Optimal. Leaf size=139 \[ \frac{10}{11} a^2 c^3 d x^{11}+\frac{10}{7} a^3 c^2 d x^7+\frac{10}{13} a^2 c^3 e x^{13}+\frac{10}{9} a^3 c^2 e x^9+\frac{5}{3} a^4 c d x^3+a^4 c e x^5-\frac{a^5 d}{x}+a^5 e x+\frac{1}{3} a c^4 d x^{15}+\frac{5}{17} a c^4 e x^{17}+\frac{1}{19} c^5 d x^{19}+\frac{1}{21} c^5 e x^{21} \]
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Rubi [A] time = 0.081904, antiderivative size = 139, 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} \[ \frac{10}{11} a^2 c^3 d x^{11}+\frac{10}{7} a^3 c^2 d x^7+\frac{10}{13} a^2 c^3 e x^{13}+\frac{10}{9} a^3 c^2 e x^9+\frac{5}{3} a^4 c d x^3+a^4 c e x^5-\frac{a^5 d}{x}+a^5 e x+\frac{1}{3} a c^4 d x^{15}+\frac{5}{17} a c^4 e x^{17}+\frac{1}{19} c^5 d x^{19}+\frac{1}{21} c^5 e x^{21} \]
Antiderivative was successfully verified.
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Rule 1262
Rubi steps
\begin{align*} \int \frac{\left (d+e x^2\right ) \left (a+c x^4\right )^5}{x^2} \, dx &=\int \left (a^5 e+\frac{a^5 d}{x^2}+5 a^4 c d x^2+5 a^4 c e x^4+10 a^3 c^2 d x^6+10 a^3 c^2 e x^8+10 a^2 c^3 d x^{10}+10 a^2 c^3 e x^{12}+5 a c^4 d x^{14}+5 a c^4 e x^{16}+c^5 d x^{18}+c^5 e x^{20}\right ) \, dx\\ &=-\frac{a^5 d}{x}+a^5 e x+\frac{5}{3} a^4 c d x^3+a^4 c e x^5+\frac{10}{7} a^3 c^2 d x^7+\frac{10}{9} a^3 c^2 e x^9+\frac{10}{11} a^2 c^3 d x^{11}+\frac{10}{13} a^2 c^3 e x^{13}+\frac{1}{3} a c^4 d x^{15}+\frac{5}{17} a c^4 e x^{17}+\frac{1}{19} c^5 d x^{19}+\frac{1}{21} c^5 e x^{21}\\ \end{align*}
Mathematica [A] time = 0.0078724, size = 139, normalized size = 1. \[ \frac{10}{11} a^2 c^3 d x^{11}+\frac{10}{7} a^3 c^2 d x^7+\frac{10}{13} a^2 c^3 e x^{13}+\frac{10}{9} a^3 c^2 e x^9+\frac{5}{3} a^4 c d x^3+a^4 c e x^5-\frac{a^5 d}{x}+a^5 e x+\frac{1}{3} a c^4 d x^{15}+\frac{5}{17} a c^4 e x^{17}+\frac{1}{19} c^5 d x^{19}+\frac{1}{21} c^5 e x^{21} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.016, size = 122, normalized size = 0.9 \begin{align*} -{\frac{{a}^{5}d}{x}}+{a}^{5}ex+{\frac{5\,{a}^{4}cd{x}^{3}}{3}}+{a}^{4}ce{x}^{5}+{\frac{10\,{a}^{3}{c}^{2}d{x}^{7}}{7}}+{\frac{10\,{a}^{3}{c}^{2}e{x}^{9}}{9}}+{\frac{10\,{a}^{2}{c}^{3}d{x}^{11}}{11}}+{\frac{10\,{a}^{2}{c}^{3}e{x}^{13}}{13}}+{\frac{a{c}^{4}d{x}^{15}}{3}}+{\frac{5\,a{c}^{4}e{x}^{17}}{17}}+{\frac{{c}^{5}d{x}^{19}}{19}}+{\frac{{c}^{5}e{x}^{21}}{21}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.939337, size = 163, normalized size = 1.17 \begin{align*} \frac{1}{21} \, c^{5} e x^{21} + \frac{1}{19} \, c^{5} d x^{19} + \frac{5}{17} \, a c^{4} e x^{17} + \frac{1}{3} \, a c^{4} d x^{15} + \frac{10}{13} \, a^{2} c^{3} e x^{13} + \frac{10}{11} \, a^{2} c^{3} d x^{11} + \frac{10}{9} \, a^{3} c^{2} e x^{9} + \frac{10}{7} \, a^{3} c^{2} d x^{7} + a^{4} c e x^{5} + \frac{5}{3} \, a^{4} c d x^{3} + a^{5} e x - \frac{a^{5} d}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.66616, size = 373, normalized size = 2.68 \begin{align*} \frac{138567 \, c^{5} e x^{22} + 153153 \, c^{5} d x^{20} + 855855 \, a c^{4} e x^{18} + 969969 \, a c^{4} d x^{16} + 2238390 \, a^{2} c^{3} e x^{14} + 2645370 \, a^{2} c^{3} d x^{12} + 3233230 \, a^{3} c^{2} e x^{10} + 4157010 \, a^{3} c^{2} d x^{8} + 2909907 \, a^{4} c e x^{6} + 4849845 \, a^{4} c d x^{4} + 2909907 \, a^{5} e x^{2} - 2909907 \, a^{5} d}{2909907 \, x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.405244, size = 143, normalized size = 1.03 \begin{align*} - \frac{a^{5} d}{x} + a^{5} e x + \frac{5 a^{4} c d x^{3}}{3} + a^{4} c e x^{5} + \frac{10 a^{3} c^{2} d x^{7}}{7} + \frac{10 a^{3} c^{2} e x^{9}}{9} + \frac{10 a^{2} c^{3} d x^{11}}{11} + \frac{10 a^{2} c^{3} e x^{13}}{13} + \frac{a c^{4} d x^{15}}{3} + \frac{5 a c^{4} e x^{17}}{17} + \frac{c^{5} d x^{19}}{19} + \frac{c^{5} e x^{21}}{21} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14078, size = 171, normalized size = 1.23 \begin{align*} \frac{1}{21} \, c^{5} x^{21} e + \frac{1}{19} \, c^{5} d x^{19} + \frac{5}{17} \, a c^{4} x^{17} e + \frac{1}{3} \, a c^{4} d x^{15} + \frac{10}{13} \, a^{2} c^{3} x^{13} e + \frac{10}{11} \, a^{2} c^{3} d x^{11} + \frac{10}{9} \, a^{3} c^{2} x^{9} e + \frac{10}{7} \, a^{3} c^{2} d x^{7} + a^{4} c x^{5} e + \frac{5}{3} \, a^{4} c d x^{3} + a^{5} x e - \frac{a^{5} d}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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