Optimal. Leaf size=45 \[ \frac {a (A b-a B)}{b^3 (a+b x)}+\frac {(A b-2 a B) \log (a+b x)}{b^3}+\frac {B x}{b^2} \]
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Rubi [A] time = 0.03, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {77} \begin {gather*} \frac {a (A b-a B)}{b^3 (a+b x)}+\frac {(A b-2 a B) \log (a+b x)}{b^3}+\frac {B x}{b^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 77
Rubi steps
\begin {align*} \int \frac {x (A+B x)}{(a+b x)^2} \, dx &=\int \left (\frac {B}{b^2}+\frac {a (-A b+a B)}{b^2 (a+b x)^2}+\frac {A b-2 a B}{b^2 (a+b x)}\right ) \, dx\\ &=\frac {B x}{b^2}+\frac {a (A b-a B)}{b^3 (a+b x)}+\frac {(A b-2 a B) \log (a+b x)}{b^3}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 41, normalized size = 0.91 \begin {gather*} \frac {\frac {a (A b-a B)}{a+b x}+(A b-2 a B) \log (a+b x)+b B x}{b^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 {x (A+B x)}{(a+b x)^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 1.42, size = 72, normalized size = 1.60 \begin {gather*} \frac {B b^{2} x^{2} + B a b x - B a^{2} + A a b - {\left (2 \, B a^{2} - A a b + {\left (2 \, B a b - A b^{2}\right )} x\right )} \log \left (b x + a\right )}{b^{4} x + a b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.96, size = 80, normalized size = 1.78 \begin {gather*} \frac {\frac {{\left (b x + a\right )} B}{b^{2}} + \frac {{\left (2 \, B a - A b\right )} \log \left (\frac {{\left | b x + a \right |}}{{\left (b x + a\right )}^{2} {\left | b \right |}}\right )}{b^{2}} - \frac {\frac {B a^{2} b}{b x + a} - \frac {A a b^{2}}{b x + a}}{b^{3}}}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 61, normalized size = 1.36 \begin {gather*} \frac {A a}{\left (b x +a \right ) b^{2}}+\frac {A \ln \left (b x +a \right )}{b^{2}}-\frac {B \,a^{2}}{\left (b x +a \right ) b^{3}}-\frac {2 B a \ln \left (b x +a \right )}{b^{3}}+\frac {B x}{b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.07, size = 53, normalized size = 1.18 \begin {gather*} -\frac {B a^{2} - A a b}{b^{4} x + a b^{3}} + \frac {B x}{b^{2}} - \frac {{\left (2 \, B a - A b\right )} \log \left (b x + a\right )}{b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.31, size = 54, normalized size = 1.20 \begin {gather*} \frac {B\,x}{b^2}-\frac {B\,a^2-A\,a\,b}{b\,\left (x\,b^3+a\,b^2\right )}+\frac {\ln \left (a+b\,x\right )\,\left (A\,b-2\,B\,a\right )}{b^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.49, size = 44, normalized size = 0.98 \begin {gather*} \frac {B x}{b^{2}} + \frac {A a b - B a^{2}}{a b^{3} + b^{4} x} - \frac {\left (- A b + 2 B a\right ) \log {\left (a + b x \right )}}{b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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