Integrand size = 27, antiderivative size = 417 \[ \int \frac {1}{(e x)^{3/2} (c+d x) \left (a-b x^2\right )^{3/2}} \, dx=\frac {b (c-d x)}{a \left (b c^2-a d^2\right ) e \sqrt {e x} \sqrt {a-b x^2}}-\frac {\left (3 b c^2-2 a d^2\right ) \sqrt {a-b x^2}}{a^2 c \left (b c^2-a d^2\right ) e \sqrt {e x}}-\frac {\sqrt [4]{b} \left (3 b c^2-2 a d^2\right ) \sqrt {1-\frac {b x^2}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )\right |-1\right )}{a^{5/4} c \left (b c^2-a d^2\right ) e^{3/2} \sqrt {a-b x^2}}+\frac {\sqrt [4]{b} \left (3 \sqrt {b} c+2 \sqrt {a} d\right ) \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{a^{5/4} c \left (\sqrt {b} c+\sqrt {a} d\right ) e^{3/2} \sqrt {a-b x^2}}+\frac {2 \sqrt [4]{a} d^3 \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{\sqrt {b} c},\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} c^2 \left (b c^2-a d^2\right ) e^{3/2} \sqrt {a-b x^2}} \] Output:
b*(-d*x+c)/a/(-a*d^2+b*c^2)/e/(e*x)^(1/2)/(-b*x^2+a)^(1/2)-(-2*a*d^2+3*b*c ^2)*(-b*x^2+a)^(1/2)/a^2/c/(-a*d^2+b*c^2)/e/(e*x)^(1/2)-b^(1/4)*(-2*a*d^2+ 3*b*c^2)*(1-b*x^2/a)^(1/2)*EllipticE(b^(1/4)*(e*x)^(1/2)/a^(1/4)/e^(1/2),I )/a^(5/4)/c/(-a*d^2+b*c^2)/e^(3/2)/(-b*x^2+a)^(1/2)+b^(1/4)*(3*b^(1/2)*c+2 *a^(1/2)*d)*(1-b*x^2/a)^(1/2)*EllipticF(b^(1/4)*(e*x)^(1/2)/a^(1/4)/e^(1/2 ),I)/a^(5/4)/c/(b^(1/2)*c+a^(1/2)*d)/e^(3/2)/(-b*x^2+a)^(1/2)+2*a^(1/4)*d^ 3*(1-b*x^2/a)^(1/2)*EllipticPi(b^(1/4)*(e*x)^(1/2)/a^(1/4)/e^(1/2),-a^(1/2 )*d/b^(1/2)/c,I)/b^(1/4)/c^2/(-a*d^2+b*c^2)/e^(3/2)/(-b*x^2+a)^(1/2)
Result contains complex when optimal does not.
Time = 24.55 (sec) , antiderivative size = 337, normalized size of antiderivative = 0.81 \[ \int \frac {1}{(e x)^{3/2} (c+d x) \left (a-b x^2\right )^{3/2}} \, dx=\frac {x \left (-i \sqrt {b} c \left (3 b c^2-2 a d^2\right ) \sqrt {1-\frac {a}{b x^2}} x^{3/2} E\left (\left .i \text {arcsinh}\left (\frac {\sqrt {-\frac {\sqrt {a}}{\sqrt {b}}}}{\sqrt {x}}\right )\right |-1\right )+i \left (3 b^{3/2} c^3+\sqrt {a} b c^2 d-2 a \sqrt {b} c d^2-2 a^{3/2} d^3\right ) \sqrt {1-\frac {a}{b x^2}} x^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-\frac {\sqrt {a}}{\sqrt {b}}}}{\sqrt {x}}\right ),-1\right )+\sqrt {a} \left (\sqrt {-\frac {\sqrt {a}}{\sqrt {b}}} b c^2 (-c+d x)+2 i a d^3 \sqrt {1-\frac {a}{b x^2}} x^{3/2} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} c}{\sqrt {a} d},i \text {arcsinh}\left (\frac {\sqrt {-\frac {\sqrt {a}}{\sqrt {b}}}}{\sqrt {x}}\right ),-1\right )\right )\right )}{a^{3/2} \sqrt {-\frac {\sqrt {a}}{\sqrt {b}}} c^2 \left (-b c^2+a d^2\right ) (e x)^{3/2} \sqrt {a-b x^2}} \] Input:
Integrate[1/((e*x)^(3/2)*(c + d*x)*(a - b*x^2)^(3/2)),x]
Output:
(x*((-I)*Sqrt[b]*c*(3*b*c^2 - 2*a*d^2)*Sqrt[1 - a/(b*x^2)]*x^(3/2)*Ellipti cE[I*ArcSinh[Sqrt[-(Sqrt[a]/Sqrt[b])]/Sqrt[x]], -1] + I*(3*b^(3/2)*c^3 + S qrt[a]*b*c^2*d - 2*a*Sqrt[b]*c*d^2 - 2*a^(3/2)*d^3)*Sqrt[1 - a/(b*x^2)]*x^ (3/2)*EllipticF[I*ArcSinh[Sqrt[-(Sqrt[a]/Sqrt[b])]/Sqrt[x]], -1] + Sqrt[a] *(Sqrt[-(Sqrt[a]/Sqrt[b])]*b*c^2*(-c + d*x) + (2*I)*a*d^3*Sqrt[1 - a/(b*x^ 2)]*x^(3/2)*EllipticPi[-((Sqrt[b]*c)/(Sqrt[a]*d)), I*ArcSinh[Sqrt[-(Sqrt[a ]/Sqrt[b])]/Sqrt[x]], -1])))/(a^(3/2)*Sqrt[-(Sqrt[a]/Sqrt[b])]*c^2*(-(b*c^ 2) + a*d^2)*(e*x)^(3/2)*Sqrt[a - b*x^2])
Time = 1.47 (sec) , antiderivative size = 423, normalized size of antiderivative = 1.01, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.593, Rules used = {616, 27, 1645, 1543, 1542, 2368, 2374, 9, 27, 1513, 27, 765, 762, 1390, 1389, 327}
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.
| \(\displaystyle \int \frac {1}{(e x)^{3/2} \left (a-b x^2\right )^{3/2} (c+d x)} \, dx\) |
| \(\Big \downarrow \) 616 |
| \(\displaystyle \frac {2 \int \frac {1}{x (c e+d x e) \left (a-b x^2\right )^{3/2}}d\sqrt {e x}}{e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \int \frac {1}{e x (c e+d x e) \left (a-b x^2\right )^{3/2}}d\sqrt {e x}\) |
| \(\Big \downarrow \) 1645 |
| \(\displaystyle 2 \left (\frac {d \int \frac {\frac {b c^2}{d}-b x c+b d x^2-a d}{e x \left (a-b x^2\right )^{3/2}}d\sqrt {e x}}{c e \left (b c^2-a d^2\right )}+\frac {d^3 \int \frac {1}{(c e+d x e) \sqrt {a-b x^2}}d\sqrt {e x}}{c e \left (b c^2-a d^2\right )}\right )\) |
| \(\Big \downarrow \) 1543 |
| \(\displaystyle 2 \left (\frac {d \int \frac {\frac {b c^2}{d}-b x c+b d x^2-a d}{e x \left (a-b x^2\right )^{3/2}}d\sqrt {e x}}{c e \left (b c^2-a d^2\right )}+\frac {d^3 \sqrt {1-\frac {b x^2}{a}} \int \frac {1}{(c e+d x e) \sqrt {1-\frac {b x^2}{a}}}d\sqrt {e x}}{c e \sqrt {a-b x^2} \left (b c^2-a d^2\right )}\right )\) |
| \(\Big \downarrow \) 1542 |
| \(\displaystyle 2 \left (\frac {d \int \frac {\frac {b c^2}{d}-b x c+b d x^2-a d}{e x \left (a-b x^2\right )^{3/2}}d\sqrt {e x}}{c e \left (b c^2-a d^2\right )}+\frac {\sqrt [4]{a} d^3 \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{\sqrt {b} c},\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} c^2 e^{3/2} \sqrt {a-b x^2} \left (b c^2-a d^2\right )}\right )\) |
| \(\Big \downarrow \) 2368 |
| \(\displaystyle 2 \left (\frac {d \left (\frac {e^2 \int \frac {-\frac {c^2 x^2 b^3}{a d e^2}-\frac {c x b^2}{e^2}+\frac {2 \left (b c^2-a d^2\right ) b}{d e^2}}{e x \sqrt {a-b x^2}}d\sqrt {e x}}{2 a b}-\frac {b c \sqrt {e x} (a d e-b c e x)}{2 a^2 d e^2 \sqrt {a-b x^2}}\right )}{c e \left (b c^2-a d^2\right )}+\frac {\sqrt [4]{a} d^3 \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{\sqrt {b} c},\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} c^2 e^{3/2} \sqrt {a-b x^2} \left (b c^2-a d^2\right )}\right )\) |
| \(\Big \downarrow \) 2374 |
| \(\displaystyle 2 \left (\frac {d \left (\frac {e^2 \left (-\frac {\int \frac {2 \left (\frac {\left (\frac {3 b c^2}{d}-2 a d\right ) (e x)^{3/2} b^2}{e^4}+\frac {a c \sqrt {e x} b^2}{e^3}\right )}{\sqrt {e x} \sqrt {a-b x^2}}d\sqrt {e x}}{2 a}-\frac {2 b \sqrt {a-b x^2} \left (b c^2-a d^2\right )}{a d e^2 \sqrt {e x}}\right )}{2 a b}-\frac {b c \sqrt {e x} (a d e-b c e x)}{2 a^2 d e^2 \sqrt {a-b x^2}}\right )}{c e \left (b c^2-a d^2\right )}+\frac {\sqrt [4]{a} d^3 \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{\sqrt {b} c},\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} c^2 e^{3/2} \sqrt {a-b x^2} \left (b c^2-a d^2\right )}\right )\) |
| \(\Big \downarrow \) 9 |
| \(\displaystyle 2 \left (\frac {d \left (\frac {e^2 \left (-\frac {\int \frac {2 b^2 \left (a c e+\left (\frac {3 b c^2}{d}-2 a d\right ) x e\right )}{e^4 \sqrt {a-b x^2}}d\sqrt {e x}}{2 a}-\frac {2 b \sqrt {a-b x^2} \left (b c^2-a d^2\right )}{a d e^2 \sqrt {e x}}\right )}{2 a b}-\frac {b c \sqrt {e x} (a d e-b c e x)}{2 a^2 d e^2 \sqrt {a-b x^2}}\right )}{c e \left (b c^2-a d^2\right )}+\frac {\sqrt [4]{a} d^3 \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{\sqrt {b} c},\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} c^2 e^{3/2} \sqrt {a-b x^2} \left (b c^2-a d^2\right )}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \left (\frac {d \left (\frac {e^2 \left (-\frac {b^2 \int \frac {a c e+\left (\frac {3 b c^2}{d}-2 a d\right ) x e}{\sqrt {a-b x^2}}d\sqrt {e x}}{a e^4}-\frac {2 b \sqrt {a-b x^2} \left (b c^2-a d^2\right )}{a d e^2 \sqrt {e x}}\right )}{2 a b}-\frac {b c \sqrt {e x} (a d e-b c e x)}{2 a^2 d e^2 \sqrt {a-b x^2}}\right )}{c e \left (b c^2-a d^2\right )}+\frac {\sqrt [4]{a} d^3 \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{\sqrt {b} c},\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} c^2 e^{3/2} \sqrt {a-b x^2} \left (b c^2-a d^2\right )}\right )\) |
| \(\Big \downarrow \) 1513 |
| \(\displaystyle 2 \left (\frac {d \left (\frac {e^2 \left (-\frac {b^2 \left (\frac {\sqrt {a} e \left (\sqrt {a} \sqrt {b} c+2 a d-\frac {3 b c^2}{d}\right ) \int \frac {1}{\sqrt {a-b x^2}}d\sqrt {e x}}{\sqrt {b}}+\frac {\sqrt {a} e \left (\frac {3 b c^2}{d}-2 a d\right ) \int \frac {\sqrt {b} x e+\sqrt {a} e}{\sqrt {a} e \sqrt {a-b x^2}}d\sqrt {e x}}{\sqrt {b}}\right )}{a e^4}-\frac {2 b \sqrt {a-b x^2} \left (b c^2-a d^2\right )}{a d e^2 \sqrt {e x}}\right )}{2 a b}-\frac {b c \sqrt {e x} (a d e-b c e x)}{2 a^2 d e^2 \sqrt {a-b x^2}}\right )}{c e \left (b c^2-a d^2\right )}+\frac {\sqrt [4]{a} d^3 \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{\sqrt {b} c},\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} c^2 e^{3/2} \sqrt {a-b x^2} \left (b c^2-a d^2\right )}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \left (\frac {d \left (\frac {e^2 \left (-\frac {b^2 \left (\frac {\sqrt {a} e \left (\sqrt {a} \sqrt {b} c+2 a d-\frac {3 b c^2}{d}\right ) \int \frac {1}{\sqrt {a-b x^2}}d\sqrt {e x}}{\sqrt {b}}+\frac {\left (\frac {3 b c^2}{d}-2 a d\right ) \int \frac {\sqrt {b} x e+\sqrt {a} e}{\sqrt {a-b x^2}}d\sqrt {e x}}{\sqrt {b}}\right )}{a e^4}-\frac {2 b \sqrt {a-b x^2} \left (b c^2-a d^2\right )}{a d e^2 \sqrt {e x}}\right )}{2 a b}-\frac {b c \sqrt {e x} (a d e-b c e x)}{2 a^2 d e^2 \sqrt {a-b x^2}}\right )}{c e \left (b c^2-a d^2\right )}+\frac {\sqrt [4]{a} d^3 \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{\sqrt {b} c},\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} c^2 e^{3/2} \sqrt {a-b x^2} \left (b c^2-a d^2\right )}\right )\) |
| \(\Big \downarrow \) 765 |
| \(\displaystyle 2 \left (\frac {d \left (\frac {e^2 \left (-\frac {b^2 \left (\frac {\left (\frac {3 b c^2}{d}-2 a d\right ) \int \frac {\sqrt {b} x e+\sqrt {a} e}{\sqrt {a-b x^2}}d\sqrt {e x}}{\sqrt {b}}+\frac {\sqrt {a} e \sqrt {1-\frac {b x^2}{a}} \left (\sqrt {a} \sqrt {b} c+2 a d-\frac {3 b c^2}{d}\right ) \int \frac {1}{\sqrt {1-\frac {b x^2}{a}}}d\sqrt {e x}}{\sqrt {b} \sqrt {a-b x^2}}\right )}{a e^4}-\frac {2 b \sqrt {a-b x^2} \left (b c^2-a d^2\right )}{a d e^2 \sqrt {e x}}\right )}{2 a b}-\frac {b c \sqrt {e x} (a d e-b c e x)}{2 a^2 d e^2 \sqrt {a-b x^2}}\right )}{c e \left (b c^2-a d^2\right )}+\frac {\sqrt [4]{a} d^3 \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{\sqrt {b} c},\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} c^2 e^{3/2} \sqrt {a-b x^2} \left (b c^2-a d^2\right )}\right )\) |
| \(\Big \downarrow \) 762 |
| \(\displaystyle 2 \left (\frac {d \left (\frac {e^2 \left (-\frac {b^2 \left (\frac {\left (\frac {3 b c^2}{d}-2 a d\right ) \int \frac {\sqrt {b} x e+\sqrt {a} e}{\sqrt {a-b x^2}}d\sqrt {e x}}{\sqrt {b}}+\frac {a^{3/4} e^{3/2} \sqrt {1-\frac {b x^2}{a}} \left (\sqrt {a} \sqrt {b} c+2 a d-\frac {3 b c^2}{d}\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{b^{3/4} \sqrt {a-b x^2}}\right )}{a e^4}-\frac {2 b \sqrt {a-b x^2} \left (b c^2-a d^2\right )}{a d e^2 \sqrt {e x}}\right )}{2 a b}-\frac {b c \sqrt {e x} (a d e-b c e x)}{2 a^2 d e^2 \sqrt {a-b x^2}}\right )}{c e \left (b c^2-a d^2\right )}+\frac {\sqrt [4]{a} d^3 \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{\sqrt {b} c},\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} c^2 e^{3/2} \sqrt {a-b x^2} \left (b c^2-a d^2\right )}\right )\) |
| \(\Big \downarrow \) 1390 |
| \(\displaystyle 2 \left (\frac {d \left (\frac {e^2 \left (-\frac {b^2 \left (\frac {\sqrt {1-\frac {b x^2}{a}} \left (\frac {3 b c^2}{d}-2 a d\right ) \int \frac {\sqrt {b} x e+\sqrt {a} e}{\sqrt {1-\frac {b x^2}{a}}}d\sqrt {e x}}{\sqrt {b} \sqrt {a-b x^2}}+\frac {a^{3/4} e^{3/2} \sqrt {1-\frac {b x^2}{a}} \left (\sqrt {a} \sqrt {b} c+2 a d-\frac {3 b c^2}{d}\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{b^{3/4} \sqrt {a-b x^2}}\right )}{a e^4}-\frac {2 b \sqrt {a-b x^2} \left (b c^2-a d^2\right )}{a d e^2 \sqrt {e x}}\right )}{2 a b}-\frac {b c \sqrt {e x} (a d e-b c e x)}{2 a^2 d e^2 \sqrt {a-b x^2}}\right )}{c e \left (b c^2-a d^2\right )}+\frac {\sqrt [4]{a} d^3 \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{\sqrt {b} c},\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} c^2 e^{3/2} \sqrt {a-b x^2} \left (b c^2-a d^2\right )}\right )\) |
| \(\Big \downarrow \) 1389 |
| \(\displaystyle 2 \left (\frac {d \left (\frac {e^2 \left (-\frac {b^2 \left (\frac {\sqrt {a} e \sqrt {1-\frac {b x^2}{a}} \left (\frac {3 b c^2}{d}-2 a d\right ) \int \frac {\sqrt {\frac {\sqrt {b} x}{\sqrt {a}}+1}}{\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}d\sqrt {e x}}{\sqrt {b} \sqrt {a-b x^2}}+\frac {a^{3/4} e^{3/2} \sqrt {1-\frac {b x^2}{a}} \left (\sqrt {a} \sqrt {b} c+2 a d-\frac {3 b c^2}{d}\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{b^{3/4} \sqrt {a-b x^2}}\right )}{a e^4}-\frac {2 b \sqrt {a-b x^2} \left (b c^2-a d^2\right )}{a d e^2 \sqrt {e x}}\right )}{2 a b}-\frac {b c \sqrt {e x} (a d e-b c e x)}{2 a^2 d e^2 \sqrt {a-b x^2}}\right )}{c e \left (b c^2-a d^2\right )}+\frac {\sqrt [4]{a} d^3 \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{\sqrt {b} c},\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} c^2 e^{3/2} \sqrt {a-b x^2} \left (b c^2-a d^2\right )}\right )\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle 2 \left (\frac {d \left (\frac {e^2 \left (-\frac {b^2 \left (\frac {a^{3/4} e^{3/2} \sqrt {1-\frac {b x^2}{a}} \left (\sqrt {a} \sqrt {b} c+2 a d-\frac {3 b c^2}{d}\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{b^{3/4} \sqrt {a-b x^2}}+\frac {a^{3/4} e^{3/2} \sqrt {1-\frac {b x^2}{a}} \left (\frac {3 b c^2}{d}-2 a d\right ) E\left (\left .\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )\right |-1\right )}{b^{3/4} \sqrt {a-b x^2}}\right )}{a e^4}-\frac {2 b \sqrt {a-b x^2} \left (b c^2-a d^2\right )}{a d e^2 \sqrt {e x}}\right )}{2 a b}-\frac {b c \sqrt {e x} (a d e-b c e x)}{2 a^2 d e^2 \sqrt {a-b x^2}}\right )}{c e \left (b c^2-a d^2\right )}+\frac {\sqrt [4]{a} d^3 \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{\sqrt {b} c},\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} c^2 e^{3/2} \sqrt {a-b x^2} \left (b c^2-a d^2\right )}\right )\) |
Input:
Int[1/((e*x)^(3/2)*(c + d*x)*(a - b*x^2)^(3/2)),x]
Output:
2*((d*(-1/2*(b*c*Sqrt[e*x]*(a*d*e - b*c*e*x))/(a^2*d*e^2*Sqrt[a - b*x^2]) + (e^2*((-2*b*(b*c^2 - a*d^2)*Sqrt[a - b*x^2])/(a*d*e^2*Sqrt[e*x]) - (b^2* ((a^(3/4)*((3*b*c^2)/d - 2*a*d)*e^(3/2)*Sqrt[1 - (b*x^2)/a]*EllipticE[ArcS in[(b^(1/4)*Sqrt[e*x])/(a^(1/4)*Sqrt[e])], -1])/(b^(3/4)*Sqrt[a - b*x^2]) + (a^(3/4)*(Sqrt[a]*Sqrt[b]*c - (3*b*c^2)/d + 2*a*d)*e^(3/2)*Sqrt[1 - (b*x ^2)/a]*EllipticF[ArcSin[(b^(1/4)*Sqrt[e*x])/(a^(1/4)*Sqrt[e])], -1])/(b^(3 /4)*Sqrt[a - b*x^2])))/(a*e^4)))/(2*a*b)))/(c*(b*c^2 - a*d^2)*e) + (a^(1/4 )*d^3*Sqrt[1 - (b*x^2)/a]*EllipticPi[-((Sqrt[a]*d)/(Sqrt[b]*c)), ArcSin[(b ^(1/4)*Sqrt[e*x])/(a^(1/4)*Sqrt[e])], -1])/(b^(1/4)*c^2*(b*c^2 - a*d^2)*e^ (3/2)*Sqrt[a - b*x^2]))
Int[(u_.)*(Px_)^(p_.)*((e_.)*(x_))^(m_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[1/e^(p*r) Int[u*(e*x)^(m + p*r)*ExpandToSum[Px/x^r, x]^p, x], x] /; IGtQ[r, 0]] /; FreeQ[{e, m}, x] && PolyQ[Px, x] && IntegerQ[p] && !MonomialQ[Px, x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ (Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) )], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k/e Subst[Int[x^(k*(m + 1) - 1)*(c + d*(x^k/e))^n*(a + b*(x^(2*k)/e^2))^p, x], x, (e*x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, p}, x] && ILtQ[n, 0] && FractionQ[m]
Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[(1/(Sqrt[a]*Rt[-b/a, 4]) )*EllipticF[ArcSin[Rt[-b/a, 4]*x], -1], x] /; FreeQ[{a, b}, x] && NegQ[b/a] && GtQ[a, 0]
Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[Sqrt[1 + b*(x^4/a)]/Sqrt [a + b*x^4] Int[1/Sqrt[1 + b*(x^4/a)], x], x] /; FreeQ[{a, b}, x] && NegQ [b/a] && !GtQ[a, 0]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> Simp[d/Sq rt[a] Int[Sqrt[1 + e*(x^2/d)]/Sqrt[1 - e*(x^2/d)], x], x] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 + a*e^2, 0] && NegQ[c/a] && GtQ[a, 0]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> Simp[Sqrt [1 + c*(x^4/a)]/Sqrt[a + c*x^4] Int[(d + e*x^2)/Sqrt[1 + c*(x^4/a)], x], x] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 + a*e^2, 0] && NegQ[c/a] && !GtQ [a, 0] && !(LtQ[a, 0] && GtQ[c, 0])
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[-c/a, 2]}, Simp[(d*q - e)/q Int[1/Sqrt[a + c*x^4], x], x] + Simp[e/q Int[(1 + q*x^2)/Sqrt[a + c*x^4], x], x]] /; FreeQ[{a, c, d, e}, x] && Neg Q[c/a] && NeQ[c*d^2 + a*e^2, 0]
Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> With[ {q = Rt[-c/a, 4]}, Simp[(1/(d*Sqrt[a]*q))*EllipticPi[-e/(d*q^2), ArcSin[q*x ], -1], x]] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] && GtQ[a, 0]
Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> Simp[ Sqrt[1 + c*(x^4/a)]/Sqrt[a + c*x^4] Int[1/((d + e*x^2)*Sqrt[1 + c*(x^4/a) ]), x], x] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] && !GtQ[a, 0]
Int[((x_)^(m_)*((a_) + (c_.)*(x_)^4)^(p_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(-d/e)^(m/2)*((c*d^2 + a*e^2)^(p + 1/2)/e^(2*p + 1)) Int[1/((d + e*x^2)*Sqrt[a + c*x^4]), x], x] + Simp[(-d/e)^(m/2)*(c*d^2 + a*e^2)^(p + 1 /2) Int[x^m*(a + c*x^4)^p*ExpandToSum[((c*d^2 + a*e^2)^(-p - 1/2)/(-d/e)^ (m/2) - (e^(-2*p - 1)*(a + c*x^4)^(-p - 1/2))/x^m)/(d + e*x^2), x], x], x] /; FreeQ[{a, c, d, e}, x] && ILtQ[p + 1/2, 0] && ILtQ[m/2, 0] && NeQ[c*d^2 - a*e^2, 0] && NegQ[c/a]
Int[(Pq_)*(x_)^(m_)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> With[{q = Expon[Pq, x]}, Module[{Q = PolynomialQuotient[a*b^(Floor[(q - 1)/n] + 1)*x^ m*Pq, a + b*x^n, x], R = PolynomialRemainder[a*b^(Floor[(q - 1)/n] + 1)*x^m *Pq, a + b*x^n, x], i}, Simp[(-x)*R*((a + b*x^n)^(p + 1)/(a^2*n*(p + 1)*b^( Floor[(q - 1)/n] + 1))), x] + Simp[1/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1)) Int[x^m*(a + b*x^n)^(p + 1)*ExpandToSum[(n*(p + 1)*Q)/x^m + Sum[((n*(p + 1) + i + 1)/a)*Coeff[R, x, i]*x^(i - m), {i, 0, n - 1}], x], x], x]]] /; F reeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[n, 0] && LtQ[p, -1] && ILtQ[m, 0]
Int[(Pq_)*((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Wit h[{Pq0 = Coeff[Pq, x, 0]}, Simp[Pq0*(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*c *(m + 1))), x] + Simp[1/(2*a*c*(m + 1)) Int[(c*x)^(m + 1)*ExpandToSum[2*a *(m + 1)*((Pq - Pq0)/x) - 2*b*Pq0*(m + n*(p + 1) + 1)*x^(n - 1), x]*(a + b* x^n)^p, x], x] /; NeQ[Pq0, 0]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] && IGtQ[n, 0] && LtQ[m, -1] && LeQ[n - 1, Expon[Pq, x]]
Time = 1.91 (sec) , antiderivative size = 619, normalized size of antiderivative = 1.48
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (-b \,x^{2}+a \right ) e x}\, \left (\frac {2 b e x \left (-\frac {b c x}{2 a^{2} e^{2} \left (a \,d^{2}-b \,c^{2}\right )}+\frac {d}{2 a \,e^{2} \left (a \,d^{2}-b \,c^{2}\right )}\right )}{\sqrt {-\left (x^{2}-\frac {a}{b}\right ) b e x}}-\frac {2 \left (-b e \,x^{2}+a e \right )}{a^{2} e^{2} c \sqrt {x \left (-b e \,x^{2}+a e \right )}}+\frac {d \sqrt {a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {b x}{\sqrt {a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right )}{2 a e \left (a \,d^{2}-b \,c^{2}\right ) \sqrt {-b e \,x^{3}+a e x}}+\frac {\left (\frac {b^{2} c}{2 a^{2} \left (a \,d^{2}-b \,c^{2}\right ) e}-\frac {b}{a^{2} e c}\right ) \sqrt {a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {b x}{\sqrt {a b}}}\, \left (-\frac {2 \sqrt {a b}\, \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {a b}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{b \sqrt {-b e \,x^{3}+a e x}}-\frac {d^{2} \sqrt {a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {b x}{\sqrt {a b}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}, -\frac {\sqrt {a b}}{b \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right )}, \frac {\sqrt {2}}{2}\right )}{\left (a \,d^{2}-b \,c^{2}\right ) e c b \sqrt {-b e \,x^{3}+a e x}\, \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right )}\right )}{\sqrt {e x}\, \sqrt {-b \,x^{2}+a}}\) | \(619\) |
| default | \(-\frac {3 \sqrt {2}\, \operatorname {EllipticF}\left (\sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right ) a^{2} b c \,d^{2} \sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}\, \sqrt {\frac {-b x +\sqrt {a b}}{\sqrt {a b}}}\, \sqrt {-\frac {b x}{\sqrt {a b}}}-2 \sqrt {2}\, \operatorname {EllipticF}\left (\sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right ) a^{2} d^{3} \sqrt {a b}\, \sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}\, \sqrt {\frac {-b x +\sqrt {a b}}{\sqrt {a b}}}\, \sqrt {-\frac {b x}{\sqrt {a b}}}-3 \sqrt {2}\, \operatorname {EllipticF}\left (\sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right ) a \,b^{2} c^{3} \sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}\, \sqrt {\frac {-b x +\sqrt {a b}}{\sqrt {a b}}}\, \sqrt {-\frac {b x}{\sqrt {a b}}}+2 \sqrt {2}\, \operatorname {EllipticF}\left (\sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right ) a b \,c^{2} d \sqrt {a b}\, \sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}\, \sqrt {\frac {-b x +\sqrt {a b}}{\sqrt {a b}}}\, \sqrt {-\frac {b x}{\sqrt {a b}}}+2 \sqrt {2}\, \operatorname {EllipticPi}\left (\sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}, \frac {\sqrt {a b}\, d}{d \sqrt {a b}-b c}, \frac {\sqrt {2}}{2}\right ) a^{2} d^{3} \sqrt {a b}\, \sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}\, \sqrt {\frac {-b x +\sqrt {a b}}{\sqrt {a b}}}\, \sqrt {-\frac {b x}{\sqrt {a b}}}-4 \sqrt {2}\, \operatorname {EllipticE}\left (\sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right ) a^{2} b c \,d^{2} \sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}\, \sqrt {\frac {-b x +\sqrt {a b}}{\sqrt {a b}}}\, \sqrt {-\frac {b x}{\sqrt {a b}}}+4 \sqrt {2}\, \operatorname {EllipticE}\left (\sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right ) a^{2} d^{3} \sqrt {a b}\, \sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}\, \sqrt {\frac {-b x +\sqrt {a b}}{\sqrt {a b}}}\, \sqrt {-\frac {b x}{\sqrt {a b}}}+6 \sqrt {2}\, \operatorname {EllipticE}\left (\sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right ) a \,b^{2} c^{3} \sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}\, \sqrt {\frac {-b x +\sqrt {a b}}{\sqrt {a b}}}\, \sqrt {-\frac {b x}{\sqrt {a b}}}-6 \sqrt {2}\, \operatorname {EllipticE}\left (\sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right ) a b \,c^{2} d \sqrt {a b}\, \sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}\, \sqrt {\frac {-b x +\sqrt {a b}}{\sqrt {a b}}}\, \sqrt {-\frac {b x}{\sqrt {a b}}}-4 a \,b^{2} c \,d^{2} x^{2}+4 a b \,d^{3} x^{2} \sqrt {a b}+6 b^{3} c^{3} x^{2}-6 b^{2} c^{2} d \,x^{2} \sqrt {a b}-2 a \,b^{2} c^{2} d x +2 a b c \,d^{2} x \sqrt {a b}+4 a^{2} b c \,d^{2}-4 a^{2} d^{3} \sqrt {a b}-4 a \,b^{2} c^{3}+4 a b \,c^{2} d \sqrt {a b}}{2 \sqrt {-b \,x^{2}+a}\, c \left (b c -d \sqrt {a b}\right ) \left (a \,d^{2}-b \,c^{2}\right ) a^{2} e \sqrt {e x}}\) | \(954\) |
Input:
int(1/(e*x)^(3/2)/(d*x+c)/(-b*x^2+a)^(3/2),x,method=_RETURNVERBOSE)
Output:
((-b*x^2+a)*e*x)^(1/2)/(e*x)^(1/2)/(-b*x^2+a)^(1/2)*(2*b*e*x*(-1/2/a^2/e^2 *b*c/(a*d^2-b*c^2)*x+1/2*d/a/e^2/(a*d^2-b*c^2))/(-(x^2-a/b)*b*e*x)^(1/2)-2 *(-b*e*x^2+a*e)/a^2/e^2/c/(x*(-b*e*x^2+a*e))^(1/2)+1/2/a/e/(a*d^2-b*c^2)*d *(a*b)^(1/2)*((x+1/b*(a*b)^(1/2))*b/(a*b)^(1/2))^(1/2)*(-2*(x-1/b*(a*b)^(1 /2))*b/(a*b)^(1/2))^(1/2)*(-b*x/(a*b)^(1/2))^(1/2)/(-b*e*x^3+a*e*x)^(1/2)* EllipticF(((x+1/b*(a*b)^(1/2))*b/(a*b)^(1/2))^(1/2),1/2*2^(1/2))+(1/2/a^2* b^2*c/(a*d^2-b*c^2)/e-b/a^2/e/c)/b*(a*b)^(1/2)*((x+1/b*(a*b)^(1/2))*b/(a*b )^(1/2))^(1/2)*(-2*(x-1/b*(a*b)^(1/2))*b/(a*b)^(1/2))^(1/2)*(-b*x/(a*b)^(1 /2))^(1/2)/(-b*e*x^3+a*e*x)^(1/2)*(-2/b*(a*b)^(1/2)*EllipticE(((x+1/b*(a*b )^(1/2))*b/(a*b)^(1/2))^(1/2),1/2*2^(1/2))+1/b*(a*b)^(1/2)*EllipticF(((x+1 /b*(a*b)^(1/2))*b/(a*b)^(1/2))^(1/2),1/2*2^(1/2)))-d^2/(a*d^2-b*c^2)/e/c/b *(a*b)^(1/2)*((x+1/b*(a*b)^(1/2))*b/(a*b)^(1/2))^(1/2)*(-2*(x-1/b*(a*b)^(1 /2))*b/(a*b)^(1/2))^(1/2)*(-b*x/(a*b)^(1/2))^(1/2)/(-b*e*x^3+a*e*x)^(1/2)/ (c/d-1/b*(a*b)^(1/2))*EllipticPi(((x+1/b*(a*b)^(1/2))*b/(a*b)^(1/2))^(1/2) ,-1/b*(a*b)^(1/2)/(c/d-1/b*(a*b)^(1/2)),1/2*2^(1/2)))
Timed out. \[ \int \frac {1}{(e x)^{3/2} (c+d x) \left (a-b x^2\right )^{3/2}} \, dx=\text {Timed out} \] Input:
integrate(1/(e*x)^(3/2)/(d*x+c)/(-b*x^2+a)^(3/2),x, algorithm="fricas")
Output:
Timed out
Timed out. \[ \int \frac {1}{(e x)^{3/2} (c+d x) \left (a-b x^2\right )^{3/2}} \, dx=\text {Timed out} \] Input:
integrate(1/(e*x)**(3/2)/(d*x+c)/(-b*x**2+a)**(3/2),x)
Output:
Timed out
\[ \int \frac {1}{(e x)^{3/2} (c+d x) \left (a-b x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (-b x^{2} + a\right )}^{\frac {3}{2}} {\left (d x + c\right )} \left (e x\right )^{\frac {3}{2}}} \,d x } \] Input:
integrate(1/(e*x)^(3/2)/(d*x+c)/(-b*x^2+a)^(3/2),x, algorithm="maxima")
Output:
integrate(1/((-b*x^2 + a)^(3/2)*(d*x + c)*(e*x)^(3/2)), x)
\[ \int \frac {1}{(e x)^{3/2} (c+d x) \left (a-b x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (-b x^{2} + a\right )}^{\frac {3}{2}} {\left (d x + c\right )} \left (e x\right )^{\frac {3}{2}}} \,d x } \] Input:
integrate(1/(e*x)^(3/2)/(d*x+c)/(-b*x^2+a)^(3/2),x, algorithm="giac")
Output:
integrate(1/((-b*x^2 + a)^(3/2)*(d*x + c)*(e*x)^(3/2)), x)
Timed out. \[ \int \frac {1}{(e x)^{3/2} (c+d x) \left (a-b x^2\right )^{3/2}} \, dx=\int \frac {1}{{\left (e\,x\right )}^{3/2}\,{\left (a-b\,x^2\right )}^{3/2}\,\left (c+d\,x\right )} \,d x \] Input:
int(1/((e*x)^(3/2)*(a - b*x^2)^(3/2)*(c + d*x)),x)
Output:
int(1/((e*x)^(3/2)*(a - b*x^2)^(3/2)*(c + d*x)), x)
\[ \int \frac {1}{(e x)^{3/2} (c+d x) \left (a-b x^2\right )^{3/2}} \, dx=\frac {\sqrt {e}\, \left (-2 \sqrt {-b \,x^{2}+a}-\sqrt {x}\, \left (\int \frac {\sqrt {-b \,x^{2}+a}}{\sqrt {x}\, a^{2} c +\sqrt {x}\, a^{2} d x -2 \sqrt {x}\, a b c \,x^{2}-2 \sqrt {x}\, a b d \,x^{3}+\sqrt {x}\, b^{2} c \,x^{4}+\sqrt {x}\, b^{2} d \,x^{5}}d x \right ) a^{2} d +\sqrt {x}\, \left (\int \frac {\sqrt {-b \,x^{2}+a}}{\sqrt {x}\, a^{2} c +\sqrt {x}\, a^{2} d x -2 \sqrt {x}\, a b c \,x^{2}-2 \sqrt {x}\, a b d \,x^{3}+\sqrt {x}\, b^{2} c \,x^{4}+\sqrt {x}\, b^{2} d \,x^{5}}d x \right ) a b d \,x^{2}+3 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {-b \,x^{2}+a}\, x}{b^{2} d \,x^{5}+b^{2} c \,x^{4}-2 a b d \,x^{3}-2 a b c \,x^{2}+a^{2} d x +a^{2} c}d x \right ) a b d -3 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {-b \,x^{2}+a}\, x}{b^{2} d \,x^{5}+b^{2} c \,x^{4}-2 a b d \,x^{3}-2 a b c \,x^{2}+a^{2} d x +a^{2} c}d x \right ) b^{2} d \,x^{2}+3 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {-b \,x^{2}+a}}{b^{2} d \,x^{5}+b^{2} c \,x^{4}-2 a b d \,x^{3}-2 a b c \,x^{2}+a^{2} d x +a^{2} c}d x \right ) a b c -3 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {-b \,x^{2}+a}}{b^{2} d \,x^{5}+b^{2} c \,x^{4}-2 a b d \,x^{3}-2 a b c \,x^{2}+a^{2} d x +a^{2} c}d x \right ) b^{2} c \,x^{2}\right )}{\sqrt {x}\, a c \,e^{2} \left (-b \,x^{2}+a \right )} \] Input:
int(1/(e*x)^(3/2)/(d*x+c)/(-b*x^2+a)^(3/2),x)
Output:
(sqrt(e)*( - 2*sqrt(a - b*x**2) - sqrt(x)*int(sqrt(a - b*x**2)/(sqrt(x)*a* *2*c + sqrt(x)*a**2*d*x - 2*sqrt(x)*a*b*c*x**2 - 2*sqrt(x)*a*b*d*x**3 + sq rt(x)*b**2*c*x**4 + sqrt(x)*b**2*d*x**5),x)*a**2*d + sqrt(x)*int(sqrt(a - b*x**2)/(sqrt(x)*a**2*c + sqrt(x)*a**2*d*x - 2*sqrt(x)*a*b*c*x**2 - 2*sqrt (x)*a*b*d*x**3 + sqrt(x)*b**2*c*x**4 + sqrt(x)*b**2*d*x**5),x)*a*b*d*x**2 + 3*sqrt(x)*int((sqrt(x)*sqrt(a - b*x**2)*x)/(a**2*c + a**2*d*x - 2*a*b*c* x**2 - 2*a*b*d*x**3 + b**2*c*x**4 + b**2*d*x**5),x)*a*b*d - 3*sqrt(x)*int( (sqrt(x)*sqrt(a - b*x**2)*x)/(a**2*c + a**2*d*x - 2*a*b*c*x**2 - 2*a*b*d*x **3 + b**2*c*x**4 + b**2*d*x**5),x)*b**2*d*x**2 + 3*sqrt(x)*int((sqrt(x)*s qrt(a - b*x**2))/(a**2*c + a**2*d*x - 2*a*b*c*x**2 - 2*a*b*d*x**3 + b**2*c *x**4 + b**2*d*x**5),x)*a*b*c - 3*sqrt(x)*int((sqrt(x)*sqrt(a - b*x**2))/( a**2*c + a**2*d*x - 2*a*b*c*x**2 - 2*a*b*d*x**3 + b**2*c*x**4 + b**2*d*x** 5),x)*b**2*c*x**2))/(sqrt(x)*a*c*e**2*(a - b*x**2))