Optimal. Leaf size=94 \[ \frac {1}{7} d x^7 (a d f+2 b c f+b d e)+\frac {1}{5} x^5 (a d (2 c f+d e)+b c (c f+2 d e))+\frac {1}{3} c x^3 (a c f+2 a d e+b c e)+a c^2 e x+\frac {1}{9} b d^2 f x^9 \]
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Rubi [A] time = 0.08, antiderivative size = 94, normalized size of antiderivative = 1.00, 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 = {521} \[ \frac {1}{7} d x^7 (a d f+2 b c f+b d e)+\frac {1}{5} x^5 (a d (2 c f+d e)+b c (c f+2 d e))+\frac {1}{3} c x^3 (a c f+2 a d e+b c e)+a c^2 e x+\frac {1}{9} b d^2 f x^9 \]
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 ) \, dx &=\int \left (a c^2 e+c (b c e+2 a d e+a c f) x^2+(b c (2 d e+c f)+a d (d e+2 c f)) x^4+d (b d e+2 b c f+a d f) x^6+b d^2 f x^8\right ) \, dx\\ &=a c^2 e x+\frac {1}{3} c (b c e+2 a d e+a c f) x^3+\frac {1}{5} (b c (2 d e+c f)+a d (d e+2 c f)) x^5+\frac {1}{7} d (b d e+2 b c f+a d f) x^7+\frac {1}{9} b d^2 f x^9\\ \end {align*}
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Mathematica [A] time = 0.03, size = 96, normalized size = 1.02 \[ \frac {1}{5} x^5 \left (2 a c d f+a d^2 e+b c^2 f+2 b c d e\right )+\frac {1}{7} d x^7 (a d f+2 b c f+b d e)+\frac {1}{3} c x^3 (a c f+2 a d e+b c e)+a c^2 e x+\frac {1}{9} b d^2 f x^9 \]
Antiderivative was successfully verified.
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fricas [A] time = 0.67, size = 114, normalized size = 1.21 \[ \frac {1}{9} x^{9} f d^{2} b + \frac {1}{7} x^{7} e d^{2} b + \frac {2}{7} x^{7} f d c b + \frac {1}{7} x^{7} f d^{2} a + \frac {2}{5} x^{5} e d c b + \frac {1}{5} x^{5} f c^{2} b + \frac {1}{5} x^{5} e d^{2} a + \frac {2}{5} x^{5} f d c a + \frac {1}{3} x^{3} e c^{2} b + \frac {2}{3} x^{3} e d c a + \frac {1}{3} x^{3} f c^{2} a + x e c^{2} a \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.29, size = 120, normalized size = 1.28 \[ \frac {1}{9} \, b d^{2} f x^{9} + \frac {2}{7} \, b c d f x^{7} + \frac {1}{7} \, a d^{2} f x^{7} + \frac {1}{7} \, b d^{2} x^{7} e + \frac {1}{5} \, b c^{2} f x^{5} + \frac {2}{5} \, a c d f x^{5} + \frac {2}{5} \, b c d x^{5} e + \frac {1}{5} \, a d^{2} x^{5} e + \frac {1}{3} \, a c^{2} f x^{3} + \frac {1}{3} \, b c^{2} x^{3} e + \frac {2}{3} \, a c d x^{3} e + a c^{2} x e \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 101, normalized size = 1.07 \[ \frac {b \,d^{2} f \,x^{9}}{9}+\frac {\left (b \,d^{2} e +\left (a \,d^{2}+2 b c d \right ) f \right ) x^{7}}{7}+a \,c^{2} e x +\frac {\left (\left (a \,d^{2}+2 b c d \right ) e +\left (2 a c d +b \,c^{2}\right ) f \right ) x^{5}}{5}+\frac {\left (a \,c^{2} f +\left (2 a c d +b \,c^{2}\right ) e \right ) x^{3}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.76, size = 100, normalized size = 1.06 \[ \frac {1}{9} \, b d^{2} f x^{9} + \frac {1}{7} \, {\left (b d^{2} e + {\left (2 \, b c d + a d^{2}\right )} f\right )} x^{7} + \frac {1}{5} \, {\left ({\left (2 \, b c d + a d^{2}\right )} e + {\left (b c^{2} + 2 \, a c d\right )} f\right )} x^{5} + a c^{2} e x + \frac {1}{3} \, {\left (a c^{2} f + {\left (b c^{2} + 2 \, a c d\right )} e\right )} x^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 99, normalized size = 1.05 \[ x^5\,\left (\frac {a\,d^2\,e}{5}+\frac {b\,c^2\,f}{5}+\frac {2\,a\,c\,d\,f}{5}+\frac {2\,b\,c\,d\,e}{5}\right )+x^3\,\left (\frac {a\,c^2\,f}{3}+\frac {b\,c^2\,e}{3}+\frac {2\,a\,c\,d\,e}{3}\right )+x^7\,\left (\frac {a\,d^2\,f}{7}+\frac {b\,d^2\,e}{7}+\frac {2\,b\,c\,d\,f}{7}\right )+a\,c^2\,e\,x+\frac {b\,d^2\,f\,x^9}{9} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.08, size = 121, normalized size = 1.29 \[ a c^{2} e x + \frac {b d^{2} f x^{9}}{9} + x^{7} \left (\frac {a d^{2} f}{7} + \frac {2 b c d f}{7} + \frac {b d^{2} e}{7}\right ) + x^{5} \left (\frac {2 a c d f}{5} + \frac {a d^{2} e}{5} + \frac {b c^{2} f}{5} + \frac {2 b c d e}{5}\right ) + x^{3} \left (\frac {a c^{2} f}{3} + \frac {2 a c d e}{3} + \frac {b c^{2} e}{3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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