\(\int (d+e x)^2 (a+b x^2+c x^4)^2 \, dx\) [230]

Optimal result

Integrand size = 22, antiderivative size = 177 \[ \int (d+e x)^2 \left (a+b x^2+c x^4\right )^2 \, dx=a^2 d^2 x+a^2 d e x^2+\frac {1}{3} a \left (2 b d^2+a e^2\right ) x^3+a b d e x^4+\frac {1}{5} \left (b^2 d^2+2 a c d^2+2 a b e^2\right ) x^5+\frac {1}{3} \left (b^2+2 a c\right ) d e x^6+\frac {1}{7} \left (2 b c d^2+b^2 e^2+2 a c e^2\right ) x^7+\frac {1}{2} b c d e x^8+\frac {1}{9} c \left (c d^2+2 b e^2\right ) x^9+\frac {1}{5} c^2 d e x^{10}+\frac {1}{11} c^2 e^2 x^{11} \] Output:

a^2*d^2*x+a^2*d*e*x^2+1/3*a*(a*e^2+2*b*d^2)*x^3+a*b*d*e*x^4+1/5*(2*a*b*e^2 
+2*a*c*d^2+b^2*d^2)*x^5+1/3*(2*a*c+b^2)*d*e*x^6+1/7*(2*a*c*e^2+b^2*e^2+2*b 
*c*d^2)*x^7+1/2*b*c*d*e*x^8+1/9*c*(2*b*e^2+c*d^2)*x^9+1/5*c^2*d*e*x^10+1/1 
1*c^2*e^2*x^11
 

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.00 \[ \int (d+e x)^2 \left (a+b x^2+c x^4\right )^2 \, dx=a^2 d^2 x+a^2 d e x^2+\frac {1}{3} a \left (2 b d^2+a e^2\right ) x^3+a b d e x^4+\frac {1}{5} \left (b^2 d^2+2 a c d^2+2 a b e^2\right ) x^5+\frac {1}{3} \left (b^2+2 a c\right ) d e x^6+\frac {1}{7} \left (2 b c d^2+b^2 e^2+2 a c e^2\right ) x^7+\frac {1}{2} b c d e x^8+\frac {1}{9} c \left (c d^2+2 b e^2\right ) x^9+\frac {1}{5} c^2 d e x^{10}+\frac {1}{11} c^2 e^2 x^{11} \] Input:

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

Output:

a^2*d^2*x + a^2*d*e*x^2 + (a*(2*b*d^2 + a*e^2)*x^3)/3 + a*b*d*e*x^4 + ((b^ 
2*d^2 + 2*a*c*d^2 + 2*a*b*e^2)*x^5)/5 + ((b^2 + 2*a*c)*d*e*x^6)/3 + ((2*b* 
c*d^2 + b^2*e^2 + 2*a*c*e^2)*x^7)/7 + (b*c*d*e*x^8)/2 + (c*(c*d^2 + 2*b*e^ 
2)*x^9)/9 + (c^2*d*e*x^10)/5 + (c^2*e^2*x^11)/11
 

Rubi [A] (verified)

Time = 0.70 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2200, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int (d+e x)^2 \left (a+b x^2+c x^4\right )^2 \, dx\)

\(\Big \downarrow \) 2200

\(\displaystyle \int \left (a^2 d^2+2 a^2 d e x+x^6 \left (2 a c e^2+b^2 e^2+2 b c d^2\right )+x^4 \left (2 a b e^2+2 a c d^2+b^2 d^2\right )+2 d e x^5 \left (2 a c+b^2\right )+a x^2 \left (a e^2+2 b d^2\right )+4 a b d e x^3+c x^8 \left (2 b e^2+c d^2\right )+4 b c d e x^7+2 c^2 d e x^9+c^2 e^2 x^{10}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle a^2 d^2 x+a^2 d e x^2+\frac {1}{7} x^7 \left (2 a c e^2+b^2 e^2+2 b c d^2\right )+\frac {1}{5} x^5 \left (2 a b e^2+2 a c d^2+b^2 d^2\right )+\frac {1}{3} d e x^6 \left (2 a c+b^2\right )+\frac {1}{3} a x^3 \left (a e^2+2 b d^2\right )+a b d e x^4+\frac {1}{9} c x^9 \left (2 b e^2+c d^2\right )+\frac {1}{2} b c d e x^8+\frac {1}{5} c^2 d e x^{10}+\frac {1}{11} c^2 e^2 x^{11}\)

Input:

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

Output:

a^2*d^2*x + a^2*d*e*x^2 + (a*(2*b*d^2 + a*e^2)*x^3)/3 + a*b*d*e*x^4 + ((b^ 
2*d^2 + 2*a*c*d^2 + 2*a*b*e^2)*x^5)/5 + ((b^2 + 2*a*c)*d*e*x^6)/3 + ((2*b* 
c*d^2 + b^2*e^2 + 2*a*c*e^2)*x^7)/7 + (b*c*d*e*x^8)/2 + (c*(c*d^2 + 2*b*e^ 
2)*x^9)/9 + (c^2*d*e*x^10)/5 + (c^2*e^2*x^11)/11
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2200

Int[(Px_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[Expa 
ndIntegrand[Px*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c}, x] && Poly 
Q[Px, x] && IGtQ[p, 0]
 

Maple [A] (verified)

Time = 0.16 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.92

method	result	size
default	\(\frac {c^{2} e^{2} x^{11}}{11}+\frac {c^{2} d e \,x^{10}}{5}+\frac {\left (2 b c \,e^{2}+c^{2} d^{2}\right ) x^{9}}{9}+\frac {b c d e \,x^{8}}{2}+\frac {\left (2 b c \,d^{2}+e^{2} \left (2 a c +b^{2}\right )\right ) x^{7}}{7}+\frac {\left (2 a c +b^{2}\right ) d e \,x^{6}}{3}+\frac {\left (d^{2} \left (2 a c +b^{2}\right )+2 a b \,e^{2}\right ) x^{5}}{5}+a b d e \,x^{4}+\frac {\left (a^{2} e^{2}+2 a b \,d^{2}\right ) x^{3}}{3}+a^{2} d e \,x^{2}+x \,a^{2} d^{2}\)	\(162\)
norman	\(\frac {c^{2} e^{2} x^{11}}{11}+\frac {c^{2} d e \,x^{10}}{5}+\left (\frac {2}{9} b c \,e^{2}+\frac {1}{9} c^{2} d^{2}\right ) x^{9}+\frac {b c d e \,x^{8}}{2}+\left (\frac {2}{7} a c \,e^{2}+\frac {1}{7} b^{2} e^{2}+\frac {2}{7} b c \,d^{2}\right ) x^{7}+\left (\frac {2}{3} a c d e +\frac {1}{3} b^{2} d e \right ) x^{6}+\left (\frac {2}{5} a b \,e^{2}+\frac {2}{5} a c \,d^{2}+\frac {1}{5} b^{2} d^{2}\right ) x^{5}+a b d e \,x^{4}+\left (\frac {1}{3} a^{2} e^{2}+\frac {2}{3} a b \,d^{2}\right ) x^{3}+a^{2} d e \,x^{2}+x \,a^{2} d^{2}\)	\(169\)
gosper	\(\frac {1}{11} c^{2} e^{2} x^{11}+\frac {1}{5} c^{2} d e \,x^{10}+\frac {2}{9} x^{9} b c \,e^{2}+\frac {1}{9} c^{2} d^{2} x^{9}+\frac {1}{2} b c d e \,x^{8}+\frac {2}{7} a c \,e^{2} x^{7}+\frac {1}{7} x^{7} b^{2} e^{2}+\frac {2}{7} x^{7} b c \,d^{2}+\frac {2}{3} a c d e \,x^{6}+\frac {1}{3} x^{6} b^{2} d e +\frac {2}{5} x^{5} a b \,e^{2}+\frac {2}{5} a c \,d^{2} x^{5}+\frac {1}{5} x^{5} b^{2} d^{2}+a b d e \,x^{4}+\frac {1}{3} x^{3} a^{2} e^{2}+\frac {2}{3} a b \,d^{2} x^{3}+a^{2} d e \,x^{2}+x \,a^{2} d^{2}\)	\(180\)
risch	\(\frac {1}{11} c^{2} e^{2} x^{11}+\frac {1}{5} c^{2} d e \,x^{10}+\frac {2}{9} x^{9} b c \,e^{2}+\frac {1}{9} c^{2} d^{2} x^{9}+\frac {1}{2} b c d e \,x^{8}+\frac {2}{7} a c \,e^{2} x^{7}+\frac {1}{7} x^{7} b^{2} e^{2}+\frac {2}{7} x^{7} b c \,d^{2}+\frac {2}{3} a c d e \,x^{6}+\frac {1}{3} x^{6} b^{2} d e +\frac {2}{5} x^{5} a b \,e^{2}+\frac {2}{5} a c \,d^{2} x^{5}+\frac {1}{5} x^{5} b^{2} d^{2}+a b d e \,x^{4}+\frac {1}{3} x^{3} a^{2} e^{2}+\frac {2}{3} a b \,d^{2} x^{3}+a^{2} d e \,x^{2}+x \,a^{2} d^{2}\)	\(180\)
parallelrisch	\(\frac {1}{11} c^{2} e^{2} x^{11}+\frac {1}{5} c^{2} d e \,x^{10}+\frac {2}{9} x^{9} b c \,e^{2}+\frac {1}{9} c^{2} d^{2} x^{9}+\frac {1}{2} b c d e \,x^{8}+\frac {2}{7} a c \,e^{2} x^{7}+\frac {1}{7} x^{7} b^{2} e^{2}+\frac {2}{7} x^{7} b c \,d^{2}+\frac {2}{3} a c d e \,x^{6}+\frac {1}{3} x^{6} b^{2} d e +\frac {2}{5} x^{5} a b \,e^{2}+\frac {2}{5} a c \,d^{2} x^{5}+\frac {1}{5} x^{5} b^{2} d^{2}+a b d e \,x^{4}+\frac {1}{3} x^{3} a^{2} e^{2}+\frac {2}{3} a b \,d^{2} x^{3}+a^{2} d e \,x^{2}+x \,a^{2} d^{2}\)	\(180\)
orering	\(\frac {x \left (630 c^{2} e^{2} x^{10}+1386 c^{2} d e \,x^{9}+1540 b c \,e^{2} x^{8}+770 c^{2} d^{2} x^{8}+3465 b c d e \,x^{7}+1980 a c \,e^{2} x^{6}+990 b^{2} e^{2} x^{6}+1980 b c \,d^{2} x^{6}+4620 a c d e \,x^{5}+2310 b^{2} d e \,x^{5}+2772 a b \,e^{2} x^{4}+2772 d^{2} x^{4} a c +1386 b^{2} d^{2} x^{4}+6930 a b d e \,x^{3}+2310 a^{2} e^{2} x^{2}+4620 a b \,d^{2} x^{2}+6930 a^{2} d e x +6930 a^{2} d^{2}\right )}{6930}\)	\(183\)

Input:

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

Output:

1/11*c^2*e^2*x^11+1/5*c^2*d*e*x^10+1/9*(2*b*c*e^2+c^2*d^2)*x^9+1/2*b*c*d*e 
*x^8+1/7*(2*b*c*d^2+e^2*(2*a*c+b^2))*x^7+1/3*(2*a*c+b^2)*d*e*x^6+1/5*(d^2* 
(2*a*c+b^2)+2*a*b*e^2)*x^5+a*b*d*e*x^4+1/3*(a^2*e^2+2*a*b*d^2)*x^3+a^2*d*e 
*x^2+x*a^2*d^2
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.91 \[ \int (d+e x)^2 \left (a+b x^2+c x^4\right )^2 \, dx=\frac {1}{11} \, c^{2} e^{2} x^{11} + \frac {1}{5} \, c^{2} d e x^{10} + \frac {1}{2} \, b c d e x^{8} + \frac {1}{9} \, {\left (c^{2} d^{2} + 2 \, b c e^{2}\right )} x^{9} + \frac {1}{3} \, {\left (b^{2} + 2 \, a c\right )} d e x^{6} + a b d e x^{4} + \frac {1}{7} \, {\left (2 \, b c d^{2} + {\left (b^{2} + 2 \, a c\right )} e^{2}\right )} x^{7} + a^{2} d e x^{2} + \frac {1}{5} \, {\left (2 \, a b e^{2} + {\left (b^{2} + 2 \, a c\right )} d^{2}\right )} x^{5} + a^{2} d^{2} x + \frac {1}{3} \, {\left (2 \, a b d^{2} + a^{2} e^{2}\right )} x^{3} \] Input:

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

Output:

1/11*c^2*e^2*x^11 + 1/5*c^2*d*e*x^10 + 1/2*b*c*d*e*x^8 + 1/9*(c^2*d^2 + 2* 
b*c*e^2)*x^9 + 1/3*(b^2 + 2*a*c)*d*e*x^6 + a*b*d*e*x^4 + 1/7*(2*b*c*d^2 + 
(b^2 + 2*a*c)*e^2)*x^7 + a^2*d*e*x^2 + 1/5*(2*a*b*e^2 + (b^2 + 2*a*c)*d^2) 
*x^5 + a^2*d^2*x + 1/3*(2*a*b*d^2 + a^2*e^2)*x^3
 

Sympy [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.10 \[ \int (d+e x)^2 \left (a+b x^2+c x^4\right )^2 \, dx=a^{2} d^{2} x + a^{2} d e x^{2} + a b d e x^{4} + \frac {b c d e x^{8}}{2} + \frac {c^{2} d e x^{10}}{5} + \frac {c^{2} e^{2} x^{11}}{11} + x^{9} \cdot \left (\frac {2 b c e^{2}}{9} + \frac {c^{2} d^{2}}{9}\right ) + x^{7} \cdot \left (\frac {2 a c e^{2}}{7} + \frac {b^{2} e^{2}}{7} + \frac {2 b c d^{2}}{7}\right ) + x^{6} \cdot \left (\frac {2 a c d e}{3} + \frac {b^{2} d e}{3}\right ) + x^{5} \cdot \left (\frac {2 a b e^{2}}{5} + \frac {2 a c d^{2}}{5} + \frac {b^{2} d^{2}}{5}\right ) + x^{3} \left (\frac {a^{2} e^{2}}{3} + \frac {2 a b d^{2}}{3}\right ) \] Input:

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

Output:

a**2*d**2*x + a**2*d*e*x**2 + a*b*d*e*x**4 + b*c*d*e*x**8/2 + c**2*d*e*x** 
10/5 + c**2*e**2*x**11/11 + x**9*(2*b*c*e**2/9 + c**2*d**2/9) + x**7*(2*a* 
c*e**2/7 + b**2*e**2/7 + 2*b*c*d**2/7) + x**6*(2*a*c*d*e/3 + b**2*d*e/3) + 
 x**5*(2*a*b*e**2/5 + 2*a*c*d**2/5 + b**2*d**2/5) + x**3*(a**2*e**2/3 + 2* 
a*b*d**2/3)
 

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.91 \[ \int (d+e x)^2 \left (a+b x^2+c x^4\right )^2 \, dx=\frac {1}{11} \, c^{2} e^{2} x^{11} + \frac {1}{5} \, c^{2} d e x^{10} + \frac {1}{2} \, b c d e x^{8} + \frac {1}{9} \, {\left (c^{2} d^{2} + 2 \, b c e^{2}\right )} x^{9} + \frac {1}{3} \, {\left (b^{2} + 2 \, a c\right )} d e x^{6} + a b d e x^{4} + \frac {1}{7} \, {\left (2 \, b c d^{2} + {\left (b^{2} + 2 \, a c\right )} e^{2}\right )} x^{7} + a^{2} d e x^{2} + \frac {1}{5} \, {\left (2 \, a b e^{2} + {\left (b^{2} + 2 \, a c\right )} d^{2}\right )} x^{5} + a^{2} d^{2} x + \frac {1}{3} \, {\left (2 \, a b d^{2} + a^{2} e^{2}\right )} x^{3} \] Input:

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

Output:

1/11*c^2*e^2*x^11 + 1/5*c^2*d*e*x^10 + 1/2*b*c*d*e*x^8 + 1/9*(c^2*d^2 + 2* 
b*c*e^2)*x^9 + 1/3*(b^2 + 2*a*c)*d*e*x^6 + a*b*d*e*x^4 + 1/7*(2*b*c*d^2 + 
(b^2 + 2*a*c)*e^2)*x^7 + a^2*d*e*x^2 + 1/5*(2*a*b*e^2 + (b^2 + 2*a*c)*d^2) 
*x^5 + a^2*d^2*x + 1/3*(2*a*b*d^2 + a^2*e^2)*x^3
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.01 \[ \int (d+e x)^2 \left (a+b x^2+c x^4\right )^2 \, dx=\frac {1}{11} \, c^{2} e^{2} x^{11} + \frac {1}{5} \, c^{2} d e x^{10} + \frac {1}{9} \, c^{2} d^{2} x^{9} + \frac {2}{9} \, b c e^{2} x^{9} + \frac {1}{2} \, b c d e x^{8} + \frac {2}{7} \, b c d^{2} x^{7} + \frac {1}{7} \, b^{2} e^{2} x^{7} + \frac {2}{7} \, a c e^{2} x^{7} + \frac {1}{3} \, b^{2} d e x^{6} + \frac {2}{3} \, a c d e x^{6} + \frac {1}{5} \, b^{2} d^{2} x^{5} + \frac {2}{5} \, a c d^{2} x^{5} + \frac {2}{5} \, a b e^{2} x^{5} + a b d e x^{4} + \frac {2}{3} \, a b d^{2} x^{3} + \frac {1}{3} \, a^{2} e^{2} x^{3} + a^{2} d e x^{2} + a^{2} d^{2} x \] Input:

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

Output:

1/11*c^2*e^2*x^11 + 1/5*c^2*d*e*x^10 + 1/9*c^2*d^2*x^9 + 2/9*b*c*e^2*x^9 + 
 1/2*b*c*d*e*x^8 + 2/7*b*c*d^2*x^7 + 1/7*b^2*e^2*x^7 + 2/7*a*c*e^2*x^7 + 1 
/3*b^2*d*e*x^6 + 2/3*a*c*d*e*x^6 + 1/5*b^2*d^2*x^5 + 2/5*a*c*d^2*x^5 + 2/5 
*a*b*e^2*x^5 + a*b*d*e*x^4 + 2/3*a*b*d^2*x^3 + 1/3*a^2*e^2*x^3 + a^2*d*e*x 
^2 + a^2*d^2*x
 

Mupad [B] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.93 \[ \int (d+e x)^2 \left (a+b x^2+c x^4\right )^2 \, dx=x^5\,\left (\frac {b^2\,d^2}{5}+\frac {2\,a\,b\,e^2}{5}+\frac {2\,a\,c\,d^2}{5}\right )+x^7\,\left (\frac {b^2\,e^2}{7}+\frac {2\,c\,b\,d^2}{7}+\frac {2\,a\,c\,e^2}{7}\right )+x^3\,\left (\frac {a^2\,e^2}{3}+\frac {2\,b\,a\,d^2}{3}\right )+x^9\,\left (\frac {c^2\,d^2}{9}+\frac {2\,b\,c\,e^2}{9}\right )+a^2\,d^2\,x+\frac {c^2\,e^2\,x^{11}}{11}+\frac {d\,e\,x^6\,\left (b^2+2\,a\,c\right )}{3}+a^2\,d\,e\,x^2+\frac {c^2\,d\,e\,x^{10}}{5}+a\,b\,d\,e\,x^4+\frac {b\,c\,d\,e\,x^8}{2} \] Input:

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

Output:

x^5*((b^2*d^2)/5 + (2*a*b*e^2)/5 + (2*a*c*d^2)/5) + x^7*((b^2*e^2)/7 + (2* 
a*c*e^2)/7 + (2*b*c*d^2)/7) + x^3*((a^2*e^2)/3 + (2*a*b*d^2)/3) + x^9*((c^ 
2*d^2)/9 + (2*b*c*e^2)/9) + a^2*d^2*x + (c^2*e^2*x^11)/11 + (d*e*x^6*(2*a* 
c + b^2))/3 + a^2*d*e*x^2 + (c^2*d*e*x^10)/5 + a*b*d*e*x^4 + (b*c*d*e*x^8) 
/2
 

Reduce [B] (verification not implemented)

Time = 0.39 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.03 \[ \int (d+e x)^2 \left (a+b x^2+c x^4\right )^2 \, dx=\frac {x \left (630 c^{2} e^{2} x^{10}+1386 c^{2} d e \,x^{9}+1540 b c \,e^{2} x^{8}+770 c^{2} d^{2} x^{8}+3465 b c d e \,x^{7}+1980 a c \,e^{2} x^{6}+990 b^{2} e^{2} x^{6}+1980 b c \,d^{2} x^{6}+4620 a c d e \,x^{5}+2310 b^{2} d e \,x^{5}+2772 a b \,e^{2} x^{4}+2772 a c \,d^{2} x^{4}+1386 b^{2} d^{2} x^{4}+6930 a b d e \,x^{3}+2310 a^{2} e^{2} x^{2}+4620 a b \,d^{2} x^{2}+6930 a^{2} d e x +6930 a^{2} d^{2}\right )}{6930} \] Input:

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

Output:

(x*(6930*a**2*d**2 + 6930*a**2*d*e*x + 2310*a**2*e**2*x**2 + 4620*a*b*d**2 
*x**2 + 6930*a*b*d*e*x**3 + 2772*a*b*e**2*x**4 + 2772*a*c*d**2*x**4 + 4620 
*a*c*d*e*x**5 + 1980*a*c*e**2*x**6 + 1386*b**2*d**2*x**4 + 2310*b**2*d*e*x 
**5 + 990*b**2*e**2*x**6 + 1980*b*c*d**2*x**6 + 3465*b*c*d*e*x**7 + 1540*b 
*c*e**2*x**8 + 770*c**2*d**2*x**8 + 1386*c**2*d*e*x**9 + 630*c**2*e**2*x** 
10))/6930