Optimal. Leaf size=57 \[ -\frac {a^5}{4 x^4}-\frac {5 a^4 b}{3 x^3}-\frac {5 a^3 b^2}{x^2}-\frac {10 a^2 b^3}{x}+5 a b^4 \log (x)+b^5 x \]
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Rubi [A] time = 0.02, antiderivative size = 57, 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 {5 a^3 b^2}{x^2}-\frac {10 a^2 b^3}{x}-\frac {5 a^4 b}{3 x^3}-\frac {a^5}{4 x^4}+5 a b^4 \log (x)+b^5 x \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin {align*} \int \frac {(a+b x)^5}{x^5} \, dx &=\int \left (b^5+\frac {a^5}{x^5}+\frac {5 a^4 b}{x^4}+\frac {10 a^3 b^2}{x^3}+\frac {10 a^2 b^3}{x^2}+\frac {5 a b^4}{x}\right ) \, dx\\ &=-\frac {a^5}{4 x^4}-\frac {5 a^4 b}{3 x^3}-\frac {5 a^3 b^2}{x^2}-\frac {10 a^2 b^3}{x}+b^5 x+5 a b^4 \log (x)\\ \end {align*}
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Mathematica [A] time = 0.01, size = 57, normalized size = 1.00 \begin {gather*} -\frac {a^5}{4 x^4}-\frac {5 a^4 b}{3 x^3}-\frac {5 a^3 b^2}{x^2}-\frac {10 a^2 b^3}{x}+5 a b^4 \log (x)+b^5 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^5} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.81, size = 59, normalized size = 1.04 \begin {gather*} \frac {12 \, b^{5} x^{5} + 60 \, a b^{4} x^{4} \log \relax (x) - 120 \, a^{2} b^{3} x^{3} - 60 \, a^{3} b^{2} x^{2} - 20 \, a^{4} b x - 3 \, a^{5}}{12 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.38, size = 55, normalized size = 0.96 \begin {gather*} b^{5} x + 5 \, a b^{4} \log \left ({\left | x \right |}\right ) - \frac {120 \, a^{2} b^{3} x^{3} + 60 \, a^{3} b^{2} x^{2} + 20 \, a^{4} b x + 3 \, a^{5}}{12 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 54, normalized size = 0.95 \begin {gather*} 5 a \,b^{4} \ln \relax (x )+b^{5} x -\frac {10 a^{2} b^{3}}{x}-\frac {5 a^{3} b^{2}}{x^{2}}-\frac {5 a^{4} b}{3 x^{3}}-\frac {a^{5}}{4 x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.37, size = 54, normalized size = 0.95 \begin {gather*} b^{5} x + 5 \, a b^{4} \log \relax (x) - \frac {120 \, a^{2} b^{3} x^{3} + 60 \, a^{3} b^{2} x^{2} + 20 \, a^{4} b x + 3 \, a^{5}}{12 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 54, normalized size = 0.95 \begin {gather*} b^5\,x-\frac {\frac {a^5}{4}+\frac {5\,a^4\,b\,x}{3}+5\,a^3\,b^2\,x^2+10\,a^2\,b^3\,x^3}{x^4}+5\,a\,b^4\,\ln \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.29, size = 58, normalized size = 1.02 \begin {gather*} 5 a b^{4} \log {\relax (x )} + b^{5} x + \frac {- 3 a^{5} - 20 a^{4} b x - 60 a^{3} b^{2} x^{2} - 120 a^{2} b^{3} x^{3}}{12 x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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