\(\int (a b^2+b^3 x+c x^2+b^2 d x^3+a d^2 x^4)^2 \, dx\) [27]

Optimal result

Integrand size = 34, antiderivative size = 139 \[ \int \left (a b^2+b^3 x+c x^2+b^2 d x^3+a d^2 x^4\right )^2 \, dx=a^2 b^4 x+a b^5 x^2+\frac {1}{3} b^2 \left (b^4+2 a c\right ) x^3+\frac {1}{2} b^3 (c+a b d) x^4+\frac {1}{5} \left (c^2+2 b^2 d \left (b^3+a^2 d\right )\right ) x^5+\frac {1}{3} b^2 d (c+a b d) x^6+\frac {1}{7} \left (b^4+2 a c\right ) d^2 x^7+\frac {1}{4} a b^2 d^3 x^8+\frac {1}{9} a^2 d^4 x^9 \] Output:

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

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.01 \[ \int \left (a b^2+b^3 x+c x^2+b^2 d x^3+a d^2 x^4\right )^2 \, dx=a^2 b^4 x+a b^5 x^2+\frac {1}{3} b^2 \left (b^4+2 a c\right ) x^3+\frac {1}{2} b^3 (c+a b d) x^4+\frac {1}{5} \left (c^2+2 b^5 d+2 a^2 b^2 d^2\right ) x^5+\frac {1}{3} b^2 d (c+a b d) x^6+\frac {1}{7} \left (b^4+2 a c\right ) d^2 x^7+\frac {1}{4} a b^2 d^3 x^8+\frac {1}{9} a^2 d^4 x^9 \] Input:

Integrate[(a*b^2 + b^3*x + c*x^2 + b^2*d*x^3 + a*d^2*x^4)^2,x]
 

Output:

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

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {2465, 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.

\(\displaystyle \int \left (a b^2+a d^2 x^4+b^3 x+b^2 d x^3+c x^2\right )^2 \, dx\)

\(\Big \downarrow \) 2465

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

\(\Big \downarrow \) 2009

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

Input:

Int[(a*b^2 + b^3*x + c*x^2 + b^2*d*x^3 + a*d^2*x^4)^2,x]
 

Output:

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

Defintions of rubi rules used

rule 2009

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

rule 2465

Int[(u_.)*(Px_)^(p_), x_Symbol] :> Int[ExpandToSum[u, Px^p, x], x] /; PolyQ 
[Px, x] && GtQ[Expon[Px, x], 2] &&  !BinomialQ[Px, x] &&  !TrinomialQ[Px, x 
] && IGtQ[p, 0]
 

Maple [A] (verified)

Time = 0.07 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.04

method	result	size
default	\(\frac {a^{2} d^{4} x^{9}}{9}+\frac {a \,b^{2} d^{3} x^{8}}{4}+\frac {\left (b^{4} d^{2}+2 a c \,d^{2}\right ) x^{7}}{7}+\frac {\left (2 b^{3} a \,d^{2}+2 b^{2} c d \right ) x^{6}}{6}+\frac {\left (2 a^{2} b^{2} d^{2}+2 b^{5} d +c^{2}\right ) x^{5}}{5}+\frac {\left (2 a \,b^{4} d +2 b^{3} c \right ) x^{4}}{4}+\frac {\left (b^{6}+2 a \,b^{2} c \right ) x^{3}}{3}+a \,b^{5} x^{2}+a^{2} b^{4} x\)	\(144\)
norman	\(\frac {a^{2} d^{4} x^{9}}{9}+\frac {a \,b^{2} d^{3} x^{8}}{4}+\left (\frac {1}{7} b^{4} d^{2}+\frac {2}{7} a c \,d^{2}\right ) x^{7}+\left (\frac {1}{3} b^{3} a \,d^{2}+\frac {1}{3} b^{2} c d \right ) x^{6}+\left (\frac {2}{5} a^{2} b^{2} d^{2}+\frac {2}{5} b^{5} d +\frac {1}{5} c^{2}\right ) x^{5}+\left (\frac {1}{2} a \,b^{4} d +\frac {1}{2} b^{3} c \right ) x^{4}+\left (\frac {1}{3} b^{6}+\frac {2}{3} a \,b^{2} c \right ) x^{3}+a \,b^{5} x^{2}+a^{2} b^{4} x\)	\(144\)
gosper	\(\frac {1}{9} a^{2} d^{4} x^{9}+\frac {1}{4} a \,b^{2} d^{3} x^{8}+\frac {1}{7} x^{7} b^{4} d^{2}+\frac {2}{7} x^{7} a c \,d^{2}+\frac {1}{3} x^{6} b^{3} a \,d^{2}+\frac {1}{3} x^{6} b^{2} c d +\frac {2}{5} x^{5} a^{2} b^{2} d^{2}+\frac {2}{5} x^{5} b^{5} d +\frac {1}{5} c^{2} x^{5}+\frac {1}{2} x^{4} a \,b^{4} d +\frac {1}{2} x^{4} b^{3} c +\frac {1}{3} x^{3} b^{6}+\frac {2}{3} x^{3} a \,b^{2} c +a \,b^{5} x^{2}+a^{2} b^{4} x\)	\(152\)
risch	\(\frac {1}{9} a^{2} d^{4} x^{9}+\frac {1}{4} a \,b^{2} d^{3} x^{8}+\frac {1}{7} x^{7} b^{4} d^{2}+\frac {2}{7} x^{7} a c \,d^{2}+\frac {1}{3} x^{6} b^{3} a \,d^{2}+\frac {1}{3} x^{6} b^{2} c d +\frac {2}{5} x^{5} a^{2} b^{2} d^{2}+\frac {2}{5} x^{5} b^{5} d +\frac {1}{5} c^{2} x^{5}+\frac {1}{2} x^{4} a \,b^{4} d +\frac {1}{2} x^{4} b^{3} c +\frac {1}{3} x^{3} b^{6}+\frac {2}{3} x^{3} a \,b^{2} c +a \,b^{5} x^{2}+a^{2} b^{4} x\)	\(152\)
parallelrisch	\(\frac {1}{9} a^{2} d^{4} x^{9}+\frac {1}{4} a \,b^{2} d^{3} x^{8}+\frac {1}{7} x^{7} b^{4} d^{2}+\frac {2}{7} x^{7} a c \,d^{2}+\frac {1}{3} x^{6} b^{3} a \,d^{2}+\frac {1}{3} x^{6} b^{2} c d +\frac {2}{5} x^{5} a^{2} b^{2} d^{2}+\frac {2}{5} x^{5} b^{5} d +\frac {1}{5} c^{2} x^{5}+\frac {1}{2} x^{4} a \,b^{4} d +\frac {1}{2} x^{4} b^{3} c +\frac {1}{3} x^{3} b^{6}+\frac {2}{3} x^{3} a \,b^{2} c +a \,b^{5} x^{2}+a^{2} b^{4} x\)	\(152\)
orering	\(\frac {x \left (140 x^{8} d^{4} a^{2}+315 b^{2} d^{3} x^{7} a +180 b^{4} d^{2} x^{6}+420 a \,b^{3} d^{2} x^{5}+504 a^{2} b^{2} d^{2} x^{4}+360 a c \,d^{2} x^{6}+504 b^{5} d \,x^{4}+630 a \,b^{4} d \,x^{3}+420 b^{2} c d \,x^{5}+420 b^{6} x^{2}+1260 b^{5} x a +630 b^{3} c \,x^{3}+1260 b^{4} a^{2}+840 a \,b^{2} c \,x^{2}+252 c^{2} x^{4}\right )}{1260}\)	\(154\)

Input:

int((a*d^2*x^4+b^2*d*x^3+b^3*x+a*b^2+c*x^2)^2,x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.97 \[ \int \left (a b^2+b^3 x+c x^2+b^2 d x^3+a d^2 x^4\right )^2 \, dx=\frac {1}{9} \, a^{2} d^{4} x^{9} + \frac {1}{4} \, a b^{2} d^{3} x^{8} + \frac {1}{7} \, {\left (b^{4} + 2 \, a c\right )} d^{2} x^{7} + a b^{5} x^{2} + a^{2} b^{4} x + \frac {1}{3} \, {\left (a b^{3} d^{2} + b^{2} c d\right )} x^{6} + \frac {1}{5} \, {\left (2 \, b^{5} d + 2 \, a^{2} b^{2} d^{2} + c^{2}\right )} x^{5} + \frac {1}{2} \, {\left (a b^{4} d + b^{3} c\right )} x^{4} + \frac {1}{3} \, {\left (b^{6} + 2 \, a b^{2} c\right )} x^{3} \] Input:

integrate((a*d^2*x^4+b^2*d*x^3+b^3*x+a*b^2+c*x^2)^2,x, algorithm="fricas")
 

Output:

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

Sympy [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.10 \[ \int \left (a b^2+b^3 x+c x^2+b^2 d x^3+a d^2 x^4\right )^2 \, dx=a^{2} b^{4} x + \frac {a^{2} d^{4} x^{9}}{9} + a b^{5} x^{2} + \frac {a b^{2} d^{3} x^{8}}{4} + x^{7} \cdot \left (\frac {2 a c d^{2}}{7} + \frac {b^{4} d^{2}}{7}\right ) + x^{6} \left (\frac {a b^{3} d^{2}}{3} + \frac {b^{2} c d}{3}\right ) + x^{5} \cdot \left (\frac {2 a^{2} b^{2} d^{2}}{5} + \frac {2 b^{5} d}{5} + \frac {c^{2}}{5}\right ) + x^{4} \left (\frac {a b^{4} d}{2} + \frac {b^{3} c}{2}\right ) + x^{3} \cdot \left (\frac {2 a b^{2} c}{3} + \frac {b^{6}}{3}\right ) \] Input:

integrate((a*d**2*x**4+b**2*d*x**3+b**3*x+a*b**2+c*x**2)**2,x)
 

Output:

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

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.08 \[ \int \left (a b^2+b^3 x+c x^2+b^2 d x^3+a d^2 x^4\right )^2 \, dx=\frac {1}{9} \, a^{2} d^{4} x^{9} + \frac {1}{4} \, a b^{2} d^{3} x^{8} + \frac {1}{7} \, b^{4} d^{2} x^{7} + \frac {1}{3} \, b^{6} x^{3} + a^{2} b^{4} x + \frac {1}{5} \, c^{2} x^{5} + \frac {1}{30} \, {\left (12 \, a d^{2} x^{5} + 15 \, b^{2} d x^{4} + 30 \, b^{3} x^{2} + 20 \, c x^{3}\right )} a b^{2} + \frac {1}{30} \, {\left (10 \, a d^{2} x^{6} + 12 \, b^{2} d x^{5} + 15 \, c x^{4}\right )} b^{3} + \frac {1}{21} \, {\left (6 \, a d^{2} x^{7} + 7 \, b^{2} d x^{6}\right )} c \] Input:

integrate((a*d^2*x^4+b^2*d*x^3+b^3*x+a*b^2+c*x^2)^2,x, algorithm="maxima")
 

Output:

1/9*a^2*d^4*x^9 + 1/4*a*b^2*d^3*x^8 + 1/7*b^4*d^2*x^7 + 1/3*b^6*x^3 + a^2* 
b^4*x + 1/5*c^2*x^5 + 1/30*(12*a*d^2*x^5 + 15*b^2*d*x^4 + 30*b^3*x^2 + 20* 
c*x^3)*a*b^2 + 1/30*(10*a*d^2*x^6 + 12*b^2*d*x^5 + 15*c*x^4)*b^3 + 1/21*(6 
*a*d^2*x^7 + 7*b^2*d*x^6)*c
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.09 \[ \int \left (a b^2+b^3 x+c x^2+b^2 d x^3+a d^2 x^4\right )^2 \, dx=\frac {1}{9} \, a^{2} d^{4} x^{9} + \frac {1}{4} \, a b^{2} d^{3} x^{8} + \frac {1}{7} \, b^{4} d^{2} x^{7} + \frac {1}{3} \, a b^{3} d^{2} x^{6} + \frac {2}{5} \, b^{5} d x^{5} + \frac {2}{5} \, a^{2} b^{2} d^{2} x^{5} + \frac {2}{7} \, a c d^{2} x^{7} + \frac {1}{2} \, a b^{4} d x^{4} + \frac {1}{3} \, b^{2} c d x^{6} + \frac {1}{3} \, b^{6} x^{3} + a b^{5} x^{2} + \frac {1}{2} \, b^{3} c x^{4} + a^{2} b^{4} x + \frac {2}{3} \, a b^{2} c x^{3} + \frac {1}{5} \, c^{2} x^{5} \] Input:

integrate((a*d^2*x^4+b^2*d*x^3+b^3*x+a*b^2+c*x^2)^2,x, algorithm="giac")
                                                                                    
                                                                                    
 

Output:

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

Mupad [B] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.93 \[ \int \left (a b^2+b^3 x+c x^2+b^2 d x^3+a d^2 x^4\right )^2 \, dx=x^3\,\left (\frac {b^6}{3}+\frac {2\,a\,c\,b^2}{3}\right )+x^5\,\left (\frac {2\,a^2\,b^2\,d^2}{5}+\frac {2\,b^5\,d}{5}+\frac {c^2}{5}\right )+\frac {d^2\,x^7\,\left (b^4+2\,a\,c\right )}{7}+\frac {b^3\,x^4\,\left (c+a\,b\,d\right )}{2}+a^2\,b^4\,x+a\,b^5\,x^2+\frac {a^2\,d^4\,x^9}{9}+\frac {a\,b^2\,d^3\,x^8}{4}+\frac {b^2\,d\,x^6\,\left (c+a\,b\,d\right )}{3} \] Input:

int((a*b^2 + b^3*x + c*x^2 + a*d^2*x^4 + b^2*d*x^3)^2,x)
 

Output:

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

Reduce [B] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.10 \[ \int \left (a b^2+b^3 x+c x^2+b^2 d x^3+a d^2 x^4\right )^2 \, dx=\frac {x \left (140 a^{2} d^{4} x^{8}+315 a \,b^{2} d^{3} x^{7}+180 b^{4} d^{2} x^{6}+420 a \,b^{3} d^{2} x^{5}+504 a^{2} b^{2} d^{2} x^{4}+360 a c \,d^{2} x^{6}+504 b^{5} d \,x^{4}+630 a \,b^{4} d \,x^{3}+420 b^{2} c d \,x^{5}+420 b^{6} x^{2}+1260 a \,b^{5} x +630 b^{3} c \,x^{3}+1260 a^{2} b^{4}+840 a \,b^{2} c \,x^{2}+252 c^{2} x^{4}\right )}{1260} \] Input:

int((a*d^2*x^4+b^2*d*x^3+b^3*x+a*b^2+c*x^2)^2,x)
 

Output:

(x*(1260*a**2*b**4 + 504*a**2*b**2*d**2*x**4 + 140*a**2*d**4*x**8 + 1260*a 
*b**5*x + 630*a*b**4*d*x**3 + 420*a*b**3*d**2*x**5 + 840*a*b**2*c*x**2 + 3 
15*a*b**2*d**3*x**7 + 360*a*c*d**2*x**6 + 420*b**6*x**2 + 504*b**5*d*x**4 
+ 180*b**4*d**2*x**6 + 630*b**3*c*x**3 + 420*b**2*c*d*x**5 + 252*c**2*x**4 
))/1260