Optimal. Leaf size=326 \[ -\frac{3 \sqrt{\pi } b^{3/2} c e^{a/b} \text{Erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{8 d^2}-\frac{3 \sqrt{\frac{\pi }{2}} b^{3/2} e^{\frac{2 a}{b}} \text{Erf}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{64 d^2}-\frac{3 \sqrt{\pi } b^{3/2} c e^{-\frac{a}{b}} \text{Erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{8 d^2}+\frac{3 \sqrt{\frac{\pi }{2}} b^{3/2} e^{-\frac{2 a}{b}} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{64 d^2}-\frac{3 b \sinh \left (2 \sinh ^{-1}(c+d x)\right ) \sqrt{a+b \sinh ^{-1}(c+d x)}}{16 d^2}+\frac{3 b c \sqrt{(c+d x)^2+1} \sqrt{a+b \sinh ^{-1}(c+d x)}}{2 d^2}-\frac{c (c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{d^2}+\frac{\cosh \left (2 \sinh ^{-1}(c+d x)\right ) \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{4 d^2} \]
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Rubi [A] time = 0.964722, antiderivative size = 326, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 10, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.625, Rules used = {5865, 5805, 6741, 6742, 5325, 5324, 5299, 2205, 2204, 5298} \[ -\frac{3 \sqrt{\pi } b^{3/2} c e^{a/b} \text{Erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{8 d^2}-\frac{3 \sqrt{\frac{\pi }{2}} b^{3/2} e^{\frac{2 a}{b}} \text{Erf}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{64 d^2}-\frac{3 \sqrt{\pi } b^{3/2} c e^{-\frac{a}{b}} \text{Erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{8 d^2}+\frac{3 \sqrt{\frac{\pi }{2}} b^{3/2} e^{-\frac{2 a}{b}} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{64 d^2}-\frac{3 b \sinh \left (2 \sinh ^{-1}(c+d x)\right ) \sqrt{a+b \sinh ^{-1}(c+d x)}}{16 d^2}+\frac{3 b c \sqrt{(c+d x)^2+1} \sqrt{a+b \sinh ^{-1}(c+d x)}}{2 d^2}-\frac{c (c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{d^2}+\frac{\cosh \left (2 \sinh ^{-1}(c+d x)\right ) \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{4 d^2} \]
Antiderivative was successfully verified.
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Rule 5865
Rule 5805
Rule 6741
Rule 6742
Rule 5325
Rule 5324
Rule 5299
Rule 2205
Rule 2204
Rule 5298
Rubi steps
\begin{align*} \int x \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2} \, dx &=\frac{\operatorname{Subst}\left (\int \left (-\frac{c}{d}+\frac{x}{d}\right ) \left (a+b \sinh ^{-1}(x)\right )^{3/2} \, dx,x,c+d x\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int (a+b x)^{3/2} \cosh (x) \left (-\frac{c}{d}+\frac{\sinh (x)}{d}\right ) \, dx,x,\sinh ^{-1}(c+d x)\right )}{d}\\ &=-\frac{2 \operatorname{Subst}\left (\int x^4 \cosh \left (\frac{a-x^2}{b}\right ) \left (c+\sinh \left (\frac{a-x^2}{b}\right )\right ) \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{b d^2}\\ &=-\frac{2 \operatorname{Subst}\left (\int x^4 \cosh \left (\frac{a}{b}-\frac{x^2}{b}\right ) \left (c+\sinh \left (\frac{a-x^2}{b}\right )\right ) \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{b d^2}\\ &=-\frac{2 \operatorname{Subst}\left (\int \left (c x^4 \cosh \left (\frac{a}{b}-\frac{x^2}{b}\right )+\frac{1}{2} x^4 \sinh \left (\frac{2 a}{b}-\frac{2 x^2}{b}\right )\right ) \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{b d^2}\\ &=-\frac{\operatorname{Subst}\left (\int x^4 \sinh \left (\frac{2 a}{b}-\frac{2 x^2}{b}\right ) \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{b d^2}-\frac{(2 c) \operatorname{Subst}\left (\int x^4 \cosh \left (\frac{a}{b}-\frac{x^2}{b}\right ) \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{b d^2}\\ &=-\frac{c (c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{d^2}+\frac{\left (a+b \sinh ^{-1}(c+d x)\right )^{3/2} \cosh \left (2 \sinh ^{-1}(c+d x)\right )}{4 d^2}-\frac{3 \operatorname{Subst}\left (\int x^2 \cosh \left (\frac{2 a}{b}-\frac{2 x^2}{b}\right ) \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{4 d^2}-\frac{(3 c) \operatorname{Subst}\left (\int x^2 \sinh \left (\frac{a}{b}-\frac{x^2}{b}\right ) \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{d^2}\\ &=\frac{3 b c \sqrt{1+(c+d x)^2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{2 d^2}-\frac{c (c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{d^2}+\frac{\left (a+b \sinh ^{-1}(c+d x)\right )^{3/2} \cosh \left (2 \sinh ^{-1}(c+d x)\right )}{4 d^2}-\frac{3 b \sqrt{a+b \sinh ^{-1}(c+d x)} \sinh \left (2 \sinh ^{-1}(c+d x)\right )}{16 d^2}-\frac{(3 b) \operatorname{Subst}\left (\int \sinh \left (\frac{2 a}{b}-\frac{2 x^2}{b}\right ) \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{16 d^2}-\frac{(3 b c) \operatorname{Subst}\left (\int \cosh \left (\frac{a}{b}-\frac{x^2}{b}\right ) \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{2 d^2}\\ &=\frac{3 b c \sqrt{1+(c+d x)^2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{2 d^2}-\frac{c (c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{d^2}+\frac{\left (a+b \sinh ^{-1}(c+d x)\right )^{3/2} \cosh \left (2 \sinh ^{-1}(c+d x)\right )}{4 d^2}-\frac{3 b \sqrt{a+b \sinh ^{-1}(c+d x)} \sinh \left (2 \sinh ^{-1}(c+d x)\right )}{16 d^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+d x)}\right )}{32 d^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+d x)}\right )}{32 d^2}-\frac{(3 b c) \operatorname{Subst}\left (\int e^{\frac{a}{b}-\frac{x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{4 d^2}-\frac{(3 b c) \operatorname{Subst}\left (\int e^{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{4 d^2}\\ &=\frac{3 b c \sqrt{1+(c+d x)^2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{2 d^2}-\frac{c (c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{d^2}+\frac{\left (a+b \sinh ^{-1}(c+d x)\right )^{3/2} \cosh \left (2 \sinh ^{-1}(c+d x)\right )}{4 d^2}-\frac{3 b^{3/2} c e^{a/b} \sqrt{\pi } \text{erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{8 d^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+d x)}}{\sqrt{b}}\right )}{64 d^2}-\frac{3 b^{3/2} c e^{-\frac{a}{b}} \sqrt{\pi } \text{erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{8 d^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+d x)}}{\sqrt{b}}\right )}{64 d^2}-\frac{3 b \sqrt{a+b \sinh ^{-1}(c+d x)} \sinh \left (2 \sinh ^{-1}(c+d x)\right )}{16 d^2}\\ \end{align*}
Mathematica [A] time = 5.19937, size = 582, normalized size = 1.79 \[ \frac{-64 a c e^{-\frac{a}{b}} \sqrt{a+b \sinh ^{-1}(c+d x)} \left (\frac{\text{Gamma}\left (\frac{3}{2},-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right )}{\sqrt{-\frac{a+b \sinh ^{-1}(c+d x)}{b}}}-\frac{e^{\frac{2 a}{b}} \text{Gamma}\left (\frac{3}{2},\frac{a}{b}+\sinh ^{-1}(c+d x)\right )}{\sqrt{\frac{a}{b}+\sinh ^{-1}(c+d x)}}\right )-16 \sqrt{b} c \left (\sqrt{\pi } (3 b-2 a) \left (\sinh \left (\frac{a}{b}\right )+\cosh \left (\frac{a}{b}\right )\right ) \text{Erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )+\sqrt{\pi } (2 a+3 b) \left (\cosh \left (\frac{a}{b}\right )-\sinh \left (\frac{a}{b}\right )\right ) \text{Erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )+4 \sqrt{b} \left (2 (c+d x) \sinh ^{-1}(c+d x)-3 \sqrt{(c+d x)^2+1}\right ) \sqrt{a+b \sinh ^{-1}(c+d x)}\right )+4 a \left (-\sqrt{2 \pi } \sqrt{b} \left (\sinh \left (\frac{2 a}{b}\right )+\cosh \left (\frac{2 a}{b}\right )\right ) \text{Erf}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )+\sqrt{2 \pi } \sqrt{b} \left (\sinh \left (\frac{2 a}{b}\right )-\cosh \left (\frac{2 a}{b}\right )\right ) \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )+8 \cosh \left (2 \sinh ^{-1}(c+d x)\right ) \sqrt{a+b \sinh ^{-1}(c+d x)}\right )+\sqrt{b} \left (\sqrt{2 \pi } (4 a-3 b) \left (\sinh \left (\frac{2 a}{b}\right )+\cosh \left (\frac{2 a}{b}\right )\right ) \text{Erf}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )+\sqrt{2 \pi } (4 a+3 b) \left (\cosh \left (\frac{2 a}{b}\right )-\sinh \left (\frac{2 a}{b}\right )\right ) \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )+8 \sqrt{b} \left (4 \sinh ^{-1}(c+d x) \cosh \left (2 \sinh ^{-1}(c+d x)\right )-3 \sinh \left (2 \sinh ^{-1}(c+d x)\right )\right ) \sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{128 d^2} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.159, size = 0, normalized size = 0. \begin{align*} \int x \left ( a+b{\it Arcsinh} \left ( dx+c \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 (d x + c\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 + d 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 (d x + c\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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