Optimal. Leaf size=49 \[ -\frac {a^2 A}{3 x^3}-\frac {a (a B+2 A b)}{2 x^2}-\frac {b (2 a B+A b)}{x}+b^2 B \log (x) \]
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Rubi [A] time = 0.02, antiderivative size = 49, 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} \[ -\frac {a^2 A}{3 x^3}-\frac {a (a B+2 A b)}{2 x^2}-\frac {b (2 a B+A b)}{x}+b^2 B \log (x) \]
Antiderivative was successfully verified.
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Rule 76
Rubi steps
\begin {align*} \int \frac {(a+b x)^2 (A+B x)}{x^4} \, dx &=\int \left (\frac {a^2 A}{x^4}+\frac {a (2 A b+a B)}{x^3}+\frac {b (A b+2 a B)}{x^2}+\frac {b^2 B}{x}\right ) \, dx\\ &=-\frac {a^2 A}{3 x^3}-\frac {a (2 A b+a B)}{2 x^2}-\frac {b (A b+2 a B)}{x}+b^2 B \log (x)\\ \end {align*}
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Mathematica [A] time = 0.03, size = 48, normalized size = 0.98 \[ b^2 B \log (x)-\frac {a^2 (2 A+3 B x)+6 a b x (A+2 B x)+6 A b^2 x^2}{6 x^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.79, size = 53, normalized size = 1.08 \[ \frac {6 \, B b^{2} x^{3} \log \relax (x) - 2 \, A a^{2} - 6 \, {\left (2 \, B a b + A b^{2}\right )} x^{2} - 3 \, {\left (B a^{2} + 2 \, A a b\right )} x}{6 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.16, size = 51, normalized size = 1.04 \[ B b^{2} \log \left ({\left | x \right |}\right ) - \frac {2 \, A a^{2} + 6 \, {\left (2 \, B a b + A b^{2}\right )} x^{2} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} x}{6 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 52, normalized size = 1.06 \[ B \,b^{2} \ln \relax (x )-\frac {A \,b^{2}}{x}-\frac {2 B a b}{x}-\frac {A a b}{x^{2}}-\frac {B \,a^{2}}{2 x^{2}}-\frac {A \,a^{2}}{3 x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.07, size = 50, normalized size = 1.02 \[ B b^{2} \log \relax (x) - \frac {2 \, A a^{2} + 6 \, {\left (2 \, B a b + A b^{2}\right )} x^{2} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} x}{6 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.32, size = 48, normalized size = 0.98 \[ B\,b^2\,\ln \relax (x)-\frac {x^2\,\left (A\,b^2+2\,B\,a\,b\right )+\frac {A\,a^2}{3}+x\,\left (\frac {B\,a^2}{2}+A\,b\,a\right )}{x^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.59, size = 54, normalized size = 1.10 \[ B b^{2} \log {\relax (x )} + \frac {- 2 A a^{2} + x^{2} \left (- 6 A b^{2} - 12 B a b\right ) + x \left (- 6 A a b - 3 B a^{2}\right )}{6 x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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