3.425 \(\int x^2 (a^2+2 a b x^2+b^2 x^4)^2 \, dx\)

Optimal. Leaf size=56 \[ \frac{6}{7} a^2 b^2 x^7+\frac{4}{5} a^3 b x^5+\frac{a^4 x^3}{3}+\frac{4}{9} a b^3 x^9+\frac{b^4 x^{11}}{11} \]

[Out]

(a^4*x^3)/3 + (4*a^3*b*x^5)/5 + (6*a^2*b^2*x^7)/7 + (4*a*b^3*x^9)/9 + (b^4*x^11)/11

________________________________________________________________________________________

Rubi [A]  time = 0.0288531, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {28, 270} \[ \frac{6}{7} a^2 b^2 x^7+\frac{4}{5} a^3 b x^5+\frac{a^4 x^3}{3}+\frac{4}{9} a b^3 x^9+\frac{b^4 x^{11}}{11} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^4*x^3)/3 + (4*a^3*b*x^5)/5 + (6*a^2*b^2*x^7)/7 + (4*a*b^3*x^9)/9 + (b^4*x^11)/11

Rule 28

Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/c^p, Int[u*(b/2 + c*x^n)^(2*
p), x], x] /; FreeQ[{a, b, c, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rubi steps

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

Mathematica [A]  time = 0.0021344, size = 56, normalized size = 1. \[ \frac{6}{7} a^2 b^2 x^7+\frac{4}{5} a^3 b x^5+\frac{a^4 x^3}{3}+\frac{4}{9} a b^3 x^9+\frac{b^4 x^{11}}{11} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^4*x^3)/3 + (4*a^3*b*x^5)/5 + (6*a^2*b^2*x^7)/7 + (4*a*b^3*x^9)/9 + (b^4*x^11)/11

________________________________________________________________________________________

Maple [A]  time = 0.04, size = 47, normalized size = 0.8 \begin{align*}{\frac{{a}^{4}{x}^{3}}{3}}+{\frac{4\,{a}^{3}b{x}^{5}}{5}}+{\frac{6\,{a}^{2}{b}^{2}{x}^{7}}{7}}+{\frac{4\,a{b}^{3}{x}^{9}}{9}}+{\frac{{b}^{4}{x}^{11}}{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*a^4*x^3+4/5*a^3*b*x^5+6/7*a^2*b^2*x^7+4/9*a*b^3*x^9+1/11*b^4*x^11

________________________________________________________________________________________

Maxima [A]  time = 0.989445, size = 62, normalized size = 1.11 \begin{align*} \frac{1}{11} \, b^{4} x^{11} + \frac{4}{9} \, a b^{3} x^{9} + \frac{6}{7} \, a^{2} b^{2} x^{7} + \frac{4}{5} \, a^{3} b x^{5} + \frac{1}{3} \, a^{4} x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/11*b^4*x^11 + 4/9*a*b^3*x^9 + 6/7*a^2*b^2*x^7 + 4/5*a^3*b*x^5 + 1/3*a^4*x^3

________________________________________________________________________________________

Fricas [A]  time = 1.51636, size = 107, normalized size = 1.91 \begin{align*} \frac{1}{11} x^{11} b^{4} + \frac{4}{9} x^{9} b^{3} a + \frac{6}{7} x^{7} b^{2} a^{2} + \frac{4}{5} x^{5} b a^{3} + \frac{1}{3} x^{3} a^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/11*x^11*b^4 + 4/9*x^9*b^3*a + 6/7*x^7*b^2*a^2 + 4/5*x^5*b*a^3 + 1/3*x^3*a^4

________________________________________________________________________________________

Sympy [A]  time = 0.070825, size = 53, normalized size = 0.95 \begin{align*} \frac{a^{4} x^{3}}{3} + \frac{4 a^{3} b x^{5}}{5} + \frac{6 a^{2} b^{2} x^{7}}{7} + \frac{4 a b^{3} x^{9}}{9} + \frac{b^{4} x^{11}}{11} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**4*x**3/3 + 4*a**3*b*x**5/5 + 6*a**2*b**2*x**7/7 + 4*a*b**3*x**9/9 + b**4*x**11/11

________________________________________________________________________________________

Giac [A]  time = 1.12835, size = 62, normalized size = 1.11 \begin{align*} \frac{1}{11} \, b^{4} x^{11} + \frac{4}{9} \, a b^{3} x^{9} + \frac{6}{7} \, a^{2} b^{2} x^{7} + \frac{4}{5} \, a^{3} b x^{5} + \frac{1}{3} \, a^{4} x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/11*b^4*x^11 + 4/9*a*b^3*x^9 + 6/7*a^2*b^2*x^7 + 4/5*a^3*b*x^5 + 1/3*a^4*x^3