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]

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

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Rubi [A]  time = 0.01, antiderivative size = 33, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.00, size = 33, normalized size = 1.00 \[ \frac {1}{3} c e^2 x^3+\frac {2}{7} c e f x^7+\frac {1}{11} c f^2 x^{11} \]

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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fricas [A]  time = 0.53, size = 27, normalized size = 0.82 \[ \frac {1}{11} x^{11} f^{2} c + \frac {2}{7} x^{7} f e c + \frac {1}{3} x^{3} e^{2} c \]

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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giac [A]  time = 0.20, size = 27, normalized size = 0.82 \[ \frac {1}{231} \, {\left (21 \, f^{2} x^{11} + 66 \, f x^{7} e + 77 \, x^{3} e^{2}\right )} c \]

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

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maple [A]  time = 0.04, size = 27, normalized size = 0.82 \[ \left (\frac {1}{11} f^{2} x^{11}+\frac {2}{7} e f \,x^{7}+\frac {1}{3} e^{2} x^{3}\right ) c \]

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 = 1.36, size = 27, normalized size = 0.82 \[ \frac {1}{231} \, {\left (21 \, f^{2} x^{11} + 66 \, e f x^{7} + 77 \, e^{2} x^{3}\right )} c \]

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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mupad [B]  time = 0.04, size = 27, normalized size = 0.82 \[ \frac {c\,x^3\,\left (77\,e^2+66\,e\,f\,x^4+21\,f^2\,x^8\right )}{231} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(c*x^2*(e + f*x^4)^2,x)

[Out]

(c*x^3*(77*e^2 + 21*f^2*x^8 + 66*e*f*x^4))/231

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sympy [A]  time = 0.13, size = 31, normalized size = 0.94 \[ \frac {c e^{2} x^{3}}{3} + \frac {2 c e f x^{7}}{7} + \frac {c f^{2} x^{11}}{11} \]

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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