Optimal. Leaf size=61 \[ -\frac{A b-2 a B}{4 b^3 (a+b x)^4}+\frac{a (A b-a B)}{5 b^3 (a+b x)^5}-\frac{B}{3 b^3 (a+b x)^3} \]
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Rubi [A] time = 0.0418753, antiderivative size = 61, 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} \[ -\frac{A b-2 a B}{4 b^3 (a+b x)^4}+\frac{a (A b-a B)}{5 b^3 (a+b x)^5}-\frac{B}{3 b^3 (a+b x)^3} \]
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 )^3} \, dx &=\int \frac{x (A+B x)}{(a+b x)^6} \, dx\\ &=\int \left (\frac{a (-A b+a B)}{b^2 (a+b x)^6}+\frac{A b-2 a B}{b^2 (a+b x)^5}+\frac{B}{b^2 (a+b x)^4}\right ) \, dx\\ &=\frac{a (A b-a B)}{5 b^3 (a+b x)^5}-\frac{A b-2 a B}{4 b^3 (a+b x)^4}-\frac{B}{3 b^3 (a+b x)^3}\\ \end{align*}
Mathematica [A] time = 0.0175527, size = 46, normalized size = 0.75 \[ -\frac{2 a^2 B+a b (3 A+10 B x)+5 b^2 x (3 A+4 B x)}{60 b^3 (a+b x)^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 56, normalized size = 0.9 \begin{align*}{\frac{a \left ( Ab-aB \right ) }{5\,{b}^{3} \left ( bx+a \right ) ^{5}}}-{\frac{Ab-2\,aB}{4\,{b}^{3} \left ( bx+a \right ) ^{4}}}-{\frac{B}{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 = 1.09885, size = 128, normalized size = 2.1 \begin{align*} -\frac{20 \, B b^{2} x^{2} + 2 \, B a^{2} + 3 \, A a b + 5 \,{\left (2 \, B a b + 3 \, A b^{2}\right )} x}{60 \,{\left (b^{8} x^{5} + 5 \, a b^{7} x^{4} + 10 \, a^{2} b^{6} x^{3} + 10 \, a^{3} b^{5} x^{2} + 5 \, a^{4} b^{4} x + a^{5} b^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.34236, size = 203, normalized size = 3.33 \begin{align*} -\frac{20 \, B b^{2} x^{2} + 2 \, B a^{2} + 3 \, A a b + 5 \,{\left (2 \, B a b + 3 \, A b^{2}\right )} x}{60 \,{\left (b^{8} x^{5} + 5 \, a b^{7} x^{4} + 10 \, a^{2} b^{6} x^{3} + 10 \, a^{3} b^{5} x^{2} + 5 \, a^{4} b^{4} x + a^{5} b^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.985209, size = 100, normalized size = 1.64 \begin{align*} - \frac{3 A a b + 2 B a^{2} + 20 B b^{2} x^{2} + x \left (15 A b^{2} + 10 B a b\right )}{60 a^{5} b^{3} + 300 a^{4} b^{4} x + 600 a^{3} b^{5} x^{2} + 600 a^{2} b^{6} x^{3} + 300 a b^{7} x^{4} + 60 b^{8} x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14248, size = 62, normalized size = 1.02 \begin{align*} -\frac{20 \, B b^{2} x^{2} + 10 \, B a b x + 15 \, A b^{2} x + 2 \, B a^{2} + 3 \, A a b}{60 \,{\left (b x + a\right )}^{5} b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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