\(\int (d+e x^2)^2 (a+c x^4)^2 \, dx\) [129]

Optimal result

Integrand size = 19, antiderivative size = 97 \[ \int \left (d+e x^2\right )^2 \left (a+c x^4\right )^2 \, dx=a^2 d^2 x+\frac {2}{3} a^2 d e x^3+\frac {1}{5} a \left (2 c d^2+a e^2\right ) x^5+\frac {4}{7} a c d e x^7+\frac {1}{9} c \left (c d^2+2 a e^2\right ) x^9+\frac {2}{11} c^2 d e x^{11}+\frac {1}{13} c^2 e^2 x^{13} \]

[Out]

a^2*d^2*x+2/3*a^2*d*e*x^3+1/5*a*(a*e^2+2*c*d^2)*x^5+4/7*a*c*d*e*x^7+1/9*c*(2*a*e^2+c*d^2)*x^9+2/11*c^2*d*e*x^1
1+1/13*c^2*e^2*x^13

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {1168} \[ \int \left (d+e x^2\right )^2 \left (a+c x^4\right )^2 \, dx=a^2 d^2 x+\frac {2}{3} a^2 d e x^3+\frac {1}{9} c x^9 \left (2 a e^2+c d^2\right )+\frac {1}{5} a x^5 \left (a e^2+2 c d^2\right )+\frac {4}{7} a c d e x^7+\frac {2}{11} c^2 d e x^{11}+\frac {1}{13} c^2 e^2 x^{13} \]

[In]

Int[(d + e*x^2)^2*(a + c*x^4)^2,x]

[Out]

a^2*d^2*x + (2*a^2*d*e*x^3)/3 + (a*(2*c*d^2 + a*e^2)*x^5)/5 + (4*a*c*d*e*x^7)/7 + (c*(c*d^2 + 2*a*e^2)*x^9)/9
+ (2*c^2*d*e*x^11)/11 + (c^2*e^2*x^13)/13

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^2 d^2+2 a^2 d e x^2+a \left (2 c d^2+a e^2\right ) x^4+4 a c d e x^6+c \left (c d^2+2 a e^2\right ) x^8+2 c^2 d e x^{10}+c^2 e^2 x^{12}\right ) \, dx \\ & = a^2 d^2 x+\frac {2}{3} a^2 d e x^3+\frac {1}{5} a \left (2 c d^2+a e^2\right ) x^5+\frac {4}{7} a c d e x^7+\frac {1}{9} c \left (c d^2+2 a e^2\right ) x^9+\frac {2}{11} c^2 d e x^{11}+\frac {1}{13} c^2 e^2 x^{13} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.00 \[ \int \left (d+e x^2\right )^2 \left (a+c x^4\right )^2 \, dx=a^2 d^2 x+\frac {2}{3} a^2 d e x^3+\frac {1}{5} a \left (2 c d^2+a e^2\right ) x^5+\frac {4}{7} a c d e x^7+\frac {1}{9} c \left (c d^2+2 a e^2\right ) x^9+\frac {2}{11} c^2 d e x^{11}+\frac {1}{13} c^2 e^2 x^{13} \]

[In]

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

[Out]

a^2*d^2*x + (2*a^2*d*e*x^3)/3 + (a*(2*c*d^2 + a*e^2)*x^5)/5 + (4*a*c*d*e*x^7)/7 + (c*(c*d^2 + 2*a*e^2)*x^9)/9
+ (2*c^2*d*e*x^11)/11 + (c^2*e^2*x^13)/13

Maple [A] (verified)

Time = 0.24 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.93

method	result	size
default	\(\frac {c^{2} e^{2} x^{13}}{13}+\frac {2 c^{2} d e \,x^{11}}{11}+\frac {\left (2 e^{2} a c +c^{2} d^{2}\right ) x^{9}}{9}+\frac {4 a c d e \,x^{7}}{7}+\frac {\left (e^{2} a^{2}+2 d^{2} a c \right ) x^{5}}{5}+\frac {2 a^{2} d e \,x^{3}}{3}+a^{2} d^{2} x\)	\(90\)
norman	\(\frac {c^{2} e^{2} x^{13}}{13}+\frac {2 c^{2} d e \,x^{11}}{11}+\left (\frac {2}{9} e^{2} a c +\frac {1}{9} c^{2} d^{2}\right ) x^{9}+\frac {4 a c d e \,x^{7}}{7}+\left (\frac {1}{5} e^{2} a^{2}+\frac {2}{5} d^{2} a c \right ) x^{5}+\frac {2 a^{2} d e \,x^{3}}{3}+a^{2} d^{2} x\)	\(90\)
gosper	\(\frac {1}{13} c^{2} e^{2} x^{13}+\frac {2}{11} c^{2} d e \,x^{11}+\frac {2}{9} x^{9} e^{2} a c +\frac {1}{9} x^{9} c^{2} d^{2}+\frac {4}{7} a c d e \,x^{7}+\frac {1}{5} x^{5} e^{2} a^{2}+\frac {2}{5} x^{5} d^{2} a c +\frac {2}{3} a^{2} d e \,x^{3}+a^{2} d^{2} x\)	\(92\)
risch	\(\frac {1}{13} c^{2} e^{2} x^{13}+\frac {2}{11} c^{2} d e \,x^{11}+\frac {2}{9} x^{9} e^{2} a c +\frac {1}{9} x^{9} c^{2} d^{2}+\frac {4}{7} a c d e \,x^{7}+\frac {1}{5} x^{5} e^{2} a^{2}+\frac {2}{5} x^{5} d^{2} a c +\frac {2}{3} a^{2} d e \,x^{3}+a^{2} d^{2} x\)	\(92\)
parallelrisch	\(\frac {1}{13} c^{2} e^{2} x^{13}+\frac {2}{11} c^{2} d e \,x^{11}+\frac {2}{9} x^{9} e^{2} a c +\frac {1}{9} x^{9} c^{2} d^{2}+\frac {4}{7} a c d e \,x^{7}+\frac {1}{5} x^{5} e^{2} a^{2}+\frac {2}{5} x^{5} d^{2} a c +\frac {2}{3} a^{2} d e \,x^{3}+a^{2} d^{2} x\)	\(92\)

[In]

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

[Out]

1/13*c^2*e^2*x^13+2/11*c^2*d*e*x^11+1/9*(2*a*c*e^2+c^2*d^2)*x^9+4/7*a*c*d*e*x^7+1/5*(a^2*e^2+2*a*c*d^2)*x^5+2/
3*a^2*d*e*x^3+a^2*d^2*x

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.92 \[ \int \left (d+e x^2\right )^2 \left (a+c x^4\right )^2 \, dx=\frac {1}{13} \, c^{2} e^{2} x^{13} + \frac {2}{11} \, c^{2} d e x^{11} + \frac {4}{7} \, a c d e x^{7} + \frac {1}{9} \, {\left (c^{2} d^{2} + 2 \, a c e^{2}\right )} x^{9} + \frac {2}{3} \, a^{2} d e x^{3} + \frac {1}{5} \, {\left (2 \, a c d^{2} + a^{2} e^{2}\right )} x^{5} + a^{2} d^{2} x \]

[In]

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

[Out]

1/13*c^2*e^2*x^13 + 2/11*c^2*d*e*x^11 + 4/7*a*c*d*e*x^7 + 1/9*(c^2*d^2 + 2*a*c*e^2)*x^9 + 2/3*a^2*d*e*x^3 + 1/
5*(2*a*c*d^2 + a^2*e^2)*x^5 + a^2*d^2*x

Sympy [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.07 \[ \int \left (d+e x^2\right )^2 \left (a+c x^4\right )^2 \, dx=a^{2} d^{2} x + \frac {2 a^{2} d e x^{3}}{3} + \frac {4 a c d e x^{7}}{7} + \frac {2 c^{2} d e x^{11}}{11} + \frac {c^{2} e^{2} x^{13}}{13} + x^{9} \cdot \left (\frac {2 a c e^{2}}{9} + \frac {c^{2} d^{2}}{9}\right ) + x^{5} \left (\frac {a^{2} e^{2}}{5} + \frac {2 a c d^{2}}{5}\right ) \]

[In]

integrate((e*x**2+d)**2*(c*x**4+a)**2,x)

[Out]

a**2*d**2*x + 2*a**2*d*e*x**3/3 + 4*a*c*d*e*x**7/7 + 2*c**2*d*e*x**11/11 + c**2*e**2*x**13/13 + x**9*(2*a*c*e*
*2/9 + c**2*d**2/9) + x**5*(a**2*e**2/5 + 2*a*c*d**2/5)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.92 \[ \int \left (d+e x^2\right )^2 \left (a+c x^4\right )^2 \, dx=\frac {1}{13} \, c^{2} e^{2} x^{13} + \frac {2}{11} \, c^{2} d e x^{11} + \frac {4}{7} \, a c d e x^{7} + \frac {1}{9} \, {\left (c^{2} d^{2} + 2 \, a c e^{2}\right )} x^{9} + \frac {2}{3} \, a^{2} d e x^{3} + \frac {1}{5} \, {\left (2 \, a c d^{2} + a^{2} e^{2}\right )} x^{5} + a^{2} d^{2} x \]

[In]

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

[Out]

1/13*c^2*e^2*x^13 + 2/11*c^2*d*e*x^11 + 4/7*a*c*d*e*x^7 + 1/9*(c^2*d^2 + 2*a*c*e^2)*x^9 + 2/3*a^2*d*e*x^3 + 1/
5*(2*a*c*d^2 + a^2*e^2)*x^5 + a^2*d^2*x

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.94 \[ \int \left (d+e x^2\right )^2 \left (a+c x^4\right )^2 \, dx=\frac {1}{13} \, c^{2} e^{2} x^{13} + \frac {2}{11} \, c^{2} d e x^{11} + \frac {1}{9} \, c^{2} d^{2} x^{9} + \frac {2}{9} \, a c e^{2} x^{9} + \frac {4}{7} \, a c d e x^{7} + \frac {2}{5} \, a c d^{2} x^{5} + \frac {1}{5} \, a^{2} e^{2} x^{5} + \frac {2}{3} \, a^{2} d e x^{3} + a^{2} d^{2} x \]

[In]

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

[Out]

1/13*c^2*e^2*x^13 + 2/11*c^2*d*e*x^11 + 1/9*c^2*d^2*x^9 + 2/9*a*c*e^2*x^9 + 4/7*a*c*d*e*x^7 + 2/5*a*c*d^2*x^5
+ 1/5*a^2*e^2*x^5 + 2/3*a^2*d*e*x^3 + a^2*d^2*x

Mupad [B] (verification not implemented)

Time = 14.07 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.92 \[ \int \left (d+e x^2\right )^2 \left (a+c x^4\right )^2 \, dx=x^5\,\left (\frac {a^2\,e^2}{5}+\frac {2\,c\,a\,d^2}{5}\right )+x^9\,\left (\frac {c^2\,d^2}{9}+\frac {2\,a\,c\,e^2}{9}\right )+a^2\,d^2\,x+\frac {c^2\,e^2\,x^{13}}{13}+\frac {2\,a^2\,d\,e\,x^3}{3}+\frac {2\,c^2\,d\,e\,x^{11}}{11}+\frac {4\,a\,c\,d\,e\,x^7}{7} \]

[In]

int((a + c*x^4)^2*(d + e*x^2)^2,x)

[Out]

x^5*((a^2*e^2)/5 + (2*a*c*d^2)/5) + x^9*((c^2*d^2)/9 + (2*a*c*e^2)/9) + a^2*d^2*x + (c^2*e^2*x^13)/13 + (2*a^2
*d*e*x^3)/3 + (2*c^2*d*e*x^11)/11 + (4*a*c*d*e*x^7)/7