Optimal. Leaf size=87 \[ -\frac {a^6 c^5}{10 x^{10}}+\frac {4 a^5 b c^5}{9 x^9}-\frac {5 a^4 b^2 c^5}{8 x^8}+\frac {5 a^2 b^4 c^5}{6 x^6}-\frac {4 a b^5 c^5}{5 x^5}+\frac {b^6 c^5}{4 x^4} \]
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Rubi [A]
time = 0.02, antiderivative size = 87, 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}{10 x^{10}}+\frac {4 a^5 b c^5}{9 x^9}-\frac {5 a^4 b^2 c^5}{8 x^8}+\frac {5 a^2 b^4 c^5}{6 x^6}-\frac {4 a b^5 c^5}{5 x^5}+\frac {b^6 c^5}{4 x^4} \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^{11}} \, dx &=\int \left (\frac {a^6 c^5}{x^{11}}-\frac {4 a^5 b c^5}{x^{10}}+\frac {5 a^4 b^2 c^5}{x^9}-\frac {5 a^2 b^4 c^5}{x^7}+\frac {4 a b^5 c^5}{x^6}-\frac {b^6 c^5}{x^5}\right ) \, dx\\ &=-\frac {a^6 c^5}{10 x^{10}}+\frac {4 a^5 b c^5}{9 x^9}-\frac {5 a^4 b^2 c^5}{8 x^8}+\frac {5 a^2 b^4 c^5}{6 x^6}-\frac {4 a b^5 c^5}{5 x^5}+\frac {b^6 c^5}{4 x^4}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 73, normalized size = 0.84 \begin {gather*} c^5 \left (-\frac {a^6}{10 x^{10}}+\frac {4 a^5 b}{9 x^9}-\frac {5 a^4 b^2}{8 x^8}+\frac {5 a^2 b^4}{6 x^6}-\frac {4 a b^5}{5 x^5}+\frac {b^6}{4 x^4}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 62, normalized size = 0.71
method | result | size |
gosper | \(-\frac {c^{5} \left (-90 b^{6} x^{6}+288 a \,b^{5} x^{5}-300 a^{2} b^{4} x^{4}+225 a^{4} b^{2} x^{2}-160 a^{5} b x +36 a^{6}\right )}{360 x^{10}}\) | \(61\) |
default | \(c^{5} \left (\frac {4 a^{5} b}{9 x^{9}}-\frac {4 a \,b^{5}}{5 x^{5}}+\frac {b^{6}}{4 x^{4}}+\frac {5 a^{2} b^{4}}{6 x^{6}}-\frac {a^{6}}{10 x^{10}}-\frac {5 a^{4} b^{2}}{8 x^{8}}\right )\) | \(62\) |
norman | \(\frac {-\frac {1}{10} a^{6} c^{5}+\frac {1}{4} b^{6} c^{5} x^{6}-\frac {4}{5} a \,b^{5} c^{5} x^{5}+\frac {5}{6} a^{2} b^{4} c^{5} x^{4}-\frac {5}{8} a^{4} b^{2} c^{5} x^{2}+\frac {4}{9} a^{5} b \,c^{5} x}{x^{10}}\) | \(75\) |
risch | \(\frac {-\frac {1}{10} a^{6} c^{5}+\frac {1}{4} b^{6} c^{5} x^{6}-\frac {4}{5} a \,b^{5} c^{5} x^{5}+\frac {5}{6} a^{2} b^{4} c^{5} x^{4}-\frac {5}{8} a^{4} b^{2} c^{5} x^{2}+\frac {4}{9} a^{5} b \,c^{5} x}{x^{10}}\) | \(75\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 75, normalized size = 0.86 \begin {gather*} \frac {90 \, b^{6} c^{5} x^{6} - 288 \, a b^{5} c^{5} x^{5} + 300 \, a^{2} b^{4} c^{5} x^{4} - 225 \, a^{4} b^{2} c^{5} x^{2} + 160 \, a^{5} b c^{5} x - 36 \, a^{6} c^{5}}{360 \, x^{10}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.34, size = 75, normalized size = 0.86 \begin {gather*} \frac {90 \, b^{6} c^{5} x^{6} - 288 \, a b^{5} c^{5} x^{5} + 300 \, a^{2} b^{4} c^{5} x^{4} - 225 \, a^{4} b^{2} c^{5} x^{2} + 160 \, a^{5} b c^{5} x - 36 \, a^{6} c^{5}}{360 \, x^{10}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.25, size = 82, normalized size = 0.94 \begin {gather*} - \frac {36 a^{6} c^{5} - 160 a^{5} b c^{5} x + 225 a^{4} b^{2} c^{5} x^{2} - 300 a^{2} b^{4} c^{5} x^{4} + 288 a b^{5} c^{5} x^{5} - 90 b^{6} c^{5} x^{6}}{360 x^{10}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.83, size = 75, normalized size = 0.86 \begin {gather*} \frac {90 \, b^{6} c^{5} x^{6} - 288 \, a b^{5} c^{5} x^{5} + 300 \, a^{2} b^{4} c^{5} x^{4} - 225 \, a^{4} b^{2} c^{5} x^{2} + 160 \, a^{5} b c^{5} x - 36 \, a^{6} c^{5}}{360 \, x^{10}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 75, normalized size = 0.86 \begin {gather*} -\frac {\frac {a^6\,c^5}{10}-\frac {4\,a^5\,b\,c^5\,x}{9}+\frac {5\,a^4\,b^2\,c^5\,x^2}{8}-\frac {5\,a^2\,b^4\,c^5\,x^4}{6}+\frac {4\,a\,b^5\,c^5\,x^5}{5}-\frac {b^6\,c^5\,x^6}{4}}{x^{10}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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