Optimal. Leaf size=56 \[ -\frac {b^2 (a+b x)^6}{168 a^3 x^6}+\frac {b (a+b x)^6}{28 a^2 x^7}-\frac {(a+b x)^6}{8 a x^8} \]
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Rubi [A] time = 0.01, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {45, 37} \begin {gather*} -\frac {b^2 (a+b x)^6}{168 a^3 x^6}+\frac {b (a+b x)^6}{28 a^2 x^7}-\frac {(a+b x)^6}{8 a x^8} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 45
Rubi steps
\begin {align*} \int \frac {(a+b x)^5}{x^9} \, dx &=-\frac {(a+b x)^6}{8 a x^8}-\frac {b \int \frac {(a+b x)^5}{x^8} \, dx}{4 a}\\ &=-\frac {(a+b x)^6}{8 a x^8}+\frac {b (a+b x)^6}{28 a^2 x^7}+\frac {b^2 \int \frac {(a+b x)^5}{x^7} \, dx}{28 a^2}\\ &=-\frac {(a+b x)^6}{8 a x^8}+\frac {b (a+b x)^6}{28 a^2 x^7}-\frac {b^2 (a+b x)^6}{168 a^3 x^6}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 67, normalized size = 1.20 \begin {gather*} -\frac {a^5}{8 x^8}-\frac {5 a^4 b}{7 x^7}-\frac {5 a^3 b^2}{3 x^6}-\frac {2 a^2 b^3}{x^5}-\frac {5 a b^4}{4 x^4}-\frac {b^5}{3 x^3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(a+b x)^5}{x^9} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 1.77, size = 57, normalized size = 1.02 \begin {gather*} -\frac {56 \, b^{5} x^{5} + 210 \, a b^{4} x^{4} + 336 \, a^{2} b^{3} x^{3} + 280 \, a^{3} b^{2} x^{2} + 120 \, a^{4} b x + 21 \, a^{5}}{168 \, x^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.10, size = 57, normalized size = 1.02 \begin {gather*} -\frac {56 \, b^{5} x^{5} + 210 \, a b^{4} x^{4} + 336 \, a^{2} b^{3} x^{3} + 280 \, a^{3} b^{2} x^{2} + 120 \, a^{4} b x + 21 \, a^{5}}{168 \, x^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 58, normalized size = 1.04 \begin {gather*} -\frac {b^{5}}{3 x^{3}}-\frac {5 a \,b^{4}}{4 x^{4}}-\frac {2 a^{2} b^{3}}{x^{5}}-\frac {5 a^{3} b^{2}}{3 x^{6}}-\frac {5 a^{4} b}{7 x^{7}}-\frac {a^{5}}{8 x^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.40, size = 57, normalized size = 1.02 \begin {gather*} -\frac {56 \, b^{5} x^{5} + 210 \, a b^{4} x^{4} + 336 \, a^{2} b^{3} x^{3} + 280 \, a^{3} b^{2} x^{2} + 120 \, a^{4} b x + 21 \, a^{5}}{168 \, x^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 57, normalized size = 1.02 \begin {gather*} -\frac {\frac {a^5}{8}+\frac {5\,a^4\,b\,x}{7}+\frac {5\,a^3\,b^2\,x^2}{3}+2\,a^2\,b^3\,x^3+\frac {5\,a\,b^4\,x^4}{4}+\frac {b^5\,x^5}{3}}{x^8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.43, size = 61, normalized size = 1.09 \begin {gather*} \frac {- 21 a^{5} - 120 a^{4} b x - 280 a^{3} b^{2} x^{2} - 336 a^{2} b^{3} x^{3} - 210 a b^{4} x^{4} - 56 b^{5} x^{5}}{168 x^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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