3.133 \(\int a (e+f x^4)^2 \, dx\)

Optimal. Leaf size=28 \[ a e^2 x+\frac{2}{5} a e f x^5+\frac{1}{9} a f^2 x^9 \]

[Out]

a*e^2*x + (2*a*e*f*x^5)/5 + (a*f^2*x^9)/9

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Rubi [A]  time = 0.0113591, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {12, 194} \[ a e^2 x+\frac{2}{5} a e f x^5+\frac{1}{9} a f^2 x^9 \]

Antiderivative was successfully verified.

[In]

Int[a*(e + f*x^4)^2,x]

[Out]

a*e^2*x + (2*a*e*f*x^5)/5 + (a*f^2*x^9)/9

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 a \left (e+f x^4\right )^2 \, dx &=a \int \left (e+f x^4\right )^2 \, dx\\ &=a \int \left (e^2+2 e f x^4+f^2 x^8\right ) \, dx\\ &=a e^2 x+\frac{2}{5} a e f x^5+\frac{1}{9} a f^2 x^9\\ \end{align*}

Mathematica [A]  time = 0.0017667, size = 27, normalized size = 0.96 \[ a \left (e^2 x+\frac{2}{5} e f x^5+\frac{f^2 x^9}{9}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[a*(e + f*x^4)^2,x]

[Out]

a*(e^2*x + (2*e*f*x^5)/5 + (f^2*x^9)/9)

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Maple [A]  time = 0.04, size = 24, normalized size = 0.9 \begin{align*} a \left ({\frac{{f}^{2}{x}^{9}}{9}}+{\frac{2\,ef{x}^{5}}{5}}+{e}^{2}x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(a*(f*x^4+e)^2,x)

[Out]

a*(1/9*f^2*x^9+2/5*e*f*x^5+e^2*x)

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Maxima [A]  time = 0.955677, size = 34, normalized size = 1.21 \begin{align*} \frac{1}{45} \,{\left (5 \, f^{2} x^{9} + 18 \, e f x^{5} + 45 \, e^{2} x\right )} a \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/45*(5*f^2*x^9 + 18*e*f*x^5 + 45*e^2*x)*a

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Fricas [A]  time = 1.05641, size = 55, normalized size = 1.96 \begin{align*} \frac{1}{9} x^{9} f^{2} a + \frac{2}{5} x^{5} f e a + x e^{2} a \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*x^9*f^2*a + 2/5*x^5*f*e*a + x*e^2*a

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Sympy [A]  time = 0.059508, size = 27, normalized size = 0.96 \begin{align*} a e^{2} x + \frac{2 a e f x^{5}}{5} + \frac{a f^{2} x^{9}}{9} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a*(f*x**4+e)**2,x)

[Out]

a*e**2*x + 2*a*e*f*x**5/5 + a*f**2*x**9/9

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Giac [A]  time = 1.06919, size = 34, normalized size = 1.21 \begin{align*} \frac{1}{45} \,{\left (5 \, f^{2} x^{9} + 18 \, f x^{5} e + 45 \, x e^{2}\right )} a \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/45*(5*f^2*x^9 + 18*f*x^5*e + 45*x*e^2)*a