\(\int (a+b \sqrt {x}+c x)^3 (d+e \sqrt {x}+f x) \, dx\) [111]

Optimal result

Integrand size = 27, antiderivative size = 242 \[ \int \left (a+b \sqrt {x}+c x\right )^3 \left (d+e \sqrt {x}+f x\right ) \, dx=a^3 d x+\frac {2}{3} a^2 (3 b d+a e) x^{3/2}+\frac {1}{2} a \left (3 b^2 d+3 a b e+a (3 c d+a f)\right ) x^2+\frac {2}{5} \left (b^3 d+3 a b^2 e+3 a^2 c e+3 a b (2 c d+a f)\right ) x^{5/2}+\frac {1}{3} \left (b^3 e+6 a b c e+3 b^2 (c d+a f)+3 a c (c d+a f)\right ) x^3+\frac {2}{7} \left (3 b^2 c e+3 a c^2 e+b^3 f+3 b c (c d+2 a f)\right ) x^{7/2}+\frac {1}{4} c \left (c^2 d+3 b^2 f+3 c (b e+a f)\right ) x^4+\frac {2}{9} c^2 (c e+3 b f) x^{9/2}+\frac {1}{5} c^3 f x^5 \] Output:

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

Mathematica [A] (verified)

Time = 0.44 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.12 \[ \int \left (a+b \sqrt {x}+c x\right )^3 \left (d+e \sqrt {x}+f x\right ) \, dx=\frac {1}{6} a^3 x \left (6 d+4 e \sqrt {x}+3 f x\right )+\frac {1}{180} c^3 x^4 \left (45 d+40 e \sqrt {x}+36 f x\right )+\frac {1}{84} b c^2 x^{7/2} \left (72 d+63 e \sqrt {x}+56 f x\right )+b^3 \left (\frac {2}{5} d x^{5/2}+\frac {e x^3}{3}+\frac {2}{7} f x^{7/2}\right )+b^2 c \left (d x^3+\frac {6}{7} e x^{7/2}+\frac {3 f x^4}{4}\right )+\frac {1}{10} a^2 \left (c x^2 \left (15 d+12 e \sqrt {x}+10 f x\right )+b x^{3/2} \left (20 d+15 e \sqrt {x}+12 f x\right )\right )+\frac {1}{140} a x^2 \left (14 b^2 \left (15 d+12 e \sqrt {x}+10 f x\right )+5 c^2 x \left (28 d+24 e \sqrt {x}+21 f x\right )+8 b c \sqrt {x} \left (42 d+35 e \sqrt {x}+30 f x\right )\right ) \] Input:

Integrate[(a + b*Sqrt[x] + c*x)^3*(d + e*Sqrt[x] + f*x),x]
 

Output:

(a^3*x*(6*d + 4*e*Sqrt[x] + 3*f*x))/6 + (c^3*x^4*(45*d + 40*e*Sqrt[x] + 36 
*f*x))/180 + (b*c^2*x^(7/2)*(72*d + 63*e*Sqrt[x] + 56*f*x))/84 + b^3*((2*d 
*x^(5/2))/5 + (e*x^3)/3 + (2*f*x^(7/2))/7) + b^2*c*(d*x^3 + (6*e*x^(7/2))/ 
7 + (3*f*x^4)/4) + (a^2*(c*x^2*(15*d + 12*e*Sqrt[x] + 10*f*x) + b*x^(3/2)* 
(20*d + 15*e*Sqrt[x] + 12*f*x)))/10 + (a*x^2*(14*b^2*(15*d + 12*e*Sqrt[x] 
+ 10*f*x) + 5*c^2*x*(28*d + 24*e*Sqrt[x] + 21*f*x) + 8*b*c*Sqrt[x]*(42*d + 
 35*e*Sqrt[x] + 30*f*x)))/140
 

Rubi [F]

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.

\(\displaystyle \int \left (a+b \sqrt {x}+c x\right )^3 \left (d+e \sqrt {x}+f x\right ) \, dx\)

\(\Big \downarrow \) 2329

\(\displaystyle \int \left (a+b \sqrt {x}+c x\right )^3 \left (d+e \sqrt {x}+f x\right )dx\)

Input:

Int[(a + b*Sqrt[x] + c*x)^3*(d + e*Sqrt[x] + f*x),x]
 

Output:

$Aborted
 

Defintions of rubi rules used

rule 2329

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.))^(p_.), x_Symbol] :> 
 Unintegrable[Pq*(a + b*x^n + c*x^(2*n))^p, x] /; FreeQ[{a, b, c, n, p}, x] 
 && EqQ[n2, 2*n] && (PolyQ[Pq, x] || PolyQ[Pq, x^n])
 

Maple [A] (verified)

Time = 8.34 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.12

method	result	size
default	\(\frac {b^{3} e \,x^{3}}{3}+b^{2} \left (\frac {2 \left (b f +3 c e \right ) x^{\frac {7}{2}}}{7}+\frac {2 \left (3 a e +b d \right ) x^{\frac {5}{2}}}{5}\right )+\frac {\left (3 b^{2} c f +3 b \,c^{2} e \right ) x^{4}}{4}+\frac {\left (3 b^{2} a f +6 a b c e +3 b^{2} c d \right ) x^{3}}{3}+\frac {\left (3 b \,a^{2} e +3 a \,b^{2} d \right ) x^{2}}{2}+\frac {2 c^{2} \left (3 b f +c e \right ) x^{\frac {9}{2}}}{9}+\frac {2 \left (2 a c \left (3 b f +c e \right )+c^{2} \left (a e +3 b d \right )\right ) x^{\frac {7}{2}}}{7}+\frac {2 \left (a^{2} \left (3 b f +c e \right )+2 a c \left (a e +3 b d \right )\right ) x^{\frac {5}{2}}}{5}+\frac {2 a^{2} \left (a e +3 b d \right ) x^{\frac {3}{2}}}{3}+\frac {c^{3} f \,x^{5}}{5}+\frac {\left (3 a \,c^{2} f +c^{3} d \right ) x^{4}}{4}+\frac {\left (3 a^{2} c f +3 a \,c^{2} d \right ) x^{3}}{3}+\frac {\left (a^{3} f +3 a^{2} c d \right ) x^{2}}{2}+a^{3} d x\)	\(271\)
derivativedivides	\(\frac {c^{3} f \,x^{5}}{5}+\frac {2 \left (3 b \,c^{2} f +e \,c^{3}\right ) x^{\frac {9}{2}}}{9}+\frac {\left (\left (a \,c^{2}+2 c \,b^{2}+c \left (2 a c +b^{2}\right )\right ) f +3 b \,c^{2} e +c^{3} d \right ) x^{4}}{4}+\frac {2 \left (\left (4 a b c +b \left (2 a c +b^{2}\right )\right ) f +\left (a \,c^{2}+2 c \,b^{2}+c \left (2 a c +b^{2}\right )\right ) e +3 b \,c^{2} d \right ) x^{\frac {7}{2}}}{7}+\frac {\left (\left (a \left (2 a c +b^{2}\right )+2 b^{2} a +a^{2} c \right ) f +\left (4 a b c +b \left (2 a c +b^{2}\right )\right ) e +\left (a \,c^{2}+2 c \,b^{2}+c \left (2 a c +b^{2}\right )\right ) d \right ) x^{3}}{3}+\frac {2 \left (3 b \,a^{2} f +\left (a \left (2 a c +b^{2}\right )+2 b^{2} a +a^{2} c \right ) e +\left (4 a b c +b \left (2 a c +b^{2}\right )\right ) d \right ) x^{\frac {5}{2}}}{5}+\frac {\left (a^{3} f +3 b \,a^{2} e +\left (a \left (2 a c +b^{2}\right )+2 b^{2} a +a^{2} c \right ) d \right ) x^{2}}{2}+\frac {2 \left (a^{3} e +3 a^{2} b d \right ) x^{\frac {3}{2}}}{3}+a^{3} d x\)	\(319\)
trager	\(\frac {\left (12 c^{3} f \,x^{4}+45 a \,c^{2} f \,x^{3}+45 b^{2} c f \,x^{3}+45 x^{3} b \,c^{2} e +15 c^{3} d \,x^{3}+12 c^{3} f \,x^{3}+60 a^{2} c f \,x^{2}+60 b^{2} a f \,x^{2}+120 x^{2} a b c e +60 a \,c^{2} d \,x^{2}+45 a \,c^{2} f \,x^{2}+20 b^{3} x^{2} e +60 b^{2} c d \,x^{2}+45 b^{2} c f \,x^{2}+45 b \,c^{2} e \,x^{2}+15 c^{3} d \,x^{2}+12 c^{3} f \,x^{2}+30 a^{3} f x +90 a^{2} b e x +90 a^{2} c d x +60 a^{2} c f x +90 a \,b^{2} d x +60 b^{2} a f x +120 a b c e x +60 a \,c^{2} d x +45 a \,c^{2} f x +20 b^{3} e x +60 b^{2} c d x +45 b^{2} c f x +45 b \,c^{2} e x +15 c^{3} d x +12 c^{3} f x +60 a^{3} d +30 a^{3} f +90 b \,a^{2} e +90 a^{2} c d +60 a^{2} c f +90 a \,b^{2} d +60 b^{2} a f +120 a b c e +60 a \,c^{2} d +45 a \,c^{2} f +20 b^{3} e +60 b^{2} c d +45 b^{2} c f +45 b \,c^{2} e +15 c^{3} d +12 c^{3} f \right ) \left (x -1\right )}{60}+\frac {2 x^{\frac {3}{2}} \left (105 b \,c^{2} f \,x^{3}+35 c^{3} e \,x^{3}+270 a b c f \,x^{2}+135 a \,c^{2} e \,x^{2}+45 b^{3} x^{2} f +135 b^{2} c e \,x^{2}+135 b \,c^{2} d \,x^{2}+189 a^{2} b f x +189 a^{2} c e x +189 a \,b^{2} e x +378 a b c d x +63 b^{3} d x +105 a^{3} e +315 a^{2} b d \right )}{315}\)	\(516\)
orering	\(\text {Expression too large to display}\)	\(2778\)

Input:

int((a+b*x^(1/2)+c*x)^3*(d+e*x^(1/2)+f*x),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.00 \[ \int \left (a+b \sqrt {x}+c x\right )^3 \left (d+e \sqrt {x}+f x\right ) \, dx=\frac {1}{5} \, c^{3} f x^{5} + a^{3} d x + \frac {1}{4} \, {\left (c^{3} d + 3 \, b c^{2} e + 3 \, {\left (b^{2} c + a c^{2}\right )} f\right )} x^{4} + \frac {1}{3} \, {\left (3 \, {\left (b^{2} c + a c^{2}\right )} d + {\left (b^{3} + 6 \, a b c\right )} e + 3 \, {\left (a b^{2} + a^{2} c\right )} f\right )} x^{3} + \frac {1}{2} \, {\left (3 \, a^{2} b e + a^{3} f + 3 \, {\left (a b^{2} + a^{2} c\right )} d\right )} x^{2} + \frac {2}{315} \, {\left (35 \, {\left (c^{3} e + 3 \, b c^{2} f\right )} x^{4} + 45 \, {\left (3 \, b c^{2} d + 3 \, {\left (b^{2} c + a c^{2}\right )} e + {\left (b^{3} + 6 \, a b c\right )} f\right )} x^{3} + 63 \, {\left (3 \, a^{2} b f + {\left (b^{3} + 6 \, a b c\right )} d + 3 \, {\left (a b^{2} + a^{2} c\right )} e\right )} x^{2} + 105 \, {\left (3 \, a^{2} b d + a^{3} e\right )} x\right )} \sqrt {x} \] Input:

integrate((a+b*x^(1/2)+c*x)^3*(d+e*x^(1/2)+f*x),x, algorithm="fricas")
 

Output:

1/5*c^3*f*x^5 + a^3*d*x + 1/4*(c^3*d + 3*b*c^2*e + 3*(b^2*c + a*c^2)*f)*x^ 
4 + 1/3*(3*(b^2*c + a*c^2)*d + (b^3 + 6*a*b*c)*e + 3*(a*b^2 + a^2*c)*f)*x^ 
3 + 1/2*(3*a^2*b*e + a^3*f + 3*(a*b^2 + a^2*c)*d)*x^2 + 2/315*(35*(c^3*e + 
 3*b*c^2*f)*x^4 + 45*(3*b*c^2*d + 3*(b^2*c + a*c^2)*e + (b^3 + 6*a*b*c)*f) 
*x^3 + 63*(3*a^2*b*f + (b^3 + 6*a*b*c)*d + 3*(a*b^2 + a^2*c)*e)*x^2 + 105* 
(3*a^2*b*d + a^3*e)*x)*sqrt(x)
 

Sympy [A] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 386, normalized size of antiderivative = 1.60 \[ \int \left (a+b \sqrt {x}+c x\right )^3 \left (d+e \sqrt {x}+f x\right ) \, dx=a^{3} d x + \frac {2 a^{3} e x^{\frac {3}{2}}}{3} + \frac {a^{3} f x^{2}}{2} + 2 a^{2} b d x^{\frac {3}{2}} + \frac {3 a^{2} b e x^{2}}{2} + \frac {6 a^{2} b f x^{\frac {5}{2}}}{5} + \frac {3 a^{2} c d x^{2}}{2} + \frac {6 a^{2} c e x^{\frac {5}{2}}}{5} + a^{2} c f x^{3} + \frac {3 a b^{2} d x^{2}}{2} + \frac {6 a b^{2} e x^{\frac {5}{2}}}{5} + a b^{2} f x^{3} + \frac {12 a b c d x^{\frac {5}{2}}}{5} + 2 a b c e x^{3} + \frac {12 a b c f x^{\frac {7}{2}}}{7} + a c^{2} d x^{3} + \frac {6 a c^{2} e x^{\frac {7}{2}}}{7} + \frac {3 a c^{2} f x^{4}}{4} + \frac {2 b^{3} d x^{\frac {5}{2}}}{5} + \frac {b^{3} e x^{3}}{3} + \frac {2 b^{3} f x^{\frac {7}{2}}}{7} + b^{2} c d x^{3} + \frac {6 b^{2} c e x^{\frac {7}{2}}}{7} + \frac {3 b^{2} c f x^{4}}{4} + \frac {6 b c^{2} d x^{\frac {7}{2}}}{7} + \frac {3 b c^{2} e x^{4}}{4} + \frac {2 b c^{2} f x^{\frac {9}{2}}}{3} + \frac {c^{3} d x^{4}}{4} + \frac {2 c^{3} e x^{\frac {9}{2}}}{9} + \frac {c^{3} f x^{5}}{5} \] Input:

integrate((a+b*x**(1/2)+c*x)**3*(d+e*x**(1/2)+f*x),x)
 

Output:

a**3*d*x + 2*a**3*e*x**(3/2)/3 + a**3*f*x**2/2 + 2*a**2*b*d*x**(3/2) + 3*a 
**2*b*e*x**2/2 + 6*a**2*b*f*x**(5/2)/5 + 3*a**2*c*d*x**2/2 + 6*a**2*c*e*x* 
*(5/2)/5 + a**2*c*f*x**3 + 3*a*b**2*d*x**2/2 + 6*a*b**2*e*x**(5/2)/5 + a*b 
**2*f*x**3 + 12*a*b*c*d*x**(5/2)/5 + 2*a*b*c*e*x**3 + 12*a*b*c*f*x**(7/2)/ 
7 + a*c**2*d*x**3 + 6*a*c**2*e*x**(7/2)/7 + 3*a*c**2*f*x**4/4 + 2*b**3*d*x 
**(5/2)/5 + b**3*e*x**3/3 + 2*b**3*f*x**(7/2)/7 + b**2*c*d*x**3 + 6*b**2*c 
*e*x**(7/2)/7 + 3*b**2*c*f*x**4/4 + 6*b*c**2*d*x**(7/2)/7 + 3*b*c**2*e*x** 
4/4 + 2*b*c**2*f*x**(9/2)/3 + c**3*d*x**4/4 + 2*c**3*e*x**(9/2)/9 + c**3*f 
*x**5/5
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 237, normalized size of antiderivative = 0.98 \[ \int \left (a+b \sqrt {x}+c x\right )^3 \left (d+e \sqrt {x}+f x\right ) \, dx=\frac {1}{5} \, c^{3} f x^{5} + \frac {2}{9} \, {\left (c^{3} e + 3 \, b c^{2} f\right )} x^{\frac {9}{2}} + a^{3} d x + \frac {1}{4} \, {\left (c^{3} d + 3 \, b c^{2} e + 3 \, {\left (b^{2} c + a c^{2}\right )} f\right )} x^{4} + \frac {2}{7} \, {\left (3 \, b c^{2} d + 3 \, {\left (b^{2} c + a c^{2}\right )} e + {\left (b^{3} + 6 \, a b c\right )} f\right )} x^{\frac {7}{2}} + \frac {1}{3} \, {\left (3 \, {\left (b^{2} c + a c^{2}\right )} d + {\left (b^{3} + 6 \, a b c\right )} e + 3 \, {\left (a b^{2} + a^{2} c\right )} f\right )} x^{3} + \frac {2}{5} \, {\left (3 \, a^{2} b f + {\left (b^{3} + 6 \, a b c\right )} d + 3 \, {\left (a b^{2} + a^{2} c\right )} e\right )} x^{\frac {5}{2}} + \frac {1}{2} \, {\left (3 \, a^{2} b e + a^{3} f + 3 \, {\left (a b^{2} + a^{2} c\right )} d\right )} x^{2} + \frac {2}{3} \, {\left (3 \, a^{2} b d + a^{3} e\right )} x^{\frac {3}{2}} \] Input:

integrate((a+b*x^(1/2)+c*x)^3*(d+e*x^(1/2)+f*x),x, algorithm="maxima")
 

Output:

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

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 282, normalized size of antiderivative = 1.17 \[ \int \left (a+b \sqrt {x}+c x\right )^3 \left (d+e \sqrt {x}+f x\right ) \, dx=\frac {1}{5} \, c^{3} f x^{5} + \frac {2}{9} \, c^{3} e x^{\frac {9}{2}} + \frac {2}{3} \, b c^{2} f x^{\frac {9}{2}} + \frac {1}{4} \, c^{3} d x^{4} + \frac {3}{4} \, b c^{2} e x^{4} + \frac {3}{4} \, b^{2} c f x^{4} + \frac {3}{4} \, a c^{2} f x^{4} + \frac {6}{7} \, b c^{2} d x^{\frac {7}{2}} + \frac {6}{7} \, b^{2} c e x^{\frac {7}{2}} + \frac {6}{7} \, a c^{2} e x^{\frac {7}{2}} + \frac {2}{7} \, b^{3} f x^{\frac {7}{2}} + \frac {12}{7} \, a b c f x^{\frac {7}{2}} + b^{2} c d x^{3} + a c^{2} d x^{3} + \frac {1}{3} \, b^{3} e x^{3} + 2 \, a b c e x^{3} + a b^{2} f x^{3} + a^{2} c f x^{3} + \frac {2}{5} \, b^{3} d x^{\frac {5}{2}} + \frac {12}{5} \, a b c d x^{\frac {5}{2}} + \frac {6}{5} \, a b^{2} e x^{\frac {5}{2}} + \frac {6}{5} \, a^{2} c e x^{\frac {5}{2}} + \frac {6}{5} \, a^{2} b f x^{\frac {5}{2}} + \frac {3}{2} \, a b^{2} d x^{2} + \frac {3}{2} \, a^{2} c d x^{2} + \frac {3}{2} \, a^{2} b e x^{2} + \frac {1}{2} \, a^{3} f x^{2} + 2 \, a^{2} b d x^{\frac {3}{2}} + \frac {2}{3} \, a^{3} e x^{\frac {3}{2}} + a^{3} d x \] Input:

integrate((a+b*x^(1/2)+c*x)^3*(d+e*x^(1/2)+f*x),x, algorithm="giac")
 

Output:

1/5*c^3*f*x^5 + 2/9*c^3*e*x^(9/2) + 2/3*b*c^2*f*x^(9/2) + 1/4*c^3*d*x^4 + 
3/4*b*c^2*e*x^4 + 3/4*b^2*c*f*x^4 + 3/4*a*c^2*f*x^4 + 6/7*b*c^2*d*x^(7/2) 
+ 6/7*b^2*c*e*x^(7/2) + 6/7*a*c^2*e*x^(7/2) + 2/7*b^3*f*x^(7/2) + 12/7*a*b 
*c*f*x^(7/2) + b^2*c*d*x^3 + a*c^2*d*x^3 + 1/3*b^3*e*x^3 + 2*a*b*c*e*x^3 + 
 a*b^2*f*x^3 + a^2*c*f*x^3 + 2/5*b^3*d*x^(5/2) + 12/5*a*b*c*d*x^(5/2) + 6/ 
5*a*b^2*e*x^(5/2) + 6/5*a^2*c*e*x^(5/2) + 6/5*a^2*b*f*x^(5/2) + 3/2*a*b^2* 
d*x^2 + 3/2*a^2*c*d*x^2 + 3/2*a^2*b*e*x^2 + 1/2*a^3*f*x^2 + 2*a^2*b*d*x^(3 
/2) + 2/3*a^3*e*x^(3/2) + a^3*d*x
                                                                                    
                                                                                    
 

Mupad [B] (verification not implemented)

Time = 21.87 (sec) , antiderivative size = 233, normalized size of antiderivative = 0.96 \[ \int \left (a+b \sqrt {x}+c x\right )^3 \left (d+e \sqrt {x}+f x\right ) \, dx=x^{3/2}\,\left (\frac {2\,e\,a^3}{3}+2\,b\,d\,a^2\right )+x^{9/2}\,\left (\frac {2\,e\,c^3}{9}+\frac {2\,b\,f\,c^2}{3}\right )+x^3\,\left (f\,a^2\,c+f\,a\,b^2+2\,e\,a\,b\,c+d\,a\,c^2+\frac {e\,b^3}{3}+d\,b^2\,c\right )+x^{5/2}\,\left (\frac {6\,f\,a^2\,b}{5}+\frac {6\,c\,e\,a^2}{5}+\frac {6\,e\,a\,b^2}{5}+\frac {12\,c\,d\,a\,b}{5}+\frac {2\,d\,b^3}{5}\right )+x^{7/2}\,\left (\frac {2\,f\,b^3}{7}+\frac {6\,e\,b^2\,c}{7}+\frac {6\,d\,b\,c^2}{7}+\frac {12\,a\,f\,b\,c}{7}+\frac {6\,a\,e\,c^2}{7}\right )+x^2\,\left (\frac {f\,a^3}{2}+\frac {3\,e\,a^2\,b}{2}+\frac {3\,c\,d\,a^2}{2}+\frac {3\,d\,a\,b^2}{2}\right )+x^4\,\left (\frac {3\,f\,b^2\,c}{4}+\frac {3\,e\,b\,c^2}{4}+\frac {d\,c^3}{4}+\frac {3\,a\,f\,c^2}{4}\right )+\frac {c^3\,f\,x^5}{5}+a^3\,d\,x \] Input:

int((a + c*x + b*x^(1/2))^3*(d + f*x + e*x^(1/2)),x)
 

Output:

x^(3/2)*((2*a^3*e)/3 + 2*a^2*b*d) + x^(9/2)*((2*c^3*e)/9 + (2*b*c^2*f)/3) 
+ x^3*((b^3*e)/3 + a*c^2*d + a*b^2*f + b^2*c*d + a^2*c*f + 2*a*b*c*e) + x^ 
(5/2)*((2*b^3*d)/5 + (6*a*b^2*e)/5 + (6*a^2*b*f)/5 + (6*a^2*c*e)/5 + (12*a 
*b*c*d)/5) + x^(7/2)*((2*b^3*f)/7 + (6*a*c^2*e)/7 + (6*b*c^2*d)/7 + (6*b^2 
*c*e)/7 + (12*a*b*c*f)/7) + x^2*((a^3*f)/2 + (3*a*b^2*d)/2 + (3*a^2*b*e)/2 
 + (3*a^2*c*d)/2) + x^4*((c^3*d)/4 + (3*a*c^2*f)/4 + (3*b*c^2*e)/4 + (3*b^ 
2*c*f)/4) + (c^3*f*x^5)/5 + a^3*d*x
 

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 293, normalized size of antiderivative = 1.21 \[ \int \left (a+b \sqrt {x}+c x\right )^3 \left (d+e \sqrt {x}+f x\right ) \, dx=\frac {x \left (3024 \sqrt {x}\, a b c d x +420 b^{3} e \,x^{2}+1080 \sqrt {x}\, b \,c^{2} d \,x^{2}+315 c^{3} d \,x^{3}+1890 a^{2} c d x +1260 a \,c^{2} d \,x^{2}+1260 b^{2} c d \,x^{2}+360 \sqrt {x}\, b^{3} f \,x^{2}+1890 a^{2} b e x +1260 a \,b^{2} f \,x^{2}+1512 \sqrt {x}\, a^{2} b f x +1512 \sqrt {x}\, a \,b^{2} e x +1260 a^{3} d +840 \sqrt {x}\, a^{3} e +630 a^{3} f x +1512 \sqrt {x}\, a^{2} c e x +1080 \sqrt {x}\, a \,c^{2} e \,x^{2}+1080 \sqrt {x}\, b^{2} c e \,x^{2}+840 \sqrt {x}\, b \,c^{2} f \,x^{3}+2520 a b c e \,x^{2}+2520 \sqrt {x}\, a^{2} b d +504 \sqrt {x}\, b^{3} d x +1890 a \,b^{2} d x +280 \sqrt {x}\, c^{3} e \,x^{3}+1260 a^{2} c f \,x^{2}+945 a \,c^{2} f \,x^{3}+945 b^{2} c f \,x^{3}+945 b \,c^{2} e \,x^{3}+2160 \sqrt {x}\, a b c f \,x^{2}+252 c^{3} f \,x^{4}\right )}{1260} \] Input:

int((a+b*x^(1/2)+c*x)^3*(d+e*x^(1/2)+f*x),x)
 

Output:

(x*(840*sqrt(x)*a**3*e + 2520*sqrt(x)*a**2*b*d + 1512*sqrt(x)*a**2*b*f*x + 
 1512*sqrt(x)*a**2*c*e*x + 1512*sqrt(x)*a*b**2*e*x + 3024*sqrt(x)*a*b*c*d* 
x + 2160*sqrt(x)*a*b*c*f*x**2 + 1080*sqrt(x)*a*c**2*e*x**2 + 504*sqrt(x)*b 
**3*d*x + 360*sqrt(x)*b**3*f*x**2 + 1080*sqrt(x)*b**2*c*e*x**2 + 1080*sqrt 
(x)*b*c**2*d*x**2 + 840*sqrt(x)*b*c**2*f*x**3 + 280*sqrt(x)*c**3*e*x**3 + 
1260*a**3*d + 630*a**3*f*x + 1890*a**2*b*e*x + 1890*a**2*c*d*x + 1260*a**2 
*c*f*x**2 + 1890*a*b**2*d*x + 1260*a*b**2*f*x**2 + 2520*a*b*c*e*x**2 + 126 
0*a*c**2*d*x**2 + 945*a*c**2*f*x**3 + 420*b**3*e*x**2 + 1260*b**2*c*d*x**2 
 + 945*b**2*c*f*x**3 + 945*b*c**2*e*x**3 + 315*c**3*d*x**3 + 252*c**3*f*x* 
*4))/1260