Optimal. Leaf size=24 \[ \text {Int}\left (\frac {a+b \text {csch}^{-1}(c x)}{x^2 (d+e x)^{5/2}},x\right ) \]
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Rubi [A] time = 0.11, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {a+b \text {csch}^{-1}(c x)}{x^2 (d+e x)^{5/2}} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {a+b \text {csch}^{-1}(c x)}{x^2 (d+e x)^{5/2}} \, dx &=\int \frac {a+b \text {csch}^{-1}(c x)}{x^2 (d+e x)^{5/2}} \, dx\\ \end {align*}
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Mathematica [A] time = 28.47, size = 0, normalized size = 0.00 \[ \int \frac {a+b \text {csch}^{-1}(c x)}{x^2 (d+e x)^{5/2}} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 2.04, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {e x + d} {\left (b \operatorname {arcsch}\left (c x\right ) + a\right )}}{e^{3} x^{5} + 3 \, d e^{2} x^{4} + 3 \, d^{2} e x^{3} + d^{3} x^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {b \operatorname {arcsch}\left (c x\right ) + a}{{\left (e x + d\right )}^{\frac {5}{2}} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F(-2)] time = 180.00, size = 0, normalized size = 0.00 \[ \int \frac {a +b \,\mathrm {arccsch}\left (c x \right )}{x^{2} \left (e x +d \right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{6} \, b {\left (\frac {{\left (\frac {2 \, {\left (15 \, {\left (e x + d\right )}^{2} e^{2} - 10 \, {\left (e x + d\right )} d e^{2} - 2 \, d^{2} e^{2}\right )}}{{\left (e x + d\right )}^{\frac {5}{2}} d^{3} - {\left (e x + d\right )}^{\frac {3}{2}} d^{4}} + \frac {15 \, e^{2} \log \left (\frac {\sqrt {e x + d} - \sqrt {d}}{\sqrt {e x + d} + \sqrt {d}}\right )}{d^{\frac {7}{2}}}\right )} \log \relax (c)}{e} - 6 \, \int \frac {\log \relax (x)}{\sqrt {e x + d} e^{2} x^{4} + 2 \, \sqrt {e x + d} d e x^{3} + \sqrt {e x + d} d^{2} x^{2}}\,{d x} + 6 \, \int \frac {\log \left (\sqrt {c^{2} x^{2} + 1} + 1\right )}{\sqrt {e x + d} e^{2} x^{4} + 2 \, \sqrt {e x + d} d e x^{3} + \sqrt {e x + d} d^{2} x^{2}}\,{d x}\right )} - \frac {1}{6} \, a {\left (\frac {2 \, {\left (15 \, {\left (e x + d\right )}^{2} e - 10 \, {\left (e x + d\right )} d e - 2 \, d^{2} e\right )}}{{\left (e x + d\right )}^{\frac {5}{2}} d^{3} - {\left (e x + d\right )}^{\frac {3}{2}} d^{4}} + \frac {15 \, e \log \left (\frac {\sqrt {e x + d} - \sqrt {d}}{\sqrt {e x + d} + \sqrt {d}}\right )}{d^{\frac {7}{2}}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )}{x^2\,{\left (d+e\,x\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-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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