Optimal. Leaf size=396 \[ \frac{3 \sqrt{\pi } c^2 \sqrt{a^2 c x^2+c} \text{Erf}\left (2 \sqrt{\sinh ^{-1}(a x)}\right )}{64 a \sqrt{a^2 x^2+1}}+\frac{15 \sqrt{\frac{\pi }{2}} c^2 \sqrt{a^2 c x^2+c} \text{Erf}\left (\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right )}{64 a \sqrt{a^2 x^2+1}}+\frac{\sqrt{\frac{\pi }{6}} c^2 \sqrt{a^2 c x^2+c} \text{Erf}\left (\sqrt{6} \sqrt{\sinh ^{-1}(a x)}\right )}{64 a \sqrt{a^2 x^2+1}}+\frac{3 \sqrt{\pi } c^2 \sqrt{a^2 c x^2+c} \text{Erfi}\left (2 \sqrt{\sinh ^{-1}(a x)}\right )}{64 a \sqrt{a^2 x^2+1}}+\frac{15 \sqrt{\frac{\pi }{2}} c^2 \sqrt{a^2 c x^2+c} \text{Erfi}\left (\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right )}{64 a \sqrt{a^2 x^2+1}}+\frac{\sqrt{\frac{\pi }{6}} c^2 \sqrt{a^2 c x^2+c} \text{Erfi}\left (\sqrt{6} \sqrt{\sinh ^{-1}(a x)}\right )}{64 a \sqrt{a^2 x^2+1}}+\frac{5 c^2 \sqrt{a^2 c x^2+c} \sqrt{\sinh ^{-1}(a x)}}{8 a \sqrt{a^2 x^2+1}} \]
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Rubi [A] time = 0.30802, antiderivative size = 396, normalized size of antiderivative = 1., number of steps used = 19, number of rules used = 7, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.304, Rules used = {5702, 5699, 3312, 3307, 2180, 2204, 2205} \[ \frac{3 \sqrt{\pi } c^2 \sqrt{a^2 c x^2+c} \text{Erf}\left (2 \sqrt{\sinh ^{-1}(a x)}\right )}{64 a \sqrt{a^2 x^2+1}}+\frac{15 \sqrt{\frac{\pi }{2}} c^2 \sqrt{a^2 c x^2+c} \text{Erf}\left (\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right )}{64 a \sqrt{a^2 x^2+1}}+\frac{\sqrt{\frac{\pi }{6}} c^2 \sqrt{a^2 c x^2+c} \text{Erf}\left (\sqrt{6} \sqrt{\sinh ^{-1}(a x)}\right )}{64 a \sqrt{a^2 x^2+1}}+\frac{3 \sqrt{\pi } c^2 \sqrt{a^2 c x^2+c} \text{Erfi}\left (2 \sqrt{\sinh ^{-1}(a x)}\right )}{64 a \sqrt{a^2 x^2+1}}+\frac{15 \sqrt{\frac{\pi }{2}} c^2 \sqrt{a^2 c x^2+c} \text{Erfi}\left (\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right )}{64 a \sqrt{a^2 x^2+1}}+\frac{\sqrt{\frac{\pi }{6}} c^2 \sqrt{a^2 c x^2+c} \text{Erfi}\left (\sqrt{6} \sqrt{\sinh ^{-1}(a x)}\right )}{64 a \sqrt{a^2 x^2+1}}+\frac{5 c^2 \sqrt{a^2 c x^2+c} \sqrt{\sinh ^{-1}(a x)}}{8 a \sqrt{a^2 x^2+1}} \]
Antiderivative was successfully verified.
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Rule 5702
Rule 5699
Rule 3312
Rule 3307
Rule 2180
Rule 2204
Rule 2205
Rubi steps
\begin{align*} \int \frac{\left (c+a^2 c x^2\right )^{5/2}}{\sqrt{\sinh ^{-1}(a x)}} \, dx &=\frac{\left (c^2 \sqrt{c+a^2 c x^2}\right ) \int \frac{\left (1+a^2 x^2\right )^{5/2}}{\sqrt{\sinh ^{-1}(a x)}} \, dx}{\sqrt{1+a^2 x^2}}\\ &=\frac{\left (c^2 \sqrt{c+a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{\cosh ^6(x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{a \sqrt{1+a^2 x^2}}\\ &=\frac{\left (c^2 \sqrt{c+a^2 c x^2}\right ) \operatorname{Subst}\left (\int \left (\frac{5}{16 \sqrt{x}}+\frac{15 \cosh (2 x)}{32 \sqrt{x}}+\frac{3 \cosh (4 x)}{16 \sqrt{x}}+\frac{\cosh (6 x)}{32 \sqrt{x}}\right ) \, dx,x,\sinh ^{-1}(a x)\right )}{a \sqrt{1+a^2 x^2}}\\ &=\frac{5 c^2 \sqrt{c+a^2 c x^2} \sqrt{\sinh ^{-1}(a x)}}{8 a \sqrt{1+a^2 x^2}}+\frac{\left (c^2 \sqrt{c+a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{\cosh (6 x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{32 a \sqrt{1+a^2 x^2}}+\frac{\left (3 c^2 \sqrt{c+a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{\cosh (4 x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{16 a \sqrt{1+a^2 x^2}}+\frac{\left (15 c^2 \sqrt{c+a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{\cosh (2 x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{32 a \sqrt{1+a^2 x^2}}\\ &=\frac{5 c^2 \sqrt{c+a^2 c x^2} \sqrt{\sinh ^{-1}(a x)}}{8 a \sqrt{1+a^2 x^2}}+\frac{\left (c^2 \sqrt{c+a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{e^{-6 x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{64 a \sqrt{1+a^2 x^2}}+\frac{\left (c^2 \sqrt{c+a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{e^{6 x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{64 a \sqrt{1+a^2 x^2}}+\frac{\left (3 c^2 \sqrt{c+a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{e^{-4 x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{32 a \sqrt{1+a^2 x^2}}+\frac{\left (3 c^2 \sqrt{c+a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{e^{4 x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{32 a \sqrt{1+a^2 x^2}}+\frac{\left (15 c^2 \sqrt{c+a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{e^{-2 x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{64 a \sqrt{1+a^2 x^2}}+\frac{\left (15 c^2 \sqrt{c+a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{e^{2 x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{64 a \sqrt{1+a^2 x^2}}\\ &=\frac{5 c^2 \sqrt{c+a^2 c x^2} \sqrt{\sinh ^{-1}(a x)}}{8 a \sqrt{1+a^2 x^2}}+\frac{\left (c^2 \sqrt{c+a^2 c x^2}\right ) \operatorname{Subst}\left (\int e^{-6 x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{32 a \sqrt{1+a^2 x^2}}+\frac{\left (c^2 \sqrt{c+a^2 c x^2}\right ) \operatorname{Subst}\left (\int e^{6 x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{32 a \sqrt{1+a^2 x^2}}+\frac{\left (3 c^2 \sqrt{c+a^2 c x^2}\right ) \operatorname{Subst}\left (\int e^{-4 x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{16 a \sqrt{1+a^2 x^2}}+\frac{\left (3 c^2 \sqrt{c+a^2 c x^2}\right ) \operatorname{Subst}\left (\int e^{4 x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{16 a \sqrt{1+a^2 x^2}}+\frac{\left (15 c^2 \sqrt{c+a^2 c x^2}\right ) \operatorname{Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{32 a \sqrt{1+a^2 x^2}}+\frac{\left (15 c^2 \sqrt{c+a^2 c x^2}\right ) \operatorname{Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{32 a \sqrt{1+a^2 x^2}}\\ &=\frac{5 c^2 \sqrt{c+a^2 c x^2} \sqrt{\sinh ^{-1}(a x)}}{8 a \sqrt{1+a^2 x^2}}+\frac{3 c^2 \sqrt{\pi } \sqrt{c+a^2 c x^2} \text{erf}\left (2 \sqrt{\sinh ^{-1}(a x)}\right )}{64 a \sqrt{1+a^2 x^2}}+\frac{15 c^2 \sqrt{\frac{\pi }{2}} \sqrt{c+a^2 c x^2} \text{erf}\left (\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right )}{64 a \sqrt{1+a^2 x^2}}+\frac{c^2 \sqrt{\frac{\pi }{6}} \sqrt{c+a^2 c x^2} \text{erf}\left (\sqrt{6} \sqrt{\sinh ^{-1}(a x)}\right )}{64 a \sqrt{1+a^2 x^2}}+\frac{3 c^2 \sqrt{\pi } \sqrt{c+a^2 c x^2} \text{erfi}\left (2 \sqrt{\sinh ^{-1}(a x)}\right )}{64 a \sqrt{1+a^2 x^2}}+\frac{15 c^2 \sqrt{\frac{\pi }{2}} \sqrt{c+a^2 c x^2} \text{erfi}\left (\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right )}{64 a \sqrt{1+a^2 x^2}}+\frac{c^2 \sqrt{\frac{\pi }{6}} \sqrt{c+a^2 c x^2} \text{erfi}\left (\sqrt{6} \sqrt{\sinh ^{-1}(a x)}\right )}{64 a \sqrt{1+a^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.321456, size = 197, normalized size = 0.5 \[ \frac{c^2 \sqrt{a^2 c x^2+c} \left (-45 \sqrt{2} \sqrt{\sinh ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},2 \sinh ^{-1}(a x)\right )-18 \sqrt{\sinh ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},4 \sinh ^{-1}(a x)\right )-\sqrt{6} \sqrt{\sinh ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},6 \sinh ^{-1}(a x)\right )+\sqrt{6} \sqrt{-\sinh ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-6 \sinh ^{-1}(a x)\right )+18 \sqrt{-\sinh ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-4 \sinh ^{-1}(a x)\right )+45 \sqrt{2} \sqrt{-\sinh ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-2 \sinh ^{-1}(a x)\right )+240 \sinh ^{-1}(a x)\right )}{384 a \sqrt{a^2 x^2+1} \sqrt{\sinh ^{-1}(a x)}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.164, size = 0, normalized size = 0. \begin{align*} \int{ \left ({a}^{2}c{x}^{2}+c \right ) ^{{\frac{5}{2}}}{\frac{1}{\sqrt{{\it Arcsinh} \left ( ax \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 \frac{{\left (a^{2} c x^{2} + c\right )}^{\frac{5}{2}}}{\sqrt{\operatorname{arsinh}\left (a x\right )}}\,{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(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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{{\left (a^{2} c x^{2} + c\right )}^{\frac{5}{2}}}{\sqrt{\operatorname{arsinh}\left (a x\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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