3.136 \(\int c x^2 (e+f x^4)^2 \, dx\)

Optimal. Leaf size=33 \[ \frac{1}{3} c e^2 x^3+\frac{2}{7} c e f x^7+\frac{1}{11} c f^2 x^{11} \]

[Out]

(c*e^2*x^3)/3 + (2*c*e*f*x^7)/7 + (c*f^2*x^11)/11

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Rubi [A]  time = 0.0132649, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {12, 270} \[ \frac{1}{3} c e^2 x^3+\frac{2}{7} c e f x^7+\frac{1}{11} c f^2 x^{11} \]

Antiderivative was successfully verified.

[In]

Int[c*x^2*(e + f*x^4)^2,x]

[Out]

(c*e^2*x^3)/3 + (2*c*e*f*x^7)/7 + (c*f^2*x^11)/11

Rule 12

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

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int c x^2 \left (e+f x^4\right )^2 \, dx &=c \int x^2 \left (e+f x^4\right )^2 \, dx\\ &=c \int \left (e^2 x^2+2 e f x^6+f^2 x^{10}\right ) \, dx\\ &=\frac{1}{3} c e^2 x^3+\frac{2}{7} c e f x^7+\frac{1}{11} c f^2 x^{11}\\ \end{align*}

Mathematica [A]  time = 0.0011796, size = 33, normalized size = 1. \[ \frac{1}{3} c e^2 x^3+\frac{2}{7} c e f x^7+\frac{1}{11} c f^2 x^{11} \]

Antiderivative was successfully verified.

[In]

Integrate[c*x^2*(e + f*x^4)^2,x]

[Out]

(c*e^2*x^3)/3 + (2*c*e*f*x^7)/7 + (c*f^2*x^11)/11

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Maple [A]  time = 0.041, size = 27, normalized size = 0.8 \begin{align*} c \left ({\frac{{f}^{2}{x}^{11}}{11}}+{\frac{2\,ef{x}^{7}}{7}}+{\frac{{e}^{2}{x}^{3}}{3}} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c*(1/11*f^2*x^11+2/7*e*f*x^7+1/3*e^2*x^3)

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Maxima [A]  time = 0.986278, size = 36, normalized size = 1.09 \begin{align*} \frac{1}{231} \,{\left (21 \, f^{2} x^{11} + 66 \, e f x^{7} + 77 \, e^{2} x^{3}\right )} c \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/231*(21*f^2*x^11 + 66*e*f*x^7 + 77*e^2*x^3)*c

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Fricas [A]  time = 1.05338, size = 66, normalized size = 2. \begin{align*} \frac{1}{11} x^{11} f^{2} c + \frac{2}{7} x^{7} f e c + \frac{1}{3} x^{3} e^{2} c \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/11*x^11*f^2*c + 2/7*x^7*f*e*c + 1/3*x^3*e^2*c

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Sympy [A]  time = 0.059401, size = 31, normalized size = 0.94 \begin{align*} \frac{c e^{2} x^{3}}{3} + \frac{2 c e f x^{7}}{7} + \frac{c f^{2} x^{11}}{11} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c*e**2*x**3/3 + 2*c*e*f*x**7/7 + c*f**2*x**11/11

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Giac [A]  time = 1.08192, size = 36, normalized size = 1.09 \begin{align*} \frac{1}{231} \,{\left (21 \, f^{2} x^{11} + 66 \, f x^{7} e + 77 \, x^{3} e^{2}\right )} c \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/231*(21*f^2*x^11 + 66*f*x^7*e + 77*x^3*e^2)*c