Optimal. Leaf size=731 \[ \frac {2 d^3 \left (a+b \text {csch}^{-1}(c x)\right )}{e^4 \sqrt {d+e x}}+\frac {6 d^2 \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^4}-\frac {2 d (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^4}-\frac {64 b d^3 \sqrt {c^2 x^2+1} \sqrt {\frac {\sqrt {-c^2} (d+e x)}{\sqrt {-c^2} d+e}} \Pi \left (2;\sin ^{-1}\left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )|\frac {2 e}{\sqrt {-c^2} d+e}\right )}{5 c e^4 x \sqrt {\frac {1}{c^2 x^2}+1} \sqrt {d+e x}}+\frac {8 b c d^2 \sqrt {c^2 x^2+1} \sqrt {\frac {c^2 (d+e x)}{c^2 d-\sqrt {-c^2} e}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )|-\frac {2 \sqrt {-c^2} e}{c^2 d-\sqrt {-c^2} e}\right )}{\left (-c^2\right )^{3/2} e^3 x \sqrt {\frac {1}{c^2 x^2}+1} \sqrt {d+e x}}-\frac {4 b c \sqrt {c^2 x^2+1} \left (2 c^2 d^2-e^2\right ) \sqrt {\frac {c^2 (d+e x)}{c^2 d-\sqrt {-c^2} e}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )|-\frac {2 \sqrt {-c^2} e}{c^2 d-\sqrt {-c^2} e}\right )}{15 \left (-c^2\right )^{5/2} e^3 x \sqrt {\frac {1}{c^2 x^2}+1} \sqrt {d+e x}}-\frac {32 b c d \sqrt {c^2 x^2+1} \sqrt {d+e x} E\left (\sin ^{-1}\left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )|-\frac {2 \sqrt {-c^2} e}{c^2 d-\sqrt {-c^2} e}\right )}{15 \left (-c^2\right )^{3/2} e^3 x \sqrt {\frac {1}{c^2 x^2}+1} \sqrt {\frac {c^2 (d+e x)}{c^2 d-\sqrt {-c^2} e}}}+\frac {4 b \left (c^2 x^2+1\right ) \sqrt {d+e x}}{15 c^3 e^2 x \sqrt {\frac {1}{c^2 x^2}+1}} \]
[Out]
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Rubi [A] time = 2.64, antiderivative size = 731, normalized size of antiderivative = 1.00, number of steps used = 23, number of rules used = 15, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {43, 6310, 12, 6721, 6742, 719, 419, 932, 168, 538, 537, 844, 424, 931, 1584} \[ \frac {2 d^3 \left (a+b \text {csch}^{-1}(c x)\right )}{e^4 \sqrt {d+e x}}+\frac {6 d^2 \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^4}-\frac {2 d (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^4}+\frac {8 b c d^2 \sqrt {c^2 x^2+1} \sqrt {\frac {c^2 (d+e x)}{c^2 d-\sqrt {-c^2} e}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )|-\frac {2 \sqrt {-c^2} e}{c^2 d-\sqrt {-c^2} e}\right )}{\left (-c^2\right )^{3/2} e^3 x \sqrt {\frac {1}{c^2 x^2}+1} \sqrt {d+e x}}-\frac {4 b c \sqrt {c^2 x^2+1} \left (2 c^2 d^2-e^2\right ) \sqrt {\frac {c^2 (d+e x)}{c^2 d-\sqrt {-c^2} e}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )|-\frac {2 \sqrt {-c^2} e}{c^2 d-\sqrt {-c^2} e}\right )}{15 \left (-c^2\right )^{5/2} e^3 x \sqrt {\frac {1}{c^2 x^2}+1} \sqrt {d+e x}}-\frac {64 b d^3 \sqrt {c^2 x^2+1} \sqrt {\frac {\sqrt {-c^2} (d+e x)}{\sqrt {-c^2} d+e}} \Pi \left (2;\sin ^{-1}\left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )|\frac {2 e}{\sqrt {-c^2} d+e}\right )}{5 c e^4 x \sqrt {\frac {1}{c^2 x^2}+1} \sqrt {d+e x}}+\frac {4 b \left (c^2 x^2+1\right ) \sqrt {d+e x}}{15 c^3 e^2 x \sqrt {\frac {1}{c^2 x^2}+1}}-\frac {32 b c d \sqrt {c^2 x^2+1} \sqrt {d+e x} E\left (\sin ^{-1}\left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )|-\frac {2 \sqrt {-c^2} e}{c^2 d-\sqrt {-c^2} e}\right )}{15 \left (-c^2\right )^{3/2} e^3 x \sqrt {\frac {1}{c^2 x^2}+1} \sqrt {\frac {c^2 (d+e x)}{c^2 d-\sqrt {-c^2} e}}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 43
Rule 168
Rule 419
Rule 424
Rule 537
Rule 538
Rule 719
Rule 844
Rule 931
Rule 932
Rule 1584
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)^{3/2}} \, dx &=\frac {2 d^3 \left (a+b \text {csch}^{-1}(c x)\right )}{e^4 \sqrt {d+e x}}+\frac {6 d^2 \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^4}-\frac {2 d (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^4}+\frac {b \int \frac {2 \left (16 d^3+8 d^2 e x-2 d e^2 x^2+e^3 x^3\right )}{5 e^4 \sqrt {1+\frac {1}{c^2 x^2}} x^2 \sqrt {d+e x}} \, dx}{c}\\ &=\frac {2 d^3 \left (a+b \text {csch}^{-1}(c x)\right )}{e^4 \sqrt {d+e x}}+\frac {6 d^2 \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^4}-\frac {2 d (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^4}+\frac {(2 b) \int \frac {16 d^3+8 d^2 e x-2 d e^2 x^2+e^3 x^3}{\sqrt {1+\frac {1}{c^2 x^2}} x^2 \sqrt {d+e x}} \, dx}{5 c e^4}\\ &=\frac {2 d^3 \left (a+b \text {csch}^{-1}(c x)\right )}{e^4 \sqrt {d+e x}}+\frac {6 d^2 \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^4}-\frac {2 d (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^4}+\frac {\left (2 b \sqrt {1+c^2 x^2}\right ) \int \frac {16 d^3+8 d^2 e x-2 d e^2 x^2+e^3 x^3}{x \sqrt {d+e x} \sqrt {1+c^2 x^2}} \, dx}{5 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 )}{e^4 \sqrt {d+e x}}+\frac {6 d^2 \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^4}-\frac {2 d (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^4}+\frac {\left (2 b \sqrt {1+c^2 x^2}\right ) \int \left (\frac {8 d^2 e}{\sqrt {d+e x} \sqrt {1+c^2 x^2}}+\frac {16 d^3}{x \sqrt {d+e x} \sqrt {1+c^2 x^2}}-\frac {2 d e^2 x}{\sqrt {d+e x} \sqrt {1+c^2 x^2}}+\frac {e^3 x^2}{\sqrt {d+e x} \sqrt {1+c^2 x^2}}\right ) \, dx}{5 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 )}{e^4 \sqrt {d+e x}}+\frac {6 d^2 \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^4}-\frac {2 d (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^4}+\frac {\left (32 b d^3 \sqrt {1+c^2 x^2}\right ) \int \frac {1}{x \sqrt {d+e x} \sqrt {1+c^2 x^2}} \, dx}{5 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}{\sqrt {d+e x} \sqrt {1+c^2 x^2}} \, dx}{5 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}{\sqrt {d+e x} \sqrt {1+c^2 x^2}} \, dx}{5 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}{\sqrt {d+e x} \sqrt {1+c^2 x^2}} \, dx}{5 c e \sqrt {1+\frac {1}{c^2 x^2}} x}\\ &=\frac {4 b \sqrt {d+e x} \left (1+c^2 x^2\right )}{15 c^3 e^2 \sqrt {1+\frac {1}{c^2 x^2}} x}+\frac {2 d^3 \left (a+b \text {csch}^{-1}(c x)\right )}{e^4 \sqrt {d+e x}}+\frac {6 d^2 \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^4}-\frac {2 d (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^4}+\frac {\left (32 b d^3 \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}{5 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 {\sqrt {d+e x}}{\sqrt {1+c^2 x^2}} \, dx}{5 c e^3 \sqrt {1+\frac {1}{c^2 x^2}} x}+\frac {\left (4 b d^2 \sqrt {1+c^2 x^2}\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {1+c^2 x^2}} \, dx}{5 c e^3 \sqrt {1+\frac {1}{c^2 x^2}} x}-\frac {\left (2 b \sqrt {1+c^2 x^2}\right ) \int \frac {e x+2 c^2 d x^2}{x \sqrt {d+e x} \sqrt {1+c^2 x^2}} \, dx}{15 c^3 e^2 \sqrt {1+\frac {1}{c^2 x^2}} x}+\frac {\left (32 b \sqrt {-c^2} d^2 \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 )}{5 c^3 e^3 \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}\\ &=\frac {4 b \sqrt {d+e x} \left (1+c^2 x^2\right )}{15 c^3 e^2 \sqrt {1+\frac {1}{c^2 x^2}} x}+\frac {2 d^3 \left (a+b \text {csch}^{-1}(c x)\right )}{e^4 \sqrt {d+e x}}+\frac {6 d^2 \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^4}-\frac {2 d (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^4}+\frac {32 b \sqrt {-c^2} d^2 \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 )}{5 c^3 e^3 \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {\left (64 b d^3 \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 )}{5 c e^4 \sqrt {1+\frac {1}{c^2 x^2}} x}-\frac {\left (2 b \sqrt {1+c^2 x^2}\right ) \int \frac {e+2 c^2 d x}{\sqrt {d+e x} \sqrt {1+c^2 x^2}} \, dx}{15 c^3 e^2 \sqrt {1+\frac {1}{c^2 x^2}} x}-\frac {\left (8 b \sqrt {-c^2} d \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 )}{5 c^3 e^3 \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^2 \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 )}{5 c^3 e^3 \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}\\ &=\frac {4 b \sqrt {d+e x} \left (1+c^2 x^2\right )}{15 c^3 e^2 \sqrt {1+\frac {1}{c^2 x^2}} x}+\frac {2 d^3 \left (a+b \text {csch}^{-1}(c x)\right )}{e^4 \sqrt {d+e x}}+\frac {6 d^2 \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^4}-\frac {2 d (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^4}-\frac {8 b \sqrt {-c^2} d \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 )}{5 c^3 e^3 \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {\frac {c^2 (d+e x)}{c^2 d-\sqrt {-c^2} e}}}+\frac {8 b \sqrt {-c^2} d^2 \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 )}{c^3 e^3 \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {\left (4 b d \sqrt {1+c^2 x^2}\right ) \int \frac {\sqrt {d+e x}}{\sqrt {1+c^2 x^2}} \, dx}{15 c e^3 \sqrt {1+\frac {1}{c^2 x^2}} x}-\frac {\left (2 b \left (-2 c^2 d^2+e^2\right ) \sqrt {1+c^2 x^2}\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {1+c^2 x^2}} \, dx}{15 c^3 e^3 \sqrt {1+\frac {1}{c^2 x^2}} x}-\frac {\left (64 b d^3 \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 )}{5 c e^4 \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}\\ &=\frac {4 b \sqrt {d+e x} \left (1+c^2 x^2\right )}{15 c^3 e^2 \sqrt {1+\frac {1}{c^2 x^2}} x}+\frac {2 d^3 \left (a+b \text {csch}^{-1}(c x)\right )}{e^4 \sqrt {d+e x}}+\frac {6 d^2 \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^4}-\frac {2 d (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^4}-\frac {8 b \sqrt {-c^2} d \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 )}{5 c^3 e^3 \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {\frac {c^2 (d+e x)}{c^2 d-\sqrt {-c^2} e}}}+\frac {8 b \sqrt {-c^2} d^2 \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 )}{c^3 e^3 \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {64 b d^3 \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 )}{5 c e^4 \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {\left (8 b \sqrt {-c^2} d \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 )}{15 c^3 e^3 \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 (-2 c^2 d^2+e^2\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 )}{15 c^5 e^3 \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}\\ &=\frac {4 b \sqrt {d+e x} \left (1+c^2 x^2\right )}{15 c^3 e^2 \sqrt {1+\frac {1}{c^2 x^2}} x}+\frac {2 d^3 \left (a+b \text {csch}^{-1}(c x)\right )}{e^4 \sqrt {d+e x}}+\frac {6 d^2 \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^4}-\frac {2 d (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^4}-\frac {32 b \sqrt {-c^2} d \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 )}{15 c^3 e^3 \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {\frac {c^2 (d+e x)}{c^2 d-\sqrt {-c^2} e}}}+\frac {8 b \sqrt {-c^2} d^2 \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 )}{c^3 e^3 \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {4 b \sqrt {-c^2} \left (2 c^2 d^2-e^2\right ) \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 )}{15 c^5 e^3 \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {64 b d^3 \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 )}{5 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 = 14.70, size = 1042, normalized size = 1.43 \[ \frac {a \left (\frac {e x}{d}+1\right )^{3/2} B_{-\frac {e x}{d}}\left (4,-\frac {1}{2}\right ) d^4}{e^4 (d+e x)^{3/2}}+\frac {b \left (-\frac {c^2 \left (\frac {d}{x}+e\right )^2 \left (\frac {2 c^2 \text {csch}^{-1}(c x) d^2}{e^3 \left (\frac {d}{x}+e\right )}-\frac {32 c^2 \text {csch}^{-1}(c x) d^2}{5 e^4}+\frac {32 c \sqrt {1+\frac {1}{c^2 x^2}} d}{15 e^3}-\frac {2 c^2 x^2 \text {csch}^{-1}(c x)}{5 e^2}-\frac {2 c x \left (2 e \sqrt {1+\frac {1}{c^2 x^2}}-9 c d \text {csch}^{-1}(c x)\right )}{15 e^3}\right ) x^2}{(d+e x)^{3/2}}-\frac {2 \left (\frac {d}{x}+e\right )^{3/2} (c x)^{3/2} \left (-\frac {\sqrt {2} \left (32 c^2 d^2 e-e^3\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 (48 c^3 d^3-8 c d e^2\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 )}{e \sqrt {1+\frac {1}{c^2 x^2}} \sqrt {\frac {d}{x}+e} (c x)^{3/2}}+\frac {16 c d e \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 )}{\sqrt {1+\frac {1}{c^2 x^2}} \sqrt {\frac {d}{x}+e} \sqrt {c x} \left (c^2 x^2+2\right )}\right )}{15 e^4 (d+e x)^{3/2}}\right )}{c^4} \]
Antiderivative was successfully verified.
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fricas [F] time = 16.60, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b x^{3} \operatorname {arcsch}\left (c x\right ) + a x^{3}\right )} \sqrt {e x + d}}{e^{2} x^{2} + 2 \, d e x + d^{2}}, x\right ) \]
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 \[ \int \frac {{\left (b \operatorname {arcsch}\left (c x\right ) + a\right )} x^{3}}{{\left (e x + d\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.08, size = 2019, normalized size = 2.76 \[ \text {result too large to display} \]
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 \[ \frac {2}{5} \, a {\left (\frac {{\left (e x + d\right )}^{\frac {5}{2}}}{e^{4}} - \frac {5 \, {\left (e x + d\right )}^{\frac {3}{2}} d}{e^{4}} + \frac {15 \, \sqrt {e x + d} d^{2}}{e^{4}} + \frac {5 \, d^{3}}{\sqrt {e x + d} e^{4}}\right )} + \frac {1}{5} \, b {\left (\frac {2 \, {\left (e^{3} x^{3} - 2 \, d e^{2} x^{2} + 8 \, d^{2} e x + 16 \, d^{3}\right )} \log \left (\sqrt {c^{2} x^{2} + 1} + 1\right )}{\sqrt {e x + d} e^{4}} + 5 \, \int \frac {2 \, {\left (c^{2} e^{3} x^{4} - 2 \, c^{2} d e^{2} x^{3} + 8 \, c^{2} d^{2} e x^{2} + 16 \, c^{2} d^{3} x\right )}}{5 \, {\left ({\left (c^{2} e^{4} x^{2} + e^{4}\right )} \sqrt {c^{2} x^{2} + 1} \sqrt {e x + d} + {\left (c^{2} e^{4} x^{2} + e^{4}\right )} \sqrt {e x + d}\right )}}\,{d x} - 5 \, \int -\frac {2 \, c^{2} d e^{3} x^{4} - 48 \, c^{2} d^{3} e x^{2} - {\left (5 \, e^{4} \log \relax (c) + 2 \, e^{4}\right )} c^{2} x^{5} - 32 \, c^{2} d^{4} x - {\left (12 \, c^{2} d^{2} e^{2} + 5 \, e^{4} \log \relax (c)\right )} x^{3} - 5 \, {\left (c^{2} e^{4} x^{5} + e^{4} x^{3}\right )} \log \relax (x)}{5 \, {\left (c^{2} e^{5} x^{3} + c^{2} d e^{4} x^{2} + e^{5} x + d e^{4}\right )} \sqrt {e x + d}}\,{d x}\right )} \]
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 \[ \int \frac {x^3\,\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right )}{{\left (d+e\,x\right )}^{3/2}} \,d x \]
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 \[ \int \frac {x^{3} \left (a + b \operatorname {acsch}{\left (c x \right )}\right )}{\left (d + e x\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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