\(\int x^4 (a+b x^2)^2 (c+d x^2)^2 \, dx\) [150]

Optimal result

Integrand size = 22, antiderivative size = 87 \[ \int x^4 \left (a+b x^2\right )^2 \left (c+d x^2\right )^2 \, dx=\frac {1}{5} a^2 c^2 x^5+\frac {2}{7} a c (b c+a d) x^7+\frac {1}{9} \left (b^2 c^2+4 a b c d+a^2 d^2\right ) x^9+\frac {2}{11} b d (b c+a d) x^{11}+\frac {1}{13} b^2 d^2 x^{13} \]

[Out]

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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {459} \[ \int x^4 \left (a+b x^2\right )^2 \left (c+d x^2\right )^2 \, dx=\frac {1}{9} x^9 \left (a^2 d^2+4 a b c d+b^2 c^2\right )+\frac {1}{5} a^2 c^2 x^5+\frac {2}{11} b d x^{11} (a d+b c)+\frac {2}{7} a c x^7 (a d+b c)+\frac {1}{13} b^2 d^2 x^{13} \]

[In]

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

[Out]

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

Rule 459

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.00 \[ \int x^4 \left (a+b x^2\right )^2 \left (c+d x^2\right )^2 \, dx=\frac {1}{5} a^2 c^2 x^5+\frac {2}{7} a c (b c+a d) x^7+\frac {1}{9} \left (b^2 c^2+4 a b c d+a^2 d^2\right ) x^9+\frac {2}{11} b d (b c+a d) x^{11}+\frac {1}{13} b^2 d^2 x^{13} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 2.61 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.02

method	result	size
norman	\(\frac {b^{2} d^{2} x^{13}}{13}+\left (\frac {2}{11} a b \,d^{2}+\frac {2}{11} b^{2} c d \right ) x^{11}+\left (\frac {1}{9} a^{2} d^{2}+\frac {4}{9} a b c d +\frac {1}{9} b^{2} c^{2}\right ) x^{9}+\left (\frac {2}{7} a^{2} c d +\frac {2}{7} b \,c^{2} a \right ) x^{7}+\frac {a^{2} c^{2} x^{5}}{5}\)	\(89\)
default	\(\frac {b^{2} d^{2} x^{13}}{13}+\frac {\left (2 a b \,d^{2}+2 b^{2} c d \right ) x^{11}}{11}+\frac {\left (a^{2} d^{2}+4 a b c d +b^{2} c^{2}\right ) x^{9}}{9}+\frac {\left (2 a^{2} c d +2 b \,c^{2} a \right ) x^{7}}{7}+\frac {a^{2} c^{2} x^{5}}{5}\)	\(90\)
gosper	\(\frac {1}{13} b^{2} d^{2} x^{13}+\frac {2}{11} x^{11} a b \,d^{2}+\frac {2}{11} x^{11} b^{2} c d +\frac {1}{9} x^{9} a^{2} d^{2}+\frac {4}{9} x^{9} a b c d +\frac {1}{9} x^{9} b^{2} c^{2}+\frac {2}{7} x^{7} a^{2} c d +\frac {2}{7} x^{7} b \,c^{2} a +\frac {1}{5} a^{2} c^{2} x^{5}\)	\(95\)
risch	\(\frac {1}{13} b^{2} d^{2} x^{13}+\frac {2}{11} x^{11} a b \,d^{2}+\frac {2}{11} x^{11} b^{2} c d +\frac {1}{9} x^{9} a^{2} d^{2}+\frac {4}{9} x^{9} a b c d +\frac {1}{9} x^{9} b^{2} c^{2}+\frac {2}{7} x^{7} a^{2} c d +\frac {2}{7} x^{7} b \,c^{2} a +\frac {1}{5} a^{2} c^{2} x^{5}\)	\(95\)
parallelrisch	\(\frac {1}{13} b^{2} d^{2} x^{13}+\frac {2}{11} x^{11} a b \,d^{2}+\frac {2}{11} x^{11} b^{2} c d +\frac {1}{9} x^{9} a^{2} d^{2}+\frac {4}{9} x^{9} a b c d +\frac {1}{9} x^{9} b^{2} c^{2}+\frac {2}{7} x^{7} a^{2} c d +\frac {2}{7} x^{7} b \,c^{2} a +\frac {1}{5} a^{2} c^{2} x^{5}\)	\(95\)

[In]

int(x^4*(b*x^2+a)^2*(d*x^2+c)^2,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.98 \[ \int x^4 \left (a+b x^2\right )^2 \left (c+d x^2\right )^2 \, dx=\frac {1}{13} \, b^{2} d^{2} x^{13} + \frac {2}{11} \, {\left (b^{2} c d + a b d^{2}\right )} x^{11} + \frac {1}{9} \, {\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{9} + \frac {1}{5} \, a^{2} c^{2} x^{5} + \frac {2}{7} \, {\left (a b c^{2} + a^{2} c d\right )} x^{7} \]

[In]

integrate(x^4*(b*x^2+a)^2*(d*x^2+c)^2,x, algorithm="fricas")

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.15 \[ \int x^4 \left (a+b x^2\right )^2 \left (c+d x^2\right )^2 \, dx=\frac {a^{2} c^{2} x^{5}}{5} + \frac {b^{2} d^{2} x^{13}}{13} + x^{11} \cdot \left (\frac {2 a b d^{2}}{11} + \frac {2 b^{2} c d}{11}\right ) + x^{9} \left (\frac {a^{2} d^{2}}{9} + \frac {4 a b c d}{9} + \frac {b^{2} c^{2}}{9}\right ) + x^{7} \cdot \left (\frac {2 a^{2} c d}{7} + \frac {2 a b c^{2}}{7}\right ) \]

[In]

integrate(x**4*(b*x**2+a)**2*(d*x**2+c)**2,x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.98 \[ \int x^4 \left (a+b x^2\right )^2 \left (c+d x^2\right )^2 \, dx=\frac {1}{13} \, b^{2} d^{2} x^{13} + \frac {2}{11} \, {\left (b^{2} c d + a b d^{2}\right )} x^{11} + \frac {1}{9} \, {\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{9} + \frac {1}{5} \, a^{2} c^{2} x^{5} + \frac {2}{7} \, {\left (a b c^{2} + a^{2} c d\right )} x^{7} \]

[In]

integrate(x^4*(b*x^2+a)^2*(d*x^2+c)^2,x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.08 \[ \int x^4 \left (a+b x^2\right )^2 \left (c+d x^2\right )^2 \, dx=\frac {1}{13} \, b^{2} d^{2} x^{13} + \frac {2}{11} \, b^{2} c d x^{11} + \frac {2}{11} \, a b d^{2} x^{11} + \frac {1}{9} \, b^{2} c^{2} x^{9} + \frac {4}{9} \, a b c d x^{9} + \frac {1}{9} \, a^{2} d^{2} x^{9} + \frac {2}{7} \, a b c^{2} x^{7} + \frac {2}{7} \, a^{2} c d x^{7} + \frac {1}{5} \, a^{2} c^{2} x^{5} \]

[In]

integrate(x^4*(b*x^2+a)^2*(d*x^2+c)^2,x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.90 \[ \int x^4 \left (a+b x^2\right )^2 \left (c+d x^2\right )^2 \, dx=x^9\,\left (\frac {a^2\,d^2}{9}+\frac {4\,a\,b\,c\,d}{9}+\frac {b^2\,c^2}{9}\right )+\frac {a^2\,c^2\,x^5}{5}+\frac {b^2\,d^2\,x^{13}}{13}+\frac {2\,a\,c\,x^7\,\left (a\,d+b\,c\right )}{7}+\frac {2\,b\,d\,x^{11}\,\left (a\,d+b\,c\right )}{11} \]

[In]

int(x^4*(a + b*x^2)^2*(c + d*x^2)^2,x)

[Out]

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