Optimal. Leaf size=96 \[ -\frac {a^4 (a x+b)^9}{6435 b^5 x^9}+\frac {a^3 (a x+b)^9}{715 b^4 x^{10}}-\frac {a^2 (a x+b)^9}{143 b^3 x^{11}}+\frac {a (a x+b)^9}{39 b^2 x^{12}}-\frac {(a x+b)^9}{13 b x^{13}} \]
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Rubi [A] time = 0.03, antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {263, 45, 37} \[ -\frac {a^4 (a x+b)^9}{6435 b^5 x^9}+\frac {a^3 (a x+b)^9}{715 b^4 x^{10}}-\frac {a^2 (a x+b)^9}{143 b^3 x^{11}}+\frac {a (a x+b)^9}{39 b^2 x^{12}}-\frac {(a x+b)^9}{13 b x^{13}} \]
Antiderivative was successfully verified.
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Rule 37
Rule 45
Rule 263
Rubi steps
\begin {align*} \int \frac {\left (a+\frac {b}{x}\right )^8}{x^6} \, dx &=\int \frac {(b+a x)^8}{x^{14}} \, dx\\ &=-\frac {(b+a x)^9}{13 b x^{13}}-\frac {(4 a) \int \frac {(b+a x)^8}{x^{13}} \, dx}{13 b}\\ &=-\frac {(b+a x)^9}{13 b x^{13}}+\frac {a (b+a x)^9}{39 b^2 x^{12}}+\frac {a^2 \int \frac {(b+a x)^8}{x^{12}} \, dx}{13 b^2}\\ &=-\frac {(b+a x)^9}{13 b x^{13}}+\frac {a (b+a x)^9}{39 b^2 x^{12}}-\frac {a^2 (b+a x)^9}{143 b^3 x^{11}}-\frac {\left (2 a^3\right ) \int \frac {(b+a x)^8}{x^{11}} \, dx}{143 b^3}\\ &=-\frac {(b+a x)^9}{13 b x^{13}}+\frac {a (b+a x)^9}{39 b^2 x^{12}}-\frac {a^2 (b+a x)^9}{143 b^3 x^{11}}+\frac {a^3 (b+a x)^9}{715 b^4 x^{10}}+\frac {a^4 \int \frac {(b+a x)^8}{x^{10}} \, dx}{715 b^4}\\ &=-\frac {(b+a x)^9}{13 b x^{13}}+\frac {a (b+a x)^9}{39 b^2 x^{12}}-\frac {a^2 (b+a x)^9}{143 b^3 x^{11}}+\frac {a^3 (b+a x)^9}{715 b^4 x^{10}}-\frac {a^4 (b+a x)^9}{6435 b^5 x^9}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 104, normalized size = 1.08 \[ -\frac {a^8}{5 x^5}-\frac {4 a^7 b}{3 x^6}-\frac {4 a^6 b^2}{x^7}-\frac {7 a^5 b^3}{x^8}-\frac {70 a^4 b^4}{9 x^9}-\frac {28 a^3 b^5}{5 x^{10}}-\frac {28 a^2 b^6}{11 x^{11}}-\frac {2 a b^7}{3 x^{12}}-\frac {b^8}{13 x^{13}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.03, size = 90, normalized size = 0.94 \[ -\frac {1287 \, a^{8} x^{8} + 8580 \, a^{7} b x^{7} + 25740 \, a^{6} b^{2} x^{6} + 45045 \, a^{5} b^{3} x^{5} + 50050 \, a^{4} b^{4} x^{4} + 36036 \, a^{3} b^{5} x^{3} + 16380 \, a^{2} b^{6} x^{2} + 4290 \, a b^{7} x + 495 \, b^{8}}{6435 \, x^{13}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 90, normalized size = 0.94 \[ -\frac {1287 \, a^{8} x^{8} + 8580 \, a^{7} b x^{7} + 25740 \, a^{6} b^{2} x^{6} + 45045 \, a^{5} b^{3} x^{5} + 50050 \, a^{4} b^{4} x^{4} + 36036 \, a^{3} b^{5} x^{3} + 16380 \, a^{2} b^{6} x^{2} + 4290 \, a b^{7} x + 495 \, b^{8}}{6435 \, x^{13}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 91, normalized size = 0.95 \[ -\frac {a^{8}}{5 x^{5}}-\frac {4 a^{7} b}{3 x^{6}}-\frac {4 a^{6} b^{2}}{x^{7}}-\frac {7 a^{5} b^{3}}{x^{8}}-\frac {70 a^{4} b^{4}}{9 x^{9}}-\frac {28 a^{3} b^{5}}{5 x^{10}}-\frac {28 a^{2} b^{6}}{11 x^{11}}-\frac {2 a \,b^{7}}{3 x^{12}}-\frac {b^{8}}{13 x^{13}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.01, size = 90, normalized size = 0.94 \[ -\frac {1287 \, a^{8} x^{8} + 8580 \, a^{7} b x^{7} + 25740 \, a^{6} b^{2} x^{6} + 45045 \, a^{5} b^{3} x^{5} + 50050 \, a^{4} b^{4} x^{4} + 36036 \, a^{3} b^{5} x^{3} + 16380 \, a^{2} b^{6} x^{2} + 4290 \, a b^{7} x + 495 \, b^{8}}{6435 \, x^{13}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.08, size = 90, normalized size = 0.94 \[ -\frac {\frac {a^8\,x^8}{5}+\frac {4\,a^7\,b\,x^7}{3}+4\,a^6\,b^2\,x^6+7\,a^5\,b^3\,x^5+\frac {70\,a^4\,b^4\,x^4}{9}+\frac {28\,a^3\,b^5\,x^3}{5}+\frac {28\,a^2\,b^6\,x^2}{11}+\frac {2\,a\,b^7\,x}{3}+\frac {b^8}{13}}{x^{13}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.89, size = 97, normalized size = 1.01 \[ \frac {- 1287 a^{8} x^{8} - 8580 a^{7} b x^{7} - 25740 a^{6} b^{2} x^{6} - 45045 a^{5} b^{3} x^{5} - 50050 a^{4} b^{4} x^{4} - 36036 a^{3} b^{5} x^{3} - 16380 a^{2} b^{6} x^{2} - 4290 a b^{7} x - 495 b^{8}}{6435 x^{13}} \]
Verification of antiderivative is not currently implemented for this CAS.
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