Optimal. Leaf size=219 \[ \frac{8 \sqrt{2 \pi } e^{\frac{2 a}{b}} \text{Erf}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{15 b^{7/2} c^2}+\frac{8 \sqrt{2 \pi } e^{-\frac{2 a}{b}} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{15 b^{7/2} c^2}-\frac{32 x \sqrt{c^2 x^2+1}}{15 b^3 c \sqrt{a+b \sinh ^{-1}(c x)}}-\frac{4}{15 b^2 c^2 \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}-\frac{8 x^2}{15 b^2 \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}-\frac{2 x \sqrt{c^2 x^2+1}}{5 b c \left (a+b \sinh ^{-1}(c x)\right )^{5/2}} \]
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Rubi [A] time = 0.505574, antiderivative size = 219, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.571, Rules used = {5667, 5774, 5665, 3307, 2180, 2204, 2205, 5675} \[ \frac{8 \sqrt{2 \pi } e^{\frac{2 a}{b}} \text{Erf}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{15 b^{7/2} c^2}+\frac{8 \sqrt{2 \pi } e^{-\frac{2 a}{b}} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{15 b^{7/2} c^2}-\frac{32 x \sqrt{c^2 x^2+1}}{15 b^3 c \sqrt{a+b \sinh ^{-1}(c x)}}-\frac{4}{15 b^2 c^2 \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}-\frac{8 x^2}{15 b^2 \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}-\frac{2 x \sqrt{c^2 x^2+1}}{5 b c \left (a+b \sinh ^{-1}(c x)\right )^{5/2}} \]
Antiderivative was successfully verified.
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Rule 5667
Rule 5774
Rule 5665
Rule 3307
Rule 2180
Rule 2204
Rule 2205
Rule 5675
Rubi steps
\begin{align*} \int \frac{x}{\left (a+b \sinh ^{-1}(c x)\right )^{7/2}} \, dx &=-\frac{2 x \sqrt{1+c^2 x^2}}{5 b c \left (a+b \sinh ^{-1}(c x)\right )^{5/2}}+\frac{2 \int \frac{1}{\sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^{5/2}} \, dx}{5 b c}+\frac{(4 c) \int \frac{x^2}{\sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^{5/2}} \, dx}{5 b}\\ &=-\frac{2 x \sqrt{1+c^2 x^2}}{5 b c \left (a+b \sinh ^{-1}(c x)\right )^{5/2}}-\frac{4}{15 b^2 c^2 \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}-\frac{8 x^2}{15 b^2 \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}+\frac{16 \int \frac{x}{\left (a+b \sinh ^{-1}(c x)\right )^{3/2}} \, dx}{15 b^2}\\ &=-\frac{2 x \sqrt{1+c^2 x^2}}{5 b c \left (a+b \sinh ^{-1}(c x)\right )^{5/2}}-\frac{4}{15 b^2 c^2 \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}-\frac{8 x^2}{15 b^2 \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}-\frac{32 x \sqrt{1+c^2 x^2}}{15 b^3 c \sqrt{a+b \sinh ^{-1}(c x)}}+\frac{32 \operatorname{Subst}\left (\int \frac{\cosh (2 x)}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{15 b^3 c^2}\\ &=-\frac{2 x \sqrt{1+c^2 x^2}}{5 b c \left (a+b \sinh ^{-1}(c x)\right )^{5/2}}-\frac{4}{15 b^2 c^2 \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}-\frac{8 x^2}{15 b^2 \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}-\frac{32 x \sqrt{1+c^2 x^2}}{15 b^3 c \sqrt{a+b \sinh ^{-1}(c x)}}+\frac{16 \operatorname{Subst}\left (\int \frac{e^{-2 x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{15 b^3 c^2}+\frac{16 \operatorname{Subst}\left (\int \frac{e^{2 x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{15 b^3 c^2}\\ &=-\frac{2 x \sqrt{1+c^2 x^2}}{5 b c \left (a+b \sinh ^{-1}(c x)\right )^{5/2}}-\frac{4}{15 b^2 c^2 \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}-\frac{8 x^2}{15 b^2 \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}-\frac{32 x \sqrt{1+c^2 x^2}}{15 b^3 c \sqrt{a+b \sinh ^{-1}(c x)}}+\frac{32 \operatorname{Subst}\left (\int e^{\frac{2 a}{b}-\frac{2 x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c x)}\right )}{15 b^4 c^2}+\frac{32 \operatorname{Subst}\left (\int e^{-\frac{2 a}{b}+\frac{2 x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c x)}\right )}{15 b^4 c^2}\\ &=-\frac{2 x \sqrt{1+c^2 x^2}}{5 b c \left (a+b \sinh ^{-1}(c x)\right )^{5/2}}-\frac{4}{15 b^2 c^2 \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}-\frac{8 x^2}{15 b^2 \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}-\frac{32 x \sqrt{1+c^2 x^2}}{15 b^3 c \sqrt{a+b \sinh ^{-1}(c x)}}+\frac{8 e^{\frac{2 a}{b}} \sqrt{2 \pi } \text{erf}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{15 b^{7/2} c^2}+\frac{8 e^{-\frac{2 a}{b}} \sqrt{2 \pi } \text{erfi}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{15 b^{7/2} c^2}\\ \end{align*}
Mathematica [A] time = 1.13849, size = 208, normalized size = 0.95 \[ -\frac{\left (a+b \sinh ^{-1}(c x)\right ) \left (e^{-\frac{2 a}{b}} \left (8 \sqrt{2} b \left (-\frac{a+b \sinh ^{-1}(c x)}{b}\right )^{3/2} \text{Gamma}\left (\frac{1}{2},-\frac{2 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )+2 e^{2 \left (\frac{a}{b}+\sinh ^{-1}(c x)\right )} \left (4 a+4 b \sinh ^{-1}(c x)+b\right )\right )+e^{-2 \sinh ^{-1}(c x)} \left (8 \sqrt{2} e^{2 \left (\frac{a}{b}+\sinh ^{-1}(c x)\right )} \sqrt{\frac{a}{b}+\sinh ^{-1}(c x)} \left (a+b \sinh ^{-1}(c x)\right ) \text{Gamma}\left (\frac{1}{2},\frac{2 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )-8 a-8 b \sinh ^{-1}(c x)+2 b\right )\right )+3 b^2 \sinh \left (2 \sinh ^{-1}(c x)\right )}{15 b^3 c^2 \left (a+b \sinh ^{-1}(c x)\right )^{5/2}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.053, size = 0, normalized size = 0. \begin{align*} \int{x \left ( a+b{\it Arcsinh} \left ( cx \right ) \right ) ^{-{\frac{7}{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{x}{{\left (b \operatorname{arsinh}\left (c x\right ) + a\right )}^{\frac{7}{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(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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{x}{{\left (b \operatorname{arsinh}\left (c x\right ) + a\right )}^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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