Optimal. Leaf size=45 \[ \frac {1}{3} x^3 \left (a+b \tan ^{-1}(c x)\right )+\frac {b \log \left (c^2 x^2+1\right )}{6 c^3}-\frac {b x^2}{6 c} \]
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Rubi [A] time = 0.03, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4852, 266, 43} \[ \frac {1}{3} x^3 \left (a+b \tan ^{-1}(c x)\right )+\frac {b \log \left (c^2 x^2+1\right )}{6 c^3}-\frac {b x^2}{6 c} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rule 4852
Rubi steps
\begin {align*} \int x^2 \left (a+b \tan ^{-1}(c x)\right ) \, dx &=\frac {1}{3} x^3 \left (a+b \tan ^{-1}(c x)\right )-\frac {1}{3} (b c) \int \frac {x^3}{1+c^2 x^2} \, dx\\ &=\frac {1}{3} x^3 \left (a+b \tan ^{-1}(c x)\right )-\frac {1}{6} (b c) \operatorname {Subst}\left (\int \frac {x}{1+c^2 x} \, dx,x,x^2\right )\\ &=\frac {1}{3} x^3 \left (a+b \tan ^{-1}(c x)\right )-\frac {1}{6} (b c) \operatorname {Subst}\left (\int \left (\frac {1}{c^2}-\frac {1}{c^2 \left (1+c^2 x\right )}\right ) \, dx,x,x^2\right )\\ &=-\frac {b x^2}{6 c}+\frac {1}{3} x^3 \left (a+b \tan ^{-1}(c x)\right )+\frac {b \log \left (1+c^2 x^2\right )}{6 c^3}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 50, normalized size = 1.11 \[ \frac {a x^3}{3}+\frac {b \log \left (c^2 x^2+1\right )}{6 c^3}+\frac {1}{3} b x^3 \tan ^{-1}(c x)-\frac {b x^2}{6 c} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 49, normalized size = 1.09 \[ \frac {2 \, b c^{3} x^{3} \arctan \left (c x\right ) + 2 \, a c^{3} x^{3} - b c^{2} x^{2} + b \log \left (c^{2} x^{2} + 1\right )}{6 \, c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 43, normalized size = 0.96 \[ \frac {x^{3} a}{3}+\frac {b \,x^{3} \arctan \left (c x \right )}{3}-\frac {b \,x^{2}}{6 c}+\frac {b \ln \left (c^{2} x^{2}+1\right )}{6 c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 46, normalized size = 1.02 \[ \frac {1}{3} \, a x^{3} + \frac {1}{6} \, {\left (2 \, x^{3} \arctan \left (c x\right ) - c {\left (\frac {x^{2}}{c^{2}} - \frac {\log \left (c^{2} x^{2} + 1\right )}{c^{4}}\right )}\right )} b \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.37, size = 42, normalized size = 0.93 \[ \frac {a\,x^3}{3}+\frac {b\,x^3\,\mathrm {atan}\left (c\,x\right )}{3}+\frac {b\,\ln \left (c^2\,x^2+1\right )}{6\,c^3}-\frac {b\,x^2}{6\,c} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.58, size = 49, normalized size = 1.09 \[ \begin {cases} \frac {a x^{3}}{3} + \frac {b x^{3} \operatorname {atan}{\left (c x \right )}}{3} - \frac {b x^{2}}{6 c} + \frac {b \log {\left (x^{2} + \frac {1}{c^{2}} \right )}}{6 c^{3}} & \text {for}\: c \neq 0 \\\frac {a x^{3}}{3} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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