Optimal. Leaf size=379 \[ \frac {2 x \sqrt {\sinh ^{-1}(a x)}}{5 a^4}-\frac {x^3 \sqrt {\sinh ^{-1}(a x)}}{15 a^2}+\frac {3}{100} x^5 \sqrt {\sinh ^{-1}(a x)}-\frac {4 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^{3/2}}{15 a^5}+\frac {2 x^2 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^{3/2}}{15 a^3}-\frac {x^4 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^{3/2}}{10 a}+\frac {1}{5} x^5 \sinh ^{-1}(a x)^{5/2}+\frac {15 \sqrt {\pi } \text {Erf}\left (\sqrt {\sinh ^{-1}(a x)}\right )}{128 a^5}-\frac {\sqrt {\frac {\pi }{3}} \text {Erf}\left (\sqrt {3} \sqrt {\sinh ^{-1}(a x)}\right )}{240 a^5}-\frac {\sqrt {3 \pi } \text {Erf}\left (\sqrt {3} \sqrt {\sinh ^{-1}(a x)}\right )}{1280 a^5}+\frac {3 \sqrt {\frac {\pi }{5}} \text {Erf}\left (\sqrt {5} \sqrt {\sinh ^{-1}(a x)}\right )}{6400 a^5}-\frac {15 \sqrt {\pi } \text {Erfi}\left (\sqrt {\sinh ^{-1}(a x)}\right )}{128 a^5}+\frac {\sqrt {\frac {\pi }{3}} \text {Erfi}\left (\sqrt {3} \sqrt {\sinh ^{-1}(a x)}\right )}{240 a^5}+\frac {\sqrt {3 \pi } \text {Erfi}\left (\sqrt {3} \sqrt {\sinh ^{-1}(a x)}\right )}{1280 a^5}-\frac {3 \sqrt {\frac {\pi }{5}} \text {Erfi}\left (\sqrt {5} \sqrt {\sinh ^{-1}(a x)}\right )}{6400 a^5} \]
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Rubi [A]
time = 0.65, antiderivative size = 379, normalized size of antiderivative = 1.00, 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 = {5777, 5812,
5798, 5772, 5819, 3389, 2211, 2235, 2236, 3393} \begin {gather*} \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 {2 x \sqrt {\sinh ^{-1}(a x)}}{5 a^4}-\frac {x^3 \sqrt {\sinh ^{-1}(a x)}}{15 a^2}-\frac {x^4 \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)^{3/2}}{10 a}-\frac {4 \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)^{3/2}}{15 a^5}+\frac {2 x^2 \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)^{3/2}}{15 a^3}+\frac {1}{5} x^5 \sinh ^{-1}(a x)^{5/2}+\frac {3}{100} x^5 \sqrt {\sinh ^{-1}(a x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2211
Rule 2235
Rule 2236
Rule 3389
Rule 3393
Rule 5772
Rule 5777
Rule 5798
Rule 5812
Rule 5819
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 \text {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) \text {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 {\text {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 \text {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 \text {Subst}\left (\int \frac {\sinh (5 x)}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{3200 a^5}+\frac {3 \text {Subst}\left (\int \frac {\sinh (3 x)}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{640 a^5}-\frac {3 \text {Subst}\left (\int \frac {\sinh (x)}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{320 a^5}-\frac {\text {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 \text {Subst}\left (\int \frac {e^{-5 x}}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{6400 a^5}-\frac {3 \text {Subst}\left (\int \frac {e^{5 x}}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{6400 a^5}-\frac {3 \text {Subst}\left (\int \frac {e^{-3 x}}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{1280 a^5}+\frac {3 \text {Subst}\left (\int \frac {e^{3 x}}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{1280 a^5}+\frac {3 \text {Subst}\left (\int \frac {e^{-x}}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{640 a^5}-\frac {3 \text {Subst}\left (\int \frac {e^x}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{640 a^5}+\frac {\text {Subst}\left (\int \frac {\sinh (3 x)}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{120 a^5}-\frac {\text {Subst}\left (\int \frac {\sinh (x)}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{40 a^5}+\frac {\text {Subst}\left (\int \frac {e^{-x}}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{10 a^5}-\frac {\text {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 \text {Subst}\left (\int e^{-5 x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{3200 a^5}-\frac {3 \text {Subst}\left (\int e^{5 x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{3200 a^5}-\frac {\text {Subst}\left (\int \frac {e^{-3 x}}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{240 a^5}+\frac {\text {Subst}\left (\int \frac {e^{3 x}}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{240 a^5}-\frac {3 \text {Subst}\left (\int e^{-3 x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{640 a^5}+\frac {3 \text {Subst}\left (\int e^{3 x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{640 a^5}+\frac {3 \text {Subst}\left (\int e^{-x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{320 a^5}-\frac {3 \text {Subst}\left (\int e^{x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{320 a^5}+\frac {\text {Subst}\left (\int \frac {e^{-x}}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{80 a^5}-\frac {\text {Subst}\left (\int \frac {e^x}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{80 a^5}+\frac {\text {Subst}\left (\int e^{-x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{5 a^5}-\frac {\text {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 {\text {Subst}\left (\int e^{-3 x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{120 a^5}+\frac {\text {Subst}\left (\int e^{3 x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{120 a^5}+\frac {\text {Subst}\left (\int e^{-x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{40 a^5}-\frac {\text {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*}
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Mathematica [A]
time = 0.08, size = 152, normalized size = 0.40 \begin {gather*} \frac {\frac {27 \sqrt {5} \sqrt {\sinh ^{-1}(a x)} \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)} \Gamma \left (\frac {7}{2},-3 \sinh ^{-1}(a x)\right )}{\sqrt {\sinh ^{-1}(a x)}}+\frac {33750 \sqrt {\sinh ^{-1}(a x)} \Gamma \left (\frac {7}{2},-\sinh ^{-1}(a x)\right )}{\sqrt {-\sinh ^{-1}(a x)}}-33750 \Gamma \left (\frac {7}{2},\sinh ^{-1}(a x)\right )+625 \sqrt {3} \Gamma \left (\frac {7}{2},3 \sinh ^{-1}(a x)\right )-27 \sqrt {5} \Gamma \left (\frac {7}{2},5 \sinh ^{-1}(a x)\right )}{540000 a^5} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 6.56, size = 0, normalized size = 0.00 \[\int x^{4} \arcsinh \left (a x \right )^{\frac {5}{2}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^4\,{\mathrm {asinh}\left (a\,x\right )}^{5/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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