Optimal. Leaf size=158 \[ \frac{1}{7} x^7 \left (2 a d f (c f+d e)+b \left (c^2 f^2+4 c d e f+d^2 e^2\right )\right )+\frac{1}{5} x^5 \left (a \left (c^2 f^2+4 c d e f+d^2 e^2\right )+2 b c e (c f+d e)\right )+\frac{1}{9} d f x^9 (a d f+2 b (c f+d e))+\frac{1}{3} c e x^3 (2 a (c f+d e)+b c e)+a c^2 e^2 x+\frac{1}{11} b d^2 f^2 x^{11} \]
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Rubi [A] time = 0.167084, antiderivative size = 158, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.038, Rules used = {521} \[ \frac{1}{7} x^7 \left (2 a d f (c f+d e)+b \left (c^2 f^2+4 c d e f+d^2 e^2\right )\right )+\frac{1}{5} x^5 \left (a \left (c^2 f^2+4 c d e f+d^2 e^2\right )+2 b c e (c f+d e)\right )+\frac{1}{9} d f x^9 (a d f+2 b (c f+d e))+\frac{1}{3} c e x^3 (2 a (c f+d e)+b c e)+a c^2 e^2 x+\frac{1}{11} b d^2 f^2 x^{11} \]
Antiderivative was successfully verified.
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Rule 521
Rubi steps
\begin{align*} \int \left (a+b x^2\right ) \left (c+d x^2\right )^2 \left (e+f x^2\right )^2 \, dx &=\int \left (a c^2 e^2+c e (b c e+2 a (d e+c f)) x^2+\left (2 b c e (d e+c f)+a \left (d^2 e^2+4 c d e f+c^2 f^2\right )\right ) x^4+\left (2 a d f (d e+c f)+b \left (d^2 e^2+4 c d e f+c^2 f^2\right )\right ) x^6+d f (a d f+2 b (d e+c f)) x^8+b d^2 f^2 x^{10}\right ) \, dx\\ &=a c^2 e^2 x+\frac{1}{3} c e (b c e+2 a (d e+c f)) x^3+\frac{1}{5} \left (2 b c e (d e+c f)+a \left (d^2 e^2+4 c d e f+c^2 f^2\right )\right ) x^5+\frac{1}{7} \left (2 a d f (d e+c f)+b \left (d^2 e^2+4 c d e f+c^2 f^2\right )\right ) x^7+\frac{1}{9} d f (a d f+2 b (d e+c f)) x^9+\frac{1}{11} b d^2 f^2 x^{11}\\ \end{align*}
Mathematica [A] time = 0.0595616, size = 158, normalized size = 1. \[ \frac{1}{7} x^7 \left (2 a d f (c f+d e)+b \left (c^2 f^2+4 c d e f+d^2 e^2\right )\right )+\frac{1}{5} x^5 \left (a \left (c^2 f^2+4 c d e f+d^2 e^2\right )+2 b c e (c f+d e)\right )+\frac{1}{9} d f x^9 (a d f+2 b (c f+d e))+\frac{1}{3} c e x^3 (2 a (c f+d e)+b c e)+a c^2 e^2 x+\frac{1}{11} b d^2 f^2 x^{11} \]
Antiderivative was successfully verified.
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Maple [A] time = 0., size = 169, normalized size = 1.1 \begin{align*}{\frac{b{d}^{2}{f}^{2}{x}^{11}}{11}}+{\frac{ \left ( \left ( a{d}^{2}+2\,bcd \right ){f}^{2}+2\,b{d}^{2}ef \right ){x}^{9}}{9}}+{\frac{ \left ( \left ( 2\,acd+b{c}^{2} \right ){f}^{2}+2\, \left ( a{d}^{2}+2\,bcd \right ) ef+b{d}^{2}{e}^{2} \right ){x}^{7}}{7}}+{\frac{ \left ( a{c}^{2}{f}^{2}+2\, \left ( 2\,acd+b{c}^{2} \right ) ef+ \left ( a{d}^{2}+2\,bcd \right ){e}^{2} \right ){x}^{5}}{5}}+{\frac{ \left ( 2\,a{c}^{2}ef+ \left ( 2\,acd+b{c}^{2} \right ){e}^{2} \right ){x}^{3}}{3}}+a{c}^{2}{e}^{2}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01531, size = 227, normalized size = 1.44 \begin{align*} \frac{1}{11} \, b d^{2} f^{2} x^{11} + \frac{1}{9} \,{\left (2 \, b d^{2} e f +{\left (2 \, b c d + a d^{2}\right )} f^{2}\right )} x^{9} + \frac{1}{7} \,{\left (b d^{2} e^{2} + 2 \,{\left (2 \, b c d + a d^{2}\right )} e f +{\left (b c^{2} + 2 \, a c d\right )} f^{2}\right )} x^{7} + a c^{2} e^{2} x + \frac{1}{5} \,{\left (a c^{2} f^{2} +{\left (2 \, b c d + a d^{2}\right )} e^{2} + 2 \,{\left (b c^{2} + 2 \, a c d\right )} e f\right )} x^{5} + \frac{1}{3} \,{\left (2 \, a c^{2} e f +{\left (b c^{2} + 2 \, a c d\right )} e^{2}\right )} x^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.27589, size = 479, normalized size = 3.03 \begin{align*} \frac{1}{11} x^{11} f^{2} d^{2} b + \frac{2}{9} x^{9} f e d^{2} b + \frac{2}{9} x^{9} f^{2} d c b + \frac{1}{9} x^{9} f^{2} d^{2} a + \frac{1}{7} x^{7} e^{2} d^{2} b + \frac{4}{7} x^{7} f e d c b + \frac{1}{7} x^{7} f^{2} c^{2} b + \frac{2}{7} x^{7} f e d^{2} a + \frac{2}{7} x^{7} f^{2} d c a + \frac{2}{5} x^{5} e^{2} d c b + \frac{2}{5} x^{5} f e c^{2} b + \frac{1}{5} x^{5} e^{2} d^{2} a + \frac{4}{5} x^{5} f e d c a + \frac{1}{5} x^{5} f^{2} c^{2} a + \frac{1}{3} x^{3} e^{2} c^{2} b + \frac{2}{3} x^{3} e^{2} d c a + \frac{2}{3} x^{3} f e c^{2} a + x e^{2} c^{2} a \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.086888, size = 216, normalized size = 1.37 \begin{align*} a c^{2} e^{2} x + \frac{b d^{2} f^{2} x^{11}}{11} + x^{9} \left (\frac{a d^{2} f^{2}}{9} + \frac{2 b c d f^{2}}{9} + \frac{2 b d^{2} e f}{9}\right ) + x^{7} \left (\frac{2 a c d f^{2}}{7} + \frac{2 a d^{2} e f}{7} + \frac{b c^{2} f^{2}}{7} + \frac{4 b c d e f}{7} + \frac{b d^{2} e^{2}}{7}\right ) + x^{5} \left (\frac{a c^{2} f^{2}}{5} + \frac{4 a c d e f}{5} + \frac{a d^{2} e^{2}}{5} + \frac{2 b c^{2} e f}{5} + \frac{2 b c d e^{2}}{5}\right ) + x^{3} \left (\frac{2 a c^{2} e f}{3} + \frac{2 a c d e^{2}}{3} + \frac{b c^{2} e^{2}}{3}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14455, size = 273, normalized size = 1.73 \begin{align*} \frac{1}{11} \, b d^{2} f^{2} x^{11} + \frac{2}{9} \, b c d f^{2} x^{9} + \frac{1}{9} \, a d^{2} f^{2} x^{9} + \frac{2}{9} \, b d^{2} f x^{9} e + \frac{1}{7} \, b c^{2} f^{2} x^{7} + \frac{2}{7} \, a c d f^{2} x^{7} + \frac{4}{7} \, b c d f x^{7} e + \frac{2}{7} \, a d^{2} f x^{7} e + \frac{1}{7} \, b d^{2} x^{7} e^{2} + \frac{1}{5} \, a c^{2} f^{2} x^{5} + \frac{2}{5} \, b c^{2} f x^{5} e + \frac{4}{5} \, a c d f x^{5} e + \frac{2}{5} \, b c d x^{5} e^{2} + \frac{1}{5} \, a d^{2} x^{5} e^{2} + \frac{2}{3} \, a c^{2} f x^{3} e + \frac{1}{3} \, b c^{2} x^{3} e^{2} + \frac{2}{3} \, a c d x^{3} e^{2} + a c^{2} x e^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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