Optimal. Leaf size=149 \[ \frac{2}{3} a^2 c^3 d x^{15}+\frac{10}{11} a^3 c^2 d x^{11}+\frac{10}{17} a^2 c^3 e x^{17}+\frac{10}{13} a^3 c^2 e x^{13}+\frac{5}{7} a^4 c d x^7+\frac{5}{9} a^4 c e x^9+\frac{1}{3} a^5 d x^3+\frac{1}{5} a^5 e x^5+\frac{5}{19} a c^4 d x^{19}+\frac{5}{21} a c^4 e x^{21}+\frac{1}{23} c^5 d x^{23}+\frac{1}{25} c^5 e x^{25} \]
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Rubi [A] time = 0.0977514, antiderivative size = 149, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {1262} \[ \frac{2}{3} a^2 c^3 d x^{15}+\frac{10}{11} a^3 c^2 d x^{11}+\frac{10}{17} a^2 c^3 e x^{17}+\frac{10}{13} a^3 c^2 e x^{13}+\frac{5}{7} a^4 c d x^7+\frac{5}{9} a^4 c e x^9+\frac{1}{3} a^5 d x^3+\frac{1}{5} a^5 e x^5+\frac{5}{19} a c^4 d x^{19}+\frac{5}{21} a c^4 e x^{21}+\frac{1}{23} c^5 d x^{23}+\frac{1}{25} c^5 e x^{25} \]
Antiderivative was successfully verified.
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Rule 1262
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*}
Mathematica [A] time = 0.0041141, size = 149, normalized size = 1. \[ \frac{2}{3} a^2 c^3 d x^{15}+\frac{10}{11} a^3 c^2 d x^{11}+\frac{10}{17} a^2 c^3 e x^{17}+\frac{10}{13} a^3 c^2 e x^{13}+\frac{5}{7} a^4 c d x^7+\frac{5}{9} a^4 c e x^9+\frac{1}{3} a^5 d x^3+\frac{1}{5} a^5 e x^5+\frac{5}{19} a c^4 d x^{19}+\frac{5}{21} a c^4 e x^{21}+\frac{1}{23} c^5 d x^{23}+\frac{1}{25} c^5 e x^{25} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 126, normalized size = 0.9 \begin{align*}{\frac{{a}^{5}d{x}^{3}}{3}}+{\frac{{a}^{5}e{x}^{5}}{5}}+{\frac{5\,{a}^{4}cd{x}^{7}}{7}}+{\frac{5\,{a}^{4}ce{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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.952864, size = 169, normalized size = 1.13 \begin{align*} \frac{1}{25} \, c^{5} e x^{25} + \frac{1}{23} \, c^{5} d x^{23} + \frac{5}{21} \, a c^{4} e x^{21} + \frac{5}{19} \, a c^{4} d x^{19} + \frac{10}{17} \, a^{2} c^{3} e x^{17} + \frac{2}{3} \, a^{2} c^{3} d x^{15} + \frac{10}{13} \, a^{3} c^{2} e x^{13} + \frac{10}{11} \, a^{3} c^{2} d x^{11} + \frac{5}{9} \, a^{4} c e x^{9} + \frac{5}{7} \, a^{4} c d x^{7} + \frac{1}{5} \, a^{5} e x^{5} + \frac{1}{3} \, a^{5} d x^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.27712, size = 315, normalized size = 2.11 \begin{align*} \frac{1}{25} x^{25} e c^{5} + \frac{1}{23} x^{23} d c^{5} + \frac{5}{21} x^{21} e c^{4} a + \frac{5}{19} x^{19} d c^{4} a + \frac{10}{17} x^{17} e c^{3} a^{2} + \frac{2}{3} x^{15} d c^{3} a^{2} + \frac{10}{13} x^{13} e c^{2} a^{3} + \frac{10}{11} x^{11} d c^{2} a^{3} + \frac{5}{9} x^{9} e c a^{4} + \frac{5}{7} x^{7} d c a^{4} + \frac{1}{5} x^{5} e a^{5} + \frac{1}{3} x^{3} d a^{5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.086129, size = 155, normalized size = 1.04 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16342, size = 177, normalized size = 1.19 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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