Optimal. Leaf size=81 \[ \frac {b^4 (a x+b)^9}{9 a^5}-\frac {2 b^3 (a x+b)^{10}}{5 a^5}+\frac {6 b^2 (a x+b)^{11}}{11 a^5}+\frac {(a x+b)^{13}}{13 a^5}-\frac {b (a x+b)^{12}}{3 a^5} \]
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Rubi [A] time = 0.04, antiderivative size = 81, 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} \[ \frac {6 b^2 (a x+b)^{11}}{11 a^5}-\frac {2 b^3 (a x+b)^{10}}{5 a^5}+\frac {b^4 (a x+b)^9}{9 a^5}+\frac {(a x+b)^{13}}{13 a^5}-\frac {b (a x+b)^{12}}{3 a^5} \]
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^{12} \, dx &=\int x^4 (b+a x)^8 \, dx\\ &=\int \left (\frac {b^4 (b+a x)^8}{a^4}-\frac {4 b^3 (b+a x)^9}{a^4}+\frac {6 b^2 (b+a x)^{10}}{a^4}-\frac {4 b (b+a x)^{11}}{a^4}+\frac {(b+a x)^{12}}{a^4}\right ) \, dx\\ &=\frac {b^4 (b+a x)^9}{9 a^5}-\frac {2 b^3 (b+a x)^{10}}{5 a^5}+\frac {6 b^2 (b+a x)^{11}}{11 a^5}-\frac {b (b+a x)^{12}}{3 a^5}+\frac {(b+a x)^{13}}{13 a^5}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 104, normalized size = 1.28 \[ \frac {a^8 x^{13}}{13}+\frac {2}{3} a^7 b x^{12}+\frac {28}{11} a^6 b^2 x^{11}+\frac {28}{5} a^5 b^3 x^{10}+\frac {70}{9} a^4 b^4 x^9+7 a^3 b^5 x^8+4 a^2 b^6 x^7+\frac {4}{3} a b^7 x^6+\frac {b^8 x^5}{5} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.93, size = 90, normalized size = 1.11 \[ \frac {1}{13} \, a^{8} x^{13} + \frac {2}{3} \, a^{7} b x^{12} + \frac {28}{11} \, a^{6} b^{2} x^{11} + \frac {28}{5} \, a^{5} b^{3} x^{10} + \frac {70}{9} \, a^{4} b^{4} x^{9} + 7 \, a^{3} b^{5} x^{8} + 4 \, a^{2} b^{6} x^{7} + \frac {4}{3} \, a b^{7} x^{6} + \frac {1}{5} \, b^{8} x^{5} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 90, normalized size = 1.11 \[ \frac {1}{13} \, a^{8} x^{13} + \frac {2}{3} \, a^{7} b x^{12} + \frac {28}{11} \, a^{6} b^{2} x^{11} + \frac {28}{5} \, a^{5} b^{3} x^{10} + \frac {70}{9} \, a^{4} b^{4} x^{9} + 7 \, a^{3} b^{5} x^{8} + 4 \, a^{2} b^{6} x^{7} + \frac {4}{3} \, a b^{7} x^{6} + \frac {1}{5} \, b^{8} x^{5} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 91, normalized size = 1.12 \[ \frac {1}{13} a^{8} x^{13}+\frac {2}{3} a^{7} b \,x^{12}+\frac {28}{11} a^{6} b^{2} x^{11}+\frac {28}{5} a^{5} b^{3} x^{10}+\frac {70}{9} a^{4} b^{4} x^{9}+7 a^{3} b^{5} x^{8}+4 a^{2} b^{6} x^{7}+\frac {4}{3} a \,b^{7} x^{6}+\frac {1}{5} b^{8} x^{5} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.99, size = 90, normalized size = 1.11 \[ \frac {1}{13} \, a^{8} x^{13} + \frac {2}{3} \, a^{7} b x^{12} + \frac {28}{11} \, a^{6} b^{2} x^{11} + \frac {28}{5} \, a^{5} b^{3} x^{10} + \frac {70}{9} \, a^{4} b^{4} x^{9} + 7 \, a^{3} b^{5} x^{8} + 4 \, a^{2} b^{6} x^{7} + \frac {4}{3} \, a b^{7} x^{6} + \frac {1}{5} \, b^{8} x^{5} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.06, size = 90, normalized size = 1.11 \[ \frac {a^8\,x^{13}}{13}+\frac {2\,a^7\,b\,x^{12}}{3}+\frac {28\,a^6\,b^2\,x^{11}}{11}+\frac {28\,a^5\,b^3\,x^{10}}{5}+\frac {70\,a^4\,b^4\,x^9}{9}+7\,a^3\,b^5\,x^8+4\,a^2\,b^6\,x^7+\frac {4\,a\,b^7\,x^6}{3}+\frac {b^8\,x^5}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.09, size = 104, normalized size = 1.28 \[ \frac {a^{8} x^{13}}{13} + \frac {2 a^{7} b x^{12}}{3} + \frac {28 a^{6} b^{2} x^{11}}{11} + \frac {28 a^{5} b^{3} x^{10}}{5} + \frac {70 a^{4} b^{4} x^{9}}{9} + 7 a^{3} b^{5} x^{8} + 4 a^{2} b^{6} x^{7} + \frac {4 a b^{7} x^{6}}{3} + \frac {b^{8} x^{5}}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
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