Optimal. Leaf size=454 \[ d^3 \text{Unintegrable}\left (\frac{\left (a+b \sinh ^{-1}(c x)\right )^n}{x^2 \sqrt{c^2 d x^2+d}},x\right )+\frac{c d^3 2^{-2 (n+3)} e^{-\frac{4 a}{b}} \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )^n \left (-\frac{a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \text{Gamma}\left (n+1,-\frac{4 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{\sqrt{c^2 d x^2+d}}+\frac{c d^3 2^{-n-2} e^{-\frac{2 a}{b}} \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )^n \left (-\frac{a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \text{Gamma}\left (n+1,-\frac{2 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{\sqrt{c^2 d x^2+d}}-\frac{c d^3 2^{-n-2} e^{\frac{2 a}{b}} \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )^n \left (\frac{a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \text{Gamma}\left (n+1,\frac{2 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{\sqrt{c^2 d x^2+d}}-\frac{c d^3 2^{-2 (n+3)} e^{\frac{4 a}{b}} \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )^n \left (\frac{a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \text{Gamma}\left (n+1,\frac{4 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{\sqrt{c^2 d x^2+d}}+\frac{15 c d^3 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )^{n+1}}{8 b (n+1) \sqrt{c^2 d x^2+d}} \]
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Rubi [A] time = 0.155804, 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 = {} \[ \int \frac{\left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^n}{x^2} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{\left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^n}{x^2} \, dx &=\int \frac{\left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^n}{x^2} \, dx\\ \end{align*}
Mathematica [A] time = 0.638545, size = 0, normalized size = 0. \[ \int \frac{\left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^n}{x^2} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.162, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b{\it Arcsinh} \left ( cx \right ) \right ) ^{n}}{{x}^{2}} \left ({c}^{2}d{x}^{2}+d \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{{\left (c^{2} d x^{2} + d\right )}^{\frac{5}{2}}{\left (b \operatorname{arsinh}\left (c x\right ) + a\right )}^{n}}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (c^{4} d^{2} x^{4} + 2 \, c^{2} d^{2} x^{2} + d^{2}\right )} \sqrt{c^{2} d x^{2} + d}{\left (b \operatorname{arsinh}\left (c x\right ) + a\right )}^{n}}{x^{2}}, x\right ) \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{{\left (c^{2} d x^{2} + d\right )}^{\frac{5}{2}}{\left (b \operatorname{arsinh}\left (c x\right ) + a\right )}^{n}}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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