3.1 \(\int (a+b x^2) (c+d x^2) (e+f x^2)^4 \, dx\)

Optimal. Leaf size=172 \[ \frac{1}{5} e^2 x^5 (2 a f (3 c f+2 d e)+b e (4 c f+d e))+\frac{1}{3} e^3 x^3 (4 a c f+a d e+b c e)+\frac{1}{11} f^3 x^{11} (a d f+b c f+4 b d e)+\frac{1}{9} f^2 x^9 (a f (c f+4 d e)+2 b e (2 c f+3 d e))+\frac{2}{7} e f x^7 (a f (2 c f+3 d e)+b e (3 c f+2 d e))+a c e^4 x+\frac{1}{13} b d f^4 x^{13} \]

[Out]

a*c*e^4*x + (e^3*(b*c*e + a*d*e + 4*a*c*f)*x^3)/3 + (e^2*(2*a*f*(2*d*e + 3*c*f) + b*e*(d*e + 4*c*f))*x^5)/5 +
(2*e*f*(a*f*(3*d*e + 2*c*f) + b*e*(2*d*e + 3*c*f))*x^7)/7 + (f^2*(a*f*(4*d*e + c*f) + 2*b*e*(3*d*e + 2*c*f))*x
^9)/9 + (f^3*(4*b*d*e + b*c*f + a*d*f)*x^11)/11 + (b*d*f^4*x^13)/13

________________________________________________________________________________________

Rubi [A]  time = 0.192214, 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, Rules used = {521} \[ \frac{1}{5} e^2 x^5 (2 a f (3 c f+2 d e)+b e (4 c f+d e))+\frac{1}{3} e^3 x^3 (4 a c f+a d e+b c e)+\frac{1}{11} f^3 x^{11} (a d f+b c f+4 b d e)+\frac{1}{9} f^2 x^9 (a f (c f+4 d e)+2 b e (2 c f+3 d e))+\frac{2}{7} e f x^7 (a f (2 c f+3 d e)+b e (3 c f+2 d e))+a c e^4 x+\frac{1}{13} b d f^4 x^{13} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)*(c + d*x^2)*(e + f*x^2)^4,x]

[Out]

a*c*e^4*x + (e^3*(b*c*e + a*d*e + 4*a*c*f)*x^3)/3 + (e^2*(2*a*f*(2*d*e + 3*c*f) + b*e*(d*e + 4*c*f))*x^5)/5 +
(2*e*f*(a*f*(3*d*e + 2*c*f) + b*e*(2*d*e + 3*c*f))*x^7)/7 + (f^2*(a*f*(4*d*e + c*f) + 2*b*e*(3*d*e + 2*c*f))*x
^9)/9 + (f^3*(4*b*d*e + b*c*f + a*d*f)*x^11)/11 + (b*d*f^4*x^13)/13

Rule 521

Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_))^(r_.), x_Symbol] :>
Int[ExpandIntegrand[(a + b*x^n)^p*(c + d*x^n)^q*(e + f*x^n)^r, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && I
GtQ[p, 0] && IGtQ[q, 0] && IGtQ[r, 0]

Rubi steps

\begin{align*} \int \left (a+b x^2\right ) \left (c+d x^2\right ) \left (e+f x^2\right )^4 \, dx &=\int \left (a c e^4+e^3 (b c e+a d e+4 a c f) x^2+e^2 (2 a f (2 d e+3 c f)+b e (d e+4 c f)) x^4+2 e f (a f (3 d e+2 c f)+b e (2 d e+3 c f)) x^6+f^2 (a f (4 d e+c f)+2 b e (3 d e+2 c f)) x^8+f^3 (4 b d e+b c f+a d f) x^{10}+b d f^4 x^{12}\right ) \, dx\\ &=a c e^4 x+\frac{1}{3} e^3 (b c e+a d e+4 a c f) x^3+\frac{1}{5} e^2 (2 a f (2 d e+3 c f)+b e (d e+4 c f)) x^5+\frac{2}{7} e f (a f (3 d e+2 c f)+b e (2 d e+3 c f)) x^7+\frac{1}{9} f^2 (a f (4 d e+c f)+2 b e (3 d e+2 c f)) x^9+\frac{1}{11} f^3 (4 b d e+b c f+a d f) x^{11}+\frac{1}{13} b d f^4 x^{13}\\ \end{align*}

Mathematica [A]  time = 0.087498, size = 172, normalized size = 1. \[ \frac{1}{5} e^2 x^5 (2 a f (3 c f+2 d e)+b e (4 c f+d e))+\frac{1}{3} e^3 x^3 (4 a c f+a d e+b c e)+\frac{1}{11} f^3 x^{11} (a d f+b c f+4 b d e)+\frac{1}{9} f^2 x^9 (a f (c f+4 d e)+2 b e (2 c f+3 d e))+\frac{2}{7} e f x^7 (a f (2 c f+3 d e)+b e (3 c f+2 d e))+a c e^4 x+\frac{1}{13} b d f^4 x^{13} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)*(c + d*x^2)*(e + f*x^2)^4,x]

[Out]

a*c*e^4*x + (e^3*(b*c*e + a*d*e + 4*a*c*f)*x^3)/3 + (e^2*(2*a*f*(2*d*e + 3*c*f) + b*e*(d*e + 4*c*f))*x^5)/5 +
(2*e*f*(a*f*(3*d*e + 2*c*f) + b*e*(2*d*e + 3*c*f))*x^7)/7 + (f^2*(a*f*(4*d*e + c*f) + 2*b*e*(3*d*e + 2*c*f))*x
^9)/9 + (f^3*(4*b*d*e + b*c*f + a*d*f)*x^11)/11 + (b*d*f^4*x^13)/13

________________________________________________________________________________________

Maple [A]  time = 0.002, size = 176, normalized size = 1. \begin{align*}{\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 \end{align*}

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]

1/13*b*d*f^4*x^13+1/11*((a*d+b*c)*f^4+4*b*d*e*f^3)*x^11+1/9*(a*c*f^4+4*(a*d+b*c)*e*f^3+6*b*d*e^2*f^2)*x^9+1/7*
(4*a*c*e*f^3+6*(a*d+b*c)*e^2*f^2+4*b*d*e^3*f)*x^7+1/5*(6*a*c*e^2*f^2+4*(a*d+b*c)*e^3*f+b*d*e^4)*x^5+1/3*(4*a*c
*e^3*f+(a*d+b*c)*e^4)*x^3+a*c*e^4*x

________________________________________________________________________________________

Maxima [A]  time = 0.986467, size = 236, normalized size = 1.37 \begin{align*} \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} \end{align*}

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]

1/13*b*d*f^4*x^13 + 1/11*(4*b*d*e*f^3 + (b*c + a*d)*f^4)*x^11 + 1/9*(6*b*d*e^2*f^2 + a*c*f^4 + 4*(b*c + a*d)*e
*f^3)*x^9 + 2/7*(2*b*d*e^3*f + 2*a*c*e*f^3 + 3*(b*c + a*d)*e^2*f^2)*x^7 + a*c*e^4*x + 1/5*(b*d*e^4 + 6*a*c*e^2
*f^2 + 4*(b*c + a*d)*e^3*f)*x^5 + 1/3*(4*a*c*e^3*f + (b*c + a*d)*e^4)*x^3

________________________________________________________________________________________

Fricas [A]  time = 1.24982, size = 531, normalized size = 3.09 \begin{align*} \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 \end{align*}

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]

1/13*x^13*f^4*d*b + 4/11*x^11*f^3*e*d*b + 1/11*x^11*f^4*c*b + 1/11*x^11*f^4*d*a + 2/3*x^9*f^2*e^2*d*b + 4/9*x^
9*f^3*e*c*b + 4/9*x^9*f^3*e*d*a + 1/9*x^9*f^4*c*a + 4/7*x^7*f*e^3*d*b + 6/7*x^7*f^2*e^2*c*b + 6/7*x^7*f^2*e^2*
d*a + 4/7*x^7*f^3*e*c*a + 1/5*x^5*e^4*d*b + 4/5*x^5*f*e^3*c*b + 4/5*x^5*f*e^3*d*a + 6/5*x^5*f^2*e^2*c*a + 1/3*
x^3*e^4*c*b + 1/3*x^3*e^4*d*a + 4/3*x^3*f*e^3*c*a + x*e^4*c*a

________________________________________________________________________________________

Sympy [A]  time = 0.088928, size = 236, normalized size = 1.37 \begin{align*} 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 ) \end{align*}

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]

a*c*e**4*x + b*d*f**4*x**13/13 + x**11*(a*d*f**4/11 + b*c*f**4/11 + 4*b*d*e*f**3/11) + x**9*(a*c*f**4/9 + 4*a*
d*e*f**3/9 + 4*b*c*e*f**3/9 + 2*b*d*e**2*f**2/3) + x**7*(4*a*c*e*f**3/7 + 6*a*d*e**2*f**2/7 + 6*b*c*e**2*f**2/
7 + 4*b*d*e**3*f/7) + x**5*(6*a*c*e**2*f**2/5 + 4*a*d*e**3*f/5 + 4*b*c*e**3*f/5 + b*d*e**4/5) + x**3*(4*a*c*e*
*3*f/3 + a*d*e**4/3 + b*c*e**4/3)

________________________________________________________________________________________

Giac [A]  time = 1.20963, size = 284, normalized size = 1.65 \begin{align*} \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} \end{align*}

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]

1/13*b*d*f^4*x^13 + 1/11*b*c*f^4*x^11 + 1/11*a*d*f^4*x^11 + 4/11*b*d*f^3*x^11*e + 1/9*a*c*f^4*x^9 + 4/9*b*c*f^
3*x^9*e + 4/9*a*d*f^3*x^9*e + 2/3*b*d*f^2*x^9*e^2 + 4/7*a*c*f^3*x^7*e + 6/7*b*c*f^2*x^7*e^2 + 6/7*a*d*f^2*x^7*
e^2 + 4/7*b*d*f*x^7*e^3 + 6/5*a*c*f^2*x^5*e^2 + 4/5*b*c*f*x^5*e^3 + 4/5*a*d*f*x^5*e^3 + 1/5*b*d*x^5*e^4 + 4/3*
a*c*f*x^3*e^3 + 1/3*b*c*x^3*e^4 + 1/3*a*d*x^3*e^4 + a*c*x*e^4