Optimal. Leaf size=318 \[ \frac {2 \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^2}+\frac {2 d \left (a+b \text {csch}^{-1}(c x)\right )}{e^2 \sqrt {d+e x}}-\frac {8 b d \sqrt {c^2 x^2+1} \sqrt {\frac {\sqrt {-c^2} (d+e x)}{\sqrt {-c^2} d+e}} \Pi \left (2;\sin ^{-1}\left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )|\frac {2 e}{\sqrt {-c^2} d+e}\right )}{c e^2 x \sqrt {\frac {1}{c^2 x^2}+1} \sqrt {d+e x}}+\frac {4 b c \sqrt {c^2 x^2+1} \sqrt {\frac {c^2 (d+e x)}{c^2 d-\sqrt {-c^2} e}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )|-\frac {2 \sqrt {-c^2} e}{c^2 d-\sqrt {-c^2} e}\right )}{\left (-c^2\right )^{3/2} e x \sqrt {\frac {1}{c^2 x^2}+1} \sqrt {d+e x}} \]
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Rubi [A] time = 1.69, antiderivative size = 318, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 11, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.579, Rules used = {43, 6310, 12, 6721, 6742, 719, 419, 932, 168, 538, 537} \[ \frac {2 \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^2}+\frac {2 d \left (a+b \text {csch}^{-1}(c x)\right )}{e^2 \sqrt {d+e x}}-\frac {8 b d \sqrt {c^2 x^2+1} \sqrt {\frac {\sqrt {-c^2} (d+e x)}{\sqrt {-c^2} d+e}} \Pi \left (2;\sin ^{-1}\left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )|\frac {2 e}{\sqrt {-c^2} d+e}\right )}{c e^2 x \sqrt {\frac {1}{c^2 x^2}+1} \sqrt {d+e x}}+\frac {4 b c \sqrt {c^2 x^2+1} \sqrt {\frac {c^2 (d+e x)}{c^2 d-\sqrt {-c^2} e}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )|-\frac {2 \sqrt {-c^2} e}{c^2 d-\sqrt {-c^2} e}\right )}{\left (-c^2\right )^{3/2} e x \sqrt {\frac {1}{c^2 x^2}+1} \sqrt {d+e x}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 43
Rule 168
Rule 419
Rule 537
Rule 538
Rule 719
Rule 932
Rule 6310
Rule 6721
Rule 6742
Rubi steps
\begin {align*} \int \frac {x \left (a+b \text {csch}^{-1}(c x)\right )}{(d+e x)^{3/2}} \, dx &=\frac {2 d \left (a+b \text {csch}^{-1}(c x)\right )}{e^2 \sqrt {d+e x}}+\frac {2 \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^2}+\frac {b \int \frac {2 (2 d+e x)}{e^2 \sqrt {1+\frac {1}{c^2 x^2}} x^2 \sqrt {d+e x}} \, dx}{c}\\ &=\frac {2 d \left (a+b \text {csch}^{-1}(c x)\right )}{e^2 \sqrt {d+e x}}+\frac {2 \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^2}+\frac {(2 b) \int \frac {2 d+e x}{\sqrt {1+\frac {1}{c^2 x^2}} x^2 \sqrt {d+e x}} \, dx}{c e^2}\\ &=\frac {2 d \left (a+b \text {csch}^{-1}(c x)\right )}{e^2 \sqrt {d+e x}}+\frac {2 \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^2}+\frac {\left (2 b \sqrt {1+c^2 x^2}\right ) \int \frac {2 d+e x}{x \sqrt {d+e x} \sqrt {1+c^2 x^2}} \, dx}{c e^2 \sqrt {1+\frac {1}{c^2 x^2}} x}\\ &=\frac {2 d \left (a+b \text {csch}^{-1}(c x)\right )}{e^2 \sqrt {d+e x}}+\frac {2 \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^2}+\frac {\left (2 b \sqrt {1+c^2 x^2}\right ) \int \left (\frac {e}{\sqrt {d+e x} \sqrt {1+c^2 x^2}}+\frac {2 d}{x \sqrt {d+e x} \sqrt {1+c^2 x^2}}\right ) \, dx}{c e^2 \sqrt {1+\frac {1}{c^2 x^2}} x}\\ &=\frac {2 d \left (a+b \text {csch}^{-1}(c x)\right )}{e^2 \sqrt {d+e x}}+\frac {2 \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^2}+\frac {\left (4 b d \sqrt {1+c^2 x^2}\right ) \int \frac {1}{x \sqrt {d+e x} \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 {1}{\sqrt {d+e x} \sqrt {1+c^2 x^2}} \, dx}{c e \sqrt {1+\frac {1}{c^2 x^2}} x}\\ &=\frac {2 d \left (a+b \text {csch}^{-1}(c x)\right )}{e^2 \sqrt {d+e x}}+\frac {2 \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^2}+\frac {\left (4 b d \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}{c e^2 \sqrt {1+\frac {1}{c^2 x^2}} x}+\frac {\left (4 b \sqrt {-c^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 )}{c^3 e \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}\\ &=\frac {2 d \left (a+b \text {csch}^{-1}(c x)\right )}{e^2 \sqrt {d+e x}}+\frac {2 \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^2}+\frac {4 b \sqrt {-c^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 \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {\left (8 b d \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 )}{c e^2 \sqrt {1+\frac {1}{c^2 x^2}} x}\\ &=\frac {2 d \left (a+b \text {csch}^{-1}(c x)\right )}{e^2 \sqrt {d+e x}}+\frac {2 \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^2}+\frac {4 b \sqrt {-c^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 \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {\left (8 b d \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 )}{c e^2 \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}\\ &=\frac {2 d \left (a+b \text {csch}^{-1}(c x)\right )}{e^2 \sqrt {d+e x}}+\frac {2 \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^2}+\frac {4 b \sqrt {-c^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 \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {8 b d \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 )}{c e^2 \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}\\ \end {align*}
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Mathematica [C] time = 1.58, size = 264, normalized size = 0.83 \[ \frac {2 \left (\frac {a (2 d+e x)}{\sqrt {d+e x}}-\frac {2 i b \sqrt {-\frac {e (c x-i)}{c d+i e}} \sqrt {-\frac {e (c x+i)}{c d-i e}} \left (F\left (i \sinh ^{-1}\left (\sqrt {-\frac {c}{c d-i e}} \sqrt {d+e x}\right )|\frac {c d-i e}{c d+i e}\right )-2 \Pi \left (1-\frac {i e}{c d};i \sinh ^{-1}\left (\sqrt {-\frac {c}{c d-i e}} \sqrt {d+e x}\right )|\frac {c d-i e}{c d+i e}\right )\right )}{c x \sqrt {\frac {1}{c^2 x^2}+1} \sqrt {-\frac {c}{c d-i e}}}+\frac {b \text {csch}^{-1}(c x) (2 d+e x)}{\sqrt {d+e x}}\right )}{e^2} \]
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
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 \[ \int \frac {{\left (b \operatorname {arcsch}\left (c x\right ) + a\right )} x}{{\left (e x + d\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.07, size = 418, normalized size = 1.31 \[ \frac {2 a \left (\sqrt {e x +d}+\frac {d}{\sqrt {e x +d}}\right )+2 b \left (\sqrt {e x +d}\, \mathrm {arccsch}\left (c x \right )+\frac {\mathrm {arccsch}\left (c x \right ) d}{\sqrt {e x +d}}+\frac {2 \sqrt {-\frac {i \left (e x +d \right ) c e +\left (e x +d \right ) c^{2} d -c^{2} d^{2}-e^{2}}{c^{2} d^{2}+e^{2}}}\, \sqrt {\frac {i \left (e x +d \right ) c e -\left (e x +d \right ) c^{2} d +c^{2} d^{2}+e^{2}}{c^{2} d^{2}+e^{2}}}\, \left (\EllipticF \left (\sqrt {e x +d}\, \sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}, \sqrt {-\frac {-c^{2} d^{2}+2 i c d e +e^{2}}{c^{2} d^{2}+e^{2}}}\right )-2 \EllipticPi \left (\sqrt {e x +d}\, \sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}, \frac {c^{2} d^{2}+e^{2}}{\left (c d +i e \right ) c d}, \frac {\sqrt {-\frac {\left (-c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}}{\sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}}\right )\right )}{c \sqrt {\frac {\left (e x +d \right )^{2} c^{2}-2 \left (e x +d \right ) c^{2} d +c^{2} d^{2}+e^{2}}{c^{2} x^{2} e^{2}}}\, x \sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}}\right )}{e^{2}} \]
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 \[ b {\left (\frac {2 \, {\left (e x + 2 \, d\right )} \log \left (\sqrt {c^{2} x^{2} + 1} + 1\right )}{\sqrt {e x + d} e^{2}} + \int \frac {2 \, {\left (c^{2} e x^{2} + 2 \, c^{2} d x\right )}}{{\left (c^{2} e^{2} x^{2} + e^{2}\right )} \sqrt {c^{2} x^{2} + 1} \sqrt {e x + d} + {\left (c^{2} e^{2} x^{2} + e^{2}\right )} \sqrt {e x + d}}\,{d x} - \int \frac {6 \, c^{2} d e x^{2} + {\left (e^{2} \log \relax (c) + 2 \, e^{2}\right )} c^{2} x^{3} + {\left (4 \, c^{2} d^{2} + e^{2} \log \relax (c)\right )} x + {\left (c^{2} e^{2} x^{3} + e^{2} x\right )} \log \relax (x)}{{\left (c^{2} e^{3} x^{3} + c^{2} d e^{2} x^{2} + e^{3} x + d e^{2}\right )} \sqrt {e x + d}}\,{d x}\right )} + 2 \, a {\left (\frac {\sqrt {e x + d}}{e^{2}} + \frac {d}{\sqrt {e x + d} e^{2}}\right )} \]
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 \[ \int \frac {x\,\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right )}{{\left (d+e\,x\right )}^{3/2}} \,d x \]
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 \[ \int \frac {x \left (a + b \operatorname {acsch}{\left (c x \right )}\right )}{\left (d + e x\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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