Optimal. Leaf size=135 \[ -\frac {a^8 x^{-n}}{n}+8 a^7 b \log (x)+\frac {28 a^6 b^2 x^n}{n}+\frac {28 a^5 b^3 x^{2 n}}{n}+\frac {70 a^4 b^4 x^{3 n}}{3 n}+\frac {14 a^3 b^5 x^{4 n}}{n}+\frac {28 a^2 b^6 x^{5 n}}{5 n}+\frac {4 a b^7 x^{6 n}}{3 n}+\frac {b^8 x^{7 n}}{7 n} \]
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Rubi [A] time = 0.06, antiderivative size = 135, 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} \[ \frac {28 a^6 b^2 x^n}{n}+\frac {28 a^5 b^3 x^{2 n}}{n}+\frac {70 a^4 b^4 x^{3 n}}{3 n}+\frac {14 a^3 b^5 x^{4 n}}{n}+\frac {28 a^2 b^6 x^{5 n}}{5 n}+8 a^7 b \log (x)-\frac {a^8 x^{-n}}{n}+\frac {4 a b^7 x^{6 n}}{3 n}+\frac {b^8 x^{7 n}}{7 n} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rubi steps
\begin {align*} \int x^{-1-n} \left (a+b x^n\right )^8 \, dx &=\frac {\operatorname {Subst}\left (\int \frac {(a+b x)^8}{x^2} \, dx,x,x^n\right )}{n}\\ &=\frac {\operatorname {Subst}\left (\int \left (28 a^6 b^2+\frac {a^8}{x^2}+\frac {8 a^7 b}{x}+56 a^5 b^3 x+70 a^4 b^4 x^2+56 a^3 b^5 x^3+28 a^2 b^6 x^4+8 a b^7 x^5+b^8 x^6\right ) \, dx,x,x^n\right )}{n}\\ &=-\frac {a^8 x^{-n}}{n}+\frac {28 a^6 b^2 x^n}{n}+\frac {28 a^5 b^3 x^{2 n}}{n}+\frac {70 a^4 b^4 x^{3 n}}{3 n}+\frac {14 a^3 b^5 x^{4 n}}{n}+\frac {28 a^2 b^6 x^{5 n}}{5 n}+\frac {4 a b^7 x^{6 n}}{3 n}+\frac {b^8 x^{7 n}}{7 n}+8 a^7 b \log (x)\\ \end {align*}
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Mathematica [A] time = 0.10, size = 116, normalized size = 0.86 \[ \frac {-a^8 x^{-n}+8 a^7 b n \log (x)+28 a^6 b^2 x^n+28 a^5 b^3 x^{2 n}+\frac {70}{3} a^4 b^4 x^{3 n}+14 a^3 b^5 x^{4 n}+\frac {28}{5} a^2 b^6 x^{5 n}+\frac {4}{3} a b^7 x^{6 n}+\frac {1}{7} b^8 x^{7 n}}{n} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.61, size = 114, normalized size = 0.84 \[ \frac {840 \, a^{7} b n x^{n} \log \relax (x) + 15 \, b^{8} x^{8 \, n} + 140 \, a b^{7} x^{7 \, n} + 588 \, a^{2} b^{6} x^{6 \, n} + 1470 \, a^{3} b^{5} x^{5 \, n} + 2450 \, a^{4} b^{4} x^{4 \, n} + 2940 \, a^{5} b^{3} x^{3 \, n} + 2940 \, a^{6} b^{2} x^{2 \, n} - 105 \, a^{8}}{105 \, n x^{n}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.28, size = 114, normalized size = 0.84 \[ \frac {840 \, a^{7} b n x^{n} \log \relax (x) + 15 \, b^{8} x^{8 \, n} + 140 \, a b^{7} x^{7 \, n} + 588 \, a^{2} b^{6} x^{6 \, n} + 1470 \, a^{3} b^{5} x^{5 \, n} + 2450 \, a^{4} b^{4} x^{4 \, n} + 2940 \, a^{5} b^{3} x^{3 \, n} + 2940 \, a^{6} b^{2} x^{2 \, n} - 105 \, a^{8}}{105 \, n x^{n}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 128, normalized size = 0.95 \[ 8 a^{7} b \ln \relax (x )-\frac {a^{8} x^{-n}}{n}+\frac {28 a^{6} b^{2} x^{n}}{n}+\frac {28 a^{5} b^{3} x^{2 n}}{n}+\frac {70 a^{4} b^{4} x^{3 n}}{3 n}+\frac {14 a^{3} b^{5} x^{4 n}}{n}+\frac {28 a^{2} b^{6} x^{5 n}}{5 n}+\frac {4 a \,b^{7} x^{6 n}}{3 n}+\frac {b^{8} x^{7 n}}{7 n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.50, size = 127, normalized size = 0.94 \[ 8 \, a^{7} b \log \relax (x) + \frac {b^{8} x^{7 \, n}}{7 \, n} + \frac {4 \, a b^{7} x^{6 \, n}}{3 \, n} + \frac {28 \, a^{2} b^{6} x^{5 \, n}}{5 \, n} + \frac {14 \, a^{3} b^{5} x^{4 \, n}}{n} + \frac {70 \, a^{4} b^{4} x^{3 \, n}}{3 \, n} + \frac {28 \, a^{5} b^{3} x^{2 \, n}}{n} + \frac {28 \, a^{6} b^{2} x^{n}}{n} - \frac {a^{8}}{n x^{n}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.72, size = 127, normalized size = 0.94 \[ 8\,a^7\,b\,\ln \relax (x)-\frac {a^8}{n\,x^n}+\frac {b^8\,x^{7\,n}}{7\,n}+\frac {28\,a^5\,b^3\,x^{2\,n}}{n}+\frac {70\,a^4\,b^4\,x^{3\,n}}{3\,n}+\frac {14\,a^3\,b^5\,x^{4\,n}}{n}+\frac {28\,a^2\,b^6\,x^{5\,n}}{5\,n}+\frac {4\,a\,b^7\,x^{6\,n}}{3\,n}+\frac {28\,a^6\,b^2\,x^n}{n} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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