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

Optimal. Leaf size=30 \[ \frac{a^2 x^5}{5}+\frac{2}{7} a b x^7+\frac{b^2 x^9}{9} \]

[Out]

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

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Rubi [A]  time = 0.0106896, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {270} \[ \frac{a^2 x^5}{5}+\frac{2}{7} a b x^7+\frac{b^2 x^9}{9} \]

Antiderivative was successfully verified.

[In]

Int[x^4*(a + b*x^2)^2,x]

[Out]

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

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

Mathematica [A]  time = 0.0007241, size = 30, normalized size = 1. \[ \frac{a^2 x^5}{5}+\frac{2}{7} a b x^7+\frac{b^2 x^9}{9} \]

Antiderivative was successfully verified.

[In]

Integrate[x^4*(a + b*x^2)^2,x]

[Out]

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

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Maple [A]  time = 0., size = 25, normalized size = 0.8 \begin{align*}{\frac{{a}^{2}{x}^{5}}{5}}+{\frac{2\,ab{x}^{7}}{7}}+{\frac{{b}^{2}{x}^{9}}{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 2.40818, size = 32, normalized size = 1.07 \begin{align*} \frac{1}{9} \, b^{2} x^{9} + \frac{2}{7} \, a b x^{7} + \frac{1}{5} \, a^{2} x^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.28446, size = 55, normalized size = 1.83 \begin{align*} \frac{1}{9} x^{9} b^{2} + \frac{2}{7} x^{7} b a + \frac{1}{5} x^{5} a^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.05982, size = 26, normalized size = 0.87 \begin{align*} \frac{a^{2} x^{5}}{5} + \frac{2 a b x^{7}}{7} + \frac{b^{2} x^{9}}{9} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**2*x**5/5 + 2*a*b*x**7/7 + b**2*x**9/9

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Giac [A]  time = 3.2839, size = 32, normalized size = 1.07 \begin{align*} \frac{1}{9} \, b^{2} x^{9} + \frac{2}{7} \, a b x^{7} + \frac{1}{5} \, a^{2} x^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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