Optimal. Leaf size=223 \[ -\frac {15 \sqrt {\frac {\pi }{2}} b^{5/2} e^{\frac {2 a}{b}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{256 c^2}-\frac {15 \sqrt {\frac {\pi }{2}} b^{5/2} e^{-\frac {2 a}{b}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{256 c^2}+\frac {15 b^2 \sqrt {a+b \sinh ^{-1}(c x)}}{64 c^2}+\frac {15}{32} b^2 x^2 \sqrt {a+b \sinh ^{-1}(c x)}-\frac {5 b x \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}{8 c}+\frac {\left (a+b \sinh ^{-1}(c x)\right )^{5/2}}{4 c^2}+\frac {1}{2} x^2 \left (a+b \sinh ^{-1}(c x)\right )^{5/2} \]
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Rubi [A] time = 0.75, antiderivative size = 223, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.643, Rules used = {5663, 5758, 5675, 5779, 3312, 3307, 2180, 2204, 2205} \[ -\frac {15 \sqrt {\frac {\pi }{2}} b^{5/2} e^{\frac {2 a}{b}} \text {Erf}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{256 c^2}-\frac {15 \sqrt {\frac {\pi }{2}} b^{5/2} e^{-\frac {2 a}{b}} \text {Erfi}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{256 c^2}+\frac {15 b^2 \sqrt {a+b \sinh ^{-1}(c x)}}{64 c^2}+\frac {15}{32} b^2 x^2 \sqrt {a+b \sinh ^{-1}(c x)}-\frac {5 b x \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}{8 c}+\frac {\left (a+b \sinh ^{-1}(c x)\right )^{5/2}}{4 c^2}+\frac {1}{2} x^2 \left (a+b \sinh ^{-1}(c x)\right )^{5/2} \]
Antiderivative was successfully verified.
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Rule 2180
Rule 2204
Rule 2205
Rule 3307
Rule 3312
Rule 5663
Rule 5675
Rule 5758
Rule 5779
Rubi steps
\begin {align*} \int x \left (a+b \sinh ^{-1}(c x)\right )^{5/2} \, dx &=\frac {1}{2} x^2 \left (a+b \sinh ^{-1}(c x)\right )^{5/2}-\frac {1}{4} (5 b c) \int \frac {x^2 \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}{\sqrt {1+c^2 x^2}} \, dx\\ &=-\frac {5 b x \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}{8 c}+\frac {1}{2} x^2 \left (a+b \sinh ^{-1}(c x)\right )^{5/2}+\frac {1}{16} \left (15 b^2\right ) \int x \sqrt {a+b \sinh ^{-1}(c x)} \, dx+\frac {(5 b) \int \frac {\left (a+b \sinh ^{-1}(c x)\right )^{3/2}}{\sqrt {1+c^2 x^2}} \, dx}{8 c}\\ &=\frac {15}{32} b^2 x^2 \sqrt {a+b \sinh ^{-1}(c x)}-\frac {5 b x \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}{8 c}+\frac {\left (a+b \sinh ^{-1}(c x)\right )^{5/2}}{4 c^2}+\frac {1}{2} x^2 \left (a+b \sinh ^{-1}(c x)\right )^{5/2}-\frac {1}{64} \left (15 b^3 c\right ) \int \frac {x^2}{\sqrt {1+c^2 x^2} \sqrt {a+b \sinh ^{-1}(c x)}} \, dx\\ &=\frac {15}{32} b^2 x^2 \sqrt {a+b \sinh ^{-1}(c x)}-\frac {5 b x \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}{8 c}+\frac {\left (a+b \sinh ^{-1}(c x)\right )^{5/2}}{4 c^2}+\frac {1}{2} x^2 \left (a+b \sinh ^{-1}(c x)\right )^{5/2}-\frac {\left (15 b^3\right ) \operatorname {Subst}\left (\int \frac {\sinh ^2(x)}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{64 c^2}\\ &=\frac {15}{32} b^2 x^2 \sqrt {a+b \sinh ^{-1}(c x)}-\frac {5 b x \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}{8 c}+\frac {\left (a+b \sinh ^{-1}(c x)\right )^{5/2}}{4 c^2}+\frac {1}{2} x^2 \left (a+b \sinh ^{-1}(c x)\right )^{5/2}+\frac {\left (15 b^3\right ) \operatorname {Subst}\left (\int \left (\frac {1}{2 \sqrt {a+b x}}-\frac {\cosh (2 x)}{2 \sqrt {a+b x}}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{64 c^2}\\ &=\frac {15 b^2 \sqrt {a+b \sinh ^{-1}(c x)}}{64 c^2}+\frac {15}{32} b^2 x^2 \sqrt {a+b \sinh ^{-1}(c x)}-\frac {5 b x \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}{8 c}+\frac {\left (a+b \sinh ^{-1}(c x)\right )^{5/2}}{4 c^2}+\frac {1}{2} x^2 \left (a+b \sinh ^{-1}(c x)\right )^{5/2}-\frac {\left (15 b^3\right ) \operatorname {Subst}\left (\int \frac {\cosh (2 x)}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{128 c^2}\\ &=\frac {15 b^2 \sqrt {a+b \sinh ^{-1}(c x)}}{64 c^2}+\frac {15}{32} b^2 x^2 \sqrt {a+b \sinh ^{-1}(c x)}-\frac {5 b x \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}{8 c}+\frac {\left (a+b \sinh ^{-1}(c x)\right )^{5/2}}{4 c^2}+\frac {1}{2} x^2 \left (a+b \sinh ^{-1}(c x)\right )^{5/2}-\frac {\left (15 b^3\right ) \operatorname {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{256 c^2}-\frac {\left (15 b^3\right ) \operatorname {Subst}\left (\int \frac {e^{2 x}}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{256 c^2}\\ &=\frac {15 b^2 \sqrt {a+b \sinh ^{-1}(c x)}}{64 c^2}+\frac {15}{32} b^2 x^2 \sqrt {a+b \sinh ^{-1}(c x)}-\frac {5 b x \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}{8 c}+\frac {\left (a+b \sinh ^{-1}(c x)\right )^{5/2}}{4 c^2}+\frac {1}{2} x^2 \left (a+b \sinh ^{-1}(c x)\right )^{5/2}-\frac {\left (15 b^2\right ) \operatorname {Subst}\left (\int e^{\frac {2 a}{b}-\frac {2 x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c x)}\right )}{128 c^2}-\frac {\left (15 b^2\right ) \operatorname {Subst}\left (\int e^{-\frac {2 a}{b}+\frac {2 x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c x)}\right )}{128 c^2}\\ &=\frac {15 b^2 \sqrt {a+b \sinh ^{-1}(c x)}}{64 c^2}+\frac {15}{32} b^2 x^2 \sqrt {a+b \sinh ^{-1}(c x)}-\frac {5 b x \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}{8 c}+\frac {\left (a+b \sinh ^{-1}(c x)\right )^{5/2}}{4 c^2}+\frac {1}{2} x^2 \left (a+b \sinh ^{-1}(c x)\right )^{5/2}-\frac {15 b^{5/2} e^{\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{256 c^2}-\frac {15 b^{5/2} e^{-\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{256 c^2}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 115, normalized size = 0.52 \[ \frac {e^{-\frac {2 a}{b}} \left (b^3 e^{\frac {4 a}{b}} \sqrt {\frac {a}{b}+\sinh ^{-1}(c x)} \Gamma \left (\frac {7}{2},\frac {2 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )-b^3 \sqrt {-\frac {a+b \sinh ^{-1}(c x)}{b}} \Gamma \left (\frac {7}{2},-\frac {2 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )\right )}{32 \sqrt {2} c^2 \sqrt {a+b \sinh ^{-1}(c x)}} \]
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(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.38, size = 0, normalized size = 0.00 \[ \int x \left (a +b \arcsinh \left (c x \right )\right )^{\frac {5}{2}}\, dx \]
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 \[ \int {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{\frac {5}{2}} x\,{d x} \]
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 \[ \int x\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^{5/2} \,d x \]
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 \[ \int x \left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{\frac {5}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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