Optimal. Leaf size=106 \[ -\frac {a^8}{6 x^6}-\frac {8 a^7 b}{7 x^7}-\frac {7 a^6 b^2}{2 x^8}-\frac {56 a^5 b^3}{9 x^9}-\frac {7 a^4 b^4}{x^{10}}-\frac {56 a^3 b^5}{11 x^{11}}-\frac {7 a^2 b^6}{3 x^{12}}-\frac {8 a b^7}{13 x^{13}}-\frac {b^8}{14 x^{14}} \]
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Rubi [A] time = 0.04, antiderivative size = 106, 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} \[ -\frac {7 a^6 b^2}{2 x^8}-\frac {56 a^5 b^3}{9 x^9}-\frac {7 a^4 b^4}{x^{10}}-\frac {56 a^3 b^5}{11 x^{11}}-\frac {7 a^2 b^6}{3 x^{12}}-\frac {8 a^7 b}{7 x^7}-\frac {a^8}{6 x^6}-\frac {8 a b^7}{13 x^{13}}-\frac {b^8}{14 x^{14}} \]
Antiderivative was successfully verified.
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Rule 43
Rule 263
Rubi steps
\begin {align*} \int \frac {\left (a+\frac {b}{x}\right )^8}{x^7} \, dx &=\int \frac {(b+a x)^8}{x^{15}} \, dx\\ &=\int \left (\frac {b^8}{x^{15}}+\frac {8 a b^7}{x^{14}}+\frac {28 a^2 b^6}{x^{13}}+\frac {56 a^3 b^5}{x^{12}}+\frac {70 a^4 b^4}{x^{11}}+\frac {56 a^5 b^3}{x^{10}}+\frac {28 a^6 b^2}{x^9}+\frac {8 a^7 b}{x^8}+\frac {a^8}{x^7}\right ) \, dx\\ &=-\frac {b^8}{14 x^{14}}-\frac {8 a b^7}{13 x^{13}}-\frac {7 a^2 b^6}{3 x^{12}}-\frac {56 a^3 b^5}{11 x^{11}}-\frac {7 a^4 b^4}{x^{10}}-\frac {56 a^5 b^3}{9 x^9}-\frac {7 a^6 b^2}{2 x^8}-\frac {8 a^7 b}{7 x^7}-\frac {a^8}{6 x^6}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 106, normalized size = 1.00 \[ -\frac {a^8}{6 x^6}-\frac {8 a^7 b}{7 x^7}-\frac {7 a^6 b^2}{2 x^8}-\frac {56 a^5 b^3}{9 x^9}-\frac {7 a^4 b^4}{x^{10}}-\frac {56 a^3 b^5}{11 x^{11}}-\frac {7 a^2 b^6}{3 x^{12}}-\frac {8 a b^7}{13 x^{13}}-\frac {b^8}{14 x^{14}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.84, size = 90, normalized size = 0.85 \[ -\frac {3003 \, a^{8} x^{8} + 20592 \, a^{7} b x^{7} + 63063 \, a^{6} b^{2} x^{6} + 112112 \, a^{5} b^{3} x^{5} + 126126 \, a^{4} b^{4} x^{4} + 91728 \, a^{3} b^{5} x^{3} + 42042 \, a^{2} b^{6} x^{2} + 11088 \, a b^{7} x + 1287 \, b^{8}}{18018 \, x^{14}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 90, normalized size = 0.85 \[ -\frac {3003 \, a^{8} x^{8} + 20592 \, a^{7} b x^{7} + 63063 \, a^{6} b^{2} x^{6} + 112112 \, a^{5} b^{3} x^{5} + 126126 \, a^{4} b^{4} x^{4} + 91728 \, a^{3} b^{5} x^{3} + 42042 \, a^{2} b^{6} x^{2} + 11088 \, a b^{7} x + 1287 \, b^{8}}{18018 \, x^{14}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 91, normalized size = 0.86 \[ -\frac {a^{8}}{6 x^{6}}-\frac {8 a^{7} b}{7 x^{7}}-\frac {7 a^{6} b^{2}}{2 x^{8}}-\frac {56 a^{5} b^{3}}{9 x^{9}}-\frac {7 a^{4} b^{4}}{x^{10}}-\frac {56 a^{3} b^{5}}{11 x^{11}}-\frac {7 a^{2} b^{6}}{3 x^{12}}-\frac {8 a \,b^{7}}{13 x^{13}}-\frac {b^{8}}{14 x^{14}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.08, size = 90, normalized size = 0.85 \[ -\frac {3003 \, a^{8} x^{8} + 20592 \, a^{7} b x^{7} + 63063 \, a^{6} b^{2} x^{6} + 112112 \, a^{5} b^{3} x^{5} + 126126 \, a^{4} b^{4} x^{4} + 91728 \, a^{3} b^{5} x^{3} + 42042 \, a^{2} b^{6} x^{2} + 11088 \, a b^{7} x + 1287 \, b^{8}}{18018 \, x^{14}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 90, normalized size = 0.85 \[ -\frac {\frac {a^8\,x^8}{6}+\frac {8\,a^7\,b\,x^7}{7}+\frac {7\,a^6\,b^2\,x^6}{2}+\frac {56\,a^5\,b^3\,x^5}{9}+7\,a^4\,b^4\,x^4+\frac {56\,a^3\,b^5\,x^3}{11}+\frac {7\,a^2\,b^6\,x^2}{3}+\frac {8\,a\,b^7\,x}{13}+\frac {b^8}{14}}{x^{14}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.93, size = 97, normalized size = 0.92 \[ \frac {- 3003 a^{8} x^{8} - 20592 a^{7} b x^{7} - 63063 a^{6} b^{2} x^{6} - 112112 a^{5} b^{3} x^{5} - 126126 a^{4} b^{4} x^{4} - 91728 a^{3} b^{5} x^{3} - 42042 a^{2} b^{6} x^{2} - 11088 a b^{7} x - 1287 b^{8}}{18018 x^{14}} \]
Verification of antiderivative is not currently implemented for this CAS.
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