Optimal. Leaf size=42 \[ \frac{2 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^{7/2}}{7 a \sqrt{a^2 c x^2+c}} \]
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Rubi [A] time = 0.0720049, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {5677, 5675} \[ \frac{2 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^{7/2}}{7 a \sqrt{a^2 c x^2+c}} \]
Antiderivative was successfully verified.
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Rule 5677
Rule 5675
Rubi steps
\begin{align*} \int \frac{\sinh ^{-1}(a x)^{5/2}}{\sqrt{c+a^2 c x^2}} \, dx &=\frac{\sqrt{1+a^2 x^2} \int \frac{\sinh ^{-1}(a x)^{5/2}}{\sqrt{1+a^2 x^2}} \, dx}{\sqrt{c+a^2 c x^2}}\\ &=\frac{2 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^{7/2}}{7 a \sqrt{c+a^2 c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0361538, size = 42, normalized size = 1. \[ \frac{2 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^{7/2}}{7 a \sqrt{a^2 c x^2+c}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.036, size = 36, normalized size = 0.9 \begin{align*}{\frac{2}{7\,a} \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{{\frac{7}{2}}}\sqrt{{a}^{2}{x}^{2}+1}{\frac{1}{\sqrt{c \left ({a}^{2}{x}^{2}+1 \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{arsinh}\left (a x\right )^{\frac{5}{2}}}{\sqrt{a^{2} c x^{2} + c}}\,{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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{arsinh}\left (a x\right )^{\frac{5}{2}}}{\sqrt{a^{2} c x^{2} + c}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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