Optimal. Leaf size=63 \[ -\frac {a^3 x^{-7 n}}{7 n}-\frac {a^2 b x^{-6 n}}{2 n}-\frac {3 a b^2 x^{-5 n}}{5 n}-\frac {b^3 x^{-4 n}}{4 n} \]
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Rubi [A]
time = 0.02, antiderivative size = 63, 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 = {272, 45}
\begin {gather*} -\frac {a^3 x^{-7 n}}{7 n}-\frac {a^2 b x^{-6 n}}{2 n}-\frac {3 a b^2 x^{-5 n}}{5 n}-\frac {b^3 x^{-4 n}}{4 n} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 272
Rubi steps
\begin {align*} \int x^{-1-7 n} \left (a+b x^n\right )^3 \, dx &=\frac {\text {Subst}\left (\int \frac {(a+b x)^3}{x^8} \, dx,x,x^n\right )}{n}\\ &=\frac {\text {Subst}\left (\int \left (\frac {a^3}{x^8}+\frac {3 a^2 b}{x^7}+\frac {3 a b^2}{x^6}+\frac {b^3}{x^5}\right ) \, dx,x,x^n\right )}{n}\\ &=-\frac {a^3 x^{-7 n}}{7 n}-\frac {a^2 b x^{-6 n}}{2 n}-\frac {3 a b^2 x^{-5 n}}{5 n}-\frac {b^3 x^{-4 n}}{4 n}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 48, normalized size = 0.76 \begin {gather*} \frac {x^{-7 n} \left (-20 a^3-70 a^2 b x^n-84 a b^2 x^{2 n}-35 b^3 x^{3 n}\right )}{140 n} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.23, size = 56, normalized size = 0.89
method | result | size |
risch | \(-\frac {b^{3} x^{-4 n}}{4 n}-\frac {3 a \,b^{2} x^{-5 n}}{5 n}-\frac {a^{2} b \,x^{-6 n}}{2 n}-\frac {a^{3} x^{-7 n}}{7 n}\) | \(56\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 63, normalized size = 1.00 \begin {gather*} -\frac {a^{3}}{7 \, n x^{7 \, n}} - \frac {a^{2} b}{2 \, n x^{6 \, n}} - \frac {3 \, a b^{2}}{5 \, n x^{5 \, n}} - \frac {b^{3}}{4 \, n x^{4 \, n}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.43, size = 48, normalized size = 0.76 \begin {gather*} -\frac {35 \, b^{3} x^{3 \, n} + 84 \, a b^{2} x^{2 \, n} + 70 \, a^{2} b x^{n} + 20 \, a^{3}}{140 \, n x^{7 \, n}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 2.18, size = 61, normalized size = 0.97 \begin {gather*} \begin {cases} - \frac {a^{3} x^{- 7 n}}{7 n} - \frac {a^{2} b x^{- 6 n}}{2 n} - \frac {3 a b^{2} x^{- 5 n}}{5 n} - \frac {b^{3} x^{- 4 n}}{4 n} & \text {for}\: n \neq 0 \\\left (a + b\right )^{3} \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 = 1.55, size = 48, normalized size = 0.76 \begin {gather*} -\frac {35 \, b^{3} x^{3 \, n} + 84 \, a b^{2} x^{2 \, n} + 70 \, a^{2} b x^{n} + 20 \, a^{3}}{140 \, n x^{7 \, n}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.37, size = 63, normalized size = 1.00 \begin {gather*} -\frac {a^3}{7\,n\,x^{7\,n}}-\frac {b^3}{4\,n\,x^{4\,n}}-\frac {3\,a\,b^2}{5\,n\,x^{5\,n}}-\frac {a^2\,b}{2\,n\,x^{6\,n}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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