Optimal. Leaf size=48 \[ \frac{1}{4} x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac{b x}{4 c^3}-\frac{b \tan ^{-1}(c x)}{4 c^4}-\frac{b x^3}{12 c} \]
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Rubi [A] time = 0.0271138, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {4852, 302, 203} \[ \frac{1}{4} x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac{b x}{4 c^3}-\frac{b \tan ^{-1}(c x)}{4 c^4}-\frac{b x^3}{12 c} \]
Antiderivative was successfully verified.
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Rule 4852
Rule 302
Rule 203
Rubi steps
\begin{align*} \int x^3 \left (a+b \tan ^{-1}(c x)\right ) \, dx &=\frac{1}{4} x^4 \left (a+b \tan ^{-1}(c x)\right )-\frac{1}{4} (b c) \int \frac{x^4}{1+c^2 x^2} \, dx\\ &=\frac{1}{4} x^4 \left (a+b \tan ^{-1}(c x)\right )-\frac{1}{4} (b c) \int \left (-\frac{1}{c^4}+\frac{x^2}{c^2}+\frac{1}{c^4 \left (1+c^2 x^2\right )}\right ) \, dx\\ &=\frac{b x}{4 c^3}-\frac{b x^3}{12 c}+\frac{1}{4} x^4 \left (a+b \tan ^{-1}(c x)\right )-\frac{b \int \frac{1}{1+c^2 x^2} \, dx}{4 c^3}\\ &=\frac{b x}{4 c^3}-\frac{b x^3}{12 c}-\frac{b \tan ^{-1}(c x)}{4 c^4}+\frac{1}{4} x^4 \left (a+b \tan ^{-1}(c x)\right )\\ \end{align*}
Mathematica [A] time = 0.0026207, size = 53, normalized size = 1.1 \[ \frac{a x^4}{4}+\frac{b x}{4 c^3}-\frac{b \tan ^{-1}(c x)}{4 c^4}-\frac{b x^3}{12 c}+\frac{1}{4} b x^4 \tan ^{-1}(c x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 44, normalized size = 0.9 \begin{align*}{\frac{{x}^{4}a}{4}}+{\frac{{x}^{4}b\arctan \left ( cx \right ) }{4}}-{\frac{b{x}^{3}}{12\,c}}+{\frac{bx}{4\,{c}^{3}}}-{\frac{b\arctan \left ( cx \right ) }{4\,{c}^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.44273, size = 65, normalized size = 1.35 \begin{align*} \frac{1}{4} \, a x^{4} + \frac{1}{12} \,{\left (3 \, x^{4} \arctan \left (c x\right ) - c{\left (\frac{c^{2} x^{3} - 3 \, x}{c^{4}} + \frac{3 \, \arctan \left (c x\right )}{c^{5}}\right )}\right )} b \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.44174, size = 105, normalized size = 2.19 \begin{align*} \frac{3 \, a c^{4} x^{4} - b c^{3} x^{3} + 3 \, b c x + 3 \,{\left (b c^{4} x^{4} - b\right )} \arctan \left (c x\right )}{12 \, c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.10828, size = 53, normalized size = 1.1 \begin{align*} \begin{cases} \frac{a x^{4}}{4} + \frac{b x^{4} \operatorname{atan}{\left (c x \right )}}{4} - \frac{b x^{3}}{12 c} + \frac{b x}{4 c^{3}} - \frac{b \operatorname{atan}{\left (c x \right )}}{4 c^{4}} & \text{for}\: c \neq 0 \\\frac{a x^{4}}{4} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.35318, size = 66, normalized size = 1.38 \begin{align*} \frac{3 \, b c^{4} x^{4} \arctan \left (c x\right ) + 3 \, a c^{4} x^{4} - b c^{3} x^{3} + 3 \, b c x - 3 \, b \arctan \left (c x\right )}{12 \, c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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