Optimal. Leaf size=72 \[ -\frac {A \log (a+b x)}{a^4}+\frac {A \log (x)}{a^4}+\frac {A}{a^3 (a+b x)}+\frac {A}{2 a^2 (a+b x)^2}+\frac {A b-a B}{3 a b (a+b x)^3} \]
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Rubi [A] time = 0.05, antiderivative size = 72, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {A}{a^3 (a+b x)}+\frac {A}{2 a^2 (a+b x)^2}-\frac {A \log (a+b x)}{a^4}+\frac {A \log (x)}{a^4}+\frac {A b-a B}{3 a b (a+b x)^3} \end {gather*}
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 )^2} \, dx &=\int \frac {A+B x}{x (a+b x)^4} \, dx\\ &=\int \left (\frac {A}{a^4 x}+\frac {-A b+a B}{a (a+b x)^4}-\frac {A b}{a^2 (a+b x)^3}-\frac {A b}{a^3 (a+b x)^2}-\frac {A b}{a^4 (a+b x)}\right ) \, dx\\ &=\frac {A b-a B}{3 a b (a+b x)^3}+\frac {A}{2 a^2 (a+b x)^2}+\frac {A}{a^3 (a+b x)}+\frac {A \log (x)}{a^4}-\frac {A \log (a+b x)}{a^4}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 65, normalized size = 0.90 \begin {gather*} \frac {\frac {a \left (-2 a^3 B+11 a^2 A b+15 a A b^2 x+6 A b^3 x^2\right )}{b (a+b x)^3}-6 A \log (a+b x)+6 A \log (x)}{6 a^4} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {A+B x}{x \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.44, size = 156, normalized size = 2.17 \begin {gather*} \frac {6 \, A a b^{3} x^{2} + 15 \, A a^{2} b^{2} x - 2 \, B a^{4} + 11 \, A a^{3} b - 6 \, {\left (A b^{4} x^{3} + 3 \, A a b^{3} x^{2} + 3 \, A a^{2} b^{2} x + A a^{3} b\right )} \log \left (b x + a\right ) + 6 \, {\left (A b^{4} x^{3} + 3 \, A a b^{3} x^{2} + 3 \, A a^{2} b^{2} x + A a^{3} b\right )} \log \relax (x)}{6 \, {\left (a^{4} b^{4} x^{3} + 3 \, a^{5} b^{3} x^{2} + 3 \, a^{6} b^{2} x + a^{7} b\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 71, normalized size = 0.99 \begin {gather*} -\frac {A \log \left ({\left | b x + a \right |}\right )}{a^{4}} + \frac {A \log \left ({\left | x \right |}\right )}{a^{4}} + \frac {6 \, A a b^{3} x^{2} + 15 \, A a^{2} b^{2} x - 2 \, B a^{4} + 11 \, A a^{3} b}{6 \, {\left (b x + a\right )}^{3} a^{4} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 72, normalized size = 1.00 \begin {gather*} \frac {A}{3 \left (b x +a \right )^{3} a}-\frac {B}{3 \left (b x +a \right )^{3} b}+\frac {A}{2 \left (b x +a \right )^{2} a^{2}}+\frac {A}{\left (b x +a \right ) a^{3}}+\frac {A \ln \relax (x )}{a^{4}}-\frac {A \ln \left (b x +a \right )}{a^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.59, size = 91, normalized size = 1.26 \begin {gather*} \frac {6 \, A b^{3} x^{2} + 15 \, A a b^{2} x - 2 \, B a^{3} + 11 \, A a^{2} b}{6 \, {\left (a^{3} b^{4} x^{3} + 3 \, a^{4} b^{3} x^{2} + 3 \, a^{5} b^{2} x + a^{6} b\right )}} - \frac {A \log \left (b x + a\right )}{a^{4}} + \frac {A \log \relax (x)}{a^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.11, size = 84, normalized size = 1.17 \begin {gather*} \frac {\frac {11\,A\,b-2\,B\,a}{6\,a\,b}+\frac {5\,A\,b\,x}{2\,a^2}+\frac {A\,b^2\,x^2}{a^3}}{a^3+3\,a^2\,b\,x+3\,a\,b^2\,x^2+b^3\,x^3}-\frac {2\,A\,\mathrm {atanh}\left (\frac {2\,b\,x}{a}+1\right )}{a^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.53, size = 90, normalized size = 1.25 \begin {gather*} \frac {A \left (\log {\relax (x )} - \log {\left (\frac {a}{b} + x \right )}\right )}{a^{4}} + \frac {11 A a^{2} b + 15 A a b^{2} x + 6 A b^{3} x^{2} - 2 B a^{3}}{6 a^{6} b + 18 a^{5} b^{2} x + 18 a^{4} b^{3} x^{2} + 6 a^{3} b^{4} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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