Optimal. Leaf size=102 \[ \frac{A}{a^5 (a+b x)}+\frac{A}{2 a^4 (a+b x)^2}+\frac{A}{3 a^3 (a+b x)^3}+\frac{A}{4 a^2 (a+b x)^4}-\frac{A \log (a+b x)}{a^6}+\frac{A \log (x)}{a^6}+\frac{A b-a B}{5 a b (a+b x)^5} \]
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Rubi [A] time = 0.0742038, antiderivative size = 102, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074, Rules used = {27, 77} \[ \frac{A}{a^5 (a+b x)}+\frac{A}{2 a^4 (a+b x)^2}+\frac{A}{3 a^3 (a+b x)^3}+\frac{A}{4 a^2 (a+b x)^4}-\frac{A \log (a+b x)}{a^6}+\frac{A \log (x)}{a^6}+\frac{A b-a B}{5 a b (a+b x)^5} \]
Antiderivative was successfully verified.
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Rule 27
Rule 77
Rubi steps
\begin{align*} \int \frac{A+B x}{x \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx &=\int \frac{A+B x}{x (a+b x)^6} \, dx\\ &=\int \left (\frac{A}{a^6 x}+\frac{-A b+a B}{a (a+b x)^6}-\frac{A b}{a^2 (a+b x)^5}-\frac{A b}{a^3 (a+b x)^4}-\frac{A b}{a^4 (a+b x)^3}-\frac{A b}{a^5 (a+b x)^2}-\frac{A b}{a^6 (a+b x)}\right ) \, dx\\ &=\frac{A b-a B}{5 a b (a+b x)^5}+\frac{A}{4 a^2 (a+b x)^4}+\frac{A}{3 a^3 (a+b x)^3}+\frac{A}{2 a^4 (a+b x)^2}+\frac{A}{a^5 (a+b x)}+\frac{A \log (x)}{a^6}-\frac{A \log (a+b x)}{a^6}\\ \end{align*}
Mathematica [A] time = 0.0627001, size = 89, normalized size = 0.87 \[ \frac{\frac{a \left (470 a^2 A b^3 x^2+385 a^3 A b^2 x+137 a^4 A b-12 a^5 B+270 a A b^4 x^3+60 A b^5 x^4\right )}{b (a+b x)^5}-60 A \log (a+b x)+60 A \log (x)}{60 a^6} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 98, normalized size = 1. \begin{align*}{\frac{A\ln \left ( x \right ) }{{a}^{6}}}+{\frac{A}{5\,a \left ( bx+a \right ) ^{5}}}-{\frac{B}{5\,b \left ( bx+a \right ) ^{5}}}-{\frac{A\ln \left ( bx+a \right ) }{{a}^{6}}}+{\frac{A}{{a}^{5} \left ( bx+a \right ) }}+{\frac{A}{2\,{a}^{4} \left ( bx+a \right ) ^{2}}}+{\frac{A}{3\,{a}^{3} \left ( bx+a \right ) ^{3}}}+{\frac{A}{4\,{a}^{2} \left ( bx+a \right ) ^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.08678, size = 185, normalized size = 1.81 \begin{align*} \frac{60 \, A b^{5} x^{4} + 270 \, A a b^{4} x^{3} + 470 \, A a^{2} b^{3} x^{2} + 385 \, A a^{3} b^{2} x - 12 \, B a^{5} + 137 \, A a^{4} b}{60 \,{\left (a^{5} b^{6} x^{5} + 5 \, a^{6} b^{5} x^{4} + 10 \, a^{7} b^{4} x^{3} + 10 \, a^{8} b^{3} x^{2} + 5 \, a^{9} b^{2} x + a^{10} b\right )}} - \frac{A \log \left (b x + a\right )}{a^{6}} + \frac{A \log \left (x\right )}{a^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.29422, size = 551, normalized size = 5.4 \begin{align*} \frac{60 \, A a b^{5} x^{4} + 270 \, A a^{2} b^{4} x^{3} + 470 \, A a^{3} b^{3} x^{2} + 385 \, A a^{4} b^{2} x - 12 \, B a^{6} + 137 \, A a^{5} b - 60 \,{\left (A b^{6} x^{5} + 5 \, A a b^{5} x^{4} + 10 \, A a^{2} b^{4} x^{3} + 10 \, A a^{3} b^{3} x^{2} + 5 \, A a^{4} b^{2} x + A a^{5} b\right )} \log \left (b x + a\right ) + 60 \,{\left (A b^{6} x^{5} + 5 \, A a b^{5} x^{4} + 10 \, A a^{2} b^{4} x^{3} + 10 \, A a^{3} b^{3} x^{2} + 5 \, A a^{4} b^{2} x + A a^{5} b\right )} \log \left (x\right )}{60 \,{\left (a^{6} b^{6} x^{5} + 5 \, a^{7} b^{5} x^{4} + 10 \, a^{8} b^{4} x^{3} + 10 \, a^{9} b^{3} x^{2} + 5 \, a^{10} b^{2} x + a^{11} b\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.18633, size = 141, normalized size = 1.38 \begin{align*} \frac{A \left (\log{\left (x \right )} - \log{\left (\frac{a}{b} + x \right )}\right )}{a^{6}} + \frac{137 A a^{4} b + 385 A a^{3} b^{2} x + 470 A a^{2} b^{3} x^{2} + 270 A a b^{4} x^{3} + 60 A b^{5} x^{4} - 12 B a^{5}}{60 a^{10} b + 300 a^{9} b^{2} x + 600 a^{8} b^{3} x^{2} + 600 a^{7} b^{4} x^{3} + 300 a^{6} b^{5} x^{4} + 60 a^{5} b^{6} x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.22947, size = 128, normalized size = 1.25 \begin{align*} -\frac{A \log \left ({\left | b x + a \right |}\right )}{a^{6}} + \frac{A \log \left ({\left | x \right |}\right )}{a^{6}} + \frac{60 \, A a b^{5} x^{4} + 270 \, A a^{2} b^{4} x^{3} + 470 \, A a^{3} b^{3} x^{2} + 385 \, A a^{4} b^{2} x - 12 \, B a^{6} + 137 \, A a^{5} b}{60 \,{\left (b x + a\right )}^{5} a^{6} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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