Optimal. Leaf size=213 \[ -\frac{\sqrt{\pi } \sqrt{b} e^{a/b} \text{Erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{16 c^3}+\frac{\sqrt{\frac{\pi }{3}} \sqrt{b} e^{\frac{3 a}{b}} \text{Erf}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{48 c^3}+\frac{\sqrt{\pi } \sqrt{b} e^{-\frac{a}{b}} \text{Erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{16 c^3}-\frac{\sqrt{\frac{\pi }{3}} \sqrt{b} e^{-\frac{3 a}{b}} \text{Erfi}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{48 c^3}+\frac{1}{3} x^3 \sqrt{a+b \sinh ^{-1}(c x)} \]
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Rubi [A] time = 0.604637, antiderivative size = 213, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 7, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.438, Rules used = {5663, 5779, 3312, 3308, 2180, 2204, 2205} \[ -\frac{\sqrt{\pi } \sqrt{b} e^{a/b} \text{Erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{16 c^3}+\frac{\sqrt{\frac{\pi }{3}} \sqrt{b} e^{\frac{3 a}{b}} \text{Erf}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{48 c^3}+\frac{\sqrt{\pi } \sqrt{b} e^{-\frac{a}{b}} \text{Erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{16 c^3}-\frac{\sqrt{\frac{\pi }{3}} \sqrt{b} e^{-\frac{3 a}{b}} \text{Erfi}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{48 c^3}+\frac{1}{3} x^3 \sqrt{a+b \sinh ^{-1}(c x)} \]
Antiderivative was successfully verified.
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Rule 5663
Rule 5779
Rule 3312
Rule 3308
Rule 2180
Rule 2204
Rule 2205
Rubi steps
\begin{align*} \int x^2 \sqrt{a+b \sinh ^{-1}(c x)} \, dx &=\frac{1}{3} x^3 \sqrt{a+b \sinh ^{-1}(c x)}-\frac{1}{6} (b c) \int \frac{x^3}{\sqrt{1+c^2 x^2} \sqrt{a+b \sinh ^{-1}(c x)}} \, dx\\ &=\frac{1}{3} x^3 \sqrt{a+b \sinh ^{-1}(c x)}-\frac{b \operatorname{Subst}\left (\int \frac{\sinh ^3(x)}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{6 c^3}\\ &=\frac{1}{3} x^3 \sqrt{a+b \sinh ^{-1}(c x)}-\frac{(i b) \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 )}{6 c^3}\\ &=\frac{1}{3} x^3 \sqrt{a+b \sinh ^{-1}(c x)}-\frac{b \operatorname{Subst}\left (\int \frac{\sinh (3 x)}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{24 c^3}+\frac{b \operatorname{Subst}\left (\int \frac{\sinh (x)}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{8 c^3}\\ &=\frac{1}{3} x^3 \sqrt{a+b \sinh ^{-1}(c x)}+\frac{b \operatorname{Subst}\left (\int \frac{e^{-3 x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{48 c^3}-\frac{b \operatorname{Subst}\left (\int \frac{e^{3 x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{48 c^3}-\frac{b \operatorname{Subst}\left (\int \frac{e^{-x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{16 c^3}+\frac{b \operatorname{Subst}\left (\int \frac{e^x}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{16 c^3}\\ &=\frac{1}{3} x^3 \sqrt{a+b \sinh ^{-1}(c x)}+\frac{\operatorname{Subst}\left (\int e^{\frac{3 a}{b}-\frac{3 x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c x)}\right )}{24 c^3}-\frac{\operatorname{Subst}\left (\int e^{-\frac{3 a}{b}+\frac{3 x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c x)}\right )}{24 c^3}-\frac{\operatorname{Subst}\left (\int e^{\frac{a}{b}-\frac{x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c x)}\right )}{8 c^3}+\frac{\operatorname{Subst}\left (\int e^{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c x)}\right )}{8 c^3}\\ &=\frac{1}{3} x^3 \sqrt{a+b \sinh ^{-1}(c x)}-\frac{\sqrt{b} e^{a/b} \sqrt{\pi } \text{erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{16 c^3}+\frac{\sqrt{b} 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 )}{48 c^3}+\frac{\sqrt{b} e^{-\frac{a}{b}} \sqrt{\pi } \text{erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{16 c^3}-\frac{\sqrt{b} 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 )}{48 c^3}\\ \end{align*}
Mathematica [A] time = 0.402476, size = 215, normalized size = 1.01 \[ \frac{e^{-\frac{3 a}{b}} \sqrt{a+b \sinh ^{-1}(c x)} \left (9 e^{\frac{4 a}{b}} \sqrt{-\frac{a+b \sinh ^{-1}(c x)}{b}} \text{Gamma}\left (\frac{3}{2},\frac{a}{b}+\sinh ^{-1}(c x)\right )+\sqrt{3} \sqrt{\frac{a}{b}+\sinh ^{-1}(c x)} \text{Gamma}\left (\frac{3}{2},-\frac{3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )-9 e^{\frac{2 a}{b}} \sqrt{\frac{a}{b}+\sinh ^{-1}(c x)} \text{Gamma}\left (\frac{3}{2},-\frac{a+b \sinh ^{-1}(c x)}{b}\right )-\sqrt{3} e^{\frac{6 a}{b}} \sqrt{-\frac{a+b \sinh ^{-1}(c x)}{b}} \text{Gamma}\left (\frac{3}{2},\frac{3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )\right )}{72 c^3 \sqrt{-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{b^2}}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.134, size = 0, normalized size = 0. \begin{align*} \int{x}^{2}\sqrt{a+b{\it Arcsinh} \left ( cx \right ) }\, 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 \sqrt{b \operatorname{arsinh}\left (c x\right ) + a} 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] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2} \sqrt{a + b \operatorname{asinh}{\left (c x \right )}}\, 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 \sqrt{b \operatorname{arsinh}\left (c x\right ) + a} x^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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