Optimal. Leaf size=64 \[ \frac{1}{3} x^3 \left (a+b \sin ^{-1}\left (\frac{c}{x}\right )\right )+\frac{1}{6} b c x^2 \sqrt{1-\frac{c^2}{x^2}}+\frac{1}{6} b c^3 \tanh ^{-1}\left (\sqrt{1-\frac{c^2}{x^2}}\right ) \]
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Rubi [A] time = 0.0426628, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {4842, 12, 266, 51, 63, 208} \[ \frac{1}{3} x^3 \left (a+b \sin ^{-1}\left (\frac{c}{x}\right )\right )+\frac{1}{6} b c x^2 \sqrt{1-\frac{c^2}{x^2}}+\frac{1}{6} b c^3 \tanh ^{-1}\left (\sqrt{1-\frac{c^2}{x^2}}\right ) \]
Antiderivative was successfully verified.
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Rule 4842
Rule 12
Rule 266
Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int x^2 \left (a+b \sin ^{-1}\left (\frac{c}{x}\right )\right ) \, dx &=\frac{1}{3} x^3 \left (a+b \sin ^{-1}\left (\frac{c}{x}\right )\right )+\frac{1}{3} b \int \frac{c x}{\sqrt{1-\frac{c^2}{x^2}}} \, dx\\ &=\frac{1}{3} x^3 \left (a+b \sin ^{-1}\left (\frac{c}{x}\right )\right )+\frac{1}{3} (b c) \int \frac{x}{\sqrt{1-\frac{c^2}{x^2}}} \, dx\\ &=\frac{1}{3} x^3 \left (a+b \sin ^{-1}\left (\frac{c}{x}\right )\right )-\frac{1}{6} (b c) \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{1-c^2 x}} \, dx,x,\frac{1}{x^2}\right )\\ &=\frac{1}{6} b c \sqrt{1-\frac{c^2}{x^2}} x^2+\frac{1}{3} x^3 \left (a+b \sin ^{-1}\left (\frac{c}{x}\right )\right )-\frac{1}{12} \left (b c^3\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-c^2 x}} \, dx,x,\frac{1}{x^2}\right )\\ &=\frac{1}{6} b c \sqrt{1-\frac{c^2}{x^2}} x^2+\frac{1}{3} x^3 \left (a+b \sin ^{-1}\left (\frac{c}{x}\right )\right )+\frac{1}{6} (b c) \operatorname{Subst}\left (\int \frac{1}{\frac{1}{c^2}-\frac{x^2}{c^2}} \, dx,x,\sqrt{1-\frac{c^2}{x^2}}\right )\\ &=\frac{1}{6} b c \sqrt{1-\frac{c^2}{x^2}} x^2+\frac{1}{3} x^3 \left (a+b \sin ^{-1}\left (\frac{c}{x}\right )\right )+\frac{1}{6} b c^3 \tanh ^{-1}\left (\sqrt{1-\frac{c^2}{x^2}}\right )\\ \end{align*}
Mathematica [A] time = 0.0335098, size = 79, normalized size = 1.23 \[ \frac{a x^3}{3}+\frac{1}{6} b c x^2 \sqrt{\frac{x^2-c^2}{x^2}}+\frac{1}{6} b c^3 \log \left (x \left (\sqrt{\frac{x^2-c^2}{x^2}}+1\right )\right )+\frac{1}{3} b x^3 \sin ^{-1}\left (\frac{c}{x}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 68, normalized size = 1.1 \begin{align*} -{c}^{3} \left ( -{\frac{{x}^{3}a}{3\,{c}^{3}}}+b \left ( -{\frac{{x}^{3}}{3\,{c}^{3}}\arcsin \left ({\frac{c}{x}} \right ) }-{\frac{{x}^{2}}{6\,{c}^{2}}\sqrt{1-{\frac{{c}^{2}}{{x}^{2}}}}}-{\frac{1}{6}{\it Artanh} \left ({\frac{1}{\sqrt{1-{\frac{{c}^{2}}{{x}^{2}}}}}} \right ) } \right ) \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.43071, size = 109, normalized size = 1.7 \begin{align*} \frac{1}{3} \, a x^{3} + \frac{1}{12} \,{\left (4 \, x^{3} \arcsin \left (\frac{c}{x}\right ) +{\left (c^{2} \log \left (\sqrt{-\frac{c^{2}}{x^{2}} + 1} + 1\right ) - c^{2} \log \left (\sqrt{-\frac{c^{2}}{x^{2}} + 1} - 1\right ) + 2 \, x^{2} \sqrt{-\frac{c^{2}}{x^{2}} + 1}\right )} c\right )} b \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.50002, size = 235, normalized size = 3.67 \begin{align*} -\frac{1}{6} \, b c^{3} \log \left (x \sqrt{-\frac{c^{2} - x^{2}}{x^{2}}} - x\right ) + \frac{1}{6} \, b c x^{2} \sqrt{-\frac{c^{2} - x^{2}}{x^{2}}} + \frac{1}{3} \, a x^{3} + \frac{1}{3} \,{\left (b x^{3} - b\right )} \arcsin \left (\frac{c}{x}\right ) - \frac{2}{3} \, b \arctan \left (\frac{x \sqrt{-\frac{c^{2} - x^{2}}{x^{2}}} - x}{c}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.79475, size = 109, normalized size = 1.7 \begin{align*} \frac{a x^{3}}{3} + \frac{b c \left (\begin{cases} \frac{c^{2} \operatorname{acosh}{\left (\frac{x}{c} \right )}}{2} + \frac{c x \sqrt{-1 + \frac{x^{2}}{c^{2}}}}{2} & \text{for}\: \frac{\left |{x^{2}}\right |}{\left |{c^{2}}\right |} > 1 \\- \frac{i c^{2} \operatorname{asin}{\left (\frac{x}{c} \right )}}{2} + \frac{i c x}{2 \sqrt{1 - \frac{x^{2}}{c^{2}}}} - \frac{i x^{3}}{2 c \sqrt{1 - \frac{x^{2}}{c^{2}}}} & \text{otherwise} \end{cases}\right )}{3} + \frac{b x^{3} \operatorname{asin}{\left (\frac{c}{x} \right )}}{3} \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{\left (b \arcsin \left (\frac{c}{x}\right ) + a\right )} x^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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