Optimal. Leaf size=488 \[ \frac {3 d^2 \text {Int}\left (\frac {x^m \left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt {c^2 d x^2+d}},x\right )}{m^2+6 m+8}+\frac {3 d x^{m+1} \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{m^2+6 m+8}-\frac {2 b c d x^{m+2} \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{\left (m^2+6 m+8\right ) \sqrt {c^2 x^2+1}}+\frac {x^{m+1} \left (c^2 d x^2+d\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{m+4}-\frac {6 b c d x^{m+2} \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{(m+2)^2 (m+4) \sqrt {c^2 x^2+1}}-\frac {2 b c^3 d x^{m+4} \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{(m+4)^2 \sqrt {c^2 x^2+1}}+\frac {2 b^2 c^2 d (3 m+10) x^{m+3} \sqrt {c^2 d x^2+d} \, _2F_1\left (\frac {1}{2},\frac {m+3}{2};\frac {m+5}{2};-c^2 x^2\right )}{(m+2) (m+3) (m+4)^3 \sqrt {c^2 x^2+1}}+\frac {6 b^2 c^2 d x^{m+3} \sqrt {c^2 d x^2+d} \, _2F_1\left (\frac {1}{2},\frac {m+3}{2};\frac {m+5}{2};-c^2 x^2\right )}{(m+2)^2 (m+3) (m+4) \sqrt {c^2 x^2+1}}+\frac {2 b^2 c^2 d x^{m+3} \sqrt {c^2 d x^2+d}}{(m+4)^3} \]
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Rubi [A] time = 0.15, 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 = {} \[ \int x^m \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx \]
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 )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx &=\int x^m \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx\\ \end {align*}
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Mathematica [A] time = 0.43, size = 0, normalized size = 0.00 \[ \int x^m \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.54, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a^{2} c^{2} d x^{2} + a^{2} d + {\left (b^{2} c^{2} d x^{2} + b^{2} d\right )} \operatorname {arsinh}\left (c x\right )^{2} + 2 \, {\left (a b c^{2} d x^{2} + a b d\right )} \operatorname {arsinh}\left (c x\right )\right )} \sqrt {c^{2} d x^{2} + d} x^{m}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.42, size = 0, normalized size = 0.00 \[ \int x^{m} \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \arcsinh \left (c x \right )\right )^{2}\, dx \]
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 \[ \int {\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2} x^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.00 \[ \int x^m\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,{\left (d\,c^2\,x^2+d\right )}^{3/2} \,d x \]
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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