Optimal. Leaf size=79 \[ \frac{\left (1-3 a^2\right ) \log \left ((a+b x)^2+1\right )}{6 b^3}-\frac{a \left (3-a^2\right ) \tan ^{-1}(a+b x)}{3 b^3}+\frac{a x}{b^2}-\frac{(a+b x)^2}{6 b^3}+\frac{1}{3} x^3 \tan ^{-1}(a+b x) \]
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Rubi [A] time = 0.0917566, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used = {5047, 4862, 702, 635, 203, 260} \[ \frac{\left (1-3 a^2\right ) \log \left ((a+b x)^2+1\right )}{6 b^3}-\frac{a \left (3-a^2\right ) \tan ^{-1}(a+b x)}{3 b^3}+\frac{a x}{b^2}-\frac{(a+b x)^2}{6 b^3}+\frac{1}{3} x^3 \tan ^{-1}(a+b x) \]
Antiderivative was successfully verified.
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Rule 5047
Rule 4862
Rule 702
Rule 635
Rule 203
Rule 260
Rubi steps
\begin{align*} \int x^2 \tan ^{-1}(a+b x) \, dx &=\frac{\operatorname{Subst}\left (\int \left (-\frac{a}{b}+\frac{x}{b}\right )^2 \tan ^{-1}(x) \, dx,x,a+b x\right )}{b}\\ &=\frac{1}{3} x^3 \tan ^{-1}(a+b x)-\frac{1}{3} \operatorname{Subst}\left (\int \frac{\left (-\frac{a}{b}+\frac{x}{b}\right )^3}{1+x^2} \, dx,x,a+b x\right )\\ &=\frac{1}{3} x^3 \tan ^{-1}(a+b x)-\frac{1}{3} \operatorname{Subst}\left (\int \left (-\frac{3 a}{b^3}+\frac{x}{b^3}+\frac{a \left (3-a^2\right )-\left (1-3 a^2\right ) x}{b^3 \left (1+x^2\right )}\right ) \, dx,x,a+b x\right )\\ &=\frac{a x}{b^2}-\frac{(a+b x)^2}{6 b^3}+\frac{1}{3} x^3 \tan ^{-1}(a+b x)-\frac{\operatorname{Subst}\left (\int \frac{a \left (3-a^2\right )-\left (1-3 a^2\right ) x}{1+x^2} \, dx,x,a+b x\right )}{3 b^3}\\ &=\frac{a x}{b^2}-\frac{(a+b x)^2}{6 b^3}+\frac{1}{3} x^3 \tan ^{-1}(a+b x)+\frac{\left (1-3 a^2\right ) \operatorname{Subst}\left (\int \frac{x}{1+x^2} \, dx,x,a+b x\right )}{3 b^3}-\frac{\left (a \left (3-a^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,a+b x\right )}{3 b^3}\\ &=\frac{a x}{b^2}-\frac{(a+b x)^2}{6 b^3}-\frac{a \left (3-a^2\right ) \tan ^{-1}(a+b x)}{3 b^3}+\frac{1}{3} x^3 \tan ^{-1}(a+b x)+\frac{\left (1-3 a^2\right ) \log \left (1+(a+b x)^2\right )}{6 b^3}\\ \end{align*}
Mathematica [C] time = 0.0483124, size = 114, normalized size = 1.44 \[ \frac{\frac{1}{3} b \left (\frac{a+b x}{b}-\frac{a}{b}\right )^3 \tan ^{-1}(a+b x)-\frac{1}{3} b \left (\frac{(a+b x)^2}{2 b^3}-\frac{3 a x}{b^2}-\frac{(1-i a)^3 \log (a+b x+i)}{2 b^3}-\frac{(1+i a)^3 \log (-a-b x+i)}{2 b^3}\right )}{b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.036, size = 95, normalized size = 1.2 \begin{align*}{\frac{{x}^{3}\arctan \left ( bx+a \right ) }{3}}+{\frac{\arctan \left ( bx+a \right ){a}^{3}}{3\,{b}^{3}}}-{\frac{{x}^{2}}{6\,b}}+{\frac{2\,ax}{3\,{b}^{2}}}+{\frac{5\,{a}^{2}}{6\,{b}^{3}}}-{\frac{\ln \left ( 1+ \left ( bx+a \right ) ^{2} \right ){a}^{2}}{2\,{b}^{3}}}+{\frac{\ln \left ( 1+ \left ( bx+a \right ) ^{2} \right ) }{6\,{b}^{3}}}-{\frac{\arctan \left ( bx+a \right ) a}{{b}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.50323, size = 115, normalized size = 1.46 \begin{align*} \frac{1}{3} \, x^{3} \arctan \left (b x + a\right ) - \frac{1}{6} \, b{\left (\frac{b x^{2} - 4 \, a x}{b^{3}} - \frac{2 \,{\left (a^{3} - 3 \, a\right )} \arctan \left (\frac{b^{2} x + a b}{b}\right )}{b^{4}} + \frac{{\left (3 \, a^{2} - 1\right )} \log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}{b^{4}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.74905, size = 161, normalized size = 2.04 \begin{align*} -\frac{b^{2} x^{2} - 4 \, a b x - 2 \,{\left (b^{3} x^{3} + a^{3} - 3 \, a\right )} \arctan \left (b x + a\right ) +{\left (3 \, a^{2} - 1\right )} \log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}{6 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.4325, size = 117, normalized size = 1.48 \begin{align*} \begin{cases} \frac{a^{3} \operatorname{atan}{\left (a + b x \right )}}{3 b^{3}} - \frac{a^{2} \log{\left (a^{2} + 2 a b x + b^{2} x^{2} + 1 \right )}}{2 b^{3}} + \frac{2 a x}{3 b^{2}} - \frac{a \operatorname{atan}{\left (a + b x \right )}}{b^{3}} + \frac{x^{3} \operatorname{atan}{\left (a + b x \right )}}{3} - \frac{x^{2}}{6 b} + \frac{\log{\left (a^{2} + 2 a b x + b^{2} x^{2} + 1 \right )}}{6 b^{3}} & \text{for}\: b \neq 0 \\\frac{x^{3} \operatorname{atan}{\left (a \right )}}{3} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09272, size = 111, normalized size = 1.41 \begin{align*} \frac{1}{3} \, x^{3} \arctan \left (b x + a\right ) + \frac{1}{6} \, b{\left (\frac{2 \,{\left (a^{3} - 3 \, a\right )} \arctan \left (b x + a\right )}{b^{4}} - \frac{{\left (3 \, a^{2} - 1\right )} \log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}{b^{4}} - \frac{b^{2} x^{2} - 4 \, a b x}{b^{4}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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