Optimal. Leaf size=155 \[ a^2 d^2 x+\frac{1}{9} x^9 \left (2 c e (a e+2 b d)+b^2 e^2+c^2 d^2\right )+\frac{2}{7} x^7 \left (a b e^2+2 a c d e+b^2 d e+b c d^2\right )+\frac{1}{5} x^5 \left (4 a b d e+a \left (a e^2+2 c d^2\right )+b^2 d^2\right )+\frac{2}{3} a d x^3 (a e+b d)+\frac{2}{11} c e x^{11} (b e+c d)+\frac{1}{13} c^2 e^2 x^{13} \]
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Rubi [A] time = 0.141042, antiderivative size = 155, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {1153} \[ a^2 d^2 x+\frac{1}{9} x^9 \left (2 c e (a e+2 b d)+b^2 e^2+c^2 d^2\right )+\frac{2}{7} x^7 \left (a b e^2+2 a c d e+b^2 d e+b c d^2\right )+\frac{1}{5} x^5 \left (4 a b d e+a \left (a e^2+2 c d^2\right )+b^2 d^2\right )+\frac{2}{3} a d x^3 (a e+b d)+\frac{2}{11} c e x^{11} (b e+c d)+\frac{1}{13} c^2 e^2 x^{13} \]
Antiderivative was successfully verified.
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Rule 1153
Rubi steps
\begin{align*} \int \left (d+e x^2\right )^2 \left (a+b x^2+c x^4\right )^2 \, dx &=\int \left (a^2 d^2+2 a d (b d+a e) x^2+\left (b^2 d^2+4 a b d e+a \left (2 c d^2+a e^2\right )\right ) x^4+2 \left (b c d^2+b^2 d e+2 a c d e+a b e^2\right ) x^6+\left (c^2 d^2+b^2 e^2+2 c e (2 b d+a e)\right ) x^8+2 c e (c d+b e) x^{10}+c^2 e^2 x^{12}\right ) \, dx\\ &=a^2 d^2 x+\frac{2}{3} a d (b d+a e) x^3+\frac{1}{5} \left (b^2 d^2+4 a b d e+a \left (2 c d^2+a e^2\right )\right ) x^5+\frac{2}{7} \left (b c d^2+b^2 d e+2 a c d e+a b e^2\right ) x^7+\frac{1}{9} \left (c^2 d^2+b^2 e^2+2 c e (2 b d+a e)\right ) x^9+\frac{2}{11} c e (c d+b e) x^{11}+\frac{1}{13} c^2 e^2 x^{13}\\ \end{align*}
Mathematica [A] time = 0.0536867, size = 156, normalized size = 1.01 \[ \frac{1}{5} x^5 \left (a^2 e^2+4 a b d e+2 a c d^2+b^2 d^2\right )+a^2 d^2 x+\frac{1}{9} x^9 \left (2 a c e^2+b^2 e^2+4 b c d e+c^2 d^2\right )+\frac{2}{7} x^7 \left (a b e^2+2 a c d e+b^2 d e+b c d^2\right )+\frac{2}{3} a d x^3 (a e+b d)+\frac{2}{11} c e x^{11} (b e+c d)+\frac{1}{13} c^2 e^2 x^{13} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 155, normalized size = 1. \begin{align*}{\frac{{c}^{2}{e}^{2}{x}^{13}}{13}}+{\frac{ \left ( 2\,{e}^{2}bc+2\,de{c}^{2} \right ){x}^{11}}{11}}+{\frac{ \left ({c}^{2}{d}^{2}+4\,debc+{e}^{2} \left ( 2\,ac+{b}^{2} \right ) \right ){x}^{9}}{9}}+{\frac{ \left ( 2\,bc{d}^{2}+2\,de \left ( 2\,ac+{b}^{2} \right ) +2\,ab{e}^{2} \right ){x}^{7}}{7}}+{\frac{ \left ({d}^{2} \left ( 2\,ac+{b}^{2} \right ) +4\,abde+{e}^{2}{a}^{2} \right ){x}^{5}}{5}}+{\frac{ \left ( 2\,de{a}^{2}+2\,{d}^{2}ab \right ){x}^{3}}{3}}+{a}^{2}{d}^{2}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.975113, size = 198, normalized size = 1.28 \begin{align*} \frac{1}{13} \, c^{2} e^{2} x^{13} + \frac{2}{11} \,{\left (c^{2} d e + b c e^{2}\right )} x^{11} + \frac{1}{9} \,{\left (c^{2} d^{2} + 4 \, b c d e +{\left (b^{2} + 2 \, a c\right )} e^{2}\right )} x^{9} + \frac{2}{7} \,{\left (b c d^{2} + a b e^{2} +{\left (b^{2} + 2 \, a c\right )} d e\right )} x^{7} + \frac{1}{5} \,{\left (4 \, a b d e + a^{2} e^{2} +{\left (b^{2} + 2 \, a c\right )} d^{2}\right )} x^{5} + a^{2} d^{2} x + \frac{2}{3} \,{\left (a b d^{2} + a^{2} d e\right )} x^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.43315, size = 436, normalized size = 2.81 \begin{align*} \frac{1}{13} x^{13} e^{2} c^{2} + \frac{2}{11} x^{11} e d c^{2} + \frac{2}{11} x^{11} e^{2} c b + \frac{1}{9} x^{9} d^{2} c^{2} + \frac{4}{9} x^{9} e d c b + \frac{1}{9} x^{9} e^{2} b^{2} + \frac{2}{9} x^{9} e^{2} c a + \frac{2}{7} x^{7} d^{2} c b + \frac{2}{7} x^{7} e d b^{2} + \frac{4}{7} x^{7} e d c a + \frac{2}{7} x^{7} e^{2} b a + \frac{1}{5} x^{5} d^{2} b^{2} + \frac{2}{5} x^{5} d^{2} c a + \frac{4}{5} x^{5} e d b a + \frac{1}{5} x^{5} e^{2} a^{2} + \frac{2}{3} x^{3} d^{2} b a + \frac{2}{3} x^{3} e d a^{2} + x d^{2} a^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.09538, size = 192, normalized size = 1.24 \begin{align*} a^{2} d^{2} x + \frac{c^{2} e^{2} x^{13}}{13} + x^{11} \left (\frac{2 b c e^{2}}{11} + \frac{2 c^{2} d e}{11}\right ) + x^{9} \left (\frac{2 a c e^{2}}{9} + \frac{b^{2} e^{2}}{9} + \frac{4 b c d e}{9} + \frac{c^{2} d^{2}}{9}\right ) + x^{7} \left (\frac{2 a b e^{2}}{7} + \frac{4 a c d e}{7} + \frac{2 b^{2} d e}{7} + \frac{2 b c d^{2}}{7}\right ) + x^{5} \left (\frac{a^{2} e^{2}}{5} + \frac{4 a b d e}{5} + \frac{2 a c d^{2}}{5} + \frac{b^{2} d^{2}}{5}\right ) + x^{3} \left (\frac{2 a^{2} d e}{3} + \frac{2 a b d^{2}}{3}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13493, size = 244, normalized size = 1.57 \begin{align*} \frac{1}{13} \, c^{2} x^{13} e^{2} + \frac{2}{11} \, c^{2} d x^{11} e + \frac{2}{11} \, b c x^{11} e^{2} + \frac{1}{9} \, c^{2} d^{2} x^{9} + \frac{4}{9} \, b c d x^{9} e + \frac{1}{9} \, b^{2} x^{9} e^{2} + \frac{2}{9} \, a c x^{9} e^{2} + \frac{2}{7} \, b c d^{2} x^{7} + \frac{2}{7} \, b^{2} d x^{7} e + \frac{4}{7} \, a c d x^{7} e + \frac{2}{7} \, a b x^{7} e^{2} + \frac{1}{5} \, b^{2} d^{2} x^{5} + \frac{2}{5} \, a c d^{2} x^{5} + \frac{4}{5} \, a b d x^{5} e + \frac{1}{5} \, a^{2} x^{5} e^{2} + \frac{2}{3} \, a b d^{2} x^{3} + \frac{2}{3} \, a^{2} d x^{3} e + a^{2} d^{2} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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