Integrand size = 28, antiderivative size = 476 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}}{(e x)^{3/2}} \, dx=-\frac {4 \left (3 b^2 c^2-13 a d (2 b c+9 a d)\right ) (e x)^{3/2} \sqrt {c+d x^2}}{195 d e^3}-\frac {8 c \left (3 b^2 c^2-13 a d (2 b c+9 a d)\right ) \sqrt {e x} \sqrt {c+d x^2}}{195 d^{3/2} e^2 \left (\sqrt {c}+\sqrt {d} x\right )}-\frac {2 \left (3 b^2 c^2-13 a d (2 b c+9 a d)\right ) (e x)^{3/2} \left (c+d x^2\right )^{3/2}}{117 c d e^3}-\frac {2 a^2 \left (c+d x^2\right )^{5/2}}{c e \sqrt {e x}}+\frac {2 b^2 (e x)^{3/2} \left (c+d x^2\right )^{5/2}}{13 d e^3}+\frac {8 c^{5/4} \left (3 b^2 c^2-13 a d (2 b c+9 a d)\right ) \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{195 d^{7/4} e^{3/2} \sqrt {c+d x^2}}-\frac {4 c^{5/4} \left (3 b^2 c^2-13 a d (2 b c+9 a d)\right ) \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),\frac {1}{2}\right )}{195 d^{7/4} e^{3/2} \sqrt {c+d x^2}} \]
[Out]
Time = 0.33 (sec) , antiderivative size = 476, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {473, 470, 285, 335, 311, 226, 1210} \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}}{(e x)^{3/2}} \, dx=-\frac {2 a^2 \left (c+d x^2\right )^{5/2}}{c e \sqrt {e x}}-\frac {4 c^{5/4} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \left (3 b^2 c^2-13 a d (9 a d+2 b c)\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),\frac {1}{2}\right )}{195 d^{7/4} e^{3/2} \sqrt {c+d x^2}}+\frac {8 c^{5/4} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \left (3 b^2 c^2-13 a d (9 a d+2 b c)\right ) E\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{195 d^{7/4} e^{3/2} \sqrt {c+d x^2}}-\frac {8 c \sqrt {e x} \sqrt {c+d x^2} \left (3 b^2 c^2-13 a d (9 a d+2 b c)\right )}{195 d^{3/2} e^2 \left (\sqrt {c}+\sqrt {d} x\right )}-\frac {2 (e x)^{3/2} \left (c+d x^2\right )^{3/2} \left (3 b^2 c^2-13 a d (9 a d+2 b c)\right )}{117 c d e^3}-\frac {4 (e x)^{3/2} \sqrt {c+d x^2} \left (3 b^2 c^2-13 a d (9 a d+2 b c)\right )}{195 d e^3}+\frac {2 b^2 (e x)^{3/2} \left (c+d x^2\right )^{5/2}}{13 d e^3} \]
[In]
[Out]
Rule 226
Rule 285
Rule 311
Rule 335
Rule 470
Rule 473
Rule 1210
Rubi steps \begin{align*} \text {integral}& = -\frac {2 a^2 \left (c+d x^2\right )^{5/2}}{c e \sqrt {e x}}+\frac {2 \int \sqrt {e x} \left (\frac {1}{2} a (2 b c+9 a d)+\frac {1}{2} b^2 c x^2\right ) \left (c+d x^2\right )^{3/2} \, dx}{c e^2} \\ & = -\frac {2 a^2 \left (c+d x^2\right )^{5/2}}{c e \sqrt {e x}}+\frac {2 b^2 (e x)^{3/2} \left (c+d x^2\right )^{5/2}}{13 d e^3}-\frac {\left (4 \left (\frac {3 b^2 c^2}{4}-\frac {13}{4} a d (2 b c+9 a d)\right )\right ) \int \sqrt {e x} \left (c+d x^2\right )^{3/2} \, dx}{13 c d e^2} \\ & = -\frac {2 \left (3 b^2 c^2-13 a d (2 b c+9 a d)\right ) (e x)^{3/2} \left (c+d x^2\right )^{3/2}}{117 c d e^3}-\frac {2 a^2 \left (c+d x^2\right )^{5/2}}{c e \sqrt {e x}}+\frac {2 b^2 (e x)^{3/2} \left (c+d x^2\right )^{5/2}}{13 d e^3}-\frac {\left (2 \left (3 b^2 c^2-13 a d (2 b c+9 a d)\right )\right ) \int \sqrt {e x} \sqrt {c+d x^2} \, dx}{39 d e^2} \\ & = -\frac {4 \left (3 b^2 c^2-13 a d (2 b c+9 a d)\right ) (e x)^{3/2} \sqrt {c+d x^2}}{195 d e^3}-\frac {2 \left (3 b^2 c^2-13 a d (2 b c+9 a d)\right ) (e x)^{3/2} \left (c+d x^2\right )^{3/2}}{117 c d e^3}-\frac {2 a^2 \left (c+d x^2\right )^{5/2}}{c e \sqrt {e x}}+\frac {2 b^2 (e x)^{3/2} \left (c+d x^2\right )^{5/2}}{13 d e^3}-\frac {\left (4 c \left (3 b^2 c^2-13 a d (2 b c+9 a d)\right )\right ) \int \frac {\sqrt {e x}}{\sqrt {c+d x^2}} \, dx}{195 d e^2} \\ & = -\frac {4 \left (3 b^2 c^2-13 a d (2 b c+9 a d)\right ) (e x)^{3/2} \sqrt {c+d x^2}}{195 d e^3}-\frac {2 \left (3 b^2 c^2-13 a d (2 b c+9 a d)\right ) (e x)^{3/2} \left (c+d x^2\right )^{3/2}}{117 c d e^3}-\frac {2 a^2 \left (c+d x^2\right )^{5/2}}{c e \sqrt {e x}}+\frac {2 b^2 (e x)^{3/2} \left (c+d x^2\right )^{5/2}}{13 d e^3}-\frac {\left (8 c \left (3 b^2 c^2-13 a d (2 b c+9 a d)\right )\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {c+\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{195 d e^3} \\ & = -\frac {4 \left (3 b^2 c^2-13 a d (2 b c+9 a d)\right ) (e x)^{3/2} \sqrt {c+d x^2}}{195 d e^3}-\frac {2 \left (3 b^2 c^2-13 a d (2 b c+9 a d)\right ) (e x)^{3/2} \left (c+d x^2\right )^{3/2}}{117 c d e^3}-\frac {2 a^2 \left (c+d x^2\right )^{5/2}}{c e \sqrt {e x}}+\frac {2 b^2 (e x)^{3/2} \left (c+d x^2\right )^{5/2}}{13 d e^3}-\frac {\left (8 c^{3/2} \left (3 b^2 c^2-13 a d (2 b c+9 a d)\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {c+\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{195 d^{3/2} e^2}+\frac {\left (8 c^{3/2} \left (3 b^2 c^2-13 a d (2 b c+9 a d)\right )\right ) \text {Subst}\left (\int \frac {1-\frac {\sqrt {d} x^2}{\sqrt {c} e}}{\sqrt {c+\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{195 d^{3/2} e^2} \\ & = -\frac {4 \left (3 b^2 c^2-13 a d (2 b c+9 a d)\right ) (e x)^{3/2} \sqrt {c+d x^2}}{195 d e^3}-\frac {8 c \left (3 b^2 c^2-13 a d (2 b c+9 a d)\right ) \sqrt {e x} \sqrt {c+d x^2}}{195 d^{3/2} e^2 \left (\sqrt {c}+\sqrt {d} x\right )}-\frac {2 \left (3 b^2 c^2-13 a d (2 b c+9 a d)\right ) (e x)^{3/2} \left (c+d x^2\right )^{3/2}}{117 c d e^3}-\frac {2 a^2 \left (c+d x^2\right )^{5/2}}{c e \sqrt {e x}}+\frac {2 b^2 (e x)^{3/2} \left (c+d x^2\right )^{5/2}}{13 d e^3}+\frac {8 c^{5/4} \left (3 b^2 c^2-13 a d (2 b c+9 a d)\right ) \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{195 d^{7/4} e^{3/2} \sqrt {c+d x^2}}-\frac {4 c^{5/4} \left (3 b^2 c^2-13 a d (2 b c+9 a d)\right ) \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{195 d^{7/4} e^{3/2} \sqrt {c+d x^2}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 11.12 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.34 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}}{(e x)^{3/2}} \, dx=\frac {x \left (2 \left (c+d x^2\right ) \left (117 a^2 d \left (-5 c+d x^2\right )+26 a b d x^2 \left (11 c+5 d x^2\right )+3 b^2 x^2 \left (4 c^2+25 c d x^2+15 d^2 x^4\right )\right )+24 c \left (-3 b^2 c^2+26 a b c d+117 a^2 d^2\right ) \sqrt {1+\frac {c}{d x^2}} x^2 \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {1}{2},\frac {3}{4},-\frac {c}{d x^2}\right )\right )}{585 d (e x)^{3/2} \sqrt {c+d x^2}} \]
[In]
[Out]
Time = 3.18 (sec) , antiderivative size = 307, normalized size of antiderivative = 0.64
method | result | size |
risch | \(-\frac {2 \sqrt {d \,x^{2}+c}\, \left (-45 b^{2} d^{2} x^{6}-130 a b \,d^{2} x^{4}-75 b^{2} c d \,x^{4}-117 a^{2} d^{2} x^{2}-286 a b c d \,x^{2}-12 b^{2} c^{2} x^{2}+585 a^{2} c d \right )}{585 d e \sqrt {e x}}+\frac {4 c \left (117 a^{2} d^{2}+26 a b c d -3 b^{2} c^{2}\right ) \sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, \left (-\frac {2 \sqrt {-c d}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}+\frac {\sqrt {-c d}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}\right ) \sqrt {e x \left (d \,x^{2}+c \right )}}{195 d^{2} \sqrt {d e \,x^{3}+c e x}\, e \sqrt {e x}\, \sqrt {d \,x^{2}+c}}\) | \(307\) |
elliptic | \(\frac {\sqrt {e x \left (d \,x^{2}+c \right )}\, \left (-\frac {2 \left (d e \,x^{2}+c e \right ) c \,a^{2}}{e^{2} \sqrt {x \left (d e \,x^{2}+c e \right )}}+\frac {2 b^{2} d \,x^{5} \sqrt {d e \,x^{3}+c e x}}{13 e^{2}}+\frac {2 \left (\frac {2 b d \left (a d +b c \right )}{e}-\frac {11 b^{2} d c}{13 e}\right ) x^{3} \sqrt {d e \,x^{3}+c e x}}{9 d e}+\frac {2 \left (\frac {a^{2} d^{2}+4 a b c d +b^{2} c^{2}}{e}-\frac {7 \left (\frac {2 b d \left (a d +b c \right )}{e}-\frac {11 b^{2} d c}{13 e}\right ) c}{9 d}\right ) x \sqrt {d e \,x^{3}+c e x}}{5 d e}+\frac {\left (\frac {2 a c \left (a d +b c \right )}{e}+\frac {d c \,a^{2}}{e}-\frac {3 \left (\frac {a^{2} d^{2}+4 a b c d +b^{2} c^{2}}{e}-\frac {7 \left (\frac {2 b d \left (a d +b c \right )}{e}-\frac {11 b^{2} d c}{13 e}\right ) c}{9 d}\right ) c}{5 d}\right ) \sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, \left (-\frac {2 \sqrt {-c d}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}+\frac {\sqrt {-c d}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}\right )}{d \sqrt {d e \,x^{3}+c e x}}\right )}{\sqrt {e x}\, \sqrt {d \,x^{2}+c}}\) | \(461\) |
default | \(\frac {\frac {2 b^{2} d^{4} x^{8}}{13}+\frac {4 a b \,d^{4} x^{6}}{9}+\frac {16 b^{2} c \,d^{3} x^{6}}{39}+\frac {24 \sqrt {2}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, E\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a^{2} c^{2} d^{2}}{5}+\frac {16 \sqrt {2}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, E\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a b \,c^{3} d}{15}-\frac {8 \sqrt {2}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, E\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) b^{2} c^{4}}{65}-\frac {12 \sqrt {2}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a^{2} c^{2} d^{2}}{5}-\frac {8 \sqrt {2}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a b \,c^{3} d}{15}+\frac {4 \sqrt {2}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) b^{2} c^{4}}{65}+\frac {2 a^{2} d^{4} x^{4}}{5}+\frac {64 c a b \,x^{4} d^{3}}{45}+\frac {58 b^{2} c^{2} d^{2} x^{4}}{195}-\frac {8 a^{2} c \,d^{3} x^{2}}{5}+\frac {44 a b \,c^{2} d^{2} x^{2}}{45}+\frac {8 b^{2} c^{3} d \,x^{2}}{195}-2 a^{2} c^{2} d^{2}}{\sqrt {d \,x^{2}+c}\, d^{2} e \sqrt {e x}}\) | \(669\) |
[In]
[Out]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.32 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}}{(e x)^{3/2}} \, dx=\frac {2 \, {\left (12 \, {\left (3 \, b^{2} c^{3} - 26 \, a b c^{2} d - 117 \, a^{2} c d^{2}\right )} \sqrt {d e} x {\rm weierstrassZeta}\left (-\frac {4 \, c}{d}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, c}{d}, 0, x\right )\right ) + {\left (45 \, b^{2} d^{3} x^{6} - 585 \, a^{2} c d^{2} + 5 \, {\left (15 \, b^{2} c d^{2} + 26 \, a b d^{3}\right )} x^{4} + {\left (12 \, b^{2} c^{2} d + 286 \, a b c d^{2} + 117 \, a^{2} d^{3}\right )} x^{2}\right )} \sqrt {d x^{2} + c} \sqrt {e x}\right )}}{585 \, d^{2} e^{2} x} \]
[In]
[Out]
Result contains complex when optimal does not.
Time = 13.68 (sec) , antiderivative size = 309, normalized size of antiderivative = 0.65 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}}{(e x)^{3/2}} \, dx=\frac {a^{2} c^{\frac {3}{2}} \Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, - \frac {1}{4} \\ \frac {3}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{2 e^{\frac {3}{2}} \sqrt {x} \Gamma \left (\frac {3}{4}\right )} + \frac {a^{2} \sqrt {c} d x^{\frac {3}{2}} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{2 e^{\frac {3}{2}} \Gamma \left (\frac {7}{4}\right )} + \frac {a b c^{\frac {3}{2}} x^{\frac {3}{2}} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{e^{\frac {3}{2}} \Gamma \left (\frac {7}{4}\right )} + \frac {a b \sqrt {c} d x^{\frac {7}{2}} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{e^{\frac {3}{2}} \Gamma \left (\frac {11}{4}\right )} + \frac {b^{2} c^{\frac {3}{2}} x^{\frac {7}{2}} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{2 e^{\frac {3}{2}} \Gamma \left (\frac {11}{4}\right )} + \frac {b^{2} \sqrt {c} d x^{\frac {11}{2}} \Gamma \left (\frac {11}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {11}{4} \\ \frac {15}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{2 e^{\frac {3}{2}} \Gamma \left (\frac {15}{4}\right )} \]
[In]
[Out]
\[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}}{(e x)^{3/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{2} {\left (d x^{2} + c\right )}^{\frac {3}{2}}}{\left (e x\right )^{\frac {3}{2}}} \,d x } \]
[In]
[Out]
\[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}}{(e x)^{3/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{2} {\left (d x^{2} + c\right )}^{\frac {3}{2}}}{\left (e x\right )^{\frac {3}{2}}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}}{(e x)^{3/2}} \, dx=\int \frac {{\left (b\,x^2+a\right )}^2\,{\left (d\,x^2+c\right )}^{3/2}}{{\left (e\,x\right )}^{3/2}} \,d x \]
[In]
[Out]