Optimal. Leaf size=149 \[ \frac {1}{3} a^5 d x^3+\frac {1}{5} a^5 e x^5+\frac {5}{7} a^4 c d x^7+\frac {5}{9} a^4 c e x^9+\frac {10}{11} a^3 c^2 d x^{11}+\frac {10}{13} a^3 c^2 e x^{13}+\frac {2}{3} a^2 c^3 d x^{15}+\frac {10}{17} a^2 c^3 e x^{17}+\frac {5}{19} a c^4 d x^{19}+\frac {5}{21} a c^4 e x^{21}+\frac {1}{23} c^5 d x^{23}+\frac {1}{25} c^5 e x^{25} \]
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Rubi [A]
time = 0.06, antiderivative size = 149, 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 = {1276}
\begin {gather*} \frac {1}{3} a^5 d x^3+\frac {1}{5} a^5 e x^5+\frac {5}{7} a^4 c d x^7+\frac {5}{9} a^4 c e x^9+\frac {10}{11} a^3 c^2 d x^{11}+\frac {10}{13} a^3 c^2 e x^{13}+\frac {2}{3} a^2 c^3 d x^{15}+\frac {10}{17} a^2 c^3 e x^{17}+\frac {5}{19} a c^4 d x^{19}+\frac {5}{21} a c^4 e x^{21}+\frac {1}{23} c^5 d x^{23}+\frac {1}{25} c^5 e x^{25} \end {gather*}
Antiderivative was successfully verified.
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Rule 1276
Rubi steps
\begin {align*} \int x^2 \left (d+e x^2\right ) \left (a+c x^4\right )^5 \, dx &=\int \left (a^5 d x^2+a^5 e x^4+5 a^4 c d x^6+5 a^4 c e x^8+10 a^3 c^2 d x^{10}+10 a^3 c^2 e x^{12}+10 a^2 c^3 d x^{14}+10 a^2 c^3 e x^{16}+5 a c^4 d x^{18}+5 a c^4 e x^{20}+c^5 d x^{22}+c^5 e x^{24}\right ) \, dx\\ &=\frac {1}{3} a^5 d x^3+\frac {1}{5} a^5 e x^5+\frac {5}{7} a^4 c d x^7+\frac {5}{9} a^4 c e x^9+\frac {10}{11} a^3 c^2 d x^{11}+\frac {10}{13} a^3 c^2 e x^{13}+\frac {2}{3} a^2 c^3 d x^{15}+\frac {10}{17} a^2 c^3 e x^{17}+\frac {5}{19} a c^4 d x^{19}+\frac {5}{21} a c^4 e x^{21}+\frac {1}{23} c^5 d x^{23}+\frac {1}{25} c^5 e x^{25}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 149, normalized size = 1.00 \begin {gather*} \frac {1}{3} a^5 d x^3+\frac {1}{5} a^5 e x^5+\frac {5}{7} a^4 c d x^7+\frac {5}{9} a^4 c e x^9+\frac {10}{11} a^3 c^2 d x^{11}+\frac {10}{13} a^3 c^2 e x^{13}+\frac {2}{3} a^2 c^3 d x^{15}+\frac {10}{17} a^2 c^3 e x^{17}+\frac {5}{19} a c^4 d x^{19}+\frac {5}{21} a c^4 e x^{21}+\frac {1}{23} c^5 d x^{23}+\frac {1}{25} c^5 e x^{25} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 126, normalized size = 0.85
method | result | size |
gosper | \(\frac {1}{3} a^{5} d \,x^{3}+\frac {1}{5} a^{5} e \,x^{5}+\frac {5}{7} a^{4} c d \,x^{7}+\frac {5}{9} a^{4} c e \,x^{9}+\frac {10}{11} a^{3} c^{2} d \,x^{11}+\frac {10}{13} a^{3} c^{2} e \,x^{13}+\frac {2}{3} a^{2} c^{3} d \,x^{15}+\frac {10}{17} a^{2} c^{3} e \,x^{17}+\frac {5}{19} a \,c^{4} d \,x^{19}+\frac {5}{21} a \,c^{4} e \,x^{21}+\frac {1}{23} c^{5} d \,x^{23}+\frac {1}{25} c^{5} e \,x^{25}\) | \(126\) |
default | \(\frac {1}{3} a^{5} d \,x^{3}+\frac {1}{5} a^{5} e \,x^{5}+\frac {5}{7} a^{4} c d \,x^{7}+\frac {5}{9} a^{4} c e \,x^{9}+\frac {10}{11} a^{3} c^{2} d \,x^{11}+\frac {10}{13} a^{3} c^{2} e \,x^{13}+\frac {2}{3} a^{2} c^{3} d \,x^{15}+\frac {10}{17} a^{2} c^{3} e \,x^{17}+\frac {5}{19} a \,c^{4} d \,x^{19}+\frac {5}{21} a \,c^{4} e \,x^{21}+\frac {1}{23} c^{5} d \,x^{23}+\frac {1}{25} c^{5} e \,x^{25}\) | \(126\) |
norman | \(\frac {1}{3} a^{5} d \,x^{3}+\frac {1}{5} a^{5} e \,x^{5}+\frac {5}{7} a^{4} c d \,x^{7}+\frac {5}{9} a^{4} c e \,x^{9}+\frac {10}{11} a^{3} c^{2} d \,x^{11}+\frac {10}{13} a^{3} c^{2} e \,x^{13}+\frac {2}{3} a^{2} c^{3} d \,x^{15}+\frac {10}{17} a^{2} c^{3} e \,x^{17}+\frac {5}{19} a \,c^{4} d \,x^{19}+\frac {5}{21} a \,c^{4} e \,x^{21}+\frac {1}{23} c^{5} d \,x^{23}+\frac {1}{25} c^{5} e \,x^{25}\) | \(126\) |
risch | \(\frac {1}{3} a^{5} d \,x^{3}+\frac {1}{5} a^{5} e \,x^{5}+\frac {5}{7} a^{4} c d \,x^{7}+\frac {5}{9} a^{4} c e \,x^{9}+\frac {10}{11} a^{3} c^{2} d \,x^{11}+\frac {10}{13} a^{3} c^{2} e \,x^{13}+\frac {2}{3} a^{2} c^{3} d \,x^{15}+\frac {10}{17} a^{2} c^{3} e \,x^{17}+\frac {5}{19} a \,c^{4} d \,x^{19}+\frac {5}{21} a \,c^{4} e \,x^{21}+\frac {1}{23} c^{5} d \,x^{23}+\frac {1}{25} c^{5} e \,x^{25}\) | \(126\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 131, normalized size = 0.88 \begin {gather*} \frac {1}{25} \, c^{5} x^{25} e + \frac {1}{23} \, c^{5} d x^{23} + \frac {5}{21} \, a c^{4} x^{21} e + \frac {5}{19} \, a c^{4} d x^{19} + \frac {10}{17} \, a^{2} c^{3} x^{17} e + \frac {2}{3} \, a^{2} c^{3} d x^{15} + \frac {10}{13} \, a^{3} c^{2} x^{13} e + \frac {10}{11} \, a^{3} c^{2} d x^{11} + \frac {5}{9} \, a^{4} c x^{9} e + \frac {5}{7} \, a^{4} c d x^{7} + \frac {1}{5} \, a^{5} x^{5} e + \frac {1}{3} \, a^{5} d x^{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 124, normalized size = 0.83 \begin {gather*} \frac {1}{23} \, c^{5} d x^{23} + \frac {5}{19} \, a c^{4} d x^{19} + \frac {2}{3} \, a^{2} c^{3} d x^{15} + \frac {10}{11} \, a^{3} c^{2} d x^{11} + \frac {5}{7} \, a^{4} c d x^{7} + \frac {1}{3} \, a^{5} d x^{3} + \frac {1}{348075} \, {\left (13923 \, c^{5} x^{25} + 82875 \, a c^{4} x^{21} + 204750 \, a^{2} c^{3} x^{17} + 267750 \, a^{3} c^{2} x^{13} + 193375 \, a^{4} c x^{9} + 69615 \, a^{5} x^{5}\right )} e \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.01, size = 155, normalized size = 1.04 \begin {gather*} \frac {a^{5} d x^{3}}{3} + \frac {a^{5} e x^{5}}{5} + \frac {5 a^{4} c d x^{7}}{7} + \frac {5 a^{4} c e x^{9}}{9} + \frac {10 a^{3} c^{2} d x^{11}}{11} + \frac {10 a^{3} c^{2} e x^{13}}{13} + \frac {2 a^{2} c^{3} d x^{15}}{3} + \frac {10 a^{2} c^{3} e x^{17}}{17} + \frac {5 a c^{4} d x^{19}}{19} + \frac {5 a c^{4} e x^{21}}{21} + \frac {c^{5} d x^{23}}{23} + \frac {c^{5} e x^{25}}{25} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 6.34, size = 131, normalized size = 0.88 \begin {gather*} \frac {1}{25} \, c^{5} x^{25} e + \frac {1}{23} \, c^{5} d x^{23} + \frac {5}{21} \, a c^{4} x^{21} e + \frac {5}{19} \, a c^{4} d x^{19} + \frac {10}{17} \, a^{2} c^{3} x^{17} e + \frac {2}{3} \, a^{2} c^{3} d x^{15} + \frac {10}{13} \, a^{3} c^{2} x^{13} e + \frac {10}{11} \, a^{3} c^{2} d x^{11} + \frac {5}{9} \, a^{4} c x^{9} e + \frac {5}{7} \, a^{4} c d x^{7} + \frac {1}{5} \, a^{5} x^{5} e + \frac {1}{3} \, a^{5} d x^{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.07, size = 125, normalized size = 0.84 \begin {gather*} \frac {e\,a^5\,x^5}{5}+\frac {d\,a^5\,x^3}{3}+\frac {5\,e\,a^4\,c\,x^9}{9}+\frac {5\,d\,a^4\,c\,x^7}{7}+\frac {10\,e\,a^3\,c^2\,x^{13}}{13}+\frac {10\,d\,a^3\,c^2\,x^{11}}{11}+\frac {10\,e\,a^2\,c^3\,x^{17}}{17}+\frac {2\,d\,a^2\,c^3\,x^{15}}{3}+\frac {5\,e\,a\,c^4\,x^{21}}{21}+\frac {5\,d\,a\,c^4\,x^{19}}{19}+\frac {e\,c^5\,x^{25}}{25}+\frac {d\,c^5\,x^{23}}{23} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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