Integrand size = 25, antiderivative size = 63 \[ \int \left (a+b \sqrt {x}+c x\right ) \left (d+e \sqrt {x}+f x\right ) \, dx=a d x+\frac {2}{3} (b d+a e) x^{3/2}+\frac {1}{2} (c d+b e+a f) x^2+\frac {2}{5} (c e+b f) x^{5/2}+\frac {1}{3} c f x^3 \] Output:
a*d*x+2/3*(a*e+b*d)*x^(3/2)+1/2*(a*f+b*e+c*d)*x^2+2/5*(b*f+c*e)*x^(5/2)+1/ 3*c*f*x^3
Time = 0.09 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.08 \[ \int \left (a+b \sqrt {x}+c x\right ) \left (d+e \sqrt {x}+f x\right ) \, dx=\frac {1}{30} x \left (30 a d+20 b d \sqrt {x}+20 a e \sqrt {x}+15 c d x+15 b e x+15 a f x+12 c e x^{3/2}+12 b f x^{3/2}+10 c f x^2\right ) \] Input:
Integrate[(a + b*Sqrt[x] + c*x)*(d + e*Sqrt[x] + f*x),x]
Output:
(x*(30*a*d + 20*b*d*Sqrt[x] + 20*a*e*Sqrt[x] + 15*c*d*x + 15*b*e*x + 15*a* f*x + 12*c*e*x^(3/2) + 12*b*f*x^(3/2) + 10*c*f*x^2))/30
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.
\(\displaystyle \int \left (a+b \sqrt {x}+c x\right ) \left (d+e \sqrt {x}+f x\right ) \, dx\) |
\(\Big \downarrow \) 2329 |
\(\displaystyle \int \left (a+b \sqrt {x}+c x\right ) \left (d+e \sqrt {x}+f x\right )dx\) |
Input:
Int[(a + b*Sqrt[x] + c*x)*(d + e*Sqrt[x] + f*x),x]
Output:
$Aborted
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])
Time = 0.49 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.83
method | result | size |
derivativedivides | \(a d x +\frac {2 \left (a e +b d \right ) x^{\frac {3}{2}}}{3}+\frac {\left (a f +e b +c d \right ) x^{2}}{2}+\frac {2 \left (b f +c e \right ) x^{\frac {5}{2}}}{5}+\frac {c f \,x^{3}}{3}\) | \(52\) |
default | \(\frac {x^{2} e b}{2}+\frac {2 \left (b f +c e \right ) x^{\frac {5}{2}}}{5}+\frac {2 \left (a e +b d \right ) x^{\frac {3}{2}}}{3}+\frac {c f \,x^{3}}{3}+\frac {\left (a f +c d \right ) x^{2}}{2}+a d x\) | \(56\) |
trager | \(\frac {\left (2 c f \,x^{2}+3 a f x +3 x e b +3 x c d +2 c f x +6 a d +3 a f +3 e b +3 c d +2 c f \right ) \left (x -1\right )}{6}+\frac {2 x^{\frac {3}{2}} \left (3 b f x +3 c e x +5 a e +5 b d \right )}{15}\) | \(79\) |
orering | \(-\frac {\left (18 b^{2} c \,f^{3} x^{4}+36 x^{4} c^{2} f^{2} b e +18 c^{3} e^{2} f \,x^{4}-21 a \,b^{2} f^{3} x^{3}+28 x^{3} c \,f^{2} a b e +49 a \,c^{2} e^{2} f \,x^{3}-21 b^{3} x^{3} f^{2} e +49 b^{2} c d \,f^{2} x^{3}-42 x^{3} c f \,b^{2} e^{2}+28 x^{3} c^{2} f b d e -21 b \,c^{2} e^{3} x^{3}-21 c^{3} d \,e^{2} x^{3}+25 a^{2} b e \,f^{2} x^{2}+25 a^{2} c \,e^{2} f \,x^{2}-65 a \,b^{2} d \,f^{2} x^{2}+25 a \,b^{2} e^{2} f \,x^{2}-130 a b c d e f \,x^{2}+25 a b c \,e^{3} x^{2}-65 a \,c^{2} d \,e^{2} x^{2}+25 b^{3} d e f \,x^{2}+25 b^{2} c \,d^{2} f \,x^{2}+25 b^{2} c d \,e^{2} x^{2}+25 b \,c^{2} d^{2} e \,x^{2}+10 a^{3} d \,e^{2}+20 a^{2} b \,d^{2} e +10 a \,b^{2} d^{3}\right ) \left (a +b \sqrt {x}+c x \right ) \left (d +e \sqrt {x}+f x \right )}{30 \left (b f +c e \right ) \left (-b c \,f^{2} x^{3}-c^{2} e f \,x^{3}+a b \,f^{2} x^{2}-2 a c e f \,x^{2}+b^{2} x^{2} e f -2 b c d f \,x^{2}+b c \,e^{2} x^{2}+c^{2} d e \,x^{2}-a^{2} e f x +2 a b d f x -a b \,e^{2} x +2 a c d e x -b^{2} d e x -b c \,d^{2} x +a^{2} d e +a b \,d^{2}\right )}+\frac {\left (2 b^{2} c \,f^{3} x^{4}+4 x^{4} c^{2} f^{2} b e +2 c^{3} e^{2} f \,x^{4}-3 a \,b^{2} f^{3} x^{3}+4 x^{3} c \,f^{2} a b e +7 a \,c^{2} e^{2} f \,x^{3}-3 b^{3} x^{3} f^{2} e +7 b^{2} c d \,f^{2} x^{3}-6 x^{3} c f \,b^{2} e^{2}+4 x^{3} c^{2} f b d e -3 b \,c^{2} e^{3} x^{3}-3 c^{3} d \,e^{2} x^{3}+5 a^{2} b e \,f^{2} x^{2}+5 a^{2} c \,e^{2} f \,x^{2}-13 a \,b^{2} d \,f^{2} x^{2}+5 a \,b^{2} e^{2} f \,x^{2}-26 a b c d e f \,x^{2}+5 a b c \,e^{3} x^{2}-13 a \,c^{2} d \,e^{2} x^{2}+5 b^{3} d e f \,x^{2}+5 b^{2} c \,d^{2} f \,x^{2}+5 b^{2} c d \,e^{2} x^{2}+5 b \,c^{2} d^{2} e \,x^{2}+10 a^{3} d \,e^{2}+20 a^{2} b \,d^{2} e +10 a \,b^{2} d^{3}\right ) x \left (\left (\frac {b}{2 \sqrt {x}}+c \right ) \left (d +e \sqrt {x}+f x \right )+\left (a +b \sqrt {x}+c x \right ) \left (\frac {e}{2 \sqrt {x}}+f \right )\right )}{15 \left (b f +c e \right ) \left (-b c \,f^{2} x^{3}-c^{2} e f \,x^{3}+a b \,f^{2} x^{2}-2 a c e f \,x^{2}+b^{2} x^{2} e f -2 b c d f \,x^{2}+b c \,e^{2} x^{2}+c^{2} d e \,x^{2}-a^{2} e f x +2 a b d f x -a b \,e^{2} x +2 a c d e x -b^{2} d e x -b c \,d^{2} x +a^{2} d e +a b \,d^{2}\right )}\) | \(982\) |
Input:
int((a+b*x^(1/2)+c*x)*(d+e*x^(1/2)+f*x),x,method=_RETURNVERBOSE)
Output:
a*d*x+2/3*(a*e+b*d)*x^(3/2)+1/2*(a*f+b*e+c*d)*x^2+2/5*(b*f+c*e)*x^(5/2)+1/ 3*c*f*x^3
Time = 0.09 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.87 \[ \int \left (a+b \sqrt {x}+c x\right ) \left (d+e \sqrt {x}+f x\right ) \, dx=\frac {1}{3} \, c f x^{3} + a d x + \frac {1}{2} \, {\left (c d + b e + a f\right )} x^{2} + \frac {2}{15} \, {\left (3 \, {\left (c e + b f\right )} x^{2} + 5 \, {\left (b d + a e\right )} x\right )} \sqrt {x} \] Input:
integrate((a+b*x^(1/2)+c*x)*(d+e*x^(1/2)+f*x),x, algorithm="fricas")
Output:
1/3*c*f*x^3 + a*d*x + 1/2*(c*d + b*e + a*f)*x^2 + 2/15*(3*(c*e + b*f)*x^2 + 5*(b*d + a*e)*x)*sqrt(x)
Time = 0.10 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.35 \[ \int \left (a+b \sqrt {x}+c x\right ) \left (d+e \sqrt {x}+f x\right ) \, dx=a d x + \frac {2 a e x^{\frac {3}{2}}}{3} + \frac {a f x^{2}}{2} + \frac {2 b d x^{\frac {3}{2}}}{3} + \frac {b e x^{2}}{2} + \frac {2 b f x^{\frac {5}{2}}}{5} + \frac {c d x^{2}}{2} + \frac {2 c e x^{\frac {5}{2}}}{5} + \frac {c f x^{3}}{3} \] Input:
integrate((a+b*x**(1/2)+c*x)*(d+e*x**(1/2)+f*x),x)
Output:
a*d*x + 2*a*e*x**(3/2)/3 + a*f*x**2/2 + 2*b*d*x**(3/2)/3 + b*e*x**2/2 + 2* b*f*x**(5/2)/5 + c*d*x**2/2 + 2*c*e*x**(5/2)/5 + c*f*x**3/3
Time = 0.04 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.81 \[ \int \left (a+b \sqrt {x}+c x\right ) \left (d+e \sqrt {x}+f x\right ) \, dx=\frac {1}{3} \, c f x^{3} + \frac {2}{5} \, {\left (c e + b f\right )} x^{\frac {5}{2}} + a d x + \frac {1}{2} \, {\left (c d + b e + a f\right )} x^{2} + \frac {2}{3} \, {\left (b d + a e\right )} x^{\frac {3}{2}} \] Input:
integrate((a+b*x^(1/2)+c*x)*(d+e*x^(1/2)+f*x),x, algorithm="maxima")
Output:
1/3*c*f*x^3 + 2/5*(c*e + b*f)*x^(5/2) + a*d*x + 1/2*(c*d + b*e + a*f)*x^2 + 2/3*(b*d + a*e)*x^(3/2)
Time = 0.12 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.97 \[ \int \left (a+b \sqrt {x}+c x\right ) \left (d+e \sqrt {x}+f x\right ) \, dx=\frac {1}{3} \, c f x^{3} + \frac {2}{5} \, c e x^{\frac {5}{2}} + \frac {2}{5} \, b f x^{\frac {5}{2}} + \frac {1}{2} \, c d x^{2} + \frac {1}{2} \, b e x^{2} + \frac {1}{2} \, a f x^{2} + \frac {2}{3} \, b d x^{\frac {3}{2}} + \frac {2}{3} \, a e x^{\frac {3}{2}} + a d x \] Input:
integrate((a+b*x^(1/2)+c*x)*(d+e*x^(1/2)+f*x),x, algorithm="giac")
Output:
1/3*c*f*x^3 + 2/5*c*e*x^(5/2) + 2/5*b*f*x^(5/2) + 1/2*c*d*x^2 + 1/2*b*e*x^ 2 + 1/2*a*f*x^2 + 2/3*b*d*x^(3/2) + 2/3*a*e*x^(3/2) + a*d*x
Time = 21.71 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.87 \[ \int \left (a+b \sqrt {x}+c x\right ) \left (d+e \sqrt {x}+f x\right ) \, dx=x^2\,\left (\frac {a\,f}{2}+\frac {b\,e}{2}+\frac {c\,d}{2}\right )+x^{3/2}\,\left (\frac {2\,a\,e}{3}+\frac {2\,b\,d}{3}\right )+x^{5/2}\,\left (\frac {2\,b\,f}{5}+\frac {2\,c\,e}{5}\right )+a\,d\,x+\frac {c\,f\,x^3}{3} \] Input:
int((a + c*x + b*x^(1/2))*(d + f*x + e*x^(1/2)),x)
Output:
x^2*((a*f)/2 + (b*e)/2 + (c*d)/2) + x^(3/2)*((2*a*e)/3 + (2*b*d)/3) + x^(5 /2)*((2*b*f)/5 + (2*c*e)/5) + a*d*x + (c*f*x^3)/3
Time = 0.23 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.89 \[ \int \left (a+b \sqrt {x}+c x\right ) \left (d+e \sqrt {x}+f x\right ) \, dx=\frac {x \left (20 \sqrt {x}\, a e +20 \sqrt {x}\, b d +12 \sqrt {x}\, b f x +12 \sqrt {x}\, c e x +30 a d +15 a f x +15 b e x +15 c d x +10 c f \,x^{2}\right )}{30} \] Input:
int((a+b*x^(1/2)+c*x)*(d+e*x^(1/2)+f*x),x)
Output:
(x*(20*sqrt(x)*a*e + 20*sqrt(x)*b*d + 12*sqrt(x)*b*f*x + 12*sqrt(x)*c*e*x + 30*a*d + 15*a*f*x + 15*b*e*x + 15*c*d*x + 10*c*f*x**2))/30