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 = 0.01, antiderivative size = 31, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\frac {a B+A b}{2 x^2}-\frac {a A}{3 x^3}-\frac {b B}{x} \end {gather*}
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*}
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Mathematica [A] time = 0.01, size = 28, normalized size = 0.90 \begin {gather*} -\frac {a (2 A+3 B x)+3 b x (A+2 B x)}{6 x^3} \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) (A+B x)}{x^4} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 1.27, size = 27, normalized size = 0.87 \begin {gather*} -\frac {6 \, B b x^{2} + 2 \, A a + 3 \, {\left (B a + A b\right )} x}{6 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.25, size = 27, normalized size = 0.87 \begin {gather*} -\frac {6 \, B b x^{2} + 3 \, B a x + 3 \, A b x + 2 \, A a}{6 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 28, normalized size = 0.90 \begin {gather*} -\frac {B b}{x}-\frac {A a}{3 x^{3}}-\frac {A b +B a}{2 x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.14, size = 27, normalized size = 0.87 \begin {gather*} -\frac {6 \, B b x^{2} + 2 \, A a + 3 \, {\left (B a + A b\right )} x}{6 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 27, normalized size = 0.87 \begin {gather*} -\frac {B\,b\,x^2+\left (\frac {A\,b}{2}+\frac {B\,a}{2}\right )\,x+\frac {A\,a}{3}}{x^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.32, size = 31, normalized size = 1.00 \begin {gather*} \frac {- 2 A a - 6 B b x^{2} + x \left (- 3 A b - 3 B a\right )}{6 x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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