Optimal. Leaf size=113 \[ -\frac {a^7 c^6}{7 x^7}+\frac {5 a^6 b c^6}{6 x^6}-\frac {9 a^5 b^2 c^6}{5 x^5}+\frac {5 a^4 b^3 c^6}{4 x^4}+\frac {5 a^3 b^4 c^6}{3 x^3}-\frac {9 a^2 b^5 c^6}{2 x^2}+\frac {5 a b^6 c^6}{x}+b^7 c^6 \log (x) \]
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Rubi [A]
time = 0.04, antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {76}
\begin {gather*} -\frac {a^7 c^6}{7 x^7}+\frac {5 a^6 b c^6}{6 x^6}-\frac {9 a^5 b^2 c^6}{5 x^5}+\frac {5 a^4 b^3 c^6}{4 x^4}+\frac {5 a^3 b^4 c^6}{3 x^3}-\frac {9 a^2 b^5 c^6}{2 x^2}+\frac {5 a b^6 c^6}{x}+b^7 c^6 \log (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 76
Rubi steps
\begin {align*} \int \frac {(a+b x) (a c-b c x)^6}{x^8} \, dx &=\int \left (\frac {a^7 c^6}{x^8}-\frac {5 a^6 b c^6}{x^7}+\frac {9 a^5 b^2 c^6}{x^6}-\frac {5 a^4 b^3 c^6}{x^5}-\frac {5 a^3 b^4 c^6}{x^4}+\frac {9 a^2 b^5 c^6}{x^3}-\frac {5 a b^6 c^6}{x^2}+\frac {b^7 c^6}{x}\right ) \, dx\\ &=-\frac {a^7 c^6}{7 x^7}+\frac {5 a^6 b c^6}{6 x^6}-\frac {9 a^5 b^2 c^6}{5 x^5}+\frac {5 a^4 b^3 c^6}{4 x^4}+\frac {5 a^3 b^4 c^6}{3 x^3}-\frac {9 a^2 b^5 c^6}{2 x^2}+\frac {5 a b^6 c^6}{x}+b^7 c^6 \log (x)\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 113, normalized size = 1.00 \begin {gather*} -\frac {a^7 c^6}{7 x^7}+\frac {5 a^6 b c^6}{6 x^6}-\frac {9 a^5 b^2 c^6}{5 x^5}+\frac {5 a^4 b^3 c^6}{4 x^4}+\frac {5 a^3 b^4 c^6}{3 x^3}-\frac {9 a^2 b^5 c^6}{2 x^2}+\frac {5 a b^6 c^6}{x}+b^7 c^6 \log (x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 82, normalized size = 0.73
method | result | size |
default | \(c^{6} \left (\frac {5 a \,b^{6}}{x}-\frac {9 a^{5} b^{2}}{5 x^{5}}+\frac {5 a^{4} b^{3}}{4 x^{4}}+\frac {5 a^{6} b}{6 x^{6}}-\frac {9 a^{2} b^{5}}{2 x^{2}}+b^{7} \ln \left (x \right )+\frac {5 a^{3} b^{4}}{3 x^{3}}-\frac {a^{7}}{7 x^{7}}\right )\) | \(82\) |
norman | \(\frac {-\frac {1}{7} a^{7} c^{6}+5 a \,b^{6} c^{6} x^{6}-\frac {9}{2} a^{2} b^{5} c^{6} x^{5}+\frac {5}{3} a^{3} b^{4} c^{6} x^{4}+\frac {5}{4} a^{4} b^{3} c^{6} x^{3}-\frac {9}{5} a^{5} b^{2} c^{6} x^{2}+\frac {5}{6} a^{6} b \,c^{6} x}{x^{7}}+b^{7} c^{6} \ln \left (x \right )\) | \(102\) |
risch | \(\frac {-\frac {1}{7} a^{7} c^{6}+5 a \,b^{6} c^{6} x^{6}-\frac {9}{2} a^{2} b^{5} c^{6} x^{5}+\frac {5}{3} a^{3} b^{4} c^{6} x^{4}+\frac {5}{4} a^{4} b^{3} c^{6} x^{3}-\frac {9}{5} a^{5} b^{2} c^{6} x^{2}+\frac {5}{6} a^{6} b \,c^{6} x}{x^{7}}+b^{7} c^{6} \ln \left (x \right )\) | \(102\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 102, normalized size = 0.90 \begin {gather*} b^{7} c^{6} \log \left (x\right ) + \frac {2100 \, a b^{6} c^{6} x^{6} - 1890 \, a^{2} b^{5} c^{6} x^{5} + 700 \, a^{3} b^{4} c^{6} x^{4} + 525 \, a^{4} b^{3} c^{6} x^{3} - 756 \, a^{5} b^{2} c^{6} x^{2} + 350 \, a^{6} b c^{6} x - 60 \, a^{7} c^{6}}{420 \, x^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.33, size = 105, normalized size = 0.93 \begin {gather*} \frac {420 \, b^{7} c^{6} x^{7} \log \left (x\right ) + 2100 \, a b^{6} c^{6} x^{6} - 1890 \, a^{2} b^{5} c^{6} x^{5} + 700 \, a^{3} b^{4} c^{6} x^{4} + 525 \, a^{4} b^{3} c^{6} x^{3} - 756 \, a^{5} b^{2} c^{6} x^{2} + 350 \, a^{6} b c^{6} x - 60 \, a^{7} c^{6}}{420 \, x^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.26, size = 109, normalized size = 0.96 \begin {gather*} b^{7} c^{6} \log {\left (x \right )} + \frac {- 60 a^{7} c^{6} + 350 a^{6} b c^{6} x - 756 a^{5} b^{2} c^{6} x^{2} + 525 a^{4} b^{3} c^{6} x^{3} + 700 a^{3} b^{4} c^{6} x^{4} - 1890 a^{2} b^{5} c^{6} x^{5} + 2100 a b^{6} c^{6} x^{6}}{420 x^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.05, size = 103, normalized size = 0.91 \begin {gather*} b^{7} c^{6} \log \left ({\left | x \right |}\right ) + \frac {2100 \, a b^{6} c^{6} x^{6} - 1890 \, a^{2} b^{5} c^{6} x^{5} + 700 \, a^{3} b^{4} c^{6} x^{4} + 525 \, a^{4} b^{3} c^{6} x^{3} - 756 \, a^{5} b^{2} c^{6} x^{2} + 350 \, a^{6} b c^{6} x - 60 \, a^{7} c^{6}}{420 \, x^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.31, size = 82, normalized size = 0.73 \begin {gather*} \frac {c^6\,\left (5\,a\,b^6\,x^6-\frac {a^7}{7}-\frac {9\,a^5\,b^2\,x^2}{5}+\frac {5\,a^4\,b^3\,x^3}{4}+\frac {5\,a^3\,b^4\,x^4}{3}-\frac {9\,a^2\,b^5\,x^5}{2}+b^7\,x^7\,\ln \left (x\right )+\frac {5\,a^6\,b\,x}{6}\right )}{x^7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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