Optimal. Leaf size=50 \[ \frac {b x^{-4 n} \left (a+b x^n\right )^4}{20 a^2 n}-\frac {x^{-5 n} \left (a+b x^n\right )^4}{5 a n} \]
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Rubi [A] time = 0.02, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {266, 45, 37} \[ \frac {b x^{-4 n} \left (a+b x^n\right )^4}{20 a^2 n}-\frac {x^{-5 n} \left (a+b x^n\right )^4}{5 a n} \]
Antiderivative was successfully verified.
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Rule 37
Rule 45
Rule 266
Rubi steps
\begin {align*} \int x^{-1-5 n} \left (a+b x^n\right )^3 \, dx &=\frac {\operatorname {Subst}\left (\int \frac {(a+b x)^3}{x^6} \, dx,x,x^n\right )}{n}\\ &=-\frac {x^{-5 n} \left (a+b x^n\right )^4}{5 a n}-\frac {b \operatorname {Subst}\left (\int \frac {(a+b x)^3}{x^5} \, dx,x,x^n\right )}{5 a n}\\ &=-\frac {x^{-5 n} \left (a+b x^n\right )^4}{5 a n}+\frac {b x^{-4 n} \left (a+b x^n\right )^4}{20 a^2 n}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 48, normalized size = 0.96 \[ -\frac {x^{-5 n} \left (4 a^3+15 a^2 b x^n+20 a b^2 x^{2 n}+10 b^3 x^{3 n}\right )}{20 n} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.83, size = 48, normalized size = 0.96 \[ -\frac {10 \, b^{3} x^{3 \, n} + 20 \, a b^{2} x^{2 \, n} + 15 \, a^{2} b x^{n} + 4 \, a^{3}}{20 \, n x^{5 \, n}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 48, normalized size = 0.96 \[ -\frac {10 \, b^{3} x^{3 \, n} + 20 \, a b^{2} x^{2 \, n} + 15 \, a^{2} b x^{n} + 4 \, a^{3}}{20 \, n x^{5 \, n}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 63, normalized size = 1.26 \[ \left (-\frac {3 a^{2} b \,{\mathrm e}^{n \ln \relax (x )}}{4 n}-\frac {a \,b^{2} {\mathrm e}^{2 n \ln \relax (x )}}{n}-\frac {b^{3} {\mathrm e}^{3 n \ln \relax (x )}}{2 n}-\frac {a^{3}}{5 n}\right ) {\mathrm e}^{-5 n \ln \relax (x )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.58, size = 63, normalized size = 1.26 \[ -\frac {a^{3}}{5 \, n x^{5 \, n}} - \frac {3 \, a^{2} b}{4 \, n x^{4 \, n}} - \frac {a b^{2}}{n x^{3 \, n}} - \frac {b^{3}}{2 \, n x^{2 \, n}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.31, size = 63, normalized size = 1.26 \[ -\frac {a^3}{5\,n\,x^{5\,n}}-\frac {b^3}{2\,n\,x^{2\,n}}-\frac {a\,b^2}{n\,x^{3\,n}}-\frac {3\,a^2\,b}{4\,n\,x^{4\,n}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 26.47, size = 60, normalized size = 1.20 \[ \begin {cases} - \frac {a^{3} x^{- 5 n}}{5 n} - \frac {3 a^{2} b x^{- 4 n}}{4 n} - \frac {a b^{2} x^{- 3 n}}{n} - \frac {b^{3} x^{- 2 n}}{2 n} & \text {for}\: n \neq 0 \\\left (a + b\right )^{3} \log {\relax (x )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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