Optimal. Leaf size=56 \[ \frac {a^4 x^5}{5}+\frac {4}{7} a^3 b x^7+\frac {2}{3} a^2 b^2 x^9+\frac {4}{11} a b^3 x^{11}+\frac {b^4 x^{13}}{13} \]
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Rubi [A]
time = 0.02, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {28, 276}
\begin {gather*} \frac {a^4 x^5}{5}+\frac {4}{7} a^3 b x^7+\frac {2}{3} a^2 b^2 x^9+\frac {4}{11} a b^3 x^{11}+\frac {b^4 x^{13}}{13} \end {gather*}
Antiderivative was successfully verified.
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Rule 28
Rule 276
Rubi steps
\begin {align*} \int x^4 \left (a^2+2 a b x^2+b^2 x^4\right )^2 \, dx &=\frac {\int x^4 \left (a b+b^2 x^2\right )^4 \, dx}{b^4}\\ &=\frac {\int \left (a^4 b^4 x^4+4 a^3 b^5 x^6+6 a^2 b^6 x^8+4 a b^7 x^{10}+b^8 x^{12}\right ) \, dx}{b^4}\\ &=\frac {a^4 x^5}{5}+\frac {4}{7} a^3 b x^7+\frac {2}{3} a^2 b^2 x^9+\frac {4}{11} a b^3 x^{11}+\frac {b^4 x^{13}}{13}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 56, normalized size = 1.00 \begin {gather*} \frac {a^4 x^5}{5}+\frac {4}{7} a^3 b x^7+\frac {2}{3} a^2 b^2 x^9+\frac {4}{11} a b^3 x^{11}+\frac {b^4 x^{13}}{13} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.05, size = 47, normalized size = 0.84
method | result | size |
default | \(\frac {1}{5} a^{4} x^{5}+\frac {4}{7} a^{3} b \,x^{7}+\frac {2}{3} a^{2} b^{2} x^{9}+\frac {4}{11} a \,b^{3} x^{11}+\frac {1}{13} b^{4} x^{13}\) | \(47\) |
norman | \(\frac {1}{5} a^{4} x^{5}+\frac {4}{7} a^{3} b \,x^{7}+\frac {2}{3} a^{2} b^{2} x^{9}+\frac {4}{11} a \,b^{3} x^{11}+\frac {1}{13} b^{4} x^{13}\) | \(47\) |
risch | \(\frac {1}{5} a^{4} x^{5}+\frac {4}{7} a^{3} b \,x^{7}+\frac {2}{3} a^{2} b^{2} x^{9}+\frac {4}{11} a \,b^{3} x^{11}+\frac {1}{13} b^{4} x^{13}\) | \(47\) |
gosper | \(\frac {x^{5} \left (1155 b^{4} x^{8}+5460 a \,b^{3} x^{6}+10010 a^{2} b^{2} x^{4}+8580 a^{3} b \,x^{2}+3003 a^{4}\right )}{15015}\) | \(49\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 46, normalized size = 0.82 \begin {gather*} \frac {1}{13} \, b^{4} x^{13} + \frac {4}{11} \, a b^{3} x^{11} + \frac {2}{3} \, a^{2} b^{2} x^{9} + \frac {4}{7} \, a^{3} b x^{7} + \frac {1}{5} \, a^{4} x^{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 46, normalized size = 0.82 \begin {gather*} \frac {1}{13} \, b^{4} x^{13} + \frac {4}{11} \, a b^{3} x^{11} + \frac {2}{3} \, a^{2} b^{2} x^{9} + \frac {4}{7} \, a^{3} b x^{7} + \frac {1}{5} \, a^{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 = 53, normalized size = 0.95 \begin {gather*} \frac {a^{4} x^{5}}{5} + \frac {4 a^{3} b x^{7}}{7} + \frac {2 a^{2} b^{2} x^{9}}{3} + \frac {4 a b^{3} x^{11}}{11} + \frac {b^{4} x^{13}}{13} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.39, size = 46, normalized size = 0.82 \begin {gather*} \frac {1}{13} \, b^{4} x^{13} + \frac {4}{11} \, a b^{3} x^{11} + \frac {2}{3} \, a^{2} b^{2} x^{9} + \frac {4}{7} \, a^{3} b x^{7} + \frac {1}{5} \, a^{4} x^{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.02, size = 46, normalized size = 0.82 \begin {gather*} \frac {a^4\,x^5}{5}+\frac {4\,a^3\,b\,x^7}{7}+\frac {2\,a^2\,b^2\,x^9}{3}+\frac {4\,a\,b^3\,x^{11}}{11}+\frac {b^4\,x^{13}}{13} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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