Optimal. Leaf size=179 \[ -\frac{3 \sqrt{\frac{\pi }{2}} b^{3/2} e^{\frac{2 a}{b}} \text{Erf}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{64 c^2}+\frac{3 \sqrt{\frac{\pi }{2}} b^{3/2} e^{-\frac{2 a}{b}} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{64 c^2}-\frac{3 b x \sqrt{c^2 x^2+1} \sqrt{a+b \sinh ^{-1}(c x)}}{8 c}+\frac{\left (a+b \sinh ^{-1}(c x)\right )^{3/2}}{4 c^2}+\frac{1}{2} x^2 \left (a+b \sinh ^{-1}(c x)\right )^{3/2} \]
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Rubi [A] time = 0.478203, antiderivative size = 179, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 10, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.714, Rules used = {5663, 5758, 5675, 5669, 5448, 12, 3308, 2180, 2204, 2205} \[ -\frac{3 \sqrt{\frac{\pi }{2}} b^{3/2} e^{\frac{2 a}{b}} \text{Erf}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{64 c^2}+\frac{3 \sqrt{\frac{\pi }{2}} b^{3/2} e^{-\frac{2 a}{b}} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{64 c^2}-\frac{3 b x \sqrt{c^2 x^2+1} \sqrt{a+b \sinh ^{-1}(c x)}}{8 c}+\frac{\left (a+b \sinh ^{-1}(c x)\right )^{3/2}}{4 c^2}+\frac{1}{2} x^2 \left (a+b \sinh ^{-1}(c x)\right )^{3/2} \]
Antiderivative was successfully verified.
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Rule 5663
Rule 5758
Rule 5675
Rule 5669
Rule 5448
Rule 12
Rule 3308
Rule 2180
Rule 2204
Rule 2205
Rubi steps
\begin{align*} \int x \left (a+b \sinh ^{-1}(c x)\right )^{3/2} \, dx &=\frac{1}{2} x^2 \left (a+b \sinh ^{-1}(c x)\right )^{3/2}-\frac{1}{4} (3 b c) \int \frac{x^2 \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{1+c^2 x^2}} \, dx\\ &=-\frac{3 b x \sqrt{1+c^2 x^2} \sqrt{a+b \sinh ^{-1}(c x)}}{8 c}+\frac{1}{2} x^2 \left (a+b \sinh ^{-1}(c x)\right )^{3/2}+\frac{1}{16} \left (3 b^2\right ) \int \frac{x}{\sqrt{a+b \sinh ^{-1}(c x)}} \, dx+\frac{(3 b) \int \frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{1+c^2 x^2}} \, dx}{8 c}\\ &=-\frac{3 b x \sqrt{1+c^2 x^2} \sqrt{a+b \sinh ^{-1}(c x)}}{8 c}+\frac{\left (a+b \sinh ^{-1}(c x)\right )^{3/2}}{4 c^2}+\frac{1}{2} x^2 \left (a+b \sinh ^{-1}(c x)\right )^{3/2}+\frac{\left (3 b^2\right ) \operatorname{Subst}\left (\int \frac{\cosh (x) \sinh (x)}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{16 c^2}\\ &=-\frac{3 b x \sqrt{1+c^2 x^2} \sqrt{a+b \sinh ^{-1}(c x)}}{8 c}+\frac{\left (a+b \sinh ^{-1}(c x)\right )^{3/2}}{4 c^2}+\frac{1}{2} x^2 \left (a+b \sinh ^{-1}(c x)\right )^{3/2}+\frac{\left (3 b^2\right ) \operatorname{Subst}\left (\int \frac{\sinh (2 x)}{2 \sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{16 c^2}\\ &=-\frac{3 b x \sqrt{1+c^2 x^2} \sqrt{a+b \sinh ^{-1}(c x)}}{8 c}+\frac{\left (a+b \sinh ^{-1}(c x)\right )^{3/2}}{4 c^2}+\frac{1}{2} x^2 \left (a+b \sinh ^{-1}(c x)\right )^{3/2}+\frac{\left (3 b^2\right ) \operatorname{Subst}\left (\int \frac{\sinh (2 x)}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{32 c^2}\\ &=-\frac{3 b x \sqrt{1+c^2 x^2} \sqrt{a+b \sinh ^{-1}(c x)}}{8 c}+\frac{\left (a+b \sinh ^{-1}(c x)\right )^{3/2}}{4 c^2}+\frac{1}{2} x^2 \left (a+b \sinh ^{-1}(c x)\right )^{3/2}-\frac{\left (3 b^2\right ) \operatorname{Subst}\left (\int \frac{e^{-2 x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{64 c^2}+\frac{\left (3 b^2\right ) \operatorname{Subst}\left (\int \frac{e^{2 x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{64 c^2}\\ &=-\frac{3 b x \sqrt{1+c^2 x^2} \sqrt{a+b \sinh ^{-1}(c x)}}{8 c}+\frac{\left (a+b \sinh ^{-1}(c x)\right )^{3/2}}{4 c^2}+\frac{1}{2} x^2 \left (a+b \sinh ^{-1}(c x)\right )^{3/2}-\frac{(3 b) \operatorname{Subst}\left (\int e^{\frac{2 a}{b}-\frac{2 x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c x)}\right )}{32 c^2}+\frac{(3 b) \operatorname{Subst}\left (\int e^{-\frac{2 a}{b}+\frac{2 x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c x)}\right )}{32 c^2}\\ &=-\frac{3 b x \sqrt{1+c^2 x^2} \sqrt{a+b \sinh ^{-1}(c x)}}{8 c}+\frac{\left (a+b \sinh ^{-1}(c x)\right )^{3/2}}{4 c^2}+\frac{1}{2} x^2 \left (a+b \sinh ^{-1}(c x)\right )^{3/2}-\frac{3 b^{3/2} e^{\frac{2 a}{b}} \sqrt{\frac{\pi }{2}} \text{erf}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{64 c^2}+\frac{3 b^{3/2} e^{-\frac{2 a}{b}} \sqrt{\frac{\pi }{2}} \text{erfi}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{64 c^2}\\ \end{align*}
Mathematica [A] time = 0.100346, size = 129, normalized size = 0.72 \[ \frac{b e^{-\frac{2 a}{b}} \sqrt{a+b \sinh ^{-1}(c x)} \left (e^{\frac{4 a}{b}} \sqrt{-\frac{a+b \sinh ^{-1}(c x)}{b}} \text{Gamma}\left (\frac{5}{2},\frac{2 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )-\sqrt{\frac{a}{b}+\sinh ^{-1}(c x)} \text{Gamma}\left (\frac{5}{2},-\frac{2 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )\right )}{16 \sqrt{2} c^2 \sqrt{-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{b^2}}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.052, size = 0, normalized size = 0. \begin{align*} \int x \left ( a+b{\it Arcsinh} \left ( cx \right ) \right ) ^{{\frac{3}{2}}}\, dx \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{3}{2}} x\,{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] time = 0., size = 0, normalized size = 0. \begin{align*} \int x \left (a + b \operatorname{asinh}{\left (c x \right )}\right )^{\frac{3}{2}}\, dx \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{3}{2}} x\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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