Optimal. Leaf size=59 \[ -\frac{A b-2 a B}{2 b^3 (a+b x)^2}+\frac{a (A b-a B)}{3 b^3 (a+b x)^3}-\frac{B}{b^3 (a+b x)} \]
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Rubi [A] time = 0.0422136, antiderivative size = 59, 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, 77} \[ -\frac{A b-2 a B}{2 b^3 (a+b x)^2}+\frac{a (A b-a B)}{3 b^3 (a+b x)^3}-\frac{B}{b^3 (a+b x)} \]
Antiderivative was successfully verified.
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Rule 27
Rule 77
Rubi steps
\begin{align*} \int \frac{x (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx &=\int \frac{x (A+B x)}{(a+b x)^4} \, dx\\ &=\int \left (\frac{a (-A b+a B)}{b^2 (a+b x)^4}+\frac{A b-2 a B}{b^2 (a+b x)^3}+\frac{B}{b^2 (a+b x)^2}\right ) \, dx\\ &=\frac{a (A b-a B)}{3 b^3 (a+b x)^3}-\frac{A b-2 a B}{2 b^3 (a+b x)^2}-\frac{B}{b^3 (a+b x)}\\ \end{align*}
Mathematica [A] time = 0.0154844, size = 42, normalized size = 0.71 \[ -\frac{2 a^2 B+a b (A+6 B x)+3 b^2 x (A+2 B x)}{6 b^3 (a+b x)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 56, normalized size = 1. \begin{align*} -{\frac{B}{{b}^{3} \left ( bx+a \right ) }}-{\frac{Ab-2\,aB}{2\,{b}^{3} \left ( bx+a \right ) ^{2}}}+{\frac{a \left ( Ab-aB \right ) }{3\,{b}^{3} \left ( bx+a \right ) ^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04415, size = 96, normalized size = 1.63 \begin{align*} -\frac{6 \, B b^{2} x^{2} + 2 \, B a^{2} + A a b + 3 \,{\left (2 \, B a b + A b^{2}\right )} x}{6 \,{\left (b^{6} x^{3} + 3 \, a b^{5} x^{2} + 3 \, a^{2} b^{4} x + a^{3} b^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.23519, size = 149, normalized size = 2.53 \begin{align*} -\frac{6 \, B b^{2} x^{2} + 2 \, B a^{2} + A a b + 3 \,{\left (2 \, B a b + A b^{2}\right )} x}{6 \,{\left (b^{6} x^{3} + 3 \, a b^{5} x^{2} + 3 \, a^{2} b^{4} x + a^{3} b^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.683322, size = 75, normalized size = 1.27 \begin{align*} - \frac{A a b + 2 B a^{2} + 6 B b^{2} x^{2} + x \left (3 A b^{2} + 6 B a b\right )}{6 a^{3} b^{3} + 18 a^{2} b^{4} x + 18 a b^{5} x^{2} + 6 b^{6} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13882, size = 61, normalized size = 1.03 \begin{align*} -\frac{6 \, B b^{2} x^{2} + 6 \, B a b x + 3 \, A b^{2} x + 2 \, B a^{2} + A a b}{6 \,{\left (b x + a\right )}^{3} b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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