Optimal. Leaf size=247 \[ -\frac{15 \sqrt{\pi } \text{Erf}\left (2 \sqrt{\sinh ^{-1}(a x)}\right )}{16384 a^4}+\frac{15 \sqrt{\frac{\pi }{2}} \text{Erf}\left (\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right )}{512 a^4}-\frac{15 \sqrt{\pi } \text{Erfi}\left (2 \sqrt{\sinh ^{-1}(a x)}\right )}{16384 a^4}+\frac{15 \sqrt{\frac{\pi }{2}} \text{Erfi}\left (\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right )}{512 a^4}-\frac{5 x^3 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^{3/2}}{32 a}-\frac{45 x^2 \sqrt{\sinh ^{-1}(a x)}}{256 a^2}+\frac{15 x \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^{3/2}}{64 a^3}-\frac{3 \sinh ^{-1}(a x)^{5/2}}{32 a^4}-\frac{225 \sqrt{\sinh ^{-1}(a x)}}{2048 a^4}+\frac{1}{4} x^4 \sinh ^{-1}(a x)^{5/2}+\frac{15}{256} x^4 \sqrt{\sinh ^{-1}(a x)} \]
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Rubi [A] time = 0.71077, antiderivative size = 247, normalized size of antiderivative = 1., number of steps used = 27, number of rules used = 9, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.75, Rules used = {5663, 5758, 5675, 5779, 3312, 3307, 2180, 2204, 2205} \[ -\frac{15 \sqrt{\pi } \text{Erf}\left (2 \sqrt{\sinh ^{-1}(a x)}\right )}{16384 a^4}+\frac{15 \sqrt{\frac{\pi }{2}} \text{Erf}\left (\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right )}{512 a^4}-\frac{15 \sqrt{\pi } \text{Erfi}\left (2 \sqrt{\sinh ^{-1}(a x)}\right )}{16384 a^4}+\frac{15 \sqrt{\frac{\pi }{2}} \text{Erfi}\left (\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right )}{512 a^4}-\frac{5 x^3 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^{3/2}}{32 a}-\frac{45 x^2 \sqrt{\sinh ^{-1}(a x)}}{256 a^2}+\frac{15 x \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^{3/2}}{64 a^3}-\frac{3 \sinh ^{-1}(a x)^{5/2}}{32 a^4}-\frac{225 \sqrt{\sinh ^{-1}(a x)}}{2048 a^4}+\frac{1}{4} x^4 \sinh ^{-1}(a x)^{5/2}+\frac{15}{256} x^4 \sqrt{\sinh ^{-1}(a x)} \]
Antiderivative was successfully verified.
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Rule 5663
Rule 5758
Rule 5675
Rule 5779
Rule 3312
Rule 3307
Rule 2180
Rule 2204
Rule 2205
Rubi steps
\begin{align*} \int x^3 \sinh ^{-1}(a x)^{5/2} \, dx &=\frac{1}{4} x^4 \sinh ^{-1}(a x)^{5/2}-\frac{1}{8} (5 a) \int \frac{x^4 \sinh ^{-1}(a x)^{3/2}}{\sqrt{1+a^2 x^2}} \, dx\\ &=-\frac{5 x^3 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^{3/2}}{32 a}+\frac{1}{4} x^4 \sinh ^{-1}(a x)^{5/2}+\frac{15}{64} \int x^3 \sqrt{\sinh ^{-1}(a x)} \, dx+\frac{15 \int \frac{x^2 \sinh ^{-1}(a x)^{3/2}}{\sqrt{1+a^2 x^2}} \, dx}{32 a}\\ &=\frac{15}{256} x^4 \sqrt{\sinh ^{-1}(a x)}+\frac{15 x \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^{3/2}}{64 a^3}-\frac{5 x^3 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^{3/2}}{32 a}+\frac{1}{4} x^4 \sinh ^{-1}(a x)^{5/2}-\frac{15 \int \frac{\sinh ^{-1}(a x)^{3/2}}{\sqrt{1+a^2 x^2}} \, dx}{64 a^3}-\frac{45 \int x \sqrt{\sinh ^{-1}(a x)} \, dx}{128 a^2}-\frac{1}{512} (15 a) \int \frac{x^4}{\sqrt{1+a^2 x^2} \sqrt{\sinh ^{-1}(a x)}} \, dx\\ &=-\frac{45 x^2 \sqrt{\sinh ^{-1}(a x)}}{256 a^2}+\frac{15}{256} x^4 \sqrt{\sinh ^{-1}(a x)}+\frac{15 x \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^{3/2}}{64 a^3}-\frac{5 x^3 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^{3/2}}{32 a}-\frac{3 \sinh ^{-1}(a x)^{5/2}}{32 a^4}+\frac{1}{4} x^4 \sinh ^{-1}(a x)^{5/2}-\frac{15 \operatorname{Subst}\left (\int \frac{\sinh ^4(x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{512 a^4}+\frac{45 \int \frac{x^2}{\sqrt{1+a^2 x^2} \sqrt{\sinh ^{-1}(a x)}} \, dx}{512 a}\\ &=-\frac{45 x^2 \sqrt{\sinh ^{-1}(a x)}}{256 a^2}+\frac{15}{256} x^4 \sqrt{\sinh ^{-1}(a x)}+\frac{15 x \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^{3/2}}{64 a^3}-\frac{5 x^3 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^{3/2}}{32 a}-\frac{3 \sinh ^{-1}(a x)^{5/2}}{32 a^4}+\frac{1}{4} x^4 \sinh ^{-1}(a x)^{5/2}-\frac{15 \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 )}{512 a^4}+\frac{45 \operatorname{Subst}\left (\int \frac{\sinh ^2(x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{512 a^4}\\ &=-\frac{45 \sqrt{\sinh ^{-1}(a x)}}{2048 a^4}-\frac{45 x^2 \sqrt{\sinh ^{-1}(a x)}}{256 a^2}+\frac{15}{256} x^4 \sqrt{\sinh ^{-1}(a x)}+\frac{15 x \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^{3/2}}{64 a^3}-\frac{5 x^3 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^{3/2}}{32 a}-\frac{3 \sinh ^{-1}(a x)^{5/2}}{32 a^4}+\frac{1}{4} x^4 \sinh ^{-1}(a x)^{5/2}-\frac{15 \operatorname{Subst}\left (\int \frac{\cosh (4 x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{4096 a^4}+\frac{15 \operatorname{Subst}\left (\int \frac{\cosh (2 x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{1024 a^4}-\frac{45 \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 )}{512 a^4}\\ &=-\frac{225 \sqrt{\sinh ^{-1}(a x)}}{2048 a^4}-\frac{45 x^2 \sqrt{\sinh ^{-1}(a x)}}{256 a^2}+\frac{15}{256} x^4 \sqrt{\sinh ^{-1}(a x)}+\frac{15 x \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^{3/2}}{64 a^3}-\frac{5 x^3 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^{3/2}}{32 a}-\frac{3 \sinh ^{-1}(a x)^{5/2}}{32 a^4}+\frac{1}{4} x^4 \sinh ^{-1}(a x)^{5/2}-\frac{15 \operatorname{Subst}\left (\int \frac{e^{-4 x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{8192 a^4}-\frac{15 \operatorname{Subst}\left (\int \frac{e^{4 x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{8192 a^4}+\frac{15 \operatorname{Subst}\left (\int \frac{e^{-2 x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{2048 a^4}+\frac{15 \operatorname{Subst}\left (\int \frac{e^{2 x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{2048 a^4}+\frac{45 \operatorname{Subst}\left (\int \frac{\cosh (2 x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{1024 a^4}\\ &=-\frac{225 \sqrt{\sinh ^{-1}(a x)}}{2048 a^4}-\frac{45 x^2 \sqrt{\sinh ^{-1}(a x)}}{256 a^2}+\frac{15}{256} x^4 \sqrt{\sinh ^{-1}(a x)}+\frac{15 x \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^{3/2}}{64 a^3}-\frac{5 x^3 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^{3/2}}{32 a}-\frac{3 \sinh ^{-1}(a x)^{5/2}}{32 a^4}+\frac{1}{4} x^4 \sinh ^{-1}(a x)^{5/2}-\frac{15 \operatorname{Subst}\left (\int e^{-4 x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{4096 a^4}-\frac{15 \operatorname{Subst}\left (\int e^{4 x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{4096 a^4}+\frac{15 \operatorname{Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{1024 a^4}+\frac{15 \operatorname{Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{1024 a^4}+\frac{45 \operatorname{Subst}\left (\int \frac{e^{-2 x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{2048 a^4}+\frac{45 \operatorname{Subst}\left (\int \frac{e^{2 x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{2048 a^4}\\ &=-\frac{225 \sqrt{\sinh ^{-1}(a x)}}{2048 a^4}-\frac{45 x^2 \sqrt{\sinh ^{-1}(a x)}}{256 a^2}+\frac{15}{256} x^4 \sqrt{\sinh ^{-1}(a x)}+\frac{15 x \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^{3/2}}{64 a^3}-\frac{5 x^3 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^{3/2}}{32 a}-\frac{3 \sinh ^{-1}(a x)^{5/2}}{32 a^4}+\frac{1}{4} x^4 \sinh ^{-1}(a x)^{5/2}-\frac{15 \sqrt{\pi } \text{erf}\left (2 \sqrt{\sinh ^{-1}(a x)}\right )}{16384 a^4}+\frac{15 \sqrt{\frac{\pi }{2}} \text{erf}\left (\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right )}{2048 a^4}-\frac{15 \sqrt{\pi } \text{erfi}\left (2 \sqrt{\sinh ^{-1}(a x)}\right )}{16384 a^4}+\frac{15 \sqrt{\frac{\pi }{2}} \text{erfi}\left (\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right )}{2048 a^4}+\frac{45 \operatorname{Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{1024 a^4}+\frac{45 \operatorname{Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{1024 a^4}\\ &=-\frac{225 \sqrt{\sinh ^{-1}(a x)}}{2048 a^4}-\frac{45 x^2 \sqrt{\sinh ^{-1}(a x)}}{256 a^2}+\frac{15}{256} x^4 \sqrt{\sinh ^{-1}(a x)}+\frac{15 x \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^{3/2}}{64 a^3}-\frac{5 x^3 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^{3/2}}{32 a}-\frac{3 \sinh ^{-1}(a x)^{5/2}}{32 a^4}+\frac{1}{4} x^4 \sinh ^{-1}(a x)^{5/2}-\frac{15 \sqrt{\pi } \text{erf}\left (2 \sqrt{\sinh ^{-1}(a x)}\right )}{16384 a^4}+\frac{15 \sqrt{\frac{\pi }{2}} \text{erf}\left (\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right )}{512 a^4}-\frac{15 \sqrt{\pi } \text{erfi}\left (2 \sqrt{\sinh ^{-1}(a x)}\right )}{16384 a^4}+\frac{15 \sqrt{\frac{\pi }{2}} \text{erfi}\left (\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right )}{512 a^4}\\ \end{align*}
Mathematica [A] time = 0.03659, size = 101, normalized size = 0.41 \[ \frac{\sqrt{\sinh ^{-1}(a x)} \text{Gamma}\left (\frac{7}{2},-4 \sinh ^{-1}(a x)\right )-16 \sqrt{2} \sqrt{\sinh ^{-1}(a x)} \text{Gamma}\left (\frac{7}{2},-2 \sinh ^{-1}(a x)\right )+\sqrt{-\sinh ^{-1}(a x)} \left (\text{Gamma}\left (\frac{7}{2},4 \sinh ^{-1}(a x)\right )-16 \sqrt{2} \text{Gamma}\left (\frac{7}{2},2 \sinh ^{-1}(a x)\right )\right )}{2048 a^4 \sqrt{-\sinh ^{-1}(a x)}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.099, size = 0, normalized size = 0. \begin{align*} \int{x}^{3} \left ({\it Arcsinh} \left ( ax \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 x^{3} \operatorname{arsinh}\left (a x\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(-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 x^{3} \operatorname{arsinh}\left (a x\right )^{\frac{5}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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