Optimal. Leaf size=68 \[ d x \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{3} e x^3 \left (a+b \tan ^{-1}(c x)\right )-\frac{b \left (3 c^2 d-e\right ) \log \left (c^2 x^2+1\right )}{6 c^3}-\frac{b e x^2}{6 c} \]
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Rubi [A] time = 0.0718078, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {4912, 1593, 444, 43} \[ d x \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{3} e x^3 \left (a+b \tan ^{-1}(c x)\right )-\frac{b \left (3 c^2 d-e\right ) \log \left (c^2 x^2+1\right )}{6 c^3}-\frac{b e x^2}{6 c} \]
Antiderivative was successfully verified.
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Rule 4912
Rule 1593
Rule 444
Rule 43
Rubi steps
\begin{align*} \int \left (d+e x^2\right ) \left (a+b \tan ^{-1}(c x)\right ) \, dx &=d x \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{3} e x^3 \left (a+b \tan ^{-1}(c x)\right )-(b c) \int \frac{d x+\frac{e x^3}{3}}{1+c^2 x^2} \, dx\\ &=d x \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{3} e x^3 \left (a+b \tan ^{-1}(c x)\right )-(b c) \int \frac{x \left (d+\frac{e x^2}{3}\right )}{1+c^2 x^2} \, dx\\ &=d x \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{3} e x^3 \left (a+b \tan ^{-1}(c x)\right )-\frac{1}{2} (b c) \operatorname{Subst}\left (\int \frac{d+\frac{e x}{3}}{1+c^2 x} \, dx,x,x^2\right )\\ &=d x \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{3} e x^3 \left (a+b \tan ^{-1}(c x)\right )-\frac{1}{2} (b c) \operatorname{Subst}\left (\int \left (\frac{e}{3 c^2}+\frac{3 c^2 d-e}{3 c^2 \left (1+c^2 x\right )}\right ) \, dx,x,x^2\right )\\ &=-\frac{b e x^2}{6 c}+d x \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{3} e x^3 \left (a+b \tan ^{-1}(c x)\right )-\frac{b \left (3 c^2 d-e\right ) \log \left (1+c^2 x^2\right )}{6 c^3}\\ \end{align*}
Mathematica [A] time = 0.0100485, size = 85, normalized size = 1.25 \[ a d x+\frac{1}{3} a e x^3-\frac{b d \log \left (c^2 x^2+1\right )}{2 c}+\frac{b e \log \left (c^2 x^2+1\right )}{6 c^3}+b d x \tan ^{-1}(c x)-\frac{b e x^2}{6 c}+\frac{1}{3} b e x^3 \tan ^{-1}(c x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.035, size = 76, normalized size = 1.1 \begin{align*}{\frac{ae{x}^{3}}{3}}+adx+{\frac{be{x}^{3}\arctan \left ( cx \right ) }{3}}+b\arctan \left ( cx \right ) dx-{\frac{be{x}^{2}}{6\,c}}-{\frac{b\ln \left ({c}^{2}{x}^{2}+1 \right ) d}{2\,c}}+{\frac{be\ln \left ({c}^{2}{x}^{2}+1 \right ) }{6\,{c}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.05791, size = 108, normalized size = 1.59 \begin{align*} \frac{1}{3} \, a e 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 e + a d x + \frac{{\left (2 \, c x \arctan \left (c x\right ) - \log \left (c^{2} x^{2} + 1\right )\right )} b d}{2 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.69946, size = 181, normalized size = 2.66 \begin{align*} \frac{2 \, a c^{3} e x^{3} + 6 \, a c^{3} d x - b c^{2} e x^{2} + 2 \,{\left (b c^{3} e x^{3} + 3 \, b c^{3} d x\right )} \arctan \left (c x\right ) -{\left (3 \, b c^{2} d - b e\right )} \log \left (c^{2} x^{2} + 1\right )}{6 \, c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.03202, size = 94, normalized size = 1.38 \begin{align*} \begin{cases} a d x + \frac{a e x^{3}}{3} + b d x \operatorname{atan}{\left (c x \right )} + \frac{b e x^{3} \operatorname{atan}{\left (c x \right )}}{3} - \frac{b d \log{\left (x^{2} + \frac{1}{c^{2}} \right )}}{2 c} - \frac{b e x^{2}}{6 c} + \frac{b e \log{\left (x^{2} + \frac{1}{c^{2}} \right )}}{6 c^{3}} & \text{for}\: c \neq 0 \\a \left (d x + \frac{e x^{3}}{3}\right ) & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10571, size = 127, normalized size = 1.87 \begin{align*} \frac{2 \, b c^{3} x^{3} \arctan \left (c x\right ) e + 2 \, a c^{3} x^{3} e + 6 \, b c^{3} d x \arctan \left (c x\right ) + 6 \, a c^{3} d x - b c^{2} x^{2} e - 3 \, b c^{2} d \log \left (c^{2} x^{2} + 1\right ) + b e \log \left (c^{2} x^{2} + 1\right )}{6 \, c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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