Optimal. Leaf size=271 \[ -\frac{\sqrt{\pi } e^{a/b} \text{Erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{6 b^{5/2} c^3}+\frac{\sqrt{3 \pi } e^{\frac{3 a}{b}} \text{Erf}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{2 b^{5/2} c^3}-\frac{\sqrt{\pi } e^{-\frac{a}{b}} \text{Erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{6 b^{5/2} c^3}+\frac{\sqrt{3 \pi } e^{-\frac{3 a}{b}} \text{Erfi}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{2 b^{5/2} c^3}-\frac{8 x}{3 b^2 c^2 \sqrt{a+b \sinh ^{-1}(c x)}}-\frac{4 x^3}{b^2 \sqrt{a+b \sinh ^{-1}(c x)}}-\frac{2 x^2 \sqrt{c^2 x^2+1}}{3 b c \left (a+b \sinh ^{-1}(c x)\right )^{3/2}} \]
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Rubi [A] time = 0.899481, antiderivative size = 271, normalized size of antiderivative = 1., number of steps used = 22, number of rules used = 9, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.562, Rules used = {5667, 5774, 5669, 5448, 3307, 2180, 2204, 2205, 5657} \[ -\frac{\sqrt{\pi } e^{a/b} \text{Erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{6 b^{5/2} c^3}+\frac{\sqrt{3 \pi } e^{\frac{3 a}{b}} \text{Erf}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{2 b^{5/2} c^3}-\frac{\sqrt{\pi } e^{-\frac{a}{b}} \text{Erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{6 b^{5/2} c^3}+\frac{\sqrt{3 \pi } e^{-\frac{3 a}{b}} \text{Erfi}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{2 b^{5/2} c^3}-\frac{8 x}{3 b^2 c^2 \sqrt{a+b \sinh ^{-1}(c x)}}-\frac{4 x^3}{b^2 \sqrt{a+b \sinh ^{-1}(c x)}}-\frac{2 x^2 \sqrt{c^2 x^2+1}}{3 b c \left (a+b \sinh ^{-1}(c x)\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 5667
Rule 5774
Rule 5669
Rule 5448
Rule 3307
Rule 2180
Rule 2204
Rule 2205
Rule 5657
Rubi steps
\begin{align*} \int \frac{x^2}{\left (a+b \sinh ^{-1}(c x)\right )^{5/2}} \, dx &=-\frac{2 x^2 \sqrt{1+c^2 x^2}}{3 b c \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}+\frac{4 \int \frac{x}{\sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^{3/2}} \, dx}{3 b c}+\frac{(2 c) \int \frac{x^3}{\sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^{3/2}} \, dx}{b}\\ &=-\frac{2 x^2 \sqrt{1+c^2 x^2}}{3 b c \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}-\frac{8 x}{3 b^2 c^2 \sqrt{a+b \sinh ^{-1}(c x)}}-\frac{4 x^3}{b^2 \sqrt{a+b \sinh ^{-1}(c x)}}+\frac{12 \int \frac{x^2}{\sqrt{a+b \sinh ^{-1}(c x)}} \, dx}{b^2}+\frac{8 \int \frac{1}{\sqrt{a+b \sinh ^{-1}(c x)}} \, dx}{3 b^2 c^2}\\ &=-\frac{2 x^2 \sqrt{1+c^2 x^2}}{3 b c \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}-\frac{8 x}{3 b^2 c^2 \sqrt{a+b \sinh ^{-1}(c x)}}-\frac{4 x^3}{b^2 \sqrt{a+b \sinh ^{-1}(c x)}}+\frac{8 \operatorname{Subst}\left (\int \frac{\cosh \left (\frac{a}{b}-\frac{x}{b}\right )}{\sqrt{x}} \, dx,x,a+b \sinh ^{-1}(c x)\right )}{3 b^3 c^3}+\frac{12 \operatorname{Subst}\left (\int \frac{\cosh (x) \sinh ^2(x)}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{b^2 c^3}\\ &=-\frac{2 x^2 \sqrt{1+c^2 x^2}}{3 b c \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}-\frac{8 x}{3 b^2 c^2 \sqrt{a+b \sinh ^{-1}(c x)}}-\frac{4 x^3}{b^2 \sqrt{a+b \sinh ^{-1}(c x)}}+\frac{4 \operatorname{Subst}\left (\int \frac{e^{-i \left (\frac{i a}{b}-\frac{i x}{b}\right )}}{\sqrt{x}} \, dx,x,a+b \sinh ^{-1}(c x)\right )}{3 b^3 c^3}+\frac{4 \operatorname{Subst}\left (\int \frac{e^{i \left (\frac{i a}{b}-\frac{i x}{b}\right )}}{\sqrt{x}} \, dx,x,a+b \sinh ^{-1}(c x)\right )}{3 b^3 c^3}+\frac{12 \operatorname{Subst}\left (\int \left (-\frac{\cosh (x)}{4 \sqrt{a+b x}}+\frac{\cosh (3 x)}{4 \sqrt{a+b x}}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{b^2 c^3}\\ &=-\frac{2 x^2 \sqrt{1+c^2 x^2}}{3 b c \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}-\frac{8 x}{3 b^2 c^2 \sqrt{a+b \sinh ^{-1}(c x)}}-\frac{4 x^3}{b^2 \sqrt{a+b \sinh ^{-1}(c x)}}+\frac{8 \operatorname{Subst}\left (\int e^{\frac{a}{b}-\frac{x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c x)}\right )}{3 b^3 c^3}+\frac{8 \operatorname{Subst}\left (\int e^{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c x)}\right )}{3 b^3 c^3}-\frac{3 \operatorname{Subst}\left (\int \frac{\cosh (x)}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{b^2 c^3}+\frac{3 \operatorname{Subst}\left (\int \frac{\cosh (3 x)}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{b^2 c^3}\\ &=-\frac{2 x^2 \sqrt{1+c^2 x^2}}{3 b c \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}-\frac{8 x}{3 b^2 c^2 \sqrt{a+b \sinh ^{-1}(c x)}}-\frac{4 x^3}{b^2 \sqrt{a+b \sinh ^{-1}(c x)}}+\frac{4 e^{a/b} \sqrt{\pi } \text{erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{3 b^{5/2} c^3}+\frac{4 e^{-\frac{a}{b}} \sqrt{\pi } \text{erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{3 b^{5/2} c^3}+\frac{3 \operatorname{Subst}\left (\int \frac{e^{-3 x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{2 b^2 c^3}-\frac{3 \operatorname{Subst}\left (\int \frac{e^{-x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{2 b^2 c^3}-\frac{3 \operatorname{Subst}\left (\int \frac{e^x}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{2 b^2 c^3}+\frac{3 \operatorname{Subst}\left (\int \frac{e^{3 x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{2 b^2 c^3}\\ &=-\frac{2 x^2 \sqrt{1+c^2 x^2}}{3 b c \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}-\frac{8 x}{3 b^2 c^2 \sqrt{a+b \sinh ^{-1}(c x)}}-\frac{4 x^3}{b^2 \sqrt{a+b \sinh ^{-1}(c x)}}+\frac{4 e^{a/b} \sqrt{\pi } \text{erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{3 b^{5/2} c^3}+\frac{4 e^{-\frac{a}{b}} \sqrt{\pi } \text{erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{3 b^{5/2} c^3}+\frac{3 \operatorname{Subst}\left (\int e^{\frac{3 a}{b}-\frac{3 x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c x)}\right )}{b^3 c^3}-\frac{3 \operatorname{Subst}\left (\int e^{\frac{a}{b}-\frac{x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c x)}\right )}{b^3 c^3}-\frac{3 \operatorname{Subst}\left (\int e^{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c x)}\right )}{b^3 c^3}+\frac{3 \operatorname{Subst}\left (\int e^{-\frac{3 a}{b}+\frac{3 x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c x)}\right )}{b^3 c^3}\\ &=-\frac{2 x^2 \sqrt{1+c^2 x^2}}{3 b c \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}-\frac{8 x}{3 b^2 c^2 \sqrt{a+b \sinh ^{-1}(c x)}}-\frac{4 x^3}{b^2 \sqrt{a+b \sinh ^{-1}(c x)}}-\frac{e^{a/b} \sqrt{\pi } \text{erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{6 b^{5/2} c^3}+\frac{e^{\frac{3 a}{b}} \sqrt{3 \pi } \text{erf}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{2 b^{5/2} c^3}-\frac{e^{-\frac{a}{b}} \sqrt{\pi } \text{erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{6 b^{5/2} c^3}+\frac{e^{-\frac{3 a}{b}} \sqrt{3 \pi } \text{erfi}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{2 b^{5/2} c^3}\\ \end{align*}
Mathematica [A] time = 1.5089, size = 340, normalized size = 1.25 \[ \frac{e^{-3 \left (\frac{a}{b}+\sinh ^{-1}(c x)\right )} \left (-6 \sqrt{3} b e^{3 \sinh ^{-1}(c x)} \left (-\frac{a+b \sinh ^{-1}(c x)}{b}\right )^{3/2} \text{Gamma}\left (\frac{1}{2},-\frac{3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )+2 b e^{\frac{2 a}{b}+3 \sinh ^{-1}(c x)} \left (-\frac{a+b \sinh ^{-1}(c x)}{b}\right )^{3/2} \text{Gamma}\left (\frac{1}{2},-\frac{a+b \sinh ^{-1}(c x)}{b}\right )+2 e^{\frac{4 a}{b}+3 \sinh ^{-1}(c x)} \sqrt{\frac{a}{b}+\sinh ^{-1}(c x)} \left (a+b \sinh ^{-1}(c x)\right ) \text{Gamma}\left (\frac{1}{2},\frac{a}{b}+\sinh ^{-1}(c x)\right )-e^{\frac{3 a}{b}} \left (6 \sqrt{3} e^{3 \left (\frac{a}{b}+\sinh ^{-1}(c x)\right )} \sqrt{\frac{a}{b}+\sinh ^{-1}(c x)} \left (a+b \sinh ^{-1}(c x)\right ) \text{Gamma}\left (\frac{1}{2},\frac{3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )+\left (e^{2 \sinh ^{-1}(c x)}-1\right ) \left (a \left (4 e^{2 \sinh ^{-1}(c x)}+6 e^{4 \sinh ^{-1}(c x)}+6\right )+b \left (e^{4 \sinh ^{-1}(c x)}-1\right )+2 b \left (2 e^{2 \sinh ^{-1}(c x)}+3 e^{4 \sinh ^{-1}(c x)}+3\right ) \sinh ^{-1}(c x)\right )\right )\right )}{12 b^2 c^3 \left (a+b \sinh ^{-1}(c x)\right )^{3/2}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.111, size = 0, normalized size = 0. \begin{align*} \int{{x}^{2} \left ( a+b{\it Arcsinh} \left ( cx \right ) \right ) ^{-{\frac{5}{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 \frac{x^{2}}{{\left (b \operatorname{arsinh}\left (c x\right ) + a\right )}^{\frac{5}{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 \frac{x^{2}}{\left (a + b \operatorname{asinh}{\left (c x \right )}\right )^{\frac{5}{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 \frac{x^{2}}{{\left (b \operatorname{arsinh}\left (c x\right ) + a\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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