Optimal. Leaf size=183 \[ -\frac {2 x \sqrt {1+c^2 x^2}}{3 b c \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}-\frac {4}{3 b^2 c^2 \sqrt {a+b \sinh ^{-1}(c x)}}-\frac {8 x^2}{3 b^2 \sqrt {a+b \sinh ^{-1}(c x)}}-\frac {2 e^{\frac {2 a}{b}} \sqrt {2 \pi } \text {Erf}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{3 b^{5/2} c^2}+\frac {2 e^{-\frac {2 a}{b}} \sqrt {2 \pi } \text {Erfi}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{3 b^{5/2} c^2} \]
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Rubi [A]
time = 0.34, antiderivative size = 183, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 10, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {5779, 5818,
5780, 5556, 12, 3389, 2211, 2236, 2235, 5783} \begin {gather*} -\frac {2 \sqrt {2 \pi } e^{\frac {2 a}{b}} \text {Erf}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{3 b^{5/2} c^2}+\frac {2 \sqrt {2 \pi } e^{-\frac {2 a}{b}} \text {Erfi}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{3 b^{5/2} c^2}-\frac {4}{3 b^2 c^2 \sqrt {a+b \sinh ^{-1}(c x)}}-\frac {8 x^2}{3 b^2 \sqrt {a+b \sinh ^{-1}(c x)}}-\frac {2 x \sqrt {c^2 x^2+1}}{3 b c \left (a+b \sinh ^{-1}(c x)\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2211
Rule 2235
Rule 2236
Rule 3389
Rule 5556
Rule 5779
Rule 5780
Rule 5783
Rule 5818
Rubi steps
\begin {align*} \int \frac {x}{\left (a+b \sinh ^{-1}(c x)\right )^{5/2}} \, dx &=-\frac {2 x \sqrt {1+c^2 x^2}}{3 b c \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}+\frac {2 \int \frac {1}{\sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^{3/2}} \, dx}{3 b c}+\frac {(4 c) \int \frac {x^2}{\sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^{3/2}} \, dx}{3 b}\\ &=-\frac {2 x \sqrt {1+c^2 x^2}}{3 b c \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}-\frac {4}{3 b^2 c^2 \sqrt {a+b \sinh ^{-1}(c x)}}-\frac {8 x^2}{3 b^2 \sqrt {a+b \sinh ^{-1}(c x)}}+\frac {16 \int \frac {x}{\sqrt {a+b \sinh ^{-1}(c x)}} \, dx}{3 b^2}\\ &=-\frac {2 x \sqrt {1+c^2 x^2}}{3 b c \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}-\frac {4}{3 b^2 c^2 \sqrt {a+b \sinh ^{-1}(c x)}}-\frac {8 x^2}{3 b^2 \sqrt {a+b \sinh ^{-1}(c x)}}+\frac {16 \text {Subst}\left (\int \frac {\cosh (x) \sinh (x)}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{3 b^2 c^2}\\ &=-\frac {2 x \sqrt {1+c^2 x^2}}{3 b c \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}-\frac {4}{3 b^2 c^2 \sqrt {a+b \sinh ^{-1}(c x)}}-\frac {8 x^2}{3 b^2 \sqrt {a+b \sinh ^{-1}(c x)}}+\frac {16 \text {Subst}\left (\int \frac {\sinh (2 x)}{2 \sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{3 b^2 c^2}\\ &=-\frac {2 x \sqrt {1+c^2 x^2}}{3 b c \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}-\frac {4}{3 b^2 c^2 \sqrt {a+b \sinh ^{-1}(c x)}}-\frac {8 x^2}{3 b^2 \sqrt {a+b \sinh ^{-1}(c x)}}+\frac {8 \text {Subst}\left (\int \frac {\sinh (2 x)}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{3 b^2 c^2}\\ &=-\frac {2 x \sqrt {1+c^2 x^2}}{3 b c \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}-\frac {4}{3 b^2 c^2 \sqrt {a+b \sinh ^{-1}(c x)}}-\frac {8 x^2}{3 b^2 \sqrt {a+b \sinh ^{-1}(c x)}}-\frac {4 \text {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{3 b^2 c^2}+\frac {4 \text {Subst}\left (\int \frac {e^{2 x}}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{3 b^2 c^2}\\ &=-\frac {2 x \sqrt {1+c^2 x^2}}{3 b c \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}-\frac {4}{3 b^2 c^2 \sqrt {a+b \sinh ^{-1}(c x)}}-\frac {8 x^2}{3 b^2 \sqrt {a+b \sinh ^{-1}(c x)}}-\frac {8 \text {Subst}\left (\int e^{\frac {2 a}{b}-\frac {2 x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c x)}\right )}{3 b^3 c^2}+\frac {8 \text {Subst}\left (\int e^{-\frac {2 a}{b}+\frac {2 x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c x)}\right )}{3 b^3 c^2}\\ &=-\frac {2 x \sqrt {1+c^2 x^2}}{3 b c \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}-\frac {4}{3 b^2 c^2 \sqrt {a+b \sinh ^{-1}(c x)}}-\frac {8 x^2}{3 b^2 \sqrt {a+b \sinh ^{-1}(c x)}}-\frac {2 e^{\frac {2 a}{b}} \sqrt {2 \pi } \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{3 b^{5/2} c^2}+\frac {2 e^{-\frac {2 a}{b}} \sqrt {2 \pi } \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{3 b^{5/2} c^2}\\ \end {align*}
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Mathematica [A]
time = 0.49, size = 200, normalized size = 1.09 \begin {gather*} \frac {e^{-2 \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )} \left (-4 \sqrt {2} b e^{2 \sinh ^{-1}(c x)} \left (-\frac {a+b \sinh ^{-1}(c x)}{b}\right )^{3/2} \Gamma \left (\frac {1}{2},-\frac {2 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )+e^{\frac {2 a}{b}} \left (-4 a+b-4 a e^{4 \sinh ^{-1}(c x)}-b e^{4 \sinh ^{-1}(c x)}-4 b \left (1+e^{4 \sinh ^{-1}(c x)}\right ) \sinh ^{-1}(c x)+4 \sqrt {2} e^{2 \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )} \sqrt {\frac {a}{b}+\sinh ^{-1}(c x)} \left (a+b \sinh ^{-1}(c x)\right ) \Gamma \left (\frac {1}{2},\frac {2 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )\right )\right )}{6 b^2 c^2 \left (a+b \sinh ^{-1}(c x)\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {x}{\left (a +b \arcsinh \left (c x \right )\right )^{\frac {5}{2}}}\, dx\]
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 \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 \frac {x}{\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]
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 \frac {x}{{\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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