Optimal. Leaf size=64 \[ -\frac {b^3 (a x+b)^9}{9 a^4}+\frac {3 b^2 (a x+b)^{10}}{10 a^4}+\frac {(a x+b)^{12}}{12 a^4}-\frac {3 b (a x+b)^{11}}{11 a^4} \]
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Rubi [A] time = 0.03, antiderivative size = 64, 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} \[ \frac {3 b^2 (a x+b)^{10}}{10 a^4}-\frac {b^3 (a x+b)^9}{9 a^4}+\frac {(a x+b)^{12}}{12 a^4}-\frac {3 b (a x+b)^{11}}{11 a^4} \]
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^{11} \, dx &=\int x^3 (b+a x)^8 \, dx\\ &=\int \left (-\frac {b^3 (b+a x)^8}{a^3}+\frac {3 b^2 (b+a x)^9}{a^3}-\frac {3 b (b+a x)^{10}}{a^3}+\frac {(b+a x)^{11}}{a^3}\right ) \, dx\\ &=-\frac {b^3 (b+a x)^9}{9 a^4}+\frac {3 b^2 (b+a x)^{10}}{10 a^4}-\frac {3 b (b+a x)^{11}}{11 a^4}+\frac {(b+a x)^{12}}{12 a^4}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 106, normalized size = 1.66 \[ \frac {a^8 x^{12}}{12}+\frac {8}{11} a^7 b x^{11}+\frac {14}{5} a^6 b^2 x^{10}+\frac {56}{9} a^5 b^3 x^9+\frac {35}{4} a^4 b^4 x^8+8 a^3 b^5 x^7+\frac {14}{3} a^2 b^6 x^6+\frac {8}{5} a b^7 x^5+\frac {b^8 x^4}{4} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.05, size = 90, normalized size = 1.41 \[ \frac {1}{12} \, a^{8} x^{12} + \frac {8}{11} \, a^{7} b x^{11} + \frac {14}{5} \, a^{6} b^{2} x^{10} + \frac {56}{9} \, a^{5} b^{3} x^{9} + \frac {35}{4} \, a^{4} b^{4} x^{8} + 8 \, a^{3} b^{5} x^{7} + \frac {14}{3} \, a^{2} b^{6} x^{6} + \frac {8}{5} \, a b^{7} x^{5} + \frac {1}{4} \, b^{8} x^{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 90, normalized size = 1.41 \[ \frac {1}{12} \, a^{8} x^{12} + \frac {8}{11} \, a^{7} b x^{11} + \frac {14}{5} \, a^{6} b^{2} x^{10} + \frac {56}{9} \, a^{5} b^{3} x^{9} + \frac {35}{4} \, a^{4} b^{4} x^{8} + 8 \, a^{3} b^{5} x^{7} + \frac {14}{3} \, a^{2} b^{6} x^{6} + \frac {8}{5} \, a b^{7} x^{5} + \frac {1}{4} \, b^{8} x^{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 91, normalized size = 1.42 \[ \frac {1}{12} a^{8} x^{12}+\frac {8}{11} a^{7} b \,x^{11}+\frac {14}{5} a^{6} b^{2} x^{10}+\frac {56}{9} a^{5} b^{3} x^{9}+\frac {35}{4} a^{4} b^{4} x^{8}+8 a^{3} b^{5} x^{7}+\frac {14}{3} a^{2} b^{6} x^{6}+\frac {8}{5} a \,b^{7} x^{5}+\frac {1}{4} b^{8} x^{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.06, size = 90, normalized size = 1.41 \[ \frac {1}{12} \, a^{8} x^{12} + \frac {8}{11} \, a^{7} b x^{11} + \frac {14}{5} \, a^{6} b^{2} x^{10} + \frac {56}{9} \, a^{5} b^{3} x^{9} + \frac {35}{4} \, a^{4} b^{4} x^{8} + 8 \, a^{3} b^{5} x^{7} + \frac {14}{3} \, a^{2} b^{6} x^{6} + \frac {8}{5} \, a b^{7} x^{5} + \frac {1}{4} \, b^{8} x^{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 90, normalized size = 1.41 \[ \frac {a^8\,x^{12}}{12}+\frac {8\,a^7\,b\,x^{11}}{11}+\frac {14\,a^6\,b^2\,x^{10}}{5}+\frac {56\,a^5\,b^3\,x^9}{9}+\frac {35\,a^4\,b^4\,x^8}{4}+8\,a^3\,b^5\,x^7+\frac {14\,a^2\,b^6\,x^6}{3}+\frac {8\,a\,b^7\,x^5}{5}+\frac {b^8\,x^4}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.10, size = 105, normalized size = 1.64 \[ \frac {a^{8} x^{12}}{12} + \frac {8 a^{7} b x^{11}}{11} + \frac {14 a^{6} b^{2} x^{10}}{5} + \frac {56 a^{5} b^{3} x^{9}}{9} + \frac {35 a^{4} b^{4} x^{8}}{4} + 8 a^{3} b^{5} x^{7} + \frac {14 a^{2} b^{6} x^{6}}{3} + \frac {8 a b^{7} x^{5}}{5} + \frac {b^{8} x^{4}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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