Optimal. Leaf size=135 \[ \frac {3 \sqrt {\pi } b^{3/2} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{8 c}+\frac {3 \sqrt {\pi } b^{3/2} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{8 c}-\frac {3 b \sqrt {c^2 x^2+1} \sqrt {a+b \sinh ^{-1}(c x)}}{2 c}+x \left (a+b \sinh ^{-1}(c x)\right )^{3/2} \]
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Rubi [A] time = 0.25, 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 = {5653, 5717, 5657, 3307, 2180, 2205, 2204} \[ \frac {3 \sqrt {\pi } b^{3/2} e^{a/b} \text {Erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{8 c}+\frac {3 \sqrt {\pi } b^{3/2} e^{-\frac {a}{b}} \text {Erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{8 c}-\frac {3 b \sqrt {c^2 x^2+1} \sqrt {a+b \sinh ^{-1}(c x)}}{2 c}+x \left (a+b \sinh ^{-1}(c x)\right )^{3/2} \]
Antiderivative was successfully verified.
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Rule 2180
Rule 2204
Rule 2205
Rule 3307
Rule 5653
Rule 5657
Rule 5717
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) \operatorname {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) \operatorname {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) \operatorname {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) \operatorname {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) \operatorname {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 = 0.64, size = 251, normalized size = 1.86 \[ \frac {\sqrt {b} \left (4 \sqrt {b} \left (2 c x \sinh ^{-1}(c x)-3 \sqrt {c^2 x^2+1}\right ) \sqrt {a+b \sinh ^{-1}(c x)}+\sqrt {\pi } (3 b-2 a) \left (\sinh \left (\frac {a}{b}\right )+\cosh \left (\frac {a}{b}\right )\right ) \text {erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )+\sqrt {\pi } (2 a+3 b) \left (\cosh \left (\frac {a}{b}\right )-\sinh \left (\frac {a}{b}\right )\right ) \text {erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )\right )}{8 c}+\frac {a e^{-\frac {a}{b}} \sqrt {a+b \sinh ^{-1}(c x)} \left (\frac {\Gamma \left (\frac {3}{2},-\frac {a+b \sinh ^{-1}(c x)}{b}\right )}{\sqrt {-\frac {a+b \sinh ^{-1}(c x)}{b}}}-\frac {e^{\frac {2 a}{b}} \Gamma \left (\frac {3}{2},\frac {a}{b}+\sinh ^{-1}(c x)\right )}{\sqrt {\frac {a}{b}+\sinh ^{-1}(c x)}}\right )}{2 c} \]
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 \[ \int {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.07, 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 \[ \int {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^{3/2} \,d x \]
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 \[ \int \left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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