3.451 \(\int x (a^2+2 a b x^2+b^2 x^4)^3 \, dx\)

Optimal. Leaf size=16 \[ \frac{\left (a+b x^2\right )^7}{14 b} \]

[Out]

(a + b*x^2)^7/(14*b)

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Rubi [A]  time = 0.0048827, 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} \[ \frac{\left (a+b x^2\right )^7}{14 b} \]

Antiderivative was successfully verified.

[In]

Int[x*(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]

[Out]

(a + b*x^2)^7/(14*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 )^3 \, dx &=\frac{\int x \left (a b+b^2 x^2\right )^6 \, dx}{b^6}\\ &=\frac{\left (a+b x^2\right )^7}{14 b}\\ \end{align*}

Mathematica [A]  time = 0.0019941, size = 16, normalized size = 1. \[ \frac{\left (a+b x^2\right )^7}{14 b} \]

Antiderivative was successfully verified.

[In]

Integrate[x*(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]

[Out]

(a + b*x^2)^7/(14*b)

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Maple [B]  time = 0.04, size = 69, normalized size = 4.3 \begin{align*}{\frac{{b}^{6}{x}^{14}}{14}}+{\frac{a{b}^{5}{x}^{12}}{2}}+{\frac{3\,{a}^{2}{b}^{4}{x}^{10}}{2}}+{\frac{5\,{a}^{3}{b}^{3}{x}^{8}}{2}}+{\frac{5\,{a}^{4}{b}^{2}{x}^{6}}{2}}+{\frac{3\,{a}^{5}b{x}^{4}}{2}}+{\frac{{x}^{2}{a}^{6}}{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)^3,x)

[Out]

1/14*b^6*x^14+1/2*a*b^5*x^12+3/2*a^2*b^4*x^10+5/2*a^3*b^3*x^8+5/2*a^4*b^2*x^6+3/2*a^5*b*x^4+1/2*x^2*a^6

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Maxima [B]  time = 0.993829, size = 92, normalized size = 5.75 \begin{align*} \frac{1}{14} \, b^{6} x^{14} + \frac{1}{2} \, a b^{5} x^{12} + \frac{3}{2} \, a^{2} b^{4} x^{10} + \frac{5}{2} \, a^{3} b^{3} x^{8} + \frac{5}{2} \, a^{4} b^{2} x^{6} + \frac{3}{2} \, a^{5} b x^{4} + \frac{1}{2} \, a^{6} 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)^3,x, algorithm="maxima")

[Out]

1/14*b^6*x^14 + 1/2*a*b^5*x^12 + 3/2*a^2*b^4*x^10 + 5/2*a^3*b^3*x^8 + 5/2*a^4*b^2*x^6 + 3/2*a^5*b*x^4 + 1/2*a^
6*x^2

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Fricas [B]  time = 1.46501, size = 158, normalized size = 9.88 \begin{align*} \frac{1}{14} x^{14} b^{6} + \frac{1}{2} x^{12} b^{5} a + \frac{3}{2} x^{10} b^{4} a^{2} + \frac{5}{2} x^{8} b^{3} a^{3} + \frac{5}{2} x^{6} b^{2} a^{4} + \frac{3}{2} x^{4} b a^{5} + \frac{1}{2} x^{2} a^{6} \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)^3,x, algorithm="fricas")

[Out]

1/14*x^14*b^6 + 1/2*x^12*b^5*a + 3/2*x^10*b^4*a^2 + 5/2*x^8*b^3*a^3 + 5/2*x^6*b^2*a^4 + 3/2*x^4*b*a^5 + 1/2*x^
2*a^6

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Sympy [B]  time = 0.076982, size = 78, normalized size = 4.88 \begin{align*} \frac{a^{6} x^{2}}{2} + \frac{3 a^{5} b x^{4}}{2} + \frac{5 a^{4} b^{2} x^{6}}{2} + \frac{5 a^{3} b^{3} x^{8}}{2} + \frac{3 a^{2} b^{4} x^{10}}{2} + \frac{a b^{5} x^{12}}{2} + \frac{b^{6} x^{14}}{14} \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)**3,x)

[Out]

a**6*x**2/2 + 3*a**5*b*x**4/2 + 5*a**4*b**2*x**6/2 + 5*a**3*b**3*x**8/2 + 3*a**2*b**4*x**10/2 + a*b**5*x**12/2
+ b**6*x**14/14

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Giac [B]  time = 1.17679, size = 92, normalized size = 5.75 \begin{align*} \frac{1}{14} \, b^{6} x^{14} + \frac{1}{2} \, a b^{5} x^{12} + \frac{3}{2} \, a^{2} b^{4} x^{10} + \frac{5}{2} \, a^{3} b^{3} x^{8} + \frac{5}{2} \, a^{4} b^{2} x^{6} + \frac{3}{2} \, a^{5} b x^{4} + \frac{1}{2} \, a^{6} 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)^3,x, algorithm="giac")

[Out]

1/14*b^6*x^14 + 1/2*a*b^5*x^12 + 3/2*a^2*b^4*x^10 + 5/2*a^3*b^3*x^8 + 5/2*a^4*b^2*x^6 + 3/2*a^5*b*x^4 + 1/2*a^
6*x^2