Optimal. Leaf size=82 \[ -\frac {a^5 x^{-4 n}}{4 n}-\frac {5 a^4 b x^{-3 n}}{3 n}-\frac {5 a^3 b^2 x^{-2 n}}{n}-\frac {10 a^2 b^3 x^{-n}}{n}+5 a b^4 \log (x)+\frac {b^5 x^n}{n} \]
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Rubi [A] time = 0.04, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {266, 43} \[ -\frac {5 a^3 b^2 x^{-2 n}}{n}-\frac {10 a^2 b^3 x^{-n}}{n}-\frac {5 a^4 b x^{-3 n}}{3 n}-\frac {a^5 x^{-4 n}}{4 n}+5 a b^4 \log (x)+\frac {b^5 x^n}{n} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rubi steps
\begin {align*} \int x^{-1-4 n} \left (a+b x^n\right )^5 \, dx &=\frac {\operatorname {Subst}\left (\int \frac {(a+b x)^5}{x^5} \, dx,x,x^n\right )}{n}\\ &=\frac {\operatorname {Subst}\left (\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,x,x^n\right )}{n}\\ &=-\frac {a^5 x^{-4 n}}{4 n}-\frac {5 a^4 b x^{-3 n}}{3 n}-\frac {5 a^3 b^2 x^{-2 n}}{n}-\frac {10 a^2 b^3 x^{-n}}{n}+\frac {b^5 x^n}{n}+5 a b^4 \log (x)\\ \end {align*}
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Mathematica [A] time = 0.05, size = 72, normalized size = 0.88 \[ \frac {-\frac {1}{4} a^5 x^{-4 n}-\frac {5}{3} a^4 b x^{-3 n}-5 a^3 b^2 x^{-2 n}-10 a^2 b^3 x^{-n}+5 a b^4 n \log (x)+b^5 x^n}{n} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.95, size = 77, normalized size = 0.94 \[ \frac {60 \, a b^{4} n x^{4 \, n} \log \relax (x) + 12 \, b^{5} x^{5 \, n} - 120 \, a^{2} b^{3} x^{3 \, n} - 60 \, a^{3} b^{2} x^{2 \, n} - 20 \, a^{4} b x^{n} - 3 \, a^{5}}{12 \, n x^{4 \, n}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 77, normalized size = 0.94 \[ \frac {60 \, a b^{4} n x^{4 \, n} \log \relax (x) + 12 \, b^{5} x^{5 \, n} - 120 \, a^{2} b^{3} x^{3 \, n} - 60 \, a^{3} b^{2} x^{2 \, n} - 20 \, a^{4} b x^{n} - 3 \, a^{5}}{12 \, n x^{4 \, n}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 97, normalized size = 1.18 \[ \left (5 a \,b^{4} {\mathrm e}^{4 n \ln \relax (x )} \ln \relax (x )-\frac {5 a^{4} b \,{\mathrm e}^{n \ln \relax (x )}}{3 n}-\frac {5 a^{3} b^{2} {\mathrm e}^{2 n \ln \relax (x )}}{n}-\frac {10 a^{2} b^{3} {\mathrm e}^{3 n \ln \relax (x )}}{n}+\frac {b^{5} {\mathrm e}^{5 n \ln \relax (x )}}{n}-\frac {a^{5}}{4 n}\right ) {\mathrm e}^{-4 n \ln \relax (x )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.71, size = 84, normalized size = 1.02 \[ 5 \, a b^{4} \log \relax (x) + \frac {b^{5} x^{n}}{n} - \frac {a^{5}}{4 \, n x^{4 \, n}} - \frac {5 \, a^{4} b}{3 \, n x^{3 \, n}} - \frac {5 \, a^{3} b^{2}}{n x^{2 \, n}} - \frac {10 \, a^{2} b^{3}}{n x^{n}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.41, size = 84, normalized size = 1.02 \[ \frac {b^5\,x^n}{n}+5\,a\,b^4\,\ln \relax (x)-\frac {a^5}{4\,n\,x^{4\,n}}-\frac {10\,a^2\,b^3}{n\,x^n}-\frac {5\,a^3\,b^2}{n\,x^{2\,n}}-\frac {5\,a^4\,b}{3\,n\,x^{3\,n}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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