\(\int c x^2 (e+f x^4)^2 \, dx\) [136]

   Optimal result
   Rubi [A] (verified)
   Mathematica [A] (verified)
   Maple [A] (verified)
   Fricas [A] (verification not implemented)
   Sympy [A] (verification not implemented)
   Maxima [A] (verification not implemented)
   Giac [A] (verification not implemented)
   Mupad [B] (verification not implemented)

Optimal result

Integrand size = 14, antiderivative size = 33 \[ \int c x^2 \left (e+f x^4\right )^2 \, dx=\frac {1}{3} c e^2 x^3+\frac {2}{7} c e f x^7+\frac {1}{11} c f^2 x^{11} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {12, 276} \[ \int c x^2 \left (e+f x^4\right )^2 \, dx=\frac {1}{3} c e^2 x^3+\frac {2}{7} c e f x^7+\frac {1}{11} c f^2 x^{11} \]

[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 276

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*} \text {integral}& = 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] (verified)

Time = 0.00 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00 \[ \int c x^2 \left (e+f x^4\right )^2 \, dx=\frac {1}{3} c e^2 x^3+\frac {2}{7} c e f x^7+\frac {1}{11} c f^2 x^{11} \]

[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

Maple [A] (verified)

Time = 1.44 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.82

method result size
default \(\left (\frac {1}{11} f^{2} x^{11}+\frac {2}{7} e f \,x^{7}+\frac {1}{3} e^{2} x^{3}\right ) c\) \(27\)
parallelrisch \(\left (\frac {1}{11} f^{2} x^{11}+\frac {2}{7} e f \,x^{7}+\frac {1}{3} e^{2} x^{3}\right ) c\) \(27\)
gosper \(\frac {x^{3} \left (21 f^{2} x^{8}+66 e f \,x^{4}+77 e^{2}\right ) c}{231}\) \(28\)
norman \(\frac {1}{3} c \,e^{2} x^{3}+\frac {2}{7} c e f \,x^{7}+\frac {1}{11} c \,f^{2} x^{11}\) \(28\)
risch \(\frac {1}{3} c \,e^{2} x^{3}+\frac {2}{7} c e f \,x^{7}+\frac {1}{11} c \,f^{2} x^{11}\) \(28\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.82 \[ \int c x^2 \left (e+f x^4\right )^2 \, dx=\frac {1}{11} \, c f^{2} x^{11} + \frac {2}{7} \, c e f x^{7} + \frac {1}{3} \, c e^{2} x^{3} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.94 \[ \int c x^2 \left (e+f x^4\right )^2 \, dx=\frac {c e^{2} x^{3}}{3} + \frac {2 c e f x^{7}}{7} + \frac {c f^{2} x^{11}}{11} \]

[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

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.82 \[ \int c x^2 \left (e+f x^4\right )^2 \, dx=\frac {1}{231} \, {\left (21 \, f^{2} x^{11} + 66 \, e f x^{7} + 77 \, e^{2} x^{3}\right )} c \]

[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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.82 \[ \int c x^2 \left (e+f x^4\right )^2 \, dx=\frac {1}{231} \, {\left (21 \, f^{2} x^{11} + 66 \, e f x^{7} + 77 \, e^{2} x^{3}\right )} c \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.82 \[ \int c x^2 \left (e+f x^4\right )^2 \, dx=\frac {c\,x^3\,\left (77\,e^2+66\,e\,f\,x^4+21\,f^2\,x^8\right )}{231} \]

[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