Optimal. Leaf size=82 \[ \frac{\left (d+e x^2\right )^2 \left (a+b \tan ^{-1}(c x)\right )}{4 e}-\frac{b x \left (2 c^2 d-e\right )}{4 c^3}-\frac{b \left (c^2 d-e\right )^2 \tan ^{-1}(c x)}{4 c^4 e}-\frac{b e x^3}{12 c} \]
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Rubi [A] time = 0.0698475, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {4974, 390, 203} \[ \frac{\left (d+e x^2\right )^2 \left (a+b \tan ^{-1}(c x)\right )}{4 e}-\frac{b x \left (2 c^2 d-e\right )}{4 c^3}-\frac{b \left (c^2 d-e\right )^2 \tan ^{-1}(c x)}{4 c^4 e}-\frac{b e x^3}{12 c} \]
Antiderivative was successfully verified.
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Rule 4974
Rule 390
Rule 203
Rubi steps
\begin{align*} \int x \left (d+e x^2\right ) \left (a+b \tan ^{-1}(c x)\right ) \, dx &=\frac{\left (d+e x^2\right )^2 \left (a+b \tan ^{-1}(c x)\right )}{4 e}-\frac{(b c) \int \frac{\left (d+e x^2\right )^2}{1+c^2 x^2} \, dx}{4 e}\\ &=\frac{\left (d+e x^2\right )^2 \left (a+b \tan ^{-1}(c x)\right )}{4 e}-\frac{(b c) \int \left (\frac{\left (2 c^2 d-e\right ) e}{c^4}+\frac{e^2 x^2}{c^2}+\frac{c^4 d^2-2 c^2 d e+e^2}{c^4 \left (1+c^2 x^2\right )}\right ) \, dx}{4 e}\\ &=-\frac{b \left (2 c^2 d-e\right ) x}{4 c^3}-\frac{b e x^3}{12 c}+\frac{\left (d+e x^2\right )^2 \left (a+b \tan ^{-1}(c x)\right )}{4 e}-\frac{\left (b \left (c^2 d-e\right )^2\right ) \int \frac{1}{1+c^2 x^2} \, dx}{4 c^3 e}\\ &=-\frac{b \left (2 c^2 d-e\right ) x}{4 c^3}-\frac{b e x^3}{12 c}-\frac{b \left (c^2 d-e\right )^2 \tan ^{-1}(c x)}{4 c^4 e}+\frac{\left (d+e x^2\right )^2 \left (a+b \tan ^{-1}(c x)\right )}{4 e}\\ \end{align*}
Mathematica [A] time = 0.0043604, size = 103, normalized size = 1.26 \[ \frac{1}{2} a d x^2+\frac{1}{4} a e x^4+\frac{b d \tan ^{-1}(c x)}{2 c^2}+\frac{b e x}{4 c^3}-\frac{b e \tan ^{-1}(c x)}{4 c^4}+\frac{1}{2} b d x^2 \tan ^{-1}(c x)-\frac{b d x}{2 c}-\frac{b e x^3}{12 c}+\frac{1}{4} b e x^4 \tan ^{-1}(c x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.037, size = 86, normalized size = 1.1 \begin{align*}{\frac{ae{x}^{4}}{4}}+{\frac{a{x}^{2}d}{2}}+{\frac{b\arctan \left ( cx \right ) e{x}^{4}}{4}}+{\frac{b\arctan \left ( cx \right ) d{x}^{2}}{2}}-{\frac{be{x}^{3}}{12\,c}}-{\frac{bdx}{2\,c}}+{\frac{bex}{4\,{c}^{3}}}+{\frac{\arctan \left ( cx \right ) bd}{2\,{c}^{2}}}-{\frac{b\arctan \left ( cx \right ) e}{4\,{c}^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.41794, size = 119, normalized size = 1.45 \begin{align*} \frac{1}{4} \, a e x^{4} + \frac{1}{2} \, a d x^{2} + \frac{1}{2} \,{\left (x^{2} \arctan \left (c x\right ) - c{\left (\frac{x}{c^{2}} - \frac{\arctan \left (c x\right )}{c^{3}}\right )}\right )} b d + \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 e \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.75088, size = 197, normalized size = 2.4 \begin{align*} \frac{3 \, a c^{4} e x^{4} + 6 \, a c^{4} d x^{2} - b c^{3} e x^{3} - 3 \,{\left (2 \, b c^{3} d - b c e\right )} x + 3 \,{\left (b c^{4} e x^{4} + 2 \, b c^{4} d x^{2} + 2 \, b c^{2} d - b e\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 = 1.56302, size = 114, normalized size = 1.39 \begin{align*} \begin{cases} \frac{a d x^{2}}{2} + \frac{a e x^{4}}{4} + \frac{b d x^{2} \operatorname{atan}{\left (c x \right )}}{2} + \frac{b e x^{4} \operatorname{atan}{\left (c x \right )}}{4} - \frac{b d x}{2 c} - \frac{b e x^{3}}{12 c} + \frac{b d \operatorname{atan}{\left (c x \right )}}{2 c^{2}} + \frac{b e x}{4 c^{3}} - \frac{b e \operatorname{atan}{\left (c x \right )}}{4 c^{4}} & \text{for}\: c \neq 0 \\a \left (\frac{d x^{2}}{2} + \frac{e x^{4}}{4}\right ) & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1876, size = 154, normalized size = 1.88 \begin{align*} \frac{3 \, b c^{4} x^{4} \arctan \left (c x\right ) e + 3 \, a c^{4} x^{4} e + 6 \, b c^{4} d x^{2} \arctan \left (c x\right ) + 6 \, a c^{4} d x^{2} - b c^{3} x^{3} e - 6 \, \pi b c^{2} d \mathrm{sgn}\left (c\right ) \mathrm{sgn}\left (x\right ) - 6 \, b c^{3} d x + 6 \, b c^{2} d \arctan \left (c x\right ) + 3 \, b c x e - 3 \, b \arctan \left (c x\right ) e}{12 \, c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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