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

Optimal. Leaf size=56 \[ \frac{2}{3} a^2 b^2 x^9+\frac{4}{7} a^3 b x^7+\frac{a^4 x^5}{5}+\frac{4}{11} a b^3 x^{11}+\frac{b^4 x^{13}}{13} \]

[Out]

(a^4*x^5)/5 + (4*a^3*b*x^7)/7 + (2*a^2*b^2*x^9)/3 + (4*a*b^3*x^11)/11 + (b^4*x^13)/13

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Rubi [A]  time = 0.0272856, 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} \[ \frac{2}{3} a^2 b^2 x^9+\frac{4}{7} a^3 b x^7+\frac{a^4 x^5}{5}+\frac{4}{11} a b^3 x^{11}+\frac{b^4 x^{13}}{13} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^4*x^5)/5 + (4*a^3*b*x^7)/7 + (2*a^2*b^2*x^9)/3 + (4*a*b^3*x^11)/11 + (b^4*x^13)/13

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^4 \left (a^2+2 a b x^2+b^2 x^4\right )^2 \, dx &=\frac{\int x^4 \left (a b+b^2 x^2\right )^4 \, dx}{b^4}\\ &=\frac{\int \left (a^4 b^4 x^4+4 a^3 b^5 x^6+6 a^2 b^6 x^8+4 a b^7 x^{10}+b^8 x^{12}\right ) \, dx}{b^4}\\ &=\frac{a^4 x^5}{5}+\frac{4}{7} a^3 b x^7+\frac{2}{3} a^2 b^2 x^9+\frac{4}{11} a b^3 x^{11}+\frac{b^4 x^{13}}{13}\\ \end{align*}

Mathematica [A]  time = 0.00262, size = 56, normalized size = 1. \[ \frac{2}{3} a^2 b^2 x^9+\frac{4}{7} a^3 b x^7+\frac{a^4 x^5}{5}+\frac{4}{11} a b^3 x^{11}+\frac{b^4 x^{13}}{13} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^4*x^5)/5 + (4*a^3*b*x^7)/7 + (2*a^2*b^2*x^9)/3 + (4*a*b^3*x^11)/11 + (b^4*x^13)/13

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Maple [A]  time = 0.043, size = 47, normalized size = 0.8 \begin{align*}{\frac{{a}^{4}{x}^{5}}{5}}+{\frac{4\,{a}^{3}b{x}^{7}}{7}}+{\frac{2\,{a}^{2}{b}^{2}{x}^{9}}{3}}+{\frac{4\,a{b}^{3}{x}^{11}}{11}}+{\frac{{b}^{4}{x}^{13}}{13}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*a^4*x^5+4/7*a^3*b*x^7+2/3*a^2*b^2*x^9+4/11*a*b^3*x^11+1/13*b^4*x^13

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Maxima [A]  time = 0.992755, size = 62, normalized size = 1.11 \begin{align*} \frac{1}{13} \, b^{4} x^{13} + \frac{4}{11} \, a b^{3} x^{11} + \frac{2}{3} \, a^{2} b^{2} x^{9} + \frac{4}{7} \, a^{3} b x^{7} + \frac{1}{5} \, a^{4} x^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/13*b^4*x^13 + 4/11*a*b^3*x^11 + 2/3*a^2*b^2*x^9 + 4/7*a^3*b*x^7 + 1/5*a^4*x^5

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Fricas [A]  time = 1.50247, size = 109, normalized size = 1.95 \begin{align*} \frac{1}{13} x^{13} b^{4} + \frac{4}{11} x^{11} b^{3} a + \frac{2}{3} x^{9} b^{2} a^{2} + \frac{4}{7} x^{7} b a^{3} + \frac{1}{5} x^{5} a^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/13*x^13*b^4 + 4/11*x^11*b^3*a + 2/3*x^9*b^2*a^2 + 4/7*x^7*b*a^3 + 1/5*x^5*a^4

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Sympy [A]  time = 0.071213, size = 53, normalized size = 0.95 \begin{align*} \frac{a^{4} x^{5}}{5} + \frac{4 a^{3} b x^{7}}{7} + \frac{2 a^{2} b^{2} x^{9}}{3} + \frac{4 a b^{3} x^{11}}{11} + \frac{b^{4} x^{13}}{13} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**4*x**5/5 + 4*a**3*b*x**7/7 + 2*a**2*b**2*x**9/3 + 4*a*b**3*x**11/11 + b**4*x**13/13

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Giac [A]  time = 1.12688, size = 62, normalized size = 1.11 \begin{align*} \frac{1}{13} \, b^{4} x^{13} + \frac{4}{11} \, a b^{3} x^{11} + \frac{2}{3} \, a^{2} b^{2} x^{9} + \frac{4}{7} \, a^{3} b x^{7} + \frac{1}{5} \, a^{4} x^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/13*b^4*x^13 + 4/11*a*b^3*x^11 + 2/3*a^2*b^2*x^9 + 4/7*a^3*b*x^7 + 1/5*a^4*x^5