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

Optimal. Leaf size=51 \[ \frac{6}{5} a^2 b^2 x^5+\frac{4}{3} a^3 b x^3+a^4 x+\frac{4}{7} a b^3 x^7+\frac{b^4 x^9}{9} \]

[Out]

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

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Rubi [A]  time = 0.0238017, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {28, 194} \[ \frac{6}{5} a^2 b^2 x^5+\frac{4}{3} a^3 b x^3+a^4 x+\frac{4}{7} a b^3 x^7+\frac{b^4 x^9}{9} \]

Antiderivative was successfully verified.

[In]

Int[(a^2 + 2*a*b*x^2 + b^2*x^4)^2,x]

[Out]

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

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 194

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

Rubi steps

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

Mathematica [A]  time = 0.0012044, size = 51, normalized size = 1. \[ \frac{6}{5} a^2 b^2 x^5+\frac{4}{3} a^3 b x^3+a^4 x+\frac{4}{7} a b^3 x^7+\frac{b^4 x^9}{9} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.04, size = 44, normalized size = 0.9 \begin{align*}{a}^{4}x+{\frac{4\,{a}^{3}b{x}^{3}}{3}}+{\frac{6\,{a}^{2}{b}^{2}{x}^{5}}{5}}+{\frac{4\,a{b}^{3}{x}^{7}}{7}}+{\frac{{b}^{4}{x}^{9}}{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.976963, size = 74, normalized size = 1.45 \begin{align*} \frac{1}{9} \, b^{4} x^{9} + \frac{4}{7} \, a b^{3} x^{7} + \frac{4}{5} \, a^{2} b^{2} x^{5} + a^{4} x + \frac{2}{15} \,{\left (3 \, b^{2} x^{5} + 10 \, a b x^{3}\right )} a^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*b^4*x^9 + 4/7*a*b^3*x^7 + 4/5*a^2*b^2*x^5 + a^4*x + 2/15*(3*b^2*x^5 + 10*a*b*x^3)*a^2

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Fricas [A]  time = 1.4885, size = 96, normalized size = 1.88 \begin{align*} \frac{1}{9} x^{9} b^{4} + \frac{4}{7} x^{7} b^{3} a + \frac{6}{5} x^{5} b^{2} a^{2} + \frac{4}{3} x^{3} b a^{3} + x a^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.068264, size = 49, normalized size = 0.96 \begin{align*} a^{4} x + \frac{4 a^{3} b x^{3}}{3} + \frac{6 a^{2} b^{2} x^{5}}{5} + \frac{4 a b^{3} x^{7}}{7} + \frac{b^{4} x^{9}}{9} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.16168, size = 58, normalized size = 1.14 \begin{align*} \frac{1}{9} \, b^{4} x^{9} + \frac{4}{7} \, a b^{3} x^{7} + \frac{6}{5} \, a^{2} b^{2} x^{5} + \frac{4}{3} \, a^{3} b x^{3} + a^{4} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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