Optimal. Leaf size=77 \[ -\frac {b^2 x^{-9 n} \left (a+b x^n\right )^9}{495 a^3 n}+\frac {b x^{-10 n} \left (a+b x^n\right )^9}{55 a^2 n}-\frac {x^{-11 n} \left (a+b x^n\right )^9}{11 a n} \]
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Rubi [A] time = 0.03, 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 = {266, 45, 37} \[ -\frac {b^2 x^{-9 n} \left (a+b x^n\right )^9}{495 a^3 n}+\frac {b x^{-10 n} \left (a+b x^n\right )^9}{55 a^2 n}-\frac {x^{-11 n} \left (a+b x^n\right )^9}{11 a n} \]
Antiderivative was successfully verified.
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Rule 37
Rule 45
Rule 266
Rubi steps
\begin {align*} \int x^{-1-11 n} \left (a+b x^n\right )^8 \, dx &=\frac {\operatorname {Subst}\left (\int \frac {(a+b x)^8}{x^{12}} \, dx,x,x^n\right )}{n}\\ &=-\frac {x^{-11 n} \left (a+b x^n\right )^9}{11 a n}-\frac {(2 b) \operatorname {Subst}\left (\int \frac {(a+b x)^8}{x^{11}} \, dx,x,x^n\right )}{11 a n}\\ &=-\frac {x^{-11 n} \left (a+b x^n\right )^9}{11 a n}+\frac {b x^{-10 n} \left (a+b x^n\right )^9}{55 a^2 n}+\frac {b^2 \operatorname {Subst}\left (\int \frac {(a+b x)^8}{x^{10}} \, dx,x,x^n\right )}{55 a^2 n}\\ &=-\frac {x^{-11 n} \left (a+b x^n\right )^9}{11 a n}+\frac {b x^{-10 n} \left (a+b x^n\right )^9}{55 a^2 n}-\frac {b^2 x^{-9 n} \left (a+b x^n\right )^9}{495 a^3 n}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 46, normalized size = 0.60 \[ -\frac {x^{-11 n} \left (a+b x^n\right )^9 \left (45 a^2-9 a b x^n+b^2 x^{2 n}\right )}{495 a^3 n} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.59, size = 113, normalized size = 1.47 \[ -\frac {165 \, b^{8} x^{8 \, n} + 990 \, a b^{7} x^{7 \, n} + 2772 \, a^{2} b^{6} x^{6 \, n} + 4620 \, a^{3} b^{5} x^{5 \, n} + 4950 \, a^{4} b^{4} x^{4 \, n} + 3465 \, a^{5} b^{3} x^{3 \, n} + 1540 \, a^{6} b^{2} x^{2 \, n} + 396 \, a^{7} b x^{n} + 45 \, a^{8}}{495 \, n x^{11 \, n}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.33, size = 113, normalized size = 1.47 \[ -\frac {165 \, b^{8} x^{8 \, n} + 990 \, a b^{7} x^{7 \, n} + 2772 \, a^{2} b^{6} x^{6 \, n} + 4620 \, a^{3} b^{5} x^{5 \, n} + 4950 \, a^{4} b^{4} x^{4 \, n} + 3465 \, a^{5} b^{3} x^{3 \, n} + 1540 \, a^{6} b^{2} x^{2 \, n} + 396 \, a^{7} b x^{n} + 45 \, a^{8}}{495 \, n x^{11 \, n}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 136, normalized size = 1.77 \[ -\frac {a^{8} x^{-11 n}}{11 n}-\frac {4 a^{7} b \,x^{-10 n}}{5 n}-\frac {28 a^{6} b^{2} x^{-9 n}}{9 n}-\frac {7 a^{5} b^{3} x^{-8 n}}{n}-\frac {10 a^{4} b^{4} x^{-7 n}}{n}-\frac {28 a^{3} b^{5} x^{-6 n}}{3 n}-\frac {28 a^{2} b^{6} x^{-5 n}}{5 n}-\frac {2 a \,b^{7} x^{-4 n}}{n}-\frac {b^{8} x^{-3 n}}{3 n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.61, size = 153, normalized size = 1.99 \[ -\frac {a^{8}}{11 \, n x^{11 \, n}} - \frac {4 \, a^{7} b}{5 \, n x^{10 \, n}} - \frac {28 \, a^{6} b^{2}}{9 \, n x^{9 \, n}} - \frac {7 \, a^{5} b^{3}}{n x^{8 \, n}} - \frac {10 \, a^{4} b^{4}}{n x^{7 \, n}} - \frac {28 \, a^{3} b^{5}}{3 \, n x^{6 \, n}} - \frac {28 \, a^{2} b^{6}}{5 \, n x^{5 \, n}} - \frac {2 \, a b^{7}}{n x^{4 \, n}} - \frac {b^{8}}{3 \, n x^{3 \, n}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.42, size = 153, normalized size = 1.99 \[ -\frac {a^8}{11\,n\,x^{11\,n}}-\frac {b^8}{3\,n\,x^{3\,n}}-\frac {28\,a^2\,b^6}{5\,n\,x^{5\,n}}-\frac {28\,a^3\,b^5}{3\,n\,x^{6\,n}}-\frac {10\,a^4\,b^4}{n\,x^{7\,n}}-\frac {7\,a^5\,b^3}{n\,x^{8\,n}}-\frac {28\,a^6\,b^2}{9\,n\,x^{9\,n}}-\frac {2\,a\,b^7}{n\,x^{4\,n}}-\frac {4\,a^7\,b}{5\,n\,x^{10\,n}} \]
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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