Optimal. Leaf size=149 \[ \frac {4 b \sqrt {c^2 x^2+1} \sqrt {\frac {\sqrt {-c^2} (d+e x)}{\sqrt {-c^2} d+e}} \Pi \left (2;\sin ^{-1}\left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )|\frac {2 e}{\sqrt {-c^2} d+e}\right )}{c e x \sqrt {\frac {1}{c^2 x^2}+1} \sqrt {d+e x}}-\frac {2 \left (a+b \text {csch}^{-1}(c x)\right )}{e \sqrt {d+e x}} \]
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Rubi [A] time = 0.28, antiderivative size = 149, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6290, 1574, 933, 168, 538, 537} \[ \frac {4 b \sqrt {c^2 x^2+1} \sqrt {\frac {\sqrt {-c^2} (d+e x)}{\sqrt {-c^2} d+e}} \Pi \left (2;\sin ^{-1}\left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )|\frac {2 e}{\sqrt {-c^2} d+e}\right )}{c e x \sqrt {\frac {1}{c^2 x^2}+1} \sqrt {d+e x}}-\frac {2 \left (a+b \text {csch}^{-1}(c x)\right )}{e \sqrt {d+e x}} \]
Antiderivative was successfully verified.
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Rule 168
Rule 537
Rule 538
Rule 933
Rule 1574
Rule 6290
Rubi steps
\begin {align*} \int \frac {a+b \text {csch}^{-1}(c x)}{(d+e x)^{3/2}} \, dx &=-\frac {2 \left (a+b \text {csch}^{-1}(c x)\right )}{e \sqrt {d+e x}}-\frac {(2 b) \int \frac {1}{\sqrt {1+\frac {1}{c^2 x^2}} x^2 \sqrt {d+e x}} \, dx}{c e}\\ &=-\frac {2 \left (a+b \text {csch}^{-1}(c x)\right )}{e \sqrt {d+e x}}-\frac {\left (2 b \sqrt {\frac {1}{c^2}+x^2}\right ) \int \frac {1}{x \sqrt {d+e x} \sqrt {\frac {1}{c^2}+x^2}} \, dx}{c e \sqrt {1+\frac {1}{c^2 x^2}} x}\\ &=-\frac {2 \left (a+b \text {csch}^{-1}(c x)\right )}{e \sqrt {d+e x}}-\frac {\left (2 b \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 \sqrt {1+\frac {1}{c^2 x^2}} x}\\ &=-\frac {2 \left (a+b \text {csch}^{-1}(c x)\right )}{e \sqrt {d+e x}}+\frac {\left (4 b \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 \sqrt {1+\frac {1}{c^2 x^2}} x}\\ &=-\frac {2 \left (a+b \text {csch}^{-1}(c x)\right )}{e \sqrt {d+e x}}+\frac {\left (4 b \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 \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}\\ &=-\frac {2 \left (a+b \text {csch}^{-1}(c x)\right )}{e \sqrt {d+e x}}+\frac {4 b \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 \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}\\ \end {align*}
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Mathematica [C] time = 0.68, size = 166, normalized size = 1.11 \[ \frac {-2 e \left (c^2 x^2+1\right ) \left (a+b \text {csch}^{-1}(c x)\right )+2 b c x \sqrt {\frac {2}{c^2 x^2}+2} \sqrt {1+i c x} (e+i c d) \sqrt {\frac {c e (c x+i) (d+e x)}{(e+i c d)^2}} \Pi \left (\frac {i c d}{e}+1;\sin ^{-1}\left (\sqrt {-\frac {e (c x+i)}{c d-i e}}\right )|\frac {i c d+e}{2 e}\right )}{e^2 \left (c^2 x^2+1\right ) \sqrt {d+e x}} \]
Antiderivative was successfully verified.
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fricas [F] time = 9.00, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {e x + d} {\left (b \operatorname {arcsch}\left (c x\right ) + a\right )}}{e^{2} x^{2} + 2 \, d e x + d^{2}}, x\right ) \]
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 \[ \int \frac {b \operatorname {arcsch}\left (c x\right ) + a}{{\left (e x + d\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.07, size = 328, normalized size = 2.20 \[ \frac {-\frac {2 a}{\sqrt {e x +d}}+2 b \left (-\frac {\mathrm {arccsch}\left (c x \right )}{\sqrt {e x +d}}+\frac {2 \sqrt {-\frac {i \left (e x +d \right ) c e +\left (e x +d \right ) c^{2} d -c^{2} d^{2}-e^{2}}{c^{2} d^{2}+e^{2}}}\, \sqrt {\frac {i \left (e x +d \right ) c e -\left (e x +d \right ) c^{2} d +c^{2} d^{2}+e^{2}}{c^{2} d^{2}+e^{2}}}\, \EllipticPi \left (\sqrt {e x +d}\, \sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}, \frac {c^{2} d^{2}+e^{2}}{\left (c d +i e \right ) c d}, \frac {\sqrt {-\frac {\left (-c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}}{\sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}}\right )}{c \sqrt {\frac {\left (e x +d \right )^{2} c^{2}-2 \left (e x +d \right ) c^{2} d +c^{2} d^{2}+e^{2}}{c^{2} x^{2} e^{2}}}\, x d \sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}}\right )}{e} \]
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 \[ -{\left (2 \, c^{2} \int \frac {x}{{\left (c^{2} e x^{2} + e\right )} \sqrt {c^{2} x^{2} + 1} \sqrt {e x + d} + {\left (c^{2} e x^{2} + e\right )} \sqrt {e x + d}}\,{d x} + \frac {2 \, \log \left (\sqrt {c^{2} x^{2} + 1} + 1\right )}{\sqrt {e x + d} e} + \int \frac {{\left (e \log \relax (c) - 2 \, e\right )} c^{2} x^{2} - 2 \, c^{2} d x + e \log \relax (c) + {\left (c^{2} e x^{2} + e\right )} \log \relax (x)}{{\left (c^{2} e^{2} x^{3} + c^{2} d e x^{2} + e^{2} x + d e\right )} \sqrt {e x + d}}\,{d x}\right )} b - \frac {2 \, a}{\sqrt {e x + d} e} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )}{{\left (d+e\,x\right )}^{3/2}} \,d x \]
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 \[ \int \frac {a + b \operatorname {acsch}{\left (c x \right )}}{\left (d + e x\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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