Optimal. Leaf size=327 \[ -\frac {5 b^2 x \sqrt {a+b \sinh ^{-1}(c x)}}{6 c^2}+\frac {5}{36} b^2 x^3 \sqrt {a+b \sinh ^{-1}(c x)}+\frac {5 b \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}{9 c^3}-\frac {5 b x^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}{18 c}+\frac {1}{3} x^3 \left (a+b \sinh ^{-1}(c x)\right )^{5/2}-\frac {15 b^{5/2} e^{a/b} \sqrt {\pi } \text {Erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{64 c^3}+\frac {5 b^{5/2} e^{\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {Erf}\left (\frac {\sqrt {3} \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{576 c^3}+\frac {15 b^{5/2} e^{-\frac {a}{b}} \sqrt {\pi } \text {Erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{64 c^3}-\frac {5 b^{5/2} e^{-\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {Erfi}\left (\frac {\sqrt {3} \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{576 c^3} \]
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Rubi [A]
time = 0.77, antiderivative size = 327, normalized size of antiderivative = 1.00, number of steps
used = 24, number of rules used = 10, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {5777, 5812,
5798, 5772, 5819, 3389, 2211, 2236, 2235, 3393} \begin {gather*} -\frac {15 \sqrt {\pi } b^{5/2} e^{a/b} \text {Erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{64 c^3}+\frac {5 \sqrt {\frac {\pi }{3}} b^{5/2} e^{\frac {3 a}{b}} \text {Erf}\left (\frac {\sqrt {3} \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{576 c^3}+\frac {15 \sqrt {\pi } b^{5/2} e^{-\frac {a}{b}} \text {Erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{64 c^3}-\frac {5 \sqrt {\frac {\pi }{3}} b^{5/2} e^{-\frac {3 a}{b}} \text {Erfi}\left (\frac {\sqrt {3} \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{576 c^3}-\frac {5 b^2 x \sqrt {a+b \sinh ^{-1}(c x)}}{6 c^2}+\frac {5}{36} b^2 x^3 \sqrt {a+b \sinh ^{-1}(c x)}-\frac {5 b x^2 \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}{18 c}+\frac {5 b \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}{9 c^3}+\frac {1}{3} x^3 \left (a+b \sinh ^{-1}(c x)\right )^{5/2} \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^2 \left (a+b \sinh ^{-1}(c x)\right )^{5/2} \, dx &=\frac {1}{3} x^3 \left (a+b \sinh ^{-1}(c x)\right )^{5/2}-\frac {1}{6} (5 b c) \int \frac {x^3 \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}{\sqrt {1+c^2 x^2}} \, dx\\ &=-\frac {5 b x^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}{18 c}+\frac {1}{3} x^3 \left (a+b \sinh ^{-1}(c x)\right )^{5/2}+\frac {1}{12} \left (5 b^2\right ) \int x^2 \sqrt {a+b \sinh ^{-1}(c x)} \, dx+\frac {(5 b) \int \frac {x \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}{\sqrt {1+c^2 x^2}} \, dx}{9 c}\\ &=\frac {5}{36} b^2 x^3 \sqrt {a+b \sinh ^{-1}(c x)}+\frac {5 b \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}{9 c^3}-\frac {5 b x^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}{18 c}+\frac {1}{3} x^3 \left (a+b \sinh ^{-1}(c x)\right )^{5/2}-\frac {\left (5 b^2\right ) \int \sqrt {a+b \sinh ^{-1}(c x)} \, dx}{6 c^2}-\frac {1}{72} \left (5 b^3 c\right ) \int \frac {x^3}{\sqrt {1+c^2 x^2} \sqrt {a+b \sinh ^{-1}(c x)}} \, dx\\ &=-\frac {5 b^2 x \sqrt {a+b \sinh ^{-1}(c x)}}{6 c^2}+\frac {5}{36} b^2 x^3 \sqrt {a+b \sinh ^{-1}(c x)}+\frac {5 b \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}{9 c^3}-\frac {5 b x^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}{18 c}+\frac {1}{3} x^3 \left (a+b \sinh ^{-1}(c x)\right )^{5/2}-\frac {\left (5 b^3\right ) \text {Subst}\left (\int \frac {\sinh ^3(x)}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{72 c^3}+\frac {\left (5 b^3\right ) \int \frac {x}{\sqrt {1+c^2 x^2} \sqrt {a+b \sinh ^{-1}(c x)}} \, dx}{12 c}\\ &=-\frac {5 b^2 x \sqrt {a+b \sinh ^{-1}(c x)}}{6 c^2}+\frac {5}{36} b^2 x^3 \sqrt {a+b \sinh ^{-1}(c x)}+\frac {5 b \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}{9 c^3}-\frac {5 b x^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}{18 c}+\frac {1}{3} x^3 \left (a+b \sinh ^{-1}(c x)\right )^{5/2}-\frac {\left (5 i b^3\right ) \text {Subst}\left (\int \left (\frac {3 i \sinh (x)}{4 \sqrt {a+b x}}-\frac {i \sinh (3 x)}{4 \sqrt {a+b x}}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{72 c^3}+\frac {\left (5 b^3\right ) \text {Subst}\left (\int \frac {\sinh (x)}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{12 c^3}\\ &=-\frac {5 b^2 x \sqrt {a+b \sinh ^{-1}(c x)}}{6 c^2}+\frac {5}{36} b^2 x^3 \sqrt {a+b \sinh ^{-1}(c x)}+\frac {5 b \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}{9 c^3}-\frac {5 b x^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}{18 c}+\frac {1}{3} x^3 \left (a+b \sinh ^{-1}(c x)\right )^{5/2}-\frac {\left (5 b^3\right ) \text {Subst}\left (\int \frac {\sinh (3 x)}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{288 c^3}+\frac {\left (5 b^3\right ) \text {Subst}\left (\int \frac {\sinh (x)}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{96 c^3}-\frac {\left (5 b^3\right ) \text {Subst}\left (\int \frac {e^{-x}}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{24 c^3}+\frac {\left (5 b^3\right ) \text {Subst}\left (\int \frac {e^x}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{24 c^3}\\ &=-\frac {5 b^2 x \sqrt {a+b \sinh ^{-1}(c x)}}{6 c^2}+\frac {5}{36} b^2 x^3 \sqrt {a+b \sinh ^{-1}(c x)}+\frac {5 b \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}{9 c^3}-\frac {5 b x^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}{18 c}+\frac {1}{3} x^3 \left (a+b \sinh ^{-1}(c x)\right )^{5/2}-\frac {\left (5 b^2\right ) \text {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c x)}\right )}{12 c^3}+\frac {\left (5 b^2\right ) \text {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c x)}\right )}{12 c^3}+\frac {\left (5 b^3\right ) \text {Subst}\left (\int \frac {e^{-3 x}}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{576 c^3}-\frac {\left (5 b^3\right ) \text {Subst}\left (\int \frac {e^{3 x}}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{576 c^3}-\frac {\left (5 b^3\right ) \text {Subst}\left (\int \frac {e^{-x}}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{192 c^3}+\frac {\left (5 b^3\right ) \text {Subst}\left (\int \frac {e^x}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{192 c^3}\\ &=-\frac {5 b^2 x \sqrt {a+b \sinh ^{-1}(c x)}}{6 c^2}+\frac {5}{36} b^2 x^3 \sqrt {a+b \sinh ^{-1}(c x)}+\frac {5 b \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}{9 c^3}-\frac {5 b x^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}{18 c}+\frac {1}{3} x^3 \left (a+b \sinh ^{-1}(c x)\right )^{5/2}-\frac {5 b^{5/2} e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{24 c^3}+\frac {5 b^{5/2} e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{24 c^3}+\frac {\left (5 b^2\right ) \text {Subst}\left (\int e^{\frac {3 a}{b}-\frac {3 x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c x)}\right )}{288 c^3}-\frac {\left (5 b^2\right ) \text {Subst}\left (\int e^{-\frac {3 a}{b}+\frac {3 x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c x)}\right )}{288 c^3}-\frac {\left (5 b^2\right ) \text {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c x)}\right )}{96 c^3}+\frac {\left (5 b^2\right ) \text {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c x)}\right )}{96 c^3}\\ &=-\frac {5 b^2 x \sqrt {a+b \sinh ^{-1}(c x)}}{6 c^2}+\frac {5}{36} b^2 x^3 \sqrt {a+b \sinh ^{-1}(c x)}+\frac {5 b \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}{9 c^3}-\frac {5 b x^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}{18 c}+\frac {1}{3} x^3 \left (a+b \sinh ^{-1}(c x)\right )^{5/2}-\frac {15 b^{5/2} e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{64 c^3}+\frac {5 b^{5/2} e^{\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{576 c^3}+\frac {15 b^{5/2} e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{64 c^3}-\frac {5 b^{5/2} e^{-\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{576 c^3}\\ \end {align*}
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Mathematica [A]
time = 0.32, size = 215, normalized size = 0.66 \begin {gather*} -\frac {e^{-\frac {3 a}{b}} \left (a+b \sinh ^{-1}(c x)\right )^{5/2} \left (81 e^{\frac {4 a}{b}} \sqrt {-\frac {a+b \sinh ^{-1}(c x)}{b}} \Gamma \left (\frac {7}{2},\frac {a}{b}+\sinh ^{-1}(c x)\right )+\sqrt {3} \sqrt {\frac {a}{b}+\sinh ^{-1}(c x)} \Gamma \left (\frac {7}{2},-\frac {3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )-81 e^{\frac {2 a}{b}} \sqrt {\frac {a}{b}+\sinh ^{-1}(c x)} \Gamma \left (\frac {7}{2},-\frac {a+b \sinh ^{-1}(c x)}{b}\right )-\sqrt {3} e^{\frac {6 a}{b}} \sqrt {-\frac {a+b \sinh ^{-1}(c x)}{b}} \Gamma \left (\frac {7}{2},\frac {3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )\right )}{648 c^3 \left (-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{b^2}\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int x^{2} \left (a +b \arcsinh \left (c x \right )\right )^{\frac {5}{2}}\, dx\]
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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{2} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{\frac {5}{2}}\, dx \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^2\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^{5/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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