Optimal. Leaf size=310 \[ \frac{3}{11} d f x^{11} \left (a d f (c f+d e)+b \left (c^2 f^2+3 c d e f+d^2 e^2\right )\right )+\frac{1}{9} x^9 \left (3 a d f \left (c^2 f^2+3 c d e f+d^2 e^2\right )+b \left (9 c^2 d e f^2+c^3 f^3+9 c d^2 e^2 f+d^3 e^3\right )\right )+\frac{1}{7} x^7 \left (a \left (9 c^2 d e f^2+c^3 f^3+9 c d^2 e^2 f+d^3 e^3\right )+3 b c e \left (c^2 f^2+3 c d e f+d^2 e^2\right )\right )+\frac{3}{5} c e x^5 \left (a \left (c^2 f^2+3 c d e f+d^2 e^2\right )+b c e (c f+d e)\right )+\frac{1}{3} c^2 e^2 x^3 (3 a (c f+d e)+b c e)+\frac{1}{13} d^2 f^2 x^{13} (a d f+3 b (c f+d e))+a c^3 e^3 x+\frac{1}{15} b d^3 f^3 x^{15} \]
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Rubi [A] time = 0.386571, antiderivative size = 310, 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} \[ \frac{3}{11} d f x^{11} \left (a d f (c f+d e)+b \left (c^2 f^2+3 c d e f+d^2 e^2\right )\right )+\frac{1}{9} x^9 \left (3 a d f \left (c^2 f^2+3 c d e f+d^2 e^2\right )+b \left (9 c^2 d e f^2+c^3 f^3+9 c d^2 e^2 f+d^3 e^3\right )\right )+\frac{1}{7} x^7 \left (a \left (9 c^2 d e f^2+c^3 f^3+9 c d^2 e^2 f+d^3 e^3\right )+3 b c e \left (c^2 f^2+3 c d e f+d^2 e^2\right )\right )+\frac{3}{5} c e x^5 \left (a \left (c^2 f^2+3 c d e f+d^2 e^2\right )+b c e (c f+d e)\right )+\frac{1}{3} c^2 e^2 x^3 (3 a (c f+d e)+b c e)+\frac{1}{13} d^2 f^2 x^{13} (a d f+3 b (c f+d e))+a c^3 e^3 x+\frac{1}{15} b d^3 f^3 x^{15} \]
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 )^3 \left (e+f x^2\right )^3 \, dx &=\int \left (a c^3 e^3+c^2 e^2 (b c e+3 a (d e+c f)) x^2+3 c e \left (b c e (d e+c f)+a \left (d^2 e^2+3 c d e f+c^2 f^2\right )\right ) x^4+\left (3 b c e \left (d^2 e^2+3 c d e f+c^2 f^2\right )+a \left (d^3 e^3+9 c d^2 e^2 f+9 c^2 d e f^2+c^3 f^3\right )\right ) x^6+\left (3 a d f \left (d^2 e^2+3 c d e f+c^2 f^2\right )+b \left (d^3 e^3+9 c d^2 e^2 f+9 c^2 d e f^2+c^3 f^3\right )\right ) x^8+3 d f \left (a d f (d e+c f)+b \left (d^2 e^2+3 c d e f+c^2 f^2\right )\right ) x^{10}+d^2 f^2 (a d f+3 b (d e+c f)) x^{12}+b d^3 f^3 x^{14}\right ) \, dx\\ &=a c^3 e^3 x+\frac{1}{3} c^2 e^2 (b c e+3 a (d e+c f)) x^3+\frac{3}{5} c e \left (b c e (d e+c f)+a \left (d^2 e^2+3 c d e f+c^2 f^2\right )\right ) x^5+\frac{1}{7} \left (3 b c e \left (d^2 e^2+3 c d e f+c^2 f^2\right )+a \left (d^3 e^3+9 c d^2 e^2 f+9 c^2 d e f^2+c^3 f^3\right )\right ) x^7+\frac{1}{9} \left (3 a d f \left (d^2 e^2+3 c d e f+c^2 f^2\right )+b \left (d^3 e^3+9 c d^2 e^2 f+9 c^2 d e f^2+c^3 f^3\right )\right ) x^9+\frac{3}{11} d f \left (a d f (d e+c f)+b \left (d^2 e^2+3 c d e f+c^2 f^2\right )\right ) x^{11}+\frac{1}{13} d^2 f^2 (a d f+3 b (d e+c f)) x^{13}+\frac{1}{15} b d^3 f^3 x^{15}\\ \end{align*}
Mathematica [A] time = 0.114383, size = 310, normalized size = 1. \[ \frac{3}{11} d f x^{11} \left (a d f (c f+d e)+b \left (c^2 f^2+3 c d e f+d^2 e^2\right )\right )+\frac{1}{9} x^9 \left (3 a d f \left (c^2 f^2+3 c d e f+d^2 e^2\right )+b \left (9 c^2 d e f^2+c^3 f^3+9 c d^2 e^2 f+d^3 e^3\right )\right )+\frac{1}{7} x^7 \left (a \left (9 c^2 d e f^2+c^3 f^3+9 c d^2 e^2 f+d^3 e^3\right )+3 b c e \left (c^2 f^2+3 c d e f+d^2 e^2\right )\right )+\frac{3}{5} c e x^5 \left (a \left (c^2 f^2+3 c d e f+d^2 e^2\right )+b c e (c f+d e)\right )+\frac{1}{3} c^2 e^2 x^3 (3 a (c f+d e)+b c e)+\frac{1}{13} d^2 f^2 x^{13} (a d f+3 b (c f+d e))+a c^3 e^3 x+\frac{1}{15} b d^3 f^3 x^{15} \]
Antiderivative was successfully verified.
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Maple [A] time = 0., size = 339, normalized size = 1.1 \begin{align*}{\frac{b{d}^{3}{f}^{3}{x}^{15}}{15}}+{\frac{ \left ( \left ( a{d}^{3}+3\,bc{d}^{2} \right ){f}^{3}+3\,b{d}^{3}e{f}^{2} \right ){x}^{13}}{13}}+{\frac{ \left ( \left ( 3\,ac{d}^{2}+3\,b{c}^{2}d \right ){f}^{3}+3\, \left ( a{d}^{3}+3\,bc{d}^{2} \right ) e{f}^{2}+3\,b{d}^{3}{e}^{2}f \right ){x}^{11}}{11}}+{\frac{ \left ( \left ( 3\,a{c}^{2}d+b{c}^{3} \right ){f}^{3}+3\, \left ( 3\,ac{d}^{2}+3\,b{c}^{2}d \right ) e{f}^{2}+3\, \left ( a{d}^{3}+3\,bc{d}^{2} \right ){e}^{2}f+b{d}^{3}{e}^{3} \right ){x}^{9}}{9}}+{\frac{ \left ( a{c}^{3}{f}^{3}+3\, \left ( 3\,a{c}^{2}d+b{c}^{3} \right ) e{f}^{2}+3\, \left ( 3\,ac{d}^{2}+3\,b{c}^{2}d \right ){e}^{2}f+ \left ( a{d}^{3}+3\,bc{d}^{2} \right ){e}^{3} \right ){x}^{7}}{7}}+{\frac{ \left ( 3\,a{c}^{3}e{f}^{2}+3\, \left ( 3\,a{c}^{2}d+b{c}^{3} \right ){e}^{2}f+ \left ( 3\,ac{d}^{2}+3\,b{c}^{2}d \right ){e}^{3} \right ){x}^{5}}{5}}+{\frac{ \left ( 3\,a{c}^{3}{e}^{2}f+ \left ( 3\,a{c}^{2}d+b{c}^{3} \right ){e}^{3} \right ){x}^{3}}{3}}+a{c}^{3}{e}^{3}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.985705, size = 440, normalized size = 1.42 \begin{align*} \frac{1}{15} \, b d^{3} f^{3} x^{15} + \frac{1}{13} \,{\left (3 \, b d^{3} e f^{2} +{\left (3 \, b c d^{2} + a d^{3}\right )} f^{3}\right )} x^{13} + \frac{3}{11} \,{\left (b d^{3} e^{2} f +{\left (3 \, b c d^{2} + a d^{3}\right )} e f^{2} +{\left (b c^{2} d + a c d^{2}\right )} f^{3}\right )} x^{11} + \frac{1}{9} \,{\left (b d^{3} e^{3} + 3 \,{\left (3 \, b c d^{2} + a d^{3}\right )} e^{2} f + 9 \,{\left (b c^{2} d + a c d^{2}\right )} e f^{2} +{\left (b c^{3} + 3 \, a c^{2} d\right )} f^{3}\right )} x^{9} + a c^{3} e^{3} x + \frac{1}{7} \,{\left (a c^{3} f^{3} +{\left (3 \, b c d^{2} + a d^{3}\right )} e^{3} + 9 \,{\left (b c^{2} d + a c d^{2}\right )} e^{2} f + 3 \,{\left (b c^{3} + 3 \, a c^{2} d\right )} e f^{2}\right )} x^{7} + \frac{3}{5} \,{\left (a c^{3} e f^{2} +{\left (b c^{2} d + a c d^{2}\right )} e^{3} +{\left (b c^{3} + 3 \, a c^{2} d\right )} e^{2} f\right )} x^{5} + \frac{1}{3} \,{\left (3 \, a c^{3} e^{2} f +{\left (b c^{3} + 3 \, a c^{2} d\right )} e^{3}\right )} x^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.27603, size = 938, normalized size = 3.03 \begin{align*} \frac{1}{15} x^{15} f^{3} d^{3} b + \frac{3}{13} x^{13} f^{2} e d^{3} b + \frac{3}{13} x^{13} f^{3} d^{2} c b + \frac{1}{13} x^{13} f^{3} d^{3} a + \frac{3}{11} x^{11} f e^{2} d^{3} b + \frac{9}{11} x^{11} f^{2} e d^{2} c b + \frac{3}{11} x^{11} f^{3} d c^{2} b + \frac{3}{11} x^{11} f^{2} e d^{3} a + \frac{3}{11} x^{11} f^{3} d^{2} c a + \frac{1}{9} x^{9} e^{3} d^{3} b + x^{9} f e^{2} d^{2} c b + x^{9} f^{2} e d c^{2} b + \frac{1}{9} x^{9} f^{3} c^{3} b + \frac{1}{3} x^{9} f e^{2} d^{3} a + x^{9} f^{2} e d^{2} c a + \frac{1}{3} x^{9} f^{3} d c^{2} a + \frac{3}{7} x^{7} e^{3} d^{2} c b + \frac{9}{7} x^{7} f e^{2} d c^{2} b + \frac{3}{7} x^{7} f^{2} e c^{3} b + \frac{1}{7} x^{7} e^{3} d^{3} a + \frac{9}{7} x^{7} f e^{2} d^{2} c a + \frac{9}{7} x^{7} f^{2} e d c^{2} a + \frac{1}{7} x^{7} f^{3} c^{3} a + \frac{3}{5} x^{5} e^{3} d c^{2} b + \frac{3}{5} x^{5} f e^{2} c^{3} b + \frac{3}{5} x^{5} e^{3} d^{2} c a + \frac{9}{5} x^{5} f e^{2} d c^{2} a + \frac{3}{5} x^{5} f^{2} e c^{3} a + \frac{1}{3} x^{3} e^{3} c^{3} b + x^{3} e^{3} d c^{2} a + x^{3} f e^{2} c^{3} a + x e^{3} c^{3} a \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.113477, size = 423, normalized size = 1.36 \begin{align*} a c^{3} e^{3} x + \frac{b d^{3} f^{3} x^{15}}{15} + x^{13} \left (\frac{a d^{3} f^{3}}{13} + \frac{3 b c d^{2} f^{3}}{13} + \frac{3 b d^{3} e f^{2}}{13}\right ) + x^{11} \left (\frac{3 a c d^{2} f^{3}}{11} + \frac{3 a d^{3} e f^{2}}{11} + \frac{3 b c^{2} d f^{3}}{11} + \frac{9 b c d^{2} e f^{2}}{11} + \frac{3 b d^{3} e^{2} f}{11}\right ) + x^{9} \left (\frac{a c^{2} d f^{3}}{3} + a c d^{2} e f^{2} + \frac{a d^{3} e^{2} f}{3} + \frac{b c^{3} f^{3}}{9} + b c^{2} d e f^{2} + b c d^{2} e^{2} f + \frac{b d^{3} e^{3}}{9}\right ) + x^{7} \left (\frac{a c^{3} f^{3}}{7} + \frac{9 a c^{2} d e f^{2}}{7} + \frac{9 a c d^{2} e^{2} f}{7} + \frac{a d^{3} e^{3}}{7} + \frac{3 b c^{3} e f^{2}}{7} + \frac{9 b c^{2} d e^{2} f}{7} + \frac{3 b c d^{2} e^{3}}{7}\right ) + x^{5} \left (\frac{3 a c^{3} e f^{2}}{5} + \frac{9 a c^{2} d e^{2} f}{5} + \frac{3 a c d^{2} e^{3}}{5} + \frac{3 b c^{3} e^{2} f}{5} + \frac{3 b c^{2} d e^{3}}{5}\right ) + x^{3} \left (a c^{3} e^{2} f + a c^{2} d e^{3} + \frac{b c^{3} e^{3}}{3}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14176, size = 541, normalized size = 1.75 \begin{align*} \frac{1}{15} \, b d^{3} f^{3} x^{15} + \frac{3}{13} \, b c d^{2} f^{3} x^{13} + \frac{1}{13} \, a d^{3} f^{3} x^{13} + \frac{3}{13} \, b d^{3} f^{2} x^{13} e + \frac{3}{11} \, b c^{2} d f^{3} x^{11} + \frac{3}{11} \, a c d^{2} f^{3} x^{11} + \frac{9}{11} \, b c d^{2} f^{2} x^{11} e + \frac{3}{11} \, a d^{3} f^{2} x^{11} e + \frac{3}{11} \, b d^{3} f x^{11} e^{2} + \frac{1}{9} \, b c^{3} f^{3} x^{9} + \frac{1}{3} \, a c^{2} d f^{3} x^{9} + b c^{2} d f^{2} x^{9} e + a c d^{2} f^{2} x^{9} e + b c d^{2} f x^{9} e^{2} + \frac{1}{3} \, a d^{3} f x^{9} e^{2} + \frac{1}{7} \, a c^{3} f^{3} x^{7} + \frac{1}{9} \, b d^{3} x^{9} e^{3} + \frac{3}{7} \, b c^{3} f^{2} x^{7} e + \frac{9}{7} \, a c^{2} d f^{2} x^{7} e + \frac{9}{7} \, b c^{2} d f x^{7} e^{2} + \frac{9}{7} \, a c d^{2} f x^{7} e^{2} + \frac{3}{7} \, b c d^{2} x^{7} e^{3} + \frac{1}{7} \, a d^{3} x^{7} e^{3} + \frac{3}{5} \, a c^{3} f^{2} x^{5} e + \frac{3}{5} \, b c^{3} f x^{5} e^{2} + \frac{9}{5} \, a c^{2} d f x^{5} e^{2} + \frac{3}{5} \, b c^{2} d x^{5} e^{3} + \frac{3}{5} \, a c d^{2} x^{5} e^{3} + a c^{3} f x^{3} e^{2} + \frac{1}{3} \, b c^{3} x^{3} e^{3} + a c^{2} d x^{3} e^{3} + a c^{3} x e^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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