Optimal. Leaf size=85 \[ \frac {b \log (x) (3 A b-2 a B)}{a^4}-\frac {b (3 A b-2 a B) \log (a+b x)}{a^4}+\frac {2 A b-a B}{a^3 x}+\frac {b (A b-a B)}{a^3 (a+b x)}-\frac {A}{2 a^2 x^2} \]
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Rubi [A] time = 0.07, antiderivative size = 85, 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 {2 A b-a B}{a^3 x}+\frac {b (A b-a B)}{a^3 (a+b x)}+\frac {b \log (x) (3 A b-2 a B)}{a^4}-\frac {b (3 A b-2 a B) \log (a+b x)}{a^4}-\frac {A}{2 a^2 x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 77
Rubi steps
\begin {align*} \int \frac {A+B x}{x^3 \left (a^2+2 a b x+b^2 x^2\right )} \, dx &=\int \frac {A+B x}{x^3 (a+b x)^2} \, dx\\ &=\int \left (\frac {A}{a^2 x^3}+\frac {-2 A b+a B}{a^3 x^2}-\frac {b (-3 A b+2 a B)}{a^4 x}+\frac {b^2 (-A b+a B)}{a^3 (a+b x)^2}+\frac {b^2 (-3 A b+2 a B)}{a^4 (a+b x)}\right ) \, dx\\ &=-\frac {A}{2 a^2 x^2}+\frac {2 A b-a B}{a^3 x}+\frac {b (A b-a B)}{a^3 (a+b x)}+\frac {b (3 A b-2 a B) \log (x)}{a^4}-\frac {b (3 A b-2 a B) \log (a+b x)}{a^4}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 85, normalized size = 1.00 \begin {gather*} \frac {-\frac {a \left (a^2 (A+2 B x)+a b x (4 B x-3 A)-6 A b^2 x^2\right )}{x^2 (a+b x)}+2 b \log (x) (3 A b-2 a B)+2 b (2 a B-3 A b) \log (a+b x)}{2 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^3 \left (a^2+2 a b x+b^2 x^2\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.41, size = 150, normalized size = 1.76 \begin {gather*} -\frac {A a^{3} + 2 \, {\left (2 \, B a^{2} b - 3 \, A a b^{2}\right )} x^{2} + {\left (2 \, B a^{3} - 3 \, A a^{2} b\right )} x - 2 \, {\left ({\left (2 \, B a b^{2} - 3 \, A b^{3}\right )} x^{3} + {\left (2 \, B a^{2} b - 3 \, A a b^{2}\right )} x^{2}\right )} \log \left (b x + a\right ) + 2 \, {\left ({\left (2 \, B a b^{2} - 3 \, A b^{3}\right )} x^{3} + {\left (2 \, B a^{2} b - 3 \, A a b^{2}\right )} x^{2}\right )} \log \relax (x)}{2 \, {\left (a^{4} b x^{3} + a^{5} x^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 106, normalized size = 1.25 \begin {gather*} -\frac {{\left (2 \, B a b - 3 \, A b^{2}\right )} \log \left ({\left | x \right |}\right )}{a^{4}} + \frac {{\left (2 \, B a b^{2} - 3 \, A b^{3}\right )} \log \left ({\left | b x + a \right |}\right )}{a^{4} b} - \frac {A a^{3} + 2 \, {\left (2 \, B a^{2} b - 3 \, A a b^{2}\right )} x^{2} + {\left (2 \, B a^{3} - 3 \, A a^{2} b\right )} x}{2 \, {\left (b x + a\right )} a^{4} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 107, normalized size = 1.26 \begin {gather*} \frac {A \,b^{2}}{\left (b x +a \right ) a^{3}}+\frac {3 A \,b^{2} \ln \relax (x )}{a^{4}}-\frac {3 A \,b^{2} \ln \left (b x +a \right )}{a^{4}}-\frac {B b}{\left (b x +a \right ) a^{2}}-\frac {2 B b \ln \relax (x )}{a^{3}}+\frac {2 B b \ln \left (b x +a \right )}{a^{3}}+\frac {2 A b}{a^{3} x}-\frac {B}{a^{2} x}-\frac {A}{2 a^{2} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.59, size = 99, normalized size = 1.16 \begin {gather*} -\frac {A a^{2} + 2 \, {\left (2 \, B a b - 3 \, A b^{2}\right )} x^{2} + {\left (2 \, B a^{2} - 3 \, A a b\right )} x}{2 \, {\left (a^{3} b x^{3} + a^{4} x^{2}\right )}} + \frac {{\left (2 \, B a b - 3 \, A b^{2}\right )} \log \left (b x + a\right )}{a^{4}} - \frac {{\left (2 \, B a b - 3 \, A b^{2}\right )} \log \relax (x)}{a^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.14, size = 104, normalized size = 1.22 \begin {gather*} \frac {\frac {x\,\left (3\,A\,b-2\,B\,a\right )}{2\,a^2}-\frac {A}{2\,a}+\frac {b\,x^2\,\left (3\,A\,b-2\,B\,a\right )}{a^3}}{b\,x^3+a\,x^2}-\frac {2\,b\,\mathrm {atanh}\left (\frac {b\,\left (3\,A\,b-2\,B\,a\right )\,\left (a+2\,b\,x\right )}{a\,\left (3\,A\,b^2-2\,B\,a\,b\right )}\right )\,\left (3\,A\,b-2\,B\,a\right )}{a^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.59, size = 184, normalized size = 2.16 \begin {gather*} \frac {- A a^{2} + x^{2} \left (6 A b^{2} - 4 B a b\right ) + x \left (3 A a b - 2 B a^{2}\right )}{2 a^{4} x^{2} + 2 a^{3} b x^{3}} - \frac {b \left (- 3 A b + 2 B a\right ) \log {\left (x + \frac {- 3 A a b^{2} + 2 B a^{2} b - a b \left (- 3 A b + 2 B a\right )}{- 6 A b^{3} + 4 B a b^{2}} \right )}}{a^{4}} + \frac {b \left (- 3 A b + 2 B a\right ) \log {\left (x + \frac {- 3 A a b^{2} + 2 B a^{2} b + a b \left (- 3 A b + 2 B a\right )}{- 6 A b^{3} + 4 B a b^{2}} \right )}}{a^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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