Optimal. Leaf size=28 \[ -\frac{(A+B x)^2}{2 (a+b x)^2 (A b-a B)} \]
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Rubi [A] time = 0.0043241, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {37} \[ -\frac{(A+B x)^2}{2 (a+b x)^2 (A b-a B)} \]
Antiderivative was successfully verified.
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Rule 37
Rubi steps
\begin{align*} \int \frac{A+B x}{(a+b x)^3} \, dx &=-\frac{(A+B x)^2}{2 (A b-a B) (a+b x)^2}\\ \end{align*}
Mathematica [A] time = 0.0180129, size = 26, normalized size = 0.93 \[ -\frac{B (a+2 b x)+A b}{2 b^2 (a+b x)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 35, normalized size = 1.3 \begin{align*} -{\frac{B}{{b}^{2} \left ( bx+a \right ) }}-{\frac{Ab-Ba}{2\,{b}^{2} \left ( bx+a \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0582, size = 51, normalized size = 1.82 \begin{align*} -\frac{2 \, B b x + B a + A b}{2 \,{\left (b^{4} x^{2} + 2 \, a b^{3} x + a^{2} b^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.93228, size = 81, normalized size = 2.89 \begin{align*} -\frac{2 \, B b x + B a + A b}{2 \,{\left (b^{4} x^{2} + 2 \, a b^{3} x + a^{2} b^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.480711, size = 39, normalized size = 1.39 \begin{align*} - \frac{A b + B a + 2 B b x}{2 a^{2} b^{2} + 4 a b^{3} x + 2 b^{4} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19295, size = 32, normalized size = 1.14 \begin{align*} -\frac{2 \, B b x + B a + A b}{2 \,{\left (b x + a\right )}^{2} b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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