Optimal. Leaf size=76 \[ \frac {a^3 (a x+b)^9}{1980 b^4 x^9}-\frac {a^2 (a x+b)^9}{220 b^3 x^{10}}+\frac {a (a x+b)^9}{44 b^2 x^{11}}-\frac {(a x+b)^9}{12 b x^{12}} \]
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Rubi [A] time = 0.02, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {263, 45, 37} \[ \frac {a^3 (a x+b)^9}{1980 b^4 x^9}-\frac {a^2 (a x+b)^9}{220 b^3 x^{10}}+\frac {a (a x+b)^9}{44 b^2 x^{11}}-\frac {(a x+b)^9}{12 b x^{12}} \]
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^5} \, dx &=\int \frac {(b+a x)^8}{x^{13}} \, dx\\ &=-\frac {(b+a x)^9}{12 b x^{12}}-\frac {a \int \frac {(b+a x)^8}{x^{12}} \, dx}{4 b}\\ &=-\frac {(b+a x)^9}{12 b x^{12}}+\frac {a (b+a x)^9}{44 b^2 x^{11}}+\frac {a^2 \int \frac {(b+a x)^8}{x^{11}} \, dx}{22 b^2}\\ &=-\frac {(b+a x)^9}{12 b x^{12}}+\frac {a (b+a x)^9}{44 b^2 x^{11}}-\frac {a^2 (b+a x)^9}{220 b^3 x^{10}}-\frac {a^3 \int \frac {(b+a x)^8}{x^{10}} \, dx}{220 b^3}\\ &=-\frac {(b+a x)^9}{12 b x^{12}}+\frac {a (b+a x)^9}{44 b^2 x^{11}}-\frac {a^2 (b+a x)^9}{220 b^3 x^{10}}+\frac {a^3 (b+a x)^9}{1980 b^4 x^9}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 106, normalized size = 1.39 \[ -\frac {a^8}{4 x^4}-\frac {8 a^7 b}{5 x^5}-\frac {14 a^6 b^2}{3 x^6}-\frac {8 a^5 b^3}{x^7}-\frac {35 a^4 b^4}{4 x^8}-\frac {56 a^3 b^5}{9 x^9}-\frac {14 a^2 b^6}{5 x^{10}}-\frac {8 a b^7}{11 x^{11}}-\frac {b^8}{12 x^{12}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.86, size = 90, normalized size = 1.18 \[ -\frac {495 \, a^{8} x^{8} + 3168 \, a^{7} b x^{7} + 9240 \, a^{6} b^{2} x^{6} + 15840 \, a^{5} b^{3} x^{5} + 17325 \, a^{4} b^{4} x^{4} + 12320 \, a^{3} b^{5} x^{3} + 5544 \, a^{2} b^{6} x^{2} + 1440 \, a b^{7} x + 165 \, b^{8}}{1980 \, x^{12}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 90, normalized size = 1.18 \[ -\frac {495 \, a^{8} x^{8} + 3168 \, a^{7} b x^{7} + 9240 \, a^{6} b^{2} x^{6} + 15840 \, a^{5} b^{3} x^{5} + 17325 \, a^{4} b^{4} x^{4} + 12320 \, a^{3} b^{5} x^{3} + 5544 \, a^{2} b^{6} x^{2} + 1440 \, a b^{7} x + 165 \, b^{8}}{1980 \, x^{12}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 91, normalized size = 1.20 \[ -\frac {a^{8}}{4 x^{4}}-\frac {8 a^{7} b}{5 x^{5}}-\frac {14 a^{6} b^{2}}{3 x^{6}}-\frac {8 a^{5} b^{3}}{x^{7}}-\frac {35 a^{4} b^{4}}{4 x^{8}}-\frac {56 a^{3} b^{5}}{9 x^{9}}-\frac {14 a^{2} b^{6}}{5 x^{10}}-\frac {8 a \,b^{7}}{11 x^{11}}-\frac {b^{8}}{12 x^{12}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.16, size = 90, normalized size = 1.18 \[ -\frac {495 \, a^{8} x^{8} + 3168 \, a^{7} b x^{7} + 9240 \, a^{6} b^{2} x^{6} + 15840 \, a^{5} b^{3} x^{5} + 17325 \, a^{4} b^{4} x^{4} + 12320 \, a^{3} b^{5} x^{3} + 5544 \, a^{2} b^{6} x^{2} + 1440 \, a b^{7} x + 165 \, b^{8}}{1980 \, x^{12}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 90, normalized size = 1.18 \[ -\frac {\frac {a^8\,x^8}{4}+\frac {8\,a^7\,b\,x^7}{5}+\frac {14\,a^6\,b^2\,x^6}{3}+8\,a^5\,b^3\,x^5+\frac {35\,a^4\,b^4\,x^4}{4}+\frac {56\,a^3\,b^5\,x^3}{9}+\frac {14\,a^2\,b^6\,x^2}{5}+\frac {8\,a\,b^7\,x}{11}+\frac {b^8}{12}}{x^{12}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.85, size = 97, normalized size = 1.28 \[ \frac {- 495 a^{8} x^{8} - 3168 a^{7} b x^{7} - 9240 a^{6} b^{2} x^{6} - 15840 a^{5} b^{3} x^{5} - 17325 a^{4} b^{4} x^{4} - 12320 a^{3} b^{5} x^{3} - 5544 a^{2} b^{6} x^{2} - 1440 a b^{7} x - 165 b^{8}}{1980 x^{12}} \]
Verification of antiderivative is not currently implemented for this CAS.
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