Optimal. Leaf size=379 \[ \frac{15 \sqrt{\pi } \text{Erf}\left (\sqrt{\sinh ^{-1}(a x)}\right )}{128 a^5}-\frac{\sqrt{3 \pi } \text{Erf}\left (\sqrt{3} \sqrt{\sinh ^{-1}(a x)}\right )}{1280 a^5}-\frac{\sqrt{\frac{\pi }{3}} \text{Erf}\left (\sqrt{3} \sqrt{\sinh ^{-1}(a x)}\right )}{240 a^5}+\frac{3 \sqrt{\frac{\pi }{5}} \text{Erf}\left (\sqrt{5} \sqrt{\sinh ^{-1}(a x)}\right )}{6400 a^5}-\frac{15 \sqrt{\pi } \text{Erfi}\left (\sqrt{\sinh ^{-1}(a x)}\right )}{128 a^5}+\frac{\sqrt{3 \pi } \text{Erfi}\left (\sqrt{3} \sqrt{\sinh ^{-1}(a x)}\right )}{1280 a^5}+\frac{\sqrt{\frac{\pi }{3}} \text{Erfi}\left (\sqrt{3} \sqrt{\sinh ^{-1}(a x)}\right )}{240 a^5}-\frac{3 \sqrt{\frac{\pi }{5}} \text{Erfi}\left (\sqrt{5} \sqrt{\sinh ^{-1}(a x)}\right )}{6400 a^5}-\frac{x^4 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^{3/2}}{10 a}-\frac{x^3 \sqrt{\sinh ^{-1}(a x)}}{15 a^2}+\frac{2 x^2 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^{3/2}}{15 a^3}-\frac{4 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^{3/2}}{15 a^5}+\frac{2 x \sqrt{\sinh ^{-1}(a x)}}{5 a^4}+\frac{1}{5} x^5 \sinh ^{-1}(a x)^{5/2}+\frac{3}{100} x^5 \sqrt{\sinh ^{-1}(a x)} \]
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Rubi [A] time = 0.996392, antiderivative size = 379, normalized size of antiderivative = 1., number of steps used = 44, number of rules used = 10, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.833, Rules used = {5663, 5758, 5717, 5653, 5779, 3308, 2180, 2204, 2205, 3312} \[ \frac{15 \sqrt{\pi } \text{Erf}\left (\sqrt{\sinh ^{-1}(a x)}\right )}{128 a^5}-\frac{\sqrt{3 \pi } \text{Erf}\left (\sqrt{3} \sqrt{\sinh ^{-1}(a x)}\right )}{1280 a^5}-\frac{\sqrt{\frac{\pi }{3}} \text{Erf}\left (\sqrt{3} \sqrt{\sinh ^{-1}(a x)}\right )}{240 a^5}+\frac{3 \sqrt{\frac{\pi }{5}} \text{Erf}\left (\sqrt{5} \sqrt{\sinh ^{-1}(a x)}\right )}{6400 a^5}-\frac{15 \sqrt{\pi } \text{Erfi}\left (\sqrt{\sinh ^{-1}(a x)}\right )}{128 a^5}+\frac{\sqrt{3 \pi } \text{Erfi}\left (\sqrt{3} \sqrt{\sinh ^{-1}(a x)}\right )}{1280 a^5}+\frac{\sqrt{\frac{\pi }{3}} \text{Erfi}\left (\sqrt{3} \sqrt{\sinh ^{-1}(a x)}\right )}{240 a^5}-\frac{3 \sqrt{\frac{\pi }{5}} \text{Erfi}\left (\sqrt{5} \sqrt{\sinh ^{-1}(a x)}\right )}{6400 a^5}-\frac{x^4 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^{3/2}}{10 a}-\frac{x^3 \sqrt{\sinh ^{-1}(a x)}}{15 a^2}+\frac{2 x^2 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^{3/2}}{15 a^3}-\frac{4 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^{3/2}}{15 a^5}+\frac{2 x \sqrt{\sinh ^{-1}(a x)}}{5 a^4}+\frac{1}{5} x^5 \sinh ^{-1}(a x)^{5/2}+\frac{3}{100} x^5 \sqrt{\sinh ^{-1}(a x)} \]
Antiderivative was successfully verified.
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Rule 5663
Rule 5758
Rule 5717
Rule 5653
Rule 5779
Rule 3308
Rule 2180
Rule 2204
Rule 2205
Rule 3312
Rubi steps
\begin{align*} \int x^4 \sinh ^{-1}(a x)^{5/2} \, dx &=\frac{1}{5} x^5 \sinh ^{-1}(a x)^{5/2}-\frac{1}{2} a \int \frac{x^5 \sinh ^{-1}(a x)^{3/2}}{\sqrt{1+a^2 x^2}} \, dx\\ &=-\frac{x^4 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^{3/2}}{10 a}+\frac{1}{5} x^5 \sinh ^{-1}(a x)^{5/2}+\frac{3}{20} \int x^4 \sqrt{\sinh ^{-1}(a x)} \, dx+\frac{2 \int \frac{x^3 \sinh ^{-1}(a x)^{3/2}}{\sqrt{1+a^2 x^2}} \, dx}{5 a}\\ &=\frac{3}{100} x^5 \sqrt{\sinh ^{-1}(a x)}+\frac{2 x^2 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^{3/2}}{15 a^3}-\frac{x^4 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^{3/2}}{10 a}+\frac{1}{5} x^5 \sinh ^{-1}(a x)^{5/2}-\frac{4 \int \frac{x \sinh ^{-1}(a x)^{3/2}}{\sqrt{1+a^2 x^2}} \, dx}{15 a^3}-\frac{\int x^2 \sqrt{\sinh ^{-1}(a x)} \, dx}{5 a^2}-\frac{1}{200} (3 a) \int \frac{x^5}{\sqrt{1+a^2 x^2} \sqrt{\sinh ^{-1}(a x)}} \, dx\\ &=-\frac{x^3 \sqrt{\sinh ^{-1}(a x)}}{15 a^2}+\frac{3}{100} x^5 \sqrt{\sinh ^{-1}(a x)}-\frac{4 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^{3/2}}{15 a^5}+\frac{2 x^2 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^{3/2}}{15 a^3}-\frac{x^4 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^{3/2}}{10 a}+\frac{1}{5} x^5 \sinh ^{-1}(a x)^{5/2}-\frac{3 \operatorname{Subst}\left (\int \frac{\sinh ^5(x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{200 a^5}+\frac{2 \int \sqrt{\sinh ^{-1}(a x)} \, dx}{5 a^4}+\frac{\int \frac{x^3}{\sqrt{1+a^2 x^2} \sqrt{\sinh ^{-1}(a x)}} \, dx}{30 a}\\ &=\frac{2 x \sqrt{\sinh ^{-1}(a x)}}{5 a^4}-\frac{x^3 \sqrt{\sinh ^{-1}(a x)}}{15 a^2}+\frac{3}{100} x^5 \sqrt{\sinh ^{-1}(a x)}-\frac{4 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^{3/2}}{15 a^5}+\frac{2 x^2 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^{3/2}}{15 a^3}-\frac{x^4 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^{3/2}}{10 a}+\frac{1}{5} x^5 \sinh ^{-1}(a x)^{5/2}+\frac{(3 i) \operatorname{Subst}\left (\int \left (\frac{5 i \sinh (x)}{8 \sqrt{x}}-\frac{5 i \sinh (3 x)}{16 \sqrt{x}}+\frac{i \sinh (5 x)}{16 \sqrt{x}}\right ) \, dx,x,\sinh ^{-1}(a x)\right )}{200 a^5}+\frac{\operatorname{Subst}\left (\int \frac{\sinh ^3(x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{30 a^5}-\frac{\int \frac{x}{\sqrt{1+a^2 x^2} \sqrt{\sinh ^{-1}(a x)}} \, dx}{5 a^3}\\ &=\frac{2 x \sqrt{\sinh ^{-1}(a x)}}{5 a^4}-\frac{x^3 \sqrt{\sinh ^{-1}(a x)}}{15 a^2}+\frac{3}{100} x^5 \sqrt{\sinh ^{-1}(a x)}-\frac{4 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^{3/2}}{15 a^5}+\frac{2 x^2 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^{3/2}}{15 a^3}-\frac{x^4 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^{3/2}}{10 a}+\frac{1}{5} x^5 \sinh ^{-1}(a x)^{5/2}+\frac{i \operatorname{Subst}\left (\int \left (\frac{3 i \sinh (x)}{4 \sqrt{x}}-\frac{i \sinh (3 x)}{4 \sqrt{x}}\right ) \, dx,x,\sinh ^{-1}(a x)\right )}{30 a^5}-\frac{3 \operatorname{Subst}\left (\int \frac{\sinh (5 x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{3200 a^5}+\frac{3 \operatorname{Subst}\left (\int \frac{\sinh (3 x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{640 a^5}-\frac{3 \operatorname{Subst}\left (\int \frac{\sinh (x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{320 a^5}-\frac{\operatorname{Subst}\left (\int \frac{\sinh (x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{5 a^5}\\ &=\frac{2 x \sqrt{\sinh ^{-1}(a x)}}{5 a^4}-\frac{x^3 \sqrt{\sinh ^{-1}(a x)}}{15 a^2}+\frac{3}{100} x^5 \sqrt{\sinh ^{-1}(a x)}-\frac{4 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^{3/2}}{15 a^5}+\frac{2 x^2 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^{3/2}}{15 a^3}-\frac{x^4 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^{3/2}}{10 a}+\frac{1}{5} x^5 \sinh ^{-1}(a x)^{5/2}+\frac{3 \operatorname{Subst}\left (\int \frac{e^{-5 x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{6400 a^5}-\frac{3 \operatorname{Subst}\left (\int \frac{e^{5 x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{6400 a^5}-\frac{3 \operatorname{Subst}\left (\int \frac{e^{-3 x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{1280 a^5}+\frac{3 \operatorname{Subst}\left (\int \frac{e^{3 x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{1280 a^5}+\frac{3 \operatorname{Subst}\left (\int \frac{e^{-x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{640 a^5}-\frac{3 \operatorname{Subst}\left (\int \frac{e^x}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{640 a^5}+\frac{\operatorname{Subst}\left (\int \frac{\sinh (3 x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{120 a^5}-\frac{\operatorname{Subst}\left (\int \frac{\sinh (x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{40 a^5}+\frac{\operatorname{Subst}\left (\int \frac{e^{-x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{10 a^5}-\frac{\operatorname{Subst}\left (\int \frac{e^x}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{10 a^5}\\ &=\frac{2 x \sqrt{\sinh ^{-1}(a x)}}{5 a^4}-\frac{x^3 \sqrt{\sinh ^{-1}(a x)}}{15 a^2}+\frac{3}{100} x^5 \sqrt{\sinh ^{-1}(a x)}-\frac{4 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^{3/2}}{15 a^5}+\frac{2 x^2 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^{3/2}}{15 a^3}-\frac{x^4 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^{3/2}}{10 a}+\frac{1}{5} x^5 \sinh ^{-1}(a x)^{5/2}+\frac{3 \operatorname{Subst}\left (\int e^{-5 x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{3200 a^5}-\frac{3 \operatorname{Subst}\left (\int e^{5 x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{3200 a^5}-\frac{\operatorname{Subst}\left (\int \frac{e^{-3 x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{240 a^5}+\frac{\operatorname{Subst}\left (\int \frac{e^{3 x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{240 a^5}-\frac{3 \operatorname{Subst}\left (\int e^{-3 x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{640 a^5}+\frac{3 \operatorname{Subst}\left (\int e^{3 x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{640 a^5}+\frac{3 \operatorname{Subst}\left (\int e^{-x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{320 a^5}-\frac{3 \operatorname{Subst}\left (\int e^{x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{320 a^5}+\frac{\operatorname{Subst}\left (\int \frac{e^{-x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{80 a^5}-\frac{\operatorname{Subst}\left (\int \frac{e^x}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{80 a^5}+\frac{\operatorname{Subst}\left (\int e^{-x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{5 a^5}-\frac{\operatorname{Subst}\left (\int e^{x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{5 a^5}\\ &=\frac{2 x \sqrt{\sinh ^{-1}(a x)}}{5 a^4}-\frac{x^3 \sqrt{\sinh ^{-1}(a x)}}{15 a^2}+\frac{3}{100} x^5 \sqrt{\sinh ^{-1}(a x)}-\frac{4 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^{3/2}}{15 a^5}+\frac{2 x^2 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^{3/2}}{15 a^3}-\frac{x^4 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^{3/2}}{10 a}+\frac{1}{5} x^5 \sinh ^{-1}(a x)^{5/2}+\frac{67 \sqrt{\pi } \text{erf}\left (\sqrt{\sinh ^{-1}(a x)}\right )}{640 a^5}-\frac{\sqrt{3 \pi } \text{erf}\left (\sqrt{3} \sqrt{\sinh ^{-1}(a x)}\right )}{1280 a^5}+\frac{3 \sqrt{\frac{\pi }{5}} \text{erf}\left (\sqrt{5} \sqrt{\sinh ^{-1}(a x)}\right )}{6400 a^5}-\frac{67 \sqrt{\pi } \text{erfi}\left (\sqrt{\sinh ^{-1}(a x)}\right )}{640 a^5}+\frac{\sqrt{3 \pi } \text{erfi}\left (\sqrt{3} \sqrt{\sinh ^{-1}(a x)}\right )}{1280 a^5}-\frac{3 \sqrt{\frac{\pi }{5}} \text{erfi}\left (\sqrt{5} \sqrt{\sinh ^{-1}(a x)}\right )}{6400 a^5}-\frac{\operatorname{Subst}\left (\int e^{-3 x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{120 a^5}+\frac{\operatorname{Subst}\left (\int e^{3 x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{120 a^5}+\frac{\operatorname{Subst}\left (\int e^{-x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{40 a^5}-\frac{\operatorname{Subst}\left (\int e^{x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{40 a^5}\\ &=\frac{2 x \sqrt{\sinh ^{-1}(a x)}}{5 a^4}-\frac{x^3 \sqrt{\sinh ^{-1}(a x)}}{15 a^2}+\frac{3}{100} x^5 \sqrt{\sinh ^{-1}(a x)}-\frac{4 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^{3/2}}{15 a^5}+\frac{2 x^2 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^{3/2}}{15 a^3}-\frac{x^4 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^{3/2}}{10 a}+\frac{1}{5} x^5 \sinh ^{-1}(a x)^{5/2}+\frac{15 \sqrt{\pi } \text{erf}\left (\sqrt{\sinh ^{-1}(a x)}\right )}{128 a^5}-\frac{\sqrt{\frac{\pi }{3}} \text{erf}\left (\sqrt{3} \sqrt{\sinh ^{-1}(a x)}\right )}{240 a^5}-\frac{\sqrt{3 \pi } \text{erf}\left (\sqrt{3} \sqrt{\sinh ^{-1}(a x)}\right )}{1280 a^5}+\frac{3 \sqrt{\frac{\pi }{5}} \text{erf}\left (\sqrt{5} \sqrt{\sinh ^{-1}(a x)}\right )}{6400 a^5}-\frac{15 \sqrt{\pi } \text{erfi}\left (\sqrt{\sinh ^{-1}(a x)}\right )}{128 a^5}+\frac{\sqrt{\frac{\pi }{3}} \text{erfi}\left (\sqrt{3} \sqrt{\sinh ^{-1}(a x)}\right )}{240 a^5}+\frac{\sqrt{3 \pi } \text{erfi}\left (\sqrt{3} \sqrt{\sinh ^{-1}(a x)}\right )}{1280 a^5}-\frac{3 \sqrt{\frac{\pi }{5}} \text{erfi}\left (\sqrt{5} \sqrt{\sinh ^{-1}(a x)}\right )}{6400 a^5}\\ \end{align*}
Mathematica [A] time = 0.109468, size = 152, normalized size = 0.4 \[ \frac{\frac{27 \sqrt{5} \sqrt{\sinh ^{-1}(a x)} \text{Gamma}\left (\frac{7}{2},-5 \sinh ^{-1}(a x)\right )}{\sqrt{-\sinh ^{-1}(a x)}}+\frac{625 \sqrt{3} \sqrt{-\sinh ^{-1}(a x)} \text{Gamma}\left (\frac{7}{2},-3 \sinh ^{-1}(a x)\right )}{\sqrt{\sinh ^{-1}(a x)}}+\frac{33750 \sqrt{\sinh ^{-1}(a x)} \text{Gamma}\left (\frac{7}{2},-\sinh ^{-1}(a x)\right )}{\sqrt{-\sinh ^{-1}(a x)}}-33750 \text{Gamma}\left (\frac{7}{2},\sinh ^{-1}(a x)\right )+625 \sqrt{3} \text{Gamma}\left (\frac{7}{2},3 \sinh ^{-1}(a x)\right )-27 \sqrt{5} \text{Gamma}\left (\frac{7}{2},5 \sinh ^{-1}(a x)\right )}{540000 a^5} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.204, size = 0, normalized size = 0. \begin{align*} \int{x}^{4} \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^{4} \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^{4} \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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