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

Optimal. Leaf size=310 \[ \frac {3}{11} d f x^{11} \left (a d f (c f+d e)+b \left (c^2 f^2+3 c d e f+d^2 e^2\right )\right )+\frac {3}{5} c e x^5 \left (a \left (c^2 f^2+3 c d e f+d^2 e^2\right )+b c e (c f+d e)\right )+\frac {1}{3} c^2 e^2 x^3 (3 a (c f+d e)+b c e)+\frac {1}{9} x^9 \left (3 a d f \left (c^2 f^2+3 c d e f+d^2 e^2\right )+b \left (c^3 f^3+9 c^2 d e f^2+9 c d^2 e^2 f+d^3 e^3\right )\right )+\frac {1}{7} x^7 \left (a \left (c^3 f^3+9 c^2 d e f^2+9 c d^2 e^2 f+d^3 e^3\right )+3 b c e \left (c^2 f^2+3 c d e f+d^2 e^2\right )\right )+\frac {1}{13} d^2 f^2 x^{13} (a d f+3 b (c f+d e))+a c^3 e^3 x+\frac {1}{15} b d^3 f^3 x^{15} \]

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Rubi [A]  time = 0.39, antiderivative size = 310, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {3}{11} d f x^{11} \left (a d f (c f+d e)+b \left (c^2 f^2+3 c d e f+d^2 e^2\right )\right )+\frac {1}{9} x^9 \left (3 a d f \left (c^2 f^2+3 c d e f+d^2 e^2\right )+b \left (9 c^2 d e f^2+c^3 f^3+9 c d^2 e^2 f+d^3 e^3\right )\right )+\frac {1}{7} x^7 \left (a \left (9 c^2 d e f^2+c^3 f^3+9 c d^2 e^2 f+d^3 e^3\right )+3 b c e \left (c^2 f^2+3 c d e f+d^2 e^2\right )\right )+\frac {3}{5} c e x^5 \left (a \left (c^2 f^2+3 c d e f+d^2 e^2\right )+b c e (c f+d e)\right )+\frac {1}{3} c^2 e^2 x^3 (3 a (c f+d e)+b c e)+\frac {1}{13} d^2 f^2 x^{13} (a d f+3 b (c f+d e))+a c^3 e^3 x+\frac {1}{15} b d^3 f^3 x^{15} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

a*c^3*e^3*x + (c^2*e^2*(b*c*e + 3*a*(d*e + c*f))*x^3)/3 + (3*c*e*(b*c*e*(d*e + c*f) + a*(d^2*e^2 + 3*c*d*e*f +
c^2*f^2))*x^5)/5 + ((3*b*c*e*(d^2*e^2 + 3*c*d*e*f + c^2*f^2) + a*(d^3*e^3 + 9*c*d^2*e^2*f + 9*c^2*d*e*f^2 + c
^3*f^3))*x^7)/7 + ((3*a*d*f*(d^2*e^2 + 3*c*d*e*f + c^2*f^2) + b*(d^3*e^3 + 9*c*d^2*e^2*f + 9*c^2*d*e*f^2 + c^3
*f^3))*x^9)/9 + (3*d*f*(a*d*f*(d*e + c*f) + b*(d^2*e^2 + 3*c*d*e*f + c^2*f^2))*x^11)/11 + (d^2*f^2*(a*d*f + 3*
b*(d*e + c*f))*x^13)/13 + (b*d^3*f^3*x^15)/15

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 )^3 \, dx &=\int \left (a c^3 e^3+c^2 e^2 (b c e+3 a (d e+c f)) x^2+3 c e \left (b c e (d e+c f)+a \left (d^2 e^2+3 c d e f+c^2 f^2\right )\right ) x^4+\left (3 b c e \left (d^2 e^2+3 c d e f+c^2 f^2\right )+a \left (d^3 e^3+9 c d^2 e^2 f+9 c^2 d e f^2+c^3 f^3\right )\right ) x^6+\left (3 a d f \left (d^2 e^2+3 c d e f+c^2 f^2\right )+b \left (d^3 e^3+9 c d^2 e^2 f+9 c^2 d e f^2+c^3 f^3\right )\right ) x^8+3 d f \left (a d f (d e+c f)+b \left (d^2 e^2+3 c d e f+c^2 f^2\right )\right ) x^{10}+d^2 f^2 (a d f+3 b (d e+c f)) x^{12}+b d^3 f^3 x^{14}\right ) \, dx\\ &=a c^3 e^3 x+\frac {1}{3} c^2 e^2 (b c e+3 a (d e+c f)) x^3+\frac {3}{5} c e \left (b c e (d e+c f)+a \left (d^2 e^2+3 c d e f+c^2 f^2\right )\right ) x^5+\frac {1}{7} \left (3 b c e \left (d^2 e^2+3 c d e f+c^2 f^2\right )+a \left (d^3 e^3+9 c d^2 e^2 f+9 c^2 d e f^2+c^3 f^3\right )\right ) x^7+\frac {1}{9} \left (3 a d f \left (d^2 e^2+3 c d e f+c^2 f^2\right )+b \left (d^3 e^3+9 c d^2 e^2 f+9 c^2 d e f^2+c^3 f^3\right )\right ) x^9+\frac {3}{11} d f \left (a d f (d e+c f)+b \left (d^2 e^2+3 c d e f+c^2 f^2\right )\right ) x^{11}+\frac {1}{13} d^2 f^2 (a d f+3 b (d e+c f)) x^{13}+\frac {1}{15} b d^3 f^3 x^{15}\\ \end {align*}

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Mathematica [A]  time = 0.13, size = 310, normalized size = 1.00 \begin {gather*} \frac {3}{11} d f x^{11} \left (a d f (c f+d e)+b \left (c^2 f^2+3 c d e f+d^2 e^2\right )\right )+\frac {3}{5} c e x^5 \left (a \left (c^2 f^2+3 c d e f+d^2 e^2\right )+b c e (c f+d e)\right )+\frac {1}{3} c^2 e^2 x^3 (3 a (c f+d e)+b c e)+\frac {1}{9} x^9 \left (3 a d f \left (c^2 f^2+3 c d e f+d^2 e^2\right )+b \left (c^3 f^3+9 c^2 d e f^2+9 c d^2 e^2 f+d^3 e^3\right )\right )+\frac {1}{7} x^7 \left (a \left (c^3 f^3+9 c^2 d e f^2+9 c d^2 e^2 f+d^3 e^3\right )+3 b c e \left (c^2 f^2+3 c d e f+d^2 e^2\right )\right )+\frac {1}{13} d^2 f^2 x^{13} (a d f+3 b (c f+d e))+a c^3 e^3 x+\frac {1}{15} b d^3 f^3 x^{15} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

a*c^3*e^3*x + (c^2*e^2*(b*c*e + 3*a*(d*e + c*f))*x^3)/3 + (3*c*e*(b*c*e*(d*e + c*f) + a*(d^2*e^2 + 3*c*d*e*f +
c^2*f^2))*x^5)/5 + ((3*b*c*e*(d^2*e^2 + 3*c*d*e*f + c^2*f^2) + a*(d^3*e^3 + 9*c*d^2*e^2*f + 9*c^2*d*e*f^2 + c
^3*f^3))*x^7)/7 + ((3*a*d*f*(d^2*e^2 + 3*c*d*e*f + c^2*f^2) + b*(d^3*e^3 + 9*c*d^2*e^2*f + 9*c^2*d*e*f^2 + c^3
*f^3))*x^9)/9 + (3*d*f*(a*d*f*(d*e + c*f) + b*(d^2*e^2 + 3*c*d*e*f + c^2*f^2))*x^11)/11 + (d^2*f^2*(a*d*f + 3*
b*(d*e + c*f))*x^13)/13 + (b*d^3*f^3*x^15)/15

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a+b x^2\right ) \left (c+d x^2\right )^3 \left (e+f x^2\right )^3 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.21, size = 409, normalized size = 1.32 \begin {gather*} \frac {1}{15} x^{15} f^{3} d^{3} b + \frac {3}{13} x^{13} f^{2} e d^{3} b + \frac {3}{13} x^{13} f^{3} d^{2} c b + \frac {1}{13} x^{13} f^{3} d^{3} a + \frac {3}{11} x^{11} f e^{2} d^{3} b + \frac {9}{11} x^{11} f^{2} e d^{2} c b + \frac {3}{11} x^{11} f^{3} d c^{2} b + \frac {3}{11} x^{11} f^{2} e d^{3} a + \frac {3}{11} x^{11} f^{3} d^{2} c a + \frac {1}{9} x^{9} e^{3} d^{3} b + x^{9} f e^{2} d^{2} c b + x^{9} f^{2} e d c^{2} b + \frac {1}{9} x^{9} f^{3} c^{3} b + \frac {1}{3} x^{9} f e^{2} d^{3} a + x^{9} f^{2} e d^{2} c a + \frac {1}{3} x^{9} f^{3} d c^{2} a + \frac {3}{7} x^{7} e^{3} d^{2} c b + \frac {9}{7} x^{7} f e^{2} d c^{2} b + \frac {3}{7} x^{7} f^{2} e c^{3} b + \frac {1}{7} x^{7} e^{3} d^{3} a + \frac {9}{7} x^{7} f e^{2} d^{2} c a + \frac {9}{7} x^{7} f^{2} e d c^{2} a + \frac {1}{7} x^{7} f^{3} c^{3} a + \frac {3}{5} x^{5} e^{3} d c^{2} b + \frac {3}{5} x^{5} f e^{2} c^{3} b + \frac {3}{5} x^{5} e^{3} d^{2} c a + \frac {9}{5} x^{5} f e^{2} d c^{2} a + \frac {3}{5} x^{5} f^{2} e c^{3} a + \frac {1}{3} x^{3} e^{3} c^{3} b + x^{3} e^{3} d c^{2} a + x^{3} f e^{2} c^{3} a + x e^{3} c^{3} a \end {gather*}

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)^3,x, algorithm="fricas")

[Out]

1/15*x^15*f^3*d^3*b + 3/13*x^13*f^2*e*d^3*b + 3/13*x^13*f^3*d^2*c*b + 1/13*x^13*f^3*d^3*a + 3/11*x^11*f*e^2*d^
3*b + 9/11*x^11*f^2*e*d^2*c*b + 3/11*x^11*f^3*d*c^2*b + 3/11*x^11*f^2*e*d^3*a + 3/11*x^11*f^3*d^2*c*a + 1/9*x^
9*e^3*d^3*b + x^9*f*e^2*d^2*c*b + x^9*f^2*e*d*c^2*b + 1/9*x^9*f^3*c^3*b + 1/3*x^9*f*e^2*d^3*a + x^9*f^2*e*d^2*
c*a + 1/3*x^9*f^3*d*c^2*a + 3/7*x^7*e^3*d^2*c*b + 9/7*x^7*f*e^2*d*c^2*b + 3/7*x^7*f^2*e*c^3*b + 1/7*x^7*e^3*d^
3*a + 9/7*x^7*f*e^2*d^2*c*a + 9/7*x^7*f^2*e*d*c^2*a + 1/7*x^7*f^3*c^3*a + 3/5*x^5*e^3*d*c^2*b + 3/5*x^5*f*e^2*
c^3*b + 3/5*x^5*e^3*d^2*c*a + 9/5*x^5*f*e^2*d*c^2*a + 3/5*x^5*f^2*e*c^3*a + 1/3*x^3*e^3*c^3*b + x^3*e^3*d*c^2*
a + x^3*f*e^2*c^3*a + x*e^3*c^3*a

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giac [A]  time = 0.38, size = 401, normalized size = 1.29 \begin {gather*} \frac {1}{15} \, b d^{3} f^{3} x^{15} + \frac {3}{13} \, b c d^{2} f^{3} x^{13} + \frac {1}{13} \, a d^{3} f^{3} x^{13} + \frac {3}{13} \, b d^{3} f^{2} x^{13} e + \frac {3}{11} \, b c^{2} d f^{3} x^{11} + \frac {3}{11} \, a c d^{2} f^{3} x^{11} + \frac {9}{11} \, b c d^{2} f^{2} x^{11} e + \frac {3}{11} \, a d^{3} f^{2} x^{11} e + \frac {3}{11} \, b d^{3} f x^{11} e^{2} + \frac {1}{9} \, b c^{3} f^{3} x^{9} + \frac {1}{3} \, a c^{2} d f^{3} x^{9} + b c^{2} d f^{2} x^{9} e + a c d^{2} f^{2} x^{9} e + b c d^{2} f x^{9} e^{2} + \frac {1}{3} \, a d^{3} f x^{9} e^{2} + \frac {1}{7} \, a c^{3} f^{3} x^{7} + \frac {1}{9} \, b d^{3} x^{9} e^{3} + \frac {3}{7} \, b c^{3} f^{2} x^{7} e + \frac {9}{7} \, a c^{2} d f^{2} x^{7} e + \frac {9}{7} \, b c^{2} d f x^{7} e^{2} + \frac {9}{7} \, a c d^{2} f x^{7} e^{2} + \frac {3}{7} \, b c d^{2} x^{7} e^{3} + \frac {1}{7} \, a d^{3} x^{7} e^{3} + \frac {3}{5} \, a c^{3} f^{2} x^{5} e + \frac {3}{5} \, b c^{3} f x^{5} e^{2} + \frac {9}{5} \, a c^{2} d f x^{5} e^{2} + \frac {3}{5} \, b c^{2} d x^{5} e^{3} + \frac {3}{5} \, a c d^{2} x^{5} e^{3} + a c^{3} f x^{3} e^{2} + \frac {1}{3} \, b c^{3} x^{3} e^{3} + a c^{2} d x^{3} e^{3} + a c^{3} x e^{3} \end {gather*}

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)^3,x, algorithm="giac")

[Out]

1/15*b*d^3*f^3*x^15 + 3/13*b*c*d^2*f^3*x^13 + 1/13*a*d^3*f^3*x^13 + 3/13*b*d^3*f^2*x^13*e + 3/11*b*c^2*d*f^3*x
^11 + 3/11*a*c*d^2*f^3*x^11 + 9/11*b*c*d^2*f^2*x^11*e + 3/11*a*d^3*f^2*x^11*e + 3/11*b*d^3*f*x^11*e^2 + 1/9*b*
c^3*f^3*x^9 + 1/3*a*c^2*d*f^3*x^9 + b*c^2*d*f^2*x^9*e + a*c*d^2*f^2*x^9*e + b*c*d^2*f*x^9*e^2 + 1/3*a*d^3*f*x^
9*e^2 + 1/7*a*c^3*f^3*x^7 + 1/9*b*d^3*x^9*e^3 + 3/7*b*c^3*f^2*x^7*e + 9/7*a*c^2*d*f^2*x^7*e + 9/7*b*c^2*d*f*x^
7*e^2 + 9/7*a*c*d^2*f*x^7*e^2 + 3/7*b*c*d^2*x^7*e^3 + 1/7*a*d^3*x^7*e^3 + 3/5*a*c^3*f^2*x^5*e + 3/5*b*c^3*f*x^
5*e^2 + 9/5*a*c^2*d*f*x^5*e^2 + 3/5*b*c^2*d*x^5*e^3 + 3/5*a*c*d^2*x^5*e^3 + a*c^3*f*x^3*e^2 + 1/3*b*c^3*x^3*e^
3 + a*c^2*d*x^3*e^3 + a*c^3*x*e^3

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maple [A]  time = 0.00, size = 339, normalized size = 1.09 \begin {gather*} \frac {b \,d^{3} f^{3} x^{15}}{15}+\frac {\left (3 b \,d^{3} e \,f^{2}+\left (a \,d^{3}+3 b c \,d^{2}\right ) f^{3}\right ) x^{13}}{13}+\frac {\left (3 b \,d^{3} e^{2} f +3 \left (a \,d^{3}+3 b c \,d^{2}\right ) e \,f^{2}+\left (3 a c \,d^{2}+3 b \,c^{2} d \right ) f^{3}\right ) x^{11}}{11}+\frac {\left (b \,d^{3} e^{3}+3 \left (a \,d^{3}+3 b c \,d^{2}\right ) e^{2} f +3 \left (3 a c \,d^{2}+3 b \,c^{2} d \right ) e \,f^{2}+\left (3 a \,c^{2} d +b \,c^{3}\right ) f^{3}\right ) x^{9}}{9}+a \,c^{3} e^{3} x +\frac {\left (a \,c^{3} f^{3}+\left (a \,d^{3}+3 b c \,d^{2}\right ) e^{3}+3 \left (3 a c \,d^{2}+3 b \,c^{2} d \right ) e^{2} f +3 \left (3 a \,c^{2} d +b \,c^{3}\right ) e \,f^{2}\right ) x^{7}}{7}+\frac {\left (3 a \,c^{3} e \,f^{2}+\left (3 a c \,d^{2}+3 b \,c^{2} d \right ) e^{3}+3 \left (3 a \,c^{2} d +b \,c^{3}\right ) e^{2} f \right ) x^{5}}{5}+\frac {\left (3 a \,c^{3} e^{2} f +\left (3 a \,c^{2} d +b \,c^{3}\right ) e^{3}\right ) x^{3}}{3} \end {gather*}

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)^3,x)

[Out]

1/15*b*d^3*f^3*x^15+1/13*((a*d^3+3*b*c*d^2)*f^3+3*b*d^3*e*f^2)*x^13+1/11*((3*a*c*d^2+3*b*c^2*d)*f^3+3*(a*d^3+3
*b*c*d^2)*e*f^2+3*b*d^3*e^2*f)*x^11+1/9*((3*a*c^2*d+b*c^3)*f^3+3*(3*a*c*d^2+3*b*c^2*d)*e*f^2+3*(a*d^3+3*b*c*d^
2)*e^2*f+b*d^3*e^3)*x^9+1/7*(a*c^3*f^3+3*(3*a*c^2*d+b*c^3)*e*f^2+3*(3*a*c*d^2+3*b*c^2*d)*e^2*f+(a*d^3+3*b*c*d^
2)*e^3)*x^7+1/5*(3*a*c^3*e*f^2+3*(3*a*c^2*d+b*c^3)*e^2*f+(3*a*c*d^2+3*b*c^2*d)*e^3)*x^5+1/3*(3*a*c^3*e^2*f+(3*
a*c^2*d+b*c^3)*e^3)*x^3+a*c^3*e^3*x

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maxima [A]  time = 0.86, size = 326, normalized size = 1.05 \begin {gather*} \frac {1}{15} \, b d^{3} f^{3} x^{15} + \frac {1}{13} \, {\left (3 \, b d^{3} e f^{2} + {\left (3 \, b c d^{2} + a d^{3}\right )} f^{3}\right )} x^{13} + \frac {3}{11} \, {\left (b d^{3} e^{2} f + {\left (3 \, b c d^{2} + a d^{3}\right )} e f^{2} + {\left (b c^{2} d + a c d^{2}\right )} f^{3}\right )} x^{11} + \frac {1}{9} \, {\left (b d^{3} e^{3} + 3 \, {\left (3 \, b c d^{2} + a d^{3}\right )} e^{2} f + 9 \, {\left (b c^{2} d + a c d^{2}\right )} e f^{2} + {\left (b c^{3} + 3 \, a c^{2} d\right )} f^{3}\right )} x^{9} + a c^{3} e^{3} x + \frac {1}{7} \, {\left (a c^{3} f^{3} + {\left (3 \, b c d^{2} + a d^{3}\right )} e^{3} + 9 \, {\left (b c^{2} d + a c d^{2}\right )} e^{2} f + 3 \, {\left (b c^{3} + 3 \, a c^{2} d\right )} e f^{2}\right )} x^{7} + \frac {3}{5} \, {\left (a c^{3} e f^{2} + {\left (b c^{2} d + a c d^{2}\right )} e^{3} + {\left (b c^{3} + 3 \, a c^{2} d\right )} e^{2} f\right )} x^{5} + \frac {1}{3} \, {\left (3 \, a c^{3} e^{2} f + {\left (b c^{3} + 3 \, a c^{2} d\right )} e^{3}\right )} x^{3} \end {gather*}

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)^3,x, algorithm="maxima")

[Out]

1/15*b*d^3*f^3*x^15 + 1/13*(3*b*d^3*e*f^2 + (3*b*c*d^2 + a*d^3)*f^3)*x^13 + 3/11*(b*d^3*e^2*f + (3*b*c*d^2 + a
*d^3)*e*f^2 + (b*c^2*d + a*c*d^2)*f^3)*x^11 + 1/9*(b*d^3*e^3 + 3*(3*b*c*d^2 + a*d^3)*e^2*f + 9*(b*c^2*d + a*c*
d^2)*e*f^2 + (b*c^3 + 3*a*c^2*d)*f^3)*x^9 + a*c^3*e^3*x + 1/7*(a*c^3*f^3 + (3*b*c*d^2 + a*d^3)*e^3 + 9*(b*c^2*
d + a*c*d^2)*e^2*f + 3*(b*c^3 + 3*a*c^2*d)*e*f^2)*x^7 + 3/5*(a*c^3*e*f^2 + (b*c^2*d + a*c*d^2)*e^3 + (b*c^3 +
3*a*c^2*d)*e^2*f)*x^5 + 1/3*(3*a*c^3*e^2*f + (b*c^3 + 3*a*c^2*d)*e^3)*x^3

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mupad [B]  time = 0.12, size = 335, normalized size = 1.08 \begin {gather*} x^5\,\left (\frac {3\,b\,c^3\,e^2\,f}{5}+\frac {3\,a\,c^3\,e\,f^2}{5}+\frac {3\,b\,c^2\,d\,e^3}{5}+\frac {9\,a\,c^2\,d\,e^2\,f}{5}+\frac {3\,a\,c\,d^2\,e^3}{5}\right )+x^{11}\,\left (\frac {3\,b\,c^2\,d\,f^3}{11}+\frac {9\,b\,c\,d^2\,e\,f^2}{11}+\frac {3\,a\,c\,d^2\,f^3}{11}+\frac {3\,b\,d^3\,e^2\,f}{11}+\frac {3\,a\,d^3\,e\,f^2}{11}\right )+x^7\,\left (\frac {3\,b\,c^3\,e\,f^2}{7}+\frac {a\,c^3\,f^3}{7}+\frac {9\,b\,c^2\,d\,e^2\,f}{7}+\frac {9\,a\,c^2\,d\,e\,f^2}{7}+\frac {3\,b\,c\,d^2\,e^3}{7}+\frac {9\,a\,c\,d^2\,e^2\,f}{7}+\frac {a\,d^3\,e^3}{7}\right )+x^9\,\left (\frac {b\,c^3\,f^3}{9}+b\,c^2\,d\,e\,f^2+\frac {a\,c^2\,d\,f^3}{3}+b\,c\,d^2\,e^2\,f+a\,c\,d^2\,e\,f^2+\frac {b\,d^3\,e^3}{9}+\frac {a\,d^3\,e^2\,f}{3}\right )+\frac {b\,d^3\,f^3\,x^{15}}{15}+\frac {c^2\,e^2\,x^3\,\left (3\,a\,c\,f+3\,a\,d\,e+b\,c\,e\right )}{3}+\frac {d^2\,f^2\,x^{13}\,\left (a\,d\,f+3\,b\,c\,f+3\,b\,d\,e\right )}{13}+a\,c^3\,e^3\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^5*((3*a*c*d^2*e^3)/5 + (3*b*c^2*d*e^3)/5 + (3*a*c^3*e*f^2)/5 + (3*b*c^3*e^2*f)/5 + (9*a*c^2*d*e^2*f)/5) + x^
11*((3*a*c*d^2*f^3)/11 + (3*b*c^2*d*f^3)/11 + (3*a*d^3*e*f^2)/11 + (3*b*d^3*e^2*f)/11 + (9*b*c*d^2*e*f^2)/11)
+ x^7*((a*c^3*f^3)/7 + (a*d^3*e^3)/7 + (3*b*c*d^2*e^3)/7 + (3*b*c^3*e*f^2)/7 + (9*a*c*d^2*e^2*f)/7 + (9*a*c^2*
d*e*f^2)/7 + (9*b*c^2*d*e^2*f)/7) + x^9*((b*c^3*f^3)/9 + (b*d^3*e^3)/9 + (a*c^2*d*f^3)/3 + (a*d^3*e^2*f)/3 + a
*c*d^2*e*f^2 + b*c*d^2*e^2*f + b*c^2*d*e*f^2) + (b*d^3*f^3*x^15)/15 + (c^2*e^2*x^3*(3*a*c*f + 3*a*d*e + b*c*e)
)/3 + (d^2*f^2*x^13*(a*d*f + 3*b*c*f + 3*b*d*e))/13 + a*c^3*e^3*x

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sympy [A]  time = 0.13, size = 423, normalized size = 1.36 \begin {gather*} a c^{3} e^{3} x + \frac {b d^{3} f^{3} x^{15}}{15} + x^{13} \left (\frac {a d^{3} f^{3}}{13} + \frac {3 b c d^{2} f^{3}}{13} + \frac {3 b d^{3} e f^{2}}{13}\right ) + x^{11} \left (\frac {3 a c d^{2} f^{3}}{11} + \frac {3 a d^{3} e f^{2}}{11} + \frac {3 b c^{2} d f^{3}}{11} + \frac {9 b c d^{2} e f^{2}}{11} + \frac {3 b d^{3} e^{2} f}{11}\right ) + x^{9} \left (\frac {a c^{2} d f^{3}}{3} + a c d^{2} e f^{2} + \frac {a d^{3} e^{2} f}{3} + \frac {b c^{3} f^{3}}{9} + b c^{2} d e f^{2} + b c d^{2} e^{2} f + \frac {b d^{3} e^{3}}{9}\right ) + x^{7} \left (\frac {a c^{3} f^{3}}{7} + \frac {9 a c^{2} d e f^{2}}{7} + \frac {9 a c d^{2} e^{2} f}{7} + \frac {a d^{3} e^{3}}{7} + \frac {3 b c^{3} e f^{2}}{7} + \frac {9 b c^{2} d e^{2} f}{7} + \frac {3 b c d^{2} e^{3}}{7}\right ) + x^{5} \left (\frac {3 a c^{3} e f^{2}}{5} + \frac {9 a c^{2} d e^{2} f}{5} + \frac {3 a c d^{2} e^{3}}{5} + \frac {3 b c^{3} e^{2} f}{5} + \frac {3 b c^{2} d e^{3}}{5}\right ) + x^{3} \left (a c^{3} e^{2} f + a c^{2} d e^{3} + \frac {b c^{3} e^{3}}{3}\right ) \end {gather*}

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)**3,x)

[Out]

a*c**3*e**3*x + b*d**3*f**3*x**15/15 + x**13*(a*d**3*f**3/13 + 3*b*c*d**2*f**3/13 + 3*b*d**3*e*f**2/13) + x**1
1*(3*a*c*d**2*f**3/11 + 3*a*d**3*e*f**2/11 + 3*b*c**2*d*f**3/11 + 9*b*c*d**2*e*f**2/11 + 3*b*d**3*e**2*f/11) +
x**9*(a*c**2*d*f**3/3 + a*c*d**2*e*f**2 + a*d**3*e**2*f/3 + b*c**3*f**3/9 + b*c**2*d*e*f**2 + b*c*d**2*e**2*f
+ b*d**3*e**3/9) + x**7*(a*c**3*f**3/7 + 9*a*c**2*d*e*f**2/7 + 9*a*c*d**2*e**2*f/7 + a*d**3*e**3/7 + 3*b*c**3
*e*f**2/7 + 9*b*c**2*d*e**2*f/7 + 3*b*c*d**2*e**3/7) + x**5*(3*a*c**3*e*f**2/5 + 9*a*c**2*d*e**2*f/5 + 3*a*c*d
**2*e**3/5 + 3*b*c**3*e**2*f/5 + 3*b*c**2*d*e**3/5) + x**3*(a*c**3*e**2*f + a*c**2*d*e**3 + b*c**3*e**3/3)

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