Optimal. Leaf size=65 \[ -\frac {5 b^2 c^5 (a-b x)^6}{168 a^2 x^6}-\frac {c^5 (a-b x)^6}{8 x^8}-\frac {5 b c^5 (a-b x)^6}{28 a x^7} \]
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Rubi [A] time = 0.02, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {78, 45, 37} \[ -\frac {5 b^2 c^5 (a-b x)^6}{168 a^2 x^6}-\frac {5 b c^5 (a-b x)^6}{28 a x^7}-\frac {c^5 (a-b x)^6}{8 x^8} \]
Antiderivative was successfully verified.
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Rule 37
Rule 45
Rule 78
Rubi steps
\begin {align*} \int \frac {(a+b x) (a c-b c x)^5}{x^9} \, dx &=-\frac {c^5 (a-b x)^6}{8 x^8}+\frac {1}{4} (5 b) \int \frac {(a c-b c x)^5}{x^8} \, dx\\ &=-\frac {c^5 (a-b x)^6}{8 x^8}-\frac {5 b c^5 (a-b x)^6}{28 a x^7}+\frac {\left (5 b^2\right ) \int \frac {(a c-b c x)^5}{x^7} \, dx}{28 a}\\ &=-\frac {c^5 (a-b x)^6}{8 x^8}-\frac {5 b c^5 (a-b x)^6}{28 a x^7}-\frac {5 b^2 c^5 (a-b x)^6}{168 a^2 x^6}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 73, normalized size = 1.12 \[ c^5 \left (-\frac {a^6}{8 x^8}+\frac {4 a^5 b}{7 x^7}-\frac {5 a^4 b^2}{6 x^6}+\frac {5 a^2 b^4}{4 x^4}-\frac {4 a b^5}{3 x^3}+\frac {b^6}{2 x^2}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.90, size = 75, normalized size = 1.15 \[ \frac {84 \, b^{6} c^{5} x^{6} - 224 \, a b^{5} c^{5} x^{5} + 210 \, a^{2} b^{4} c^{5} x^{4} - 140 \, a^{4} b^{2} c^{5} x^{2} + 96 \, a^{5} b c^{5} x - 21 \, a^{6} c^{5}}{168 \, x^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.02, size = 75, normalized size = 1.15 \[ \frac {84 \, b^{6} c^{5} x^{6} - 224 \, a b^{5} c^{5} x^{5} + 210 \, a^{2} b^{4} c^{5} x^{4} - 140 \, a^{4} b^{2} c^{5} x^{2} + 96 \, a^{5} b c^{5} x - 21 \, a^{6} c^{5}}{168 \, x^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 62, normalized size = 0.95 \[ \left (\frac {b^{6}}{2 x^{2}}-\frac {4 a \,b^{5}}{3 x^{3}}+\frac {5 a^{2} b^{4}}{4 x^{4}}-\frac {5 a^{4} b^{2}}{6 x^{6}}+\frac {4 a^{5} b}{7 x^{7}}-\frac {a^{6}}{8 x^{8}}\right ) c^{5} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.99, size = 75, normalized size = 1.15 \[ \frac {84 \, b^{6} c^{5} x^{6} - 224 \, a b^{5} c^{5} x^{5} + 210 \, a^{2} b^{4} c^{5} x^{4} - 140 \, a^{4} b^{2} c^{5} x^{2} + 96 \, a^{5} b c^{5} x - 21 \, a^{6} c^{5}}{168 \, x^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 75, normalized size = 1.15 \[ -\frac {\frac {a^6\,c^5}{8}-\frac {4\,a^5\,b\,c^5\,x}{7}+\frac {5\,a^4\,b^2\,c^5\,x^2}{6}-\frac {5\,a^2\,b^4\,c^5\,x^4}{4}+\frac {4\,a\,b^5\,c^5\,x^5}{3}-\frac {b^6\,c^5\,x^6}{2}}{x^8} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.49, size = 82, normalized size = 1.26 \[ - \frac {21 a^{6} c^{5} - 96 a^{5} b c^{5} x + 140 a^{4} b^{2} c^{5} x^{2} - 210 a^{2} b^{4} c^{5} x^{4} + 224 a b^{5} c^{5} x^{5} - 84 b^{6} c^{5} x^{6}}{168 x^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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