Integrand size = 27, antiderivative size = 427 \[ \int \frac {(e x)^{9/2}}{(c+d x) \left (a-b x^2\right )^{3/2}} \, dx=-\frac {a e^4 \sqrt {e x} (a d-b c x)}{b^2 \left (b c^2-a d^2\right ) \sqrt {a-b x^2}}+\frac {2 e^4 \sqrt {e x} \sqrt {a-b x^2}}{3 b^2 d}+\frac {a^{3/4} c \left (2 b c^2-3 a d^2\right ) e^{9/2} \sqrt {1-\frac {b x^2}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )\right |-1\right )}{b^{7/4} d^2 \left (b c^2-a d^2\right ) \sqrt {a-b x^2}}-\frac {\sqrt [4]{a} \left (6 b^{3/2} c^3+12 \sqrt {a} b c^2 d+14 a \sqrt {b} c d^2+5 a^{3/2} d^3\right ) e^{9/2} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{3 b^{9/4} d^3 \left (\sqrt {b} c+\sqrt {a} d\right ) \sqrt {a-b x^2}}+\frac {2 \sqrt [4]{a} c^4 e^{9/2} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{\sqrt {b} c},\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} d^3 \left (b c^2-a d^2\right ) \sqrt {a-b x^2}} \] Output:
-a*e^4*(e*x)^(1/2)*(-b*c*x+a*d)/b^2/(-a*d^2+b*c^2)/(-b*x^2+a)^(1/2)+2/3*e^ 4*(e*x)^(1/2)*(-b*x^2+a)^(1/2)/b^2/d+a^(3/4)*c*(-3*a*d^2+2*b*c^2)*e^(9/2)* (1-b*x^2/a)^(1/2)*EllipticE(b^(1/4)*(e*x)^(1/2)/a^(1/4)/e^(1/2),I)/b^(7/4) /d^2/(-a*d^2+b*c^2)/(-b*x^2+a)^(1/2)-1/3*a^(1/4)*(6*b^(3/2)*c^3+12*a^(1/2) *b*c^2*d+14*a*b^(1/2)*c*d^2+5*a^(3/2)*d^3)*e^(9/2)*(1-b*x^2/a)^(1/2)*Ellip ticF(b^(1/4)*(e*x)^(1/2)/a^(1/4)/e^(1/2),I)/b^(9/4)/d^3/(b^(1/2)*c+a^(1/2) *d)/(-b*x^2+a)^(1/2)+2*a^(1/4)*c^4*e^(9/2)*(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)/d^3/(-a* d^2+b*c^2)/(-b*x^2+a)^(1/2)
Result contains complex when optimal does not.
Time = 24.24 (sec) , antiderivative size = 528, normalized size of antiderivative = 1.24 \[ \int \frac {(e x)^{9/2}}{(c+d x) \left (a-b x^2\right )^{3/2}} \, dx=\frac {(e x)^{9/2} \sqrt {a-b x^2} \left (\frac {2 \sqrt {x}}{3 b^2 d}-\frac {-a^2 d \sqrt {x}+a b c x^{3/2}}{b^2 \left (b c^2-a d^2\right ) \left (-a+b x^2\right )}\right )}{x^{9/2}}-\frac {(e x)^{9/2} \sqrt {a-b x^2} \left (6 \sqrt {-\frac {\sqrt {a}}{\sqrt {b}}} b^2 c^3 d-9 a \sqrt {-\frac {\sqrt {a}}{\sqrt {b}}} b c d^3-\frac {6 a \sqrt {-\frac {\sqrt {a}}{\sqrt {b}}} b c^3 d}{x^2}+\frac {9 a^2 \sqrt {-\frac {\sqrt {a}}{\sqrt {b}}} c d^3}{x^2}-\frac {3 i \sqrt {a} \sqrt {b} c d \left (2 b c^2-3 a d^2\right ) \sqrt {1-\frac {a}{b x^2}} E\left (\left .i \text {arcsinh}\left (\frac {\sqrt {-\frac {\sqrt {a}}{\sqrt {b}}}}{\sqrt {x}}\right )\right |-1\right )}{\sqrt {x}}+\frac {i \sqrt {a} d \left (6 b^{3/2} c^3-2 \sqrt {a} b c^2 d-9 a \sqrt {b} c d^2+5 a^{3/2} d^3\right ) \sqrt {1-\frac {a}{b x^2}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-\frac {\sqrt {a}}{\sqrt {b}}}}{\sqrt {x}}\right ),-1\right )}{\sqrt {x}}-\frac {6 i b^2 c^4 \sqrt {1-\frac {a}{b x^2}} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} c}{\sqrt {a} d},i \text {arcsinh}\left (\frac {\sqrt {-\frac {\sqrt {a}}{\sqrt {b}}}}{\sqrt {x}}\right ),-1\right )}{\sqrt {x}}\right )}{3 \sqrt {-\frac {\sqrt {a}}{\sqrt {b}}} b^2 d^3 \left (-b c^2+a d^2\right ) \left (-b+\frac {a}{x^2}\right ) x^5} \] Input:
Integrate[(e*x)^(9/2)/((c + d*x)*(a - b*x^2)^(3/2)),x]
Output:
((e*x)^(9/2)*Sqrt[a - b*x^2]*((2*Sqrt[x])/(3*b^2*d) - (-(a^2*d*Sqrt[x]) + a*b*c*x^(3/2))/(b^2*(b*c^2 - a*d^2)*(-a + b*x^2))))/x^(9/2) - ((e*x)^(9/2) *Sqrt[a - b*x^2]*(6*Sqrt[-(Sqrt[a]/Sqrt[b])]*b^2*c^3*d - 9*a*Sqrt[-(Sqrt[a ]/Sqrt[b])]*b*c*d^3 - (6*a*Sqrt[-(Sqrt[a]/Sqrt[b])]*b*c^3*d)/x^2 + (9*a^2* Sqrt[-(Sqrt[a]/Sqrt[b])]*c*d^3)/x^2 - ((3*I)*Sqrt[a]*Sqrt[b]*c*d*(2*b*c^2 - 3*a*d^2)*Sqrt[1 - a/(b*x^2)]*EllipticE[I*ArcSinh[Sqrt[-(Sqrt[a]/Sqrt[b]) ]/Sqrt[x]], -1])/Sqrt[x] + (I*Sqrt[a]*d*(6*b^(3/2)*c^3 - 2*Sqrt[a]*b*c^2*d - 9*a*Sqrt[b]*c*d^2 + 5*a^(3/2)*d^3)*Sqrt[1 - a/(b*x^2)]*EllipticF[I*ArcS inh[Sqrt[-(Sqrt[a]/Sqrt[b])]/Sqrt[x]], -1])/Sqrt[x] - ((6*I)*b^2*c^4*Sqrt[ 1 - a/(b*x^2)]*EllipticPi[-((Sqrt[b]*c)/(Sqrt[a]*d)), I*ArcSinh[Sqrt[-(Sqr t[a]/Sqrt[b])]/Sqrt[x]], -1])/Sqrt[x]))/(3*Sqrt[-(Sqrt[a]/Sqrt[b])]*b^2*d^ 3*(-(b*c^2) + a*d^2)*(-b + a/x^2)*x^5)
Time = 1.56 (sec) , antiderivative size = 442, normalized size of antiderivative = 1.04, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.630, Rules used = {616, 27, 1641, 1543, 1542, 2397, 25, 2427, 25, 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.
| \(\displaystyle \int \frac {(e x)^{9/2}}{\left (a-b x^2\right )^{3/2} (c+d x)} \, dx\) |
| \(\Big \downarrow \) 616 |
| \(\displaystyle \frac {2 \int \frac {e^6 x^5}{(c e+d x e) \left (a-b x^2\right )^{3/2}}d\sqrt {e x}}{e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \int \frac {e^5 x^5}{(c e+d x e) \left (a-b x^2\right )^{3/2}}d\sqrt {e x}\) |
| \(\Big \downarrow \) 1641 |
| \(\displaystyle 2 \left (\frac {c^5 e^5 \int \frac {1}{(c e+d x e) \sqrt {a-b x^2}}d\sqrt {e x}}{d^3 \left (b c^2-a d^2\right )}-\frac {\int \frac {-\frac {\left (b c^2-a d^2\right ) x^4 e^4}{d}+\frac {c \left (b c^2-a d^2\right ) x^3 e^4}{d^2}-\frac {c^2 \left (b c^2-a d^2\right ) x^2 e^4}{d^3}-\frac {a c^3 x e^4}{d^2}+\frac {a c^4 e^4}{d^3}}{\left (a-b x^2\right )^{3/2}}d\sqrt {e x}}{b c^2-a d^2}\right )\) |
| \(\Big \downarrow \) 1543 |
| \(\displaystyle 2 \left (\frac {c^5 e^5 \sqrt {1-\frac {b x^2}{a}} \int \frac {1}{(c e+d x e) \sqrt {1-\frac {b x^2}{a}}}d\sqrt {e x}}{d^3 \sqrt {a-b x^2} \left (b c^2-a d^2\right )}-\frac {\int \frac {-\frac {\left (b c^2-a d^2\right ) x^4 e^4}{d}+\frac {c \left (b c^2-a d^2\right ) x^3 e^4}{d^2}-\frac {c^2 \left (b c^2-a d^2\right ) x^2 e^4}{d^3}-\frac {a c^3 x e^4}{d^2}+\frac {a c^4 e^4}{d^3}}{\left (a-b x^2\right )^{3/2}}d\sqrt {e x}}{b c^2-a d^2}\right )\) |
| \(\Big \downarrow \) 1542 |
| \(\displaystyle 2 \left (\frac {\sqrt [4]{a} c^4 e^{9/2} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{\sqrt {b} c},\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} d^3 \sqrt {a-b x^2} \left (b c^2-a d^2\right )}-\frac {\int \frac {-\frac {\left (b c^2-a d^2\right ) x^4 e^4}{d}+\frac {c \left (b c^2-a d^2\right ) x^3 e^4}{d^2}-\frac {c^2 \left (b c^2-a d^2\right ) x^2 e^4}{d^3}-\frac {a c^3 x e^4}{d^2}+\frac {a c^4 e^4}{d^3}}{\left (a-b x^2\right )^{3/2}}d\sqrt {e x}}{b c^2-a d^2}\right )\) |
| \(\Big \downarrow \) 2397 |
| \(\displaystyle 2 \left (\frac {\sqrt [4]{a} c^4 e^{9/2} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{\sqrt {b} c},\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} d^3 \sqrt {a-b x^2} \left (b c^2-a d^2\right )}-\frac {\frac {a e^3 \sqrt {e x} (a d e-b c e x)}{2 b^2 \sqrt {a-b x^2}}-\frac {e^4 \int -\frac {2 a b \left (\frac {b c^2}{d}-a d\right ) x^2+a b c \left (3 a-\frac {2 b c^2}{d^2}\right ) x+a \left (\frac {2 b^2 c^4}{d^3}-a^2 d\right )}{\sqrt {a-b x^2}}d\sqrt {e x}}{2 a b^2}}{b c^2-a d^2}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 2 \left (\frac {\sqrt [4]{a} c^4 e^{9/2} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{\sqrt {b} c},\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} d^3 \sqrt {a-b x^2} \left (b c^2-a d^2\right )}-\frac {\frac {e^4 \int \frac {2 a b \left (\frac {b c^2}{d}-a d\right ) x^2+a b c \left (3 a-\frac {2 b c^2}{d^2}\right ) x+a \left (\frac {2 b^2 c^4}{d^3}-a^2 d\right )}{\sqrt {a-b x^2}}d\sqrt {e x}}{2 a b^2}+\frac {a e^3 \sqrt {e x} (a d e-b c e x)}{2 b^2 \sqrt {a-b x^2}}}{b c^2-a d^2}\right )\) |
| \(\Big \downarrow \) 2427 |
| \(\displaystyle 2 \left (\frac {\sqrt [4]{a} c^4 e^{9/2} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{\sqrt {b} c},\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} d^3 \sqrt {a-b x^2} \left (b c^2-a d^2\right )}-\frac {\frac {e^4 \left (-\frac {e^2 \int -\frac {a b \left (3 b c \left (3 a-\frac {2 b c^2}{d^2}\right ) e x d^3+\left (6 b^2 c^4+2 a b d^2 c^2-5 a^2 d^4\right ) e\right )}{d^3 e^3 \sqrt {a-b x^2}}d\sqrt {e x}}{3 b}-\frac {2}{3} a \sqrt {e x} \sqrt {a-b x^2} \left (\frac {b c^2}{d}-a d\right )\right )}{2 a b^2}+\frac {a e^3 \sqrt {e x} (a d e-b c e x)}{2 b^2 \sqrt {a-b x^2}}}{b c^2-a d^2}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 2 \left (\frac {\sqrt [4]{a} c^4 e^{9/2} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{\sqrt {b} c},\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} d^3 \sqrt {a-b x^2} \left (b c^2-a d^2\right )}-\frac {\frac {e^4 \left (\frac {e^2 \int \frac {a b \left (\left (6 b^2 c^4+2 a b d^2 c^2-5 a^2 d^4\right ) e-3 b c d \left (2 b c^2-3 a d^2\right ) e x\right )}{d^3 e^3 \sqrt {a-b x^2}}d\sqrt {e x}}{3 b}-\frac {2}{3} a \sqrt {e x} \sqrt {a-b x^2} \left (\frac {b c^2}{d}-a d\right )\right )}{2 a b^2}+\frac {a e^3 \sqrt {e x} (a d e-b c e x)}{2 b^2 \sqrt {a-b x^2}}}{b c^2-a d^2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \left (\frac {\sqrt [4]{a} c^4 e^{9/2} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{\sqrt {b} c},\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} d^3 \sqrt {a-b x^2} \left (b c^2-a d^2\right )}-\frac {\frac {e^4 \left (\frac {a \int \frac {\left (6 b^2 c^4+2 a b d^2 c^2-5 a^2 d^4\right ) e-3 b c d \left (2 b c^2-3 a d^2\right ) e x}{\sqrt {a-b x^2}}d\sqrt {e x}}{3 d^3 e}-\frac {2}{3} a \sqrt {e x} \sqrt {a-b x^2} \left (\frac {b c^2}{d}-a d\right )\right )}{2 a b^2}+\frac {a e^3 \sqrt {e x} (a d e-b c e x)}{2 b^2 \sqrt {a-b x^2}}}{b c^2-a d^2}\right )\) |
| \(\Big \downarrow \) 1513 |
| \(\displaystyle 2 \left (\frac {\sqrt [4]{a} c^4 e^{9/2} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{\sqrt {b} c},\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} d^3 \sqrt {a-b x^2} \left (b c^2-a d^2\right )}-\frac {\frac {e^4 \left (\frac {a \left (e \left (-9 a^{3/2} \sqrt {b} c d^3-5 a^2 d^4+6 \sqrt {a} b^{3/2} c^3 d+2 a b c^2 d^2+6 b^2 c^4\right ) \int \frac {1}{\sqrt {a-b x^2}}d\sqrt {e x}-3 \sqrt {a} \sqrt {b} c d e \left (2 b c^2-3 a d^2\right ) \int \frac {\sqrt {b} x e+\sqrt {a} e}{\sqrt {a} e \sqrt {a-b x^2}}d\sqrt {e x}\right )}{3 d^3 e}-\frac {2}{3} a \sqrt {e x} \sqrt {a-b x^2} \left (\frac {b c^2}{d}-a d\right )\right )}{2 a b^2}+\frac {a e^3 \sqrt {e x} (a d e-b c e x)}{2 b^2 \sqrt {a-b x^2}}}{b c^2-a d^2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \left (\frac {\sqrt [4]{a} c^4 e^{9/2} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{\sqrt {b} c},\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} d^3 \sqrt {a-b x^2} \left (b c^2-a d^2\right )}-\frac {\frac {e^4 \left (\frac {a \left (e \left (-9 a^{3/2} \sqrt {b} c d^3-5 a^2 d^4+6 \sqrt {a} b^{3/2} c^3 d+2 a b c^2 d^2+6 b^2 c^4\right ) \int \frac {1}{\sqrt {a-b x^2}}d\sqrt {e x}-3 \sqrt {b} c d \left (2 b c^2-3 a d^2\right ) \int \frac {\sqrt {b} x e+\sqrt {a} e}{\sqrt {a-b x^2}}d\sqrt {e x}\right )}{3 d^3 e}-\frac {2}{3} a \sqrt {e x} \sqrt {a-b x^2} \left (\frac {b c^2}{d}-a d\right )\right )}{2 a b^2}+\frac {a e^3 \sqrt {e x} (a d e-b c e x)}{2 b^2 \sqrt {a-b x^2}}}{b c^2-a d^2}\right )\) |
| \(\Big \downarrow \) 765 |
| \(\displaystyle 2 \left (\frac {\sqrt [4]{a} c^4 e^{9/2} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{\sqrt {b} c},\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} d^3 \sqrt {a-b x^2} \left (b c^2-a d^2\right )}-\frac {\frac {e^4 \left (\frac {a \left (\frac {e \sqrt {1-\frac {b x^2}{a}} \left (-9 a^{3/2} \sqrt {b} c d^3-5 a^2 d^4+6 \sqrt {a} b^{3/2} c^3 d+2 a b c^2 d^2+6 b^2 c^4\right ) \int \frac {1}{\sqrt {1-\frac {b x^2}{a}}}d\sqrt {e x}}{\sqrt {a-b x^2}}-3 \sqrt {b} c d \left (2 b c^2-3 a d^2\right ) \int \frac {\sqrt {b} x e+\sqrt {a} e}{\sqrt {a-b x^2}}d\sqrt {e x}\right )}{3 d^3 e}-\frac {2}{3} a \sqrt {e x} \sqrt {a-b x^2} \left (\frac {b c^2}{d}-a d\right )\right )}{2 a b^2}+\frac {a e^3 \sqrt {e x} (a d e-b c e x)}{2 b^2 \sqrt {a-b x^2}}}{b c^2-a d^2}\right )\) |
| \(\Big \downarrow \) 762 |
| \(\displaystyle 2 \left (\frac {\sqrt [4]{a} c^4 e^{9/2} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{\sqrt {b} c},\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} d^3 \sqrt {a-b x^2} \left (b c^2-a d^2\right )}-\frac {\frac {e^4 \left (\frac {a \left (\frac {\sqrt [4]{a} e^{3/2} \sqrt {1-\frac {b x^2}{a}} \left (-9 a^{3/2} \sqrt {b} c d^3-5 a^2 d^4+6 \sqrt {a} b^{3/2} c^3 d+2 a b c^2 d^2+6 b^2 c^4\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} \sqrt {a-b x^2}}-3 \sqrt {b} c d \left (2 b c^2-3 a d^2\right ) \int \frac {\sqrt {b} x e+\sqrt {a} e}{\sqrt {a-b x^2}}d\sqrt {e x}\right )}{3 d^3 e}-\frac {2}{3} a \sqrt {e x} \sqrt {a-b x^2} \left (\frac {b c^2}{d}-a d\right )\right )}{2 a b^2}+\frac {a e^3 \sqrt {e x} (a d e-b c e x)}{2 b^2 \sqrt {a-b x^2}}}{b c^2-a d^2}\right )\) |
| \(\Big \downarrow \) 1390 |
| \(\displaystyle 2 \left (\frac {\sqrt [4]{a} c^4 e^{9/2} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{\sqrt {b} c},\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} d^3 \sqrt {a-b x^2} \left (b c^2-a d^2\right )}-\frac {\frac {e^4 \left (\frac {a \left (\frac {\sqrt [4]{a} e^{3/2} \sqrt {1-\frac {b x^2}{a}} \left (-9 a^{3/2} \sqrt {b} c d^3-5 a^2 d^4+6 \sqrt {a} b^{3/2} c^3 d+2 a b c^2 d^2+6 b^2 c^4\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} \sqrt {a-b x^2}}-\frac {3 \sqrt {b} c d \sqrt {1-\frac {b x^2}{a}} \left (2 b c^2-3 a d^2\right ) \int \frac {\sqrt {b} x e+\sqrt {a} e}{\sqrt {1-\frac {b x^2}{a}}}d\sqrt {e x}}{\sqrt {a-b x^2}}\right )}{3 d^3 e}-\frac {2}{3} a \sqrt {e x} \sqrt {a-b x^2} \left (\frac {b c^2}{d}-a d\right )\right )}{2 a b^2}+\frac {a e^3 \sqrt {e x} (a d e-b c e x)}{2 b^2 \sqrt {a-b x^2}}}{b c^2-a d^2}\right )\) |
| \(\Big \downarrow \) 1389 |
| \(\displaystyle 2 \left (\frac {\sqrt [4]{a} c^4 e^{9/2} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{\sqrt {b} c},\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} d^3 \sqrt {a-b x^2} \left (b c^2-a d^2\right )}-\frac {\frac {e^4 \left (\frac {a \left (\frac {\sqrt [4]{a} e^{3/2} \sqrt {1-\frac {b x^2}{a}} \left (-9 a^{3/2} \sqrt {b} c d^3-5 a^2 d^4+6 \sqrt {a} b^{3/2} c^3 d+2 a b c^2 d^2+6 b^2 c^4\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} \sqrt {a-b x^2}}-\frac {3 \sqrt {a} \sqrt {b} c d e \sqrt {1-\frac {b x^2}{a}} \left (2 b c^2-3 a d^2\right ) \int \frac {\sqrt {\frac {\sqrt {b} x}{\sqrt {a}}+1}}{\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}d\sqrt {e x}}{\sqrt {a-b x^2}}\right )}{3 d^3 e}-\frac {2}{3} a \sqrt {e x} \sqrt {a-b x^2} \left (\frac {b c^2}{d}-a d\right )\right )}{2 a b^2}+\frac {a e^3 \sqrt {e x} (a d e-b c e x)}{2 b^2 \sqrt {a-b x^2}}}{b c^2-a d^2}\right )\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle 2 \left (\frac {\sqrt [4]{a} c^4 e^{9/2} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{\sqrt {b} c},\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} d^3 \sqrt {a-b x^2} \left (b c^2-a d^2\right )}-\frac {\frac {e^4 \left (\frac {a \left (\frac {\sqrt [4]{a} e^{3/2} \sqrt {1-\frac {b x^2}{a}} \left (-9 a^{3/2} \sqrt {b} c d^3-5 a^2 d^4+6 \sqrt {a} b^{3/2} c^3 d+2 a b c^2 d^2+6 b^2 c^4\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} \sqrt {a-b x^2}}-\frac {3 a^{3/4} \sqrt [4]{b} c d e^{3/2} \sqrt {1-\frac {b x^2}{a}} \left (2 b c^2-3 a d^2\right ) E\left (\left .\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )\right |-1\right )}{\sqrt {a-b x^2}}\right )}{3 d^3 e}-\frac {2}{3} a \sqrt {e x} \sqrt {a-b x^2} \left (\frac {b c^2}{d}-a d\right )\right )}{2 a b^2}+\frac {a e^3 \sqrt {e x} (a d e-b c e x)}{2 b^2 \sqrt {a-b x^2}}}{b c^2-a d^2}\right )\) |
Input:
Int[(e*x)^(9/2)/((c + d*x)*(a - b*x^2)^(3/2)),x]
Output:
2*(-(((a*e^3*Sqrt[e*x]*(a*d*e - b*c*e*x))/(2*b^2*Sqrt[a - b*x^2]) + (e^4*( (-2*a*((b*c^2)/d - a*d)*Sqrt[e*x]*Sqrt[a - b*x^2])/3 + (a*((-3*a^(3/4)*b^( 1/4)*c*d*(2*b*c^2 - 3*a*d^2)*e^(3/2)*Sqrt[1 - (b*x^2)/a]*EllipticE[ArcSin[ (b^(1/4)*Sqrt[e*x])/(a^(1/4)*Sqrt[e])], -1])/Sqrt[a - b*x^2] + (a^(1/4)*(6 *b^2*c^4 + 6*Sqrt[a]*b^(3/2)*c^3*d + 2*a*b*c^2*d^2 - 9*a^(3/2)*Sqrt[b]*c*d ^3 - 5*a^2*d^4)*e^(3/2)*Sqrt[1 - (b*x^2)/a]*EllipticF[ArcSin[(b^(1/4)*Sqrt [e*x])/(a^(1/4)*Sqrt[e])], -1])/(b^(1/4)*Sqrt[a - b*x^2])))/(3*d^3*e)))/(2 *a*b^2))/(b*c^2 - a*d^2)) + (a^(1/4)*c^4*e^(9/2)*Sqrt[1 - (b*x^2)/a]*Ellip ticPi[-((Sqrt[a]*d)/(Sqrt[b]*c)), ArcSin[(b^(1/4)*Sqrt[e*x])/(a^(1/4)*Sqrt [e])], -1])/(b^(1/4)*d^3*(b*c^2 - a*d^2)*Sqrt[a - b*x^2]))
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[(c*d^2 + a*e^2)^(p + 1/2) Int[(a + c*x^4)^p*ExpandToSum[((c*d^2 + a*e^2)^(-p - 1/2)*x^m - e^(-2*p - 1)*(-d/e )^(m/2)*(a + c*x^4)^(-p - 1/2))/(d + e*x^2), x], x], x] /; FreeQ[{a, c, d, e}, x] && ILtQ[p + 1/2, 0] && IGtQ[m/2, 0] && NeQ[c*d^2 - a*e^2, 0] && NegQ [c/a]
Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> With[{q = Expon[Pq, x]}, Module[{Q = PolynomialQuotient[b^(Floor[(q - 1)/n] + 1)*Pq, a + b*x^n, x], R = PolynomialRemainder[b^(Floor[(q - 1)/n] + 1)*Pq, a + b*x^n, x]}, S imp[(-x)*R*((a + b*x^n)^(p + 1)/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1))), x] + Simp[1/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1)) Int[(a + b*x^n)^(p + 1)* ExpandToSum[a*n*(p + 1)*Q + n*(p + 1)*R + D[x*R, x], x], x], x]] /; GeQ[q, n]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[n, 0] && LtQ[p, -1]
Int[(Pq_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{q = Expon[Pq, x ]}, With[{Pqq = Coeff[Pq, x, q]}, Simp[Pqq*x^(q - n + 1)*((a + b*x^n)^(p + 1)/(b*(q + n*p + 1))), x] + Simp[1/(b*(q + n*p + 1)) Int[ExpandToSum[b*(q + n*p + 1)*(Pq - Pqq*x^q) - a*Pqq*(q - n + 1)*x^(q - n), x]*(a + b*x^n)^p, x], x]] /; NeQ[q + n*p + 1, 0] && q - n >= 0 && (IntegerQ[2*p] || IntegerQ [p + (q + 1)/(2*n)])] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x] && IGtQ[n, 0]
Time = 2.91 (sec) , antiderivative size = 655, normalized size of antiderivative = 1.53
| method | result | size |
| elliptic | \(\frac {\sqrt {e x}\, \sqrt {\left (-b \,x^{2}+a \right ) e x}\, \left (\frac {2 b e x \left (-\frac {e^{4} a c x}{2 b^{2} \left (a \,d^{2}-b \,c^{2}\right )}+\frac {a^{2} d \,e^{4}}{2 b^{3} \left (a \,d^{2}-b \,c^{2}\right )}\right )}{\sqrt {-\left (x^{2}-\frac {a}{b}\right ) b e x}}+\frac {2 e^{4} \sqrt {-b e \,x^{3}+a e x}}{3 b^{2} d}+\frac {\left (-\frac {\left (a \,d^{2}+b \,c^{2}\right ) e^{5}}{b^{2} d^{3}}+\frac {a^{2} e^{5} d}{2 b^{2} \left (a \,d^{2}-b \,c^{2}\right )}-\frac {e^{5} a}{3 b^{2} d}\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}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right )}{b \sqrt {-b e \,x^{3}+a e x}}+\frac {\left (\frac {c \,e^{5}}{d^{2} b}+\frac {a \,e^{5} c}{2 b \left (a \,d^{2}-b \,c^{2}\right )}\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 {c^{5} e^{5} \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 )}{d^{4} \left (a \,d^{2}-b \,c^{2}\right ) b \sqrt {-b e \,x^{3}+a e x}\, \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right )}\right )}{e x \sqrt {-b \,x^{2}+a}}\) | \(655\) |
| risch | \(\frac {2 \sqrt {-b \,x^{2}+a}\, x \,e^{5}}{3 d \,b^{2} \sqrt {e x}}+\frac {\left (\frac {-\frac {4 a \,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 {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right )}{b \sqrt {-b e \,x^{3}+a e x}}-\frac {3 c^{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 {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right )}{\sqrt {-b e \,x^{3}+a e x}}+\frac {3 d c \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 )}{\sqrt {-b e \,x^{3}+a e x}}}{d^{2}}-\frac {3 b \,c^{5} \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 )}{d^{3} \left (a \,d^{2}-b \,c^{2}\right ) \sqrt {-b e \,x^{3}+a e x}\, \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right )}+\frac {3 a^{2} d \left (\frac {2 x b e \left (-\frac {c x}{2 a e}+\frac {d}{2 e b}\right )}{\sqrt {-\left (x^{2}-\frac {a}{b}\right ) b e x}}+\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 b \sqrt {-b e \,x^{3}+a e x}}+\frac {c \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 )}{2 a \sqrt {-b e \,x^{3}+a e x}}\right )}{a \,d^{2}-b \,c^{2}}\right ) e^{5} \sqrt {\left (-b \,x^{2}+a \right ) e x}}{3 d \,b^{2} \sqrt {e x}\, \sqrt {-b \,x^{2}+a}}\) | \(916\) |
| default | \(\text {Expression too large to display}\) | \(1154\) |
Input:
int((e*x)^(9/2)/(d*x+c)/(-b*x^2+a)^(3/2),x,method=_RETURNVERBOSE)
Output:
1/e/x*(e*x)^(1/2)/(-b*x^2+a)^(1/2)*((-b*x^2+a)*e*x)^(1/2)*(2*b*e*x*(-1/2*e ^4*a/b^2*c/(a*d^2-b*c^2)*x+1/2*a^2*d*e^4/b^3/(a*d^2-b*c^2))/(-(x^2-a/b)*b* e*x)^(1/2)+2/3/b^2/d*e^4*(-b*e*x^3+a*e*x)^(1/2)+(-(a*d^2+b*c^2)*e^5/b^2/d^ 3+1/2*a^2/b^2*e^5*d/(a*d^2-b*c^2)-1/3/b^2/d*e^5*a)/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)*EllipticF(((x+1/b*(a*b )^(1/2))*b/(a*b)^(1/2))^(1/2),1/2*2^(1/2))+(1/d^2/b*c*e^5+1/2*a/b*e^5*c/(a *d^2-b*c^2))/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)))-c^5*e^5/d^4/(a*d^2-b*c^2)/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. \[ \int \frac {(e x)^{9/2}}{(c+d x) \left (a-b x^2\right )^{3/2}} \, dx=\text {Timed out} \] Input:
integrate((e*x)^(9/2)/(d*x+c)/(-b*x^2+a)^(3/2),x, algorithm="fricas")
Output:
Timed out
Timed out. \[ \int \frac {(e x)^{9/2}}{(c+d x) \left (a-b x^2\right )^{3/2}} \, dx=\text {Timed out} \] Input:
integrate((e*x)**(9/2)/(d*x+c)/(-b*x**2+a)**(3/2),x)
Output:
Timed out
\[ \int \frac {(e x)^{9/2}}{(c+d x) \left (a-b x^2\right )^{3/2}} \, dx=\int { \frac {\left (e x\right )^{\frac {9}{2}}}{{\left (-b x^{2} + a\right )}^{\frac {3}{2}} {\left (d x + c\right )}} \,d x } \] Input:
integrate((e*x)^(9/2)/(d*x+c)/(-b*x^2+a)^(3/2),x, algorithm="maxima")
Output:
integrate((e*x)^(9/2)/((-b*x^2 + a)^(3/2)*(d*x + c)), x)
\[ \int \frac {(e x)^{9/2}}{(c+d x) \left (a-b x^2\right )^{3/2}} \, dx=\int { \frac {\left (e x\right )^{\frac {9}{2}}}{{\left (-b x^{2} + a\right )}^{\frac {3}{2}} {\left (d x + c\right )}} \,d x } \] Input:
integrate((e*x)^(9/2)/(d*x+c)/(-b*x^2+a)^(3/2),x, algorithm="giac")
Output:
integrate((e*x)^(9/2)/((-b*x^2 + a)^(3/2)*(d*x + c)), x)
Timed out. \[ \int \frac {(e x)^{9/2}}{(c+d x) \left (a-b x^2\right )^{3/2}} \, dx=\int \frac {{\left (e\,x\right )}^{9/2}}{{\left (a-b\,x^2\right )}^{3/2}\,\left (c+d\,x\right )} \,d x \] Input:
int((e*x)^(9/2)/((a - b*x^2)^(3/2)*(c + d*x)),x)
Output:
int((e*x)^(9/2)/((a - b*x^2)^(3/2)*(c + d*x)), x)
\[ \int \frac {(e x)^{9/2}}{(c+d x) \left (a-b x^2\right )^{3/2}} \, dx=\frac {\sqrt {e}\, e^{4} \left (-8 \sqrt {x}\, \sqrt {-b \,x^{2}+a}\, a d +6 \sqrt {x}\, \sqrt {-b \,x^{2}+a}\, b c x -2 \sqrt {x}\, \sqrt {-b \,x^{2}+a}\, b d \,x^{2}+4 \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^{3} c d -4 \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} b c d \,x^{2}+9 \left (\int \frac {\sqrt {x}\, \sqrt {-b \,x^{2}+a}\, x^{2}}{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^{2} b \,d^{2}+3 \left (\int \frac {\sqrt {x}\, \sqrt {-b \,x^{2}+a}\, x^{2}}{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^{2} c^{2}-9 \left (\int \frac {\sqrt {x}\, \sqrt {-b \,x^{2}+a}\, x^{2}}{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^{2} d^{2} x^{2}-3 \left (\int \frac {\sqrt {x}\, \sqrt {-b \,x^{2}+a}\, x^{2}}{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^{3} c^{2} x^{2}+4 \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^{3} d^{2}-9 \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^{2} b \,c^{2}-4 \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^{2} b \,d^{2} x^{2}+9 \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^{2} c^{2} x^{2}\right )}{3 b^{2} d^{2} \left (-b \,x^{2}+a \right )} \] Input:
int((e*x)^(9/2)/(d*x+c)/(-b*x^2+a)^(3/2),x)
Output:
(sqrt(e)*e**4*( - 8*sqrt(x)*sqrt(a - b*x**2)*a*d + 6*sqrt(x)*sqrt(a - b*x* *2)*b*c*x - 2*sqrt(x)*sqrt(a - b*x**2)*b*d*x**2 + 4*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**3*c*d - 4*int(sqr t(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**2* b*c*d*x**2 + 9*int((sqrt(x)*sqrt(a - b*x**2)*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**2*b*d**2 + 3* int((sqrt(x)*sqrt(a - b*x**2)*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**2*c**2 - 9*int((sqrt(x)*sq rt(a - b*x**2)*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**2*d**2*x**2 - 3*int((sqrt(x)*sqrt(a - b*x **2)*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**3*c**2*x**2 + 4*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) *a**3*d**2 - 9*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)*a**2*b*c**2 - 4*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)*a**2*b*d**2*x**2 + 9*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*...