Optimal. Leaf size=36 \[ \frac{b (a+b x)^4}{20 a^2 x^4}-\frac{(a+b x)^4}{5 a x^5} \]
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Rubi [A] time = 0.005332, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {45, 37} \[ \frac{b (a+b x)^4}{20 a^2 x^4}-\frac{(a+b x)^4}{5 a x^5} \]
Antiderivative was successfully verified.
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Rule 45
Rule 37
Rubi steps
\begin{align*} \int \frac{(a+b x)^3}{x^6} \, dx &=-\frac{(a+b x)^4}{5 a x^5}-\frac{b \int \frac{(a+b x)^3}{x^5} \, dx}{5 a}\\ &=-\frac{(a+b x)^4}{5 a x^5}+\frac{b (a+b x)^4}{20 a^2 x^4}\\ \end{align*}
Mathematica [A] time = 0.006572, size = 41, normalized size = 1.14 \[ -\frac{3 a^2 b}{4 x^4}-\frac{a^3}{5 x^5}-\frac{a b^2}{x^3}-\frac{b^3}{2 x^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 36, normalized size = 1. \begin{align*} -{\frac{{b}^{2}a}{{x}^{3}}}-{\frac{{a}^{3}}{5\,{x}^{5}}}-{\frac{3\,{a}^{2}b}{4\,{x}^{4}}}-{\frac{{b}^{3}}{2\,{x}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.07295, size = 47, normalized size = 1.31 \begin{align*} -\frac{10 \, b^{3} x^{3} + 20 \, a b^{2} x^{2} + 15 \, a^{2} b x + 4 \, a^{3}}{20 \, x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.5316, size = 81, normalized size = 2.25 \begin{align*} -\frac{10 \, b^{3} x^{3} + 20 \, a b^{2} x^{2} + 15 \, a^{2} b x + 4 \, a^{3}}{20 \, x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.438314, size = 37, normalized size = 1.03 \begin{align*} - \frac{4 a^{3} + 15 a^{2} b x + 20 a b^{2} x^{2} + 10 b^{3} x^{3}}{20 x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13052, size = 47, normalized size = 1.31 \begin{align*} -\frac{10 \, b^{3} x^{3} + 20 \, a b^{2} x^{2} + 15 \, a^{2} b x + 4 \, a^{3}}{20 \, x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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