\(\int (d+e x^2)^2 (a+b x^2+c x^4)^2 \, dx\) [253]

Optimal result

Integrand size = 24, antiderivative size = 155 \[ \int \left (d+e x^2\right )^2 \left (a+b x^2+c x^4\right )^2 \, dx=a^2 d^2 x+\frac {2}{3} a d (b d+a e) x^3+\frac {1}{5} \left (b^2 d^2+4 a b d e+a \left (2 c d^2+a e^2\right )\right ) x^5+\frac {2}{7} \left (b c d^2+b^2 d e+2 a c d e+a b e^2\right ) x^7+\frac {1}{9} \left (c^2 d^2+b^2 e^2+2 c e (2 b d+a e)\right ) x^9+\frac {2}{11} c e (c d+b e) x^{11}+\frac {1}{13} c^2 e^2 x^{13} \]

[Out]

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

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {1167} \[ \int \left (d+e x^2\right )^2 \left (a+b x^2+c x^4\right )^2 \, dx=a^2 d^2 x+\frac {1}{9} x^9 \left (2 c e (a e+2 b d)+b^2 e^2+c^2 d^2\right )+\frac {2}{7} x^7 \left (a b e^2+2 a c d e+b^2 d e+b c d^2\right )+\frac {1}{5} x^5 \left (4 a b d e+a \left (a e^2+2 c d^2\right )+b^2 d^2\right )+\frac {2}{3} a d x^3 (a e+b d)+\frac {2}{11} c e x^{11} (b e+c d)+\frac {1}{13} c^2 e^2 x^{13} \]

[In]

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

[Out]

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

Rule 1167

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

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

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.01 \[ \int \left (d+e x^2\right )^2 \left (a+b x^2+c x^4\right )^2 \, dx=a^2 d^2 x+\frac {2}{3} a d (b d+a e) x^3+\frac {1}{5} \left (b^2 d^2+2 a c d^2+4 a b d e+a^2 e^2\right ) x^5+\frac {2}{7} \left (b c d^2+b^2 d e+2 a c d e+a b e^2\right ) x^7+\frac {1}{9} \left (c^2 d^2+4 b c d e+b^2 e^2+2 a c e^2\right ) x^9+\frac {2}{11} c e (c d+b e) x^{11}+\frac {1}{13} c^2 e^2 x^{13} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.26 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.00

method	result	size
default	\(\frac {c^{2} e^{2} x^{13}}{13}+\frac {\left (2 b c \,e^{2}+2 e d \,c^{2}\right ) x^{11}}{11}+\frac {\left (c^{2} d^{2}+4 b c d e +e^{2} \left (2 a c +b^{2}\right )\right ) x^{9}}{9}+\frac {\left (2 b c \,d^{2}+2 e d \left (2 a c +b^{2}\right )+2 a b \,e^{2}\right ) x^{7}}{7}+\frac {\left (d^{2} \left (2 a c +b^{2}\right )+4 a b d e +e^{2} a^{2}\right ) x^{5}}{5}+\frac {\left (2 e d \,a^{2}+2 d^{2} a b \right ) x^{3}}{3}+a^{2} d^{2} x\)	\(155\)
norman	\(\frac {c^{2} e^{2} x^{13}}{13}+\left (\frac {2}{11} b c \,e^{2}+\frac {2}{11} e d \,c^{2}\right ) x^{11}+\left (\frac {2}{9} e^{2} a c +\frac {1}{9} b^{2} e^{2}+\frac {4}{9} b c d e +\frac {1}{9} c^{2} d^{2}\right ) x^{9}+\left (\frac {2}{7} a b \,e^{2}+\frac {4}{7} a c d e +\frac {2}{7} b^{2} d e +\frac {2}{7} b c \,d^{2}\right ) x^{7}+\left (\frac {1}{5} e^{2} a^{2}+\frac {4}{5} a b d e +\frac {2}{5} d^{2} a c +\frac {1}{5} b^{2} d^{2}\right ) x^{5}+\left (\frac {2}{3} e d \,a^{2}+\frac {2}{3} d^{2} a b \right ) x^{3}+a^{2} d^{2} x\)	\(159\)
gosper	\(\frac {1}{13} c^{2} e^{2} x^{13}+\frac {2}{11} x^{11} b c \,e^{2}+\frac {2}{11} c^{2} d e \,x^{11}+\frac {2}{9} x^{9} e^{2} a c +\frac {1}{9} x^{9} b^{2} e^{2}+\frac {4}{9} x^{9} b c d e +\frac {1}{9} x^{9} c^{2} d^{2}+\frac {2}{7} x^{7} a b \,e^{2}+\frac {4}{7} a c d e \,x^{7}+\frac {2}{7} x^{7} b^{2} d e +\frac {2}{7} x^{7} b c \,d^{2}+\frac {1}{5} x^{5} e^{2} a^{2}+\frac {4}{5} x^{5} a b d e +\frac {2}{5} x^{5} d^{2} a c +\frac {1}{5} x^{5} b^{2} d^{2}+\frac {2}{3} a^{2} d e \,x^{3}+\frac {2}{3} x^{3} d^{2} a b +a^{2} d^{2} x\)	\(182\)
risch	\(\frac {1}{13} c^{2} e^{2} x^{13}+\frac {2}{11} x^{11} b c \,e^{2}+\frac {2}{11} c^{2} d e \,x^{11}+\frac {2}{9} x^{9} e^{2} a c +\frac {1}{9} x^{9} b^{2} e^{2}+\frac {4}{9} x^{9} b c d e +\frac {1}{9} x^{9} c^{2} d^{2}+\frac {2}{7} x^{7} a b \,e^{2}+\frac {4}{7} a c d e \,x^{7}+\frac {2}{7} x^{7} b^{2} d e +\frac {2}{7} x^{7} b c \,d^{2}+\frac {1}{5} x^{5} e^{2} a^{2}+\frac {4}{5} x^{5} a b d e +\frac {2}{5} x^{5} d^{2} a c +\frac {1}{5} x^{5} b^{2} d^{2}+\frac {2}{3} a^{2} d e \,x^{3}+\frac {2}{3} x^{3} d^{2} a b +a^{2} d^{2} x\)	\(182\)
parallelrisch	\(\frac {1}{13} c^{2} e^{2} x^{13}+\frac {2}{11} x^{11} b c \,e^{2}+\frac {2}{11} c^{2} d e \,x^{11}+\frac {2}{9} x^{9} e^{2} a c +\frac {1}{9} x^{9} b^{2} e^{2}+\frac {4}{9} x^{9} b c d e +\frac {1}{9} x^{9} c^{2} d^{2}+\frac {2}{7} x^{7} a b \,e^{2}+\frac {4}{7} a c d e \,x^{7}+\frac {2}{7} x^{7} b^{2} d e +\frac {2}{7} x^{7} b c \,d^{2}+\frac {1}{5} x^{5} e^{2} a^{2}+\frac {4}{5} x^{5} a b d e +\frac {2}{5} x^{5} d^{2} a c +\frac {1}{5} x^{5} b^{2} d^{2}+\frac {2}{3} a^{2} d e \,x^{3}+\frac {2}{3} x^{3} d^{2} a b +a^{2} d^{2} x\)	\(182\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.95 \[ \int \left (d+e x^2\right )^2 \left (a+b x^2+c x^4\right )^2 \, dx=\frac {1}{13} \, c^{2} e^{2} x^{13} + \frac {2}{11} \, {\left (c^{2} d e + b c e^{2}\right )} x^{11} + \frac {1}{9} \, {\left (c^{2} d^{2} + 4 \, b c d e + {\left (b^{2} + 2 \, a c\right )} e^{2}\right )} x^{9} + \frac {2}{7} \, {\left (b c d^{2} + a b e^{2} + {\left (b^{2} + 2 \, a c\right )} d e\right )} x^{7} + \frac {1}{5} \, {\left (4 \, a b d e + a^{2} e^{2} + {\left (b^{2} + 2 \, a c\right )} d^{2}\right )} x^{5} + a^{2} d^{2} x + \frac {2}{3} \, {\left (a b d^{2} + a^{2} d e\right )} x^{3} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.24 \[ \int \left (d+e x^2\right )^2 \left (a+b x^2+c x^4\right )^2 \, dx=a^{2} d^{2} x + \frac {c^{2} e^{2} x^{13}}{13} + x^{11} \cdot \left (\frac {2 b c e^{2}}{11} + \frac {2 c^{2} d e}{11}\right ) + x^{9} \cdot \left (\frac {2 a c e^{2}}{9} + \frac {b^{2} e^{2}}{9} + \frac {4 b c d e}{9} + \frac {c^{2} d^{2}}{9}\right ) + x^{7} \cdot \left (\frac {2 a b e^{2}}{7} + \frac {4 a c d e}{7} + \frac {2 b^{2} d e}{7} + \frac {2 b c d^{2}}{7}\right ) + x^{5} \left (\frac {a^{2} e^{2}}{5} + \frac {4 a b d e}{5} + \frac {2 a c d^{2}}{5} + \frac {b^{2} d^{2}}{5}\right ) + x^{3} \cdot \left (\frac {2 a^{2} d e}{3} + \frac {2 a b d^{2}}{3}\right ) \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.95 \[ \int \left (d+e x^2\right )^2 \left (a+b x^2+c x^4\right )^2 \, dx=\frac {1}{13} \, c^{2} e^{2} x^{13} + \frac {2}{11} \, {\left (c^{2} d e + b c e^{2}\right )} x^{11} + \frac {1}{9} \, {\left (c^{2} d^{2} + 4 \, b c d e + {\left (b^{2} + 2 \, a c\right )} e^{2}\right )} x^{9} + \frac {2}{7} \, {\left (b c d^{2} + a b e^{2} + {\left (b^{2} + 2 \, a c\right )} d e\right )} x^{7} + \frac {1}{5} \, {\left (4 \, a b d e + a^{2} e^{2} + {\left (b^{2} + 2 \, a c\right )} d^{2}\right )} x^{5} + a^{2} d^{2} x + \frac {2}{3} \, {\left (a b d^{2} + a^{2} d e\right )} x^{3} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.17 \[ \int \left (d+e x^2\right )^2 \left (a+b x^2+c x^4\right )^2 \, dx=\frac {1}{13} \, c^{2} e^{2} x^{13} + \frac {2}{11} \, c^{2} d e x^{11} + \frac {2}{11} \, b c e^{2} x^{11} + \frac {1}{9} \, c^{2} d^{2} x^{9} + \frac {4}{9} \, b c d e x^{9} + \frac {1}{9} \, b^{2} e^{2} x^{9} + \frac {2}{9} \, a c e^{2} x^{9} + \frac {2}{7} \, b c d^{2} x^{7} + \frac {2}{7} \, b^{2} d e x^{7} + \frac {4}{7} \, a c d e x^{7} + \frac {2}{7} \, a b e^{2} x^{7} + \frac {1}{5} \, b^{2} d^{2} x^{5} + \frac {2}{5} \, a c d^{2} x^{5} + \frac {4}{5} \, a b d e x^{5} + \frac {1}{5} \, a^{2} e^{2} x^{5} + \frac {2}{3} \, a b d^{2} x^{3} + \frac {2}{3} \, a^{2} d e x^{3} + a^{2} d^{2} x \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 7.64 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.95 \[ \int \left (d+e x^2\right )^2 \left (a+b x^2+c x^4\right )^2 \, dx=x^5\,\left (\frac {a^2\,e^2}{5}+\frac {4\,a\,b\,d\,e}{5}+\frac {2\,c\,a\,d^2}{5}+\frac {b^2\,d^2}{5}\right )+x^9\,\left (\frac {b^2\,e^2}{9}+\frac {4\,b\,c\,d\,e}{9}+\frac {c^2\,d^2}{9}+\frac {2\,a\,c\,e^2}{9}\right )+x^7\,\left (\frac {2\,b^2\,d\,e}{7}+\frac {2\,c\,b\,d^2}{7}+\frac {2\,a\,b\,e^2}{7}+\frac {4\,a\,c\,d\,e}{7}\right )+a^2\,d^2\,x+\frac {c^2\,e^2\,x^{13}}{13}+\frac {2\,a\,d\,x^3\,\left (a\,e+b\,d\right )}{3}+\frac {2\,c\,e\,x^{11}\,\left (b\,e+c\,d\right )}{11} \]

[In]

int((d + e*x^2)^2*(a + b*x^2 + c*x^4)^2,x)

[Out]

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