Optimal. Leaf size=50 \[ \frac {b x^{-9 n} \left (a+b x^n\right )^9}{90 a^2 n}-\frac {x^{-10 n} \left (a+b x^n\right )^9}{10 a n} \]
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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} \[ \frac {b x^{-9 n} \left (a+b x^n\right )^9}{90 a^2 n}-\frac {x^{-10 n} \left (a+b x^n\right )^9}{10 a n} \]
Antiderivative was successfully verified.
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Rule 37
Rule 45
Rule 266
Rubi steps
\begin {align*} \int x^{-1-10 n} \left (a+b x^n\right )^8 \, dx &=\frac {\operatorname {Subst}\left (\int \frac {(a+b x)^8}{x^{11}} \, dx,x,x^n\right )}{n}\\ &=-\frac {x^{-10 n} \left (a+b x^n\right )^9}{10 a n}-\frac {b \operatorname {Subst}\left (\int \frac {(a+b x)^8}{x^{10}} \, dx,x,x^n\right )}{10 a n}\\ &=-\frac {x^{-10 n} \left (a+b x^n\right )^9}{10 a n}+\frac {b x^{-9 n} \left (a+b x^n\right )^9}{90 a^2 n}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 33, normalized size = 0.66 \[ \frac {x^{-10 n} \left (b x^n-9 a\right ) \left (a+b x^n\right )^9}{90 a^2 n} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.61, size = 113, normalized size = 2.26 \[ -\frac {45 \, b^{8} x^{8 \, n} + 240 \, a b^{7} x^{7 \, n} + 630 \, a^{2} b^{6} x^{6 \, n} + 1008 \, a^{3} b^{5} x^{5 \, n} + 1050 \, a^{4} b^{4} x^{4 \, n} + 720 \, a^{5} b^{3} x^{3 \, n} + 315 \, a^{6} b^{2} x^{2 \, n} + 80 \, a^{7} b x^{n} + 9 \, a^{8}}{90 \, n x^{10 \, n}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.31, size = 113, normalized size = 2.26 \[ -\frac {45 \, b^{8} x^{8 \, n} + 240 \, a b^{7} x^{7 \, n} + 630 \, a^{2} b^{6} x^{6 \, n} + 1008 \, a^{3} b^{5} x^{5 \, n} + 1050 \, a^{4} b^{4} x^{4 \, n} + 720 \, a^{5} b^{3} x^{3 \, n} + 315 \, a^{6} b^{2} x^{2 \, n} + 80 \, a^{7} b x^{n} + 9 \, a^{8}}{90 \, n x^{10 \, n}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.03, size = 136, normalized size = 2.72 \[ -\frac {a^{8} x^{-10 n}}{10 n}-\frac {8 a^{7} b \,x^{-9 n}}{9 n}-\frac {7 a^{6} b^{2} x^{-8 n}}{2 n}-\frac {8 a^{5} b^{3} x^{-7 n}}{n}-\frac {35 a^{4} b^{4} x^{-6 n}}{3 n}-\frac {56 a^{3} b^{5} x^{-5 n}}{5 n}-\frac {7 a^{2} b^{6} x^{-4 n}}{n}-\frac {8 a \,b^{7} x^{-3 n}}{3 n}-\frac {b^{8} x^{-2 n}}{2 n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.58, size = 153, normalized size = 3.06 \[ -\frac {a^{8}}{10 \, n x^{10 \, n}} - \frac {8 \, a^{7} b}{9 \, n x^{9 \, n}} - \frac {7 \, a^{6} b^{2}}{2 \, n x^{8 \, n}} - \frac {8 \, a^{5} b^{3}}{n x^{7 \, n}} - \frac {35 \, a^{4} b^{4}}{3 \, n x^{6 \, n}} - \frac {56 \, a^{3} b^{5}}{5 \, n x^{5 \, n}} - \frac {7 \, a^{2} b^{6}}{n x^{4 \, n}} - \frac {8 \, a b^{7}}{3 \, n x^{3 \, n}} - \frac {b^{8}}{2 \, n x^{2 \, n}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.46, size = 153, normalized size = 3.06 \[ -\frac {a^8}{10\,n\,x^{10\,n}}-\frac {b^8}{2\,n\,x^{2\,n}}-\frac {7\,a^2\,b^6}{n\,x^{4\,n}}-\frac {56\,a^3\,b^5}{5\,n\,x^{5\,n}}-\frac {35\,a^4\,b^4}{3\,n\,x^{6\,n}}-\frac {8\,a^5\,b^3}{n\,x^{7\,n}}-\frac {7\,a^6\,b^2}{2\,n\,x^{8\,n}}-\frac {8\,a\,b^7}{3\,n\,x^{3\,n}}-\frac {8\,a^7\,b}{9\,n\,x^{9\,n}} \]
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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