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

Optimal. Leaf size=226 \[ \frac{1}{9} d x^9 \left (a d f (3 c f+2 d e)+b \left (3 c^2 f^2+6 c d e f+d^2 e^2\right )\right )+\frac{1}{7} x^7 \left (a d \left (3 c^2 f^2+6 c d e f+d^2 e^2\right )+b c \left (c^2 f^2+6 c d e f+3 d^2 e^2\right )\right )+\frac{1}{5} c x^5 \left (a \left (c^2 f^2+6 c d e f+3 d^2 e^2\right )+b c e (2 c f+3 d e)\right )+\frac{1}{3} c^2 e x^3 (2 a c f+3 a d e+b c e)+\frac{1}{11} d^2 f x^{11} (a d f+3 b c f+2 b d e)+a c^3 e^2 x+\frac{1}{13} b d^3 f^2 x^{13} \]

[Out]

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

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Rubi [A]  time = 0.216233, antiderivative size = 226, 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} \[ \frac{1}{9} d x^9 \left (a d f (3 c f+2 d e)+b \left (3 c^2 f^2+6 c d e f+d^2 e^2\right )\right )+\frac{1}{7} x^7 \left (a d \left (3 c^2 f^2+6 c d e f+d^2 e^2\right )+b c \left (c^2 f^2+6 c d e f+3 d^2 e^2\right )\right )+\frac{1}{5} c x^5 \left (a \left (c^2 f^2+6 c d e f+3 d^2 e^2\right )+b c e (2 c f+3 d e)\right )+\frac{1}{3} c^2 e x^3 (2 a c f+3 a d e+b c e)+\frac{1}{11} d^2 f x^{11} (a d f+3 b c f+2 b d e)+a c^3 e^2 x+\frac{1}{13} b d^3 f^2 x^{13} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0829819, size = 226, normalized size = 1. \[ \frac{1}{9} d x^9 \left (a d f (3 c f+2 d e)+b \left (3 c^2 f^2+6 c d e f+d^2 e^2\right )\right )+\frac{1}{7} x^7 \left (a d \left (3 c^2 f^2+6 c d e f+d^2 e^2\right )+b c \left (c^2 f^2+6 c d e f+3 d^2 e^2\right )\right )+\frac{1}{5} c x^5 \left (a \left (c^2 f^2+6 c d e f+3 d^2 e^2\right )+b c e (2 c f+3 d e)\right )+\frac{1}{3} c^2 e x^3 (2 a c f+3 a d e+b c e)+\frac{1}{11} d^2 f x^{11} (a d f+3 b c f+2 b d e)+a c^3 e^2 x+\frac{1}{13} b d^3 f^2 x^{13} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0., size = 244, normalized size = 1.1 \begin{align*}{\frac{b{d}^{3}{f}^{2}{x}^{13}}{13}}+{\frac{ \left ( \left ( a{d}^{3}+3\,bc{d}^{2} \right ){f}^{2}+2\,b{d}^{3}ef \right ){x}^{11}}{11}}+{\frac{ \left ( \left ( 3\,ac{d}^{2}+3\,b{c}^{2}d \right ){f}^{2}+2\, \left ( a{d}^{3}+3\,bc{d}^{2} \right ) ef+b{d}^{3}{e}^{2} \right ){x}^{9}}{9}}+{\frac{ \left ( \left ( 3\,a{c}^{2}d+b{c}^{3} \right ){f}^{2}+2\, \left ( 3\,ac{d}^{2}+3\,b{c}^{2}d \right ) ef+ \left ( a{d}^{3}+3\,bc{d}^{2} \right ){e}^{2} \right ){x}^{7}}{7}}+{\frac{ \left ( a{c}^{3}{f}^{2}+2\, \left ( 3\,a{c}^{2}d+b{c}^{3} \right ) ef+ \left ( 3\,ac{d}^{2}+3\,b{c}^{2}d \right ){e}^{2} \right ){x}^{5}}{5}}+{\frac{ \left ( 2\,a{c}^{3}ef+ \left ( 3\,a{c}^{2}d+b{c}^{3} \right ){e}^{2} \right ){x}^{3}}{3}}+a{c}^{3}{e}^{2}x \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)^2,x)

[Out]

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

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Maxima [A]  time = 1.01063, size = 323, normalized size = 1.43 \begin{align*} \frac{1}{13} \, b d^{3} f^{2} x^{13} + \frac{1}{11} \,{\left (2 \, b d^{3} e f +{\left (3 \, b c d^{2} + a d^{3}\right )} f^{2}\right )} x^{11} + \frac{1}{9} \,{\left (b d^{3} e^{2} + 2 \,{\left (3 \, b c d^{2} + a d^{3}\right )} e f + 3 \,{\left (b c^{2} d + a c d^{2}\right )} f^{2}\right )} x^{9} + \frac{1}{7} \,{\left ({\left (3 \, b c d^{2} + a d^{3}\right )} e^{2} + 6 \,{\left (b c^{2} d + a c d^{2}\right )} e f +{\left (b c^{3} + 3 \, a c^{2} d\right )} f^{2}\right )} x^{7} + a c^{3} e^{2} x + \frac{1}{5} \,{\left (a c^{3} f^{2} + 3 \,{\left (b c^{2} d + a c d^{2}\right )} e^{2} + 2 \,{\left (b c^{3} + 3 \, a c^{2} d\right )} e f\right )} x^{5} + \frac{1}{3} \,{\left (2 \, a c^{3} e f +{\left (b c^{3} + 3 \, a c^{2} d\right )} e^{2}\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)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.24677, size = 676, normalized size = 2.99 \begin{align*} \frac{1}{13} x^{13} f^{2} d^{3} b + \frac{2}{11} x^{11} f e d^{3} b + \frac{3}{11} x^{11} f^{2} d^{2} c b + \frac{1}{11} x^{11} f^{2} d^{3} a + \frac{1}{9} x^{9} e^{2} d^{3} b + \frac{2}{3} x^{9} f e d^{2} c b + \frac{1}{3} x^{9} f^{2} d c^{2} b + \frac{2}{9} x^{9} f e d^{3} a + \frac{1}{3} x^{9} f^{2} d^{2} c a + \frac{3}{7} x^{7} e^{2} d^{2} c b + \frac{6}{7} x^{7} f e d c^{2} b + \frac{1}{7} x^{7} f^{2} c^{3} b + \frac{1}{7} x^{7} e^{2} d^{3} a + \frac{6}{7} x^{7} f e d^{2} c a + \frac{3}{7} x^{7} f^{2} d c^{2} a + \frac{3}{5} x^{5} e^{2} d c^{2} b + \frac{2}{5} x^{5} f e c^{3} b + \frac{3}{5} x^{5} e^{2} d^{2} c a + \frac{6}{5} x^{5} f e d c^{2} a + \frac{1}{5} x^{5} f^{2} c^{3} a + \frac{1}{3} x^{3} e^{2} c^{3} b + x^{3} e^{2} d c^{2} a + \frac{2}{3} x^{3} f e c^{3} a + x e^{2} 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)^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.098346, size = 304, normalized size = 1.35 \begin{align*} a c^{3} e^{2} x + \frac{b d^{3} f^{2} x^{13}}{13} + x^{11} \left (\frac{a d^{3} f^{2}}{11} + \frac{3 b c d^{2} f^{2}}{11} + \frac{2 b d^{3} e f}{11}\right ) + x^{9} \left (\frac{a c d^{2} f^{2}}{3} + \frac{2 a d^{3} e f}{9} + \frac{b c^{2} d f^{2}}{3} + \frac{2 b c d^{2} e f}{3} + \frac{b d^{3} e^{2}}{9}\right ) + x^{7} \left (\frac{3 a c^{2} d f^{2}}{7} + \frac{6 a c d^{2} e f}{7} + \frac{a d^{3} e^{2}}{7} + \frac{b c^{3} f^{2}}{7} + \frac{6 b c^{2} d e f}{7} + \frac{3 b c d^{2} e^{2}}{7}\right ) + x^{5} \left (\frac{a c^{3} f^{2}}{5} + \frac{6 a c^{2} d e f}{5} + \frac{3 a c d^{2} e^{2}}{5} + \frac{2 b c^{3} e f}{5} + \frac{3 b c^{2} d e^{2}}{5}\right ) + x^{3} \left (\frac{2 a c^{3} e f}{3} + a c^{2} d e^{2} + \frac{b c^{3} e^{2}}{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)**2,x)

[Out]

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

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Giac [A]  time = 1.15961, size = 390, normalized size = 1.73 \begin{align*} \frac{1}{13} \, b d^{3} f^{2} x^{13} + \frac{3}{11} \, b c d^{2} f^{2} x^{11} + \frac{1}{11} \, a d^{3} f^{2} x^{11} + \frac{2}{11} \, b d^{3} f x^{11} e + \frac{1}{3} \, b c^{2} d f^{2} x^{9} + \frac{1}{3} \, a c d^{2} f^{2} x^{9} + \frac{2}{3} \, b c d^{2} f x^{9} e + \frac{2}{9} \, a d^{3} f x^{9} e + \frac{1}{9} \, b d^{3} x^{9} e^{2} + \frac{1}{7} \, b c^{3} f^{2} x^{7} + \frac{3}{7} \, a c^{2} d f^{2} x^{7} + \frac{6}{7} \, b c^{2} d f x^{7} e + \frac{6}{7} \, a c d^{2} f x^{7} e + \frac{3}{7} \, b c d^{2} x^{7} e^{2} + \frac{1}{7} \, a d^{3} x^{7} e^{2} + \frac{1}{5} \, a c^{3} f^{2} x^{5} + \frac{2}{5} \, b c^{3} f x^{5} e + \frac{6}{5} \, a c^{2} d f x^{5} e + \frac{3}{5} \, b c^{2} d x^{5} e^{2} + \frac{3}{5} \, a c d^{2} x^{5} e^{2} + \frac{2}{3} \, a c^{3} f x^{3} e + \frac{1}{3} \, b c^{3} x^{3} e^{2} + a c^{2} d x^{3} e^{2} + a c^{3} x e^{2} \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)^2,x, algorithm="giac")

[Out]

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