Optimal. Leaf size=935 \[ \frac{\left (c^2 d x^2+d\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2 x^{m+1}}{m+6}+\frac{5 d \left (c^2 d x^2+d\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2 x^{m+1}}{(m+4) (m+6)}+\frac{15 d^2 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2 x^{m+1}}{(m+6) \left (m^2+6 m+8\right )}-\frac{30 b c d^2 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right ) x^{m+2}}{(m+2)^2 (m+4) (m+6) \sqrt{c^2 x^2+1}}-\frac{10 b c d^2 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right ) x^{m+2}}{(m+6) \left (m^2+6 m+8\right ) \sqrt{c^2 x^2+1}}-\frac{2 b c d^2 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right ) x^{m+2}}{\left (m^2+8 m+12\right ) \sqrt{c^2 x^2+1}}+\frac{10 b^2 c^2 d^2 (3 m+10) \sqrt{c^2 d x^2+d} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{m+3}{2},\frac{m+5}{2},-c^2 x^2\right ) x^{m+3}}{(m+2) (m+3) (m+4)^3 (m+6) \sqrt{c^2 x^2+1}}+\frac{2 b^2 c^2 d^2 \left (15 m^2+130 m+264\right ) \sqrt{c^2 d x^2+d} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{m+3}{2},\frac{m+5}{2},-c^2 x^2\right ) x^{m+3}}{(m+2) (m+3) (m+4)^2 (m+6)^3 \sqrt{c^2 x^2+1}}+\frac{30 b^2 c^2 d^2 \sqrt{c^2 d x^2+d} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{m+3}{2},\frac{m+5}{2},-c^2 x^2\right ) x^{m+3}}{(m+2)^2 (m+3) (m+4) (m+6) \sqrt{c^2 x^2+1}}+\frac{2 b^2 c^2 d^2 \left (m^2+15 m+52\right ) \sqrt{c^2 d x^2+d} x^{m+3}}{(m+4)^2 (m+6)^3}+\frac{10 b^2 c^2 d^2 \sqrt{c^2 d x^2+d} x^{m+3}}{(m+4)^3 (m+6)}-\frac{4 b c^3 d^2 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right ) x^{m+4}}{(m+4) (m+6) \sqrt{c^2 x^2+1}}-\frac{10 b c^3 d^2 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right ) x^{m+4}}{(m+4)^2 (m+6) \sqrt{c^2 x^2+1}}+\frac{2 b^2 c^4 d^2 \sqrt{c^2 d x^2+d} x^{m+5}}{(m+6)^3}-\frac{2 b c^5 d^2 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right ) x^{m+6}}{(m+6)^2 \sqrt{c^2 x^2+1}}+\frac{15 d^3 \text{Unintegrable}\left (\frac{x^m \left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt{c^2 d x^2+d}},x\right )}{(m+6) \left (m^2+6 m+8\right )} \]
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Rubi [A] time = 0.152192, 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 = {} \[ \int x^m \left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int x^m \left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx &=\int x^m \left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx\\ \end{align*}
Mathematica [A] time = 2.72439, size = 0, normalized size = 0. \[ \int x^m \left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 1.248, size = 0, normalized size = 0. \begin{align*} \int{x}^{m} \left ({c}^{2}d{x}^{2}+d \right ) ^{{\frac{5}{2}}} \left ( a+b{\it Arcsinh} \left ( cx \right ) \right ) ^{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{\left (c^{2} d x^{2} + d\right )}^{\frac{5}{2}}{\left (b \operatorname{arsinh}\left (c x\right ) + a\right )}^{2} x^{m}\,{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 ({\left (a^{2} c^{4} d^{2} x^{4} + 2 \, a^{2} c^{2} d^{2} x^{2} + a^{2} d^{2} +{\left (b^{2} c^{4} d^{2} x^{4} + 2 \, b^{2} c^{2} d^{2} x^{2} + b^{2} d^{2}\right )} \operatorname{arsinh}\left (c x\right )^{2} + 2 \,{\left (a b c^{4} d^{2} x^{4} + 2 \, a b c^{2} d^{2} x^{2} + a b d^{2}\right )} \operatorname{arsinh}\left (c x\right )\right )} \sqrt{c^{2} d x^{2} + d} x^{m}, 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 [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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