Optimal. Leaf size=155 \[ \frac {15}{4} b^2 x \sqrt {a+b \sinh ^{-1}(c x)}-\frac {5 b \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}{2 c}+x \left (a+b \sinh ^{-1}(c x)\right )^{5/2}+\frac {15 b^{5/2} e^{a/b} \sqrt {\pi } \text {Erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{16 c}-\frac {15 b^{5/2} e^{-\frac {a}{b}} \sqrt {\pi } \text {Erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{16 c} \]
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Rubi [A]
time = 0.26, antiderivative size = 155, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 7, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {5772, 5798,
5819, 3389, 2211, 2236, 2235} \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 )}{16 c}-\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 )}{16 c}+\frac {15}{4} b^2 x \sqrt {a+b \sinh ^{-1}(c x)}-\frac {5 b \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}{2 c}+x \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 5772
Rule 5798
Rule 5819
Rubi steps
\begin {align*} \int \left (a+b \sinh ^{-1}(c x)\right )^{5/2} \, dx &=x \left (a+b \sinh ^{-1}(c x)\right )^{5/2}-\frac {1}{2} (5 b c) \int \frac {x \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}{\sqrt {1+c^2 x^2}} \, dx\\ &=-\frac {5 b \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}{2 c}+x \left (a+b \sinh ^{-1}(c x)\right )^{5/2}+\frac {1}{4} \left (15 b^2\right ) \int \sqrt {a+b \sinh ^{-1}(c x)} \, dx\\ &=\frac {15}{4} b^2 x \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}}{2 c}+x \left (a+b \sinh ^{-1}(c x)\right )^{5/2}-\frac {1}{8} \left (15 b^3 c\right ) \int \frac {x}{\sqrt {1+c^2 x^2} \sqrt {a+b \sinh ^{-1}(c x)}} \, dx\\ &=\frac {15}{4} b^2 x \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}}{2 c}+x \left (a+b \sinh ^{-1}(c x)\right )^{5/2}-\frac {\left (15 b^3\right ) \text {Subst}\left (\int \frac {\sinh (x)}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{8 c}\\ &=\frac {15}{4} b^2 x \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}}{2 c}+x \left (a+b \sinh ^{-1}(c x)\right )^{5/2}+\frac {\left (15 b^3\right ) \text {Subst}\left (\int \frac {e^{-x}}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{16 c}-\frac {\left (15 b^3\right ) \text {Subst}\left (\int \frac {e^x}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{16 c}\\ &=\frac {15}{4} b^2 x \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}}{2 c}+x \left (a+b \sinh ^{-1}(c x)\right )^{5/2}+\frac {\left (15 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 )}{8 c}-\frac {\left (15 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 )}{8 c}\\ &=\frac {15}{4} b^2 x \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}}{2 c}+x \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 )}{16 c}-\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 )}{16 c}\\ \end {align*}
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Mathematica [A]
time = 2.36, size = 282, normalized size = 1.82 \begin {gather*} \frac {\sqrt {b} e^{-\frac {a}{b}} \left (-\left (\left (4 a^2-15 b^2\right ) e^{\frac {2 a}{b}} \sqrt {\pi } \text {Erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )\right )+\left (4 a^2-15 b^2\right ) \sqrt {\pi } \text {Erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )+\frac {4 \sqrt {b} \left (e^{a/b} \left (a+b \sinh ^{-1}(c x)\right ) \left (5 \left (3 b c x-2 a \sqrt {1+c^2 x^2}\right )+2 \left (4 a c x-5 b \sqrt {1+c^2 x^2}\right ) \sinh ^{-1}(c x)+4 b c x \sinh ^{-1}(c x)^2\right )-2 a^2 e^{\frac {2 a}{b}} \sqrt {\frac {a}{b}+\sinh ^{-1}(c x)} \Gamma \left (\frac {3}{2},\frac {a}{b}+\sinh ^{-1}(c x)\right )-2 a^2 \sqrt {-\frac {a+b \sinh ^{-1}(c x)}{b}} \Gamma \left (\frac {3}{2},-\frac {a+b \sinh ^{-1}(c x)}{b}\right )\right )}{\sqrt {a+b \sinh ^{-1}(c x)}}\right )}{16 c} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \left (a +b \arcsinh \left (c x \right )\right )^{\frac {5}{2}}\, dx\]
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 \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 = -0.01 \begin {gather*} \int {\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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