Integrand size = 21, antiderivative size = 777 \[ \int \frac {x^3 \left (a+b \text {csch}^{-1}(c x)\right )}{(d+e x)^{5/2}} \, dx=\frac {4 b d^2 \left (1+c^2 x^2\right )}{3 c e^2 \left (c^2 d^2+e^2\right ) \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {2 d^3 \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^4 (d+e x)^{3/2}}-\frac {6 d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{e^4 \sqrt {d+e x}}-\frac {6 d \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^4}-\frac {8 b \sqrt {-c^2} d^2 \sqrt {d+e x} \sqrt {1+c^2 x^2} E\left (\arcsin \left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )|-\frac {2 \sqrt {-c^2} e}{c^2 d-\sqrt {-c^2} e}\right )}{3 c e^3 \left (c^2 d^2+e^2\right ) \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {\frac {c^2 (d+e x)}{c^2 d-\sqrt {-c^2} e}}}+\frac {4 b c \left (2 c^2 d^2+e^2\right ) \sqrt {d+e x} \sqrt {1+c^2 x^2} E\left (\arcsin \left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )|-\frac {2 \sqrt {-c^2} e}{c^2 d-\sqrt {-c^2} e}\right )}{3 \left (-c^2\right )^{3/2} e^3 \left (c^2 d^2+e^2\right ) \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {\frac {c^2 (d+e x)}{c^2 d-\sqrt {-c^2} e}}}-\frac {32 b c d \sqrt {\frac {c^2 (d+e x)}{c^2 d-\sqrt {-c^2} e}} \sqrt {1+c^2 x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right ),-\frac {2 \sqrt {-c^2} e}{c^2 d-\sqrt {-c^2} e}\right )}{3 \left (-c^2\right )^{3/2} e^3 \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {64 b d^2 \sqrt {\frac {\sqrt {-c^2} (d+e x)}{\sqrt {-c^2} d+e}} \sqrt {1+c^2 x^2} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right ),\frac {2 e}{\sqrt {-c^2} d+e}\right )}{3 c e^4 \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}} \]
[Out]
Time = 2.25 (sec) , antiderivative size = 777, normalized size of antiderivative = 1.00, number of steps used = 31, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.857, Rules used = {45, 6445, 12, 6853, 6874, 759, 21, 733, 435, 972, 946, 174, 552, 551, 849, 858, 430, 1665} \[ \int \frac {x^3 \left (a+b \text {csch}^{-1}(c x)\right )}{(d+e x)^{5/2}} \, dx=\frac {2 d^3 \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^4 (d+e x)^{3/2}}-\frac {6 d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{e^4 \sqrt {d+e x}}-\frac {6 d \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^4}+\frac {64 b d^2 \sqrt {c^2 x^2+1} \sqrt {\frac {\sqrt {-c^2} (d+e x)}{\sqrt {-c^2} d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right ),\frac {2 e}{\sqrt {-c^2} d+e}\right )}{3 c e^4 x \sqrt {\frac {1}{c^2 x^2}+1} \sqrt {d+e x}}-\frac {8 b \sqrt {-c^2} d^2 \sqrt {c^2 x^2+1} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )|-\frac {2 \sqrt {-c^2} e}{c^2 d-\sqrt {-c^2} e}\right )}{3 c e^3 x \sqrt {\frac {1}{c^2 x^2}+1} \left (c^2 d^2+e^2\right ) \sqrt {\frac {c^2 (d+e x)}{c^2 d-\sqrt {-c^2} e}}}+\frac {4 b c \sqrt {c^2 x^2+1} \left (2 c^2 d^2+e^2\right ) \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )|-\frac {2 \sqrt {-c^2} e}{c^2 d-\sqrt {-c^2} e}\right )}{3 \left (-c^2\right )^{3/2} e^3 x \sqrt {\frac {1}{c^2 x^2}+1} \left (c^2 d^2+e^2\right ) \sqrt {\frac {c^2 (d+e x)}{c^2 d-\sqrt {-c^2} e}}}-\frac {32 b c d \sqrt {c^2 x^2+1} \sqrt {\frac {c^2 (d+e x)}{c^2 d-\sqrt {-c^2} e}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right ),-\frac {2 \sqrt {-c^2} e}{c^2 d-\sqrt {-c^2} e}\right )}{3 \left (-c^2\right )^{3/2} e^3 x \sqrt {\frac {1}{c^2 x^2}+1} \sqrt {d+e x}}+\frac {4 b d^2 \left (c^2 x^2+1\right )}{3 c e^2 x \sqrt {\frac {1}{c^2 x^2}+1} \left (c^2 d^2+e^2\right ) \sqrt {d+e x}} \]
[In]
[Out]
Rule 12
Rule 21
Rule 45
Rule 174
Rule 430
Rule 435
Rule 551
Rule 552
Rule 733
Rule 759
Rule 849
Rule 858
Rule 946
Rule 972
Rule 1665
Rule 6445
Rule 6853
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \frac {2 d^3 \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^4 (d+e x)^{3/2}}-\frac {6 d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{e^4 \sqrt {d+e x}}-\frac {6 d \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^4}+\frac {b \int \frac {2 \left (-16 d^3-24 d^2 e x-6 d e^2 x^2+e^3 x^3\right )}{3 e^4 \sqrt {1+\frac {1}{c^2 x^2}} x^2 (d+e x)^{3/2}} \, dx}{c} \\ & = \frac {2 d^3 \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^4 (d+e x)^{3/2}}-\frac {6 d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{e^4 \sqrt {d+e x}}-\frac {6 d \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^4}+\frac {(2 b) \int \frac {-16 d^3-24 d^2 e x-6 d e^2 x^2+e^3 x^3}{\sqrt {1+\frac {1}{c^2 x^2}} x^2 (d+e x)^{3/2}} \, dx}{3 c e^4} \\ & = \frac {2 d^3 \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^4 (d+e x)^{3/2}}-\frac {6 d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{e^4 \sqrt {d+e x}}-\frac {6 d \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^4}+\frac {\left (2 b \sqrt {1+c^2 x^2}\right ) \int \frac {-16 d^3-24 d^2 e x-6 d e^2 x^2+e^3 x^3}{x (d+e x)^{3/2} \sqrt {1+c^2 x^2}} \, dx}{3 c e^4 \sqrt {1+\frac {1}{c^2 x^2}} x} \\ & = \frac {2 d^3 \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^4 (d+e x)^{3/2}}-\frac {6 d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{e^4 \sqrt {d+e x}}-\frac {6 d \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^4}+\frac {\left (2 b \sqrt {1+c^2 x^2}\right ) \int \left (-\frac {24 d^2 e}{(d+e x)^{3/2} \sqrt {1+c^2 x^2}}-\frac {16 d^3}{x (d+e x)^{3/2} \sqrt {1+c^2 x^2}}-\frac {6 d e^2 x}{(d+e x)^{3/2} \sqrt {1+c^2 x^2}}+\frac {e^3 x^2}{(d+e x)^{3/2} \sqrt {1+c^2 x^2}}\right ) \, dx}{3 c e^4 \sqrt {1+\frac {1}{c^2 x^2}} x} \\ & = \frac {2 d^3 \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^4 (d+e x)^{3/2}}-\frac {6 d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{e^4 \sqrt {d+e x}}-\frac {6 d \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^4}-\frac {\left (32 b d^3 \sqrt {1+c^2 x^2}\right ) \int \frac {1}{x (d+e x)^{3/2} \sqrt {1+c^2 x^2}} \, dx}{3 c e^4 \sqrt {1+\frac {1}{c^2 x^2}} x}-\frac {\left (16 b d^2 \sqrt {1+c^2 x^2}\right ) \int \frac {1}{(d+e x)^{3/2} \sqrt {1+c^2 x^2}} \, dx}{c e^3 \sqrt {1+\frac {1}{c^2 x^2}} x}-\frac {\left (4 b d \sqrt {1+c^2 x^2}\right ) \int \frac {x}{(d+e x)^{3/2} \sqrt {1+c^2 x^2}} \, dx}{c e^2 \sqrt {1+\frac {1}{c^2 x^2}} x}+\frac {\left (2 b \sqrt {1+c^2 x^2}\right ) \int \frac {x^2}{(d+e x)^{3/2} \sqrt {1+c^2 x^2}} \, dx}{3 c e \sqrt {1+\frac {1}{c^2 x^2}} x} \\ & = \frac {68 b d^2 \left (1+c^2 x^2\right )}{3 c e^2 \left (c^2 d^2+e^2\right ) \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {2 d^3 \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^4 (d+e x)^{3/2}}-\frac {6 d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{e^4 \sqrt {d+e x}}-\frac {6 d \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^4}-\frac {\left (32 b d^3 \sqrt {1+c^2 x^2}\right ) \int \left (-\frac {e}{d (d+e x)^{3/2} \sqrt {1+c^2 x^2}}+\frac {1}{d x \sqrt {d+e x} \sqrt {1+c^2 x^2}}\right ) \, dx}{3 c e^4 \sqrt {1+\frac {1}{c^2 x^2}} x}+\frac {\left (32 b c d^2 \sqrt {1+c^2 x^2}\right ) \int \frac {-\frac {d}{2}-\frac {e x}{2}}{\sqrt {d+e x} \sqrt {1+c^2 x^2}} \, dx}{e^3 \left (c^2 d^2+e^2\right ) \sqrt {1+\frac {1}{c^2 x^2}} x}+\frac {\left (8 b d \sqrt {1+c^2 x^2}\right ) \int \frac {-\frac {e}{2}+\frac {1}{2} c^2 d x}{\sqrt {d+e x} \sqrt {1+c^2 x^2}} \, dx}{c e^2 \left (c^2 d^2+e^2\right ) \sqrt {1+\frac {1}{c^2 x^2}} x}-\frac {\left (4 b \sqrt {1+c^2 x^2}\right ) \int \frac {\frac {d}{2}-\frac {\left (2 c^2 d^2+e^2\right ) x}{2 e}}{\sqrt {d+e x} \sqrt {1+c^2 x^2}} \, dx}{3 c e \left (c^2 d^2+e^2\right ) \sqrt {1+\frac {1}{c^2 x^2}} x} \\ & = \frac {68 b d^2 \left (1+c^2 x^2\right )}{3 c e^2 \left (c^2 d^2+e^2\right ) \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {2 d^3 \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^4 (d+e x)^{3/2}}-\frac {6 d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{e^4 \sqrt {d+e x}}-\frac {6 d \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^4}-\frac {\left (32 b d^2 \sqrt {1+c^2 x^2}\right ) \int \frac {1}{x \sqrt {d+e x} \sqrt {1+c^2 x^2}} \, dx}{3 c e^4 \sqrt {1+\frac {1}{c^2 x^2}} x}-\frac {\left (4 b d \sqrt {1+c^2 x^2}\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {1+c^2 x^2}} \, dx}{c e^3 \sqrt {1+\frac {1}{c^2 x^2}} x}+\frac {\left (32 b d^2 \sqrt {1+c^2 x^2}\right ) \int \frac {1}{(d+e x)^{3/2} \sqrt {1+c^2 x^2}} \, dx}{3 c e^3 \sqrt {1+\frac {1}{c^2 x^2}} x}+\frac {\left (4 b c d^2 \sqrt {1+c^2 x^2}\right ) \int \frac {\sqrt {d+e x}}{\sqrt {1+c^2 x^2}} \, dx}{e^3 \left (c^2 d^2+e^2\right ) \sqrt {1+\frac {1}{c^2 x^2}} x}-\frac {\left (16 b c d^2 \sqrt {1+c^2 x^2}\right ) \int \frac {\sqrt {d+e x}}{\sqrt {1+c^2 x^2}} \, dx}{e^3 \left (c^2 d^2+e^2\right ) \sqrt {1+\frac {1}{c^2 x^2}} x}+\frac {\left (2 b \left (1+\frac {2 c^2 d^2}{e^2}\right ) \sqrt {1+c^2 x^2}\right ) \int \frac {\sqrt {d+e x}}{\sqrt {1+c^2 x^2}} \, dx}{3 c e \left (c^2 d^2+e^2\right ) \sqrt {1+\frac {1}{c^2 x^2}} x}-\frac {\left (4 b \left (\frac {d e}{2}+\frac {d \left (2 c^2 d^2+e^2\right )}{2 e}\right ) \sqrt {1+c^2 x^2}\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {1+c^2 x^2}} \, dx}{3 c e^2 \left (c^2 d^2+e^2\right ) \sqrt {1+\frac {1}{c^2 x^2}} x} \\ & = \frac {4 b d^2 \left (1+c^2 x^2\right )}{3 c e^2 \left (c^2 d^2+e^2\right ) \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {2 d^3 \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^4 (d+e x)^{3/2}}-\frac {6 d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{e^4 \sqrt {d+e x}}-\frac {6 d \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^4}-\frac {\left (32 b d^2 \sqrt {1+c^2 x^2}\right ) \int \frac {1}{x \sqrt {1-\sqrt {-c^2} x} \sqrt {1+\sqrt {-c^2} x} \sqrt {d+e x}} \, dx}{3 c e^4 \sqrt {1+\frac {1}{c^2 x^2}} x}-\frac {\left (64 b c d^2 \sqrt {1+c^2 x^2}\right ) \int \frac {-\frac {d}{2}-\frac {e x}{2}}{\sqrt {d+e x} \sqrt {1+c^2 x^2}} \, dx}{3 e^3 \left (c^2 d^2+e^2\right ) \sqrt {1+\frac {1}{c^2 x^2}} x}+\frac {\left (8 b \sqrt {-c^2} d^2 \sqrt {d+e x} \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {2 \sqrt {-c^2} e x^2}{c^2 d-\sqrt {-c^2} e}}}{\sqrt {1-x^2}} \, dx,x,\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )}{c e^3 \left (c^2 d^2+e^2\right ) \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {\frac {c^2 (d+e x)}{c^2 d-\sqrt {-c^2} e}}}-\frac {\left (32 b \sqrt {-c^2} d^2 \sqrt {d+e x} \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {2 \sqrt {-c^2} e x^2}{c^2 d-\sqrt {-c^2} e}}}{\sqrt {1-x^2}} \, dx,x,\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )}{c e^3 \left (c^2 d^2+e^2\right ) \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {\frac {c^2 (d+e x)}{c^2 d-\sqrt {-c^2} e}}}+\frac {\left (4 b \sqrt {-c^2} \left (1+\frac {2 c^2 d^2}{e^2}\right ) \sqrt {d+e x} \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {2 \sqrt {-c^2} e x^2}{c^2 d-\sqrt {-c^2} e}}}{\sqrt {1-x^2}} \, dx,x,\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )}{3 c^3 e \left (c^2 d^2+e^2\right ) \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {\frac {c^2 (d+e x)}{c^2 d-\sqrt {-c^2} e}}}-\frac {\left (8 b \sqrt {-c^2} d \sqrt {\frac {c^2 (d+e x)}{c^2 d-\sqrt {-c^2} e}} \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+\frac {2 \sqrt {-c^2} e x^2}{c^2 d-\sqrt {-c^2} e}}} \, dx,x,\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )}{c^3 e^3 \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {\left (8 b \sqrt {-c^2} \left (\frac {d e}{2}+\frac {d \left (2 c^2 d^2+e^2\right )}{2 e}\right ) \sqrt {\frac {c^2 (d+e x)}{c^2 d-\sqrt {-c^2} e}} \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+\frac {2 \sqrt {-c^2} e x^2}{c^2 d-\sqrt {-c^2} e}}} \, dx,x,\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )}{3 c^3 e^2 \left (c^2 d^2+e^2\right ) \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}} \\ & = \frac {4 b d^2 \left (1+c^2 x^2\right )}{3 c e^2 \left (c^2 d^2+e^2\right ) \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {2 d^3 \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^4 (d+e x)^{3/2}}-\frac {6 d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{e^4 \sqrt {d+e x}}-\frac {6 d \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^4}-\frac {24 b \sqrt {-c^2} d^2 \sqrt {d+e x} \sqrt {1+c^2 x^2} E\left (\arcsin \left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )|-\frac {2 \sqrt {-c^2} e}{c^2 d-\sqrt {-c^2} e}\right )}{c e^3 \left (c^2 d^2+e^2\right ) \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {\frac {c^2 (d+e x)}{c^2 d-\sqrt {-c^2} e}}}+\frac {4 b \sqrt {-c^2} \left (2 c^2 d^2+e^2\right ) \sqrt {d+e x} \sqrt {1+c^2 x^2} E\left (\arcsin \left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )|-\frac {2 \sqrt {-c^2} e}{c^2 d-\sqrt {-c^2} e}\right )}{3 c^3 e^3 \left (c^2 d^2+e^2\right ) \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {\frac {c^2 (d+e x)}{c^2 d-\sqrt {-c^2} e}}}-\frac {32 b \sqrt {-c^2} d \sqrt {\frac {c^2 (d+e x)}{c^2 d-\sqrt {-c^2} e}} \sqrt {1+c^2 x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right ),-\frac {2 \sqrt {-c^2} e}{c^2 d-\sqrt {-c^2} e}\right )}{3 c^3 e^3 \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {\left (64 b d^2 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \sqrt {2-x^2} \sqrt {d+\frac {e}{\sqrt {-c^2}}-\frac {e x^2}{\sqrt {-c^2}}}} \, dx,x,\sqrt {1-\sqrt {-c^2} x}\right )}{3 c e^4 \sqrt {1+\frac {1}{c^2 x^2}} x}+\frac {\left (32 b c d^2 \sqrt {1+c^2 x^2}\right ) \int \frac {\sqrt {d+e x}}{\sqrt {1+c^2 x^2}} \, dx}{3 e^3 \left (c^2 d^2+e^2\right ) \sqrt {1+\frac {1}{c^2 x^2}} x} \\ & = \frac {4 b d^2 \left (1+c^2 x^2\right )}{3 c e^2 \left (c^2 d^2+e^2\right ) \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {2 d^3 \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^4 (d+e x)^{3/2}}-\frac {6 d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{e^4 \sqrt {d+e x}}-\frac {6 d \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^4}-\frac {24 b \sqrt {-c^2} d^2 \sqrt {d+e x} \sqrt {1+c^2 x^2} E\left (\arcsin \left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )|-\frac {2 \sqrt {-c^2} e}{c^2 d-\sqrt {-c^2} e}\right )}{c e^3 \left (c^2 d^2+e^2\right ) \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {\frac {c^2 (d+e x)}{c^2 d-\sqrt {-c^2} e}}}+\frac {4 b \sqrt {-c^2} \left (2 c^2 d^2+e^2\right ) \sqrt {d+e x} \sqrt {1+c^2 x^2} E\left (\arcsin \left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )|-\frac {2 \sqrt {-c^2} e}{c^2 d-\sqrt {-c^2} e}\right )}{3 c^3 e^3 \left (c^2 d^2+e^2\right ) \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {\frac {c^2 (d+e x)}{c^2 d-\sqrt {-c^2} e}}}-\frac {32 b \sqrt {-c^2} d \sqrt {\frac {c^2 (d+e x)}{c^2 d-\sqrt {-c^2} e}} \sqrt {1+c^2 x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right ),-\frac {2 \sqrt {-c^2} e}{c^2 d-\sqrt {-c^2} e}\right )}{3 c^3 e^3 \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {\left (64 b \sqrt {-c^2} d^2 \sqrt {d+e x} \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {2 \sqrt {-c^2} e x^2}{c^2 d-\sqrt {-c^2} e}}}{\sqrt {1-x^2}} \, dx,x,\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )}{3 c e^3 \left (c^2 d^2+e^2\right ) \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {\frac {c^2 (d+e x)}{c^2 d-\sqrt {-c^2} e}}}+\frac {\left (64 b d^2 \sqrt {1+c^2 x^2} \sqrt {1+\frac {e \left (-1+\sqrt {-c^2} x\right )}{\sqrt {-c^2} d+e}}\right ) \text {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \sqrt {2-x^2} \sqrt {1-\frac {e x^2}{\sqrt {-c^2} \left (d+\frac {e}{\sqrt {-c^2}}\right )}}} \, dx,x,\sqrt {1-\sqrt {-c^2} x}\right )}{3 c e^4 \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}} \\ & = \frac {4 b d^2 \left (1+c^2 x^2\right )}{3 c e^2 \left (c^2 d^2+e^2\right ) \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {2 d^3 \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^4 (d+e x)^{3/2}}-\frac {6 d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{e^4 \sqrt {d+e x}}-\frac {6 d \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^4}-\frac {8 b \sqrt {-c^2} d^2 \sqrt {d+e x} \sqrt {1+c^2 x^2} E\left (\arcsin \left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )|-\frac {2 \sqrt {-c^2} e}{c^2 d-\sqrt {-c^2} e}\right )}{3 c e^3 \left (c^2 d^2+e^2\right ) \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {\frac {c^2 (d+e x)}{c^2 d-\sqrt {-c^2} e}}}+\frac {4 b \sqrt {-c^2} \left (2 c^2 d^2+e^2\right ) \sqrt {d+e x} \sqrt {1+c^2 x^2} E\left (\arcsin \left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )|-\frac {2 \sqrt {-c^2} e}{c^2 d-\sqrt {-c^2} e}\right )}{3 c^3 e^3 \left (c^2 d^2+e^2\right ) \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {\frac {c^2 (d+e x)}{c^2 d-\sqrt {-c^2} e}}}-\frac {32 b \sqrt {-c^2} d \sqrt {\frac {c^2 (d+e x)}{c^2 d-\sqrt {-c^2} e}} \sqrt {1+c^2 x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right ),-\frac {2 \sqrt {-c^2} e}{c^2 d-\sqrt {-c^2} e}\right )}{3 c^3 e^3 \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {64 b d^2 \sqrt {1+c^2 x^2} \sqrt {1-\frac {e \left (1-\sqrt {-c^2} x\right )}{\sqrt {-c^2} d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right ),\frac {2 e}{\sqrt {-c^2} d+e}\right )}{3 c e^4 \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}} \\ \end{align*}
Result contains higher order function than in optimal. Order 9 vs. order 4 in optimal.
Time = 34.88 (sec) , antiderivative size = 1108, normalized size of antiderivative = 1.43 \[ \int \frac {x^3 \left (a+b \text {csch}^{-1}(c x)\right )}{(d+e x)^{5/2}} \, dx=\frac {a d^4 \left (1+\frac {e x}{d}\right )^{5/2} B_{-\frac {e x}{d}}\left (4,-\frac {3}{2}\right )}{e^4 (d+e x)^{5/2}}+\frac {b \left (-\frac {c^3 \left (e+\frac {d}{x}\right )^3 x^3 \left (-\frac {4 \sqrt {1+\frac {1}{c^2 x^2}}}{3 e \left (c^2 d^2+e^2\right )}+\frac {32 c d \text {csch}^{-1}(c x)}{3 e^4}-\frac {2 c d \text {csch}^{-1}(c x)}{3 e^2 \left (e+\frac {d}{x}\right )^2}-\frac {2 c x \text {csch}^{-1}(c x)}{3 e^3}-\frac {2 \left (2 c^2 d^2 e \sqrt {1+\frac {1}{c^2 x^2}}+7 c^3 d^3 \text {csch}^{-1}(c x)+7 c d e^2 \text {csch}^{-1}(c x)\right )}{3 e^3 \left (c^2 d^2+e^2\right ) \left (e+\frac {d}{x}\right )}\right )}{(d+e x)^{5/2}}+\frac {2 \left (e+\frac {d}{x}\right )^{5/2} (c x)^{5/2} \left (-\frac {\sqrt {2} \left (8 c^3 d^3 e+8 c d e^3\right ) \sqrt {1+i c x} (i+c x) \sqrt {\frac {c d+c e x}{c d-i e}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {-\frac {e (i+c x)}{c d-i e}}\right ),\frac {i c d+e}{2 e}\right )}{\sqrt {1+\frac {1}{c^2 x^2}} \sqrt {e+\frac {d}{x}} (c x)^{3/2} \sqrt {\frac {e (1-i c x)}{i c d+e}}}+\frac {i \sqrt {2} (c d-i e) \left (16 c^4 d^4+16 c^2 d^2 e^2-e^4\right ) \sqrt {1+i c x} \sqrt {\frac {e (i+c x) (c d+c e x)}{(i c d+e)^2}} \operatorname {EllipticPi}\left (1+\frac {i c d}{e},\arcsin \left (\sqrt {-\frac {e (i+c x)}{c d-i e}}\right ),\frac {i c d+e}{2 e}\right )}{e \sqrt {1+\frac {1}{c^2 x^2}} \sqrt {e+\frac {d}{x}} (c x)^{3/2}}+\frac {2 e^3 \cosh \left (2 \text {csch}^{-1}(c x)\right ) \left (-\left ((c d+c e x) \left (1+c^2 x^2\right )\right )+\frac {c x \left (c d \sqrt {2+2 i c x} (i+c x) \sqrt {\frac {c d+c e x}{c d-i e}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {-\frac {e (i+c x)}{c d-i e}}\right ),\frac {i c d+e}{2 e}\right )+2 \sqrt {-\frac {e (-i+c x)}{c d+i e}} (i+c x) \sqrt {\frac {c d+c e x}{c d-i e}} \left ((c d+i e) E\left (\arcsin \left (\sqrt {\frac {c d+c e x}{c d-i e}}\right )|\frac {c d-i e}{c d+i e}\right )-i e \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {c d+c e x}{c d-i e}}\right ),\frac {c d-i e}{c d+i e}\right )\right )+(i c d+e) \sqrt {2+2 i c x} \sqrt {-\frac {e (i+c x)}{c d-i e}} \sqrt {\frac {e (i+c x) (c d+c e x)}{(i c d+e)^2}} \operatorname {EllipticPi}\left (1+\frac {i c d}{e},\arcsin \left (\sqrt {-\frac {e (i+c x)}{c d-i e}}\right ),\frac {i c d+e}{2 e}\right )\right )}{2 \sqrt {-\frac {e (i+c x)}{c d-i e}}}\right )}{\sqrt {1+\frac {1}{c^2 x^2}} \sqrt {e+\frac {d}{x}} \sqrt {c x} \left (2+c^2 x^2\right )}\right )}{3 e^4 \left (c^2 d^2+e^2\right ) (d+e x)^{5/2}}\right )}{c^4} \]
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Result contains complex when optimal does not.
Time = 9.29 (sec) , antiderivative size = 2728, normalized size of antiderivative = 3.51
method | result | size |
derivativedivides | \(\text {Expression too large to display}\) | \(2728\) |
default | \(\text {Expression too large to display}\) | \(2728\) |
parts | \(\text {Expression too large to display}\) | \(2729\) |
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Timed out. \[ \int \frac {x^3 \left (a+b \text {csch}^{-1}(c x)\right )}{(d+e x)^{5/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {x^3 \left (a+b \text {csch}^{-1}(c x)\right )}{(d+e x)^{5/2}} \, dx=\int \frac {x^{3} \left (a + b \operatorname {acsch}{\left (c x \right )}\right )}{\left (d + e x\right )^{\frac {5}{2}}}\, dx \]
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\[ \int \frac {x^3 \left (a+b \text {csch}^{-1}(c x)\right )}{(d+e x)^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arcsch}\left (c x\right ) + a\right )} x^{3}}{{\left (e x + d\right )}^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {x^3 \left (a+b \text {csch}^{-1}(c x)\right )}{(d+e x)^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arcsch}\left (c x\right ) + a\right )} x^{3}}{{\left (e x + d\right )}^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {x^3 \left (a+b \text {csch}^{-1}(c x)\right )}{(d+e x)^{5/2}} \, dx=\int \frac {x^3\,\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right )}{{\left (d+e\,x\right )}^{5/2}} \,d x \]
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