Optimal. Leaf size=44 \[ \frac {(a+b x)^3 (A b-4 a B)}{12 a^2 x^3}-\frac {A (a+b x)^3}{4 a x^4} \]
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Rubi [A] time = 0.01, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {78, 37} \begin {gather*} \frac {(a+b x)^3 (A b-4 a B)}{12 a^2 x^3}-\frac {A (a+b x)^3}{4 a x^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 78
Rubi steps
\begin {align*} \int \frac {(a+b x)^2 (A+B x)}{x^5} \, dx &=-\frac {A (a+b x)^3}{4 a x^4}+\frac {(-A b+4 a B) \int \frac {(a+b x)^2}{x^4} \, dx}{4 a}\\ &=-\frac {A (a+b x)^3}{4 a x^4}+\frac {(A b-4 a B) (a+b x)^3}{12 a^2 x^3}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 47, normalized size = 1.07 \begin {gather*} -\frac {a^2 (3 A+4 B x)+4 a b x (2 A+3 B x)+6 b^2 x^2 (A+2 B 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)^2 (A+B x)}{x^5} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 1.32, size = 51, normalized size = 1.16 \begin {gather*} -\frac {12 \, B b^{2} x^{3} + 3 \, A a^{2} + 6 \, {\left (2 \, B a b + A b^{2}\right )} x^{2} + 4 \, {\left (B a^{2} + 2 \, 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 = 1.26, size = 51, normalized size = 1.16 \begin {gather*} -\frac {12 \, B b^{2} x^{3} + 12 \, B a b x^{2} + 6 \, A b^{2} x^{2} + 4 \, B a^{2} x + 8 \, A a b x + 3 \, A a^{2}}{12 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 48, normalized size = 1.09 \begin {gather*} -\frac {B \,b^{2}}{x}-\frac {A \,a^{2}}{4 x^{4}}-\frac {\left (A b +2 B a \right ) b}{2 x^{2}}-\frac {\left (2 A b +B a \right ) a}{3 x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.97, size = 51, normalized size = 1.16 \begin {gather*} -\frac {12 \, B b^{2} x^{3} + 3 \, A a^{2} + 6 \, {\left (2 \, B a b + A b^{2}\right )} x^{2} + 4 \, {\left (B a^{2} + 2 \, 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 = 0.04, size = 49, normalized size = 1.11 \begin {gather*} -\frac {x^2\,\left (\frac {A\,b^2}{2}+B\,a\,b\right )+\frac {A\,a^2}{4}+x\,\left (\frac {B\,a^2}{3}+\frac {2\,A\,b\,a}{3}\right )+B\,b^2\,x^3}{x^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.98, size = 56, normalized size = 1.27 \begin {gather*} \frac {- 3 A a^{2} - 12 B b^{2} x^{3} + x^{2} \left (- 6 A b^{2} - 12 B a b\right ) + x \left (- 8 A a b - 4 B a^{2}\right )}{12 x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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