3.140 \(\int d x^3 (e+f x^4)^2 \, dx\)

Optimal. Leaf size=17 \[ \frac{d \left (e+f x^4\right )^3}{12 f} \]

[Out]

(d*(e + f*x^4)^3)/(12*f)

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Rubi [A]  time = 0.0050034, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {12, 261} \[ \frac{d \left (e+f x^4\right )^3}{12 f} \]

Antiderivative was successfully verified.

[In]

Int[d*x^3*(e + f*x^4)^2,x]

[Out]

(d*(e + f*x^4)^3)/(12*f)

Rule 12

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

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rubi steps

\begin{align*} \int d x^3 \left (e+f x^4\right )^2 \, dx &=d \int x^3 \left (e+f x^4\right )^2 \, dx\\ &=\frac{d \left (e+f x^4\right )^3}{12 f}\\ \end{align*}

Mathematica [A]  time = 0.0008476, size = 33, normalized size = 1.94 \[ \frac{1}{4} d e^2 x^4+\frac{1}{4} d e f x^8+\frac{1}{12} d f^2 x^{12} \]

Antiderivative was successfully verified.

[In]

Integrate[d*x^3*(e + f*x^4)^2,x]

[Out]

(d*e^2*x^4)/4 + (d*e*f*x^8)/4 + (d*f^2*x^12)/12

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Maple [A]  time = 0.039, size = 27, normalized size = 1.6 \begin{align*} d \left ({\frac{{f}^{2}{x}^{12}}{12}}+{\frac{ef{x}^{8}}{4}}+{\frac{{e}^{2}{x}^{4}}{4}} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(d*x^3*(f*x^4+e)^2,x)

[Out]

d*(1/12*f^2*x^12+1/4*e*f*x^8+1/4*e^2*x^4)

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Maxima [A]  time = 1.01359, size = 20, normalized size = 1.18 \begin{align*} \frac{{\left (f x^{4} + e\right )}^{3} d}{12 \, f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(d*x^3*(f*x^4+e)^2,x, algorithm="maxima")

[Out]

1/12*(f*x^4 + e)^3*d/f

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Fricas [A]  time = 1.05472, size = 66, normalized size = 3.88 \begin{align*} \frac{1}{12} x^{12} f^{2} d + \frac{1}{4} x^{8} f e d + \frac{1}{4} x^{4} e^{2} d \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(d*x^3*(f*x^4+e)^2,x, algorithm="fricas")

[Out]

1/12*x^12*f^2*d + 1/4*x^8*f*e*d + 1/4*x^4*e^2*d

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Sympy [B]  time = 0.059406, size = 29, normalized size = 1.71 \begin{align*} \frac{d e^{2} x^{4}}{4} + \frac{d e f x^{8}}{4} + \frac{d f^{2} x^{12}}{12} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(d*x**3*(f*x**4+e)**2,x)

[Out]

d*e**2*x**4/4 + d*e*f*x**8/4 + d*f**2*x**12/12

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Giac [A]  time = 1.06087, size = 22, normalized size = 1.29 \begin{align*} \frac{{\left (f x^{4} + e\right )}^{3} d}{12 \, f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(d*x^3*(f*x^4+e)^2,x, algorithm="giac")

[Out]

1/12*(f*x^4 + e)^3*d/f