\(\int x^3 (a+b x^2)^2 (c+d x^2)^2 \, dx\) [151]

Optimal result

Integrand size = 22, antiderivative size = 87 \[ \int x^3 \left (a+b x^2\right )^2 \left (c+d x^2\right )^2 \, dx=\frac {1}{4} a^2 c^2 x^4+\frac {1}{3} a c (b c+a d) x^6+\frac {1}{8} \left (b^2 c^2+4 a b c d+a^2 d^2\right ) x^8+\frac {1}{5} b d (b c+a d) x^{10}+\frac {1}{12} b^2 d^2 x^{12} \]

[Out]

1/4*a^2*c^2*x^4+1/3*a*c*(a*d+b*c)*x^6+1/8*(a^2*d^2+4*a*b*c*d+b^2*c^2)*x^8+1/5*b*d*(a*d+b*c)*x^10+1/12*b^2*d^2*
x^12

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {457, 78} \[ \int x^3 \left (a+b x^2\right )^2 \left (c+d x^2\right )^2 \, dx=\frac {1}{8} x^8 \left (a^2 d^2+4 a b c d+b^2 c^2\right )+\frac {1}{4} a^2 c^2 x^4+\frac {1}{5} b d x^{10} (a d+b c)+\frac {1}{3} a c x^6 (a d+b c)+\frac {1}{12} b^2 d^2 x^{12} \]

[In]

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

[Out]

(a^2*c^2*x^4)/4 + (a*c*(b*c + a*d)*x^6)/3 + ((b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^8)/8 + (b*d*(b*c + a*d)*x^10)/5
+ (b^2*d^2*x^12)/12

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rule 457

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int x (a+b x)^2 (c+d x)^2 \, dx,x,x^2\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (a^2 c^2 x+2 a c (b c+a d) x^2+\left (b^2 c^2+4 a b c d+a^2 d^2\right ) x^3+2 b d (b c+a d) x^4+b^2 d^2 x^5\right ) \, dx,x,x^2\right ) \\ & = \frac {1}{4} a^2 c^2 x^4+\frac {1}{3} a c (b c+a d) x^6+\frac {1}{8} \left (b^2 c^2+4 a b c d+a^2 d^2\right ) x^8+\frac {1}{5} b d (b c+a d) x^{10}+\frac {1}{12} b^2 d^2 x^{12} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.93 \[ \int x^3 \left (a+b x^2\right )^2 \left (c+d x^2\right )^2 \, dx=\frac {1}{120} x^4 \left (30 a^2 c^2+40 a c (b c+a d) x^2+15 \left (b^2 c^2+4 a b c d+a^2 d^2\right ) x^4+24 b d (b c+a d) x^6+10 b^2 d^2 x^8\right ) \]

[In]

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

[Out]

(x^4*(30*a^2*c^2 + 40*a*c*(b*c + a*d)*x^2 + 15*(b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^4 + 24*b*d*(b*c + a*d)*x^6 +
10*b^2*d^2*x^8))/120

Maple [A] (verified)

Time = 2.58 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.02

method	result	size
norman	\(\frac {b^{2} d^{2} x^{12}}{12}+\left (\frac {1}{5} a b \,d^{2}+\frac {1}{5} b^{2} c d \right ) x^{10}+\left (\frac {1}{8} a^{2} d^{2}+\frac {1}{2} a b c d +\frac {1}{8} b^{2} c^{2}\right ) x^{8}+\left (\frac {1}{3} a^{2} c d +\frac {1}{3} b \,c^{2} a \right ) x^{6}+\frac {a^{2} c^{2} x^{4}}{4}\)	\(89\)
default	\(\frac {b^{2} d^{2} x^{12}}{12}+\frac {\left (2 a b \,d^{2}+2 b^{2} c d \right ) x^{10}}{10}+\frac {\left (a^{2} d^{2}+4 a b c d +b^{2} c^{2}\right ) x^{8}}{8}+\frac {\left (2 a^{2} c d +2 b \,c^{2} a \right ) x^{6}}{6}+\frac {a^{2} c^{2} x^{4}}{4}\)	\(90\)
gosper	\(\frac {1}{12} b^{2} d^{2} x^{12}+\frac {1}{5} x^{10} a b \,d^{2}+\frac {1}{5} x^{10} b^{2} c d +\frac {1}{8} x^{8} a^{2} d^{2}+\frac {1}{2} x^{8} a b c d +\frac {1}{8} x^{8} b^{2} c^{2}+\frac {1}{3} x^{6} a^{2} c d +\frac {1}{3} x^{6} b \,c^{2} a +\frac {1}{4} a^{2} c^{2} x^{4}\)	\(95\)
risch	\(\frac {1}{12} b^{2} d^{2} x^{12}+\frac {1}{5} x^{10} a b \,d^{2}+\frac {1}{5} x^{10} b^{2} c d +\frac {1}{8} x^{8} a^{2} d^{2}+\frac {1}{2} x^{8} a b c d +\frac {1}{8} x^{8} b^{2} c^{2}+\frac {1}{3} x^{6} a^{2} c d +\frac {1}{3} x^{6} b \,c^{2} a +\frac {1}{4} a^{2} c^{2} x^{4}\)	\(95\)
parallelrisch	\(\frac {1}{12} b^{2} d^{2} x^{12}+\frac {1}{5} x^{10} a b \,d^{2}+\frac {1}{5} x^{10} b^{2} c d +\frac {1}{8} x^{8} a^{2} d^{2}+\frac {1}{2} x^{8} a b c d +\frac {1}{8} x^{8} b^{2} c^{2}+\frac {1}{3} x^{6} a^{2} c d +\frac {1}{3} x^{6} b \,c^{2} a +\frac {1}{4} a^{2} c^{2} x^{4}\)	\(95\)

[In]

int(x^3*(b*x^2+a)^2*(d*x^2+c)^2,x,method=_RETURNVERBOSE)

[Out]

1/12*b^2*d^2*x^12+(1/5*a*b*d^2+1/5*b^2*c*d)*x^10+(1/8*a^2*d^2+1/2*a*b*c*d+1/8*b^2*c^2)*x^8+(1/3*a^2*c*d+1/3*b*
c^2*a)*x^6+1/4*a^2*c^2*x^4

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.98 \[ \int x^3 \left (a+b x^2\right )^2 \left (c+d x^2\right )^2 \, dx=\frac {1}{12} \, b^{2} d^{2} x^{12} + \frac {1}{5} \, {\left (b^{2} c d + a b d^{2}\right )} x^{10} + \frac {1}{8} \, {\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{8} + \frac {1}{4} \, a^{2} c^{2} x^{4} + \frac {1}{3} \, {\left (a b c^{2} + a^{2} c d\right )} x^{6} \]

[In]

integrate(x^3*(b*x^2+a)^2*(d*x^2+c)^2,x, algorithm="fricas")

[Out]

1/12*b^2*d^2*x^12 + 1/5*(b^2*c*d + a*b*d^2)*x^10 + 1/8*(b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^8 + 1/4*a^2*c^2*x^4 +
1/3*(a*b*c^2 + a^2*c*d)*x^6

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.06 \[ \int x^3 \left (a+b x^2\right )^2 \left (c+d x^2\right )^2 \, dx=\frac {a^{2} c^{2} x^{4}}{4} + \frac {b^{2} d^{2} x^{12}}{12} + x^{10} \left (\frac {a b d^{2}}{5} + \frac {b^{2} c d}{5}\right ) + x^{8} \left (\frac {a^{2} d^{2}}{8} + \frac {a b c d}{2} + \frac {b^{2} c^{2}}{8}\right ) + x^{6} \left (\frac {a^{2} c d}{3} + \frac {a b c^{2}}{3}\right ) \]

[In]

integrate(x**3*(b*x**2+a)**2*(d*x**2+c)**2,x)

[Out]

a**2*c**2*x**4/4 + b**2*d**2*x**12/12 + x**10*(a*b*d**2/5 + b**2*c*d/5) + x**8*(a**2*d**2/8 + a*b*c*d/2 + b**2
*c**2/8) + x**6*(a**2*c*d/3 + a*b*c**2/3)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.98 \[ \int x^3 \left (a+b x^2\right )^2 \left (c+d x^2\right )^2 \, dx=\frac {1}{12} \, b^{2} d^{2} x^{12} + \frac {1}{5} \, {\left (b^{2} c d + a b d^{2}\right )} x^{10} + \frac {1}{8} \, {\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{8} + \frac {1}{4} \, a^{2} c^{2} x^{4} + \frac {1}{3} \, {\left (a b c^{2} + a^{2} c d\right )} x^{6} \]

[In]

integrate(x^3*(b*x^2+a)^2*(d*x^2+c)^2,x, algorithm="maxima")

[Out]

1/12*b^2*d^2*x^12 + 1/5*(b^2*c*d + a*b*d^2)*x^10 + 1/8*(b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^8 + 1/4*a^2*c^2*x^4 +
1/3*(a*b*c^2 + a^2*c*d)*x^6

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.08 \[ \int x^3 \left (a+b x^2\right )^2 \left (c+d x^2\right )^2 \, dx=\frac {1}{12} \, b^{2} d^{2} x^{12} + \frac {1}{5} \, b^{2} c d x^{10} + \frac {1}{5} \, a b d^{2} x^{10} + \frac {1}{8} \, b^{2} c^{2} x^{8} + \frac {1}{2} \, a b c d x^{8} + \frac {1}{8} \, a^{2} d^{2} x^{8} + \frac {1}{3} \, a b c^{2} x^{6} + \frac {1}{3} \, a^{2} c d x^{6} + \frac {1}{4} \, a^{2} c^{2} x^{4} \]

[In]

integrate(x^3*(b*x^2+a)^2*(d*x^2+c)^2,x, algorithm="giac")

[Out]

1/12*b^2*d^2*x^12 + 1/5*b^2*c*d*x^10 + 1/5*a*b*d^2*x^10 + 1/8*b^2*c^2*x^8 + 1/2*a*b*c*d*x^8 + 1/8*a^2*d^2*x^8
+ 1/3*a*b*c^2*x^6 + 1/3*a^2*c*d*x^6 + 1/4*a^2*c^2*x^4

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.90 \[ \int x^3 \left (a+b x^2\right )^2 \left (c+d x^2\right )^2 \, dx=x^8\,\left (\frac {a^2\,d^2}{8}+\frac {a\,b\,c\,d}{2}+\frac {b^2\,c^2}{8}\right )+\frac {a^2\,c^2\,x^4}{4}+\frac {b^2\,d^2\,x^{12}}{12}+\frac {a\,c\,x^6\,\left (a\,d+b\,c\right )}{3}+\frac {b\,d\,x^{10}\,\left (a\,d+b\,c\right )}{5} \]

[In]

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

[Out]

x^8*((a^2*d^2)/8 + (b^2*c^2)/8 + (a*b*c*d)/2) + (a^2*c^2*x^4)/4 + (b^2*d^2*x^12)/12 + (a*c*x^6*(a*d + b*c))/3
+ (b*d*x^10*(a*d + b*c))/5