Optimal. Leaf size=117 \[ \frac{1}{5} x^5 \left (a+b \tan ^{-1}\left (c x^3\right )\right )+\frac{b \log \left (c^{2/3} x^2+1\right )}{10 c^{5/3}}-\frac{b \log \left (c^{4/3} x^4-c^{2/3} x^2+1\right )}{20 c^{5/3}}-\frac{\sqrt{3} b \tan ^{-1}\left (\frac{1-2 c^{2/3} x^2}{\sqrt{3}}\right )}{10 c^{5/3}}-\frac{3 b x^2}{10 c} \]
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Rubi [A] time = 0.0955969, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.643, Rules used = {5033, 275, 321, 200, 31, 634, 617, 204, 628} \[ \frac{1}{5} x^5 \left (a+b \tan ^{-1}\left (c x^3\right )\right )+\frac{b \log \left (c^{2/3} x^2+1\right )}{10 c^{5/3}}-\frac{b \log \left (c^{4/3} x^4-c^{2/3} x^2+1\right )}{20 c^{5/3}}-\frac{\sqrt{3} b \tan ^{-1}\left (\frac{1-2 c^{2/3} x^2}{\sqrt{3}}\right )}{10 c^{5/3}}-\frac{3 b x^2}{10 c} \]
Antiderivative was successfully verified.
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Rule 5033
Rule 275
Rule 321
Rule 200
Rule 31
Rule 634
Rule 617
Rule 204
Rule 628
Rubi steps
\begin{align*} \int x^4 \left (a+b \tan ^{-1}\left (c x^3\right )\right ) \, dx &=\frac{1}{5} x^5 \left (a+b \tan ^{-1}\left (c x^3\right )\right )-\frac{1}{5} (3 b c) \int \frac{x^7}{1+c^2 x^6} \, dx\\ &=\frac{1}{5} x^5 \left (a+b \tan ^{-1}\left (c x^3\right )\right )-\frac{1}{10} (3 b c) \operatorname{Subst}\left (\int \frac{x^3}{1+c^2 x^3} \, dx,x,x^2\right )\\ &=-\frac{3 b x^2}{10 c}+\frac{1}{5} x^5 \left (a+b \tan ^{-1}\left (c x^3\right )\right )+\frac{(3 b) \operatorname{Subst}\left (\int \frac{1}{1+c^2 x^3} \, dx,x,x^2\right )}{10 c}\\ &=-\frac{3 b x^2}{10 c}+\frac{1}{5} x^5 \left (a+b \tan ^{-1}\left (c x^3\right )\right )+\frac{b \operatorname{Subst}\left (\int \frac{1}{1+c^{2/3} x} \, dx,x,x^2\right )}{10 c}+\frac{b \operatorname{Subst}\left (\int \frac{2-c^{2/3} x}{1-c^{2/3} x+c^{4/3} x^2} \, dx,x,x^2\right )}{10 c}\\ &=-\frac{3 b x^2}{10 c}+\frac{1}{5} x^5 \left (a+b \tan ^{-1}\left (c x^3\right )\right )+\frac{b \log \left (1+c^{2/3} x^2\right )}{10 c^{5/3}}-\frac{b \operatorname{Subst}\left (\int \frac{-c^{2/3}+2 c^{4/3} x}{1-c^{2/3} x+c^{4/3} x^2} \, dx,x,x^2\right )}{20 c^{5/3}}+\frac{(3 b) \operatorname{Subst}\left (\int \frac{1}{1-c^{2/3} x+c^{4/3} x^2} \, dx,x,x^2\right )}{20 c}\\ &=-\frac{3 b x^2}{10 c}+\frac{1}{5} x^5 \left (a+b \tan ^{-1}\left (c x^3\right )\right )+\frac{b \log \left (1+c^{2/3} x^2\right )}{10 c^{5/3}}-\frac{b \log \left (1-c^{2/3} x^2+c^{4/3} x^4\right )}{20 c^{5/3}}+\frac{(3 b) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-2 c^{2/3} x^2\right )}{10 c^{5/3}}\\ &=-\frac{3 b x^2}{10 c}+\frac{1}{5} x^5 \left (a+b \tan ^{-1}\left (c x^3\right )\right )-\frac{\sqrt{3} b \tan ^{-1}\left (\frac{1-2 c^{2/3} x^2}{\sqrt{3}}\right )}{10 c^{5/3}}+\frac{b \log \left (1+c^{2/3} x^2\right )}{10 c^{5/3}}-\frac{b \log \left (1-c^{2/3} x^2+c^{4/3} x^4\right )}{20 c^{5/3}}\\ \end{align*}
Mathematica [A] time = 0.0266782, size = 185, normalized size = 1.58 \[ \frac{a x^5}{5}+\frac{b \log \left (c^{2/3} x^2+1\right )}{10 c^{5/3}}-\frac{b \log \left (c^{2/3} x^2-\sqrt{3} \sqrt [3]{c} x+1\right )}{20 c^{5/3}}-\frac{b \log \left (c^{2/3} x^2+\sqrt{3} \sqrt [3]{c} x+1\right )}{20 c^{5/3}}-\frac{\sqrt{3} b \tan ^{-1}\left (\sqrt{3}-2 \sqrt [3]{c} x\right )}{10 c^{5/3}}-\frac{\sqrt{3} b \tan ^{-1}\left (2 \sqrt [3]{c} x+\sqrt{3}\right )}{10 c^{5/3}}-\frac{3 b x^2}{10 c}+\frac{1}{5} b x^5 \tan ^{-1}\left (c x^3\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.028, size = 113, normalized size = 1. \begin{align*}{\frac{a{x}^{5}}{5}}+{\frac{b{x}^{5}\arctan \left ( c{x}^{3} \right ) }{5}}-{\frac{3\,b{x}^{2}}{10\,c}}+{\frac{b}{10\,{c}^{3}}\ln \left ({x}^{2}+\sqrt [3]{{c}^{-2}} \right ) \left ({c}^{-2} \right ) ^{-{\frac{2}{3}}}}-{\frac{b}{20\,{c}^{3}}\ln \left ({x}^{4}-\sqrt [3]{{c}^{-2}}{x}^{2}+ \left ({c}^{-2} \right ) ^{{\frac{2}{3}}} \right ) \left ({c}^{-2} \right ) ^{-{\frac{2}{3}}}}+{\frac{b\sqrt{3}}{10\,{c}^{3}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{\frac{{x}^{2}}{\sqrt [3]{{c}^{-2}}}}-1 \right ) } \right ) \left ({c}^{-2} \right ) ^{-{\frac{2}{3}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.52126, size = 162, normalized size = 1.38 \begin{align*} \frac{1}{5} \, a x^{5} + \frac{1}{20} \,{\left (4 \, x^{5} \arctan \left (c x^{3}\right ) - c{\left (\frac{6 \, x^{2}}{c^{2}} - \frac{2 \, \sqrt{3}{\left (c^{2}\right )}^{\frac{2}{3}} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (c^{2}\right )}^{\frac{1}{3}}{\left (2 \, x^{2} - \frac{1}{c^{2}}^{\frac{1}{3}}\right )}\right )}{c^{4}} + \frac{{\left (c^{2}\right )}^{\frac{2}{3}} \log \left (x^{4} - \frac{1}{c^{2}}^{\frac{1}{3}} x^{2} + \frac{1}{c^{2}}^{\frac{2}{3}}\right )}{c^{4}} - \frac{2 \,{\left (c^{2}\right )}^{\frac{2}{3}} \log \left (x^{2} + \frac{1}{c^{2}}^{\frac{1}{3}}\right )}{c^{4}}\right )}\right )} b \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 3.2461, size = 362, normalized size = 3.09 \begin{align*} \frac{4 \, b c^{3} x^{5} \arctan \left (c x^{3}\right ) + 4 \, a c^{3} x^{5} - 6 \, b c^{2} x^{2} + 2 \, \sqrt{3} b{\left (c^{2}\right )}^{\frac{1}{6}} c \arctan \left (\frac{\sqrt{3}{\left (2 \,{\left (c^{2}\right )}^{\frac{2}{3}} x^{2} -{\left (c^{2}\right )}^{\frac{1}{3}}\right )}{\left (c^{2}\right )}^{\frac{1}{6}}}{3 \, c}\right ) - b{\left (c^{2}\right )}^{\frac{2}{3}} \log \left (c^{2} x^{4} -{\left (c^{2}\right )}^{\frac{2}{3}} x^{2} +{\left (c^{2}\right )}^{\frac{1}{3}}\right ) + 2 \, b{\left (c^{2}\right )}^{\frac{2}{3}} \log \left (c^{2} x^{2} +{\left (c^{2}\right )}^{\frac{2}{3}}\right )}{20 \, c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 160.661, size = 1640, normalized size = 14.02 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.2201, size = 161, normalized size = 1.38 \begin{align*} \frac{1}{20} \, b c^{9}{\left (\frac{2 \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{2} - \frac{1}{{\left | c \right |}^{\frac{2}{3}}}\right )}{\left | c \right |}^{\frac{2}{3}}\right )}{c^{10}{\left | c \right |}^{\frac{2}{3}}} - \frac{\log \left (x^{4} - \frac{x^{2}}{{\left | c \right |}^{\frac{2}{3}}} + \frac{1}{{\left | c \right |}^{\frac{4}{3}}}\right )}{c^{10}{\left | c \right |}^{\frac{2}{3}}} + \frac{2 \, \log \left (x^{2} + \frac{1}{{\left | c \right |}^{\frac{2}{3}}}\right )}{c^{10}{\left | c \right |}^{\frac{2}{3}}}\right )} + \frac{2 \, b c x^{5} \arctan \left (c x^{3}\right ) + 2 \, a c x^{5} - 3 \, b x^{2}}{10 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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