Optimal. Leaf size=38 \[ -\frac{A b-a B}{3 b^2 (a+b x)^3}-\frac{B}{2 b^2 (a+b x)^2} \]
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Rubi [A] time = 0.0233002, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {27, 43} \[ -\frac{A b-a B}{3 b^2 (a+b x)^3}-\frac{B}{2 b^2 (a+b x)^2} \]
Antiderivative was successfully verified.
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Rule 27
Rule 43
Rubi steps
\begin{align*} \int \frac{A+B x}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx &=\int \frac{A+B x}{(a+b x)^4} \, dx\\ &=\int \left (\frac{A b-a B}{b (a+b x)^4}+\frac{B}{b (a+b x)^3}\right ) \, dx\\ &=-\frac{A b-a B}{3 b^2 (a+b x)^3}-\frac{B}{2 b^2 (a+b x)^2}\\ \end{align*}
Mathematica [A] time = 0.0103597, size = 27, normalized size = 0.71 \[ -\frac{B (a+3 b x)+2 A b}{6 b^2 (a+b x)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 35, normalized size = 0.9 \begin{align*} -{\frac{Ab-aB}{3\,{b}^{2} \left ( bx+a \right ) ^{3}}}-{\frac{B}{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 = 1.18199, size = 68, normalized size = 1.79 \begin{align*} -\frac{3 \, B b x + B a + 2 \, A b}{6 \,{\left (b^{5} x^{3} + 3 \, a b^{4} x^{2} + 3 \, a^{2} b^{3} x + a^{3} b^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.47451, size = 105, normalized size = 2.76 \begin{align*} -\frac{3 \, B b x + B a + 2 \, A b}{6 \,{\left (b^{5} x^{3} + 3 \, a b^{4} x^{2} + 3 \, a^{2} b^{3} x + a^{3} b^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.535453, size = 53, normalized size = 1.39 \begin{align*} - \frac{2 A b + B a + 3 B b x}{6 a^{3} b^{2} + 18 a^{2} b^{3} x + 18 a b^{4} x^{2} + 6 b^{5} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1962, size = 34, normalized size = 0.89 \begin{align*} -\frac{3 \, B b x + B a + 2 \, A b}{6 \,{\left (b x + a\right )}^{3} b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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