Optimal. Leaf size=135 \[ -\frac {3 b \sqrt {1+c^2 x^2} \sqrt {a+b \sinh ^{-1}(c x)}}{2 c}+x \left (a+b \sinh ^{-1}(c x)\right )^{3/2}+\frac {3 b^{3/2} e^{a/b} \sqrt {\pi } \text {Erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{8 c}+\frac {3 b^{3/2} e^{-\frac {a}{b}} \sqrt {\pi } \text {Erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{8 c} \]
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Rubi [A]
time = 0.18, antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 7, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {5772, 5798,
5774, 3388, 2211, 2236, 2235} \begin {gather*} \frac {3 \sqrt {\pi } b^{3/2} e^{a/b} \text {Erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{8 c}+\frac {3 \sqrt {\pi } b^{3/2} e^{-\frac {a}{b}} \text {Erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{8 c}-\frac {3 b \sqrt {c^2 x^2+1} \sqrt {a+b \sinh ^{-1}(c x)}}{2 c}+x \left (a+b \sinh ^{-1}(c x)\right )^{3/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 2211
Rule 2235
Rule 2236
Rule 3388
Rule 5772
Rule 5774
Rule 5798
Rubi steps
\begin {align*} \int \left (a+b \sinh ^{-1}(c x)\right )^{3/2} \, dx &=x \left (a+b \sinh ^{-1}(c x)\right )^{3/2}-\frac {1}{2} (3 b c) \int \frac {x \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {1+c^2 x^2}} \, dx\\ &=-\frac {3 b \sqrt {1+c^2 x^2} \sqrt {a+b \sinh ^{-1}(c x)}}{2 c}+x \left (a+b \sinh ^{-1}(c x)\right )^{3/2}+\frac {1}{4} \left (3 b^2\right ) \int \frac {1}{\sqrt {a+b \sinh ^{-1}(c x)}} \, dx\\ &=-\frac {3 b \sqrt {1+c^2 x^2} \sqrt {a+b \sinh ^{-1}(c x)}}{2 c}+x \left (a+b \sinh ^{-1}(c x)\right )^{3/2}+\frac {(3 b) \text {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \sinh ^{-1}(c x)\right )}{4 c}\\ &=-\frac {3 b \sqrt {1+c^2 x^2} \sqrt {a+b \sinh ^{-1}(c x)}}{2 c}+x \left (a+b \sinh ^{-1}(c x)\right )^{3/2}+\frac {(3 b) \text {Subst}\left (\int \frac {e^{-i \left (\frac {i a}{b}-\frac {i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \sinh ^{-1}(c x)\right )}{8 c}+\frac {(3 b) \text {Subst}\left (\int \frac {e^{i \left (\frac {i a}{b}-\frac {i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \sinh ^{-1}(c x)\right )}{8 c}\\ &=-\frac {3 b \sqrt {1+c^2 x^2} \sqrt {a+b \sinh ^{-1}(c x)}}{2 c}+x \left (a+b \sinh ^{-1}(c x)\right )^{3/2}+\frac {(3 b) \text {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c x)}\right )}{4 c}+\frac {(3 b) \text {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c x)}\right )}{4 c}\\ &=-\frac {3 b \sqrt {1+c^2 x^2} \sqrt {a+b \sinh ^{-1}(c x)}}{2 c}+x \left (a+b \sinh ^{-1}(c x)\right )^{3/2}+\frac {3 b^{3/2} e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{8 c}+\frac {3 b^{3/2} e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{8 c}\\ \end {align*}
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Mathematica [A]
time = 0.37, size = 251, normalized size = 1.86 \begin {gather*} \frac {a e^{-\frac {a}{b}} \sqrt {a+b \sinh ^{-1}(c x)} \left (-\frac {e^{\frac {2 a}{b}} \Gamma \left (\frac {3}{2},\frac {a}{b}+\sinh ^{-1}(c x)\right )}{\sqrt {\frac {a}{b}+\sinh ^{-1}(c x)}}+\frac {\Gamma \left (\frac {3}{2},-\frac {a+b \sinh ^{-1}(c x)}{b}\right )}{\sqrt {-\frac {a+b \sinh ^{-1}(c x)}{b}}}\right )}{2 c}+\frac {\sqrt {b} \left (4 \sqrt {b} \sqrt {a+b \sinh ^{-1}(c x)} \left (-3 \sqrt {1+c^2 x^2}+2 c x \sinh ^{-1}(c x)\right )+(2 a+3 b) \sqrt {\pi } \text {Erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {a}{b}\right )-\sinh \left (\frac {a}{b}\right )\right )+(-2 a+3 b) \sqrt {\pi } \text {Erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {a}{b}\right )+\sinh \left (\frac {a}{b}\right )\right )\right )}{8 c} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.11, size = 0, normalized size = 0.00 \[\int \left (a +b \arcsinh \left (c x \right )\right )^{\frac {3}{2}}\, dx\]
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 \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 {3}{2}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \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 )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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