\(\int x^4 (a+b x^2)^2 (c+d x^2) \, dx\) [141]

Optimal result

Integrand size = 20, antiderivative size = 55 \[ \int x^4 \left (a+b x^2\right )^2 \left (c+d x^2\right ) \, dx=\frac {1}{5} a^2 c x^5+\frac {1}{7} a (2 b c+a d) x^7+\frac {1}{9} b (b c+2 a d) x^9+\frac {1}{11} b^2 d x^{11} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {459} \[ \int x^4 \left (a+b x^2\right )^2 \left (c+d x^2\right ) \, dx=\frac {1}{5} a^2 c x^5+\frac {1}{9} b x^9 (2 a d+b c)+\frac {1}{7} a x^7 (a d+2 b c)+\frac {1}{11} b^2 d x^{11} \]

[In]

Int[x^4*(a + b*x^2)^2*(c + d*x^2),x]

[Out]

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

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00 \[ \int x^4 \left (a+b x^2\right )^2 \left (c+d x^2\right ) \, dx=\frac {1}{5} a^2 c x^5+\frac {1}{7} a (2 b c+a d) x^7+\frac {1}{9} b (b c+2 a d) x^9+\frac {1}{11} b^2 d x^{11} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 2.61 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.95

method	result	size
default	\(\frac {b^{2} d \,x^{11}}{11}+\frac {\left (2 a b d +b^{2} c \right ) x^{9}}{9}+\frac {\left (a^{2} d +2 a b c \right ) x^{7}}{7}+\frac {a^{2} c \,x^{5}}{5}\)	\(52\)
norman	\(\frac {b^{2} d \,x^{11}}{11}+\left (\frac {2}{9} a b d +\frac {1}{9} b^{2} c \right ) x^{9}+\left (\frac {1}{7} a^{2} d +\frac {2}{7} a b c \right ) x^{7}+\frac {a^{2} c \,x^{5}}{5}\)	\(52\)
gosper	\(\frac {1}{11} b^{2} d \,x^{11}+\frac {2}{9} x^{9} a b d +\frac {1}{9} x^{9} b^{2} c +\frac {1}{7} x^{7} a^{2} d +\frac {2}{7} x^{7} a b c +\frac {1}{5} a^{2} c \,x^{5}\)	\(54\)
risch	\(\frac {1}{11} b^{2} d \,x^{11}+\frac {2}{9} x^{9} a b d +\frac {1}{9} x^{9} b^{2} c +\frac {1}{7} x^{7} a^{2} d +\frac {2}{7} x^{7} a b c +\frac {1}{5} a^{2} c \,x^{5}\)	\(54\)
parallelrisch	\(\frac {1}{11} b^{2} d \,x^{11}+\frac {2}{9} x^{9} a b d +\frac {1}{9} x^{9} b^{2} c +\frac {1}{7} x^{7} a^{2} d +\frac {2}{7} x^{7} a b c +\frac {1}{5} a^{2} c \,x^{5}\)	\(54\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.93 \[ \int x^4 \left (a+b x^2\right )^2 \left (c+d x^2\right ) \, dx=\frac {1}{11} \, b^{2} d x^{11} + \frac {1}{9} \, {\left (b^{2} c + 2 \, a b d\right )} x^{9} + \frac {1}{5} \, a^{2} c x^{5} + \frac {1}{7} \, {\left (2 \, a b c + a^{2} d\right )} x^{7} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.02 \[ \int x^4 \left (a+b x^2\right )^2 \left (c+d x^2\right ) \, dx=\frac {a^{2} c x^{5}}{5} + \frac {b^{2} d x^{11}}{11} + x^{9} \cdot \left (\frac {2 a b d}{9} + \frac {b^{2} c}{9}\right ) + x^{7} \left (\frac {a^{2} d}{7} + \frac {2 a b c}{7}\right ) \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.93 \[ \int x^4 \left (a+b x^2\right )^2 \left (c+d x^2\right ) \, dx=\frac {1}{11} \, b^{2} d x^{11} + \frac {1}{9} \, {\left (b^{2} c + 2 \, a b d\right )} x^{9} + \frac {1}{5} \, a^{2} c x^{5} + \frac {1}{7} \, {\left (2 \, a b c + a^{2} d\right )} x^{7} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.96 \[ \int x^4 \left (a+b x^2\right )^2 \left (c+d x^2\right ) \, dx=\frac {1}{11} \, b^{2} d x^{11} + \frac {1}{9} \, b^{2} c x^{9} + \frac {2}{9} \, a b d x^{9} + \frac {2}{7} \, a b c x^{7} + \frac {1}{7} \, a^{2} d x^{7} + \frac {1}{5} \, a^{2} c x^{5} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 5.18 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.93 \[ \int x^4 \left (a+b x^2\right )^2 \left (c+d x^2\right ) \, dx=x^7\,\left (\frac {d\,a^2}{7}+\frac {2\,b\,c\,a}{7}\right )+x^9\,\left (\frac {c\,b^2}{9}+\frac {2\,a\,d\,b}{9}\right )+\frac {a^2\,c\,x^5}{5}+\frac {b^2\,d\,x^{11}}{11} \]

[In]

int(x^4*(a + b*x^2)^2*(c + d*x^2),x)

[Out]

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