Optimal. Leaf size=143 \[ \frac{2 \sqrt{\pi } e^{a/b} \text{Erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{3 b^{5/2} c}+\frac{2 \sqrt{\pi } e^{-\frac{a}{b}} \text{Erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{3 b^{5/2} c}-\frac{4 x}{3 b^2 \sqrt{a+b \sinh ^{-1}(c x)}}-\frac{2 \sqrt{c^2 x^2+1}}{3 b c \left (a+b \sinh ^{-1}(c x)\right )^{3/2}} \]
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Rubi [A] time = 0.272854, antiderivative size = 143, normalized size of antiderivative = 1., 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 = {5655, 5774, 5657, 3307, 2180, 2205, 2204} \[ \frac{2 \sqrt{\pi } e^{a/b} \text{Erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{3 b^{5/2} c}+\frac{2 \sqrt{\pi } e^{-\frac{a}{b}} \text{Erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{3 b^{5/2} c}-\frac{4 x}{3 b^2 \sqrt{a+b \sinh ^{-1}(c x)}}-\frac{2 \sqrt{c^2 x^2+1}}{3 b c \left (a+b \sinh ^{-1}(c x)\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 5655
Rule 5774
Rule 5657
Rule 3307
Rule 2180
Rule 2205
Rule 2204
Rubi steps
\begin{align*} \int \frac{1}{\left (a+b \sinh ^{-1}(c x)\right )^{5/2}} \, dx &=-\frac{2 \sqrt{1+c^2 x^2}}{3 b c \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}+\frac{(2 c) \int \frac{x}{\sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^{3/2}} \, dx}{3 b}\\ &=-\frac{2 \sqrt{1+c^2 x^2}}{3 b c \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}-\frac{4 x}{3 b^2 \sqrt{a+b \sinh ^{-1}(c x)}}+\frac{4 \int \frac{1}{\sqrt{a+b \sinh ^{-1}(c x)}} \, dx}{3 b^2}\\ &=-\frac{2 \sqrt{1+c^2 x^2}}{3 b c \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}-\frac{4 x}{3 b^2 \sqrt{a+b \sinh ^{-1}(c x)}}+\frac{4 \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 )}{3 b^3 c}\\ &=-\frac{2 \sqrt{1+c^2 x^2}}{3 b c \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}-\frac{4 x}{3 b^2 \sqrt{a+b \sinh ^{-1}(c x)}}+\frac{2 \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 )}{3 b^3 c}+\frac{2 \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 )}{3 b^3 c}\\ &=-\frac{2 \sqrt{1+c^2 x^2}}{3 b c \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}-\frac{4 x}{3 b^2 \sqrt{a+b \sinh ^{-1}(c x)}}+\frac{4 \operatorname{Subst}\left (\int e^{\frac{a}{b}-\frac{x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c x)}\right )}{3 b^3 c}+\frac{4 \operatorname{Subst}\left (\int e^{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c x)}\right )}{3 b^3 c}\\ &=-\frac{2 \sqrt{1+c^2 x^2}}{3 b c \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}-\frac{4 x}{3 b^2 \sqrt{a+b \sinh ^{-1}(c x)}}+\frac{2 e^{a/b} \sqrt{\pi } \text{erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{3 b^{5/2} c}+\frac{2 e^{-\frac{a}{b}} \sqrt{\pi } \text{erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{3 b^{5/2} c}\\ \end{align*}
Mathematica [A] time = 0.577197, size = 181, normalized size = 1.27 \[ \frac{e^{-\frac{a+b \sinh ^{-1}(c x)}{b}} \left (-2 b e^{\sinh ^{-1}(c x)} \left (-\frac{a+b \sinh ^{-1}(c x)}{b}\right )^{3/2} \text{Gamma}\left (\frac{1}{2},-\frac{a+b \sinh ^{-1}(c x)}{b}\right )-2 e^{\frac{2 a}{b}+\sinh ^{-1}(c x)} \sqrt{\frac{a}{b}+\sinh ^{-1}(c x)} \left (a+b \sinh ^{-1}(c x)\right ) \text{Gamma}\left (\frac{1}{2},\frac{a}{b}+\sinh ^{-1}(c x)\right )-e^{a/b} \left (2 a \left (e^{2 \sinh ^{-1}(c x)}-1\right )-2 b \sinh ^{-1}(c x)+b e^{2 \sinh ^{-1}(c x)} \left (2 \sinh ^{-1}(c x)+1\right )+b\right )\right )}{3 b^2 c \left (a+b \sinh ^{-1}(c x)\right )^{3/2}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.04, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b{\it Arcsinh} \left ( cx \right ) \right ) ^{-{\frac{5}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b \operatorname{arsinh}\left (c x\right ) + a\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a + b \operatorname{asinh}{\left (c x \right )}\right )^{\frac{5}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b \operatorname{arsinh}\left (c x\right ) + a\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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