Optimal. Leaf size=77 \[ -\frac {x^{-8 n} \left (a+b x^n\right )^6}{8 a n}+\frac {b x^{-7 n} \left (a+b x^n\right )^6}{28 a^2 n}-\frac {b^2 x^{-6 n} \left (a+b x^n\right )^6}{168 a^3 n} \]
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Rubi [A]
time = 0.02, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {272, 47, 37}
\begin {gather*} -\frac {b^2 x^{-6 n} \left (a+b x^n\right )^6}{168 a^3 n}+\frac {b x^{-7 n} \left (a+b x^n\right )^6}{28 a^2 n}-\frac {x^{-8 n} \left (a+b x^n\right )^6}{8 a n} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 47
Rule 272
Rubi steps
\begin {align*} \int x^{-1-8 n} \left (a+b x^n\right )^5 \, dx &=\frac {\text {Subst}\left (\int \frac {(a+b x)^5}{x^9} \, dx,x,x^n\right )}{n}\\ &=-\frac {x^{-8 n} \left (a+b x^n\right )^6}{8 a n}-\frac {b \text {Subst}\left (\int \frac {(a+b x)^5}{x^8} \, dx,x,x^n\right )}{4 a n}\\ &=-\frac {x^{-8 n} \left (a+b x^n\right )^6}{8 a n}+\frac {b x^{-7 n} \left (a+b x^n\right )^6}{28 a^2 n}+\frac {b^2 \text {Subst}\left (\int \frac {(a+b x)^5}{x^7} \, dx,x,x^n\right )}{28 a^2 n}\\ &=-\frac {x^{-8 n} \left (a+b x^n\right )^6}{8 a n}+\frac {b x^{-7 n} \left (a+b x^n\right )^6}{28 a^2 n}-\frac {b^2 x^{-6 n} \left (a+b x^n\right )^6}{168 a^3 n}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 74, normalized size = 0.96 \begin {gather*} \frac {x^{-8 n} \left (-21 a^5-120 a^4 b x^n-280 a^3 b^2 x^{2 n}-336 a^2 b^3 x^{3 n}-210 a b^4 x^{4 n}-56 b^5 x^{5 n}\right )}{168 n} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.24, size = 88, normalized size = 1.14
method | result | size |
risch | \(-\frac {b^{5} x^{-3 n}}{3 n}-\frac {5 a \,b^{4} x^{-4 n}}{4 n}-\frac {2 a^{2} b^{3} x^{-5 n}}{n}-\frac {5 a^{3} b^{2} x^{-6 n}}{3 n}-\frac {5 a^{4} b \,x^{-7 n}}{7 n}-\frac {a^{5} x^{-8 n}}{8 n}\) | \(88\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 99, normalized size = 1.29 \begin {gather*} -\frac {a^{5}}{8 \, n x^{8 \, n}} - \frac {5 \, a^{4} b}{7 \, n x^{7 \, n}} - \frac {5 \, a^{3} b^{2}}{3 \, n x^{6 \, n}} - \frac {2 \, a^{2} b^{3}}{n x^{5 \, n}} - \frac {5 \, a b^{4}}{4 \, n x^{4 \, n}} - \frac {b^{5}}{3 \, n x^{3 \, n}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 74, normalized size = 0.96 \begin {gather*} -\frac {56 \, b^{5} x^{5 \, n} + 210 \, a b^{4} x^{4 \, n} + 336 \, a^{2} b^{3} x^{3 \, n} + 280 \, a^{3} b^{2} x^{2 \, n} + 120 \, a^{4} b x^{n} + 21 \, a^{5}}{168 \, n x^{8 \, n}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 6.84, size = 95, normalized size = 1.23 \begin {gather*} \begin {cases} - \frac {a^{5} x^{- 8 n}}{8 n} - \frac {5 a^{4} b x^{- 7 n}}{7 n} - \frac {5 a^{3} b^{2} x^{- 6 n}}{3 n} - \frac {2 a^{2} b^{3} x^{- 5 n}}{n} - \frac {5 a b^{4} x^{- 4 n}}{4 n} - \frac {b^{5} x^{- 3 n}}{3 n} & \text {for}\: n \neq 0 \\\left (a + b\right )^{5} \log {\left (x \right )} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.48, size = 74, normalized size = 0.96 \begin {gather*} -\frac {56 \, b^{5} x^{5 \, n} + 210 \, a b^{4} x^{4 \, n} + 336 \, a^{2} b^{3} x^{3 \, n} + 280 \, a^{3} b^{2} x^{2 \, n} + 120 \, a^{4} b x^{n} + 21 \, a^{5}}{168 \, n x^{8 \, n}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.36, size = 99, normalized size = 1.29 \begin {gather*} -\frac {a^5}{8\,n\,x^{8\,n}}-\frac {b^5}{3\,n\,x^{3\,n}}-\frac {2\,a^2\,b^3}{n\,x^{5\,n}}-\frac {5\,a^3\,b^2}{3\,n\,x^{6\,n}}-\frac {5\,a\,b^4}{4\,n\,x^{4\,n}}-\frac {5\,a^4\,b}{7\,n\,x^{7\,n}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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