Optimal. Leaf size=44 \[ -\frac {a^2 A}{2 x^2}-\frac {a (a B+2 A b)}{x}+b \log (x) (2 a B+A b)+b^2 B x \]
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Rubi [A] time = 0.02, antiderivative size = 44, 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 = {76} \begin {gather*} -\frac {a^2 A}{2 x^2}-\frac {a (a B+2 A b)}{x}+b \log (x) (2 a B+A b)+b^2 B x \end {gather*}
Antiderivative was successfully verified.
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Rule 76
Rubi steps
\begin {align*} \int \frac {(a+b x)^2 (A+B x)}{x^3} \, dx &=\int \left (b^2 B+\frac {a^2 A}{x^3}+\frac {a (2 A b+a B)}{x^2}+\frac {b (A b+2 a B)}{x}\right ) \, dx\\ &=-\frac {a^2 A}{2 x^2}-\frac {a (2 A b+a B)}{x}+b^2 B x+b (A b+2 a B) \log (x)\\ \end {align*}
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Mathematica [A] time = 0.03, size = 43, normalized size = 0.98 \begin {gather*} -\frac {a^2 (A+2 B x)}{2 x^2}+b \log (x) (2 a B+A b)-\frac {2 a A b}{x}+b^2 B x \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)^2 (A+B x)}{x^3} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 1.42, size = 53, normalized size = 1.20 \begin {gather*} \frac {2 \, B b^{2} x^{3} + 2 \, {\left (2 \, B a b + A b^{2}\right )} x^{2} \log \relax (x) - A a^{2} - 2 \, {\left (B a^{2} + 2 \, A a b\right )} x}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.19, size = 47, normalized size = 1.07 \begin {gather*} B b^{2} x + {\left (2 \, B a b + A b^{2}\right )} \log \left ({\left | x \right |}\right ) - \frac {A a^{2} + 2 \, {\left (B a^{2} + 2 \, A a b\right )} x}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 48, normalized size = 1.09 \begin {gather*} A \,b^{2} \ln \relax (x )+2 B a b \ln \relax (x )+B \,b^{2} x -\frac {2 A a b}{x}-\frac {B \,a^{2}}{x}-\frac {A \,a^{2}}{2 x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.93, size = 46, normalized size = 1.05 \begin {gather*} B b^{2} x + {\left (2 \, B a b + A b^{2}\right )} \log \relax (x) - \frac {A a^{2} + 2 \, {\left (B a^{2} + 2 \, A a b\right )} x}{2 \, x^{2}} \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 = 1.05 \begin {gather*} \ln \relax (x)\,\left (A\,b^2+2\,B\,a\,b\right )-\frac {\frac {A\,a^2}{2}+x\,\left (B\,a^2+2\,A\,b\,a\right )}{x^2}+B\,b^2\,x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.49, size = 46, normalized size = 1.05 \begin {gather*} B b^{2} x + b \left (A b + 2 B a\right ) \log {\relax (x )} + \frac {- A a^{2} + x \left (- 4 A a b - 2 B a^{2}\right )}{2 x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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