Optimal. Leaf size=45 \[ -\frac {a (A b-a B) \log (a+b x)}{b^3}+\frac {x (A b-a B)}{b^2}+\frac {B x^2}{2 b} \]
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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 {x (A b-a B)}{b^2}-\frac {a (A b-a B) \log (a+b x)}{b^3}+\frac {B x^2}{2 b} \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} \, dx &=\int \left (\frac {A b-a B}{b^2}+\frac {B x}{b}+\frac {a (-A b+a B)}{b^2 (a+b x)}\right ) \, dx\\ &=\frac {(A b-a B) x}{b^2}+\frac {B x^2}{2 b}-\frac {a (A b-a B) \log (a+b x)}{b^3}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 41, normalized size = 0.91 \begin {gather*} \frac {b x (-2 a B+2 A b+b B x)+2 a (a B-A b) \log (a+b x)}{2 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} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.74, size = 47, normalized size = 1.04 \begin {gather*} \frac {B b^{2} x^{2} - 2 \, {\left (B a b - A b^{2}\right )} x + 2 \, {\left (B a^{2} - A a b\right )} \log \left (b x + a\right )}{2 \, b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.91, size = 45, normalized size = 1.00 \begin {gather*} \frac {B b x^{2} - 2 \, B a x + 2 \, A b x}{2 \, b^{2}} + \frac {{\left (B a^{2} - A a b\right )} \log \left ({\left | b x + a \right |}\right )}{b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 52, normalized size = 1.16 \begin {gather*} \frac {B \,x^{2}}{2 b}-\frac {A a \ln \left (b x +a \right )}{b^{2}}+\frac {A x}{b}+\frac {B \,a^{2} \ln \left (b x +a \right )}{b^{3}}-\frac {B a x}{b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.10, size = 45, normalized size = 1.00 \begin {gather*} \frac {B b x^{2} - 2 \, {\left (B a - A b\right )} x}{2 \, b^{2}} + \frac {{\left (B a^{2} - 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 = 0.06, size = 46, normalized size = 1.02 \begin {gather*} x\,\left (\frac {A}{b}-\frac {B\,a}{b^2}\right )+\frac {B\,x^2}{2\,b}+\frac {\ln \left (a+b\,x\right )\,\left (B\,a^2-A\,a\,b\right )}{b^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.21, size = 37, normalized size = 0.82 \begin {gather*} \frac {B x^{2}}{2 b} + \frac {a \left (- A b + B a\right ) \log {\left (a + b x \right )}}{b^{3}} + x \left (\frac {A}{b} - \frac {B a}{b^{2}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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