Optimal. Leaf size=40 \[ \frac {\left (a+b x^n\right )^{10}}{10 b^2 n}-\frac {a \left (a+b x^n\right )^9}{9 b^2 n} \]
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Rubi [A] time = 0.02, antiderivative size = 40, 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} \[ \frac {\left (a+b x^n\right )^{10}}{10 b^2 n}-\frac {a \left (a+b x^n\right )^9}{9 b^2 n} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rubi steps
\begin {align*} \int x^{-1+2 n} \left (a+b x^n\right )^8 \, dx &=\frac {\operatorname {Subst}\left (\int x (a+b x)^8 \, dx,x,x^n\right )}{n}\\ &=\frac {\operatorname {Subst}\left (\int \left (-\frac {a (a+b x)^8}{b}+\frac {(a+b x)^9}{b}\right ) \, dx,x,x^n\right )}{n}\\ &=-\frac {a \left (a+b x^n\right )^9}{9 b^2 n}+\frac {\left (a+b x^n\right )^{10}}{10 b^2 n}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 27, normalized size = 0.68 \[ -\frac {\left (a-9 b x^n\right ) \left (a+b x^n\right )^9}{90 b^2 n} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.68, size = 113, normalized size = 2.82 \[ \frac {9 \, b^{8} x^{10 \, n} + 80 \, a b^{7} x^{9 \, n} + 315 \, a^{2} b^{6} x^{8 \, n} + 720 \, a^{3} b^{5} x^{7 \, n} + 1050 \, a^{4} b^{4} x^{6 \, n} + 1008 \, a^{5} b^{3} x^{5 \, n} + 630 \, a^{6} b^{2} x^{4 \, n} + 240 \, a^{7} b x^{3 \, n} + 45 \, a^{8} x^{2 \, n}}{90 \, n} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b x^{n} + a\right )}^{8} x^{2 \, n - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.03, size = 136, normalized size = 3.40 \[ \frac {a^{8} x^{2 n}}{2 n}+\frac {8 a^{7} b \,x^{3 n}}{3 n}+\frac {7 a^{6} b^{2} x^{4 n}}{n}+\frac {56 a^{5} b^{3} x^{5 n}}{5 n}+\frac {35 a^{4} b^{4} x^{6 n}}{3 n}+\frac {8 a^{3} b^{5} x^{7 n}}{n}+\frac {7 a^{2} b^{6} x^{8 n}}{2 n}+\frac {8 a \,b^{7} x^{9 n}}{9 n}+\frac {b^{8} x^{10 n}}{10 n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.50, size = 135, normalized size = 3.38 \[ \frac {b^{8} x^{10 \, n}}{10 \, n} + \frac {8 \, a b^{7} x^{9 \, n}}{9 \, n} + \frac {7 \, a^{2} b^{6} x^{8 \, n}}{2 \, n} + \frac {8 \, a^{3} b^{5} x^{7 \, n}}{n} + \frac {35 \, a^{4} b^{4} x^{6 \, n}}{3 \, n} + \frac {56 \, a^{5} b^{3} x^{5 \, n}}{5 \, n} + \frac {7 \, a^{6} b^{2} x^{4 \, n}}{n} + \frac {8 \, a^{7} b x^{3 \, n}}{3 \, n} + \frac {a^{8} x^{2 \, n}}{2 \, n} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.45, size = 135, normalized size = 3.38 \[ \frac {a^8\,x^{2\,n}}{2\,n}+\frac {b^8\,x^{10\,n}}{10\,n}+\frac {7\,a^6\,b^2\,x^{4\,n}}{n}+\frac {56\,a^5\,b^3\,x^{5\,n}}{5\,n}+\frac {35\,a^4\,b^4\,x^{6\,n}}{3\,n}+\frac {8\,a^3\,b^5\,x^{7\,n}}{n}+\frac {7\,a^2\,b^6\,x^{8\,n}}{2\,n}+\frac {8\,a^7\,b\,x^{3\,n}}{3\,n}+\frac {8\,a\,b^7\,x^{9\,n}}{9\,n} \]
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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