Optimal. Leaf size=66 \[ \frac{a^2 (A b-a B) \log (a+b x)}{b^4}+\frac{x^2 (A b-a B)}{2 b^2}-\frac{a x (A b-a B)}{b^3}+\frac{B x^3}{3 b} \]
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Rubi [A] time = 0.0467123, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {77} \[ \frac{a^2 (A b-a B) \log (a+b x)}{b^4}+\frac{x^2 (A b-a B)}{2 b^2}-\frac{a x (A b-a B)}{b^3}+\frac{B x^3}{3 b} \]
Antiderivative was successfully verified.
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Rule 77
Rubi steps
\begin{align*} \int \frac{x^2 (A+B x)}{a+b x} \, dx &=\int \left (\frac{a (-A b+a B)}{b^3}+\frac{(A b-a B) x}{b^2}+\frac{B x^2}{b}-\frac{a^2 (-A b+a B)}{b^3 (a+b x)}\right ) \, dx\\ &=-\frac{a (A b-a B) x}{b^3}+\frac{(A b-a B) x^2}{2 b^2}+\frac{B x^3}{3 b}+\frac{a^2 (A b-a B) \log (a+b x)}{b^4}\\ \end{align*}
Mathematica [A] time = 0.0211556, size = 61, normalized size = 0.92 \[ \frac{b x \left (6 a^2 B-3 a b (2 A+B x)+b^2 x (3 A+2 B x)\right )+6 a^2 (A b-a B) \log (a+b x)}{6 b^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 76, normalized size = 1.2 \begin{align*}{\frac{B{x}^{3}}{3\,b}}+{\frac{A{x}^{2}}{2\,b}}-{\frac{B{x}^{2}a}{2\,{b}^{2}}}-{\frac{aAx}{{b}^{2}}}+{\frac{{a}^{2}Bx}{{b}^{3}}}+{\frac{{a}^{2}\ln \left ( bx+a \right ) A}{{b}^{3}}}-{\frac{{a}^{3}\ln \left ( bx+a \right ) B}{{b}^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01062, size = 95, normalized size = 1.44 \begin{align*} \frac{2 \, B b^{2} x^{3} - 3 \,{\left (B a b - A b^{2}\right )} x^{2} + 6 \,{\left (B a^{2} - A a b\right )} x}{6 \, b^{3}} - \frac{{\left (B a^{3} - A a^{2} b\right )} \log \left (b x + a\right )}{b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.43559, size = 149, normalized size = 2.26 \begin{align*} \frac{2 \, B b^{3} x^{3} - 3 \,{\left (B a b^{2} - A b^{3}\right )} x^{2} + 6 \,{\left (B a^{2} b - A a b^{2}\right )} x - 6 \,{\left (B a^{3} - A a^{2} b\right )} \log \left (b x + a\right )}{6 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.427378, size = 58, normalized size = 0.88 \begin{align*} \frac{B x^{3}}{3 b} - \frac{a^{2} \left (- A b + B a\right ) \log{\left (a + b x \right )}}{b^{4}} - \frac{x^{2} \left (- A b + B a\right )}{2 b^{2}} + \frac{x \left (- A a b + B a^{2}\right )}{b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.41364, size = 96, normalized size = 1.45 \begin{align*} \frac{2 \, B b^{2} x^{3} - 3 \, B a b x^{2} + 3 \, A b^{2} x^{2} + 6 \, B a^{2} x - 6 \, A a b x}{6 \, b^{3}} - \frac{{\left (B a^{3} - A a^{2} b\right )} \log \left ({\left | b x + a \right |}\right )}{b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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