3.152 \(\int x^2 (a+b x^2)^2 (c+d x^2)^2 \, dx\)

Optimal. Leaf size=87 \[ \frac {1}{7} x^7 \left (a^2 d^2+4 a b c d+b^2 c^2\right )+\frac {1}{3} a^2 c^2 x^3+\frac {2}{9} b d x^9 (a d+b c)+\frac {2}{5} a c x^5 (a d+b c)+\frac {1}{11} b^2 d^2 x^{11} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.05, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {448} \[ \frac {1}{7} x^7 \left (a^2 d^2+4 a b c d+b^2 c^2\right )+\frac {1}{3} a^2 c^2 x^3+\frac {2}{9} b d x^9 (a d+b c)+\frac {2}{5} a c x^5 (a d+b c)+\frac {1}{11} b^2 d^2 x^{11} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 448

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Int[ExpandI
ntegrand[(e*x)^m*(a + b*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] &
& IGtQ[p, 0] && IGtQ[q, 0]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.02, size = 87, normalized size = 1.00 \[ \frac {1}{7} x^7 \left (a^2 d^2+4 a b c d+b^2 c^2\right )+\frac {1}{3} a^2 c^2 x^3+\frac {2}{9} b d x^9 (a d+b c)+\frac {2}{5} a c x^5 (a d+b c)+\frac {1}{11} b^2 d^2 x^{11} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.37, size = 94, normalized size = 1.08 \[ \frac {1}{11} x^{11} d^{2} b^{2} + \frac {2}{9} x^{9} d c b^{2} + \frac {2}{9} x^{9} d^{2} b a + \frac {1}{7} x^{7} c^{2} b^{2} + \frac {4}{7} x^{7} d c b a + \frac {1}{7} x^{7} d^{2} a^{2} + \frac {2}{5} x^{5} c^{2} b a + \frac {2}{5} x^{5} d c a^{2} + \frac {1}{3} x^{3} c^{2} a^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

giac [A]  time = 0.26, size = 94, normalized size = 1.08 \[ \frac {1}{11} \, b^{2} d^{2} x^{11} + \frac {2}{9} \, b^{2} c d x^{9} + \frac {2}{9} \, a b d^{2} x^{9} + \frac {1}{7} \, b^{2} c^{2} x^{7} + \frac {4}{7} \, a b c d x^{7} + \frac {1}{7} \, a^{2} d^{2} x^{7} + \frac {2}{5} \, a b c^{2} x^{5} + \frac {2}{5} \, a^{2} c d x^{5} + \frac {1}{3} \, a^{2} c^{2} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.00, size = 90, normalized size = 1.03 \[ \frac {b^{2} d^{2} x^{11}}{11}+\frac {\left (2 a b \,d^{2}+2 b^{2} c d \right ) x^{9}}{9}+\frac {a^{2} c^{2} x^{3}}{3}+\frac {\left (a^{2} d^{2}+4 a b c d +b^{2} c^{2}\right ) x^{7}}{7}+\frac {\left (2 a^{2} c d +2 a b \,c^{2}\right ) x^{5}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 1.07, size = 85, normalized size = 0.98 \[ \frac {1}{11} \, b^{2} d^{2} x^{11} + \frac {2}{9} \, {\left (b^{2} c d + a b d^{2}\right )} x^{9} + \frac {1}{7} \, {\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{7} + \frac {1}{3} \, a^{2} c^{2} x^{3} + \frac {2}{5} \, {\left (a b c^{2} + a^{2} c d\right )} x^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(b*x^2+a)^2*(d*x^2+c)^2,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 0.03, size = 78, normalized size = 0.90 \[ x^7\,\left (\frac {a^2\,d^2}{7}+\frac {4\,a\,b\,c\,d}{7}+\frac {b^2\,c^2}{7}\right )+\frac {a^2\,c^2\,x^3}{3}+\frac {b^2\,d^2\,x^{11}}{11}+\frac {2\,a\,c\,x^5\,\left (a\,d+b\,c\right )}{5}+\frac {2\,b\,d\,x^9\,\left (a\,d+b\,c\right )}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 0.08, size = 100, normalized size = 1.15 \[ \frac {a^{2} c^{2} x^{3}}{3} + \frac {b^{2} d^{2} x^{11}}{11} + x^{9} \left (\frac {2 a b d^{2}}{9} + \frac {2 b^{2} c d}{9}\right ) + x^{7} \left (\frac {a^{2} d^{2}}{7} + \frac {4 a b c d}{7} + \frac {b^{2} c^{2}}{7}\right ) + x^{5} \left (\frac {2 a^{2} c d}{5} + \frac {2 a b c^{2}}{5}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________