Integrand size = 24, antiderivative size = 116 \[ \int \left (a+b \sqrt {x}\right )^3 \left (d+e \sqrt {x}+f x\right ) \, dx=-\frac {a \left (b^2 d-a b e+a^2 f\right ) \left (a+b \sqrt {x}\right )^4}{2 b^4}+\frac {2 \left (b^2 d-2 a b e+3 a^2 f\right ) \left (a+b \sqrt {x}\right )^5}{5 b^4}+\frac {(b e-3 a f) \left (a+b \sqrt {x}\right )^6}{3 b^4}+\frac {2 f \left (a+b \sqrt {x}\right )^7}{7 b^4} \] Output:
-1/2*a*(a^2*f-a*b*e+b^2*d)*(a+b*x^(1/2))^4/b^4+2/5*(3*a^2*f-2*a*b*e+b^2*d) *(a+b*x^(1/2))^5/b^4+1/3*(-3*a*f+b*e)*(a+b*x^(1/2))^6/b^4+2/7*f*(a+b*x^(1/ 2))^7/b^4
Time = 0.15 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.89 \[ \int \left (a+b \sqrt {x}\right )^3 \left (d+e \sqrt {x}+f x\right ) \, dx=\frac {1}{210} x \left (35 a^3 \left (6 d+4 e \sqrt {x}+3 f x\right )+21 a b^2 x \left (15 d+12 e \sqrt {x}+10 f x\right )+21 a^2 b \sqrt {x} \left (20 d+15 e \sqrt {x}+12 f x\right )+2 b^3 x^{3/2} \left (42 d+35 e \sqrt {x}+30 f x\right )\right ) \] Input:
Integrate[(a + b*Sqrt[x])^3*(d + e*Sqrt[x] + f*x),x]
Output:
(x*(35*a^3*(6*d + 4*e*Sqrt[x] + 3*f*x) + 21*a*b^2*x*(15*d + 12*e*Sqrt[x] + 10*f*x) + 21*a^2*b*Sqrt[x]*(20*d + 15*e*Sqrt[x] + 12*f*x) + 2*b^3*x^(3/2) *(42*d + 35*e*Sqrt[x] + 30*f*x)))/210
Time = 0.49 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.02, 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.
\(\displaystyle \int \left (a+b \sqrt {x}\right )^3 \left (d+e \sqrt {x}+f x\right ) \, dx\) |
\(\Big \downarrow \) 1731 |
\(\displaystyle 2 \int \left (a+b \sqrt {x}\right )^3 \sqrt {x} \left (d+f x+e \sqrt {x}\right )d\sqrt {x}\) |
\(\Big \downarrow \) 1195 |
\(\displaystyle 2 \int \left (\frac {f \left (a+b \sqrt {x}\right )^6}{b^3}+\frac {(b e-3 a f) \left (a+b \sqrt {x}\right )^5}{b^3}+\frac {\left (3 f a^2-2 b e a+b^2 d\right ) \left (a+b \sqrt {x}\right )^4}{b^3}-\frac {a \left (f a^2-b e a+b^2 d\right ) \left (a+b \sqrt {x}\right )^3}{b^3}\right )d\sqrt {x}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 2 \left (\frac {\left (a+b \sqrt {x}\right )^5 \left (3 a^2 f-2 a b e+b^2 d\right )}{5 b^4}-\frac {a \left (a+b \sqrt {x}\right )^4 \left (a^2 f-a b e+b^2 d\right )}{4 b^4}+\frac {\left (a+b \sqrt {x}\right )^6 (b e-3 a f)}{6 b^4}+\frac {f \left (a+b \sqrt {x}\right )^7}{7 b^4}\right )\) |
Input:
Int[(a + b*Sqrt[x])^3*(d + e*Sqrt[x] + f*x),x]
Output:
2*(-1/4*(a*(b^2*d - a*b*e + a^2*f)*(a + b*Sqrt[x])^4)/b^4 + ((b^2*d - 2*a* b*e + 3*a^2*f)*(a + b*Sqrt[x])^5)/(5*b^4) + ((b*e - 3*a*f)*(a + b*Sqrt[x]) ^6)/(6*b^4) + (f*(a + b*Sqrt[x])^7)/(7*b^4))
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 = 0.89 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.89
method | result | size |
derivativedivides | \(\frac {2 b^{3} f \,x^{\frac {7}{2}}}{7}+\frac {\left (3 b^{2} a f +b^{3} e \right ) x^{3}}{3}+\frac {2 \left (3 b \,a^{2} f +3 a \,b^{2} e +b^{3} d \right ) x^{\frac {5}{2}}}{5}+\frac {\left (a^{3} f +3 b \,a^{2} e +3 a \,b^{2} d \right ) x^{2}}{2}+\frac {2 \left (a^{3} e +3 a^{2} b d \right ) x^{\frac {3}{2}}}{3}+a^{3} d x\) | \(103\) |
default | \(\frac {b^{3} e \,x^{3}}{3}+b^{2} \left (\frac {2 b f \,x^{\frac {7}{2}}}{7}+\frac {2 \left (3 a e +b d \right ) x^{\frac {5}{2}}}{5}\right )+b^{2} a f \,x^{3}+\frac {\left (3 b \,a^{2} e +3 a \,b^{2} d \right ) x^{2}}{2}+a^{2} \left (\frac {6 b f \,x^{\frac {5}{2}}}{5}+\frac {2 \left (a e +3 b d \right ) x^{\frac {3}{2}}}{3}\right )+a^{3} \left (\frac {1}{2} f \,x^{2}+d x \right )\) | \(104\) |
trager | \(\frac {\left (6 b^{2} a f \,x^{2}+2 b^{3} x^{2} e +3 a^{3} f x +9 a^{2} b e x +9 a \,b^{2} d x +6 b^{2} a f x +2 b^{3} e x +6 a^{3} d +3 a^{3} f +9 b \,a^{2} e +9 a \,b^{2} d +6 b^{2} a f +2 b^{3} e \right ) \left (x -1\right )}{6}+\frac {2 x^{\frac {3}{2}} \left (15 b^{3} x^{2} f +63 a^{2} b f x +63 a \,b^{2} e x +21 b^{3} d x +35 a^{3} e +105 a^{2} b d \right )}{105}\) | \(155\) |
orering | \(\text {Expression too large to display}\) | \(1043\) |
Input:
int((a+b*x^(1/2))^3*(d+e*x^(1/2)+f*x),x,method=_RETURNVERBOSE)
Output:
2/7*b^3*f*x^(7/2)+1/3*(3*a*b^2*f+b^3*e)*x^3+2/5*(3*a^2*b*f+3*a*b^2*e+b^3*d )*x^(5/2)+1/2*(a^3*f+3*a^2*b*e+3*a*b^2*d)*x^2+2/3*(a^3*e+3*a^2*b*d)*x^(3/2 )+a^3*d*x
Time = 0.09 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.91 \[ \int \left (a+b \sqrt {x}\right )^3 \left (d+e \sqrt {x}+f x\right ) \, dx=a^{3} d x + \frac {1}{3} \, {\left (b^{3} e + 3 \, a b^{2} f\right )} x^{3} + \frac {1}{2} \, {\left (3 \, a b^{2} d + 3 \, a^{2} b e + a^{3} f\right )} x^{2} + \frac {2}{105} \, {\left (15 \, b^{3} f x^{3} + 21 \, {\left (b^{3} d + 3 \, a b^{2} e + 3 \, a^{2} b f\right )} x^{2} + 35 \, {\left (3 \, a^{2} b d + a^{3} e\right )} x\right )} \sqrt {x} \] Input:
integrate((a+b*x^(1/2))^3*(d+e*x^(1/2)+f*x),x, algorithm="fricas")
Output:
a^3*d*x + 1/3*(b^3*e + 3*a*b^2*f)*x^3 + 1/2*(3*a*b^2*d + 3*a^2*b*e + a^3*f )*x^2 + 2/105*(15*b^3*f*x^3 + 21*(b^3*d + 3*a*b^2*e + 3*a^2*b*f)*x^2 + 35* (3*a^2*b*d + a^3*e)*x)*sqrt(x)
Time = 0.62 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.01 \[ \int \left (a+b \sqrt {x}\right )^3 \left (d+e \sqrt {x}+f x\right ) \, dx=a^{3} d x + \frac {2 b^{3} f x^{\frac {7}{2}}}{7} + \frac {2 x^{\frac {5}{2}} \cdot \left (3 a^{2} b f + 3 a b^{2} e + b^{3} d\right )}{5} + \frac {2 x^{\frac {3}{2}} \left (a^{3} e + 3 a^{2} b d\right )}{3} + \frac {x^{3} \cdot \left (3 a b^{2} f + b^{3} e\right )}{3} + \frac {x^{2} \left (a^{3} f + 3 a^{2} b e + 3 a b^{2} d\right )}{2} \] Input:
integrate((a+b*x**(1/2))**3*(d+e*x**(1/2)+f*x),x)
Output:
a**3*d*x + 2*b**3*f*x**(7/2)/7 + 2*x**(5/2)*(3*a**2*b*f + 3*a*b**2*e + b** 3*d)/5 + 2*x**(3/2)*(a**3*e + 3*a**2*b*d)/3 + x**3*(3*a*b**2*f + b**3*e)/3 + x**2*(a**3*f + 3*a**2*b*e + 3*a*b**2*d)/2
Time = 0.04 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.88 \[ \int \left (a+b \sqrt {x}\right )^3 \left (d+e \sqrt {x}+f x\right ) \, dx=\frac {2}{7} \, b^{3} f x^{\frac {7}{2}} + a^{3} d x + \frac {1}{3} \, {\left (b^{3} e + 3 \, a b^{2} f\right )} x^{3} + \frac {2}{5} \, {\left (b^{3} d + 3 \, a b^{2} e + 3 \, a^{2} b f\right )} x^{\frac {5}{2}} + \frac {1}{2} \, {\left (3 \, a b^{2} d + 3 \, a^{2} b e + a^{3} f\right )} x^{2} + \frac {2}{3} \, {\left (3 \, a^{2} b d + a^{3} e\right )} x^{\frac {3}{2}} \] Input:
integrate((a+b*x^(1/2))^3*(d+e*x^(1/2)+f*x),x, algorithm="maxima")
Output:
2/7*b^3*f*x^(7/2) + a^3*d*x + 1/3*(b^3*e + 3*a*b^2*f)*x^3 + 2/5*(b^3*d + 3 *a*b^2*e + 3*a^2*b*f)*x^(5/2) + 1/2*(3*a*b^2*d + 3*a^2*b*e + a^3*f)*x^2 + 2/3*(3*a^2*b*d + a^3*e)*x^(3/2)
Time = 0.13 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.96 \[ \int \left (a+b \sqrt {x}\right )^3 \left (d+e \sqrt {x}+f x\right ) \, dx=\frac {2}{7} \, b^{3} f x^{\frac {7}{2}} + \frac {1}{3} \, b^{3} e x^{3} + a b^{2} f x^{3} + \frac {2}{5} \, b^{3} d x^{\frac {5}{2}} + \frac {6}{5} \, a b^{2} e x^{\frac {5}{2}} + \frac {6}{5} \, a^{2} b f x^{\frac {5}{2}} + \frac {3}{2} \, a b^{2} d x^{2} + \frac {3}{2} \, a^{2} b e x^{2} + \frac {1}{2} \, a^{3} f x^{2} + 2 \, a^{2} b d x^{\frac {3}{2}} + \frac {2}{3} \, a^{3} e x^{\frac {3}{2}} + a^{3} d x \] Input:
integrate((a+b*x^(1/2))^3*(d+e*x^(1/2)+f*x),x, algorithm="giac")
Output:
2/7*b^3*f*x^(7/2) + 1/3*b^3*e*x^3 + a*b^2*f*x^3 + 2/5*b^3*d*x^(5/2) + 6/5* a*b^2*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*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
Time = 0.05 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.87 \[ \int \left (a+b \sqrt {x}\right )^3 \left (d+e \sqrt {x}+f x\right ) \, dx=x^{3/2}\,\left (\frac {2\,e\,a^3}{3}+2\,b\,d\,a^2\right )+x^3\,\left (\frac {e\,b^3}{3}+a\,f\,b^2\right )+x^2\,\left (\frac {f\,a^3}{2}+\frac {3\,e\,a^2\,b}{2}+\frac {3\,d\,a\,b^2}{2}\right )+x^{5/2}\,\left (\frac {6\,f\,a^2\,b}{5}+\frac {6\,e\,a\,b^2}{5}+\frac {2\,d\,b^3}{5}\right )+\frac {2\,b^3\,f\,x^{7/2}}{7}+a^3\,d\,x \] Input:
int((a + 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^3*((b^3*e)/3 + a*b^2*f) + x^2*((a^3* f)/2 + (3*a*b^2*d)/2 + (3*a^2*b*e)/2) + x^(5/2)*((2*b^3*d)/5 + (6*a*b^2*e) /5 + (6*a^2*b*f)/5) + (2*b^3*f*x^(7/2))/7 + a^3*d*x
Time = 0.23 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.94 \[ \int \left (a+b \sqrt {x}\right )^3 \left (d+e \sqrt {x}+f x\right ) \, dx=\frac {x \left (140 \sqrt {x}\, a^{3} e +420 \sqrt {x}\, a^{2} b d +252 \sqrt {x}\, a^{2} b f x +252 \sqrt {x}\, a \,b^{2} e x +84 \sqrt {x}\, b^{3} d x +60 \sqrt {x}\, b^{3} f \,x^{2}+210 a^{3} d +105 a^{3} f x +315 a^{2} b e x +315 a \,b^{2} d x +210 a \,b^{2} f \,x^{2}+70 b^{3} e \,x^{2}\right )}{210} \] Input:
int((a+b*x^(1/2))^3*(d+e*x^(1/2)+f*x),x)
Output:
(x*(140*sqrt(x)*a**3*e + 420*sqrt(x)*a**2*b*d + 252*sqrt(x)*a**2*b*f*x + 2 52*sqrt(x)*a*b**2*e*x + 84*sqrt(x)*b**3*d*x + 60*sqrt(x)*b**3*f*x**2 + 210 *a**3*d + 105*a**3*f*x + 315*a**2*b*e*x + 315*a*b**2*d*x + 210*a*b**2*f*x* *2 + 70*b**3*e*x**2))/210