Optimal. Leaf size=327 \[ -\frac{15 \sqrt{\pi } b^{5/2} e^{a/b} \text{Erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{64 c^3}+\frac{5 \sqrt{\frac{\pi }{3}} b^{5/2} e^{\frac{3 a}{b}} \text{Erf}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{576 c^3}+\frac{15 \sqrt{\pi } b^{5/2} e^{-\frac{a}{b}} \text{Erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{64 c^3}-\frac{5 \sqrt{\frac{\pi }{3}} b^{5/2} e^{-\frac{3 a}{b}} \text{Erfi}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{576 c^3}-\frac{5 b^2 x \sqrt{a+b \sinh ^{-1}(c x)}}{6 c^2}+\frac{5}{36} b^2 x^3 \sqrt{a+b \sinh ^{-1}(c x)}-\frac{5 b x^2 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}{18 c}+\frac{5 b \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}{9 c^3}+\frac{1}{3} x^3 \left (a+b \sinh ^{-1}(c x)\right )^{5/2} \]
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Rubi [A] time = 1.25032, antiderivative size = 327, normalized size of antiderivative = 1., number of steps used = 24, number of rules used = 10, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.625, Rules used = {5663, 5758, 5717, 5653, 5779, 3308, 2180, 2204, 2205, 3312} \[ -\frac{15 \sqrt{\pi } b^{5/2} e^{a/b} \text{Erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{64 c^3}+\frac{5 \sqrt{\frac{\pi }{3}} b^{5/2} e^{\frac{3 a}{b}} \text{Erf}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{576 c^3}+\frac{15 \sqrt{\pi } b^{5/2} e^{-\frac{a}{b}} \text{Erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{64 c^3}-\frac{5 \sqrt{\frac{\pi }{3}} b^{5/2} e^{-\frac{3 a}{b}} \text{Erfi}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{576 c^3}-\frac{5 b^2 x \sqrt{a+b \sinh ^{-1}(c x)}}{6 c^2}+\frac{5}{36} b^2 x^3 \sqrt{a+b \sinh ^{-1}(c x)}-\frac{5 b x^2 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}{18 c}+\frac{5 b \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}{9 c^3}+\frac{1}{3} x^3 \left (a+b \sinh ^{-1}(c x)\right )^{5/2} \]
Antiderivative was successfully verified.
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Rule 5663
Rule 5758
Rule 5717
Rule 5653
Rule 5779
Rule 3308
Rule 2180
Rule 2204
Rule 2205
Rule 3312
Rubi steps
\begin{align*} \int x^2 \left (a+b \sinh ^{-1}(c x)\right )^{5/2} \, dx &=\frac{1}{3} x^3 \left (a+b \sinh ^{-1}(c x)\right )^{5/2}-\frac{1}{6} (5 b c) \int \frac{x^3 \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}{\sqrt{1+c^2 x^2}} \, dx\\ &=-\frac{5 b x^2 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}{18 c}+\frac{1}{3} x^3 \left (a+b \sinh ^{-1}(c x)\right )^{5/2}+\frac{1}{12} \left (5 b^2\right ) \int x^2 \sqrt{a+b \sinh ^{-1}(c x)} \, dx+\frac{(5 b) \int \frac{x \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}{\sqrt{1+c^2 x^2}} \, dx}{9 c}\\ &=\frac{5}{36} b^2 x^3 \sqrt{a+b \sinh ^{-1}(c x)}+\frac{5 b \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}{9 c^3}-\frac{5 b x^2 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}{18 c}+\frac{1}{3} x^3 \left (a+b \sinh ^{-1}(c x)\right )^{5/2}-\frac{\left (5 b^2\right ) \int \sqrt{a+b \sinh ^{-1}(c x)} \, dx}{6 c^2}-\frac{1}{72} \left (5 b^3 c\right ) \int \frac{x^3}{\sqrt{1+c^2 x^2} \sqrt{a+b \sinh ^{-1}(c x)}} \, dx\\ &=-\frac{5 b^2 x \sqrt{a+b \sinh ^{-1}(c x)}}{6 c^2}+\frac{5}{36} b^2 x^3 \sqrt{a+b \sinh ^{-1}(c x)}+\frac{5 b \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}{9 c^3}-\frac{5 b x^2 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}{18 c}+\frac{1}{3} x^3 \left (a+b \sinh ^{-1}(c x)\right )^{5/2}-\frac{\left (5 b^3\right ) \operatorname{Subst}\left (\int \frac{\sinh ^3(x)}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{72 c^3}+\frac{\left (5 b^3\right ) \int \frac{x}{\sqrt{1+c^2 x^2} \sqrt{a+b \sinh ^{-1}(c x)}} \, dx}{12 c}\\ &=-\frac{5 b^2 x \sqrt{a+b \sinh ^{-1}(c x)}}{6 c^2}+\frac{5}{36} b^2 x^3 \sqrt{a+b \sinh ^{-1}(c x)}+\frac{5 b \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}{9 c^3}-\frac{5 b x^2 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}{18 c}+\frac{1}{3} x^3 \left (a+b \sinh ^{-1}(c x)\right )^{5/2}-\frac{\left (5 i b^3\right ) \operatorname{Subst}\left (\int \left (\frac{3 i \sinh (x)}{4 \sqrt{a+b x}}-\frac{i \sinh (3 x)}{4 \sqrt{a+b x}}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{72 c^3}+\frac{\left (5 b^3\right ) \operatorname{Subst}\left (\int \frac{\sinh (x)}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{12 c^3}\\ &=-\frac{5 b^2 x \sqrt{a+b \sinh ^{-1}(c x)}}{6 c^2}+\frac{5}{36} b^2 x^3 \sqrt{a+b \sinh ^{-1}(c x)}+\frac{5 b \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}{9 c^3}-\frac{5 b x^2 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}{18 c}+\frac{1}{3} x^3 \left (a+b \sinh ^{-1}(c x)\right )^{5/2}-\frac{\left (5 b^3\right ) \operatorname{Subst}\left (\int \frac{\sinh (3 x)}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{288 c^3}+\frac{\left (5 b^3\right ) \operatorname{Subst}\left (\int \frac{\sinh (x)}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{96 c^3}-\frac{\left (5 b^3\right ) \operatorname{Subst}\left (\int \frac{e^{-x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{24 c^3}+\frac{\left (5 b^3\right ) \operatorname{Subst}\left (\int \frac{e^x}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{24 c^3}\\ &=-\frac{5 b^2 x \sqrt{a+b \sinh ^{-1}(c x)}}{6 c^2}+\frac{5}{36} b^2 x^3 \sqrt{a+b \sinh ^{-1}(c x)}+\frac{5 b \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}{9 c^3}-\frac{5 b x^2 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}{18 c}+\frac{1}{3} x^3 \left (a+b \sinh ^{-1}(c x)\right )^{5/2}-\frac{\left (5 b^2\right ) \operatorname{Subst}\left (\int e^{\frac{a}{b}-\frac{x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c x)}\right )}{12 c^3}+\frac{\left (5 b^2\right ) \operatorname{Subst}\left (\int e^{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c x)}\right )}{12 c^3}+\frac{\left (5 b^3\right ) \operatorname{Subst}\left (\int \frac{e^{-3 x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{576 c^3}-\frac{\left (5 b^3\right ) \operatorname{Subst}\left (\int \frac{e^{3 x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{576 c^3}-\frac{\left (5 b^3\right ) \operatorname{Subst}\left (\int \frac{e^{-x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{192 c^3}+\frac{\left (5 b^3\right ) \operatorname{Subst}\left (\int \frac{e^x}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{192 c^3}\\ &=-\frac{5 b^2 x \sqrt{a+b \sinh ^{-1}(c x)}}{6 c^2}+\frac{5}{36} b^2 x^3 \sqrt{a+b \sinh ^{-1}(c x)}+\frac{5 b \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}{9 c^3}-\frac{5 b x^2 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}{18 c}+\frac{1}{3} x^3 \left (a+b \sinh ^{-1}(c x)\right )^{5/2}-\frac{5 b^{5/2} e^{a/b} \sqrt{\pi } \text{erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{24 c^3}+\frac{5 b^{5/2} e^{-\frac{a}{b}} \sqrt{\pi } \text{erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{24 c^3}+\frac{\left (5 b^2\right ) \operatorname{Subst}\left (\int e^{\frac{3 a}{b}-\frac{3 x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c x)}\right )}{288 c^3}-\frac{\left (5 b^2\right ) \operatorname{Subst}\left (\int e^{-\frac{3 a}{b}+\frac{3 x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c x)}\right )}{288 c^3}-\frac{\left (5 b^2\right ) \operatorname{Subst}\left (\int e^{\frac{a}{b}-\frac{x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c x)}\right )}{96 c^3}+\frac{\left (5 b^2\right ) \operatorname{Subst}\left (\int e^{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c x)}\right )}{96 c^3}\\ &=-\frac{5 b^2 x \sqrt{a+b \sinh ^{-1}(c x)}}{6 c^2}+\frac{5}{36} b^2 x^3 \sqrt{a+b \sinh ^{-1}(c x)}+\frac{5 b \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}{9 c^3}-\frac{5 b x^2 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}{18 c}+\frac{1}{3} x^3 \left (a+b \sinh ^{-1}(c x)\right )^{5/2}-\frac{15 b^{5/2} e^{a/b} \sqrt{\pi } \text{erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{64 c^3}+\frac{5 b^{5/2} e^{\frac{3 a}{b}} \sqrt{\frac{\pi }{3}} \text{erf}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{576 c^3}+\frac{15 b^{5/2} e^{-\frac{a}{b}} \sqrt{\pi } \text{erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{64 c^3}-\frac{5 b^{5/2} e^{-\frac{3 a}{b}} \sqrt{\frac{\pi }{3}} \text{erfi}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{576 c^3}\\ \end{align*}
Mathematica [A] time = 0.431815, size = 215, normalized size = 0.66 \[ -\frac{e^{-\frac{3 a}{b}} \left (a+b \sinh ^{-1}(c x)\right )^{5/2} \left (81 e^{\frac{4 a}{b}} \sqrt{-\frac{a+b \sinh ^{-1}(c x)}{b}} \text{Gamma}\left (\frac{7}{2},\frac{a}{b}+\sinh ^{-1}(c x)\right )+\sqrt{3} \sqrt{\frac{a}{b}+\sinh ^{-1}(c x)} \text{Gamma}\left (\frac{7}{2},-\frac{3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )-81 e^{\frac{2 a}{b}} \sqrt{\frac{a}{b}+\sinh ^{-1}(c x)} \text{Gamma}\left (\frac{7}{2},-\frac{a+b \sinh ^{-1}(c x)}{b}\right )-\sqrt{3} e^{\frac{6 a}{b}} \sqrt{-\frac{a+b \sinh ^{-1}(c x)}{b}} \text{Gamma}\left (\frac{7}{2},\frac{3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )\right )}{648 c^3 \left (-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{b^2}\right )^{3/2}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 180., size = 0, normalized size = 0. \begin{align*} \text{hanged} \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{\left (b \operatorname{arsinh}\left (c x\right ) + a\right )}^{\frac{5}{2}} x^{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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \operatorname{arsinh}\left (c x\right ) + a\right )}^{\frac{5}{2}} x^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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