\(\int (a+c x^2) (e+f x^4)^2 \, dx\) [137]

Optimal result

Integrand size = 17, antiderivative size = 60 \[ \int \left (a+c x^2\right ) \left (e+f x^4\right )^2 \, dx=a e^2 x+\frac {1}{3} c e^2 x^3+\frac {2}{5} a e f x^5+\frac {2}{7} c e f x^7+\frac {1}{9} a f^2 x^9+\frac {1}{11} c f^2 x^{11} \]

[Out]

a*e^2*x+1/3*c*e^2*x^3+2/5*a*e*f*x^5+2/7*c*e*f*x^7+1/9*a*f^2*x^9+1/11*c*f^2*x^11

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {1168} \[ \int \left (a+c x^2\right ) \left (e+f x^4\right )^2 \, dx=a e^2 x+\frac {2}{5} a e f x^5+\frac {1}{9} a f^2 x^9+\frac {1}{3} c e^2 x^3+\frac {2}{7} c e f x^7+\frac {1}{11} c f^2 x^{11} \]

[In]

Int[(a + c*x^2)*(e + f*x^4)^2,x]

[Out]

a*e^2*x + (c*e^2*x^3)/3 + (2*a*e*f*x^5)/5 + (2*c*e*f*x^7)/7 + (a*f^2*x^9)/9 + (c*f^2*x^11)/11

Rule 1168

Int[((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x^2)^q*(a
+ c*x^4)^p, x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[p, 0] && IGtQ[q, -2]

Rubi steps \begin{align*} \text {integral}& = \int \left (a e^2+c e^2 x^2+2 a e f x^4+2 c e f x^6+a f^2 x^8+c f^2 x^{10}\right ) \, dx \\ & = a e^2 x+\frac {1}{3} c e^2 x^3+\frac {2}{5} a e f x^5+\frac {2}{7} c e f x^7+\frac {1}{9} a f^2 x^9+\frac {1}{11} c f^2 x^{11} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.00 \[ \int \left (a+c x^2\right ) \left (e+f x^4\right )^2 \, dx=a e^2 x+\frac {1}{3} c e^2 x^3+\frac {2}{5} a e f x^5+\frac {2}{7} c e f x^7+\frac {1}{9} a f^2 x^9+\frac {1}{11} c f^2 x^{11} \]

[In]

Integrate[(a + c*x^2)*(e + f*x^4)^2,x]

[Out]

a*e^2*x + (c*e^2*x^3)/3 + (2*a*e*f*x^5)/5 + (2*c*e*f*x^7)/7 + (a*f^2*x^9)/9 + (c*f^2*x^11)/11

Maple [A] (verified)

Time = 1.50 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.85

method	result	size
gosper	\(a \,e^{2} x +\frac {1}{3} c \,e^{2} x^{3}+\frac {2}{5} a e f \,x^{5}+\frac {2}{7} c e f \,x^{7}+\frac {1}{9} a \,f^{2} x^{9}+\frac {1}{11} c \,f^{2} x^{11}\)	\(51\)
default	\(a \,e^{2} x +\frac {1}{3} c \,e^{2} x^{3}+\frac {2}{5} a e f \,x^{5}+\frac {2}{7} c e f \,x^{7}+\frac {1}{9} a \,f^{2} x^{9}+\frac {1}{11} c \,f^{2} x^{11}\)	\(51\)
norman	\(a \,e^{2} x +\frac {1}{3} c \,e^{2} x^{3}+\frac {2}{5} a e f \,x^{5}+\frac {2}{7} c e f \,x^{7}+\frac {1}{9} a \,f^{2} x^{9}+\frac {1}{11} c \,f^{2} x^{11}\)	\(51\)
risch	\(a \,e^{2} x +\frac {1}{3} c \,e^{2} x^{3}+\frac {2}{5} a e f \,x^{5}+\frac {2}{7} c e f \,x^{7}+\frac {1}{9} a \,f^{2} x^{9}+\frac {1}{11} c \,f^{2} x^{11}\)	\(51\)
parallelrisch	\(a \,e^{2} x +\frac {1}{3} c \,e^{2} x^{3}+\frac {2}{5} a e f \,x^{5}+\frac {2}{7} c e f \,x^{7}+\frac {1}{9} a \,f^{2} x^{9}+\frac {1}{11} c \,f^{2} x^{11}\)	\(51\)

[In]

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

[Out]

a*e^2*x+1/3*c*e^2*x^3+2/5*a*e*f*x^5+2/7*c*e*f*x^7+1/9*a*f^2*x^9+1/11*c*f^2*x^11

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.83 \[ \int \left (a+c x^2\right ) \left (e+f x^4\right )^2 \, dx=\frac {1}{11} \, c f^{2} x^{11} + \frac {1}{9} \, a f^{2} x^{9} + \frac {2}{7} \, c e f x^{7} + \frac {2}{5} \, a e f x^{5} + \frac {1}{3} \, c e^{2} x^{3} + a e^{2} x \]

[In]

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

[Out]

1/11*c*f^2*x^11 + 1/9*a*f^2*x^9 + 2/7*c*e*f*x^7 + 2/5*a*e*f*x^5 + 1/3*c*e^2*x^3 + a*e^2*x

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.00 \[ \int \left (a+c x^2\right ) \left (e+f x^4\right )^2 \, dx=a e^{2} x + \frac {2 a e f x^{5}}{5} + \frac {a f^{2} x^{9}}{9} + \frac {c e^{2} x^{3}}{3} + \frac {2 c e f x^{7}}{7} + \frac {c f^{2} x^{11}}{11} \]

[In]

integrate((c*x**2+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 + 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 = 0.22 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.83 \[ \int \left (a+c x^2\right ) \left (e+f x^4\right )^2 \, dx=\frac {1}{11} \, c f^{2} x^{11} + \frac {1}{9} \, a f^{2} x^{9} + \frac {2}{7} \, c e f x^{7} + \frac {2}{5} \, a e f x^{5} + \frac {1}{3} \, c e^{2} x^{3} + a e^{2} x \]

[In]

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

[Out]

1/11*c*f^2*x^11 + 1/9*a*f^2*x^9 + 2/7*c*e*f*x^7 + 2/5*a*e*f*x^5 + 1/3*c*e^2*x^3 + a*e^2*x

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.83 \[ \int \left (a+c x^2\right ) \left (e+f x^4\right )^2 \, dx=\frac {1}{11} \, c f^{2} x^{11} + \frac {1}{9} \, a f^{2} x^{9} + \frac {2}{7} \, c e f x^{7} + \frac {2}{5} \, a e f x^{5} + \frac {1}{3} \, c e^{2} x^{3} + a e^{2} x \]

[In]

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

[Out]

1/11*c*f^2*x^11 + 1/9*a*f^2*x^9 + 2/7*c*e*f*x^7 + 2/5*a*e*f*x^5 + 1/3*c*e^2*x^3 + a*e^2*x

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.83 \[ \int \left (a+c x^2\right ) \left (e+f x^4\right )^2 \, dx=\frac {c\,e^2\,x^3}{3}+a\,e^2\,x+\frac {2\,c\,e\,f\,x^7}{7}+\frac {2\,a\,e\,f\,x^5}{5}+\frac {c\,f^2\,x^{11}}{11}+\frac {a\,f^2\,x^9}{9} \]

[In]

int((a + c*x^2)*(e + f*x^4)^2,x)

[Out]

(a*f^2*x^9)/9 + (c*e^2*x^3)/3 + (c*f^2*x^11)/11 + a*e^2*x + (2*a*e*f*x^5)/5 + (2*c*e*f*x^7)/7