Optimal. Leaf size=346 \[ \frac{\sqrt{\pi } e^{a/b} \text{Erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{15 b^{7/2} c^3}-\frac{3 \sqrt{3 \pi } e^{\frac{3 a}{b}} \text{Erf}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{5 b^{7/2} c^3}-\frac{\sqrt{\pi } e^{-\frac{a}{b}} \text{Erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{15 b^{7/2} c^3}+\frac{3 \sqrt{3 \pi } e^{-\frac{3 a}{b}} \text{Erfi}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{5 b^{7/2} c^3}-\frac{24 x^2 \sqrt{c^2 x^2+1}}{5 b^3 c \sqrt{a+b \sinh ^{-1}(c x)}}-\frac{16 \sqrt{c^2 x^2+1}}{15 b^3 c^3 \sqrt{a+b \sinh ^{-1}(c x)}}-\frac{8 x}{15 b^2 c^2 \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}-\frac{4 x^3}{5 b^2 \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}-\frac{2 x^2 \sqrt{c^2 x^2+1}}{5 b c \left (a+b \sinh ^{-1}(c x)\right )^{5/2}} \]
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Rubi [A] time = 1.03265, antiderivative size = 346, normalized size of antiderivative = 1., number of steps used = 22, number of rules used = 9, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.562, Rules used = {5667, 5774, 5665, 3308, 2180, 2204, 2205, 5655, 5779} \[ \frac{\sqrt{\pi } e^{a/b} \text{Erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{15 b^{7/2} c^3}-\frac{3 \sqrt{3 \pi } e^{\frac{3 a}{b}} \text{Erf}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{5 b^{7/2} c^3}-\frac{\sqrt{\pi } e^{-\frac{a}{b}} \text{Erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{15 b^{7/2} c^3}+\frac{3 \sqrt{3 \pi } e^{-\frac{3 a}{b}} \text{Erfi}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{5 b^{7/2} c^3}-\frac{24 x^2 \sqrt{c^2 x^2+1}}{5 b^3 c \sqrt{a+b \sinh ^{-1}(c x)}}-\frac{16 \sqrt{c^2 x^2+1}}{15 b^3 c^3 \sqrt{a+b \sinh ^{-1}(c x)}}-\frac{8 x}{15 b^2 c^2 \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}-\frac{4 x^3}{5 b^2 \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}-\frac{2 x^2 \sqrt{c^2 x^2+1}}{5 b c \left (a+b \sinh ^{-1}(c x)\right )^{5/2}} \]
Antiderivative was successfully verified.
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Rule 5667
Rule 5774
Rule 5665
Rule 3308
Rule 2180
Rule 2204
Rule 2205
Rule 5655
Rule 5779
Rubi steps
\begin{align*} \int \frac{x^2}{\left (a+b \sinh ^{-1}(c x)\right )^{7/2}} \, dx &=-\frac{2 x^2 \sqrt{1+c^2 x^2}}{5 b c \left (a+b \sinh ^{-1}(c x)\right )^{5/2}}+\frac{4 \int \frac{x}{\sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^{5/2}} \, dx}{5 b c}+\frac{(6 c) \int \frac{x^3}{\sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^{5/2}} \, dx}{5 b}\\ &=-\frac{2 x^2 \sqrt{1+c^2 x^2}}{5 b c \left (a+b \sinh ^{-1}(c x)\right )^{5/2}}-\frac{8 x}{15 b^2 c^2 \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}-\frac{4 x^3}{5 b^2 \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}+\frac{12 \int \frac{x^2}{\left (a+b \sinh ^{-1}(c x)\right )^{3/2}} \, dx}{5 b^2}+\frac{8 \int \frac{1}{\left (a+b \sinh ^{-1}(c x)\right )^{3/2}} \, dx}{15 b^2 c^2}\\ &=-\frac{2 x^2 \sqrt{1+c^2 x^2}}{5 b c \left (a+b \sinh ^{-1}(c x)\right )^{5/2}}-\frac{8 x}{15 b^2 c^2 \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}-\frac{4 x^3}{5 b^2 \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}-\frac{16 \sqrt{1+c^2 x^2}}{15 b^3 c^3 \sqrt{a+b \sinh ^{-1}(c x)}}-\frac{24 x^2 \sqrt{1+c^2 x^2}}{5 b^3 c \sqrt{a+b \sinh ^{-1}(c x)}}+\frac{24 \operatorname{Subst}\left (\int \left (-\frac{\sinh (x)}{4 \sqrt{a+b x}}+\frac{3 \sinh (3 x)}{4 \sqrt{a+b x}}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{5 b^3 c^3}+\frac{16 \int \frac{x}{\sqrt{1+c^2 x^2} \sqrt{a+b \sinh ^{-1}(c x)}} \, dx}{15 b^3 c}\\ &=-\frac{2 x^2 \sqrt{1+c^2 x^2}}{5 b c \left (a+b \sinh ^{-1}(c x)\right )^{5/2}}-\frac{8 x}{15 b^2 c^2 \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}-\frac{4 x^3}{5 b^2 \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}-\frac{16 \sqrt{1+c^2 x^2}}{15 b^3 c^3 \sqrt{a+b \sinh ^{-1}(c x)}}-\frac{24 x^2 \sqrt{1+c^2 x^2}}{5 b^3 c \sqrt{a+b \sinh ^{-1}(c x)}}+\frac{16 \operatorname{Subst}\left (\int \frac{\sinh (x)}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{15 b^3 c^3}-\frac{6 \operatorname{Subst}\left (\int \frac{\sinh (x)}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{5 b^3 c^3}+\frac{18 \operatorname{Subst}\left (\int \frac{\sinh (3 x)}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{5 b^3 c^3}\\ &=-\frac{2 x^2 \sqrt{1+c^2 x^2}}{5 b c \left (a+b \sinh ^{-1}(c x)\right )^{5/2}}-\frac{8 x}{15 b^2 c^2 \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}-\frac{4 x^3}{5 b^2 \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}-\frac{16 \sqrt{1+c^2 x^2}}{15 b^3 c^3 \sqrt{a+b \sinh ^{-1}(c x)}}-\frac{24 x^2 \sqrt{1+c^2 x^2}}{5 b^3 c \sqrt{a+b \sinh ^{-1}(c x)}}-\frac{8 \operatorname{Subst}\left (\int \frac{e^{-x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{15 b^3 c^3}+\frac{8 \operatorname{Subst}\left (\int \frac{e^x}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{15 b^3 c^3}+\frac{3 \operatorname{Subst}\left (\int \frac{e^{-x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{5 b^3 c^3}-\frac{3 \operatorname{Subst}\left (\int \frac{e^x}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{5 b^3 c^3}-\frac{9 \operatorname{Subst}\left (\int \frac{e^{-3 x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{5 b^3 c^3}+\frac{9 \operatorname{Subst}\left (\int \frac{e^{3 x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{5 b^3 c^3}\\ &=-\frac{2 x^2 \sqrt{1+c^2 x^2}}{5 b c \left (a+b \sinh ^{-1}(c x)\right )^{5/2}}-\frac{8 x}{15 b^2 c^2 \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}-\frac{4 x^3}{5 b^2 \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}-\frac{16 \sqrt{1+c^2 x^2}}{15 b^3 c^3 \sqrt{a+b \sinh ^{-1}(c x)}}-\frac{24 x^2 \sqrt{1+c^2 x^2}}{5 b^3 c \sqrt{a+b \sinh ^{-1}(c x)}}-\frac{16 \operatorname{Subst}\left (\int e^{\frac{a}{b}-\frac{x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c x)}\right )}{15 b^4 c^3}+\frac{16 \operatorname{Subst}\left (\int e^{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c x)}\right )}{15 b^4 c^3}+\frac{6 \operatorname{Subst}\left (\int e^{\frac{a}{b}-\frac{x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c x)}\right )}{5 b^4 c^3}-\frac{6 \operatorname{Subst}\left (\int e^{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c x)}\right )}{5 b^4 c^3}-\frac{18 \operatorname{Subst}\left (\int e^{\frac{3 a}{b}-\frac{3 x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c x)}\right )}{5 b^4 c^3}+\frac{18 \operatorname{Subst}\left (\int e^{-\frac{3 a}{b}+\frac{3 x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c x)}\right )}{5 b^4 c^3}\\ &=-\frac{2 x^2 \sqrt{1+c^2 x^2}}{5 b c \left (a+b \sinh ^{-1}(c x)\right )^{5/2}}-\frac{8 x}{15 b^2 c^2 \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}-\frac{4 x^3}{5 b^2 \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}-\frac{16 \sqrt{1+c^2 x^2}}{15 b^3 c^3 \sqrt{a+b \sinh ^{-1}(c x)}}-\frac{24 x^2 \sqrt{1+c^2 x^2}}{5 b^3 c \sqrt{a+b \sinh ^{-1}(c x)}}+\frac{e^{a/b} \sqrt{\pi } \text{erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{15 b^{7/2} c^3}-\frac{3 e^{\frac{3 a}{b}} \sqrt{3 \pi } \text{erf}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{5 b^{7/2} c^3}-\frac{e^{-\frac{a}{b}} \sqrt{\pi } \text{erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{15 b^{7/2} c^3}+\frac{3 e^{-\frac{3 a}{b}} \sqrt{3 \pi } \text{erfi}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{5 b^{7/2} c^3}\\ \end{align*}
Mathematica [A] time = 1.56026, size = 417, normalized size = 1.21 \[ \frac{e^{-\sinh ^{-1}(c x)} \left (-4 e^{\frac{a}{b}+\sinh ^{-1}(c x)} \sqrt{\frac{a}{b}+\sinh ^{-1}(c x)} \left (a+b \sinh ^{-1}(c x)\right )^2 \text{Gamma}\left (\frac{1}{2},\frac{a}{b}+\sinh ^{-1}(c x)\right )+4 a^2+2 b (4 a-b) \sinh ^{-1}(c x)-2 a b+4 b^2 \sinh ^{-1}(c x)^2+3 b^2\right )-3 \left (2 e^{-\frac{3 a}{b}} \left (a+b \sinh ^{-1}(c x)\right ) \left (6 \sqrt{3} b \left (-\frac{a+b \sinh ^{-1}(c x)}{b}\right )^{3/2} \text{Gamma}\left (\frac{1}{2},-\frac{3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )+e^{3 \left (\frac{a}{b}+\sinh ^{-1}(c x)\right )} \left (6 a+6 b \sinh ^{-1}(c x)+b\right )\right )+b^2 e^{3 \sinh ^{-1}(c x)}\right )-3 e^{-3 \sinh ^{-1}(c x)} \left (2 \left (a+b \sinh ^{-1}(c x)\right ) \left (-6 \sqrt{3} e^{3 \left (\frac{a}{b}+\sinh ^{-1}(c x)\right )} \sqrt{\frac{a}{b}+\sinh ^{-1}(c x)} \left (a+b \sinh ^{-1}(c x)\right ) \text{Gamma}\left (\frac{1}{2},\frac{3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )+6 a+6 b \sinh ^{-1}(c x)-b\right )+b^2\right )+2 e^{-\frac{a}{b}} \left (a+b \sinh ^{-1}(c x)\right ) \left (2 b \left (-\frac{a+b \sinh ^{-1}(c x)}{b}\right )^{3/2} \text{Gamma}\left (\frac{1}{2},-\frac{a+b \sinh ^{-1}(c x)}{b}\right )+e^{\frac{a}{b}+\sinh ^{-1}(c x)} \left (2 a+2 b \sinh ^{-1}(c x)+b\right )\right )+3 b^2 e^{\sinh ^{-1}(c x)}}{60 b^3 c^3 \left (a+b \sinh ^{-1}(c x)\right )^{5/2}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.109, size = 0, normalized size = 0. \begin{align*} \int{{x}^{2} \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^{2}}{{\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^{2}}{{\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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