Optimal. Leaf size=57 \[ -\frac {A \log (a+b x)}{a^3}+\frac {A \log (x)}{a^3}+\frac {A}{a^2 (a+b x)}+\frac {A b-a B}{2 a b (a+b x)^2} \]
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Rubi [A] time = 0.04, antiderivative size = 57, 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} \begin {gather*} \frac {A}{a^2 (a+b x)}-\frac {A \log (a+b x)}{a^3}+\frac {A \log (x)}{a^3}+\frac {A b-a B}{2 a b (a+b x)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 77
Rubi steps
\begin {align*} \int \frac {A+B x}{x (a+b x)^3} \, dx &=\int \left (\frac {A}{a^3 x}+\frac {-A b+a B}{a (a+b x)^3}-\frac {A b}{a^2 (a+b x)^2}-\frac {A b}{a^3 (a+b x)}\right ) \, dx\\ &=\frac {A b-a B}{2 a b (a+b x)^2}+\frac {A}{a^2 (a+b x)}+\frac {A \log (x)}{a^3}-\frac {A \log (a+b x)}{a^3}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 53, normalized size = 0.93 \begin {gather*} \frac {\frac {a \left (a^2 (-B)+3 a A b+2 A b^2 x\right )}{b (a+b x)^2}-2 A \log (a+b x)+2 A \log (x)}{2 a^3} \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 (a+b x)^3} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 1.18, size = 109, normalized size = 1.91 \begin {gather*} \frac {2 \, A a b^{2} x - B a^{3} + 3 \, A a^{2} b - 2 \, {\left (A b^{3} x^{2} + 2 \, A a b^{2} x + A a^{2} b\right )} \log \left (b x + a\right ) + 2 \, {\left (A b^{3} x^{2} + 2 \, A a b^{2} x + A a^{2} b\right )} \log \relax (x)}{2 \, {\left (a^{3} b^{3} x^{2} + 2 \, a^{4} b^{2} x + a^{5} b\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.76, size = 59, normalized size = 1.04 \begin {gather*} -\frac {A \log \left ({\left | b x + a \right |}\right )}{a^{3}} + \frac {A \log \left ({\left | x \right |}\right )}{a^{3}} + \frac {2 \, A a b^{2} x - B a^{3} + 3 \, A a^{2} b}{2 \, {\left (b x + a\right )}^{2} a^{3} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 59, normalized size = 1.04 \begin {gather*} \frac {A}{2 \left (b x +a \right )^{2} a}-\frac {B}{2 \left (b x +a \right )^{2} b}+\frac {A}{\left (b x +a \right ) a^{2}}+\frac {A \ln \relax (x )}{a^{3}}-\frac {A \ln \left (b x +a \right )}{a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.09, size = 68, normalized size = 1.19 \begin {gather*} \frac {2 \, A b^{2} x - B a^{2} + 3 \, A a b}{2 \, {\left (a^{2} b^{3} x^{2} + 2 \, a^{3} b^{2} x + a^{4} b\right )}} - \frac {A \log \left (b x + a\right )}{a^{3}} + \frac {A \log \relax (x)}{a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.32, size = 61, normalized size = 1.07 \begin {gather*} \frac {\frac {3\,A\,b-B\,a}{2\,a\,b}+\frac {A\,b\,x}{a^2}}{a^2+2\,a\,b\,x+b^2\,x^2}-\frac {2\,A\,\mathrm {atanh}\left (\frac {2\,b\,x}{a}+1\right )}{a^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.58, size = 63, normalized size = 1.11 \begin {gather*} \frac {A \left (\log {\relax (x )} - \log {\left (\frac {a}{b} + x \right )}\right )}{a^{3}} + \frac {3 A a b + 2 A b^{2} x - B a^{2}}{2 a^{4} b + 4 a^{3} b^{2} x + 2 a^{2} b^{3} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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