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

Optimal. Leaf size=16 \[ \frac{\left (a+b x^2\right )^5}{10 b} \]

[Out]

(a + b*x^2)^5/(10*b)

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Rubi [A]  time = 0.0048408, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {28, 261} \[ \frac{\left (a+b x^2\right )^5}{10 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a + b*x^2)^5/(10*b)

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 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rubi steps

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

Mathematica [A]  time = 0.0022879, size = 16, normalized size = 1. \[ \frac{\left (a+b x^2\right )^5}{10 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a + b*x^2)^5/(10*b)

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Maple [B]  time = 0.039, size = 45, normalized size = 2.8 \begin{align*}{\frac{{b}^{4}{x}^{10}}{10}}+{\frac{a{b}^{3}{x}^{8}}{2}}+{a}^{2}{b}^{2}{x}^{6}+{a}^{3}b{x}^{4}+{\frac{{a}^{4}{x}^{2}}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*b^4*x^10+1/2*a*b^3*x^8+a^2*b^2*x^6+a^3*b*x^4+1/2*a^4*x^2

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Maxima [B]  time = 0.982767, size = 59, normalized size = 3.69 \begin{align*} \frac{1}{10} \, b^{4} x^{10} + \frac{1}{2} \, a b^{3} x^{8} + a^{2} b^{2} x^{6} + a^{3} b x^{4} + \frac{1}{2} \, a^{4} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*b^4*x^10 + 1/2*a*b^3*x^8 + a^2*b^2*x^6 + a^3*b*x^4 + 1/2*a^4*x^2

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Fricas [B]  time = 1.466, size = 96, normalized size = 6. \begin{align*} \frac{1}{10} x^{10} b^{4} + \frac{1}{2} x^{8} b^{3} a + x^{6} b^{2} a^{2} + x^{4} b a^{3} + \frac{1}{2} x^{2} a^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*x^10*b^4 + 1/2*x^8*b^3*a + x^6*b^2*a^2 + x^4*b*a^3 + 1/2*x^2*a^4

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Sympy [B]  time = 0.072371, size = 44, normalized size = 2.75 \begin{align*} \frac{a^{4} x^{2}}{2} + a^{3} b x^{4} + a^{2} b^{2} x^{6} + \frac{a b^{3} x^{8}}{2} + \frac{b^{4} x^{10}}{10} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**4*x**2/2 + a**3*b*x**4 + a**2*b**2*x**6 + a*b**3*x**8/2 + b**4*x**10/10

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Giac [B]  time = 1.14335, size = 59, normalized size = 3.69 \begin{align*} \frac{1}{10} \, b^{4} x^{10} + \frac{1}{2} \, a b^{3} x^{8} + a^{2} b^{2} x^{6} + a^{3} b x^{4} + \frac{1}{2} \, a^{4} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*b^4*x^10 + 1/2*a*b^3*x^8 + a^2*b^2*x^6 + a^3*b*x^4 + 1/2*a^4*x^2