Optimal. Leaf size=89 \[ \frac{1}{5} x^5 \left (a+b \sin ^{-1}\left (\frac{c}{x}\right )\right )+\frac{3}{40} b c^3 x^2 \sqrt{1-\frac{c^2}{x^2}}+\frac{1}{20} b c x^4 \sqrt{1-\frac{c^2}{x^2}}+\frac{3}{40} b c^5 \tanh ^{-1}\left (\sqrt{1-\frac{c^2}{x^2}}\right ) \]
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Rubi [A] time = 0.0587458, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 7, 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} \[ \frac{1}{5} x^5 \left (a+b \sin ^{-1}\left (\frac{c}{x}\right )\right )+\frac{3}{40} b c^3 x^2 \sqrt{1-\frac{c^2}{x^2}}+\frac{1}{20} b c x^4 \sqrt{1-\frac{c^2}{x^2}}+\frac{3}{40} b c^5 \tanh ^{-1}\left (\sqrt{1-\frac{c^2}{x^2}}\right ) \]
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^4 \left (a+b \sin ^{-1}\left (\frac{c}{x}\right )\right ) \, dx &=\frac{1}{5} x^5 \left (a+b \sin ^{-1}\left (\frac{c}{x}\right )\right )+\frac{1}{5} b \int \frac{c x^3}{\sqrt{1-\frac{c^2}{x^2}}} \, dx\\ &=\frac{1}{5} x^5 \left (a+b \sin ^{-1}\left (\frac{c}{x}\right )\right )+\frac{1}{5} (b c) \int \frac{x^3}{\sqrt{1-\frac{c^2}{x^2}}} \, dx\\ &=\frac{1}{5} x^5 \left (a+b \sin ^{-1}\left (\frac{c}{x}\right )\right )-\frac{1}{10} (b c) \operatorname{Subst}\left (\int \frac{1}{x^3 \sqrt{1-c^2 x}} \, dx,x,\frac{1}{x^2}\right )\\ &=\frac{1}{20} b c \sqrt{1-\frac{c^2}{x^2}} x^4+\frac{1}{5} x^5 \left (a+b \sin ^{-1}\left (\frac{c}{x}\right )\right )-\frac{1}{40} \left (3 b c^3\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{1-c^2 x}} \, dx,x,\frac{1}{x^2}\right )\\ &=\frac{3}{40} b c^3 \sqrt{1-\frac{c^2}{x^2}} x^2+\frac{1}{20} b c \sqrt{1-\frac{c^2}{x^2}} x^4+\frac{1}{5} x^5 \left (a+b \sin ^{-1}\left (\frac{c}{x}\right )\right )-\frac{1}{80} \left (3 b c^5\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-c^2 x}} \, dx,x,\frac{1}{x^2}\right )\\ &=\frac{3}{40} b c^3 \sqrt{1-\frac{c^2}{x^2}} x^2+\frac{1}{20} b c \sqrt{1-\frac{c^2}{x^2}} x^4+\frac{1}{5} x^5 \left (a+b \sin ^{-1}\left (\frac{c}{x}\right )\right )+\frac{1}{40} \left (3 b c^3\right ) \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{3}{40} b c^3 \sqrt{1-\frac{c^2}{x^2}} x^2+\frac{1}{20} b c \sqrt{1-\frac{c^2}{x^2}} x^4+\frac{1}{5} x^5 \left (a+b \sin ^{-1}\left (\frac{c}{x}\right )\right )+\frac{3}{40} b c^5 \tanh ^{-1}\left (\sqrt{1-\frac{c^2}{x^2}}\right )\\ \end{align*}
Mathematica [A] time = 0.0711722, size = 91, normalized size = 1.02 \[ \frac{a x^5}{5}+b \sqrt{\frac{x^2-c^2}{x^2}} \left (\frac{3 c^3 x^2}{40}+\frac{c x^4}{20}\right )+\frac{3}{40} b c^5 \log \left (x \left (\sqrt{\frac{x^2-c^2}{x^2}}+1\right )\right )+\frac{1}{5} b x^5 \sin ^{-1}\left (\frac{c}{x}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.018, size = 88, normalized size = 1. \begin{align*} -{c}^{5} \left ( -{\frac{a{x}^{5}}{5\,{c}^{5}}}+b \left ( -{\frac{{x}^{5}}{5\,{c}^{5}}\arcsin \left ({\frac{c}{x}} \right ) }-{\frac{{x}^{4}}{20\,{c}^{4}}\sqrt{1-{\frac{{c}^{2}}{{x}^{2}}}}}-{\frac{3\,{x}^{2}}{40\,{c}^{2}}\sqrt{1-{\frac{{c}^{2}}{{x}^{2}}}}}-{\frac{3}{40}{\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 = 1.43042, size = 169, normalized size = 1.9 \begin{align*} \frac{1}{5} \, a x^{5} + \frac{1}{80} \,{\left (16 \, x^{5} \arcsin \left (\frac{c}{x}\right ) +{\left (3 \, c^{4} \log \left (\sqrt{-\frac{c^{2}}{x^{2}} + 1} + 1\right ) - 3 \, c^{4} \log \left (\sqrt{-\frac{c^{2}}{x^{2}} + 1} - 1\right ) - \frac{2 \,{\left (3 \, c^{4}{\left (-\frac{c^{2}}{x^{2}} + 1\right )}^{\frac{3}{2}} - 5 \, c^{4} \sqrt{-\frac{c^{2}}{x^{2}} + 1}\right )}}{{\left (\frac{c^{2}}{x^{2}} - 1\right )}^{2} + \frac{2 \, 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 = 2.59502, size = 262, normalized size = 2.94 \begin{align*} -\frac{3}{40} \, b c^{5} \log \left (x \sqrt{-\frac{c^{2} - x^{2}}{x^{2}}} - x\right ) + \frac{1}{5} \, a x^{5} + \frac{1}{5} \,{\left (b x^{5} - b\right )} \arcsin \left (\frac{c}{x}\right ) - \frac{2}{5} \, b \arctan \left (\frac{x \sqrt{-\frac{c^{2} - x^{2}}{x^{2}}} - x}{c}\right ) + \frac{1}{40} \,{\left (3 \, b c^{3} x^{2} + 2 \, b c x^{4}\right )} \sqrt{-\frac{c^{2} - x^{2}}{x^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 10.2723, size = 177, normalized size = 1.99 \begin{align*} \frac{a x^{5}}{5} + \frac{b c \left (\begin{cases} \frac{3 c^{4} \operatorname{acosh}{\left (\frac{x}{c} \right )}}{8} - \frac{3 c^{3} x}{8 \sqrt{-1 + \frac{x^{2}}{c^{2}}}} + \frac{c x^{3}}{8 \sqrt{-1 + \frac{x^{2}}{c^{2}}}} + \frac{x^{5}}{4 c \sqrt{-1 + \frac{x^{2}}{c^{2}}}} & \text{for}\: \frac{\left |{x^{2}}\right |}{\left |{c^{2}}\right |} > 1 \\- \frac{3 i c^{4} \operatorname{asin}{\left (\frac{x}{c} \right )}}{8} + \frac{3 i c^{3} x}{8 \sqrt{1 - \frac{x^{2}}{c^{2}}}} - \frac{i c x^{3}}{8 \sqrt{1 - \frac{x^{2}}{c^{2}}}} - \frac{i x^{5}}{4 c \sqrt{1 - \frac{x^{2}}{c^{2}}}} & \text{otherwise} \end{cases}\right )}{5} + \frac{b x^{5} \operatorname{asin}{\left (\frac{c}{x} \right )}}{5} \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^{4}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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