Optimal. Leaf size=133 \[ -\frac {a^8 x^{-3 n}}{3 n}-\frac {4 a^7 b x^{-2 n}}{n}-\frac {28 a^6 b^2 x^{-n}}{n}+\frac {70 a^4 b^4 x^n}{n}+\frac {28 a^3 b^5 x^{2 n}}{n}+\frac {28 a^2 b^6 x^{3 n}}{3 n}+\frac {2 a b^7 x^{4 n}}{n}+\frac {b^8 x^{5 n}}{5 n}+56 a^5 b^3 \log (x) \]
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Rubi [A]
time = 0.04, antiderivative size = 133, 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^8 x^{-3 n}}{3 n}-\frac {4 a^7 b x^{-2 n}}{n}-\frac {28 a^6 b^2 x^{-n}}{n}+56 a^5 b^3 \log (x)+\frac {70 a^4 b^4 x^n}{n}+\frac {28 a^3 b^5 x^{2 n}}{n}+\frac {28 a^2 b^6 x^{3 n}}{3 n}+\frac {2 a b^7 x^{4 n}}{n}+\frac {b^8 x^{5 n}}{5 n} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 272
Rubi steps
\begin {align*} \int x^{-1-3 n} \left (a+b x^n\right )^8 \, dx &=\frac {\text {Subst}\left (\int \frac {(a+b x)^8}{x^4} \, dx,x,x^n\right )}{n}\\ &=\frac {\text {Subst}\left (\int \left (70 a^4 b^4+\frac {a^8}{x^4}+\frac {8 a^7 b}{x^3}+\frac {28 a^6 b^2}{x^2}+\frac {56 a^5 b^3}{x}+56 a^3 b^5 x+28 a^2 b^6 x^2+8 a b^7 x^3+b^8 x^4\right ) \, dx,x,x^n\right )}{n}\\ &=-\frac {a^8 x^{-3 n}}{3 n}-\frac {4 a^7 b x^{-2 n}}{n}-\frac {28 a^6 b^2 x^{-n}}{n}+\frac {70 a^4 b^4 x^n}{n}+\frac {28 a^3 b^5 x^{2 n}}{n}+\frac {28 a^2 b^6 x^{3 n}}{3 n}+\frac {2 a b^7 x^{4 n}}{n}+\frac {b^8 x^{5 n}}{5 n}+56 a^5 b^3 \log (x)\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 116, normalized size = 0.87 \begin {gather*} \frac {x^{-3 n} \left (-5 a^8-60 a^7 b x^n-420 a^6 b^2 x^{2 n}+1050 a^4 b^4 x^{4 n}+420 a^3 b^5 x^{5 n}+140 a^2 b^6 x^{6 n}+30 a b^7 x^{7 n}+3 b^8 x^{8 n}\right )}{15 n}+\frac {56 a^5 b^3 \log \left (x^n\right )}{n} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.24, size = 128, normalized size = 0.96
method | result | size |
risch | \(56 a^{5} b^{3} \ln \left (x \right )+\frac {b^{8} x^{5 n}}{5 n}+\frac {2 a \,b^{7} x^{4 n}}{n}+\frac {28 a^{2} b^{6} x^{3 n}}{3 n}+\frac {28 a^{3} b^{5} x^{2 n}}{n}+\frac {70 a^{4} b^{4} x^{n}}{n}-\frac {28 a^{6} b^{2} x^{-n}}{n}-\frac {4 a^{7} b \,x^{-2 n}}{n}-\frac {a^{8} x^{-3 n}}{3 n}\) | \(128\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 131, normalized size = 0.98 \begin {gather*} 56 \, a^{5} b^{3} \log \left (x\right ) + \frac {b^{8} x^{5 \, n}}{5 \, n} + \frac {2 \, a b^{7} x^{4 \, n}}{n} + \frac {28 \, a^{2} b^{6} x^{3 \, n}}{3 \, n} + \frac {28 \, a^{3} b^{5} x^{2 \, n}}{n} + \frac {70 \, a^{4} b^{4} x^{n}}{n} - \frac {a^{8}}{3 \, n x^{3 \, n}} - \frac {4 \, a^{7} b}{n x^{2 \, n}} - \frac {28 \, a^{6} b^{2}}{n x^{n}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.44, size = 116, normalized size = 0.87 \begin {gather*} \frac {840 \, a^{5} b^{3} n x^{3 \, n} \log \left (x\right ) + 3 \, b^{8} x^{8 \, n} + 30 \, a b^{7} x^{7 \, n} + 140 \, a^{2} b^{6} x^{6 \, n} + 420 \, a^{3} b^{5} x^{5 \, n} + 1050 \, a^{4} b^{4} x^{4 \, n} - 420 \, a^{6} b^{2} x^{2 \, n} - 60 \, a^{7} b x^{n} - 5 \, a^{8}}{15 \, n x^{3 \, n}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.21, size = 116, normalized size = 0.87 \begin {gather*} \frac {840 \, a^{5} b^{3} n x^{3 \, n} \log \left (x\right ) + 3 \, b^{8} x^{8 \, n} + 30 \, a b^{7} x^{7 \, n} + 140 \, a^{2} b^{6} x^{6 \, n} + 420 \, a^{3} b^{5} x^{5 \, n} + 1050 \, a^{4} b^{4} x^{4 \, n} - 420 \, a^{6} b^{2} x^{2 \, n} - 60 \, a^{7} b x^{n} - 5 \, a^{8}}{15 \, n x^{3 \, n}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.60, size = 131, normalized size = 0.98 \begin {gather*} \frac {b^8\,x^{5\,n}}{5\,n}-\frac {a^8}{3\,n\,x^{3\,n}}+56\,a^5\,b^3\,\ln \left (x\right )-\frac {28\,a^6\,b^2}{n\,x^n}+\frac {28\,a^3\,b^5\,x^{2\,n}}{n}+\frac {28\,a^2\,b^6\,x^{3\,n}}{3\,n}-\frac {4\,a^7\,b}{n\,x^{2\,n}}+\frac {2\,a\,b^7\,x^{4\,n}}{n}+\frac {70\,a^4\,b^4\,x^n}{n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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