Optimal. Leaf size=45 \[ -\frac {a B+A b}{3 x^3}-\frac {a A}{4 x^4}-\frac {A c+b B}{2 x^2}-\frac {B c}{x} \]
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Rubi [A] time = 0.02, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {765} \begin {gather*} -\frac {a B+A b}{3 x^3}-\frac {a A}{4 x^4}-\frac {A c+b B}{2 x^2}-\frac {B c}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 765
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (a+b x+c x^2\right )}{x^5} \, dx &=\int \left (\frac {a A}{x^5}+\frac {A b+a B}{x^4}+\frac {b B+A c}{x^3}+\frac {B c}{x^2}\right ) \, dx\\ &=-\frac {a A}{4 x^4}-\frac {A b+a B}{3 x^3}-\frac {b B+A c}{2 x^2}-\frac {B c}{x}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 42, normalized size = 0.93 \begin {gather*} -\frac {a (3 A+4 B x)+2 x (A (2 b+3 c x)+3 B x (b+2 c x))}{12 x^4} \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) \left (a+b x+c x^2\right )}{x^5} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.39, size = 39, normalized size = 0.87 \begin {gather*} -\frac {12 \, B c x^{3} + 6 \, {\left (B b + A c\right )} x^{2} + 3 \, A a + 4 \, {\left (B a + A b\right )} x}{12 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 41, normalized size = 0.91 \begin {gather*} -\frac {12 \, B c x^{3} + 6 \, B b x^{2} + 6 \, A c x^{2} + 4 \, B a x + 4 \, A b x + 3 \, A a}{12 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 40, normalized size = 0.89 \begin {gather*} -\frac {B c}{x}-\frac {A a}{4 x^{4}}-\frac {A c +b B}{2 x^{2}}-\frac {A b +B a}{3 x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.65, size = 39, normalized size = 0.87 \begin {gather*} -\frac {12 \, B c x^{3} + 6 \, {\left (B b + A c\right )} x^{2} + 3 \, A a + 4 \, {\left (B a + A b\right )} x}{12 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 40, normalized size = 0.89 \begin {gather*} -\frac {B\,c\,x^3+\left (\frac {A\,c}{2}+\frac {B\,b}{2}\right )\,x^2+\left (\frac {A\,b}{3}+\frac {B\,a}{3}\right )\,x+\frac {A\,a}{4}}{x^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.04, size = 46, normalized size = 1.02 \begin {gather*} \frac {- 3 A a - 12 B c x^{3} + x^{2} \left (- 6 A c - 6 B b\right ) + x \left (- 4 A b - 4 B a\right )}{12 x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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