Optimal. Leaf size=87 \[ \frac {1}{5} a^5 c^4 x^5-\frac {1}{2} a^4 b c^4 x^6+\frac {2}{7} a^3 b^2 c^4 x^7+\frac {1}{4} a^2 b^3 c^4 x^8-\frac {1}{3} a b^4 c^4 x^9+\frac {1}{10} b^5 c^4 x^{10} \]
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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^5 c^4 x^5-\frac {1}{2} a^4 b c^4 x^6+\frac {2}{7} a^3 b^2 c^4 x^7+\frac {1}{4} a^2 b^3 c^4 x^8-\frac {1}{3} a b^4 c^4 x^9+\frac {1}{10} b^5 c^4 x^{10} \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)^4 \, dx &=\int \left (a^5 c^4 x^4-3 a^4 b c^4 x^5+2 a^3 b^2 c^4 x^6+2 a^2 b^3 c^4 x^7-3 a b^4 c^4 x^8+b^5 c^4 x^9\right ) \, dx\\ &=\frac {1}{5} a^5 c^4 x^5-\frac {1}{2} a^4 b c^4 x^6+\frac {2}{7} a^3 b^2 c^4 x^7+\frac {1}{4} a^2 b^3 c^4 x^8-\frac {1}{3} a b^4 c^4 x^9+\frac {1}{10} b^5 c^4 x^{10}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 87, normalized size = 1.00 \begin {gather*} \frac {1}{5} a^5 c^4 x^5-\frac {1}{2} a^4 b c^4 x^6+\frac {2}{7} a^3 b^2 c^4 x^7+\frac {1}{4} a^2 b^3 c^4 x^8-\frac {1}{3} a b^4 c^4 x^9+\frac {1}{10} b^5 c^4 x^{10} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.05, size = 76, normalized size = 0.87
method | result | size |
gosper | \(\frac {x^{5} \left (42 b^{5} x^{5}-140 a \,b^{4} x^{4}+105 a^{2} b^{3} x^{3}+120 a^{3} b^{2} x^{2}-210 a^{4} b x +84 a^{5}\right ) c^{4}}{420}\) | \(61\) |
default | \(\frac {1}{5} a^{5} c^{4} x^{5}-\frac {1}{2} a^{4} b \,c^{4} x^{6}+\frac {2}{7} a^{3} b^{2} c^{4} x^{7}+\frac {1}{4} a^{2} b^{3} c^{4} x^{8}-\frac {1}{3} a \,b^{4} c^{4} x^{9}+\frac {1}{10} b^{5} c^{4} x^{10}\) | \(76\) |
norman | \(\frac {1}{5} a^{5} c^{4} x^{5}-\frac {1}{2} a^{4} b \,c^{4} x^{6}+\frac {2}{7} a^{3} b^{2} c^{4} x^{7}+\frac {1}{4} a^{2} b^{3} c^{4} x^{8}-\frac {1}{3} a \,b^{4} c^{4} x^{9}+\frac {1}{10} b^{5} c^{4} x^{10}\) | \(76\) |
risch | \(\frac {1}{5} a^{5} c^{4} x^{5}-\frac {1}{2} a^{4} b \,c^{4} x^{6}+\frac {2}{7} a^{3} b^{2} c^{4} x^{7}+\frac {1}{4} a^{2} b^{3} c^{4} x^{8}-\frac {1}{3} a \,b^{4} c^{4} x^{9}+\frac {1}{10} b^{5} c^{4} x^{10}\) | \(76\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 75, normalized size = 0.86 \begin {gather*} \frac {1}{10} \, b^{5} c^{4} x^{10} - \frac {1}{3} \, a b^{4} c^{4} x^{9} + \frac {1}{4} \, a^{2} b^{3} c^{4} x^{8} + \frac {2}{7} \, a^{3} b^{2} c^{4} x^{7} - \frac {1}{2} \, a^{4} b c^{4} x^{6} + \frac {1}{5} \, a^{5} c^{4} x^{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.05, size = 75, normalized size = 0.86 \begin {gather*} \frac {1}{10} \, b^{5} c^{4} x^{10} - \frac {1}{3} \, a b^{4} c^{4} x^{9} + \frac {1}{4} \, a^{2} b^{3} c^{4} x^{8} + \frac {2}{7} \, a^{3} b^{2} c^{4} x^{7} - \frac {1}{2} \, a^{4} b c^{4} x^{6} + \frac {1}{5} \, a^{5} c^{4} x^{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.01, size = 82, normalized size = 0.94 \begin {gather*} \frac {a^{5} c^{4} x^{5}}{5} - \frac {a^{4} b c^{4} x^{6}}{2} + \frac {2 a^{3} b^{2} c^{4} x^{7}}{7} + \frac {a^{2} b^{3} c^{4} x^{8}}{4} - \frac {a b^{4} c^{4} x^{9}}{3} + \frac {b^{5} c^{4} x^{10}}{10} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.62, size = 75, normalized size = 0.86 \begin {gather*} \frac {1}{10} \, b^{5} c^{4} x^{10} - \frac {1}{3} \, a b^{4} c^{4} x^{9} + \frac {1}{4} \, a^{2} b^{3} c^{4} x^{8} + \frac {2}{7} \, a^{3} b^{2} c^{4} x^{7} - \frac {1}{2} \, a^{4} b c^{4} x^{6} + \frac {1}{5} \, a^{5} c^{4} 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^5\,c^4\,x^5}{5}-\frac {a^4\,b\,c^4\,x^6}{2}+\frac {2\,a^3\,b^2\,c^4\,x^7}{7}+\frac {a^2\,b^3\,c^4\,x^8}{4}-\frac {a\,b^4\,c^4\,x^9}{3}+\frac {b^5\,c^4\,x^{10}}{10} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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