Optimal. Leaf size=319 \[ \frac{\sqrt{\pi } c \sqrt{a^2 c x^2+c} \text{Erf}\left (2 \sqrt{\sinh ^{-1}(a x)}\right )}{256 a \sqrt{a^2 x^2+1}}+\frac{\sqrt{\frac{\pi }{2}} c \sqrt{a^2 c x^2+c} \text{Erf}\left (\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right )}{16 a \sqrt{a^2 x^2+1}}-\frac{\sqrt{\pi } c \sqrt{a^2 c x^2+c} \text{Erfi}\left (2 \sqrt{\sinh ^{-1}(a x)}\right )}{256 a \sqrt{a^2 x^2+1}}-\frac{\sqrt{\frac{\pi }{2}} c \sqrt{a^2 c x^2+c} \text{Erfi}\left (\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right )}{16 a \sqrt{a^2 x^2+1}}+\frac{1}{4} x \left (a^2 c x^2+c\right )^{3/2} \sqrt{\sinh ^{-1}(a x)}+\frac{c \sqrt{a^2 c x^2+c} \sinh ^{-1}(a x)^{3/2}}{4 a \sqrt{a^2 x^2+1}}+\frac{3}{8} c x \sqrt{a^2 c x^2+c} \sqrt{\sinh ^{-1}(a x)} \]
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Rubi [A] time = 0.408109, antiderivative size = 319, normalized size of antiderivative = 1., number of steps used = 24, number of rules used = 11, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.478, Rules used = {5684, 5682, 5675, 5669, 5448, 12, 3308, 2180, 2204, 2205, 5779} \[ \frac{\sqrt{\pi } c \sqrt{a^2 c x^2+c} \text{Erf}\left (2 \sqrt{\sinh ^{-1}(a x)}\right )}{256 a \sqrt{a^2 x^2+1}}+\frac{\sqrt{\frac{\pi }{2}} c \sqrt{a^2 c x^2+c} \text{Erf}\left (\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right )}{16 a \sqrt{a^2 x^2+1}}-\frac{\sqrt{\pi } c \sqrt{a^2 c x^2+c} \text{Erfi}\left (2 \sqrt{\sinh ^{-1}(a x)}\right )}{256 a \sqrt{a^2 x^2+1}}-\frac{\sqrt{\frac{\pi }{2}} c \sqrt{a^2 c x^2+c} \text{Erfi}\left (\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right )}{16 a \sqrt{a^2 x^2+1}}+\frac{1}{4} x \left (a^2 c x^2+c\right )^{3/2} \sqrt{\sinh ^{-1}(a x)}+\frac{c \sqrt{a^2 c x^2+c} \sinh ^{-1}(a x)^{3/2}}{4 a \sqrt{a^2 x^2+1}}+\frac{3}{8} c x \sqrt{a^2 c x^2+c} \sqrt{\sinh ^{-1}(a x)} \]
Antiderivative was successfully verified.
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Rule 5684
Rule 5682
Rule 5675
Rule 5669
Rule 5448
Rule 12
Rule 3308
Rule 2180
Rule 2204
Rule 2205
Rule 5779
Rubi steps
\begin{align*} \int \left (c+a^2 c x^2\right )^{3/2} \sqrt{\sinh ^{-1}(a x)} \, dx &=\frac{1}{4} x \left (c+a^2 c x^2\right )^{3/2} \sqrt{\sinh ^{-1}(a x)}+\frac{1}{4} (3 c) \int \sqrt{c+a^2 c x^2} \sqrt{\sinh ^{-1}(a x)} \, dx-\frac{\left (a c \sqrt{c+a^2 c x^2}\right ) \int \frac{x \left (1+a^2 x^2\right )}{\sqrt{\sinh ^{-1}(a x)}} \, dx}{8 \sqrt{1+a^2 x^2}}\\ &=\frac{3}{8} c x \sqrt{c+a^2 c x^2} \sqrt{\sinh ^{-1}(a x)}+\frac{1}{4} x \left (c+a^2 c x^2\right )^{3/2} \sqrt{\sinh ^{-1}(a x)}+\frac{\left (3 c \sqrt{c+a^2 c x^2}\right ) \int \frac{\sqrt{\sinh ^{-1}(a x)}}{\sqrt{1+a^2 x^2}} \, dx}{8 \sqrt{1+a^2 x^2}}-\frac{\left (c \sqrt{c+a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{\cosh ^3(x) \sinh (x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{8 a \sqrt{1+a^2 x^2}}-\frac{\left (3 a c \sqrt{c+a^2 c x^2}\right ) \int \frac{x}{\sqrt{\sinh ^{-1}(a x)}} \, dx}{16 \sqrt{1+a^2 x^2}}\\ &=\frac{3}{8} c x \sqrt{c+a^2 c x^2} \sqrt{\sinh ^{-1}(a x)}+\frac{1}{4} x \left (c+a^2 c x^2\right )^{3/2} \sqrt{\sinh ^{-1}(a x)}+\frac{c \sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2}}{4 a \sqrt{1+a^2 x^2}}-\frac{\left (c \sqrt{c+a^2 c x^2}\right ) \operatorname{Subst}\left (\int \left (\frac{\sinh (2 x)}{4 \sqrt{x}}+\frac{\sinh (4 x)}{8 \sqrt{x}}\right ) \, dx,x,\sinh ^{-1}(a x)\right )}{8 a \sqrt{1+a^2 x^2}}-\frac{\left (3 c \sqrt{c+a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{\cosh (x) \sinh (x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{16 a \sqrt{1+a^2 x^2}}\\ &=\frac{3}{8} c x \sqrt{c+a^2 c x^2} \sqrt{\sinh ^{-1}(a x)}+\frac{1}{4} x \left (c+a^2 c x^2\right )^{3/2} \sqrt{\sinh ^{-1}(a x)}+\frac{c \sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2}}{4 a \sqrt{1+a^2 x^2}}-\frac{\left (c \sqrt{c+a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{\sinh (4 x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{64 a \sqrt{1+a^2 x^2}}-\frac{\left (c \sqrt{c+a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{\sinh (2 x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{32 a \sqrt{1+a^2 x^2}}-\frac{\left (3 c \sqrt{c+a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{\sinh (2 x)}{2 \sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{16 a \sqrt{1+a^2 x^2}}\\ &=\frac{3}{8} c x \sqrt{c+a^2 c x^2} \sqrt{\sinh ^{-1}(a x)}+\frac{1}{4} x \left (c+a^2 c x^2\right )^{3/2} \sqrt{\sinh ^{-1}(a x)}+\frac{c \sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2}}{4 a \sqrt{1+a^2 x^2}}+\frac{\left (c \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 )}{128 a \sqrt{1+a^2 x^2}}-\frac{\left (c \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 )}{128 a \sqrt{1+a^2 x^2}}+\frac{\left (c \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 (c \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 (3 c \sqrt{c+a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{\sinh (2 x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{32 a \sqrt{1+a^2 x^2}}\\ &=\frac{3}{8} c x \sqrt{c+a^2 c x^2} \sqrt{\sinh ^{-1}(a x)}+\frac{1}{4} x \left (c+a^2 c x^2\right )^{3/2} \sqrt{\sinh ^{-1}(a x)}+\frac{c \sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2}}{4 a \sqrt{1+a^2 x^2}}+\frac{\left (c \sqrt{c+a^2 c x^2}\right ) \operatorname{Subst}\left (\int e^{-4 x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{64 a \sqrt{1+a^2 x^2}}-\frac{\left (c \sqrt{c+a^2 c x^2}\right ) \operatorname{Subst}\left (\int e^{4 x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{64 a \sqrt{1+a^2 x^2}}+\frac{\left (c \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 (c \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 (3 c \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 (3 c \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{3}{8} c x \sqrt{c+a^2 c x^2} \sqrt{\sinh ^{-1}(a x)}+\frac{1}{4} x \left (c+a^2 c x^2\right )^{3/2} \sqrt{\sinh ^{-1}(a x)}+\frac{c \sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2}}{4 a \sqrt{1+a^2 x^2}}+\frac{c \sqrt{\pi } \sqrt{c+a^2 c x^2} \text{erf}\left (2 \sqrt{\sinh ^{-1}(a x)}\right )}{256 a \sqrt{1+a^2 x^2}}+\frac{c \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 \sqrt{\pi } \sqrt{c+a^2 c x^2} \text{erfi}\left (2 \sqrt{\sinh ^{-1}(a x)}\right )}{256 a \sqrt{1+a^2 x^2}}-\frac{c \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{\left (3 c \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 (3 c \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{3}{8} c x \sqrt{c+a^2 c x^2} \sqrt{\sinh ^{-1}(a x)}+\frac{1}{4} x \left (c+a^2 c x^2\right )^{3/2} \sqrt{\sinh ^{-1}(a x)}+\frac{c \sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2}}{4 a \sqrt{1+a^2 x^2}}+\frac{c \sqrt{\pi } \sqrt{c+a^2 c x^2} \text{erf}\left (2 \sqrt{\sinh ^{-1}(a x)}\right )}{256 a \sqrt{1+a^2 x^2}}+\frac{c \sqrt{\frac{\pi }{2}} \sqrt{c+a^2 c x^2} \text{erf}\left (\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right )}{16 a \sqrt{1+a^2 x^2}}-\frac{c \sqrt{\pi } \sqrt{c+a^2 c x^2} \text{erfi}\left (2 \sqrt{\sinh ^{-1}(a x)}\right )}{256 a \sqrt{1+a^2 x^2}}-\frac{c \sqrt{\frac{\pi }{2}} \sqrt{c+a^2 c x^2} \text{erfi}\left (\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right )}{16 a \sqrt{1+a^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.189342, size = 142, normalized size = 0.45 \[ \frac{c \sqrt{a^2 c x^2+c} \left (-\sqrt{-\sinh ^{-1}(a x)} \text{Gamma}\left (\frac{3}{2},-4 \sinh ^{-1}(a x)\right )-8 \sqrt{2} \sqrt{-\sinh ^{-1}(a x)} \text{Gamma}\left (\frac{3}{2},-2 \sinh ^{-1}(a x)\right )+\sqrt{\sinh ^{-1}(a x)} \left (-8 \sqrt{2} \text{Gamma}\left (\frac{3}{2},2 \sinh ^{-1}(a x)\right )-\text{Gamma}\left (\frac{3}{2},4 \sinh ^{-1}(a x)\right )+32 \sinh ^{-1}(a x)^{3/2}\right )\right )}{128 a \sqrt{a^2 x^2+1} \sqrt{\sinh ^{-1}(a x)}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.187, size = 0, normalized size = 0. \begin{align*} \int \left ({a}^{2}c{x}^{2}+c \right ) ^{{\frac{3}{2}}}\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{\left (a^{2} c x^{2} + c\right )}^{\frac{3}{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{\left (a^{2} c x^{2} + c\right )}^{\frac{3}{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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