3.131 \(\int (a+c x^4)^2 \, dx\)

Optimal. Leaf size=25 \[ a^2 x+\frac{2}{5} a c x^5+\frac{c^2 x^9}{9} \]

[Out]

a^2*x + (2*a*c*x^5)/5 + (c^2*x^9)/9

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Rubi [A]  time = 0.0081181, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {194} \[ a^2 x+\frac{2}{5} a c x^5+\frac{c^2 x^9}{9} \]

Antiderivative was successfully verified.

[In]

Int[(a + c*x^4)^2,x]

[Out]

a^2*x + (2*a*c*x^5)/5 + (c^2*x^9)/9

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

Mathematica [A]  time = 0.0009166, size = 25, normalized size = 1. \[ a^2 x+\frac{2}{5} a c x^5+\frac{c^2 x^9}{9} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + c*x^4)^2,x]

[Out]

a^2*x + (2*a*c*x^5)/5 + (c^2*x^9)/9

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Maple [A]  time = 0.043, size = 22, normalized size = 0.9 \begin{align*}{a}^{2}x+{\frac{2\,ac{x}^{5}}{5}}+{\frac{{c}^{2}{x}^{9}}{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a^2*x+2/5*a*c*x^5+1/9*c^2*x^9

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Maxima [A]  time = 1.00009, size = 28, normalized size = 1.12 \begin{align*} \frac{1}{9} \, c^{2} x^{9} + \frac{2}{5} \, a c x^{5} + a^{2} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*c^2*x^9 + 2/5*a*c*x^5 + a^2*x

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Fricas [A]  time = 1.6794, size = 47, normalized size = 1.88 \begin{align*} \frac{1}{9} x^{9} c^{2} + \frac{2}{5} x^{5} c a + x a^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*x^9*c^2 + 2/5*x^5*c*a + x*a^2

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Sympy [A]  time = 0.060069, size = 22, normalized size = 0.88 \begin{align*} a^{2} x + \frac{2 a c x^{5}}{5} + \frac{c^{2} x^{9}}{9} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**2*x + 2*a*c*x**5/5 + c**2*x**9/9

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Giac [A]  time = 1.13115, size = 28, normalized size = 1.12 \begin{align*} \frac{1}{9} \, c^{2} x^{9} + \frac{2}{5} \, a c x^{5} + a^{2} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*c^2*x^9 + 2/5*a*c*x^5 + a^2*x