Optimal. Leaf size=107 \[ \frac{1}{4} d x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{6} e x^6 \left (a+b \tan ^{-1}(c x)\right )-\frac{b x^3 \left (3 c^2 d-2 e\right )}{36 c^3}+\frac{b x \left (3 c^2 d-2 e\right )}{12 c^5}-\frac{b \left (3 c^2 d-2 e\right ) \tan ^{-1}(c x)}{12 c^6}-\frac{b e x^5}{30 c} \]
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Rubi [A] time = 0.110298, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used = {14, 4976, 459, 302, 203} \[ \frac{1}{4} d x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{6} e x^6 \left (a+b \tan ^{-1}(c x)\right )-\frac{b x^3 \left (3 c^2 d-2 e\right )}{36 c^3}+\frac{b x \left (3 c^2 d-2 e\right )}{12 c^5}-\frac{b \left (3 c^2 d-2 e\right ) \tan ^{-1}(c x)}{12 c^6}-\frac{b e x^5}{30 c} \]
Antiderivative was successfully verified.
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Rule 14
Rule 4976
Rule 459
Rule 302
Rule 203
Rubi steps
\begin{align*} \int x^3 \left (d+e x^2\right ) \left (a+b \tan ^{-1}(c x)\right ) \, dx &=\frac{1}{4} d x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{6} e x^6 \left (a+b \tan ^{-1}(c x)\right )-(b c) \int \frac{x^4 \left (3 d+2 e x^2\right )}{12+12 c^2 x^2} \, dx\\ &=-\frac{b e x^5}{30 c}+\frac{1}{4} d x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{6} e x^6 \left (a+b \tan ^{-1}(c x)\right )+\left (b c \left (-3 d+\frac{2 e}{c^2}\right )\right ) \int \frac{x^4}{12+12 c^2 x^2} \, dx\\ &=-\frac{b e x^5}{30 c}+\frac{1}{4} d x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{6} e x^6 \left (a+b \tan ^{-1}(c x)\right )+\left (b c \left (-3 d+\frac{2 e}{c^2}\right )\right ) \int \left (-\frac{1}{12 c^4}+\frac{x^2}{12 c^2}+\frac{1}{c^4 \left (12+12 c^2 x^2\right )}\right ) \, dx\\ &=\frac{b \left (3 d-\frac{2 e}{c^2}\right ) x}{12 c^3}-\frac{b \left (3 d-\frac{2 e}{c^2}\right ) x^3}{36 c}-\frac{b e x^5}{30 c}+\frac{1}{4} d x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{6} e x^6 \left (a+b \tan ^{-1}(c x)\right )+\frac{\left (b \left (-3 d+\frac{2 e}{c^2}\right )\right ) \int \frac{1}{12+12 c^2 x^2} \, dx}{c^3}\\ &=\frac{b \left (3 d-\frac{2 e}{c^2}\right ) x}{12 c^3}-\frac{b \left (3 d-\frac{2 e}{c^2}\right ) x^3}{36 c}-\frac{b e x^5}{30 c}-\frac{b \left (3 d-\frac{2 e}{c^2}\right ) \tan ^{-1}(c x)}{12 c^4}+\frac{1}{4} d x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{6} e x^6 \left (a+b \tan ^{-1}(c x)\right )\\ \end{align*}
Mathematica [A] time = 0.0054455, size = 127, normalized size = 1.19 \[ \frac{1}{4} a d x^4+\frac{1}{6} a e x^6+\frac{b d x}{4 c^3}-\frac{b d \tan ^{-1}(c x)}{4 c^4}+\frac{b e x^3}{18 c^3}-\frac{b e x}{6 c^5}+\frac{b e \tan ^{-1}(c x)}{6 c^6}-\frac{b d x^3}{12 c}+\frac{1}{4} b d x^4 \tan ^{-1}(c x)-\frac{b e x^5}{30 c}+\frac{1}{6} b e x^6 \tan ^{-1}(c x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.038, size = 106, normalized size = 1. \begin{align*}{\frac{ae{x}^{6}}{6}}+{\frac{a{x}^{4}d}{4}}+{\frac{b\arctan \left ( cx \right ) e{x}^{6}}{6}}+{\frac{b\arctan \left ( cx \right ){x}^{4}d}{4}}-{\frac{be{x}^{5}}{30\,c}}-{\frac{bd{x}^{3}}{12\,c}}+{\frac{b{x}^{3}e}{18\,{c}^{3}}}+{\frac{bdx}{4\,{c}^{3}}}-{\frac{bex}{6\,{c}^{5}}}-{\frac{\arctan \left ( cx \right ) bd}{4\,{c}^{4}}}+{\frac{b\arctan \left ( cx \right ) e}{6\,{c}^{6}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.47799, size = 146, normalized size = 1.36 \begin{align*} \frac{1}{6} \, a e x^{6} + \frac{1}{4} \, a d 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 d + \frac{1}{90} \,{\left (15 \, x^{6} \arctan \left (c x\right ) - c{\left (\frac{3 \, c^{4} x^{5} - 5 \, c^{2} x^{3} + 15 \, x}{c^{6}} - \frac{15 \, \arctan \left (c x\right )}{c^{7}}\right )}\right )} b e \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.64274, size = 258, normalized size = 2.41 \begin{align*} \frac{30 \, a c^{6} e x^{6} + 45 \, a c^{6} d x^{4} - 6 \, b c^{5} e x^{5} - 5 \,{\left (3 \, b c^{5} d - 2 \, b c^{3} e\right )} x^{3} + 15 \,{\left (3 \, b c^{3} d - 2 \, b c e\right )} x + 15 \,{\left (2 \, b c^{6} e x^{6} + 3 \, b c^{6} d x^{4} - 3 \, b c^{2} d + 2 \, b e\right )} \arctan \left (c x\right )}{180 \, c^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.86752, size = 138, normalized size = 1.29 \begin{align*} \begin{cases} \frac{a d x^{4}}{4} + \frac{a e x^{6}}{6} + \frac{b d x^{4} \operatorname{atan}{\left (c x \right )}}{4} + \frac{b e x^{6} \operatorname{atan}{\left (c x \right )}}{6} - \frac{b d x^{3}}{12 c} - \frac{b e x^{5}}{30 c} + \frac{b d x}{4 c^{3}} + \frac{b e x^{3}}{18 c^{3}} - \frac{b d \operatorname{atan}{\left (c x \right )}}{4 c^{4}} - \frac{b e x}{6 c^{5}} + \frac{b e \operatorname{atan}{\left (c x \right )}}{6 c^{6}} & \text{for}\: c \neq 0 \\a \left (\frac{d x^{4}}{4} + \frac{e x^{6}}{6}\right ) & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18206, size = 180, normalized size = 1.68 \begin{align*} \frac{30 \, b c^{6} x^{6} \arctan \left (c x\right ) e + 30 \, a c^{6} x^{6} e + 45 \, b c^{6} d x^{4} \arctan \left (c x\right ) + 45 \, a c^{6} d x^{4} - 6 \, b c^{5} x^{5} e - 15 \, b c^{5} d x^{3} + 10 \, b c^{3} x^{3} e + 45 \, b c^{3} d x - 45 \, b c^{2} d \arctan \left (c x\right ) - 30 \, \pi b e \mathrm{sgn}\left (c\right ) \mathrm{sgn}\left (x\right ) - 30 \, b c x e + 30 \, b \arctan \left (c x\right ) e}{180 \, c^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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