3.50 \(\int (a+c x^2)^3 \, dx\)

Optimal. Leaf size=35 \[ a^2 c x^3+a^3 x+\frac{3}{5} a c^2 x^5+\frac{c^3 x^7}{7} \]

[Out]

a^3*x + a^2*c*x^3 + (3*a*c^2*x^5)/5 + (c^3*x^7)/7

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Rubi [A]  time = 0.0114554, antiderivative size = 35, 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} \[ a^2 c x^3+a^3 x+\frac{3}{5} a c^2 x^5+\frac{c^3 x^7}{7} \]

Antiderivative was successfully verified.

[In]

Int[(a + c*x^2)^3,x]

[Out]

a^3*x + a^2*c*x^3 + (3*a*c^2*x^5)/5 + (c^3*x^7)/7

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 )^3 \, dx &=\int \left (a^3+3 a^2 c x^2+3 a c^2 x^4+c^3 x^6\right ) \, dx\\ &=a^3 x+a^2 c x^3+\frac{3}{5} a c^2 x^5+\frac{c^3 x^7}{7}\\ \end{align*}

Mathematica [A]  time = 0.0012346, size = 35, normalized size = 1. \[ a^2 c x^3+a^3 x+\frac{3}{5} a c^2 x^5+\frac{c^3 x^7}{7} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + c*x^2)^3,x]

[Out]

a^3*x + a^2*c*x^3 + (3*a*c^2*x^5)/5 + (c^3*x^7)/7

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Maple [A]  time = 0.046, size = 32, normalized size = 0.9 \begin{align*} x{a}^{3}+{a}^{2}c{x}^{3}+{\frac{3\,a{c}^{2}{x}^{5}}{5}}+{\frac{{c}^{3}{x}^{7}}{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+a)^3,x)

[Out]

x*a^3+a^2*c*x^3+3/5*a*c^2*x^5+1/7*c^3*x^7

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Maxima [A]  time = 1.14035, size = 42, normalized size = 1.2 \begin{align*} \frac{1}{7} \, c^{3} x^{7} + \frac{3}{5} \, a c^{2} x^{5} + a^{2} c x^{3} + a^{3} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)^3,x, algorithm="maxima")

[Out]

1/7*c^3*x^7 + 3/5*a*c^2*x^5 + a^2*c*x^3 + a^3*x

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Fricas [A]  time = 1.85996, size = 66, normalized size = 1.89 \begin{align*} \frac{1}{7} x^{7} c^{3} + \frac{3}{5} x^{5} c^{2} a + x^{3} c a^{2} + x a^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)^3,x, algorithm="fricas")

[Out]

1/7*x^7*c^3 + 3/5*x^5*c^2*a + x^3*c*a^2 + x*a^3

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Sympy [A]  time = 0.066568, size = 32, normalized size = 0.91 \begin{align*} a^{3} x + a^{2} c x^{3} + \frac{3 a c^{2} x^{5}}{5} + \frac{c^{3} x^{7}}{7} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+a)**3,x)

[Out]

a**3*x + a**2*c*x**3 + 3*a*c**2*x**5/5 + c**3*x**7/7

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Giac [A]  time = 1.19568, size = 42, normalized size = 1.2 \begin{align*} \frac{1}{7} \, c^{3} x^{7} + \frac{3}{5} \, a c^{2} x^{5} + a^{2} c x^{3} + a^{3} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)^3,x, algorithm="giac")

[Out]

1/7*c^3*x^7 + 3/5*a*c^2*x^5 + a^2*c*x^3 + a^3*x