Optimal. Leaf size=71 \[ \frac{1}{3} x^3 \left (e (a e+2 b d)+c d^2\right )+d x (2 a e+b d)-\frac{a d^2}{x}+\frac{1}{5} e x^5 (b e+2 c d)+\frac{1}{7} c e^2 x^7 \]
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Rubi [A] time = 0.0483196, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.04, Rules used = {1261} \[ \frac{1}{3} x^3 \left (e (a e+2 b d)+c d^2\right )+d x (2 a e+b d)-\frac{a d^2}{x}+\frac{1}{5} e x^5 (b e+2 c d)+\frac{1}{7} c e^2 x^7 \]
Antiderivative was successfully verified.
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Rule 1261
Rubi steps
\begin{align*} \int \frac{\left (d+e x^2\right )^2 \left (a+b x^2+c x^4\right )}{x^2} \, dx &=\int \left (d (b d+2 a e)+\frac{a d^2}{x^2}+\left (c d^2+e (2 b d+a e)\right ) x^2+e (2 c d+b e) x^4+c e^2 x^6\right ) \, dx\\ &=-\frac{a d^2}{x}+d (b d+2 a e) x+\frac{1}{3} \left (c d^2+e (2 b d+a e)\right ) x^3+\frac{1}{5} e (2 c d+b e) x^5+\frac{1}{7} c e^2 x^7\\ \end{align*}
Mathematica [A] time = 0.0332107, size = 71, normalized size = 1. \[ \frac{1}{3} x^3 \left (a e^2+2 b d e+c d^2\right )+d x (2 a e+b d)-\frac{a d^2}{x}+\frac{1}{5} e x^5 (b e+2 c d)+\frac{1}{7} c e^2 x^7 \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 75, normalized size = 1.1 \begin{align*}{\frac{c{e}^{2}{x}^{7}}{7}}+{\frac{{x}^{5}b{e}^{2}}{5}}+{\frac{2\,{x}^{5}cde}{5}}+{\frac{{x}^{3}a{e}^{2}}{3}}+{\frac{2\,{x}^{3}bde}{3}}+{\frac{{x}^{3}c{d}^{2}}{3}}+2\,deax+{d}^{2}bx-{\frac{a{d}^{2}}{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.947018, size = 93, normalized size = 1.31 \begin{align*} \frac{1}{7} \, c e^{2} x^{7} + \frac{1}{5} \,{\left (2 \, c d e + b e^{2}\right )} x^{5} + \frac{1}{3} \,{\left (c d^{2} + 2 \, b d e + a e^{2}\right )} x^{3} - \frac{a d^{2}}{x} +{\left (b d^{2} + 2 \, a d e\right )} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.6627, size = 170, normalized size = 2.39 \begin{align*} \frac{15 \, c e^{2} x^{8} + 21 \,{\left (2 \, c d e + b e^{2}\right )} x^{6} + 35 \,{\left (c d^{2} + 2 \, b d e + a e^{2}\right )} x^{4} - 105 \, a d^{2} + 105 \,{\left (b d^{2} + 2 \, a d e\right )} x^{2}}{105 \, x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.315117, size = 73, normalized size = 1.03 \begin{align*} - \frac{a d^{2}}{x} + \frac{c e^{2} x^{7}}{7} + x^{5} \left (\frac{b e^{2}}{5} + \frac{2 c d e}{5}\right ) + x^{3} \left (\frac{a e^{2}}{3} + \frac{2 b d e}{3} + \frac{c d^{2}}{3}\right ) + x \left (2 a d e + b d^{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.08033, size = 100, normalized size = 1.41 \begin{align*} \frac{1}{7} \, c x^{7} e^{2} + \frac{2}{5} \, c d x^{5} e + \frac{1}{5} \, b x^{5} e^{2} + \frac{1}{3} \, c d^{2} x^{3} + \frac{2}{3} \, b d x^{3} e + \frac{1}{3} \, a x^{3} e^{2} + b d^{2} x + 2 \, a d x e - \frac{a d^{2}}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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