Optimal. Leaf size=44 \[ -\frac{a^2 A}{2 x^2}-\frac{a (a B+2 A b)}{x}+b \log (x) (2 a B+A b)+b^2 B x \]
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Rubi [A] time = 0.0244103, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {27, 76} \[ -\frac{a^2 A}{2 x^2}-\frac{a (a B+2 A b)}{x}+b \log (x) (2 a B+A b)+b^2 B x \]
Antiderivative was successfully verified.
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Rule 27
Rule 76
Rubi steps
\begin{align*} \int \frac{(A+B x) \left (a^2+2 a b x+b^2 x^2\right )}{x^3} \, dx &=\int \frac{(a+b x)^2 (A+B x)}{x^3} \, dx\\ &=\int \left (b^2 B+\frac{a^2 A}{x^3}+\frac{a (2 A b+a B)}{x^2}+\frac{b (A b+2 a B)}{x}\right ) \, dx\\ &=-\frac{a^2 A}{2 x^2}-\frac{a (2 A b+a B)}{x}+b^2 B x+b (A b+2 a B) \log (x)\\ \end{align*}
Mathematica [A] time = 0.0246844, size = 43, normalized size = 0.98 \[ -\frac{a^2 (A+2 B x)}{2 x^2}+b \log (x) (2 a B+A b)-\frac{2 a A b}{x}+b^2 B x \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 48, normalized size = 1.1 \begin{align*}{b}^{2}Bx+A{b}^{2}\ln \left ( x \right ) +2\,B\ln \left ( x \right ) ab-{\frac{A{a}^{2}}{2\,{x}^{2}}}-2\,{\frac{Aab}{x}}-{\frac{B{a}^{2}}{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02099, size = 62, normalized size = 1.41 \begin{align*} B b^{2} x +{\left (2 \, B a b + A b^{2}\right )} \log \left (x\right ) - \frac{A a^{2} + 2 \,{\left (B a^{2} + 2 \, A a b\right )} x}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.30302, size = 119, normalized size = 2.7 \begin{align*} \frac{2 \, B b^{2} x^{3} + 2 \,{\left (2 \, B a b + A b^{2}\right )} x^{2} \log \left (x\right ) - A a^{2} - 2 \,{\left (B a^{2} + 2 \, A a b\right )} x}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.515756, size = 44, normalized size = 1. \begin{align*} B b^{2} x + b \left (A b + 2 B a\right ) \log{\left (x \right )} - \frac{A a^{2} + x \left (4 A a b + 2 B a^{2}\right )}{2 x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14319, size = 63, normalized size = 1.43 \begin{align*} B b^{2} x +{\left (2 \, B a b + A b^{2}\right )} \log \left ({\left | x \right |}\right ) - \frac{A a^{2} + 2 \,{\left (B a^{2} + 2 \, A a b\right )} x}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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