Optimal. Leaf size=777 \[ \frac {2 d^3 \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^4 (d+e x)^{3/2}}-\frac {6 d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{e^4 \sqrt {d+e x}}-\frac {6 d \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^4}+\frac {64 b d^2 \sqrt {c^2 x^2+1} \sqrt {\frac {\sqrt {-c^2} (d+e x)}{\sqrt {-c^2} d+e}} \Pi \left (2;\sin ^{-1}\left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )|\frac {2 e}{\sqrt {-c^2} d+e}\right )}{3 c e^4 x \sqrt {\frac {1}{c^2 x^2}+1} \sqrt {d+e x}}+\frac {4 b d^2 \left (c^2 x^2+1\right )}{3 c e^2 x \sqrt {\frac {1}{c^2 x^2}+1} \left (c^2 d^2+e^2\right ) \sqrt {d+e x}}-\frac {8 b \sqrt {-c^2} d^2 \sqrt {c^2 x^2+1} \sqrt {d+e x} E\left (\sin ^{-1}\left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )|-\frac {2 \sqrt {-c^2} e}{c^2 d-\sqrt {-c^2} e}\right )}{3 c e^3 x \sqrt {\frac {1}{c^2 x^2}+1} \left (c^2 d^2+e^2\right ) \sqrt {\frac {c^2 (d+e x)}{c^2 d-\sqrt {-c^2} e}}}+\frac {4 b c \sqrt {c^2 x^2+1} \left (2 c^2 d^2+e^2\right ) \sqrt {d+e x} E\left (\sin ^{-1}\left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )|-\frac {2 \sqrt {-c^2} e}{c^2 d-\sqrt {-c^2} e}\right )}{3 \left (-c^2\right )^{3/2} e^3 x \sqrt {\frac {1}{c^2 x^2}+1} \left (c^2 d^2+e^2\right ) \sqrt {\frac {c^2 (d+e x)}{c^2 d-\sqrt {-c^2} e}}}-\frac {32 b c d \sqrt {c^2 x^2+1} \sqrt {\frac {c^2 (d+e x)}{c^2 d-\sqrt {-c^2} e}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )|-\frac {2 \sqrt {-c^2} e}{c^2 d-\sqrt {-c^2} e}\right )}{3 \left (-c^2\right )^{3/2} e^3 x \sqrt {\frac {1}{c^2 x^2}+1} \sqrt {d+e x}} \]
[Out]
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Rubi [A] time = 3.19, antiderivative size = 777, normalized size of antiderivative = 1.00, number of steps used = 31, number of rules used = 18, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.857, Rules used = {43, 6310, 12, 6721, 6742, 745, 21, 719, 424, 958, 932, 168, 538, 537, 835, 844, 419, 1651} \[ \frac {2 d^3 \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^4 (d+e x)^{3/2}}-\frac {6 d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{e^4 \sqrt {d+e x}}-\frac {6 d \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^4}+\frac {4 b d^2 \left (c^2 x^2+1\right )}{3 c e^2 x \sqrt {\frac {1}{c^2 x^2}+1} \left (c^2 d^2+e^2\right ) \sqrt {d+e x}}-\frac {8 b \sqrt {-c^2} d^2 \sqrt {c^2 x^2+1} \sqrt {d+e x} E\left (\sin ^{-1}\left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )|-\frac {2 \sqrt {-c^2} e}{c^2 d-\sqrt {-c^2} e}\right )}{3 c e^3 x \sqrt {\frac {1}{c^2 x^2}+1} \left (c^2 d^2+e^2\right ) \sqrt {\frac {c^2 (d+e x)}{c^2 d-\sqrt {-c^2} e}}}+\frac {4 b c \sqrt {c^2 x^2+1} \left (2 c^2 d^2+e^2\right ) \sqrt {d+e x} E\left (\sin ^{-1}\left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )|-\frac {2 \sqrt {-c^2} e}{c^2 d-\sqrt {-c^2} e}\right )}{3 \left (-c^2\right )^{3/2} e^3 x \sqrt {\frac {1}{c^2 x^2}+1} \left (c^2 d^2+e^2\right ) \sqrt {\frac {c^2 (d+e x)}{c^2 d-\sqrt {-c^2} e}}}+\frac {64 b d^2 \sqrt {c^2 x^2+1} \sqrt {\frac {\sqrt {-c^2} (d+e x)}{\sqrt {-c^2} d+e}} \Pi \left (2;\sin ^{-1}\left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )|\frac {2 e}{\sqrt {-c^2} d+e}\right )}{3 c e^4 x \sqrt {\frac {1}{c^2 x^2}+1} \sqrt {d+e x}}-\frac {32 b c d \sqrt {c^2 x^2+1} \sqrt {\frac {c^2 (d+e x)}{c^2 d-\sqrt {-c^2} e}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )|-\frac {2 \sqrt {-c^2} e}{c^2 d-\sqrt {-c^2} e}\right )}{3 \left (-c^2\right )^{3/2} e^3 x \sqrt {\frac {1}{c^2 x^2}+1} \sqrt {d+e x}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 21
Rule 43
Rule 168
Rule 419
Rule 424
Rule 537
Rule 538
Rule 719
Rule 745
Rule 835
Rule 844
Rule 932
Rule 958
Rule 1651
Rule 6310
Rule 6721
Rule 6742
Rubi steps
\begin {align*} \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 {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 (\sin ^{-1}\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 (\sin ^{-1}\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} F\left (\sin ^{-1}\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 ) \operatorname {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 (\sin ^{-1}\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 (\sin ^{-1}\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} F\left (\sin ^{-1}\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 ) \operatorname {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 ) \operatorname {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 (\sin ^{-1}\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 (\sin ^{-1}\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} F\left (\sin ^{-1}\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}} \Pi \left (2;\sin ^{-1}\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*}
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Mathematica [C] time = 14.49, size = 1108, normalized size = 1.43 \[ \frac {a \left (\frac {e x}{d}+1\right )^{5/2} B_{-\frac {e x}{d}}\left (4,-\frac {3}{2}\right ) d^4}{e^4 (d+e x)^{5/2}}+\frac {b \left (\frac {2 \left (\frac {d}{x}+e\right )^{5/2} (c x)^{5/2} \left (\frac {2 \cosh \left (2 \text {csch}^{-1}(c x)\right ) \left (\frac {c x \left (c d \sqrt {2 i c x+2} (c x+i) \sqrt {\frac {c d+c e x}{c d-i e}} F\left (\sin ^{-1}\left (\sqrt {-\frac {e (c x+i)}{c d-i e}}\right )|\frac {i c d+e}{2 e}\right )+2 \sqrt {-\frac {e (c x-i)}{c d+i e}} (c x+i) \sqrt {\frac {c d+c e x}{c d-i e}} \left ((c d+i e) E\left (\sin ^{-1}\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 F\left (\sin ^{-1}\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 i c x+2} \sqrt {-\frac {e (c x+i)}{c d-i e}} \sqrt {\frac {e (c x+i) (c d+c e x)}{(i c d+e)^2}} \Pi \left (\frac {i c d}{e}+1;\sin ^{-1}\left (\sqrt {-\frac {e (c x+i)}{c d-i e}}\right )|\frac {i c d+e}{2 e}\right )\right )}{2 \sqrt {-\frac {e (c x+i)}{c d-i e}}}-(c d+c e x) \left (c^2 x^2+1\right )\right ) e^3}{\sqrt {1+\frac {1}{c^2 x^2}} \sqrt {\frac {d}{x}+e} \sqrt {c x} \left (c^2 x^2+2\right )}-\frac {\sqrt {2} \left (8 c^3 e d^3+8 c e^3 d\right ) \sqrt {i c x+1} (c x+i) \sqrt {\frac {c d+c e x}{c d-i e}} F\left (\sin ^{-1}\left (\sqrt {-\frac {e (c x+i)}{c d-i e}}\right )|\frac {i c d+e}{2 e}\right )}{\sqrt {1+\frac {1}{c^2 x^2}} \sqrt {\frac {d}{x}+e} (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 e^2 d^2-e^4\right ) \sqrt {i c x+1} \sqrt {\frac {e (c x+i) (c d+c e x)}{(i c d+e)^2}} \Pi \left (\frac {i c d}{e}+1;\sin ^{-1}\left (\sqrt {-\frac {e (c x+i)}{c d-i e}}\right )|\frac {i c d+e}{2 e}\right )}{\sqrt {1+\frac {1}{c^2 x^2}} \sqrt {\frac {d}{x}+e} (c x)^{3/2} e}\right )}{3 e^4 \left (c^2 d^2+e^2\right ) (d+e x)^{5/2}}-\frac {c^3 \left (\frac {d}{x}+e\right )^3 x^3 \left (-\frac {2 c x \text {csch}^{-1}(c x)}{3 e^3}-\frac {2 c d \text {csch}^{-1}(c x)}{3 e^2 \left (\frac {d}{x}+e\right )^2}+\frac {32 c d \text {csch}^{-1}(c x)}{3 e^4}-\frac {2 \left (7 c^3 \text {csch}^{-1}(c x) d^3+2 c^2 e \sqrt {1+\frac {1}{c^2 x^2}} d^2+7 c e^2 \text {csch}^{-1}(c x) d\right )}{3 e^3 \left (c^2 d^2+e^2\right ) \left (\frac {d}{x}+e\right )}-\frac {4 \sqrt {1+\frac {1}{c^2 x^2}}}{3 e \left (c^2 d^2+e^2\right )}\right )}{(d+e x)^{5/2}}\right )}{c^4} \]
Antiderivative was successfully verified.
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fricas [F] time = 16.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b x^{3} \operatorname {arcsch}\left (c x\right ) + a x^{3}\right )} \sqrt {e x + d}}{e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \operatorname {arcsch}\left (c x\right ) + a\right )} x^{3}}{{\left (e x + d\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.09, size = 2726, normalized size = 3.51 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{3} \, b {\left (\frac {2 \, {\left (e^{3} x^{3} - 6 \, d e^{2} x^{2} - 24 \, d^{2} e x - 16 \, d^{3}\right )} \log \left (\sqrt {c^{2} x^{2} + 1} + 1\right )}{{\left (e^{5} x + d e^{4}\right )} \sqrt {e x + d}} + 3 \, \int \frac {2 \, {\left (c^{2} e^{3} x^{4} - 6 \, c^{2} d e^{2} x^{3} - 24 \, c^{2} d^{2} e x^{2} - 16 \, c^{2} d^{3} x\right )}}{3 \, {\left ({\left (c^{2} e^{5} x^{3} + c^{2} d e^{4} x^{2} + e^{5} x + d e^{4}\right )} \sqrt {c^{2} x^{2} + 1} \sqrt {e x + d} + {\left (c^{2} e^{5} x^{3} + c^{2} d e^{4} x^{2} + e^{5} x + d e^{4}\right )} \sqrt {e x + d}\right )}}\,{d x} - 3 \, \int -\frac {10 \, c^{2} d e^{3} x^{4} + 80 \, c^{2} d^{3} e x^{2} - {\left (3 \, e^{4} \log \relax (c) + 2 \, e^{4}\right )} c^{2} x^{5} + 32 \, c^{2} d^{4} x + 3 \, {\left (20 \, c^{2} d^{2} e^{2} - e^{4} \log \relax (c)\right )} x^{3} - 3 \, {\left (c^{2} e^{4} x^{5} + e^{4} x^{3}\right )} \log \relax (x)}{3 \, {\left (c^{2} e^{6} x^{4} + 2 \, c^{2} d e^{5} x^{3} + 2 \, d e^{5} x + d^{2} e^{4} + {\left (c^{2} d^{2} e^{4} + e^{6}\right )} x^{2}\right )} \sqrt {e x + d}}\,{d x}\right )} + \frac {2}{3} \, a {\left (\frac {{\left (e x + d\right )}^{\frac {3}{2}}}{e^{4}} - \frac {9 \, \sqrt {e x + d} d}{e^{4}} - \frac {9 \, d^{2}}{\sqrt {e x + d} e^{4}} + \frac {d^{3}}{{\left (e x + d\right )}^{\frac {3}{2}} e^{4}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^3\,\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right )}{{\left (d+e\,x\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{3} \left (a + b \operatorname {acsch}{\left (c x \right )}\right )}{\left (d + e x\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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