Optimal. Leaf size=56 \[ -\frac {a^2 (a x+b)^9}{495 b^3 x^9}+\frac {a (a x+b)^9}{55 b^2 x^{10}}-\frac {(a x+b)^9}{11 b x^{11}} \]
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Rubi [A] time = 0.01, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {263, 45, 37} \[ -\frac {a^2 (a x+b)^9}{495 b^3 x^9}+\frac {a (a x+b)^9}{55 b^2 x^{10}}-\frac {(a x+b)^9}{11 b x^{11}} \]
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^4} \, dx &=\int \frac {(b+a x)^8}{x^{12}} \, dx\\ &=-\frac {(b+a x)^9}{11 b x^{11}}-\frac {(2 a) \int \frac {(b+a x)^8}{x^{11}} \, dx}{11 b}\\ &=-\frac {(b+a x)^9}{11 b x^{11}}+\frac {a (b+a x)^9}{55 b^2 x^{10}}+\frac {a^2 \int \frac {(b+a x)^8}{x^{10}} \, dx}{55 b^2}\\ &=-\frac {(b+a x)^9}{11 b x^{11}}+\frac {a (b+a x)^9}{55 b^2 x^{10}}-\frac {a^2 (b+a x)^9}{495 b^3 x^9}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 102, normalized size = 1.82 \[ -\frac {a^8}{3 x^3}-\frac {2 a^7 b}{x^4}-\frac {28 a^6 b^2}{5 x^5}-\frac {28 a^5 b^3}{3 x^6}-\frac {10 a^4 b^4}{x^7}-\frac {7 a^3 b^5}{x^8}-\frac {28 a^2 b^6}{9 x^9}-\frac {4 a b^7}{5 x^{10}}-\frac {b^8}{11 x^{11}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.24, size = 90, normalized size = 1.61 \[ -\frac {165 \, a^{8} x^{8} + 990 \, a^{7} b x^{7} + 2772 \, a^{6} b^{2} x^{6} + 4620 \, a^{5} b^{3} x^{5} + 4950 \, a^{4} b^{4} x^{4} + 3465 \, a^{3} b^{5} x^{3} + 1540 \, a^{2} b^{6} x^{2} + 396 \, a b^{7} x + 45 \, b^{8}}{495 \, x^{11}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 90, normalized size = 1.61 \[ -\frac {165 \, a^{8} x^{8} + 990 \, a^{7} b x^{7} + 2772 \, a^{6} b^{2} x^{6} + 4620 \, a^{5} b^{3} x^{5} + 4950 \, a^{4} b^{4} x^{4} + 3465 \, a^{3} b^{5} x^{3} + 1540 \, a^{2} b^{6} x^{2} + 396 \, a b^{7} x + 45 \, b^{8}}{495 \, x^{11}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 91, normalized size = 1.62 \[ -\frac {a^{8}}{3 x^{3}}-\frac {2 a^{7} b}{x^{4}}-\frac {28 a^{6} b^{2}}{5 x^{5}}-\frac {28 a^{5} b^{3}}{3 x^{6}}-\frac {10 a^{4} b^{4}}{x^{7}}-\frac {7 a^{3} b^{5}}{x^{8}}-\frac {28 a^{2} b^{6}}{9 x^{9}}-\frac {4 a \,b^{7}}{5 x^{10}}-\frac {b^{8}}{11 x^{11}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.05, size = 90, normalized size = 1.61 \[ -\frac {165 \, a^{8} x^{8} + 990 \, a^{7} b x^{7} + 2772 \, a^{6} b^{2} x^{6} + 4620 \, a^{5} b^{3} x^{5} + 4950 \, a^{4} b^{4} x^{4} + 3465 \, a^{3} b^{5} x^{3} + 1540 \, a^{2} b^{6} x^{2} + 396 \, a b^{7} x + 45 \, b^{8}}{495 \, x^{11}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.08, size = 90, normalized size = 1.61 \[ -\frac {\frac {a^8\,x^8}{3}+2\,a^7\,b\,x^7+\frac {28\,a^6\,b^2\,x^6}{5}+\frac {28\,a^5\,b^3\,x^5}{3}+10\,a^4\,b^4\,x^4+7\,a^3\,b^5\,x^3+\frac {28\,a^2\,b^6\,x^2}{9}+\frac {4\,a\,b^7\,x}{5}+\frac {b^8}{11}}{x^{11}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.76, size = 97, normalized size = 1.73 \[ \frac {- 165 a^{8} x^{8} - 990 a^{7} b x^{7} - 2772 a^{6} b^{2} x^{6} - 4620 a^{5} b^{3} x^{5} - 4950 a^{4} b^{4} x^{4} - 3465 a^{3} b^{5} x^{3} - 1540 a^{2} b^{6} x^{2} - 396 a b^{7} x - 45 b^{8}}{495 x^{11}} \]
Verification of antiderivative is not currently implemented for this CAS.
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