Optimal. Leaf size=87 \[ \frac {1}{5} a^6 c^5 x^5-\frac {2}{3} a^5 b c^5 x^6+\frac {5}{7} a^4 b^2 c^5 x^7-\frac {5}{9} a^2 b^4 c^5 x^9+\frac {2}{5} a b^5 c^5 x^{10}-\frac {1}{11} b^6 c^5 x^{11} \]
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Rubi [A]
time = 0.03, 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 {1}{5} a^6 c^5 x^5-\frac {2}{3} a^5 b c^5 x^6+\frac {5}{7} a^4 b^2 c^5 x^7-\frac {5}{9} a^2 b^4 c^5 x^9+\frac {2}{5} a b^5 c^5 x^{10}-\frac {1}{11} b^6 c^5 x^{11} \end {gather*}
Antiderivative was successfully verified.
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Rule 76
Rubi steps
\begin {align*} \int x^4 (a+b x) (a c-b c x)^5 \, dx &=\int \left (a^6 c^5 x^4-4 a^5 b c^5 x^5+5 a^4 b^2 c^5 x^6-5 a^2 b^4 c^5 x^8+4 a b^5 c^5 x^9-b^6 c^5 x^{10}\right ) \, dx\\ &=\frac {1}{5} a^6 c^5 x^5-\frac {2}{3} a^5 b c^5 x^6+\frac {5}{7} a^4 b^2 c^5 x^7-\frac {5}{9} a^2 b^4 c^5 x^9+\frac {2}{5} a b^5 c^5 x^{10}-\frac {1}{11} b^6 c^5 x^{11}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 73, normalized size = 0.84 \begin {gather*} c^5 \left (\frac {a^6 x^5}{5}-\frac {2}{3} a^5 b x^6+\frac {5}{7} a^4 b^2 x^7-\frac {5}{9} a^2 b^4 x^9+\frac {2}{5} a b^5 x^{10}-\frac {b^6 x^{11}}{11}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 76, normalized size = 0.87
method | result | size |
gosper | \(\frac {x^{5} \left (-315 b^{6} x^{6}+1386 a \,b^{5} x^{5}-1925 a^{2} b^{4} x^{4}+2475 a^{4} b^{2} x^{2}-2310 a^{5} b x +693 a^{6}\right ) c^{5}}{3465}\) | \(61\) |
default | \(\frac {1}{5} a^{6} c^{5} x^{5}-\frac {2}{3} a^{5} b \,c^{5} x^{6}+\frac {5}{7} a^{4} b^{2} c^{5} x^{7}-\frac {5}{9} a^{2} b^{4} c^{5} x^{9}+\frac {2}{5} a \,b^{5} c^{5} x^{10}-\frac {1}{11} b^{6} c^{5} x^{11}\) | \(76\) |
norman | \(\frac {1}{5} a^{6} c^{5} x^{5}-\frac {2}{3} a^{5} b \,c^{5} x^{6}+\frac {5}{7} a^{4} b^{2} c^{5} x^{7}-\frac {5}{9} a^{2} b^{4} c^{5} x^{9}+\frac {2}{5} a \,b^{5} c^{5} x^{10}-\frac {1}{11} b^{6} c^{5} x^{11}\) | \(76\) |
risch | \(\frac {1}{5} a^{6} c^{5} x^{5}-\frac {2}{3} a^{5} b \,c^{5} x^{6}+\frac {5}{7} a^{4} b^{2} c^{5} x^{7}-\frac {5}{9} a^{2} b^{4} c^{5} x^{9}+\frac {2}{5} a \,b^{5} c^{5} x^{10}-\frac {1}{11} b^{6} c^{5} x^{11}\) | \(76\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 75, normalized size = 0.86 \begin {gather*} -\frac {1}{11} \, b^{6} c^{5} x^{11} + \frac {2}{5} \, a b^{5} c^{5} x^{10} - \frac {5}{9} \, a^{2} b^{4} c^{5} x^{9} + \frac {5}{7} \, a^{4} b^{2} c^{5} x^{7} - \frac {2}{3} \, a^{5} b c^{5} x^{6} + \frac {1}{5} \, a^{6} c^{5} x^{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.08, size = 75, normalized size = 0.86 \begin {gather*} -\frac {1}{11} \, b^{6} c^{5} x^{11} + \frac {2}{5} \, a b^{5} c^{5} x^{10} - \frac {5}{9} \, a^{2} b^{4} c^{5} x^{9} + \frac {5}{7} \, a^{4} b^{2} c^{5} x^{7} - \frac {2}{3} \, a^{5} b c^{5} x^{6} + \frac {1}{5} \, a^{6} c^{5} x^{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.02, size = 87, normalized size = 1.00 \begin {gather*} \frac {a^{6} c^{5} x^{5}}{5} - \frac {2 a^{5} b c^{5} x^{6}}{3} + \frac {5 a^{4} b^{2} c^{5} x^{7}}{7} - \frac {5 a^{2} b^{4} c^{5} x^{9}}{9} + \frac {2 a b^{5} c^{5} x^{10}}{5} - \frac {b^{6} c^{5} x^{11}}{11} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.94, size = 75, normalized size = 0.86 \begin {gather*} -\frac {1}{11} \, b^{6} c^{5} x^{11} + \frac {2}{5} \, a b^{5} c^{5} x^{10} - \frac {5}{9} \, a^{2} b^{4} c^{5} x^{9} + \frac {5}{7} \, a^{4} b^{2} c^{5} x^{7} - \frac {2}{3} \, a^{5} b c^{5} x^{6} + \frac {1}{5} \, a^{6} c^{5} x^{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 75, normalized size = 0.86 \begin {gather*} \frac {a^6\,c^5\,x^5}{5}-\frac {2\,a^5\,b\,c^5\,x^6}{3}+\frac {5\,a^4\,b^2\,c^5\,x^7}{7}-\frac {5\,a^2\,b^4\,c^5\,x^9}{9}+\frac {2\,a\,b^5\,c^5\,x^{10}}{5}-\frac {b^6\,c^5\,x^{11}}{11} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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