Optimal. Leaf size=44 \[ \frac {(a+b x)^4 (A b-5 a B)}{20 a^2 x^4}-\frac {A (a+b x)^4}{5 a x^5} \]
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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)^4 (A b-5 a B)}{20 a^2 x^4}-\frac {A (a+b x)^4}{5 a x^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 78
Rubi steps
\begin {align*} \int \frac {(a+b x)^3 (A+B x)}{x^6} \, dx &=-\frac {A (a+b x)^4}{5 a x^5}+\frac {(-A b+5 a B) \int \frac {(a+b x)^3}{x^5} \, dx}{5 a}\\ &=-\frac {A (a+b x)^4}{5 a x^5}+\frac {(A b-5 a B) (a+b x)^4}{20 a^2 x^4}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 66, normalized size = 1.50 \begin {gather*} -\frac {a^3 (4 A+5 B x)+5 a^2 b x (3 A+4 B x)+10 a b^2 x^2 (2 A+3 B x)+10 b^3 x^3 (A+2 B x)}{20 x^5} \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)^3 (A+B x)}{x^6} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 1.22, size = 73, normalized size = 1.66 \begin {gather*} -\frac {20 \, B b^{3} x^{4} + 4 \, A a^{3} + 10 \, {\left (3 \, B a b^{2} + A b^{3}\right )} x^{3} + 20 \, {\left (B a^{2} b + A a b^{2}\right )} x^{2} + 5 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x}{20 \, x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.25, size = 75, normalized size = 1.70 \begin {gather*} -\frac {20 \, B b^{3} x^{4} + 30 \, B a b^{2} x^{3} + 10 \, A b^{3} x^{3} + 20 \, B a^{2} b x^{2} + 20 \, A a b^{2} x^{2} + 5 \, B a^{3} x + 15 \, A a^{2} b x + 4 \, A a^{3}}{20 \, x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 66, normalized size = 1.50 \begin {gather*} -\frac {B \,b^{3}}{x}-\frac {\left (A b +3 B a \right ) b^{2}}{2 x^{2}}-\frac {A \,a^{3}}{5 x^{5}}-\frac {\left (A b +B a \right ) a b}{x^{3}}-\frac {\left (3 A b +B a \right ) a^{2}}{4 x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.99, size = 73, normalized size = 1.66 \begin {gather*} -\frac {20 \, B b^{3} x^{4} + 4 \, A a^{3} + 10 \, {\left (3 \, B a b^{2} + A b^{3}\right )} x^{3} + 20 \, {\left (B a^{2} b + A a b^{2}\right )} x^{2} + 5 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x}{20 \, x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.31, size = 71, normalized size = 1.61 \begin {gather*} -\frac {x^2\,\left (B\,a^2\,b+A\,a\,b^2\right )+x\,\left (\frac {B\,a^3}{4}+\frac {3\,A\,b\,a^2}{4}\right )+\frac {A\,a^3}{5}+x^3\,\left (\frac {A\,b^3}{2}+\frac {3\,B\,a\,b^2}{2}\right )+B\,b^3\,x^4}{x^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.72, size = 82, normalized size = 1.86 \begin {gather*} \frac {- 4 A a^{3} - 20 B b^{3} x^{4} + x^{3} \left (- 10 A b^{3} - 30 B a b^{2}\right ) + x^{2} \left (- 20 A a b^{2} - 20 B a^{2} b\right ) + x \left (- 15 A a^{2} b - 5 B a^{3}\right )}{20 x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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