Optimal. Leaf size=449 \[ \frac{3 \sqrt{\pi } c \sqrt{a^2 c x^2+c} \text{Erf}\left (2 \sqrt{\sinh ^{-1}(a x)}\right )}{2048 a \sqrt{a^2 x^2+1}}+\frac{3 \sqrt{\frac{\pi }{2}} c \sqrt{a^2 c x^2+c} \text{Erf}\left (\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right )}{64 a \sqrt{a^2 x^2+1}}+\frac{3 \sqrt{\pi } c \sqrt{a^2 c x^2+c} \text{Erfi}\left (2 \sqrt{\sinh ^{-1}(a x)}\right )}{2048 a \sqrt{a^2 x^2+1}}+\frac{3 \sqrt{\frac{\pi }{2}} c \sqrt{a^2 c x^2+c} \text{Erfi}\left (\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right )}{64 a \sqrt{a^2 x^2+1}}+\frac{3 c \sqrt{a^2 c x^2+c} \sinh ^{-1}(a x)^{5/2}}{20 a \sqrt{a^2 x^2+1}}+\frac{1}{4} x \left (a^2 c x^2+c\right )^{3/2} \sinh ^{-1}(a x)^{3/2}+\frac{3}{8} c x \sqrt{a^2 c x^2+c} \sinh ^{-1}(a x)^{3/2}-\frac{3 c \left (a^2 x^2+1\right )^{3/2} \sqrt{a^2 c x^2+c} \sqrt{\sinh ^{-1}(a x)}}{32 a}-\frac{9 a c x^2 \sqrt{a^2 c x^2+c} \sqrt{\sinh ^{-1}(a x)}}{32 \sqrt{a^2 x^2+1}}-\frac{27 c \sqrt{a^2 c x^2+c} \sqrt{\sinh ^{-1}(a x)}}{256 a \sqrt{a^2 x^2+1}} \]
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Rubi [A] time = 0.564976, antiderivative size = 449, normalized size of antiderivative = 1., number of steps used = 26, number of rules used = 12, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.522, Rules used = {5684, 5682, 5675, 5663, 5779, 3312, 3307, 2180, 2204, 2205, 5717, 5699} \[ \frac{3 \sqrt{\pi } c \sqrt{a^2 c x^2+c} \text{Erf}\left (2 \sqrt{\sinh ^{-1}(a x)}\right )}{2048 a \sqrt{a^2 x^2+1}}+\frac{3 \sqrt{\frac{\pi }{2}} c \sqrt{a^2 c x^2+c} \text{Erf}\left (\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right )}{64 a \sqrt{a^2 x^2+1}}+\frac{3 \sqrt{\pi } c \sqrt{a^2 c x^2+c} \text{Erfi}\left (2 \sqrt{\sinh ^{-1}(a x)}\right )}{2048 a \sqrt{a^2 x^2+1}}+\frac{3 \sqrt{\frac{\pi }{2}} c \sqrt{a^2 c x^2+c} \text{Erfi}\left (\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right )}{64 a \sqrt{a^2 x^2+1}}+\frac{3 c \sqrt{a^2 c x^2+c} \sinh ^{-1}(a x)^{5/2}}{20 a \sqrt{a^2 x^2+1}}+\frac{1}{4} x \left (a^2 c x^2+c\right )^{3/2} \sinh ^{-1}(a x)^{3/2}+\frac{3}{8} c x \sqrt{a^2 c x^2+c} \sinh ^{-1}(a x)^{3/2}-\frac{3 c \left (a^2 x^2+1\right )^{3/2} \sqrt{a^2 c x^2+c} \sqrt{\sinh ^{-1}(a x)}}{32 a}-\frac{9 a c x^2 \sqrt{a^2 c x^2+c} \sqrt{\sinh ^{-1}(a x)}}{32 \sqrt{a^2 x^2+1}}-\frac{27 c \sqrt{a^2 c x^2+c} \sqrt{\sinh ^{-1}(a x)}}{256 a \sqrt{a^2 x^2+1}} \]
Antiderivative was successfully verified.
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Rule 5684
Rule 5682
Rule 5675
Rule 5663
Rule 5779
Rule 3312
Rule 3307
Rule 2180
Rule 2204
Rule 2205
Rule 5717
Rule 5699
Rubi steps
\begin{align*} \int \left (c+a^2 c x^2\right )^{3/2} \sinh ^{-1}(a x)^{3/2} \, dx &=\frac{1}{4} x \left (c+a^2 c x^2\right )^{3/2} \sinh ^{-1}(a x)^{3/2}+\frac{1}{4} (3 c) \int \sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2} \, dx-\frac{\left (3 a c \sqrt{c+a^2 c x^2}\right ) \int x \left (1+a^2 x^2\right ) \sqrt{\sinh ^{-1}(a x)} \, dx}{8 \sqrt{1+a^2 x^2}}\\ &=-\frac{3 c \left (1+a^2 x^2\right )^{3/2} \sqrt{c+a^2 c x^2} \sqrt{\sinh ^{-1}(a x)}}{32 a}+\frac{3}{8} c x \sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2}+\frac{1}{4} x \left (c+a^2 c x^2\right )^{3/2} \sinh ^{-1}(a x)^{3/2}+\frac{\left (3 c \sqrt{c+a^2 c x^2}\right ) \int \frac{\left (1+a^2 x^2\right )^{3/2}}{\sqrt{\sinh ^{-1}(a x)}} \, dx}{64 \sqrt{1+a^2 x^2}}+\frac{\left (3 c \sqrt{c+a^2 c x^2}\right ) \int \frac{\sinh ^{-1}(a x)^{3/2}}{\sqrt{1+a^2 x^2}} \, dx}{8 \sqrt{1+a^2 x^2}}-\frac{\left (9 a c \sqrt{c+a^2 c x^2}\right ) \int x \sqrt{\sinh ^{-1}(a x)} \, dx}{16 \sqrt{1+a^2 x^2}}\\ &=-\frac{9 a c x^2 \sqrt{c+a^2 c x^2} \sqrt{\sinh ^{-1}(a x)}}{32 \sqrt{1+a^2 x^2}}-\frac{3 c \left (1+a^2 x^2\right )^{3/2} \sqrt{c+a^2 c x^2} \sqrt{\sinh ^{-1}(a x)}}{32 a}+\frac{3}{8} c x \sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2}+\frac{1}{4} x \left (c+a^2 c x^2\right )^{3/2} \sinh ^{-1}(a x)^{3/2}+\frac{3 c \sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{5/2}}{20 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 ^4(x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{64 a \sqrt{1+a^2 x^2}}+\frac{\left (9 a^2 c \sqrt{c+a^2 c x^2}\right ) \int \frac{x^2}{\sqrt{1+a^2 x^2} \sqrt{\sinh ^{-1}(a x)}} \, dx}{64 \sqrt{1+a^2 x^2}}\\ &=-\frac{9 a c x^2 \sqrt{c+a^2 c x^2} \sqrt{\sinh ^{-1}(a x)}}{32 \sqrt{1+a^2 x^2}}-\frac{3 c \left (1+a^2 x^2\right )^{3/2} \sqrt{c+a^2 c x^2} \sqrt{\sinh ^{-1}(a x)}}{32 a}+\frac{3}{8} c x \sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2}+\frac{1}{4} x \left (c+a^2 c x^2\right )^{3/2} \sinh ^{-1}(a x)^{3/2}+\frac{3 c \sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{5/2}}{20 a \sqrt{1+a^2 x^2}}+\frac{\left (3 c \sqrt{c+a^2 c x^2}\right ) \operatorname{Subst}\left (\int \left (\frac{3}{8 \sqrt{x}}+\frac{\cosh (2 x)}{2 \sqrt{x}}+\frac{\cosh (4 x)}{8 \sqrt{x}}\right ) \, dx,x,\sinh ^{-1}(a x)\right )}{64 a \sqrt{1+a^2 x^2}}+\frac{\left (9 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 )}{64 a \sqrt{1+a^2 x^2}}\\ &=\frac{9 c \sqrt{c+a^2 c x^2} \sqrt{\sinh ^{-1}(a x)}}{256 a \sqrt{1+a^2 x^2}}-\frac{9 a c x^2 \sqrt{c+a^2 c x^2} \sqrt{\sinh ^{-1}(a x)}}{32 \sqrt{1+a^2 x^2}}-\frac{3 c \left (1+a^2 x^2\right )^{3/2} \sqrt{c+a^2 c x^2} \sqrt{\sinh ^{-1}(a x)}}{32 a}+\frac{3}{8} c x \sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2}+\frac{1}{4} x \left (c+a^2 c x^2\right )^{3/2} \sinh ^{-1}(a x)^{3/2}+\frac{3 c \sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{5/2}}{20 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 (4 x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{512 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 (2 x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{128 a \sqrt{1+a^2 x^2}}-\frac{\left (9 c \sqrt{c+a^2 c x^2}\right ) \operatorname{Subst}\left (\int \left (\frac{1}{2 \sqrt{x}}-\frac{\cosh (2 x)}{2 \sqrt{x}}\right ) \, dx,x,\sinh ^{-1}(a x)\right )}{64 a \sqrt{1+a^2 x^2}}\\ &=-\frac{27 c \sqrt{c+a^2 c x^2} \sqrt{\sinh ^{-1}(a x)}}{256 a \sqrt{1+a^2 x^2}}-\frac{9 a c x^2 \sqrt{c+a^2 c x^2} \sqrt{\sinh ^{-1}(a x)}}{32 \sqrt{1+a^2 x^2}}-\frac{3 c \left (1+a^2 x^2\right )^{3/2} \sqrt{c+a^2 c x^2} \sqrt{\sinh ^{-1}(a x)}}{32 a}+\frac{3}{8} c x \sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2}+\frac{1}{4} x \left (c+a^2 c x^2\right )^{3/2} \sinh ^{-1}(a x)^{3/2}+\frac{3 c \sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{5/2}}{20 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^{-4 x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{1024 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^{4 x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{1024 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 )}{256 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 )}{256 a \sqrt{1+a^2 x^2}}+\frac{\left (9 c \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 )}{128 a \sqrt{1+a^2 x^2}}\\ &=-\frac{27 c \sqrt{c+a^2 c x^2} \sqrt{\sinh ^{-1}(a x)}}{256 a \sqrt{1+a^2 x^2}}-\frac{9 a c x^2 \sqrt{c+a^2 c x^2} \sqrt{\sinh ^{-1}(a x)}}{32 \sqrt{1+a^2 x^2}}-\frac{3 c \left (1+a^2 x^2\right )^{3/2} \sqrt{c+a^2 c x^2} \sqrt{\sinh ^{-1}(a x)}}{32 a}+\frac{3}{8} c x \sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2}+\frac{1}{4} x \left (c+a^2 c x^2\right )^{3/2} \sinh ^{-1}(a x)^{3/2}+\frac{3 c \sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{5/2}}{20 a \sqrt{1+a^2 x^2}}+\frac{\left (3 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 )}{512 a \sqrt{1+a^2 x^2}}+\frac{\left (3 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 )}{512 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 )}{128 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 )}{128 a \sqrt{1+a^2 x^2}}+\frac{\left (9 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 )}{256 a \sqrt{1+a^2 x^2}}+\frac{\left (9 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 )}{256 a \sqrt{1+a^2 x^2}}\\ &=-\frac{27 c \sqrt{c+a^2 c x^2} \sqrt{\sinh ^{-1}(a x)}}{256 a \sqrt{1+a^2 x^2}}-\frac{9 a c x^2 \sqrt{c+a^2 c x^2} \sqrt{\sinh ^{-1}(a x)}}{32 \sqrt{1+a^2 x^2}}-\frac{3 c \left (1+a^2 x^2\right )^{3/2} \sqrt{c+a^2 c x^2} \sqrt{\sinh ^{-1}(a x)}}{32 a}+\frac{3}{8} c x \sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2}+\frac{1}{4} x \left (c+a^2 c x^2\right )^{3/2} \sinh ^{-1}(a x)^{3/2}+\frac{3 c \sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{5/2}}{20 a \sqrt{1+a^2 x^2}}+\frac{3 c \sqrt{\pi } \sqrt{c+a^2 c x^2} \text{erf}\left (2 \sqrt{\sinh ^{-1}(a x)}\right )}{2048 a \sqrt{1+a^2 x^2}}+\frac{3 c \sqrt{\frac{\pi }{2}} \sqrt{c+a^2 c x^2} \text{erf}\left (\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right )}{256 a \sqrt{1+a^2 x^2}}+\frac{3 c \sqrt{\pi } \sqrt{c+a^2 c x^2} \text{erfi}\left (2 \sqrt{\sinh ^{-1}(a x)}\right )}{2048 a \sqrt{1+a^2 x^2}}+\frac{3 c \sqrt{\frac{\pi }{2}} \sqrt{c+a^2 c x^2} \text{erfi}\left (\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right )}{256 a \sqrt{1+a^2 x^2}}+\frac{\left (9 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 )}{128 a \sqrt{1+a^2 x^2}}+\frac{\left (9 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 )}{128 a \sqrt{1+a^2 x^2}}\\ &=-\frac{27 c \sqrt{c+a^2 c x^2} \sqrt{\sinh ^{-1}(a x)}}{256 a \sqrt{1+a^2 x^2}}-\frac{9 a c x^2 \sqrt{c+a^2 c x^2} \sqrt{\sinh ^{-1}(a x)}}{32 \sqrt{1+a^2 x^2}}-\frac{3 c \left (1+a^2 x^2\right )^{3/2} \sqrt{c+a^2 c x^2} \sqrt{\sinh ^{-1}(a x)}}{32 a}+\frac{3}{8} c x \sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2}+\frac{1}{4} x \left (c+a^2 c x^2\right )^{3/2} \sinh ^{-1}(a x)^{3/2}+\frac{3 c \sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{5/2}}{20 a \sqrt{1+a^2 x^2}}+\frac{3 c \sqrt{\pi } \sqrt{c+a^2 c x^2} \text{erf}\left (2 \sqrt{\sinh ^{-1}(a x)}\right )}{2048 a \sqrt{1+a^2 x^2}}+\frac{3 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{3 c \sqrt{\pi } \sqrt{c+a^2 c x^2} \text{erfi}\left (2 \sqrt{\sinh ^{-1}(a x)}\right )}{2048 a \sqrt{1+a^2 x^2}}+\frac{3 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}}\\ \end{align*}
Mathematica [A] time = 0.319373, size = 186, normalized size = 0.41 \[ \frac{c \sqrt{a^2 c x^2+c} \left (-5 \sqrt{\sinh ^{-1}(a x)} \text{Gamma}\left (\frac{5}{2},4 \sinh ^{-1}(a x)\right )+5 \sqrt{-\sinh ^{-1}(a x)} \text{Gamma}\left (\frac{5}{2},-4 \sinh ^{-1}(a x)\right )+60 \sqrt{2 \pi } \sqrt{\sinh ^{-1}(a x)} \text{Erf}\left (\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right )+60 \sqrt{2 \pi } \sqrt{\sinh ^{-1}(a x)} \text{Erfi}\left (\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right )+384 \sinh ^{-1}(a x)^3+640 \sinh \left (2 \sinh ^{-1}(a x)\right ) \sinh ^{-1}(a x)^2-480 \sinh ^{-1}(a x) \cosh \left (2 \sinh ^{-1}(a x)\right )\right )}{2560 a \sqrt{a^2 x^2+1} \sqrt{\sinh ^{-1}(a x)}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.155, size = 0, normalized size = 0. \begin{align*} \int \left ({a}^{2}c{x}^{2}+c \right ) ^{{\frac{3}{2}}} \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{{\frac{3}{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{\left (a^{2} c x^{2} + c\right )}^{\frac{3}{2}} \operatorname{arsinh}\left (a x\right )^{\frac{3}{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(-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}} \operatorname{arsinh}\left (a x\right )^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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