Optimal. Leaf size=65 \[ -\frac {a^3 A}{x}+a^2 \log (x) (a B+3 A b)+\frac {1}{2} b^2 x^2 (3 a B+A b)+3 a b x (a B+A b)+\frac {1}{3} b^3 B x^3 \]
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Rubi [A] time = 0.04, antiderivative size = 65, 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} \[ a^2 \log (x) (a B+3 A b)-\frac {a^3 A}{x}+\frac {1}{2} b^2 x^2 (3 a B+A b)+3 a b x (a B+A b)+\frac {1}{3} b^3 B x^3 \]
Antiderivative was successfully verified.
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Rule 76
Rubi steps
\begin {align*} \int \frac {(a+b x)^3 (A+B x)}{x^2} \, dx &=\int \left (3 a b (A b+a B)+\frac {a^3 A}{x^2}+\frac {a^2 (3 A b+a B)}{x}+b^2 (A b+3 a B) x+b^3 B x^2\right ) \, dx\\ &=-\frac {a^3 A}{x}+3 a b (A b+a B) x+\frac {1}{2} b^2 (A b+3 a B) x^2+\frac {1}{3} b^3 B x^3+a^2 (3 A b+a B) \log (x)\\ \end {align*}
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Mathematica [A] time = 0.03, size = 67, normalized size = 1.03 \[ -\frac {a^3 A}{x}+\log (x) \left (a^3 B+3 a^2 A b\right )+\frac {1}{2} b^2 x^2 (3 a B+A b)+3 a b x (a B+A b)+\frac {1}{3} b^3 B x^3 \]
Antiderivative was successfully verified.
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fricas [A] time = 0.82, size = 75, normalized size = 1.15 \[ \frac {2 \, B b^{3} x^{4} - 6 \, A a^{3} + 3 \, {\left (3 \, B a b^{2} + A b^{3}\right )} x^{3} + 18 \, {\left (B a^{2} b + A a b^{2}\right )} x^{2} + 6 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x \log \relax (x)}{6 \, x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.27, size = 71, normalized size = 1.09 \[ \frac {1}{3} \, B b^{3} x^{3} + \frac {3}{2} \, B a b^{2} x^{2} + \frac {1}{2} \, A b^{3} x^{2} + 3 \, B a^{2} b x + 3 \, A a b^{2} x - \frac {A a^{3}}{x} + {\left (B a^{3} + 3 \, A a^{2} b\right )} \log \left ({\left | x \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 71, normalized size = 1.09 \[ \frac {B \,b^{3} x^{3}}{3}+\frac {A \,b^{3} x^{2}}{2}+\frac {3 B a \,b^{2} x^{2}}{2}+3 A \,a^{2} b \ln \relax (x )+3 A a \,b^{2} x +B \,a^{3} \ln \relax (x )+3 B \,a^{2} b x -\frac {A \,a^{3}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.07, size = 69, normalized size = 1.06 \[ \frac {1}{3} \, B b^{3} x^{3} - \frac {A a^{3}}{x} + \frac {1}{2} \, {\left (3 \, B a b^{2} + A b^{3}\right )} x^{2} + 3 \, {\left (B a^{2} b + A a b^{2}\right )} x + {\left (B a^{3} + 3 \, A a^{2} b\right )} \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 65, normalized size = 1.00 \[ x^2\,\left (\frac {A\,b^3}{2}+\frac {3\,B\,a\,b^2}{2}\right )+\ln \relax (x)\,\left (B\,a^3+3\,A\,b\,a^2\right )-\frac {A\,a^3}{x}+\frac {B\,b^3\,x^3}{3}+3\,a\,b\,x\,\left (A\,b+B\,a\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.30, size = 70, normalized size = 1.08 \[ - \frac {A a^{3}}{x} + \frac {B b^{3} x^{3}}{3} + a^{2} \left (3 A b + B a\right ) \log {\relax (x )} + x^{2} \left (\frac {A b^{3}}{2} + \frac {3 B a b^{2}}{2}\right ) + x \left (3 A a b^{2} + 3 B a^{2} b\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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