Optimal. Leaf size=177 \[ -\frac{a \sqrt{a^2 x^2+1} \text{Unintegrable}\left (\frac{x}{\left (a^2 x^2+1\right )^2 \sqrt{\sinh ^{-1}(a x)}},x\right )}{6 c^2 \sqrt{a^2 c x^2+c}}-\frac{a \sqrt{a^2 x^2+1} \text{Unintegrable}\left (\frac{x}{\left (a^2 x^2+1\right ) \sqrt{\sinh ^{-1}(a x)}},x\right )}{3 c^2 \sqrt{a^2 c x^2+c}}+\frac{2 x \sqrt{\sinh ^{-1}(a x)}}{3 c^2 \sqrt{a^2 c x^2+c}}+\frac{x \sqrt{\sinh ^{-1}(a x)}}{3 c \left (a^2 c x^2+c\right )^{3/2}} \]
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Rubi [A] time = 0.198965, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{\sqrt{\sinh ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^{5/2}} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{\sqrt{\sinh ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^{5/2}} \, dx &=\frac{x \sqrt{\sinh ^{-1}(a x)}}{3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac{2 \int \frac{\sqrt{\sinh ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{3 c}-\frac{\left (a \sqrt{1+a^2 x^2}\right ) \int \frac{x}{\left (1+a^2 x^2\right )^2 \sqrt{\sinh ^{-1}(a x)}} \, dx}{6 c^2 \sqrt{c+a^2 c x^2}}\\ &=\frac{x \sqrt{\sinh ^{-1}(a x)}}{3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac{2 x \sqrt{\sinh ^{-1}(a x)}}{3 c^2 \sqrt{c+a^2 c x^2}}-\frac{\left (a \sqrt{1+a^2 x^2}\right ) \int \frac{x}{\left (1+a^2 x^2\right )^2 \sqrt{\sinh ^{-1}(a x)}} \, dx}{6 c^2 \sqrt{c+a^2 c x^2}}-\frac{\left (a \sqrt{1+a^2 x^2}\right ) \int \frac{x}{\left (1+a^2 x^2\right ) \sqrt{\sinh ^{-1}(a x)}} \, dx}{3 c^2 \sqrt{c+a^2 c x^2}}\\ \end{align*}
Mathematica [A] time = 1.36561, size = 0, normalized size = 0. \[ \int \frac{\sqrt{\sinh ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^{5/2}} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.234, size = 0, normalized size = 0. \begin{align*} \int{\sqrt{{\it Arcsinh} \left ( ax \right ) } \left ({a}^{2}c{x}^{2}+c \right ) ^{-{\frac{5}{2}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{\operatorname{arsinh}\left (a x\right )}}{{\left (a^{2} c x^{2} + c\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{\operatorname{arsinh}\left (a x\right )}}{{\left (a^{2} c x^{2} + c\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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