Optimal. Leaf size=106 \[ \frac{\left (1-6 a^2\right ) x}{4 b^3}-\frac{a \left (1-a^2\right ) \log \left ((a+b x)^2+1\right )}{2 b^4}-\frac{\left (a^4-6 a^2+1\right ) \tan ^{-1}(a+b x)}{4 b^4}-\frac{(a+b x)^3}{12 b^4}+\frac{a (a+b x)^2}{2 b^4}+\frac{1}{4} x^4 \tan ^{-1}(a+b x) \]
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Rubi [A] time = 0.110634, antiderivative size = 106, 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} \[ \frac{\left (1-6 a^2\right ) x}{4 b^3}-\frac{a \left (1-a^2\right ) \log \left ((a+b x)^2+1\right )}{2 b^4}-\frac{\left (a^4-6 a^2+1\right ) \tan ^{-1}(a+b x)}{4 b^4}-\frac{(a+b x)^3}{12 b^4}+\frac{a (a+b x)^2}{2 b^4}+\frac{1}{4} x^4 \tan ^{-1}(a+b x) \]
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^3 \tan ^{-1}(a+b x) \, dx &=\frac{\operatorname{Subst}\left (\int \left (-\frac{a}{b}+\frac{x}{b}\right )^3 \tan ^{-1}(x) \, dx,x,a+b x\right )}{b}\\ &=\frac{1}{4} x^4 \tan ^{-1}(a+b x)-\frac{1}{4} \operatorname{Subst}\left (\int \frac{\left (-\frac{a}{b}+\frac{x}{b}\right )^4}{1+x^2} \, dx,x,a+b x\right )\\ &=\frac{1}{4} x^4 \tan ^{-1}(a+b x)-\frac{1}{4} \operatorname{Subst}\left (\int \left (-\frac{1-6 a^2}{b^4}-\frac{4 a x}{b^4}+\frac{x^2}{b^4}+\frac{1-6 a^2+a^4+4 a \left (1-a^2\right ) x}{b^4 \left (1+x^2\right )}\right ) \, dx,x,a+b x\right )\\ &=\frac{\left (1-6 a^2\right ) x}{4 b^3}+\frac{a (a+b x)^2}{2 b^4}-\frac{(a+b x)^3}{12 b^4}+\frac{1}{4} x^4 \tan ^{-1}(a+b x)-\frac{\operatorname{Subst}\left (\int \frac{1-6 a^2+a^4+4 a \left (1-a^2\right ) x}{1+x^2} \, dx,x,a+b x\right )}{4 b^4}\\ &=\frac{\left (1-6 a^2\right ) x}{4 b^3}+\frac{a (a+b x)^2}{2 b^4}-\frac{(a+b x)^3}{12 b^4}+\frac{1}{4} x^4 \tan ^{-1}(a+b x)-\frac{\left (a \left (1-a^2\right )\right ) \operatorname{Subst}\left (\int \frac{x}{1+x^2} \, dx,x,a+b x\right )}{b^4}-\frac{\left (1-6 a^2+a^4\right ) \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,a+b x\right )}{4 b^4}\\ &=\frac{\left (1-6 a^2\right ) x}{4 b^3}+\frac{a (a+b x)^2}{2 b^4}-\frac{(a+b x)^3}{12 b^4}-\frac{\left (1-6 a^2+a^4\right ) \tan ^{-1}(a+b x)}{4 b^4}+\frac{1}{4} x^4 \tan ^{-1}(a+b x)-\frac{a \left (1-a^2\right ) \log \left (1+(a+b x)^2\right )}{2 b^4}\\ \end{align*}
Mathematica [C] time = 0.0699706, size = 95, normalized size = 0.9 \[ \frac{6 \left (1-6 a^2\right ) b x+6 b^4 x^4 \tan ^{-1}(a+b x)-2 (a+b x)^3+12 a (a+b x)^2+3 i (a-i)^4 \log (-a-b x+i)-3 i (a+i)^4 \log (a+b x+i)}{24 b^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.039, size = 132, normalized size = 1.3 \begin{align*}{\frac{{x}^{4}\arctan \left ( bx+a \right ) }{4}}-{\frac{\arctan \left ( bx+a \right ){a}^{4}}{4\,{b}^{4}}}-{\frac{{x}^{3}}{12\,b}}+{\frac{a{x}^{2}}{4\,{b}^{2}}}-{\frac{3\,{a}^{2}x}{4\,{b}^{3}}}-{\frac{13\,{a}^{3}}{12\,{b}^{4}}}+{\frac{x}{4\,{b}^{3}}}+{\frac{a}{4\,{b}^{4}}}+{\frac{\ln \left ( 1+ \left ( bx+a \right ) ^{2} \right ){a}^{3}}{2\,{b}^{4}}}-{\frac{\ln \left ( 1+ \left ( bx+a \right ) ^{2} \right ) a}{2\,{b}^{4}}}+{\frac{3\,\arctan \left ( bx+a \right ){a}^{2}}{2\,{b}^{4}}}-{\frac{\arctan \left ( bx+a \right ) }{4\,{b}^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.50389, size = 140, normalized size = 1.32 \begin{align*} \frac{1}{4} \, x^{4} \arctan \left (b x + a\right ) - \frac{1}{12} \, b{\left (\frac{b^{2} x^{3} - 3 \, a b x^{2} + 3 \,{\left (3 \, a^{2} - 1\right )} x}{b^{4}} + \frac{3 \,{\left (a^{4} - 6 \, a^{2} + 1\right )} \arctan \left (\frac{b^{2} x + a b}{b}\right )}{b^{5}} - \frac{6 \,{\left (a^{3} - a\right )} \log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}{b^{5}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.68514, size = 203, normalized size = 1.92 \begin{align*} -\frac{b^{3} x^{3} - 3 \, a b^{2} x^{2} + 3 \,{\left (3 \, a^{2} - 1\right )} b x - 3 \,{\left (b^{4} x^{4} - a^{4} + 6 \, a^{2} - 1\right )} \arctan \left (b x + a\right ) - 6 \,{\left (a^{3} - a\right )} \log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}{12 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.36794, size = 155, normalized size = 1.46 \begin{align*} \begin{cases} - \frac{a^{4} \operatorname{atan}{\left (a + b x \right )}}{4 b^{4}} + \frac{a^{3} \log{\left (a^{2} + 2 a b x + b^{2} x^{2} + 1 \right )}}{2 b^{4}} - \frac{3 a^{2} x}{4 b^{3}} + \frac{3 a^{2} \operatorname{atan}{\left (a + b x \right )}}{2 b^{4}} + \frac{a x^{2}}{4 b^{2}} - \frac{a \log{\left (a^{2} + 2 a b x + b^{2} x^{2} + 1 \right )}}{2 b^{4}} + \frac{x^{4} \operatorname{atan}{\left (a + b x \right )}}{4} - \frac{x^{3}}{12 b} + \frac{x}{4 b^{3}} - \frac{\operatorname{atan}{\left (a + b x \right )}}{4 b^{4}} & \text{for}\: b \neq 0 \\\frac{x^{4} \operatorname{atan}{\left (a \right )}}{4} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10813, size = 139, normalized size = 1.31 \begin{align*} \frac{1}{4} \, x^{4} \arctan \left (b x + a\right ) - \frac{1}{12} \, b{\left (\frac{3 \,{\left (a^{4} - 6 \, a^{2} + 1\right )} \arctan \left (b x + a\right )}{b^{5}} - \frac{6 \,{\left (a^{3} - a\right )} \log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}{b^{5}} + \frac{b^{4} x^{3} - 3 \, a b^{3} x^{2} + 9 \, a^{2} b^{2} x - 3 \, b^{2} x}{b^{6}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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