Optimal. Leaf size=88 \[ -\frac {\log (x) (3 A b-a B)}{a^4}+\frac {(3 A b-a B) \log (a+b x)}{a^4}-\frac {2 A b-a B}{a^3 (a+b x)}-\frac {A}{a^3 x}-\frac {A b-a B}{2 a^2 (a+b x)^2} \]
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Rubi [A] time = 0.07, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {77} \[ -\frac {2 A b-a B}{a^3 (a+b x)}-\frac {A b-a B}{2 a^2 (a+b x)^2}-\frac {\log (x) (3 A b-a B)}{a^4}+\frac {(3 A b-a B) \log (a+b x)}{a^4}-\frac {A}{a^3 x} \]
Antiderivative was successfully verified.
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Rule 77
Rubi steps
\begin {align*} \int \frac {A+B x}{x^2 (a+b x)^3} \, dx &=\int \left (\frac {A}{a^3 x^2}+\frac {-3 A b+a B}{a^4 x}-\frac {b (-A b+a B)}{a^2 (a+b x)^3}-\frac {b (-2 A b+a B)}{a^3 (a+b x)^2}-\frac {b (-3 A b+a B)}{a^4 (a+b x)}\right ) \, dx\\ &=-\frac {A}{a^3 x}-\frac {A b-a B}{2 a^2 (a+b x)^2}-\frac {2 A b-a B}{a^3 (a+b x)}-\frac {(3 A b-a B) \log (x)}{a^4}+\frac {(3 A b-a B) \log (a+b x)}{a^4}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 81, normalized size = 0.92 \[ \frac {\frac {a^2 (a B-A b)}{(a+b x)^2}+\frac {2 a (a B-2 A b)}{a+b x}+2 \log (x) (a B-3 A b)+2 (3 A b-a B) \log (a+b x)-\frac {2 a A}{x}}{2 a^4} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.88, size = 187, normalized size = 2.12 \[ -\frac {2 \, A a^{3} - 2 \, {\left (B a^{2} b - 3 \, A a b^{2}\right )} x^{2} - 3 \, {\left (B a^{3} - 3 \, A a^{2} b\right )} x + 2 \, {\left ({\left (B a b^{2} - 3 \, A b^{3}\right )} x^{3} + 2 \, {\left (B a^{2} b - 3 \, A a b^{2}\right )} x^{2} + {\left (B a^{3} - 3 \, A a^{2} b\right )} x\right )} \log \left (b x + a\right ) - 2 \, {\left ({\left (B a b^{2} - 3 \, A b^{3}\right )} x^{3} + 2 \, {\left (B a^{2} b - 3 \, A a b^{2}\right )} x^{2} + {\left (B a^{3} - 3 \, A a^{2} b\right )} x\right )} \log \relax (x)}{2 \, {\left (a^{4} b^{2} x^{3} + 2 \, a^{5} b x^{2} + a^{6} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.09, size = 99, normalized size = 1.12 \[ \frac {{\left (B a - 3 \, A b\right )} \log \left ({\left | x \right |}\right )}{a^{4}} - \frac {{\left (B a b - 3 \, A b^{2}\right )} \log \left ({\left | b x + a \right |}\right )}{a^{4} b} - \frac {2 \, A a^{3} - 2 \, {\left (B a^{2} b - 3 \, A a b^{2}\right )} x^{2} - 3 \, {\left (B a^{3} - 3 \, A a^{2} b\right )} x}{2 \, {\left (b x + a\right )}^{2} a^{4} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 105, normalized size = 1.19 \[ -\frac {A b}{2 \left (b x +a \right )^{2} a^{2}}+\frac {B}{2 \left (b x +a \right )^{2} a}-\frac {2 A b}{\left (b x +a \right ) a^{3}}-\frac {3 A b \ln \relax (x )}{a^{4}}+\frac {3 A b \ln \left (b x +a \right )}{a^{4}}+\frac {B}{\left (b x +a \right ) a^{2}}+\frac {B \ln \relax (x )}{a^{3}}-\frac {B \ln \left (b x +a \right )}{a^{3}}-\frac {A}{a^{3} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.11, size = 100, normalized size = 1.14 \[ -\frac {2 \, A a^{2} - 2 \, {\left (B a b - 3 \, A b^{2}\right )} x^{2} - 3 \, {\left (B a^{2} - 3 \, A a b\right )} x}{2 \, {\left (a^{3} b^{2} x^{3} + 2 \, a^{4} b x^{2} + a^{5} x\right )}} - \frac {{\left (B a - 3 \, A b\right )} \log \left (b x + a\right )}{a^{4}} + \frac {{\left (B a - 3 \, A b\right )} \log \relax (x)}{a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.10, size = 87, normalized size = 0.99 \[ \frac {2\,\mathrm {atanh}\left (\frac {2\,b\,x}{a}+1\right )\,\left (3\,A\,b-B\,a\right )}{a^4}-\frac {\frac {A}{a}+\frac {3\,x\,\left (3\,A\,b-B\,a\right )}{2\,a^2}+\frac {b\,x^2\,\left (3\,A\,b-B\,a\right )}{a^3}}{a^2\,x+2\,a\,b\,x^2+b^2\,x^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.25, size = 168, normalized size = 1.91 \[ \frac {- 2 A a^{2} + x^{2} \left (- 6 A b^{2} + 2 B a b\right ) + x \left (- 9 A a b + 3 B a^{2}\right )}{2 a^{5} x + 4 a^{4} b x^{2} + 2 a^{3} b^{2} x^{3}} + \frac {\left (- 3 A b + B a\right ) \log {\left (x + \frac {- 3 A a b + B a^{2} - a \left (- 3 A b + B a\right )}{- 6 A b^{2} + 2 B a b} \right )}}{a^{4}} - \frac {\left (- 3 A b + B a\right ) \log {\left (x + \frac {- 3 A a b + B a^{2} + a \left (- 3 A b + B a\right )}{- 6 A b^{2} + 2 B a b} \right )}}{a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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