Optimal. Leaf size=106 \[ \frac {a^8 x^{17}}{17}+\frac {1}{2} a^7 b x^{16}+\frac {28}{15} a^6 b^2 x^{15}+4 a^5 b^3 x^{14}+\frac {70}{13} a^4 b^4 x^{13}+\frac {14}{3} a^3 b^5 x^{12}+\frac {28}{11} a^2 b^6 x^{11}+\frac {4}{5} a b^7 x^{10}+\frac {b^8 x^9}{9} \]
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Rubi [A] time = 0.06, antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {263, 43} \[ \frac {28}{15} a^6 b^2 x^{15}+4 a^5 b^3 x^{14}+\frac {70}{13} a^4 b^4 x^{13}+\frac {14}{3} a^3 b^5 x^{12}+\frac {28}{11} a^2 b^6 x^{11}+\frac {1}{2} a^7 b x^{16}+\frac {a^8 x^{17}}{17}+\frac {4}{5} a b^7 x^{10}+\frac {b^8 x^9}{9} \]
Antiderivative was successfully verified.
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Rule 43
Rule 263
Rubi steps
\begin {align*} \int \left (a+\frac {b}{x}\right )^8 x^{16} \, dx &=\int x^8 (b+a x)^8 \, dx\\ &=\int \left (b^8 x^8+8 a b^7 x^9+28 a^2 b^6 x^{10}+56 a^3 b^5 x^{11}+70 a^4 b^4 x^{12}+56 a^5 b^3 x^{13}+28 a^6 b^2 x^{14}+8 a^7 b x^{15}+a^8 x^{16}\right ) \, dx\\ &=\frac {b^8 x^9}{9}+\frac {4}{5} a b^7 x^{10}+\frac {28}{11} a^2 b^6 x^{11}+\frac {14}{3} a^3 b^5 x^{12}+\frac {70}{13} a^4 b^4 x^{13}+4 a^5 b^3 x^{14}+\frac {28}{15} a^6 b^2 x^{15}+\frac {1}{2} a^7 b x^{16}+\frac {a^8 x^{17}}{17}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 106, normalized size = 1.00 \[ \frac {a^8 x^{17}}{17}+\frac {1}{2} a^7 b x^{16}+\frac {28}{15} a^6 b^2 x^{15}+4 a^5 b^3 x^{14}+\frac {70}{13} a^4 b^4 x^{13}+\frac {14}{3} a^3 b^5 x^{12}+\frac {28}{11} a^2 b^6 x^{11}+\frac {4}{5} a b^7 x^{10}+\frac {b^8 x^9}{9} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.67, size = 90, normalized size = 0.85 \[ \frac {1}{17} \, a^{8} x^{17} + \frac {1}{2} \, a^{7} b x^{16} + \frac {28}{15} \, a^{6} b^{2} x^{15} + 4 \, a^{5} b^{3} x^{14} + \frac {70}{13} \, a^{4} b^{4} x^{13} + \frac {14}{3} \, a^{3} b^{5} x^{12} + \frac {28}{11} \, a^{2} b^{6} x^{11} + \frac {4}{5} \, a b^{7} x^{10} + \frac {1}{9} \, b^{8} x^{9} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 90, normalized size = 0.85 \[ \frac {1}{17} \, a^{8} x^{17} + \frac {1}{2} \, a^{7} b x^{16} + \frac {28}{15} \, a^{6} b^{2} x^{15} + 4 \, a^{5} b^{3} x^{14} + \frac {70}{13} \, a^{4} b^{4} x^{13} + \frac {14}{3} \, a^{3} b^{5} x^{12} + \frac {28}{11} \, a^{2} b^{6} x^{11} + \frac {4}{5} \, a b^{7} x^{10} + \frac {1}{9} \, b^{8} x^{9} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 91, normalized size = 0.86 \[ \frac {1}{17} a^{8} x^{17}+\frac {1}{2} a^{7} b \,x^{16}+\frac {28}{15} a^{6} b^{2} x^{15}+4 a^{5} b^{3} x^{14}+\frac {70}{13} a^{4} b^{4} x^{13}+\frac {14}{3} a^{3} b^{5} x^{12}+\frac {28}{11} a^{2} b^{6} x^{11}+\frac {4}{5} a \,b^{7} x^{10}+\frac {1}{9} b^{8} x^{9} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.95, size = 90, normalized size = 0.85 \[ \frac {1}{17} \, a^{8} x^{17} + \frac {1}{2} \, a^{7} b x^{16} + \frac {28}{15} \, a^{6} b^{2} x^{15} + 4 \, a^{5} b^{3} x^{14} + \frac {70}{13} \, a^{4} b^{4} x^{13} + \frac {14}{3} \, a^{3} b^{5} x^{12} + \frac {28}{11} \, a^{2} b^{6} x^{11} + \frac {4}{5} \, a b^{7} x^{10} + \frac {1}{9} \, b^{8} x^{9} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 90, normalized size = 0.85 \[ \frac {a^8\,x^{17}}{17}+\frac {a^7\,b\,x^{16}}{2}+\frac {28\,a^6\,b^2\,x^{15}}{15}+4\,a^5\,b^3\,x^{14}+\frac {70\,a^4\,b^4\,x^{13}}{13}+\frac {14\,a^3\,b^5\,x^{12}}{3}+\frac {28\,a^2\,b^6\,x^{11}}{11}+\frac {4\,a\,b^7\,x^{10}}{5}+\frac {b^8\,x^9}{9} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.09, size = 104, normalized size = 0.98 \[ \frac {a^{8} x^{17}}{17} + \frac {a^{7} b x^{16}}{2} + \frac {28 a^{6} b^{2} x^{15}}{15} + 4 a^{5} b^{3} x^{14} + \frac {70 a^{4} b^{4} x^{13}}{13} + \frac {14 a^{3} b^{5} x^{12}}{3} + \frac {28 a^{2} b^{6} x^{11}}{11} + \frac {4 a b^{7} x^{10}}{5} + \frac {b^{8} x^{9}}{9} \]
Verification of antiderivative is not currently implemented for this CAS.
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