Integrand size = 17, antiderivative size = 98 \[ \int \left (a+b \sqrt {x}\right )^3 (d+e x) \, dx=-\frac {a \left (b^2 d+a^2 e\right ) \left (a+b \sqrt {x}\right )^4}{2 b^4}+\frac {2 \left (b^2 d+3 a^2 e\right ) \left (a+b \sqrt {x}\right )^5}{5 b^4}-\frac {a e \left (a+b \sqrt {x}\right )^6}{b^4}+\frac {2 e \left (a+b \sqrt {x}\right )^7}{7 b^4} \] Output:
-1/2*a*(a^2*e+b^2*d)*(a+b*x^(1/2))^4/b^4+2/5*(3*a^2*e+b^2*d)*(a+b*x^(1/2)) ^5/b^4-a*e*(a+b*x^(1/2))^6/b^4+2/7*e*(a+b*x^(1/2))^7/b^4
Time = 0.10 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.71 \[ \int \left (a+b \sqrt {x}\right )^3 (d+e x) \, dx=\frac {1}{70} x \left (35 a^3 (2 d+e x)+35 a b^2 x (3 d+2 e x)+28 a^2 b \sqrt {x} (5 d+3 e x)+4 b^3 x^{3/2} (7 d+5 e x)\right ) \] Input:
Integrate[(a + b*Sqrt[x])^3*(d + e*x),x]
Output:
(x*(35*a^3*(2*d + e*x) + 35*a*b^2*x*(3*d + 2*e*x) + 28*a^2*b*Sqrt[x]*(5*d + 3*e*x) + 4*b^3*x^(3/2)*(7*d + 5*e*x)))/70
Time = 0.42 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.04, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {1732, 522, 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.
\(\displaystyle \int \left (a+b \sqrt {x}\right )^3 (d+e x) \, dx\) |
\(\Big \downarrow \) 1732 |
\(\displaystyle 2 \int \left (a+b \sqrt {x}\right )^3 \sqrt {x} (d+e x)d\sqrt {x}\) |
\(\Big \downarrow \) 522 |
\(\displaystyle 2 \int \left (\frac {e \left (a+b \sqrt {x}\right )^6}{b^3}-\frac {3 a e \left (a+b \sqrt {x}\right )^5}{b^3}+\frac {\left (3 e a^2+b^2 d\right ) \left (a+b \sqrt {x}\right )^4}{b^3}+\frac {a \left (-e a^2-b^2 d\right ) \left (a+b \sqrt {x}\right )^3}{b^3}\right )d\sqrt {x}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 2 \left (\frac {\left (a+b \sqrt {x}\right )^5 \left (3 a^2 e+b^2 d\right )}{5 b^4}-\frac {a \left (a+b \sqrt {x}\right )^4 \left (a^2 e+b^2 d\right )}{4 b^4}+\frac {e \left (a+b \sqrt {x}\right )^7}{7 b^4}-\frac {a e \left (a+b \sqrt {x}\right )^6}{2 b^4}\right )\) |
Input:
Int[(a + b*Sqrt[x])^3*(d + e*x),x]
Output:
2*(-1/4*(a*(b^2*d + a^2*e)*(a + b*Sqrt[x])^4)/b^4 + ((b^2*d + 3*a^2*e)*(a + b*Sqrt[x])^5)/(5*b^4) - (a*e*(a + b*Sqrt[x])^6)/(2*b^4) + (e*(a + b*Sqrt [x])^7)/(7*b^4))
Int[((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_. ), x_Symbol] :> Int[ExpandIntegrand[(e*x)^m*(c + d*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 0]
Int[((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Symb ol] :> With[{g = Denominator[n]}, Simp[g Subst[Int[x^(g - 1)*(d + e*x^(g* n))^q*(a + c*x^(2*g*n))^p, x], x, x^(1/g)], x]] /; FreeQ[{a, c, d, e, p, q} , x] && EqQ[n2, 2*n] && FractionQ[n]
Time = 0.34 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.72
method | result | size |
default | \(b^{3} \left (\frac {2 e \,x^{\frac {7}{2}}}{7}+\frac {2 d \,x^{\frac {5}{2}}}{5}\right )+3 b^{2} a \left (\frac {1}{3} e \,x^{3}+\frac {1}{2} d \,x^{2}\right )+3 b \,a^{2} \left (\frac {2 e \,x^{\frac {5}{2}}}{5}+\frac {2 d \,x^{\frac {3}{2}}}{3}\right )+a^{3} \left (\frac {1}{2} e \,x^{2}+d x \right )\) | \(71\) |
derivativedivides | \(\frac {2 b^{3} e \,x^{\frac {7}{2}}}{7}+a \,b^{2} e \,x^{3}+\frac {2 \left (3 b \,a^{2} e +b^{3} d \right ) x^{\frac {5}{2}}}{5}+\frac {\left (a^{3} e +3 a \,b^{2} d \right ) x^{2}}{2}+2 a^{2} b d \,x^{\frac {3}{2}}+a^{3} d x\) | \(72\) |
trager | \(\frac {a \left (2 b^{2} e \,x^{2}+a^{2} e x +3 b^{2} d x +2 b^{2} e x +2 a^{2} d +a^{2} e +3 d \,b^{2}+2 b^{2} e \right ) \left (x -1\right )}{2}+\frac {2 b \,x^{\frac {3}{2}} \left (5 b^{2} e \,x^{2}+21 a^{2} e x +7 b^{2} d x +35 a^{2} d \right )}{35}\) | \(97\) |
orering | \(-\frac {\left (-110 b^{6} e^{2} x^{5}+243 a^{2} b^{4} e^{2} x^{4}-279 b^{6} d e \,x^{4}-147 a^{4} b^{2} e^{2} x^{3}+630 a^{2} b^{4} d e \,x^{3}-147 b^{6} d^{2} x^{3}-105 a^{4} b^{2} d e \,x^{2}+315 a^{2} b^{4} d^{2} x^{2}+630 a^{6} d e x +210 a^{6} d^{2}\right ) \left (a +b \sqrt {x}\right )^{3}}{210 b^{2} \left (-b^{2} x +a^{2}\right )^{2} \left (e x +d \right )}+\frac {\left (-10 x^{4} b^{6} e +27 x^{3} b^{4} a^{2} e -21 x^{3} b^{6} d -21 x^{2} b^{2} a^{4} e +63 x^{2} b^{4} a^{2} d +210 a^{6} d \right ) x \left (\frac {3 \left (a +b \sqrt {x}\right )^{2} \left (e x +d \right ) b}{2 \sqrt {x}}+\left (a +b \sqrt {x}\right )^{3} e \right )}{105 b^{2} \left (-b^{2} x +a^{2}\right )^{2} \left (e x +d \right )}\) | \(270\) |
Input:
int((a+b*x^(1/2))^3*(e*x+d),x,method=_RETURNVERBOSE)
Output:
b^3*(2/7*e*x^(7/2)+2/5*d*x^(5/2))+3*b^2*a*(1/3*e*x^3+1/2*d*x^2)+3*b*a^2*(2 /5*e*x^(5/2)+2/3*d*x^(3/2))+a^3*(1/2*e*x^2+d*x)
Time = 0.08 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.77 \[ \int \left (a+b \sqrt {x}\right )^3 (d+e x) \, dx=a b^{2} e x^{3} + a^{3} d x + \frac {1}{2} \, {\left (3 \, a b^{2} d + a^{3} e\right )} x^{2} + \frac {2}{35} \, {\left (5 \, b^{3} e x^{3} + 35 \, a^{2} b d x + 7 \, {\left (b^{3} d + 3 \, a^{2} b e\right )} x^{2}\right )} \sqrt {x} \] Input:
integrate((a+b*x^(1/2))^3*(e*x+d),x, algorithm="fricas")
Output:
a*b^2*e*x^3 + a^3*d*x + 1/2*(3*a*b^2*d + a^3*e)*x^2 + 2/35*(5*b^3*e*x^3 + 35*a^2*b*d*x + 7*(b^3*d + 3*a^2*b*e)*x^2)*sqrt(x)
Time = 0.58 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.85 \[ \int \left (a+b \sqrt {x}\right )^3 (d+e x) \, dx=a^{3} d x + 2 a^{2} b d x^{\frac {3}{2}} + a b^{2} e x^{3} + \frac {2 b^{3} e x^{\frac {7}{2}}}{7} + \frac {2 x^{\frac {5}{2}} \cdot \left (3 a^{2} b e + b^{3} d\right )}{5} + \frac {x^{2} \left (a^{3} e + 3 a b^{2} d\right )}{2} \] Input:
integrate((a+b*x**(1/2))**3*(e*x+d),x)
Output:
a**3*d*x + 2*a**2*b*d*x**(3/2) + a*b**2*e*x**3 + 2*b**3*e*x**(7/2)/7 + 2*x **(5/2)*(3*a**2*b*e + b**3*d)/5 + x**2*(a**3*e + 3*a*b**2*d)/2
Time = 0.04 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.72 \[ \int \left (a+b \sqrt {x}\right )^3 (d+e x) \, dx=\frac {2}{7} \, b^{3} e x^{\frac {7}{2}} + a b^{2} e x^{3} + 2 \, a^{2} b d x^{\frac {3}{2}} + a^{3} d x + \frac {2}{5} \, {\left (b^{3} d + 3 \, a^{2} b e\right )} x^{\frac {5}{2}} + \frac {1}{2} \, {\left (3 \, a b^{2} d + a^{3} e\right )} x^{2} \] Input:
integrate((a+b*x^(1/2))^3*(e*x+d),x, algorithm="maxima")
Output:
2/7*b^3*e*x^(7/2) + a*b^2*e*x^3 + 2*a^2*b*d*x^(3/2) + a^3*d*x + 2/5*(b^3*d + 3*a^2*b*e)*x^(5/2) + 1/2*(3*a*b^2*d + a^3*e)*x^2
Time = 0.13 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.74 \[ \int \left (a+b \sqrt {x}\right )^3 (d+e x) \, dx=\frac {2}{7} \, b^{3} e x^{\frac {7}{2}} + a b^{2} e x^{3} + \frac {2}{5} \, b^{3} d x^{\frac {5}{2}} + \frac {6}{5} \, a^{2} b e x^{\frac {5}{2}} + \frac {3}{2} \, a b^{2} d x^{2} + \frac {1}{2} \, a^{3} e x^{2} + 2 \, a^{2} b d x^{\frac {3}{2}} + a^{3} d x \] Input:
integrate((a+b*x^(1/2))^3*(e*x+d),x, algorithm="giac")
Output:
2/7*b^3*e*x^(7/2) + a*b^2*e*x^3 + 2/5*b^3*d*x^(5/2) + 6/5*a^2*b*e*x^(5/2) + 3/2*a*b^2*d*x^2 + 1/2*a^3*e*x^2 + 2*a^2*b*d*x^(3/2) + a^3*d*x
Time = 0.05 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.72 \[ \int \left (a+b \sqrt {x}\right )^3 (d+e x) \, dx=x^2\,\left (\frac {e\,a^3}{2}+\frac {3\,d\,a\,b^2}{2}\right )+x^{5/2}\,\left (\frac {6\,e\,a^2\,b}{5}+\frac {2\,d\,b^3}{5}\right )+\frac {2\,b^3\,e\,x^{7/2}}{7}+a^3\,d\,x+2\,a^2\,b\,d\,x^{3/2}+a\,b^2\,e\,x^3 \] Input:
int((a + b*x^(1/2))^3*(d + e*x),x)
Output:
x^2*((a^3*e)/2 + (3*a*b^2*d)/2) + x^(5/2)*((2*b^3*d)/5 + (6*a^2*b*e)/5) + (2*b^3*e*x^(7/2))/7 + a^3*d*x + 2*a^2*b*d*x^(3/2) + a*b^2*e*x^3
Time = 0.16 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.76 \[ \int \left (a+b \sqrt {x}\right )^3 (d+e x) \, dx=\frac {x \left (140 \sqrt {x}\, a^{2} b d +84 \sqrt {x}\, a^{2} b e x +28 \sqrt {x}\, b^{3} d x +20 \sqrt {x}\, b^{3} e \,x^{2}+70 a^{3} d +35 a^{3} e x +105 a \,b^{2} d x +70 a \,b^{2} e \,x^{2}\right )}{70} \] Input:
int((a+b*x^(1/2))^3*(e*x+d),x)
Output:
(x*(140*sqrt(x)*a**2*b*d + 84*sqrt(x)*a**2*b*e*x + 28*sqrt(x)*b**3*d*x + 2 0*sqrt(x)*b**3*e*x**2 + 70*a**3*d + 35*a**3*e*x + 105*a*b**2*d*x + 70*a*b* *2*e*x**2))/70