Optimal. Leaf size=151 \[ \frac {a^8 x^{8 n}}{8 n}+\frac {8 a^7 b x^{9 n}}{9 n}+\frac {14 a^6 b^2 x^{10 n}}{5 n}+\frac {56 a^5 b^3 x^{11 n}}{11 n}+\frac {35 a^4 b^4 x^{12 n}}{6 n}+\frac {56 a^3 b^5 x^{13 n}}{13 n}+\frac {2 a^2 b^6 x^{14 n}}{n}+\frac {8 a b^7 x^{15 n}}{15 n}+\frac {b^8 x^{16 n}}{16 n} \]
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Rubi [A] time = 0.07, antiderivative size = 151, 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} \[ \frac {14 a^6 b^2 x^{10 n}}{5 n}+\frac {56 a^5 b^3 x^{11 n}}{11 n}+\frac {35 a^4 b^4 x^{12 n}}{6 n}+\frac {56 a^3 b^5 x^{13 n}}{13 n}+\frac {2 a^2 b^6 x^{14 n}}{n}+\frac {8 a^7 b x^{9 n}}{9 n}+\frac {a^8 x^{8 n}}{8 n}+\frac {8 a b^7 x^{15 n}}{15 n}+\frac {b^8 x^{16 n}}{16 n} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rubi steps
\begin {align*} \int x^{-1+8 n} \left (a+b x^n\right )^8 \, dx &=\frac {\operatorname {Subst}\left (\int x^7 (a+b x)^8 \, dx,x,x^n\right )}{n}\\ &=\frac {\operatorname {Subst}\left (\int \left (a^8 x^7+8 a^7 b x^8+28 a^6 b^2 x^9+56 a^5 b^3 x^{10}+70 a^4 b^4 x^{11}+56 a^3 b^5 x^{12}+28 a^2 b^6 x^{13}+8 a b^7 x^{14}+b^8 x^{15}\right ) \, dx,x,x^n\right )}{n}\\ &=\frac {a^8 x^{8 n}}{8 n}+\frac {8 a^7 b x^{9 n}}{9 n}+\frac {14 a^6 b^2 x^{10 n}}{5 n}+\frac {56 a^5 b^3 x^{11 n}}{11 n}+\frac {35 a^4 b^4 x^{12 n}}{6 n}+\frac {56 a^3 b^5 x^{13 n}}{13 n}+\frac {2 a^2 b^6 x^{14 n}}{n}+\frac {8 a b^7 x^{15 n}}{15 n}+\frac {b^8 x^{16 n}}{16 n}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 128, normalized size = 0.85 \[ \frac {\frac {1}{8} a^8 x^{8 n}+\frac {8}{9} a^7 b x^{9 n}+\frac {14}{5} a^6 b^2 x^{10 n}+\frac {56}{11} a^5 b^3 x^{11 n}+\frac {35}{6} a^4 b^4 x^{12 n}+\frac {56}{13} a^3 b^5 x^{13 n}+2 a^2 b^6 x^{14 n}+\frac {8}{15} a b^7 x^{15 n}+\frac {1}{16} b^8 x^{16 n}}{n} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.58, size = 113, normalized size = 0.75 \[ \frac {6435 \, b^{8} x^{16 \, n} + 54912 \, a b^{7} x^{15 \, n} + 205920 \, a^{2} b^{6} x^{14 \, n} + 443520 \, a^{3} b^{5} x^{13 \, n} + 600600 \, a^{4} b^{4} x^{12 \, n} + 524160 \, a^{5} b^{3} x^{11 \, n} + 288288 \, a^{6} b^{2} x^{10 \, n} + 91520 \, a^{7} b x^{9 \, n} + 12870 \, a^{8} x^{8 \, n}}{102960 \, n} \]
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 \[ \int {\left (b x^{n} + a\right )}^{8} x^{8 \, n - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 136, normalized size = 0.90 \[ \frac {a^{8} x^{8 n}}{8 n}+\frac {8 a^{7} b \,x^{9 n}}{9 n}+\frac {14 a^{6} b^{2} x^{10 n}}{5 n}+\frac {56 a^{5} b^{3} x^{11 n}}{11 n}+\frac {35 a^{4} b^{4} x^{12 n}}{6 n}+\frac {56 a^{3} b^{5} x^{13 n}}{13 n}+\frac {2 a^{2} b^{6} x^{14 n}}{n}+\frac {8 a \,b^{7} x^{15 n}}{15 n}+\frac {b^{8} x^{16 n}}{16 n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.54, size = 135, normalized size = 0.89 \[ \frac {b^{8} x^{16 \, n}}{16 \, n} + \frac {8 \, a b^{7} x^{15 \, n}}{15 \, n} + \frac {2 \, a^{2} b^{6} x^{14 \, n}}{n} + \frac {56 \, a^{3} b^{5} x^{13 \, n}}{13 \, n} + \frac {35 \, a^{4} b^{4} x^{12 \, n}}{6 \, n} + \frac {56 \, a^{5} b^{3} x^{11 \, n}}{11 \, n} + \frac {14 \, a^{6} b^{2} x^{10 \, n}}{5 \, n} + \frac {8 \, a^{7} b x^{9 \, n}}{9 \, n} + \frac {a^{8} x^{8 \, n}}{8 \, n} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.49, size = 135, normalized size = 0.89 \[ \frac {a^8\,x^{8\,n}}{8\,n}+\frac {b^8\,x^{16\,n}}{16\,n}+\frac {14\,a^6\,b^2\,x^{10\,n}}{5\,n}+\frac {56\,a^5\,b^3\,x^{11\,n}}{11\,n}+\frac {35\,a^4\,b^4\,x^{12\,n}}{6\,n}+\frac {56\,a^3\,b^5\,x^{13\,n}}{13\,n}+\frac {2\,a^2\,b^6\,x^{14\,n}}{n}+\frac {8\,a^7\,b\,x^{9\,n}}{9\,n}+\frac {8\,a\,b^7\,x^{15\,n}}{15\,n} \]
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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