Optimal. Leaf size=496 \[ \frac{\sqrt{\pi } \sqrt{b} c^2 e^{a/b} \text{Erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{4 d^3}-\frac{\sqrt{\pi } \sqrt{b} c^2 e^{-\frac{a}{b}} \text{Erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{4 d^3}+\frac{c^2 (c+d x) \sqrt{a+b \sinh ^{-1}(c+d x)}}{d^3}-\frac{\sqrt{\pi } \sqrt{b} e^{a/b} \text{Erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{16 d^3}+\frac{\sqrt{\frac{\pi }{2}} \sqrt{b} c e^{\frac{2 a}{b}} \text{Erf}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{8 d^3}+\frac{\sqrt{\frac{\pi }{3}} \sqrt{b} e^{\frac{3 a}{b}} \text{Erf}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{48 d^3}+\frac{\sqrt{\pi } \sqrt{b} e^{-\frac{a}{b}} \text{Erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{16 d^3}+\frac{\sqrt{\frac{\pi }{2}} \sqrt{b} c e^{-\frac{2 a}{b}} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{8 d^3}-\frac{\sqrt{\frac{\pi }{3}} \sqrt{b} e^{-\frac{3 a}{b}} \text{Erfi}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{48 d^3}+\frac{(c+d x)^3 \sqrt{a+b \sinh ^{-1}(c+d x)}}{3 d^3}-\frac{c \cosh \left (2 \sinh ^{-1}(c+d x)\right ) \sqrt{a+b \sinh ^{-1}(c+d x)}}{2 d^3} \]
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Rubi [A] time = 1.8469, antiderivative size = 496, normalized size of antiderivative = 1., number of steps used = 23, number of rules used = 12, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.667, Rules used = {5865, 5805, 6741, 6742, 5325, 5298, 2205, 2204, 5324, 5299, 5372, 5300} \[ \frac{\sqrt{\pi } \sqrt{b} c^2 e^{a/b} \text{Erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{4 d^3}-\frac{\sqrt{\pi } \sqrt{b} c^2 e^{-\frac{a}{b}} \text{Erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{4 d^3}+\frac{c^2 (c+d x) \sqrt{a+b \sinh ^{-1}(c+d x)}}{d^3}-\frac{\sqrt{\pi } \sqrt{b} e^{a/b} \text{Erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{16 d^3}+\frac{\sqrt{\frac{\pi }{2}} \sqrt{b} c e^{\frac{2 a}{b}} \text{Erf}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{8 d^3}+\frac{\sqrt{\frac{\pi }{3}} \sqrt{b} e^{\frac{3 a}{b}} \text{Erf}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{48 d^3}+\frac{\sqrt{\pi } \sqrt{b} e^{-\frac{a}{b}} \text{Erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{16 d^3}+\frac{\sqrt{\frac{\pi }{2}} \sqrt{b} c e^{-\frac{2 a}{b}} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{8 d^3}-\frac{\sqrt{\frac{\pi }{3}} \sqrt{b} e^{-\frac{3 a}{b}} \text{Erfi}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{48 d^3}+\frac{(c+d x)^3 \sqrt{a+b \sinh ^{-1}(c+d x)}}{3 d^3}-\frac{c \cosh \left (2 \sinh ^{-1}(c+d x)\right ) \sqrt{a+b \sinh ^{-1}(c+d x)}}{2 d^3} \]
Antiderivative was successfully verified.
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Rule 5865
Rule 5805
Rule 6741
Rule 6742
Rule 5325
Rule 5298
Rule 2205
Rule 2204
Rule 5324
Rule 5299
Rule 5372
Rule 5300
Rubi steps
\begin{align*} \int x^2 \sqrt{a+b \sinh ^{-1}(c+d x)} \, dx &=\frac{\operatorname{Subst}\left (\int \left (-\frac{c}{d}+\frac{x}{d}\right )^2 \sqrt{a+b \sinh ^{-1}(x)} \, dx,x,c+d x\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \sqrt{a+b x} \cosh (x) \left (-\frac{c}{d}+\frac{\sinh (x)}{d}\right )^2 \, dx,x,\sinh ^{-1}(c+d x)\right )}{d}\\ &=\frac{2 \operatorname{Subst}\left (\int x^2 \cosh \left (\frac{a-x^2}{b}\right ) \left (c+\sinh \left (\frac{a-x^2}{b}\right )\right )^2 \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{b d^3}\\ &=\frac{2 \operatorname{Subst}\left (\int x^2 \cosh \left (\frac{a}{b}-\frac{x^2}{b}\right ) \left (c+\sinh \left (\frac{a-x^2}{b}\right )\right )^2 \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{b d^3}\\ &=\frac{2 \operatorname{Subst}\left (\int \left (c^2 x^2 \cosh \left (\frac{a}{b}-\frac{x^2}{b}\right )+c x^2 \sinh \left (\frac{2 a}{b}-\frac{2 x^2}{b}\right )+x^2 \cosh \left (\frac{a}{b}-\frac{x^2}{b}\right ) \sinh ^2\left (\frac{a}{b}-\frac{x^2}{b}\right )\right ) \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{b d^3}\\ &=\frac{2 \operatorname{Subst}\left (\int x^2 \cosh \left (\frac{a}{b}-\frac{x^2}{b}\right ) \sinh ^2\left (\frac{a}{b}-\frac{x^2}{b}\right ) \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{b d^3}+\frac{(2 c) \operatorname{Subst}\left (\int x^2 \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^3}+\frac{\left (2 c^2\right ) \operatorname{Subst}\left (\int x^2 \cosh \left (\frac{a}{b}-\frac{x^2}{b}\right ) \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{b d^3}\\ &=\frac{c^2 (c+d x) \sqrt{a+b \sinh ^{-1}(c+d x)}}{d^3}+\frac{(c+d x)^3 \sqrt{a+b \sinh ^{-1}(c+d x)}}{3 d^3}-\frac{c \sqrt{a+b \sinh ^{-1}(c+d x)} \cosh \left (2 \sinh ^{-1}(c+d x)\right )}{2 d^3}+\frac{\operatorname{Subst}\left (\int \sinh ^3\left (\frac{a}{b}-\frac{x^2}{b}\right ) \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{3 d^3}+\frac{c \operatorname{Subst}\left (\int \cosh \left (\frac{2 a}{b}-\frac{2 x^2}{b}\right ) \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{2 d^3}+\frac{c^2 \operatorname{Subst}\left (\int \sinh \left (\frac{a}{b}-\frac{x^2}{b}\right ) \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{d^3}\\ &=\frac{c^2 (c+d x) \sqrt{a+b \sinh ^{-1}(c+d x)}}{d^3}+\frac{(c+d x)^3 \sqrt{a+b \sinh ^{-1}(c+d x)}}{3 d^3}-\frac{c \sqrt{a+b \sinh ^{-1}(c+d x)} \cosh \left (2 \sinh ^{-1}(c+d x)\right )}{2 d^3}+\frac{\operatorname{Subst}\left (\int \left (\frac{1}{4} \sinh \left (\frac{3 a}{b}-\frac{3 x^2}{b}\right )-\frac{3}{4} \sinh \left (\frac{a}{b}-\frac{x^2}{b}\right )\right ) \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{3 d^3}+\frac{c \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 )}{4 d^3}+\frac{c \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 )}{4 d^3}+\frac{c^2 \operatorname{Subst}\left (\int e^{\frac{a}{b}-\frac{x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{2 d^3}-\frac{c^2 \operatorname{Subst}\left (\int e^{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{2 d^3}\\ &=\frac{c^2 (c+d x) \sqrt{a+b \sinh ^{-1}(c+d x)}}{d^3}+\frac{(c+d x)^3 \sqrt{a+b \sinh ^{-1}(c+d x)}}{3 d^3}-\frac{c \sqrt{a+b \sinh ^{-1}(c+d x)} \cosh \left (2 \sinh ^{-1}(c+d x)\right )}{2 d^3}+\frac{\sqrt{b} c^2 e^{a/b} \sqrt{\pi } \text{erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{4 d^3}+\frac{\sqrt{b} c 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 )}{8 d^3}-\frac{\sqrt{b} c^2 e^{-\frac{a}{b}} \sqrt{\pi } \text{erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{4 d^3}+\frac{\sqrt{b} c 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 )}{8 d^3}+\frac{\operatorname{Subst}\left (\int \sinh \left (\frac{3 a}{b}-\frac{3 x^2}{b}\right ) \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{12 d^3}-\frac{\operatorname{Subst}\left (\int \sinh \left (\frac{a}{b}-\frac{x^2}{b}\right ) \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{4 d^3}\\ &=\frac{c^2 (c+d x) \sqrt{a+b \sinh ^{-1}(c+d x)}}{d^3}+\frac{(c+d x)^3 \sqrt{a+b \sinh ^{-1}(c+d x)}}{3 d^3}-\frac{c \sqrt{a+b \sinh ^{-1}(c+d x)} \cosh \left (2 \sinh ^{-1}(c+d x)\right )}{2 d^3}+\frac{\sqrt{b} c^2 e^{a/b} \sqrt{\pi } \text{erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{4 d^3}+\frac{\sqrt{b} c 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 )}{8 d^3}-\frac{\sqrt{b} c^2 e^{-\frac{a}{b}} \sqrt{\pi } \text{erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{4 d^3}+\frac{\sqrt{b} c 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 )}{8 d^3}+\frac{\operatorname{Subst}\left (\int e^{\frac{3 a}{b}-\frac{3 x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{24 d^3}-\frac{\operatorname{Subst}\left (\int e^{-\frac{3 a}{b}+\frac{3 x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{24 d^3}-\frac{\operatorname{Subst}\left (\int e^{\frac{a}{b}-\frac{x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{8 d^3}+\frac{\operatorname{Subst}\left (\int e^{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{8 d^3}\\ &=\frac{c^2 (c+d x) \sqrt{a+b \sinh ^{-1}(c+d x)}}{d^3}+\frac{(c+d x)^3 \sqrt{a+b \sinh ^{-1}(c+d x)}}{3 d^3}-\frac{c \sqrt{a+b \sinh ^{-1}(c+d x)} \cosh \left (2 \sinh ^{-1}(c+d x)\right )}{2 d^3}-\frac{\sqrt{b} e^{a/b} \sqrt{\pi } \text{erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{16 d^3}+\frac{\sqrt{b} c^2 e^{a/b} \sqrt{\pi } \text{erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{4 d^3}+\frac{\sqrt{b} c 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 )}{8 d^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+d x)}}{\sqrt{b}}\right )}{48 d^3}+\frac{\sqrt{b} e^{-\frac{a}{b}} \sqrt{\pi } \text{erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{16 d^3}-\frac{\sqrt{b} c^2 e^{-\frac{a}{b}} \sqrt{\pi } \text{erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{4 d^3}+\frac{\sqrt{b} c 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 )}{8 d^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+d x)}}{\sqrt{b}}\right )}{48 d^3}\\ \end{align*}
Mathematica [A] time = 1.80739, size = 656, normalized size = 1.32 \[ \frac{9 \sqrt{\pi } \sqrt{b} \left (4 c^2-1\right ) \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 )+36 \sqrt{\pi } \sqrt{b} c^2 \sinh \left (\frac{a}{b}\right ) \text{Erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )-36 \sqrt{\pi } \sqrt{b} c^2 \cosh \left (\frac{a}{b}\right ) \text{Erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )+144 c^2 (c+d x) \sqrt{a+b \sinh ^{-1}(c+d x)}+\sqrt{3 \pi } \sqrt{b} \sinh \left (\frac{3 a}{b}\right ) \text{Erf}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )+9 \sqrt{2 \pi } \sqrt{b} c \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{3 \pi } \sqrt{b} \cosh \left (\frac{3 a}{b}\right ) \text{Erf}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )-9 \sqrt{2 \pi } \sqrt{b} c \sinh \left (\frac{2 a}{b}\right ) \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )-9 \sqrt{\pi } \sqrt{b} \sinh \left (\frac{a}{b}\right ) \text{Erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )+\sqrt{3 \pi } \sqrt{b} \sinh \left (\frac{3 a}{b}\right ) \text{Erfi}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )+9 \sqrt{2 \pi } \sqrt{b} c \cosh \left (\frac{2 a}{b}\right ) \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )+9 \sqrt{\pi } \sqrt{b} \cosh \left (\frac{a}{b}\right ) \text{Erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )-\sqrt{3 \pi } \sqrt{b} \cosh \left (\frac{3 a}{b}\right ) \text{Erfi}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )+12 \sinh \left (3 \sinh ^{-1}(c+d x)\right ) \sqrt{a+b \sinh ^{-1}(c+d x)}-36 (c+d x) \sqrt{a+b \sinh ^{-1}(c+d x)}-72 c \cosh \left (2 \sinh ^{-1}(c+d x)\right ) \sqrt{a+b \sinh ^{-1}(c+d x)}}{144 d^3} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.28, size = 0, normalized size = 0. \begin{align*} \int{x}^{2}\sqrt{a+b{\it Arcsinh} \left ( dx+c \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 (d x + c\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 + d 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 (d x + c\right ) + a} x^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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