Integrand size = 24, antiderivative size = 77 \[ \int \left (a+b \sqrt {x}\right )^2 \left (d+e \sqrt {x}+f x\right ) \, dx=a^2 d x+\frac {2}{3} a (2 b d+a e) x^{3/2}+\frac {1}{2} \left (b^2 d+2 a b e+a^2 f\right ) x^2+\frac {2}{5} b (b e+2 a f) x^{5/2}+\frac {1}{3} b^2 f x^3 \] Output:
a^2*d*x+2/3*a*(a*e+2*b*d)*x^(3/2)+1/2*(a^2*f+2*a*b*e+b^2*d)*x^2+2/5*b*(2*a *f+b*e)*x^(5/2)+1/3*b^2*f*x^3
Time = 0.08 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.95 \[ \int \left (a+b \sqrt {x}\right )^2 \left (d+e \sqrt {x}+f x\right ) \, dx=\frac {1}{30} x \left (5 a^2 \left (6 d+4 e \sqrt {x}+3 f x\right )+b^2 x \left (15 d+12 e \sqrt {x}+10 f x\right )+2 a b \sqrt {x} \left (20 d+15 e \sqrt {x}+12 f x\right )\right ) \] Input:
Integrate[(a + b*Sqrt[x])^2*(d + e*Sqrt[x] + f*x),x]
Output:
(x*(5*a^2*(6*d + 4*e*Sqrt[x] + 3*f*x) + b^2*x*(15*d + 12*e*Sqrt[x] + 10*f* x) + 2*a*b*Sqrt[x]*(20*d + 15*e*Sqrt[x] + 12*f*x)))/30
Time = 0.41 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.06, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {1731, 1195, 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.
\(\displaystyle \int \left (a+b \sqrt {x}\right )^2 \left (d+e \sqrt {x}+f x\right ) \, dx\) |
\(\Big \downarrow \) 1731 |
\(\displaystyle 2 \int \left (a+b \sqrt {x}\right )^2 \sqrt {x} \left (d+f x+e \sqrt {x}\right )d\sqrt {x}\) |
\(\Big \downarrow \) 1195 |
\(\displaystyle 2 \int \left (b^2 f x^{5/2}+b (b e+2 a f) x^2+\left (f a^2+2 b e a+b^2 d\right ) x^{3/2}+a (2 b d+a e) x+a^2 d \sqrt {x}\right )d\sqrt {x}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 2 \left (\frac {1}{4} x^2 \left (a^2 f+2 a b e+b^2 d\right )+\frac {1}{2} a^2 d x+\frac {1}{3} a x^{3/2} (a e+2 b d)+\frac {1}{5} b x^{5/2} (2 a f+b e)+\frac {1}{6} b^2 f x^3\right )\) |
Input:
Int[(a + b*Sqrt[x])^2*(d + e*Sqrt[x] + f*x),x]
Output:
2*((a^2*d*x)/2 + (a*(2*b*d + a*e)*x^(3/2))/3 + ((b^2*d + 2*a*b*e + a^2*f)* x^2)/4 + (b*(b*e + 2*a*f)*x^(5/2))/5 + (b^2*f*x^3)/6)
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x _) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x ] && IGtQ[p, 0]
Int[((a_.) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.)*((d_) + (e_.)*(x_)^ (n_))^(q_.), x_Symbol] :> With[{g = Denominator[n]}, Simp[g Subst[Int[x^( g - 1)*(d + e*x^(g*n))^q*(a + b*x^(g*n) + c*x^(2*g*n))^p, x], x, x^(1/g)], x]] /; FreeQ[{a, b, c, d, e, p, q}, x] && EqQ[n2, 2*n] && FractionQ[n]
Time = 0.58 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.91
method | result | size |
derivativedivides | \(\frac {b^{2} x^{3} f}{3}+\frac {2 \left (2 a b f +b^{2} e \right ) x^{\frac {5}{2}}}{5}+\frac {\left (a^{2} f +2 a b e +d \,b^{2}\right ) x^{2}}{2}+\frac {2 \left (a^{2} e +2 a b d \right ) x^{\frac {3}{2}}}{3}+a^{2} d x\) | \(70\) |
default | \(\frac {2 x^{\frac {5}{2}} b^{2} e}{5}+\frac {b^{2} x^{3} f}{3}+\frac {\left (2 a b e +d \,b^{2}\right ) x^{2}}{2}+a \left (\frac {4 b f \,x^{\frac {5}{2}}}{5}+\frac {2 \left (a e +2 b d \right ) x^{\frac {3}{2}}}{3}\right )+a^{2} \left (\frac {1}{2} f \,x^{2}+d x \right )\) | \(73\) |
trager | \(\frac {\left (2 b^{2} f \,x^{2}+3 a^{2} f x +6 a b e x +3 b^{2} d x +2 b^{2} f x +6 a^{2} d +3 a^{2} f +6 a b e +3 d \,b^{2}+2 b^{2} f \right ) \left (x -1\right )}{6}+\frac {2 x^{\frac {3}{2}} \left (6 b a f x +3 b^{2} e x +5 a^{2} e +10 a b d \right )}{15}\) | \(103\) |
orering | \(-\frac {\left (72 x^{4} b^{4} f^{3} a^{2}+72 x^{4} b^{5} f^{2} a e +18 b^{6} e^{2} f \,x^{4}-84 x^{3} b^{2} f^{3} a^{4}-112 x^{3} b^{3} f^{2} a^{3} e +196 x^{3} b^{4} f^{2} a^{2} d -119 x^{3} b^{4} f \,a^{2} e^{2}+56 x^{3} b^{5} f a d e -42 a \,b^{5} e^{3} x^{3}-21 b^{6} d \,e^{2} x^{3}+50 a^{5} b e \,f^{2} x^{2}-260 a^{4} b^{2} d \,f^{2} x^{2}+125 a^{4} b^{2} e^{2} f \,x^{2}-60 a^{3} b^{3} d e f \,x^{2}+50 a^{3} b^{3} e^{3} x^{2}+100 a^{2} b^{4} d^{2} f \,x^{2}+35 a^{2} b^{4} d \,e^{2} x^{2}+50 a \,b^{5} d^{2} e \,x^{2}+10 d \,e^{2} a^{6}+40 b \,d^{2} e \,a^{5}+40 b^{2} d^{3} a^{4}\right ) \left (a +b \sqrt {x}\right )^{2} \left (d +e \sqrt {x}+f x \right )}{30 \left (-2 a \,b^{3} f^{2} x^{3}-b^{4} e f \,x^{3}+2 a^{3} b \,f^{2} x^{2}+2 a^{2} b^{2} e f \,x^{2}-4 a \,b^{3} d f \,x^{2}+2 a \,b^{3} e^{2} x^{2}+b^{4} d e \,x^{2}-a^{4} e f x +4 a^{3} b d f x -2 a^{3} b \,e^{2} x -2 a^{2} b^{2} d e x -2 a \,b^{3} d^{2} x +a^{4} d e +2 a^{3} b \,d^{2}\right ) b \left (2 a f +e b \right )}+\frac {\left (8 x^{4} b^{4} f^{3} a^{2}+8 x^{4} b^{5} f^{2} a e +2 b^{6} e^{2} f \,x^{4}-12 x^{3} b^{2} f^{3} a^{4}-16 x^{3} b^{3} f^{2} a^{3} e +28 x^{3} b^{4} f^{2} a^{2} d -17 x^{3} b^{4} f \,a^{2} e^{2}+8 x^{3} b^{5} f a d e -6 a \,b^{5} e^{3} x^{3}-3 b^{6} d \,e^{2} x^{3}+10 a^{5} b e \,f^{2} x^{2}-52 a^{4} b^{2} d \,f^{2} x^{2}+25 a^{4} b^{2} e^{2} f \,x^{2}-12 a^{3} b^{3} d e f \,x^{2}+10 a^{3} b^{3} e^{3} x^{2}+20 a^{2} b^{4} d^{2} f \,x^{2}+7 a^{2} b^{4} d \,e^{2} x^{2}+10 a \,b^{5} d^{2} e \,x^{2}+10 d \,e^{2} a^{6}+40 b \,d^{2} e \,a^{5}+40 b^{2} d^{3} a^{4}\right ) x \left (\frac {\left (a +b \sqrt {x}\right ) \left (d +e \sqrt {x}+f x \right ) b}{\sqrt {x}}+\left (a +b \sqrt {x}\right )^{2} \left (\frac {e}{2 \sqrt {x}}+f \right )\right )}{15 b \left (2 a f +e b \right ) \left (2 a b \,f^{2} x^{2}+b^{2} x^{2} e f -a^{2} e f x +4 a b d f x -2 a b \,e^{2} x -b^{2} d e x +a^{2} d e +2 a b \,d^{2}\right ) \left (-b^{2} x +a^{2}\right )}\) | \(874\) |
Input:
int((a+b*x^(1/2))^2*(d+e*x^(1/2)+f*x),x,method=_RETURNVERBOSE)
Output:
1/3*b^2*x^3*f+2/5*(2*a*b*f+b^2*e)*x^(5/2)+1/2*(a^2*f+2*a*b*e+b^2*d)*x^2+2/ 3*(a^2*e+2*a*b*d)*x^(3/2)+a^2*d*x
Time = 0.09 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.95 \[ \int \left (a+b \sqrt {x}\right )^2 \left (d+e \sqrt {x}+f x\right ) \, dx=\frac {1}{3} \, b^{2} f x^{3} + a^{2} d x + \frac {1}{2} \, {\left (b^{2} d + 2 \, a b e + a^{2} f\right )} x^{2} + \frac {2}{15} \, {\left (3 \, {\left (b^{2} e + 2 \, a b f\right )} x^{2} + 5 \, {\left (2 \, a b d + a^{2} e\right )} x\right )} \sqrt {x} \] Input:
integrate((a+b*x^(1/2))^2*(d+e*x^(1/2)+f*x),x, algorithm="fricas")
Output:
1/3*b^2*f*x^3 + a^2*d*x + 1/2*(b^2*d + 2*a*b*e + a^2*f)*x^2 + 2/15*(3*(b^2 *e + 2*a*b*f)*x^2 + 5*(2*a*b*d + a^2*e)*x)*sqrt(x)
Time = 0.11 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.29 \[ \int \left (a+b \sqrt {x}\right )^2 \left (d+e \sqrt {x}+f x\right ) \, dx=a^{2} d x + \frac {2 a^{2} e x^{\frac {3}{2}}}{3} + \frac {a^{2} f x^{2}}{2} + \frac {4 a b d x^{\frac {3}{2}}}{3} + a b e x^{2} + \frac {4 a b f x^{\frac {5}{2}}}{5} + \frac {b^{2} d x^{2}}{2} + \frac {2 b^{2} e x^{\frac {5}{2}}}{5} + \frac {b^{2} f x^{3}}{3} \] Input:
integrate((a+b*x**(1/2))**2*(d+e*x**(1/2)+f*x),x)
Output:
a**2*d*x + 2*a**2*e*x**(3/2)/3 + a**2*f*x**2/2 + 4*a*b*d*x**(3/2)/3 + a*b* e*x**2 + 4*a*b*f*x**(5/2)/5 + b**2*d*x**2/2 + 2*b**2*e*x**(5/2)/5 + b**2*f *x**3/3
Time = 0.03 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.90 \[ \int \left (a+b \sqrt {x}\right )^2 \left (d+e \sqrt {x}+f x\right ) \, dx=\frac {1}{3} \, b^{2} f x^{3} + a^{2} d x + \frac {2}{5} \, {\left (b^{2} e + 2 \, a b f\right )} x^{\frac {5}{2}} + \frac {1}{2} \, {\left (b^{2} d + 2 \, a b e + a^{2} f\right )} x^{2} + \frac {2}{3} \, {\left (2 \, a b d + a^{2} e\right )} x^{\frac {3}{2}} \] Input:
integrate((a+b*x^(1/2))^2*(d+e*x^(1/2)+f*x),x, algorithm="maxima")
Output:
1/3*b^2*f*x^3 + a^2*d*x + 2/5*(b^2*e + 2*a*b*f)*x^(5/2) + 1/2*(b^2*d + 2*a *b*e + a^2*f)*x^2 + 2/3*(2*a*b*d + a^2*e)*x^(3/2)
Time = 0.14 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.97 \[ \int \left (a+b \sqrt {x}\right )^2 \left (d+e \sqrt {x}+f x\right ) \, dx=\frac {1}{3} \, b^{2} f x^{3} + \frac {2}{5} \, b^{2} e x^{\frac {5}{2}} + \frac {4}{5} \, a b f x^{\frac {5}{2}} + \frac {1}{2} \, b^{2} d x^{2} + a b e x^{2} + \frac {1}{2} \, a^{2} f x^{2} + \frac {4}{3} \, a b d x^{\frac {3}{2}} + \frac {2}{3} \, a^{2} e x^{\frac {3}{2}} + a^{2} d x \] Input:
integrate((a+b*x^(1/2))^2*(d+e*x^(1/2)+f*x),x, algorithm="giac")
Output:
1/3*b^2*f*x^3 + 2/5*b^2*e*x^(5/2) + 4/5*a*b*f*x^(5/2) + 1/2*b^2*d*x^2 + a* b*e*x^2 + 1/2*a^2*f*x^2 + 4/3*a*b*d*x^(3/2) + 2/3*a^2*e*x^(3/2) + a^2*d*x
Time = 0.04 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.90 \[ \int \left (a+b \sqrt {x}\right )^2 \left (d+e \sqrt {x}+f x\right ) \, dx=x^2\,\left (\frac {f\,a^2}{2}+e\,a\,b+\frac {d\,b^2}{2}\right )+x^{3/2}\,\left (\frac {2\,e\,a^2}{3}+\frac {4\,b\,d\,a}{3}\right )+x^{5/2}\,\left (\frac {2\,e\,b^2}{5}+\frac {4\,a\,f\,b}{5}\right )+\frac {b^2\,f\,x^3}{3}+a^2\,d\,x \] Input:
int((a + b*x^(1/2))^2*(d + f*x + e*x^(1/2)),x)
Output:
x^2*((b^2*d)/2 + (a^2*f)/2 + a*b*e) + x^(3/2)*((2*a^2*e)/3 + (4*a*b*d)/3) + x^(5/2)*((2*b^2*e)/5 + (4*a*b*f)/5) + (b^2*f*x^3)/3 + a^2*d*x
Time = 0.23 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.92 \[ \int \left (a+b \sqrt {x}\right )^2 \left (d+e \sqrt {x}+f x\right ) \, dx=\frac {x \left (20 \sqrt {x}\, a^{2} e +40 \sqrt {x}\, a b d +24 \sqrt {x}\, a b f x +12 \sqrt {x}\, b^{2} e x +30 a^{2} d +15 a^{2} f x +30 a b e x +15 b^{2} d x +10 b^{2} f \,x^{2}\right )}{30} \] Input:
int((a+b*x^(1/2))^2*(d+e*x^(1/2)+f*x),x)
Output:
(x*(20*sqrt(x)*a**2*e + 40*sqrt(x)*a*b*d + 24*sqrt(x)*a*b*f*x + 12*sqrt(x) *b**2*e*x + 30*a**2*d + 15*a**2*f*x + 30*a*b*e*x + 15*b**2*d*x + 10*b**2*f *x**2))/30