Optimal. Leaf size=61 \[ -\frac {a^5}{5 x^5}-\frac {5 a^4 b}{4 x^4}-\frac {10 a^3 b^2}{3 x^3}-\frac {5 a^2 b^3}{x^2}-\frac {5 a b^4}{x}+b^5 \log (x) \]
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Rubi [A] time = 0.02, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {43} \begin {gather*} -\frac {10 a^3 b^2}{3 x^3}-\frac {5 a^2 b^3}{x^2}-\frac {5 a^4 b}{4 x^4}-\frac {a^5}{5 x^5}-\frac {5 a b^4}{x}+b^5 \log (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin {align*} \int \frac {(a+b x)^5}{x^6} \, dx &=\int \left (\frac {a^5}{x^6}+\frac {5 a^4 b}{x^5}+\frac {10 a^3 b^2}{x^4}+\frac {10 a^2 b^3}{x^3}+\frac {5 a b^4}{x^2}+\frac {b^5}{x}\right ) \, dx\\ &=-\frac {a^5}{5 x^5}-\frac {5 a^4 b}{4 x^4}-\frac {10 a^3 b^2}{3 x^3}-\frac {5 a^2 b^3}{x^2}-\frac {5 a b^4}{x}+b^5 \log (x)\\ \end {align*}
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Mathematica [A] time = 0.00, size = 61, normalized size = 1.00 \begin {gather*} -\frac {a^5}{5 x^5}-\frac {5 a^4 b}{4 x^4}-\frac {10 a^3 b^2}{3 x^3}-\frac {5 a^2 b^3}{x^2}-\frac {5 a b^4}{x}+b^5 \log (x) \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)^5}{x^6} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 1.93, size = 59, normalized size = 0.97 \begin {gather*} \frac {60 \, b^{5} x^{5} \log \relax (x) - 300 \, a b^{4} x^{4} - 300 \, a^{2} b^{3} x^{3} - 200 \, a^{3} b^{2} x^{2} - 75 \, a^{4} b x - 12 \, a^{5}}{60 \, x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.37, size = 57, normalized size = 0.93 \begin {gather*} b^{5} \log \left ({\left | x \right |}\right ) - \frac {300 \, a b^{4} x^{4} + 300 \, a^{2} b^{3} x^{3} + 200 \, a^{3} b^{2} x^{2} + 75 \, a^{4} b x + 12 \, a^{5}}{60 \, x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 56, normalized size = 0.92 \begin {gather*} b^{5} \ln \relax (x )-\frac {5 a \,b^{4}}{x}-\frac {5 a^{2} b^{3}}{x^{2}}-\frac {10 a^{3} b^{2}}{3 x^{3}}-\frac {5 a^{4} b}{4 x^{4}}-\frac {a^{5}}{5 x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.35, size = 56, normalized size = 0.92 \begin {gather*} b^{5} \log \relax (x) - \frac {300 \, a b^{4} x^{4} + 300 \, a^{2} b^{3} x^{3} + 200 \, a^{3} b^{2} x^{2} + 75 \, a^{4} b x + 12 \, a^{5}}{60 \, x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 56, normalized size = 0.92 \begin {gather*} b^5\,\ln \relax (x)-\frac {\frac {a^5}{5}+\frac {5\,a^4\,b\,x}{4}+\frac {10\,a^3\,b^2\,x^2}{3}+5\,a^2\,b^3\,x^3+5\,a\,b^4\,x^4}{x^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.36, size = 60, normalized size = 0.98 \begin {gather*} b^{5} \log {\relax (x )} + \frac {- 12 a^{5} - 75 a^{4} b x - 200 a^{3} b^{2} x^{2} - 300 a^{2} b^{3} x^{3} - 300 a b^{4} x^{4}}{60 x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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