-3*b*d + a*e))*Ellipti cF[ArcSin[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]]/2, 2] - (3*(-b^2 + 4*a*c)^(3/4)*( -2*c*d + b*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]))/(4*(b + 2*c*x))))/(3*Sqrt[2]*c*e^4 *(a + x*(b + c*x))^(3/4))
Time = 1.82 (sec) , antiderivative size = 1082, normalized size of antiderivative = 1.21, number of steps used = 22, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.955, Rules used = {1161, 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.
| \(\displaystyle \int \frac {\left (a+b x+c x^2\right )^{5/4}}{(d+e x)^2} \, dx\) |
| \(\Big \downarrow \) 1161 |
| \(\displaystyle \frac {5 \int \frac {(b+2 c x) \sqrt [4]{c x^2+b x+a}}{d+e x}dx}{4 e}-\frac {\left (a+b x+c x^2\right )^{5/4}}{e (d+e x)}\) |
| \(\Big \downarrow \) 1231 |
| \(\displaystyle \frac {5 \left (-\frac {\int \frac {c \left (2 d e b^2-3 \left (c d^2+a e^2\right ) b+4 a c d e-\left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right ) x\right )}{(d+e x) \left (c x^2+b x+a\right )^{3/4}}dx}{3 c e^2}-\frac {4 \sqrt [4]{a+b x+c x^2} (-2 b e+3 c d-c e x)}{3 e^2}\right )}{4 e}-\frac {\left (a+b x+c x^2\right )^{5/4}}{e (d+e x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5 \left (-\frac {\int \frac {2 d e b^2-3 \left (c d^2+a e^2\right ) b+4 a c d e-\left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right ) x}{(d+e x) \left (c x^2+b x+a\right )^{3/4}}dx}{3 e^2}-\frac {4 \sqrt [4]{a+b x+c x^2} (-2 b e+3 c d-c e x)}{3 e^2}\right )}{4 e}-\frac {\left (a+b x+c x^2\right )^{5/4}}{e (d+e x)}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {5 \left (-\frac {\frac {3 (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \int \frac {1}{(d+e x) \left (c x^2+b x+a\right )^{3/4}}dx}{e}-\frac {\left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right ) \int \frac {1}{\left (c x^2+b x+a\right )^{3/4}}dx}{e}}{3 e^2}-\frac {4 \sqrt [4]{a+b x+c x^2} (-2 b e+3 c d-c e x)}{3 e^2}\right )}{4 e}-\frac {\left (a+b x+c x^2\right )^{5/4}}{e (d+e x)}\) |
| \(\Big \downarrow \) 1094 |
| \(\displaystyle \frac {5 \left (-\frac {\frac {3 (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \int \frac {1}{(d+e x) \left (c x^2+b x+a\right )^{3/4}}dx}{e}-\frac {4 \sqrt {(b+2 c x)^2} \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right ) \int \frac {1}{\sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}d\sqrt [4]{c x^2+b x+a}}{e (b+2 c x)}}{3 e^2}-\frac {4 \sqrt [4]{a+b x+c x^2} (-2 b e+3 c d-c e x)}{3 e^2}\right )}{4 e}-\frac {\left (a+b x+c x^2\right )^{5/4}}{e (d+e x)}\) |
| \(\Big \downarrow \) 761 |
| \(\displaystyle \frac {5 \left (-\frac {\frac {3 (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \int \frac {1}{(d+e x) \left (c x^2+b x+a\right )^{3/4}}dx}{e}-\frac {\sqrt {2} \sqrt [4]{b^2-4 a c} \sqrt {(b+2 c x)^2} \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right ) \sqrt {\frac {4 c \left (a+b x+c x^2\right )-4 a c+b^2}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right )^2}} \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right ),\frac {1}{2}\right )}{\sqrt [4]{c} e (b+2 c x) \sqrt {4 c \left (a+b x+c x^2\right )-4 a c+b^2}}}{3 e^2}-\frac {4 \sqrt [4]{a+b x+c x^2} (-2 b e+3 c d-c e x)}{3 e^2}\right )}{4 e}-\frac {\left (a+b x+c x^2\right )^{5/4}}{e (d+e x)}\) |
| \(\Big \downarrow \) 1174 |
| \(\displaystyle \frac {5 \left (-\frac {\frac {3 \left (-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4} (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \int \frac {1}{(d+e x) \left (-\frac {c^2 x^2}{b^2-4 a c}-\frac {b c x}{b^2-4 a c}-\frac {a c}{b^2-4 a c}\right )^{3/4}}dx}{e \left (a+b x+c x^2\right )^{3/4}}-\frac {\sqrt {2} \sqrt [4]{b^2-4 a c} \sqrt {(b+2 c x)^2} \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right ) \sqrt {\frac {4 c \left (a+b x+c x^2\right )-4 a c+b^2}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right )^2}} \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right ),\frac {1}{2}\right )}{\sqrt [4]{c} e (b+2 c x) \sqrt {4 c \left (a+b x+c x^2\right )-4 a c+b^2}}}{3 e^2}-\frac {4 \sqrt [4]{a+b x+c x^2} (-2 b e+3 c d-c e x)}{3 e^2}\right )}{4 e}-\frac {\left (a+b x+c x^2\right )^{5/4}}{e (d+e x)}\) |
| \(\Big \downarrow \) 1173 |
| \(\displaystyle \frac {5 \left (-\frac {\frac {6 \sqrt {2} \left (-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4} (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \int -\frac {1}{\left (\frac {c (2 c d-b e)}{b^2-4 a c}-e \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )\right ) \left (1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}\right )^{3/4}}d\left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )}{e \left (a+b x+c x^2\right )^{3/4}}-\frac {\sqrt {2} \sqrt [4]{b^2-4 a c} \sqrt {(b+2 c x)^2} \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right ) \sqrt {\frac {4 c \left (a+b x+c x^2\right )-4 a c+b^2}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right )^2}} \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right ),\frac {1}{2}\right )}{\sqrt [4]{c} e (b+2 c x) \sqrt {4 c \left (a+b x+c x^2\right )-4 a c+b^2}}}{3 e^2}-\frac {4 \sqrt [4]{a+b x+c x^2} (-2 b e+3 c d-c e x)}{3 e^2}\right )}{4 e}-\frac {\left (a+b x+c x^2\right )^{5/4}}{e (d+e x)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {5 \left (-\frac {-\frac {6 \sqrt {2} \left (-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4} (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \int \frac {1}{\left (\frac {c (2 c d-b e)}{b^2-4 a c}-e \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )\right ) \left (1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}\right )^{3/4}}d\left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )}{e \left (a+b x+c x^2\right )^{3/4}}-\frac {\sqrt {2} \sqrt [4]{b^2-4 a c} \sqrt {(b+2 c x)^2} \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right ) \sqrt {\frac {4 c \left (a+b x+c x^2\right )-4 a c+b^2}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right )^2}} \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right ),\frac {1}{2}\right )}{\sqrt [4]{c} e (b+2 c x) \sqrt {4 c \left (a+b x+c x^2\right )-4 a c+b^2}}}{3 e^2}-\frac {4 \sqrt [4]{a+b x+c x^2} (-2 b e+3 c d-c e x)}{3 e^2}\right )}{4 e}-\frac {\left (a+b x+c x^2\right )^{5/4}}{e (d+e x)}\) |
| \(\Big \downarrow \) 504 |
| \(\displaystyle \frac {5 \left (-\frac {\frac {6 \sqrt {2} \left (-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4} (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-\frac {c (2 c d-b e) \int \frac {1}{\left (1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}\right )^{3/4} \left (\frac {c^2 (2 c d-b e)^2}{\left (b^2-4 a c\right )^2}-e^2 \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2\right )}d\left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )}{b^2-4 a c}-e \int \frac {-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}}{\left (1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}\right )^{3/4} \left (\frac {c^2 (2 c d-b e)^2}{\left (b^2-4 a c\right )^2}-e^2 \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2\right )}d\left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )\right )}{e \left (a+b x+c x^2\right )^{3/4}}-\frac {\sqrt {2} \sqrt [4]{b^2-4 a c} \sqrt {(b+2 c x)^2} \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right ) \sqrt {\frac {4 c \left (a+b x+c x^2\right )-4 a c+b^2}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right )^2}} \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right ),\frac {1}{2}\right )}{\sqrt [4]{c} e (b+2 c x) \sqrt {4 c \left (a+b x+c x^2\right )-4 a c+b^2}}}{3 e^2}-\frac {4 \sqrt [4]{a+b x+c x^2} (-2 b e+3 c d-c e x)}{3 e^2}\right )}{4 e}-\frac {\left (a+b x+c x^2\right )^{5/4}}{e (d+e x)}\) |
| \(\Big \downarrow \) 312 |
| \(\displaystyle \frac {5 \left (-\frac {4 \sqrt [4]{c x^2+b x+a} (3 c d-2 b e-c e x)}{3 e^2}-\frac {\frac {6 \sqrt {2} (2 c d-b e) \left (c d^2-b e d+a e^2\right ) \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} \left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-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 )}}}{3 e^2}\right )}{4 e}-\frac {\left (c x^2+b x+a\right )^{5/4}}{e (d+e x)}\) |
| \(\Big \downarrow \) 118 |
| \(\displaystyle \frac {5 \left (-\frac {4 \sqrt [4]{c x^2+b x+a} (3 c d-2 b e-c e x)}{3 e^2}-\frac {\frac {6 \sqrt {2} (2 c d-b e) \left (c d^2-b e d+a e^2\right ) \left (-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}\right )^{3/4} \left (\frac {2 c (2 c d-b e) \sqrt {\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}} \int \frac {1}{\sqrt {1-\left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^8} \left (e^2 \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^8+\frac {4 c \left (c d^2-b e d+a e^2\right )}{b^2-4 a c}\right )}d\sqrt [4]{1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}}}{\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )}-e \int \frac {-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}}{\left (1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}\right )^{3/4} \left (\frac {c^2 (2 c d-b e)^2}{\left (b^2-4 a c\right )^2}-e^2 \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2\right )}d\left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )\right )}{e \left (c x^2+b x+a\right )^{3/4}}-\frac {\sqrt {2} \sqrt [4]{b^2-4 a c} \left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right ) \sqrt {(b+2 c x)^2} \left (\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}+1\right ) \sqrt {\frac {b^2-4 a c+4 c \left (c x^2+b x+a\right )}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right ),\frac {1}{2}\right )}{\sqrt [4]{c} e (b+2 c x) \sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}}{3 e^2}\right )}{4 e}-\frac {\left (c x^2+b x+a\right )^{5/4}}{e (d+e x)}\) |
| \(\Big \downarrow \) 353 |
| \(\displaystyle \frac {5 \left (-\frac {4 \sqrt [4]{c x^2+b x+a} (3 c d-2 b e-c e x)}{3 e^2}-\frac {\frac {6 \sqrt {2} (2 c d-b e) \left (c d^2-b e d+a e^2\right ) \left (-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}\right )^{3/4} \left (\frac {2 c (2 c d-b e) \sqrt {\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}} \int \frac {1}{\sqrt {1-\left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^8} \left (e^2 \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^8+\frac {4 c \left (c d^2-b e d+a e^2\right )}{b^2-4 a c}\right )}d\sqrt [4]{1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}}}{\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )}-\frac {1}{2} e \int \frac {1}{\left (1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}\right )^{3/4} \left (\frac {c^2 (2 c d-b e)^2}{\left (b^2-4 a c\right )^2}-e^2 \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2\right )}d\left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2\right )}{e \left (c x^2+b x+a\right )^{3/4}}-\frac {\sqrt {2} \sqrt [4]{b^2-4 a c} \left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right ) \sqrt {(b+2 c x)^2} \left (\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}+1\right ) \sqrt {\frac {b^2-4 a c+4 c \left (c x^2+b x+a\right )}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right ),\frac {1}{2}\right )}{\sqrt [4]{c} e (b+2 c x) \sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}}{3 e^2}\right )}{4 e}-\frac {\left (c x^2+b x+a\right )^{5/4}}{e (d+e x)}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {5 \left (-\frac {4 \sqrt [4]{c x^2+b x+a} (3 c d-2 b e-c e x)}{3 e^2}-\frac {\frac {6 \sqrt {2} (2 c d-b e) \left (c d^2-b e d+a e^2\right ) \left (-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}\right )^{3/4} \left (\frac {2 e \int \frac {1}{\frac {c^2 e^2 \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^8}{b^2-4 a c}+\frac {4 c^3 \left (c d^2-b e d+a e^2\right )}{\left (b^2-4 a c\right )^2}}d\sqrt [4]{1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}} c^2}{b^2-4 a c}+\frac {2 (2 c d-b e) \sqrt {\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}} \int \frac {1}{\sqrt {1-\left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^8} \left (e^2 \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^8+\frac {4 c \left (c d^2-b e d+a e^2\right )}{b^2-4 a c}\right )}d\sqrt [4]{1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}} c}{\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )}\right )}{e \left (c x^2+b x+a\right )^{3/4}}-\frac {\sqrt {2} \sqrt [4]{b^2-4 a c} \left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right ) \sqrt {(b+2 c x)^2} \left (\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}+1\right ) \sqrt {\frac {b^2-4 a c+4 c \left (c x^2+b x+a\right )}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right ),\frac {1}{2}\right )}{\sqrt [4]{c} e (b+2 c x) \sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}}{3 e^2}\right )}{4 e}-\frac {\left (c x^2+b x+a\right )^{5/4}}{e (d+e x)}\) |
| \(\Big \downarrow \) 756 |
| \(\displaystyle \frac {5 \left (-\frac {4 \sqrt [4]{c x^2+b x+a} (3 c d-2 b e-c e x)}{3 e^2}-\frac {\frac {6 \sqrt {2} (2 c d-b e) \left (c d^2-b e d+a e^2\right ) \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} \left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-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 )}}}{3 e^2}\right )}{4 e}-\frac {\left (c x^2+b x+a\right )^{5/4}}{e (d+e x)}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {5 \left (-\frac {4 \sqrt [4]{c x^2+b x+a} (3 c d-2 b e-c e x)}{3 e^2}-\frac {\frac {6 \sqrt {2} (2 c d-b e) \left (c d^2-b e d+a e^2\right ) \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} \left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-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 )}}}{3 e^2}\right )}{4 e}-\frac {\left (c x^2+b x+a\right )^{5/4}}{e (d+e x)}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {5 \left (-\frac {4 \sqrt [4]{c x^2+b x+a} (3 c d-2 b e-c e x)}{3 e^2}-\frac {\frac {6 \sqrt {2} (2 c d-b e) \left (c d^2-b e d+a e^2\right ) \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} \left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-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 )}}}{3 e^2}\right )}{4 e}-\frac {\left (c x^2+b x+a\right )^{5/4}}{e (d+e x)}\) |
| \(\Big \downarrow \) 925 |
| \(\displaystyle \frac {5 \left (-\frac {4 \sqrt [4]{c x^2+b x+a} (3 c d-2 b e-c e x)}{3 e^2}-\frac {\frac {6 \sqrt {2} (2 c d-b e) \left (c d^2-b e d+a e^2\right ) \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} \left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-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 )}}}{3 e^2}\right )}{4 e}-\frac {\left (c x^2+b x+a\right )^{5/4}}{e (d+e x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5 \left (-\frac {4 \sqrt [4]{c x^2+b x+a} (3 c d-2 b e-c e x)}{3 e^2}-\frac {\frac {6 \sqrt {2} (2 c d-b e) \left (c d^2-b e d+a e^2\right ) \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} \left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-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 )}}}{3 e^2}\right )}{4 e}-\frac {\left (c x^2+b x+a\right )^{5/4}}{e (d+e x)}\) |
| \(\Big \downarrow \) 1537 |
| \(\displaystyle \frac {5 \left (-\frac {4 \sqrt [4]{c x^2+b x+a} (3 c d-2 b e-c e x)}{3 e^2}-\frac {\frac {6 \sqrt {2} (2 c d-b e) \left (c d^2-b e d+a e^2\right ) \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} \left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-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 )}}}{3 e^2}\right )}{4 e}-\frac {\left (c x^2+b x+a\right )^{5/4}}{e (d+e x)}\) |
| \(\Big \downarrow \) 412 |
| \(\displaystyle \frac {5 \left (-\frac {4 \sqrt [4]{c x^2+b x+a} (3 c d-2 b e-c e x)}{3 e^2}-\frac {\frac {6 \sqrt {2} (2 c d-b e) \left (c d^2-b e d+a e^2\right ) \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} \left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-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 )}}}{3 e^2}\right )}{4 e}-\frac {\left (c x^2+b x+a\right )^{5/4}}{e (d+e x)}\) |
Input:
Int[(a + b*x + c*x^2)^(5/4)/(d + e*x)^2,x]
Output:
-((a + b*x + c*x^2)^(5/4)/(e*(d + e*x))) + (5*((-4*(3*c*d - 2*b*e - c*e*x) *(a + b*x + c*x^2)^(1/4))/(3*e^2) - (-((Sqrt[2]*(b^2 - 4*a*c)^(1/4)*(6*c^2 *d^2 + b^2*e^2 - 2*c*e*(3*b*d - 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)]*EllipticF[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)])) + (6*Sqrt[2]*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^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)*Ell ipticPi[-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 ...
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
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]
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]
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]
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]
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]
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])
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]
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]
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]
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]
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]
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 + 1))), x] - Si mp[p/(e*(m + 1)) Int[(d + e*x)^(m + 1)*(b + 2*c*x)*(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && GtQ[p, 0] && (IntegerQ[p] || LtQ[m, -1]) && NeQ[m, -1] && !ILtQ[m + 2*p + 1, 0] && IntQuadraticQ[a, b, c, d, e, m, p, x]
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]
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]
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])
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]
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]
\[\int \frac {\left (c \,x^{2}+b x +a \right )^{\frac {5}{4}}}{\left (e x +d \right )^{2}}d x\]
Input:
int((c*x^2+b*x+a)^(5/4)/(e*x+d)^2,x)
Output:
int((c*x^2+b*x+a)^(5/4)/(e*x+d)^2,x)
Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^{5/4}}{(d+e x)^2} \, dx=\text {Timed out} \] Input:
integrate((c*x^2+b*x+a)^(5/4)/(e*x+d)^2,x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {\left (a+b x+c x^2\right )^{5/4}}{(d+e x)^2} \, dx=\int \frac {\left (a + b x + c x^{2}\right )^{\frac {5}{4}}}{\left (d + e x\right )^{2}}\, dx \] Input:
integrate((c*x**2+b*x+a)**(5/4)/(e*x+d)**2,x)
Output:
Integral((a + b*x + c*x**2)**(5/4)/(d + e*x)**2, x)
\[ \int \frac {\left (a+b x+c x^2\right )^{5/4}}{(d+e x)^2} \, dx=\int { \frac {{\left (c x^{2} + b x + a\right )}^{\frac {5}{4}}}{{\left (e x + d\right )}^{2}} \,d x } \] Input:
integrate((c*x^2+b*x+a)^(5/4)/(e*x+d)^2,x, algorithm="maxima")
Output:
integrate((c*x^2 + b*x + a)^(5/4)/(e*x + d)^2, x)
Exception generated. \[ \int \frac {\left (a+b x+c x^2\right )^{5/4}}{(d+e x)^2} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((c*x^2+b*x+a)^(5/4)/(e*x+d)^2,x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro unding error%%%{1,[0,1,1,2,0,0,0]%%%}+%%%{-1,[0,1,0,1,1,1,0]%%%}+%%%{1,[0, 1,0,0,2,0
Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^{5/4}}{(d+e x)^2} \, dx=\int \frac {{\left (c\,x^2+b\,x+a\right )}^{5/4}}{{\left (d+e\,x\right )}^2} \,d x \] Input:
int((a + b*x + c*x^2)^(5/4)/(d + e*x)^2,x)
Output:
int((a + b*x + c*x^2)^(5/4)/(d + e*x)^2, x)
\[ \int \frac {\left (a+b x+c x^2\right )^{5/4}}{(d+e x)^2} \, dx=\text {too large to display} \] Input:
int((c*x^2+b*x+a)^(5/4)/(e*x+d)^2,x)
Output:
( - 96*(a + b*x + c*x**2)**(1/4)*a*b*e**2 + 64*(a + b*x + c*x**2)**(1/4)*a *c*d*e + 140*(a + b*x + c*x**2)**(1/4)*b**2*d*e + 84*(a + b*x + c*x**2)**( 1/4)*b**2*e**2*x - 200*(a + b*x + c*x**2)**(1/4)*b*c*d**2 - 176*(a + b*x + c*x**2)**(1/4)*b*c*d*e*x + 24*(a + b*x + c*x**2)**(1/4)*b*c*e**2*x**2 + 8 0*(a + b*x + c*x**2)**(1/4)*c**2*d**2*x - 16*(a + b*x + c*x**2)**(1/4)*c** 2*d*e*x**2 - 180*int((a + b*x + c*x**2)**(1/4)/(3*a*b*d**2*e + 6*a*b*d*e** 2*x + 3*a*b*e**3*x**2 - 2*a*c*d**3 - 4*a*c*d**2*e*x - 2*a*c*d*e**2*x**2 + 3*b**2*d**2*e*x + 6*b**2*d*e**2*x**2 + 3*b**2*e**3*x**3 - 2*b*c*d**3*x - b *c*d**2*e*x**2 + 4*b*c*d*e**2*x**3 + 3*b*c*e**3*x**4 - 2*c**2*d**3*x**2 - 4*c**2*d**2*e*x**3 - 2*c**2*d*e**2*x**4),x)*a**2*b**2*d*e**4 - 180*int((a + b*x + c*x**2)**(1/4)/(3*a*b*d**2*e + 6*a*b*d*e**2*x + 3*a*b*e**3*x**2 - 2*a*c*d**3 - 4*a*c*d**2*e*x - 2*a*c*d*e**2*x**2 + 3*b**2*d**2*e*x + 6*b**2 *d*e**2*x**2 + 3*b**2*e**3*x**3 - 2*b*c*d**3*x - b*c*d**2*e*x**2 + 4*b*c*d *e**2*x**3 + 3*b*c*e**3*x**4 - 2*c**2*d**3*x**2 - 4*c**2*d**2*e*x**3 - 2*c **2*d*e**2*x**4),x)*a**2*b**2*e**5*x + 240*int((a + b*x + c*x**2)**(1/4)/( 3*a*b*d**2*e + 6*a*b*d*e**2*x + 3*a*b*e**3*x**2 - 2*a*c*d**3 - 4*a*c*d**2* e*x - 2*a*c*d*e**2*x**2 + 3*b**2*d**2*e*x + 6*b**2*d*e**2*x**2 + 3*b**2*e* *3*x**3 - 2*b*c*d**3*x - b*c*d**2*e*x**2 + 4*b*c*d*e**2*x**3 + 3*b*c*e**3* x**4 - 2*c**2*d**3*x**2 - 4*c**2*d**2*e*x**3 - 2*c**2*d*e**2*x**4),x)*a**2 *b*c*d**2*e**3 + 240*int((a + b*x + c*x**2)**(1/4)/(3*a*b*d**2*e + 6*a*...