Optimal. Leaf size=75 \[ -\frac {a^6 c^5}{x}+5 a^4 b^2 c^5 x-\frac {5}{3} a^2 b^4 c^5 x^3+a b^5 c^5 x^4-\frac {1}{5} b^6 c^5 x^5-4 a^5 b c^5 \log (x) \]
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Rubi [A]
time = 0.02, antiderivative size = 75, 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^6 c^5}{x}-4 a^5 b c^5 \log (x)+5 a^4 b^2 c^5 x-\frac {5}{3} a^2 b^4 c^5 x^3+a b^5 c^5 x^4-\frac {1}{5} b^6 c^5 x^5 \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)^5}{x^2} \, dx &=\int \left (5 a^4 b^2 c^5+\frac {a^6 c^5}{x^2}-\frac {4 a^5 b c^5}{x}-5 a^2 b^4 c^5 x^2+4 a b^5 c^5 x^3-b^6 c^5 x^4\right ) \, dx\\ &=-\frac {a^6 c^5}{x}+5 a^4 b^2 c^5 x-\frac {5}{3} a^2 b^4 c^5 x^3+a b^5 c^5 x^4-\frac {1}{5} b^6 c^5 x^5-4 a^5 b c^5 \log (x)\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 61, normalized size = 0.81 \begin {gather*} c^5 \left (-\frac {a^6}{x}+5 a^4 b^2 x-\frac {5}{3} a^2 b^4 x^3+a b^5 x^4-\frac {b^6 x^5}{5}-4 a^5 b \log (x)\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 58, normalized size = 0.77
method | result | size |
default | \(c^{5} \left (-\frac {b^{6} x^{5}}{5}+a \,b^{5} x^{4}-\frac {5 a^{2} b^{4} x^{3}}{3}+5 a^{4} b^{2} x -\frac {a^{6}}{x}-4 a^{5} b \ln \left (x \right )\right )\) | \(58\) |
risch | \(-\frac {a^{6} c^{5}}{x}+5 a^{4} b^{2} c^{5} x -\frac {5 a^{2} b^{4} c^{5} x^{3}}{3}+a \,b^{5} c^{5} x^{4}-\frac {b^{6} c^{5} x^{5}}{5}-4 a^{5} b \,c^{5} \ln \left (x \right )\) | \(72\) |
norman | \(\frac {a \,b^{5} c^{5} x^{5}-a^{6} c^{5}-\frac {1}{5} b^{6} c^{5} x^{6}-\frac {5}{3} a^{2} b^{4} c^{5} x^{4}+5 a^{4} b^{2} c^{5} x^{2}}{x}-4 a^{5} b \,c^{5} \ln \left (x \right )\) | \(76\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.38, size = 71, normalized size = 0.95 \begin {gather*} -\frac {1}{5} \, b^{6} c^{5} x^{5} + a b^{5} c^{5} x^{4} - \frac {5}{3} \, a^{2} b^{4} c^{5} x^{3} + 5 \, a^{4} b^{2} c^{5} x - 4 \, a^{5} b c^{5} \log \left (x\right ) - \frac {a^{6} c^{5}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.87, size = 77, normalized size = 1.03 \begin {gather*} -\frac {3 \, b^{6} c^{5} x^{6} - 15 \, a b^{5} c^{5} x^{5} + 25 \, a^{2} b^{4} c^{5} x^{4} - 75 \, a^{4} b^{2} c^{5} x^{2} + 60 \, a^{5} b c^{5} x \log \left (x\right ) + 15 \, a^{6} c^{5}}{15 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.06, size = 75, normalized size = 1.00 \begin {gather*} - \frac {a^{6} c^{5}}{x} - 4 a^{5} b c^{5} \log {\left (x \right )} + 5 a^{4} b^{2} c^{5} x - \frac {5 a^{2} b^{4} c^{5} x^{3}}{3} + a b^{5} c^{5} x^{4} - \frac {b^{6} c^{5} x^{5}}{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.39, size = 72, normalized size = 0.96 \begin {gather*} -\frac {1}{5} \, b^{6} c^{5} x^{5} + a b^{5} c^{5} x^{4} - \frac {5}{3} \, a^{2} b^{4} c^{5} x^{3} + 5 \, a^{4} b^{2} c^{5} x - 4 \, a^{5} b c^{5} \log \left ({\left | x \right |}\right ) - \frac {a^{6} c^{5}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 71, normalized size = 0.95 \begin {gather*} 5\,a^4\,b^2\,c^5\,x-\frac {b^6\,c^5\,x^5}{5}-\frac {a^6\,c^5}{x}+a\,b^5\,c^5\,x^4-4\,a^5\,b\,c^5\,\ln \left (x\right )-\frac {5\,a^2\,b^4\,c^5\,x^3}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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