Optimal. Leaf size=31 \[ -\frac{a B+A b}{2 x^2}-\frac{a A}{3 x^3}-\frac{b B}{x} \]
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Rubi [A] time = 0.0133023, antiderivative size = 31, normalized size of antiderivative = 1., 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 = {76} \[ -\frac{a B+A b}{2 x^2}-\frac{a A}{3 x^3}-\frac{b B}{x} \]
Antiderivative was successfully verified.
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Rule 76
Rubi steps
\begin{align*} \int \frac{(a+b x) (A+B x)}{x^4} \, dx &=\int \left (\frac{a A}{x^4}+\frac{A b+a B}{x^3}+\frac{b B}{x^2}\right ) \, dx\\ &=-\frac{a A}{3 x^3}-\frac{A b+a B}{2 x^2}-\frac{b B}{x}\\ \end{align*}
Mathematica [A] time = 0.0089094, size = 28, normalized size = 0.9 \[ -\frac{a (2 A+3 B x)+3 b x (A+2 B x)}{6 x^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 28, normalized size = 0.9 \begin{align*} -{\frac{Aa}{3\,{x}^{3}}}-{\frac{Ab+Ba}{2\,{x}^{2}}}-{\frac{Bb}{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.15249, size = 36, normalized size = 1.16 \begin{align*} -\frac{6 \, B b x^{2} + 2 \, A a + 3 \,{\left (B a + A b\right )} x}{6 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.81821, size = 65, normalized size = 2.1 \begin{align*} -\frac{6 \, B b x^{2} + 2 \, A a + 3 \,{\left (B a + A b\right )} x}{6 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.465642, size = 31, normalized size = 1. \begin{align*} - \frac{2 A a + 6 B b x^{2} + x \left (3 A b + 3 B a\right )}{6 x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17973, size = 36, normalized size = 1.16 \begin{align*} -\frac{6 \, B b x^{2} + 3 \, B a x + 3 \, A b x + 2 \, A a}{6 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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