Optimal. Leaf size=97 \[ -\frac {a^5 x^{-9 n}}{9 n}-\frac {5 a^4 b x^{-8 n}}{8 n}-\frac {10 a^3 b^2 x^{-7 n}}{7 n}-\frac {5 a^2 b^3 x^{-6 n}}{3 n}-\frac {a b^4 x^{-5 n}}{n}-\frac {b^5 x^{-4 n}}{4 n} \]
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Rubi [A] time = 0.04, antiderivative size = 97, 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} \[ -\frac {10 a^3 b^2 x^{-7 n}}{7 n}-\frac {5 a^2 b^3 x^{-6 n}}{3 n}-\frac {5 a^4 b x^{-8 n}}{8 n}-\frac {a^5 x^{-9 n}}{9 n}-\frac {a b^4 x^{-5 n}}{n}-\frac {b^5 x^{-4 n}}{4 n} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rubi steps
\begin {align*} \int x^{-1-9 n} \left (a+b x^n\right )^5 \, dx &=\frac {\operatorname {Subst}\left (\int \frac {(a+b x)^5}{x^{10}} \, dx,x,x^n\right )}{n}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {a^5}{x^{10}}+\frac {5 a^4 b}{x^9}+\frac {10 a^3 b^2}{x^8}+\frac {10 a^2 b^3}{x^7}+\frac {5 a b^4}{x^6}+\frac {b^5}{x^5}\right ) \, dx,x,x^n\right )}{n}\\ &=-\frac {a^5 x^{-9 n}}{9 n}-\frac {5 a^4 b x^{-8 n}}{8 n}-\frac {10 a^3 b^2 x^{-7 n}}{7 n}-\frac {5 a^2 b^3 x^{-6 n}}{3 n}-\frac {a b^4 x^{-5 n}}{n}-\frac {b^5 x^{-4 n}}{4 n}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 74, normalized size = 0.76 \[ -\frac {x^{-9 n} \left (56 a^5+315 a^4 b x^n+720 a^3 b^2 x^{2 n}+840 a^2 b^3 x^{3 n}+504 a b^4 x^{4 n}+126 b^5 x^{5 n}\right )}{504 n} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.77, size = 74, normalized size = 0.76 \[ -\frac {126 \, b^{5} x^{5 \, n} + 504 \, a b^{4} x^{4 \, n} + 840 \, a^{2} b^{3} x^{3 \, n} + 720 \, a^{3} b^{2} x^{2 \, n} + 315 \, a^{4} b x^{n} + 56 \, a^{5}}{504 \, n x^{9 \, n}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.32, size = 74, normalized size = 0.76 \[ -\frac {126 \, b^{5} x^{5 \, n} + 504 \, a b^{4} x^{4 \, n} + 840 \, a^{2} b^{3} x^{3 \, n} + 720 \, a^{3} b^{2} x^{2 \, n} + 315 \, a^{4} b x^{n} + 56 \, a^{5}}{504 \, n x^{9 \, n}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 88, normalized size = 0.91 \[ -\frac {a^{5} x^{-9 n}}{9 n}-\frac {5 a^{4} b \,x^{-8 n}}{8 n}-\frac {10 a^{3} b^{2} x^{-7 n}}{7 n}-\frac {5 a^{2} b^{3} x^{-6 n}}{3 n}-\frac {a \,b^{4} x^{-5 n}}{n}-\frac {b^{5} x^{-4 n}}{4 n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.55, size = 99, normalized size = 1.02 \[ -\frac {a^{5}}{9 \, n x^{9 \, n}} - \frac {5 \, a^{4} b}{8 \, n x^{8 \, n}} - \frac {10 \, a^{3} b^{2}}{7 \, n x^{7 \, n}} - \frac {5 \, a^{2} b^{3}}{3 \, n x^{6 \, n}} - \frac {a b^{4}}{n x^{5 \, n}} - \frac {b^{5}}{4 \, n x^{4 \, n}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.34, size = 99, normalized size = 1.02 \[ -\frac {a^5}{9\,n\,x^{9\,n}}-\frac {b^5}{4\,n\,x^{4\,n}}-\frac {5\,a^2\,b^3}{3\,n\,x^{6\,n}}-\frac {10\,a^3\,b^2}{7\,n\,x^{7\,n}}-\frac {a\,b^4}{n\,x^{5\,n}}-\frac {5\,a^4\,b}{8\,n\,x^{8\,n}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 124.33, size = 94, normalized size = 0.97 \[ \begin {cases} - \frac {a^{5} x^{- 9 n}}{9 n} - \frac {5 a^{4} b x^{- 8 n}}{8 n} - \frac {10 a^{3} b^{2} x^{- 7 n}}{7 n} - \frac {5 a^{2} b^{3} x^{- 6 n}}{3 n} - \frac {a b^{4} x^{- 5 n}}{n} - \frac {b^{5} x^{- 4 n}}{4 n} & \text {for}\: n \neq 0 \\\left (a + b\right )^{5} \log {\relax (x )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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