Optimal. Leaf size=65 \[ -\frac {a^3 A}{2 x^2}-\frac {a^2 (a B+3 A b)}{x}+b^2 x (3 a B+A b)+3 a b \log (x) (a B+A b)+\frac {1}{2} b^3 B x^2 \]
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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} \[ -\frac {a^2 (a B+3 A b)}{x}-\frac {a^3 A}{2 x^2}+b^2 x (3 a B+A b)+3 a b \log (x) (a B+A b)+\frac {1}{2} b^3 B x^2 \]
Antiderivative was successfully verified.
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Rule 76
Rubi steps
\begin {align*} \int \frac {(a+b x)^3 (A+B x)}{x^3} \, dx &=\int \left (b^2 (A b+3 a B)+\frac {a^3 A}{x^3}+\frac {a^2 (3 A b+a B)}{x^2}+\frac {3 a b (A b+a B)}{x}+b^3 B x\right ) \, dx\\ &=-\frac {a^3 A}{2 x^2}-\frac {a^2 (3 A b+a B)}{x}+b^2 (A b+3 a B) x+\frac {1}{2} b^3 B x^2+3 a b (A b+a B) \log (x)\\ \end {align*}
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Mathematica [A] time = 0.03, size = 62, normalized size = 0.95 \[ \frac {1}{2} \left (-\frac {a^3 (A+2 B x)}{x^2}-\frac {6 a^2 A b}{x}+6 a b \log (x) (a B+A b)+6 a b^2 B x+b^3 x (2 A+B x)\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.75, size = 74, normalized size = 1.14 \[ \frac {B b^{3} x^{4} - A a^{3} + 2 \, {\left (3 \, B a b^{2} + A b^{3}\right )} x^{3} + 6 \, {\left (B a^{2} b + A a b^{2}\right )} x^{2} \log \relax (x) - 2 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.23, size = 69, normalized size = 1.06 \[ \frac {1}{2} \, B b^{3} x^{2} + 3 \, B a b^{2} x + A b^{3} x + 3 \, {\left (B a^{2} b + A a b^{2}\right )} \log \left ({\left | x \right |}\right ) - \frac {A a^{3} + 2 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x}{2 \, x^{2}} \]
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 \[ \frac {B \,b^{3} x^{2}}{2}+3 A a \,b^{2} \ln \relax (x )+A \,b^{3} x +3 B \,a^{2} b \ln \relax (x )+3 B a \,b^{2} x -\frac {3 A \,a^{2} b}{x}-\frac {B \,a^{3}}{x}-\frac {A \,a^{3}}{2 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.01, size = 69, normalized size = 1.06 \[ \frac {1}{2} \, B b^{3} x^{2} + {\left (3 \, B a b^{2} + A b^{3}\right )} x + 3 \, {\left (B a^{2} b + A a b^{2}\right )} \log \relax (x) - \frac {A a^{3} + 2 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 70, normalized size = 1.08 \[ \ln \relax (x)\,\left (3\,B\,a^2\,b+3\,A\,a\,b^2\right )-\frac {x\,\left (B\,a^3+3\,A\,b\,a^2\right )+\frac {A\,a^3}{2}}{x^2}+x\,\left (A\,b^3+3\,B\,a\,b^2\right )+\frac {B\,b^3\,x^2}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.65, size = 68, normalized size = 1.05 \[ \frac {B b^{3} x^{2}}{2} + 3 a b \left (A b + B a\right ) \log {\relax (x )} + x \left (A b^{3} + 3 B a b^{2}\right ) + \frac {- A a^{3} + x \left (- 6 A a^{2} b - 2 B a^{3}\right )}{2 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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