Optimal. Leaf size=69 \[ -\frac{a^2 (A b-a B)}{b^4 (a+b x)}+\frac{x (A b-2 a B)}{b^3}-\frac{a (2 A b-3 a B) \log (a+b x)}{b^4}+\frac{B x^2}{2 b^2} \]
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Rubi [A] time = 0.0560725, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074, Rules used = {27, 77} \[ -\frac{a^2 (A b-a B)}{b^4 (a+b x)}+\frac{x (A b-2 a B)}{b^3}-\frac{a (2 A b-3 a B) \log (a+b x)}{b^4}+\frac{B x^2}{2 b^2} \]
Antiderivative was successfully verified.
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Rule 27
Rule 77
Rubi steps
\begin{align*} \int \frac{x^2 (A+B x)}{a^2+2 a b x+b^2 x^2} \, dx &=\int \frac{x^2 (A+B x)}{(a+b x)^2} \, dx\\ &=\int \left (\frac{A b-2 a B}{b^3}+\frac{B x}{b^2}-\frac{a^2 (-A b+a B)}{b^3 (a+b x)^2}+\frac{a (-2 A b+3 a B)}{b^3 (a+b x)}\right ) \, dx\\ &=\frac{(A b-2 a B) x}{b^3}+\frac{B x^2}{2 b^2}-\frac{a^2 (A b-a B)}{b^4 (a+b x)}-\frac{a (2 A b-3 a B) \log (a+b x)}{b^4}\\ \end{align*}
Mathematica [A] time = 0.0573055, size = 66, normalized size = 0.96 \[ \frac{\frac{2 a^2 (a B-A b)}{a+b x}+2 b x (A b-2 a B)+2 a (3 a B-2 A b) \log (a+b x)+b^2 B x^2}{2 b^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 84, normalized size = 1.2 \begin{align*}{\frac{B{x}^{2}}{2\,{b}^{2}}}+{\frac{Ax}{{b}^{2}}}-2\,{\frac{aBx}{{b}^{3}}}-{\frac{A{a}^{2}}{{b}^{3} \left ( bx+a \right ) }}+{\frac{B{a}^{3}}{{b}^{4} \left ( bx+a \right ) }}-2\,{\frac{a\ln \left ( bx+a \right ) A}{{b}^{3}}}+3\,{\frac{{a}^{2}\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 = 0.988096, size = 100, normalized size = 1.45 \begin{align*} \frac{B a^{3} - A a^{2} b}{b^{5} x + a b^{4}} + \frac{B b x^{2} - 2 \,{\left (2 \, B a - A b\right )} x}{2 \, b^{3}} + \frac{{\left (3 \, B a^{2} - 2 \, A a 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 = 1.32588, size = 240, normalized size = 3.48 \begin{align*} \frac{B b^{3} x^{3} + 2 \, B a^{3} - 2 \, A a^{2} b -{\left (3 \, B a b^{2} - 2 \, A b^{3}\right )} x^{2} - 2 \,{\left (2 \, B a^{2} b - A a b^{2}\right )} x + 2 \,{\left (3 \, B a^{3} - 2 \, A a^{2} b +{\left (3 \, B a^{2} b - 2 \, A a b^{2}\right )} x\right )} \log \left (b x + a\right )}{2 \,{\left (b^{5} x + a b^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.524613, size = 66, normalized size = 0.96 \begin{align*} \frac{B x^{2}}{2 b^{2}} + \frac{a \left (- 2 A b + 3 B a\right ) \log{\left (a + b x \right )}}{b^{4}} + \frac{- A a^{2} b + B a^{3}}{a b^{4} + b^{5} x} - \frac{x \left (- A b + 2 B a\right )}{b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09924, size = 101, normalized size = 1.46 \begin{align*} \frac{{\left (3 \, B a^{2} - 2 \, A a b\right )} \log \left ({\left | b x + a \right |}\right )}{b^{4}} + \frac{B b^{2} x^{2} - 4 \, B a b x + 2 \, A b^{2} x}{2 \, b^{4}} + \frac{B a^{3} - A a^{2} b}{{\left (b x + a\right )} b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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