Optimal. Leaf size=330 \[ \frac {3 \sqrt {\pi } \text {erf}\left (\sqrt {\sinh ^{-1}(a x)}\right )}{64 a^5}-\frac {3 \sqrt {3 \pi } \text {erf}\left (\sqrt {3} \sqrt {\sinh ^{-1}(a x)}\right )}{3200 a^5}-\frac {\sqrt {\frac {\pi }{3}} \text {erf}\left (\sqrt {3} \sqrt {\sinh ^{-1}(a x)}\right )}{200 a^5}+\frac {3 \sqrt {\frac {\pi }{5}} \text {erf}\left (\sqrt {5} \sqrt {\sinh ^{-1}(a x)}\right )}{3200 a^5}+\frac {3 \sqrt {\pi } \text {erfi}\left (\sqrt {\sinh ^{-1}(a x)}\right )}{64 a^5}-\frac {3 \sqrt {3 \pi } \text {erfi}\left (\sqrt {3} \sqrt {\sinh ^{-1}(a x)}\right )}{3200 a^5}-\frac {\sqrt {\frac {\pi }{3}} \text {erfi}\left (\sqrt {3} \sqrt {\sinh ^{-1}(a x)}\right )}{200 a^5}+\frac {3 \sqrt {\frac {\pi }{5}} \text {erfi}\left (\sqrt {5} \sqrt {\sinh ^{-1}(a x)}\right )}{3200 a^5}-\frac {3 x^4 \sqrt {a^2 x^2+1} \sqrt {\sinh ^{-1}(a x)}}{50 a}-\frac {4 \sqrt {a^2 x^2+1} \sqrt {\sinh ^{-1}(a x)}}{25 a^5}+\frac {2 x^2 \sqrt {a^2 x^2+1} \sqrt {\sinh ^{-1}(a x)}}{25 a^3}+\frac {1}{5} x^5 \sinh ^{-1}(a x)^{3/2} \]
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Rubi [A] time = 0.71, antiderivative size = 330, normalized size of antiderivative = 1.00, number of steps used = 41, number of rules used = 10, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.833, Rules used = {5663, 5758, 5717, 5657, 3307, 2180, 2204, 2205, 5669, 5448} \[ \frac {3 \sqrt {\pi } \text {Erf}\left (\sqrt {\sinh ^{-1}(a x)}\right )}{64 a^5}-\frac {3 \sqrt {3 \pi } \text {Erf}\left (\sqrt {3} \sqrt {\sinh ^{-1}(a x)}\right )}{3200 a^5}-\frac {\sqrt {\frac {\pi }{3}} \text {Erf}\left (\sqrt {3} \sqrt {\sinh ^{-1}(a x)}\right )}{200 a^5}+\frac {3 \sqrt {\frac {\pi }{5}} \text {Erf}\left (\sqrt {5} \sqrt {\sinh ^{-1}(a x)}\right )}{3200 a^5}+\frac {3 \sqrt {\pi } \text {Erfi}\left (\sqrt {\sinh ^{-1}(a x)}\right )}{64 a^5}-\frac {3 \sqrt {3 \pi } \text {Erfi}\left (\sqrt {3} \sqrt {\sinh ^{-1}(a x)}\right )}{3200 a^5}-\frac {\sqrt {\frac {\pi }{3}} \text {Erfi}\left (\sqrt {3} \sqrt {\sinh ^{-1}(a x)}\right )}{200 a^5}+\frac {3 \sqrt {\frac {\pi }{5}} \text {Erfi}\left (\sqrt {5} \sqrt {\sinh ^{-1}(a x)}\right )}{3200 a^5}-\frac {3 x^4 \sqrt {a^2 x^2+1} \sqrt {\sinh ^{-1}(a x)}}{50 a}+\frac {2 x^2 \sqrt {a^2 x^2+1} \sqrt {\sinh ^{-1}(a x)}}{25 a^3}-\frac {4 \sqrt {a^2 x^2+1} \sqrt {\sinh ^{-1}(a x)}}{25 a^5}+\frac {1}{5} x^5 \sinh ^{-1}(a x)^{3/2} \]
Antiderivative was successfully verified.
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Rule 2180
Rule 2204
Rule 2205
Rule 3307
Rule 5448
Rule 5657
Rule 5663
Rule 5669
Rule 5717
Rule 5758
Rubi steps
\begin {align*} \int x^4 \sinh ^{-1}(a x)^{3/2} \, dx &=\frac {1}{5} x^5 \sinh ^{-1}(a x)^{3/2}-\frac {1}{10} (3 a) \int \frac {x^5 \sqrt {\sinh ^{-1}(a x)}}{\sqrt {1+a^2 x^2}} \, dx\\ &=-\frac {3 x^4 \sqrt {1+a^2 x^2} \sqrt {\sinh ^{-1}(a x)}}{50 a}+\frac {1}{5} x^5 \sinh ^{-1}(a x)^{3/2}+\frac {3}{100} \int \frac {x^4}{\sqrt {\sinh ^{-1}(a x)}} \, dx+\frac {6 \int \frac {x^3 \sqrt {\sinh ^{-1}(a x)}}{\sqrt {1+a^2 x^2}} \, dx}{25 a}\\ &=\frac {2 x^2 \sqrt {1+a^2 x^2} \sqrt {\sinh ^{-1}(a x)}}{25 a^3}-\frac {3 x^4 \sqrt {1+a^2 x^2} \sqrt {\sinh ^{-1}(a x)}}{50 a}+\frac {1}{5} x^5 \sinh ^{-1}(a x)^{3/2}+\frac {3 \operatorname {Subst}\left (\int \frac {\cosh (x) \sinh ^4(x)}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{100 a^5}-\frac {4 \int \frac {x \sqrt {\sinh ^{-1}(a x)}}{\sqrt {1+a^2 x^2}} \, dx}{25 a^3}-\frac {\int \frac {x^2}{\sqrt {\sinh ^{-1}(a x)}} \, dx}{25 a^2}\\ &=-\frac {4 \sqrt {1+a^2 x^2} \sqrt {\sinh ^{-1}(a x)}}{25 a^5}+\frac {2 x^2 \sqrt {1+a^2 x^2} \sqrt {\sinh ^{-1}(a x)}}{25 a^3}-\frac {3 x^4 \sqrt {1+a^2 x^2} \sqrt {\sinh ^{-1}(a x)}}{50 a}+\frac {1}{5} x^5 \sinh ^{-1}(a x)^{3/2}+\frac {3 \operatorname {Subst}\left (\int \left (\frac {\cosh (x)}{8 \sqrt {x}}-\frac {3 \cosh (3 x)}{16 \sqrt {x}}+\frac {\cosh (5 x)}{16 \sqrt {x}}\right ) \, dx,x,\sinh ^{-1}(a x)\right )}{100 a^5}-\frac {\operatorname {Subst}\left (\int \frac {\cosh (x) \sinh ^2(x)}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{25 a^5}+\frac {2 \int \frac {1}{\sqrt {\sinh ^{-1}(a x)}} \, dx}{25 a^4}\\ &=-\frac {4 \sqrt {1+a^2 x^2} \sqrt {\sinh ^{-1}(a x)}}{25 a^5}+\frac {2 x^2 \sqrt {1+a^2 x^2} \sqrt {\sinh ^{-1}(a x)}}{25 a^3}-\frac {3 x^4 \sqrt {1+a^2 x^2} \sqrt {\sinh ^{-1}(a x)}}{50 a}+\frac {1}{5} x^5 \sinh ^{-1}(a x)^{3/2}+\frac {3 \operatorname {Subst}\left (\int \frac {\cosh (5 x)}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{1600 a^5}+\frac {3 \operatorname {Subst}\left (\int \frac {\cosh (x)}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{800 a^5}-\frac {9 \operatorname {Subst}\left (\int \frac {\cosh (3 x)}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{1600 a^5}-\frac {\operatorname {Subst}\left (\int \left (-\frac {\cosh (x)}{4 \sqrt {x}}+\frac {\cosh (3 x)}{4 \sqrt {x}}\right ) \, dx,x,\sinh ^{-1}(a x)\right )}{25 a^5}+\frac {2 \operatorname {Subst}\left (\int \frac {\cosh (x)}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{25 a^5}\\ &=-\frac {4 \sqrt {1+a^2 x^2} \sqrt {\sinh ^{-1}(a x)}}{25 a^5}+\frac {2 x^2 \sqrt {1+a^2 x^2} \sqrt {\sinh ^{-1}(a x)}}{25 a^3}-\frac {3 x^4 \sqrt {1+a^2 x^2} \sqrt {\sinh ^{-1}(a x)}}{50 a}+\frac {1}{5} x^5 \sinh ^{-1}(a x)^{3/2}+\frac {3 \operatorname {Subst}\left (\int \frac {e^{-5 x}}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{3200 a^5}+\frac {3 \operatorname {Subst}\left (\int \frac {e^{5 x}}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{3200 a^5}+\frac {3 \operatorname {Subst}\left (\int \frac {e^{-x}}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{1600 a^5}+\frac {3 \operatorname {Subst}\left (\int \frac {e^x}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{1600 a^5}-\frac {9 \operatorname {Subst}\left (\int \frac {e^{-3 x}}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{3200 a^5}-\frac {9 \operatorname {Subst}\left (\int \frac {e^{3 x}}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{3200 a^5}+\frac {\operatorname {Subst}\left (\int \frac {\cosh (x)}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{100 a^5}-\frac {\operatorname {Subst}\left (\int \frac {\cosh (3 x)}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{100 a^5}+\frac {\operatorname {Subst}\left (\int \frac {e^{-x}}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{25 a^5}+\frac {\operatorname {Subst}\left (\int \frac {e^x}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{25 a^5}\\ &=-\frac {4 \sqrt {1+a^2 x^2} \sqrt {\sinh ^{-1}(a x)}}{25 a^5}+\frac {2 x^2 \sqrt {1+a^2 x^2} \sqrt {\sinh ^{-1}(a x)}}{25 a^3}-\frac {3 x^4 \sqrt {1+a^2 x^2} \sqrt {\sinh ^{-1}(a x)}}{50 a}+\frac {1}{5} x^5 \sinh ^{-1}(a x)^{3/2}+\frac {3 \operatorname {Subst}\left (\int e^{-5 x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{1600 a^5}+\frac {3 \operatorname {Subst}\left (\int e^{5 x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{1600 a^5}+\frac {3 \operatorname {Subst}\left (\int e^{-x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{800 a^5}+\frac {3 \operatorname {Subst}\left (\int e^{x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{800 a^5}-\frac {\operatorname {Subst}\left (\int \frac {e^{-3 x}}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{200 a^5}+\frac {\operatorname {Subst}\left (\int \frac {e^{-x}}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{200 a^5}+\frac {\operatorname {Subst}\left (\int \frac {e^x}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{200 a^5}-\frac {\operatorname {Subst}\left (\int \frac {e^{3 x}}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{200 a^5}-\frac {9 \operatorname {Subst}\left (\int e^{-3 x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{1600 a^5}-\frac {9 \operatorname {Subst}\left (\int e^{3 x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{1600 a^5}+\frac {2 \operatorname {Subst}\left (\int e^{-x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{25 a^5}+\frac {2 \operatorname {Subst}\left (\int e^{x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{25 a^5}\\ &=-\frac {4 \sqrt {1+a^2 x^2} \sqrt {\sinh ^{-1}(a x)}}{25 a^5}+\frac {2 x^2 \sqrt {1+a^2 x^2} \sqrt {\sinh ^{-1}(a x)}}{25 a^3}-\frac {3 x^4 \sqrt {1+a^2 x^2} \sqrt {\sinh ^{-1}(a x)}}{50 a}+\frac {1}{5} x^5 \sinh ^{-1}(a x)^{3/2}+\frac {67 \sqrt {\pi } \text {erf}\left (\sqrt {\sinh ^{-1}(a x)}\right )}{1600 a^5}-\frac {3 \sqrt {3 \pi } \text {erf}\left (\sqrt {3} \sqrt {\sinh ^{-1}(a x)}\right )}{3200 a^5}+\frac {3 \sqrt {\frac {\pi }{5}} \text {erf}\left (\sqrt {5} \sqrt {\sinh ^{-1}(a x)}\right )}{3200 a^5}+\frac {67 \sqrt {\pi } \text {erfi}\left (\sqrt {\sinh ^{-1}(a x)}\right )}{1600 a^5}-\frac {3 \sqrt {3 \pi } \text {erfi}\left (\sqrt {3} \sqrt {\sinh ^{-1}(a x)}\right )}{3200 a^5}+\frac {3 \sqrt {\frac {\pi }{5}} \text {erfi}\left (\sqrt {5} \sqrt {\sinh ^{-1}(a x)}\right )}{3200 a^5}-\frac {\operatorname {Subst}\left (\int e^{-3 x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{100 a^5}+\frac {\operatorname {Subst}\left (\int e^{-x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{100 a^5}+\frac {\operatorname {Subst}\left (\int e^{x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{100 a^5}-\frac {\operatorname {Subst}\left (\int e^{3 x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{100 a^5}\\ &=-\frac {4 \sqrt {1+a^2 x^2} \sqrt {\sinh ^{-1}(a x)}}{25 a^5}+\frac {2 x^2 \sqrt {1+a^2 x^2} \sqrt {\sinh ^{-1}(a x)}}{25 a^3}-\frac {3 x^4 \sqrt {1+a^2 x^2} \sqrt {\sinh ^{-1}(a x)}}{50 a}+\frac {1}{5} x^5 \sinh ^{-1}(a x)^{3/2}+\frac {3 \sqrt {\pi } \text {erf}\left (\sqrt {\sinh ^{-1}(a x)}\right )}{64 a^5}-\frac {\sqrt {\frac {\pi }{3}} \text {erf}\left (\sqrt {3} \sqrt {\sinh ^{-1}(a x)}\right )}{200 a^5}-\frac {3 \sqrt {3 \pi } \text {erf}\left (\sqrt {3} \sqrt {\sinh ^{-1}(a x)}\right )}{3200 a^5}+\frac {3 \sqrt {\frac {\pi }{5}} \text {erf}\left (\sqrt {5} \sqrt {\sinh ^{-1}(a x)}\right )}{3200 a^5}+\frac {3 \sqrt {\pi } \text {erfi}\left (\sqrt {\sinh ^{-1}(a x)}\right )}{64 a^5}-\frac {\sqrt {\frac {\pi }{3}} \text {erfi}\left (\sqrt {3} \sqrt {\sinh ^{-1}(a x)}\right )}{200 a^5}-\frac {3 \sqrt {3 \pi } \text {erfi}\left (\sqrt {3} \sqrt {\sinh ^{-1}(a x)}\right )}{3200 a^5}+\frac {3 \sqrt {\frac {\pi }{5}} \text {erfi}\left (\sqrt {5} \sqrt {\sinh ^{-1}(a x)}\right )}{3200 a^5}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 152, normalized size = 0.46 \[ \frac {\frac {9 \sqrt {5} \sqrt {-\sinh ^{-1}(a x)} \Gamma \left (\frac {5}{2},-5 \sinh ^{-1}(a x)\right )}{\sqrt {\sinh ^{-1}(a x)}}+\frac {125 \sqrt {3} \sqrt {\sinh ^{-1}(a x)} \Gamma \left (\frac {5}{2},-3 \sinh ^{-1}(a x)\right )}{\sqrt {-\sinh ^{-1}(a x)}}+\frac {2250 \sqrt {-\sinh ^{-1}(a x)} \Gamma \left (\frac {5}{2},-\sinh ^{-1}(a x)\right )}{\sqrt {\sinh ^{-1}(a x)}}-2250 \Gamma \left (\frac {5}{2},\sinh ^{-1}(a x)\right )+125 \sqrt {3} \Gamma \left (\frac {5}{2},3 \sinh ^{-1}(a x)\right )-9 \sqrt {5} \Gamma \left (\frac {5}{2},5 \sinh ^{-1}(a x)\right )}{36000 a^5} \]
Warning: Unable to verify antiderivative.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{4} \operatorname {arsinh}\left (a x\right )^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F(-2)] time = 180.00, size = 0, normalized size = 0.00 \[ \int x^{4} \arcsinh \left (a x \right )^{\frac {3}{2}}\, dx \]
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 \[ \int x^{4} \operatorname {arsinh}\left (a x\right )^{\frac {3}{2}}\,{d x} \]
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 \[ \int x^4\,{\mathrm {asinh}\left (a\,x\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{4} \operatorname {asinh}^{\frac {3}{2}}{\left (a x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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