3.18 \(\int (a+b x^2) (c+d x^2)^3 (e+f x^2) \, dx\)

Optimal. Leaf size=130 \[ \frac{1}{3} c^2 x^3 (a c f+3 a d e+b c e)+\frac{1}{9} d^2 x^9 (a d f+3 b c f+b d e)+\frac{1}{7} d x^7 (a d (3 c f+d e)+3 b c (c f+d e))+\frac{1}{5} c x^5 (3 a d (c f+d e)+b c (c f+3 d e))+a c^3 e x+\frac{1}{11} b d^3 f x^{11} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.13479, antiderivative size = 130, 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} \[ \frac{1}{3} c^2 x^3 (a c f+3 a d e+b c e)+\frac{1}{9} d^2 x^9 (a d f+3 b c f+b d e)+\frac{1}{7} d x^7 (a d (3 c f+d e)+3 b c (c f+d e))+\frac{1}{5} c x^5 (3 a d (c f+d e)+b c (c f+3 d e))+a c^3 e x+\frac{1}{11} b d^3 f x^{11} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)*(c + d*x^2)^3*(e + f*x^2),x]

[Out]

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

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

Mathematica [A]  time = 0.0497375, size = 130, normalized size = 1. \[ \frac{1}{3} c^2 x^3 (a c f+3 a d e+b c e)+\frac{1}{9} d^2 x^9 (a d f+3 b c f+b d e)+\frac{1}{7} d x^7 (a d (3 c f+d e)+3 b c (c f+d e))+\frac{1}{5} c x^5 (3 a d (c f+d e)+b c (c f+3 d e))+a c^3 e x+\frac{1}{11} b d^3 f x^{11} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)*(c + d*x^2)^3*(e + f*x^2),x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.002, size = 149, normalized size = 1.2 \begin{align*}{\frac{b{d}^{3}f{x}^{11}}{11}}+{\frac{ \left ( \left ( a{d}^{3}+3\,bc{d}^{2} \right ) f+b{d}^{3}e \right ){x}^{9}}{9}}+{\frac{ \left ( \left ( 3\,ac{d}^{2}+3\,b{c}^{2}d \right ) f+ \left ( a{d}^{3}+3\,bc{d}^{2} \right ) e \right ){x}^{7}}{7}}+{\frac{ \left ( \left ( 3\,a{c}^{2}d+b{c}^{3} \right ) f+ \left ( 3\,ac{d}^{2}+3\,b{c}^{2}d \right ) e \right ){x}^{5}}{5}}+{\frac{ \left ( a{c}^{3}f+ \left ( 3\,a{c}^{2}d+b{c}^{3} \right ) e \right ){x}^{3}}{3}}+a{c}^{3}ex \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)*(d*x^2+c)^3*(f*x^2+e),x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 0.995879, size = 197, normalized size = 1.52 \begin{align*} \frac{1}{11} \, b d^{3} f x^{11} + \frac{1}{9} \,{\left (b d^{3} e +{\left (3 \, b c d^{2} + a d^{3}\right )} f\right )} x^{9} + \frac{1}{7} \,{\left ({\left (3 \, b c d^{2} + a d^{3}\right )} e + 3 \,{\left (b c^{2} d + a c d^{2}\right )} f\right )} x^{7} + a c^{3} e x + \frac{1}{5} \,{\left (3 \,{\left (b c^{2} d + a c d^{2}\right )} e +{\left (b c^{3} + 3 \, a c^{2} d\right )} f\right )} x^{5} + \frac{1}{3} \,{\left (a c^{3} f +{\left (b c^{3} + 3 \, a c^{2} d\right )} e\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)^3*(f*x^2+e),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.23814, size = 398, normalized size = 3.06 \begin{align*} \frac{1}{11} x^{11} f d^{3} b + \frac{1}{9} x^{9} e d^{3} b + \frac{1}{3} x^{9} f d^{2} c b + \frac{1}{9} x^{9} f d^{3} a + \frac{3}{7} x^{7} e d^{2} c b + \frac{3}{7} x^{7} f d c^{2} b + \frac{1}{7} x^{7} e d^{3} a + \frac{3}{7} x^{7} f d^{2} c a + \frac{3}{5} x^{5} e d c^{2} b + \frac{1}{5} x^{5} f c^{3} b + \frac{3}{5} x^{5} e d^{2} c a + \frac{3}{5} x^{5} f d c^{2} a + \frac{1}{3} x^{3} e c^{3} b + x^{3} e d c^{2} a + \frac{1}{3} x^{3} f c^{3} a + x e c^{3} a \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)*(d*x^2+c)^3*(f*x^2+e),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 0.085075, size = 173, normalized size = 1.33 \begin{align*} a c^{3} e x + \frac{b d^{3} f x^{11}}{11} + x^{9} \left (\frac{a d^{3} f}{9} + \frac{b c d^{2} f}{3} + \frac{b d^{3} e}{9}\right ) + x^{7} \left (\frac{3 a c d^{2} f}{7} + \frac{a d^{3} e}{7} + \frac{3 b c^{2} d f}{7} + \frac{3 b c d^{2} e}{7}\right ) + x^{5} \left (\frac{3 a c^{2} d f}{5} + \frac{3 a c d^{2} e}{5} + \frac{b c^{3} f}{5} + \frac{3 b c^{2} d e}{5}\right ) + x^{3} \left (\frac{a c^{3} f}{3} + a c^{2} d e + \frac{b c^{3} e}{3}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)*(d*x**2+c)**3*(f*x**2+e),x)

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.13893, size = 234, normalized size = 1.8 \begin{align*} \frac{1}{11} \, b d^{3} f x^{11} + \frac{1}{3} \, b c d^{2} f x^{9} + \frac{1}{9} \, a d^{3} f x^{9} + \frac{1}{9} \, b d^{3} x^{9} e + \frac{3}{7} \, b c^{2} d f x^{7} + \frac{3}{7} \, a c d^{2} f x^{7} + \frac{3}{7} \, b c d^{2} x^{7} e + \frac{1}{7} \, a d^{3} x^{7} e + \frac{1}{5} \, b c^{3} f x^{5} + \frac{3}{5} \, a c^{2} d f x^{5} + \frac{3}{5} \, b c^{2} d x^{5} e + \frac{3}{5} \, a c d^{2} x^{5} e + \frac{1}{3} \, a c^{3} f x^{3} + \frac{1}{3} \, b c^{3} x^{3} e + a c^{2} d x^{3} e + a c^{3} x e \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)*(d*x^2+c)^3*(f*x^2+e),x, algorithm="giac")

[Out]

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