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

Optimal. Leaf size=25 \[ a^2 x+\frac{2}{3} a c x^3+\frac{c^2 x^5}{5} \]

[Out]

a^2*x + (2*a*c*x^3)/3 + (c^2*x^5)/5

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Rubi [A]  time = 0.0071073, 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} \[ a^2 x+\frac{2}{3} a c x^3+\frac{c^2 x^5}{5} \]

Antiderivative was successfully verified.

[In]

Int[(a + c*x^2)^2,x]

[Out]

a^2*x + (2*a*c*x^3)/3 + (c^2*x^5)/5

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^2\right )^2 \, dx &=\int \left (a^2+2 a c x^2+c^2 x^4\right ) \, dx\\ &=a^2 x+\frac{2}{3} a c x^3+\frac{c^2 x^5}{5}\\ \end{align*}

Mathematica [A]  time = 0.001019, size = 25, normalized size = 1. \[ a^2 x+\frac{2}{3} a c x^3+\frac{c^2 x^5}{5} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + c*x^2)^2,x]

[Out]

a^2*x + (2*a*c*x^3)/3 + (c^2*x^5)/5

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Maple [A]  time = 0.051, size = 22, normalized size = 0.9 \begin{align*}{a}^{2}x+{\frac{2\,a{x}^{3}c}{3}}+{\frac{{c}^{2}{x}^{5}}{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a^2*x+2/3*a*x^3*c+1/5*c^2*x^5

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Maxima [A]  time = 1.23573, size = 28, normalized size = 1.12 \begin{align*} \frac{1}{5} \, c^{2} x^{5} + \frac{2}{3} \, a c x^{3} + a^{2} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*c^2*x^5 + 2/3*a*c*x^3 + a^2*x

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Fricas [A]  time = 1.8333, size = 47, normalized size = 1.88 \begin{align*} \frac{1}{5} x^{5} c^{2} + \frac{2}{3} x^{3} c a + x a^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*x^5*c^2 + 2/3*x^3*c*a + x*a^2

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Sympy [A]  time = 0.083565, size = 22, normalized size = 0.88 \begin{align*} a^{2} x + \frac{2 a c x^{3}}{3} + \frac{c^{2} x^{5}}{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**2*x + 2*a*c*x**3/3 + c**2*x**5/5

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Giac [A]  time = 1.31485, size = 28, normalized size = 1.12 \begin{align*} \frac{1}{5} \, c^{2} x^{5} + \frac{2}{3} \, a c x^{3} + a^{2} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*c^2*x^5 + 2/3*a*c*x^3 + a^2*x