Optimal. Leaf size=142 \[ a^2 c^3 d x^{10}+\frac{5}{3} a^3 c^2 d x^6+\frac{5}{6} a^2 c^3 e x^{12}+\frac{5}{4} a^3 c^2 e x^8+\frac{5}{2} a^4 c d x^2+\frac{5}{4} a^4 c e x^4-\frac{a^5 d}{2 x^2}+a^5 e \log (x)+\frac{5}{14} a c^4 d x^{14}+\frac{5}{16} a c^4 e x^{16}+\frac{1}{18} c^5 d x^{18}+\frac{1}{20} c^5 e x^{20} \]
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Rubi [A] time = 0.121243, antiderivative size = 142, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {1252, 766} \[ a^2 c^3 d x^{10}+\frac{5}{3} a^3 c^2 d x^6+\frac{5}{6} a^2 c^3 e x^{12}+\frac{5}{4} a^3 c^2 e x^8+\frac{5}{2} a^4 c d x^2+\frac{5}{4} a^4 c e x^4-\frac{a^5 d}{2 x^2}+a^5 e \log (x)+\frac{5}{14} a c^4 d x^{14}+\frac{5}{16} a c^4 e x^{16}+\frac{1}{18} c^5 d x^{18}+\frac{1}{20} c^5 e x^{20} \]
Antiderivative was successfully verified.
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Rule 1252
Rule 766
Rubi steps
\begin{align*} \int \frac{\left (d+e x^2\right ) \left (a+c x^4\right )^5}{x^3} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(d+e x) \left (a+c x^2\right )^5}{x^2} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (5 a^4 c d+\frac{a^5 d}{x^2}+\frac{a^5 e}{x}+5 a^4 c e x+10 a^3 c^2 d x^2+10 a^3 c^2 e x^3+10 a^2 c^3 d x^4+10 a^2 c^3 e x^5+5 a c^4 d x^6+5 a c^4 e x^7+c^5 d x^8+c^5 e x^9\right ) \, dx,x,x^2\right )\\ &=-\frac{a^5 d}{2 x^2}+\frac{5}{2} a^4 c d x^2+\frac{5}{4} a^4 c e x^4+\frac{5}{3} a^3 c^2 d x^6+\frac{5}{4} a^3 c^2 e x^8+a^2 c^3 d x^{10}+\frac{5}{6} a^2 c^3 e x^{12}+\frac{5}{14} a c^4 d x^{14}+\frac{5}{16} a c^4 e x^{16}+\frac{1}{18} c^5 d x^{18}+\frac{1}{20} c^5 e x^{20}+a^5 e \log (x)\\ \end{align*}
Mathematica [A] time = 0.0082041, size = 142, normalized size = 1. \[ a^2 c^3 d x^{10}+\frac{5}{3} a^3 c^2 d x^6+\frac{5}{6} a^2 c^3 e x^{12}+\frac{5}{4} a^3 c^2 e x^8+\frac{5}{2} a^4 c d x^2+\frac{5}{4} a^4 c e x^4-\frac{a^5 d}{2 x^2}+a^5 e \log (x)+\frac{5}{14} a c^4 d x^{14}+\frac{5}{16} a c^4 e x^{16}+\frac{1}{18} c^5 d x^{18}+\frac{1}{20} c^5 e x^{20} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 123, normalized size = 0.9 \begin{align*} -{\frac{{a}^{5}d}{2\,{x}^{2}}}+{\frac{5\,{a}^{4}cd{x}^{2}}{2}}+{\frac{5\,{a}^{4}ce{x}^{4}}{4}}+{\frac{5\,{a}^{3}{c}^{2}d{x}^{6}}{3}}+{\frac{5\,{a}^{3}{c}^{2}e{x}^{8}}{4}}+{a}^{2}{c}^{3}d{x}^{10}+{\frac{5\,{a}^{2}{c}^{3}e{x}^{12}}{6}}+{\frac{5\,a{c}^{4}d{x}^{14}}{14}}+{\frac{5\,a{c}^{4}e{x}^{16}}{16}}+{\frac{{c}^{5}d{x}^{18}}{18}}+{\frac{{c}^{5}e{x}^{20}}{20}}+{a}^{5}e\ln \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.952209, size = 169, normalized size = 1.19 \begin{align*} \frac{1}{20} \, c^{5} e x^{20} + \frac{1}{18} \, c^{5} d x^{18} + \frac{5}{16} \, a c^{4} e x^{16} + \frac{5}{14} \, a c^{4} d x^{14} + \frac{5}{6} \, a^{2} c^{3} e x^{12} + a^{2} c^{3} d x^{10} + \frac{5}{4} \, a^{3} c^{2} e x^{8} + \frac{5}{3} \, a^{3} c^{2} d x^{6} + \frac{5}{4} \, a^{4} c e x^{4} + \frac{5}{2} \, a^{4} c d x^{2} + \frac{1}{2} \, a^{5} e \log \left (x^{2}\right ) - \frac{a^{5} d}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.68025, size = 336, normalized size = 2.37 \begin{align*} \frac{252 \, c^{5} e x^{22} + 280 \, c^{5} d x^{20} + 1575 \, a c^{4} e x^{18} + 1800 \, a c^{4} d x^{16} + 4200 \, a^{2} c^{3} e x^{14} + 5040 \, a^{2} c^{3} d x^{12} + 6300 \, a^{3} c^{2} e x^{10} + 8400 \, a^{3} c^{2} d x^{8} + 6300 \, a^{4} c e x^{6} + 12600 \, a^{4} c d x^{4} + 5040 \, a^{5} e x^{2} \log \left (x\right ) - 2520 \, a^{5} d}{5040 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.436451, size = 150, normalized size = 1.06 \begin{align*} - \frac{a^{5} d}{2 x^{2}} + a^{5} e \log{\left (x \right )} + \frac{5 a^{4} c d x^{2}}{2} + \frac{5 a^{4} c e x^{4}}{4} + \frac{5 a^{3} c^{2} d x^{6}}{3} + \frac{5 a^{3} c^{2} e x^{8}}{4} + a^{2} c^{3} d x^{10} + \frac{5 a^{2} c^{3} e x^{12}}{6} + \frac{5 a c^{4} d x^{14}}{14} + \frac{5 a c^{4} e x^{16}}{16} + \frac{c^{5} d x^{18}}{18} + \frac{c^{5} e x^{20}}{20} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14471, size = 192, normalized size = 1.35 \begin{align*} \frac{1}{20} \, c^{5} x^{20} e + \frac{1}{18} \, c^{5} d x^{18} + \frac{5}{16} \, a c^{4} x^{16} e + \frac{5}{14} \, a c^{4} d x^{14} + \frac{5}{6} \, a^{2} c^{3} x^{12} e + a^{2} c^{3} d x^{10} + \frac{5}{4} \, a^{3} c^{2} x^{8} e + \frac{5}{3} \, a^{3} c^{2} d x^{6} + \frac{5}{4} \, a^{4} c x^{4} e + \frac{5}{2} \, a^{4} c d x^{2} + \frac{1}{2} \, a^{5} e \log \left (x^{2}\right ) - \frac{a^{5} x^{2} e + a^{5} d}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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