Optimal. Leaf size=172 \[ \frac{1}{3} e^3 x^3 (4 a c f+a d e+b c e)+\frac{1}{5} e^2 x^5 (2 a f (3 c f+2 d e)+b e (4 c f+d e))+\frac{1}{11} f^3 x^{11} (a d f+b c f+4 b d e)+\frac{1}{9} f^2 x^9 (a f (c f+4 d e)+2 b e (2 c f+3 d e))+\frac{2}{7} e f x^7 (a f (2 c f+3 d e)+b e (3 c f+2 d e))+a c e^4 x+\frac{1}{13} b d f^4 x^{13} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.540615, antiderivative size = 172, 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 \[ \frac{1}{3} e^3 x^3 (4 a c f+a d e+b c e)+\frac{1}{5} e^2 x^5 (2 a f (3 c f+2 d e)+b e (4 c f+d e))+\frac{1}{11} f^3 x^{11} (a d f+b c f+4 b d e)+\frac{1}{9} f^2 x^9 (a f (c f+4 d e)+2 b e (2 c f+3 d e))+\frac{2}{7} e f x^7 (a f (2 c f+3 d e)+b e (3 c f+2 d e))+a c e^4 x+\frac{1}{13} b d f^4 x^{13} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x^2)*(c + d*x^2)*(e + f*x^2)^4,x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{b d f^{4} x^{13}}{13} + c e^{4} \int a\, dx + \frac{e^{3} x^{3} \left (4 a c f + a d e + b c e\right )}{3} + \frac{e^{2} x^{5} \left (6 a c f^{2} + 4 a d e f + 4 b c e f + b d e^{2}\right )}{5} + \frac{2 e f x^{7} \left (2 a c f^{2} + 3 a d e f + 3 b c e f + 2 b d e^{2}\right )}{7} + \frac{f^{3} x^{11} \left (a d f + b c f + 4 b d e\right )}{11} + \frac{f^{2} x^{9} \left (a c f^{2} + 4 a d e f + 4 b c e f + 6 b d e^{2}\right )}{9} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**2+a)*(d*x**2+c)*(f*x**2+e)**4,x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.185213, size = 172, normalized size = 1. \[ \frac{1}{3} e^3 x^3 (4 a c f+a d e+b c e)+\frac{1}{5} e^2 x^5 (2 a f (3 c f+2 d e)+b e (4 c f+d e))+\frac{1}{11} f^3 x^{11} (a d f+b c f+4 b d e)+\frac{1}{9} f^2 x^9 (a f (c f+4 d e)+2 b e (2 c f+3 d e))+\frac{2}{7} e f x^7 (a f (2 c f+3 d e)+b e (3 c f+2 d e))+a c e^4 x+\frac{1}{13} b d f^4 x^{13} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x^2)*(c + d*x^2)*(e + f*x^2)^4,x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.002, size = 176, normalized size = 1. \[{\frac{bd{f}^{4}{x}^{13}}{13}}+{\frac{ \left ( \left ( ad+bc \right ){f}^{4}+4\,bde{f}^{3} \right ){x}^{11}}{11}}+{\frac{ \left ( ac{f}^{4}+4\, \left ( ad+bc \right ) e{f}^{3}+6\,bd{e}^{2}{f}^{2} \right ){x}^{9}}{9}}+{\frac{ \left ( 4\,ace{f}^{3}+6\, \left ( ad+bc \right ){e}^{2}{f}^{2}+4\,bd{e}^{3}f \right ){x}^{7}}{7}}+{\frac{ \left ( 6\,ac{e}^{2}{f}^{2}+4\, \left ( ad+bc \right ){e}^{3}f+bd{e}^{4} \right ){x}^{5}}{5}}+{\frac{ \left ( 4\,ac{e}^{3}f+ \left ( ad+bc \right ){e}^{4} \right ){x}^{3}}{3}}+ac{e}^{4}x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^2+a)*(d*x^2+c)*(f*x^2+e)^4,x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 1.35518, size = 236, normalized size = 1.37 \[ \frac{1}{13} \, b d f^{4} x^{13} + \frac{1}{11} \,{\left (4 \, b d e f^{3} +{\left (b c + a d\right )} f^{4}\right )} x^{11} + \frac{1}{9} \,{\left (6 \, b d e^{2} f^{2} + a c f^{4} + 4 \,{\left (b c + a d\right )} e f^{3}\right )} x^{9} + \frac{2}{7} \,{\left (2 \, b d e^{3} f + 2 \, a c e f^{3} + 3 \,{\left (b c + a d\right )} e^{2} f^{2}\right )} x^{7} + a c e^{4} x + \frac{1}{5} \,{\left (b d e^{4} + 6 \, a c e^{2} f^{2} + 4 \,{\left (b c + a d\right )} e^{3} f\right )} x^{5} + \frac{1}{3} \,{\left (4 \, a c e^{3} f +{\left (b c + a d\right )} e^{4}\right )} x^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)*(d*x^2 + c)*(f*x^2 + e)^4,x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.18254, size = 1, normalized size = 0.01 \[ \frac{1}{13} x^{13} f^{4} d b + \frac{4}{11} x^{11} f^{3} e d b + \frac{1}{11} x^{11} f^{4} c b + \frac{1}{11} x^{11} f^{4} d a + \frac{2}{3} x^{9} f^{2} e^{2} d b + \frac{4}{9} x^{9} f^{3} e c b + \frac{4}{9} x^{9} f^{3} e d a + \frac{1}{9} x^{9} f^{4} c a + \frac{4}{7} x^{7} f e^{3} d b + \frac{6}{7} x^{7} f^{2} e^{2} c b + \frac{6}{7} x^{7} f^{2} e^{2} d a + \frac{4}{7} x^{7} f^{3} e c a + \frac{1}{5} x^{5} e^{4} d b + \frac{4}{5} x^{5} f e^{3} c b + \frac{4}{5} x^{5} f e^{3} d a + \frac{6}{5} x^{5} f^{2} e^{2} c a + \frac{1}{3} x^{3} e^{4} c b + \frac{1}{3} x^{3} e^{4} d a + \frac{4}{3} x^{3} f e^{3} c a + x e^{4} c a \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)*(d*x^2 + c)*(f*x^2 + e)^4,x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 0.112258, size = 236, normalized size = 1.37 \[ a c e^{4} x + \frac{b d f^{4} x^{13}}{13} + x^{11} \left (\frac{a d f^{4}}{11} + \frac{b c f^{4}}{11} + \frac{4 b d e f^{3}}{11}\right ) + x^{9} \left (\frac{a c f^{4}}{9} + \frac{4 a d e f^{3}}{9} + \frac{4 b c e f^{3}}{9} + \frac{2 b d e^{2} f^{2}}{3}\right ) + x^{7} \left (\frac{4 a c e f^{3}}{7} + \frac{6 a d e^{2} f^{2}}{7} + \frac{6 b c e^{2} f^{2}}{7} + \frac{4 b d e^{3} f}{7}\right ) + x^{5} \left (\frac{6 a c e^{2} f^{2}}{5} + \frac{4 a d e^{3} f}{5} + \frac{4 b c e^{3} f}{5} + \frac{b d e^{4}}{5}\right ) + x^{3} \left (\frac{4 a c e^{3} f}{3} + \frac{a d e^{4}}{3} + \frac{b c e^{4}}{3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x**2+a)*(d*x**2+c)*(f*x**2+e)**4,x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.228793, size = 284, normalized size = 1.65 \[ \frac{1}{13} \, b d f^{4} x^{13} + \frac{1}{11} \, b c f^{4} x^{11} + \frac{1}{11} \, a d f^{4} x^{11} + \frac{4}{11} \, b d f^{3} x^{11} e + \frac{1}{9} \, a c f^{4} x^{9} + \frac{4}{9} \, b c f^{3} x^{9} e + \frac{4}{9} \, a d f^{3} x^{9} e + \frac{2}{3} \, b d f^{2} x^{9} e^{2} + \frac{4}{7} \, a c f^{3} x^{7} e + \frac{6}{7} \, b c f^{2} x^{7} e^{2} + \frac{6}{7} \, a d f^{2} x^{7} e^{2} + \frac{4}{7} \, b d f x^{7} e^{3} + \frac{6}{5} \, a c f^{2} x^{5} e^{2} + \frac{4}{5} \, b c f x^{5} e^{3} + \frac{4}{5} \, a d f x^{5} e^{3} + \frac{1}{5} \, b d x^{5} e^{4} + \frac{4}{3} \, a c f x^{3} e^{3} + \frac{1}{3} \, b c x^{3} e^{4} + \frac{1}{3} \, a d x^{3} e^{4} + a c x e^{4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)*(d*x^2 + c)*(f*x^2 + e)^4,x, algorithm="giac")
[Out]