Optimal. Leaf size=62 \[ \frac {b \log (x) (A b-a B)}{a^3}-\frac {b (A b-a B) \log (a+b x)}{a^3}+\frac {A b-a B}{a^2 x}-\frac {A}{2 a x^2} \]
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Rubi [A] time = 0.04, antiderivative size = 62, 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} \[ \frac {A b-a B}{a^2 x}+\frac {b \log (x) (A b-a B)}{a^3}-\frac {b (A b-a B) \log (a+b x)}{a^3}-\frac {A}{2 a x^2} \]
Antiderivative was successfully verified.
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Rule 77
Rubi steps
\begin {align*} \int \frac {A+B x}{x^3 (a+b x)} \, dx &=\int \left (\frac {A}{a x^3}+\frac {-A b+a B}{a^2 x^2}-\frac {b (-A b+a B)}{a^3 x}+\frac {b^2 (-A b+a B)}{a^3 (a+b x)}\right ) \, dx\\ &=-\frac {A}{2 a x^2}+\frac {A b-a B}{a^2 x}+\frac {b (A b-a B) \log (x)}{a^3}-\frac {b (A b-a B) \log (a+b x)}{a^3}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 58, normalized size = 0.94 \[ \frac {-\frac {a (a A+2 a B x-2 A b x)}{x^2}+2 b \log (x) (A b-a B)+2 b (a B-A b) \log (a+b x)}{2 a^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.94, size = 69, normalized size = 1.11 \[ \frac {2 \, {\left (B a b - A b^{2}\right )} x^{2} \log \left (b x + a\right ) - 2 \, {\left (B a b - A b^{2}\right )} x^{2} \log \relax (x) - A a^{2} - 2 \, {\left (B a^{2} - A a b\right )} x}{2 \, a^{3} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.90, size = 75, normalized size = 1.21 \[ -\frac {{\left (B a b - A b^{2}\right )} \log \left ({\left | x \right |}\right )}{a^{3}} + \frac {{\left (B a b^{2} - A b^{3}\right )} \log \left ({\left | b x + a \right |}\right )}{a^{3} b} - \frac {A a^{2} + 2 \, {\left (B a^{2} - A a b\right )} x}{2 \, a^{3} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 75, normalized size = 1.21 \[ \frac {A \,b^{2} \ln \relax (x )}{a^{3}}-\frac {A \,b^{2} \ln \left (b x +a \right )}{a^{3}}-\frac {B b \ln \relax (x )}{a^{2}}+\frac {B b \ln \left (b x +a \right )}{a^{2}}+\frac {A b}{a^{2} x}-\frac {B}{a x}-\frac {A}{2 a \,x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.03, size = 63, normalized size = 1.02 \[ \frac {{\left (B a b - A b^{2}\right )} \log \left (b x + a\right )}{a^{3}} - \frac {{\left (B a b - A b^{2}\right )} \log \relax (x)}{a^{3}} - \frac {A a + 2 \, {\left (B a - A b\right )} x}{2 \, a^{2} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.36, size = 74, normalized size = 1.19 \[ -\frac {\frac {A}{2\,a}-\frac {x\,\left (A\,b-B\,a\right )}{a^2}}{x^2}-\frac {2\,b\,\mathrm {atanh}\left (\frac {b\,\left (A\,b-B\,a\right )\,\left (a+2\,b\,x\right )}{a\,\left (A\,b^2-B\,a\,b\right )}\right )\,\left (A\,b-B\,a\right )}{a^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.56, size = 131, normalized size = 2.11 \[ \frac {- A a + x \left (2 A b - 2 B a\right )}{2 a^{2} x^{2}} - \frac {b \left (- A b + B a\right ) \log {\left (x + \frac {- A a b^{2} + B a^{2} b - a b \left (- A b + B a\right )}{- 2 A b^{3} + 2 B a b^{2}} \right )}}{a^{3}} + \frac {b \left (- A b + B a\right ) \log {\left (x + \frac {- A a b^{2} + B a^{2} b + a b \left (- A b + B a\right )}{- 2 A b^{3} + 2 B a b^{2}} \right )}}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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