\(\int \sqrt {2+3 x} (f+g x)^3 \sqrt {1+\frac {5 x}{6}-x^2} \, dx\) [295]

Optimal result

Integrand size = 33, antiderivative size = 148 \[ \int \sqrt {2+3 x} (f+g x)^3 \sqrt {1+\frac {5 x}{6}-x^2} \, dx=-\frac {13 (2 f+3 g)^3 (3-2 x)^{3/2}}{48 \sqrt {6}}+\frac {1}{40} \sqrt {\frac {3}{2}} (2 f+3 g)^2 (f+8 g) (3-2 x)^{5/2}-\frac {1}{56} \sqrt {\frac {3}{2}} g (2 f+3 g) (3 f+11 g) (3-2 x)^{7/2}+\frac {g^2 (9 f+20 g) (3-2 x)^{9/2}}{72 \sqrt {6}}-\frac {1}{176} \sqrt {\frac {3}{2}} g^3 (3-2 x)^{11/2} \] Output:

-13/288*(2*f+3*g)^3*(3-2*x)^(3/2)*6^(1/2)+1/80*6^(1/2)*(2*f+3*g)^2*(f+8*g) 
*(3-2*x)^(5/2)-1/112*6^(1/2)*g*(2*f+3*g)*(3*f+11*g)*(3-2*x)^(7/2)+1/432*g^ 
2*(9*f+20*g)*(3-2*x)^(9/2)*6^(1/2)-1/352*6^(1/2)*g^3*(3-2*x)^(11/2)
 

Mathematica [A] (verified)

Time = 1.82 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.74 \[ \int \sqrt {2+3 x} (f+g x)^3 \sqrt {1+\frac {5 x}{6}-x^2} \, dx=\frac {(-3+2 x) \sqrt {6+5 x-6 x^2} \left (231 f^3 (19+9 x)+297 f^2 g \left (32+32 x+15 x^2\right )+495 f g^2 \left (18+18 x+15 x^2+7 x^3\right )+g^3 \left (3132+3132 x+2610 x^2+2030 x^3+945 x^4\right )\right )}{3465 \sqrt {6} \sqrt {2+3 x}} \] Input:

Integrate[Sqrt[2 + 3*x]*(f + g*x)^3*Sqrt[1 + (5*x)/6 - x^2],x]
 

Output:

((-3 + 2*x)*Sqrt[6 + 5*x - 6*x^2]*(231*f^3*(19 + 9*x) + 297*f^2*g*(32 + 32 
*x + 15*x^2) + 495*f*g^2*(18 + 18*x + 15*x^2 + 7*x^3) + g^3*(3132 + 3132*x 
 + 2610*x^2 + 2030*x^3 + 945*x^4)))/(3465*Sqrt[6]*Sqrt[2 + 3*x])
 

Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {1245, 86, 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.

\(\displaystyle \int \sqrt {3 x+2} \sqrt {-x^2+\frac {5 x}{6}+1} (f+g x)^3 \, dx\)

\(\Big \downarrow \) 1245

\(\displaystyle \int \sqrt {\frac {1}{2}-\frac {x}{3}} (3 x+2) (f+g x)^3dx\)

\(\Big \downarrow \) 86

\(\displaystyle \int \left (\frac {27}{2} g \left (\frac {1}{2}-\frac {x}{3}\right )^{5/2} \left (6 f^2+31 f g+33 g^2\right )-27 g^2 \left (\frac {1}{2}-\frac {x}{3}\right )^{7/2} (9 f+20 g)-\frac {9}{4} \left (\frac {1}{2}-\frac {x}{3}\right )^{3/2} (2 f+3 g)^2 (f+8 g)+\frac {13}{16} \sqrt {\frac {1}{2}-\frac {x}{3}} (2 f+3 g)^3+243 g^3 \left (\frac {1}{2}-\frac {x}{3}\right )^{9/2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {g^2 (3-2 x)^{9/2} (9 f+20 g)}{72 \sqrt {6}}-\frac {1}{56} \sqrt {\frac {3}{2}} g (3-2 x)^{7/2} (2 f+3 g) (3 f+11 g)+\frac {1}{40} \sqrt {\frac {3}{2}} (3-2 x)^{5/2} (2 f+3 g)^2 (f+8 g)-\frac {13 (3-2 x)^{3/2} (2 f+3 g)^3}{48 \sqrt {6}}-\frac {1}{176} \sqrt {\frac {3}{2}} g^3 (3-2 x)^{11/2}\)

Input:

Int[Sqrt[2 + 3*x]*(f + g*x)^3*Sqrt[1 + (5*x)/6 - x^2],x]
 

Output:

(-13*(2*f + 3*g)^3*(3 - 2*x)^(3/2))/(48*Sqrt[6]) + (Sqrt[3/2]*(2*f + 3*g)^ 
2*(f + 8*g)*(3 - 2*x)^(5/2))/40 - (Sqrt[3/2]*g*(2*f + 3*g)*(3*f + 11*g)*(3 
 - 2*x)^(7/2))/56 + (g^2*(9*f + 20*g)*(3 - 2*x)^(9/2))/(72*Sqrt[6]) - (Sqr 
t[3/2]*g^3*(3 - 2*x)^(11/2))/176
 

Defintions of rubi rules used

rule 86

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_ 
.), x_] :> Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; 
 FreeQ[{a, b, c, d, e, f, n}, x] && ((ILtQ[n, 0] && ILtQ[p, 0]) || EqQ[p, 1 
] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p 
+ 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
 

rule 1245

Int[((d_) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_) + (b_.)*(x_) 
 + (c_.)*(x_)^2)^(p_), x_Symbol] :> Int[(d + e*x)^(m + p)*(f + g*x)^n*(a/d 
+ (c/e)*x)^p, x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && EqQ[c*d^2 - 
 b*d*e + a*e^2, 0] && GtQ[a, 0] && GtQ[d, 0] && LtQ[c, 0]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [A] (verified)

Time = 1.17 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.86

method	result	size
gosper	\(\frac {\left (2 x -3\right ) \left (945 g^{3} x^{4}+3465 x^{3} f \,g^{2}+2030 x^{3} g^{3}+4455 x^{2} f^{2} g +7425 f \,g^{2} x^{2}+2610 x^{2} g^{3}+2079 x \,f^{3}+9504 x \,f^{2} g +8910 x f \,g^{2}+3132 x \,g^{3}+4389 f^{3}+9504 f^{2} g +8910 f \,g^{2}+3132 g^{3}\right ) \sqrt {-36 x^{2}+30 x +36}}{20790 \sqrt {3 x +2}}\)	\(127\)
default	\(\frac {\left (2 x -3\right ) \left (945 g^{3} x^{4}+3465 x^{3} f \,g^{2}+2030 x^{3} g^{3}+4455 x^{2} f^{2} g +7425 f \,g^{2} x^{2}+2610 x^{2} g^{3}+2079 x \,f^{3}+9504 x \,f^{2} g +8910 x f \,g^{2}+3132 x \,g^{3}+4389 f^{3}+9504 f^{2} g +8910 f \,g^{2}+3132 g^{3}\right ) \sqrt {-36 x^{2}+30 x +36}}{20790 \sqrt {3 x +2}}\)	\(127\)
orering	\(\frac {\left (2 x -3\right ) \left (945 g^{3} x^{4}+3465 x^{3} f \,g^{2}+2030 x^{3} g^{3}+4455 x^{2} f^{2} g +7425 f \,g^{2} x^{2}+2610 x^{2} g^{3}+2079 x \,f^{3}+9504 x \,f^{2} g +8910 x f \,g^{2}+3132 x \,g^{3}+4389 f^{3}+9504 f^{2} g +8910 f \,g^{2}+3132 g^{3}\right ) \sqrt {-36 x^{2}+30 x +36}}{20790 \sqrt {3 x +2}}\)	\(127\)
risch	\(-\frac {\sqrt {\frac {-36 x^{2}+30 x +36}{3 x +2}}\, \sqrt {3 x +2}\, \left (1890 g^{3} x^{5}+6930 f \,g^{2} x^{4}+1225 g^{3} x^{4}+8910 f^{2} g \,x^{3}+4455 x^{3} f \,g^{2}-870 x^{3} g^{3}+4158 f^{3} x^{2}+5643 x^{2} f^{2} g -4455 f \,g^{2} x^{2}-1566 x^{2} g^{3}+2541 x \,f^{3}-9504 x \,f^{2} g -8910 x f \,g^{2}-3132 x \,g^{3}-13167 f^{3}-28512 f^{2} g -26730 f \,g^{2}-9396 g^{3}\right ) \left (2 x -3\right )}{3465 \sqrt {-36 x^{2}+30 x +36}\, \sqrt {-12 x +18}}\)	\(188\)

Input:

int(1/6*(3*x+2)^(1/2)*(g*x+f)^3*(-36*x^2+30*x+36)^(1/2),x,method=_RETURNVE 
RBOSE)
 

Output:

1/20790*(2*x-3)*(945*g^3*x^4+3465*f*g^2*x^3+2030*g^3*x^3+4455*f^2*g*x^2+74 
25*f*g^2*x^2+2610*g^3*x^2+2079*f^3*x+9504*f^2*g*x+8910*f*g^2*x+3132*g^3*x+ 
4389*f^3+9504*f^2*g+8910*f*g^2+3132*g^3)*(-36*x^2+30*x+36)^(1/2)/(3*x+2)^( 
1/2)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.99 \[ \int \sqrt {2+3 x} (f+g x)^3 \sqrt {1+\frac {5 x}{6}-x^2} \, dx=\frac {{\left (1890 \, g^{3} x^{5} + 35 \, {\left (198 \, f g^{2} + 35 \, g^{3}\right )} x^{4} + 15 \, {\left (594 \, f^{2} g + 297 \, f g^{2} - 58 \, g^{3}\right )} x^{3} - 13167 \, f^{3} - 28512 \, f^{2} g - 26730 \, f g^{2} - 9396 \, g^{3} + 27 \, {\left (154 \, f^{3} + 209 \, f^{2} g - 165 \, f g^{2} - 58 \, g^{3}\right )} x^{2} + 3 \, {\left (847 \, f^{3} - 3168 \, f^{2} g - 2970 \, f g^{2} - 1044 \, g^{3}\right )} x\right )} \sqrt {-36 \, x^{2} + 30 \, x + 36}}{20790 \, \sqrt {3 \, x + 2}} \] Input:

integrate(1/6*(2+3*x)^(1/2)*(g*x+f)^3*(-36*x^2+30*x+36)^(1/2),x, algorithm 
="fricas")
 

Output:

1/20790*(1890*g^3*x^5 + 35*(198*f*g^2 + 35*g^3)*x^4 + 15*(594*f^2*g + 297* 
f*g^2 - 58*g^3)*x^3 - 13167*f^3 - 28512*f^2*g - 26730*f*g^2 - 9396*g^3 + 2 
7*(154*f^3 + 209*f^2*g - 165*f*g^2 - 58*g^3)*x^2 + 3*(847*f^3 - 3168*f^2*g 
 - 2970*f*g^2 - 1044*g^3)*x)*sqrt(-36*x^2 + 30*x + 36)/sqrt(3*x + 2)
 

Sympy [F]

\[ \int \sqrt {2+3 x} (f+g x)^3 \sqrt {1+\frac {5 x}{6}-x^2} \, dx=\frac {\sqrt {6} \left (\int f^{3} \sqrt {3 x + 2} \sqrt {- 6 x^{2} + 5 x + 6}\, dx + \int g^{3} x^{3} \sqrt {3 x + 2} \sqrt {- 6 x^{2} + 5 x + 6}\, dx + \int 3 f g^{2} x^{2} \sqrt {3 x + 2} \sqrt {- 6 x^{2} + 5 x + 6}\, dx + \int 3 f^{2} g x \sqrt {3 x + 2} \sqrt {- 6 x^{2} + 5 x + 6}\, dx\right )}{6} \] Input:

integrate(1/6*(2+3*x)**(1/2)*(g*x+f)**3*(-36*x**2+30*x+36)**(1/2),x)
 

Output:

sqrt(6)*(Integral(f**3*sqrt(3*x + 2)*sqrt(-6*x**2 + 5*x + 6), x) + Integra 
l(g**3*x**3*sqrt(3*x + 2)*sqrt(-6*x**2 + 5*x + 6), x) + Integral(3*f*g**2* 
x**2*sqrt(3*x + 2)*sqrt(-6*x**2 + 5*x + 6), x) + Integral(3*f**2*g*x*sqrt( 
3*x + 2)*sqrt(-6*x**2 + 5*x + 6), x))/6
 

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.17 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.57 \[ \int \sqrt {2+3 x} (f+g x)^3 \sqrt {1+\frac {5 x}{6}-x^2} \, dx=\frac {1}{90} \, {\left (18 i \, \sqrt {3} \sqrt {2} x^{2} + 11 i \, \sqrt {3} \sqrt {2} x - 57 i \, \sqrt {3} \sqrt {2}\right )} f^{3} \sqrt {2 \, x - 3} + \frac {1}{70} \, {\left (30 i \, \sqrt {3} \sqrt {2} x^{3} + 19 i \, \sqrt {3} \sqrt {2} x^{2} - 32 i \, \sqrt {3} \sqrt {2} x - 96 i \, \sqrt {3} \sqrt {2}\right )} f^{2} g \sqrt {2 \, x - 3} + \frac {1}{42} \, {\left (14 i \, \sqrt {3} \sqrt {2} x^{4} + 9 i \, \sqrt {3} \sqrt {2} x^{3} - 9 i \, \sqrt {3} \sqrt {2} x^{2} - 18 i \, \sqrt {3} \sqrt {2} x - 54 i \, \sqrt {3} \sqrt {2}\right )} f g^{2} \sqrt {2 \, x - 3} + \frac {1}{20790} \, {\left (1890 i \, \sqrt {3} \sqrt {2} x^{5} + 1225 i \, \sqrt {3} \sqrt {2} x^{4} - 870 i \, \sqrt {3} \sqrt {2} x^{3} - 1566 i \, \sqrt {3} \sqrt {2} x^{2} - 3132 i \, \sqrt {3} \sqrt {2} x - 9396 i \, \sqrt {3} \sqrt {2}\right )} g^{3} \sqrt {2 \, x - 3} \] Input:

integrate(1/6*(2+3*x)^(1/2)*(g*x+f)^3*(-36*x^2+30*x+36)^(1/2),x, algorithm 
="maxima")
 

Output:

1/90*(18*I*sqrt(3)*sqrt(2)*x^2 + 11*I*sqrt(3)*sqrt(2)*x - 57*I*sqrt(3)*sqr 
t(2))*f^3*sqrt(2*x - 3) + 1/70*(30*I*sqrt(3)*sqrt(2)*x^3 + 19*I*sqrt(3)*sq 
rt(2)*x^2 - 32*I*sqrt(3)*sqrt(2)*x - 96*I*sqrt(3)*sqrt(2))*f^2*g*sqrt(2*x 
- 3) + 1/42*(14*I*sqrt(3)*sqrt(2)*x^4 + 9*I*sqrt(3)*sqrt(2)*x^3 - 9*I*sqrt 
(3)*sqrt(2)*x^2 - 18*I*sqrt(3)*sqrt(2)*x - 54*I*sqrt(3)*sqrt(2))*f*g^2*sqr 
t(2*x - 3) + 1/20790*(1890*I*sqrt(3)*sqrt(2)*x^5 + 1225*I*sqrt(3)*sqrt(2)* 
x^4 - 870*I*sqrt(3)*sqrt(2)*x^3 - 1566*I*sqrt(3)*sqrt(2)*x^2 - 3132*I*sqrt 
(3)*sqrt(2)*x - 9396*I*sqrt(3)*sqrt(2))*g^3*sqrt(2*x - 3)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 585 vs. \(2 (112) = 224\).

Time = 0.50 (sec) , antiderivative size = 585, normalized size of antiderivative = 3.95 \[ \int \sqrt {2+3 x} (f+g x)^3 \sqrt {1+\frac {5 x}{6}-x^2} \, dx =\text {Too large to display} \] Input:

integrate(1/6*(2+3*x)^(1/2)*(g*x+f)^3*(-36*x^2+30*x+36)^(1/2),x, algorithm 
="giac")
 

Output:

1/332640*sqrt(6)*(16632*((2*x - 3)^2*sqrt(-2*x + 3) - 10*(-2*x + 3)^(3/2) 
+ 45*sqrt(-2*x + 3))*f^3 + 46200*((-2*x + 3)^(3/2) - 9*sqrt(-2*x + 3))*f^3 
 + 3564*(5*(2*x - 3)^3*sqrt(-2*x + 3) + 63*(2*x - 3)^2*sqrt(-2*x + 3) - 31 
5*(-2*x + 3)^(3/2) + 945*sqrt(-2*x + 3))*f^2*g - 41580*((2*x - 3)^2*sqrt(- 
2*x + 3) - 10*(-2*x + 3)^(3/2) + 45*sqrt(-2*x + 3))*f^2*g + 166320*((-2*x 
+ 3)^(3/2) - 9*sqrt(-2*x + 3))*f^2*g + 198*(35*(2*x - 3)^4*sqrt(-2*x + 3) 
+ 540*(2*x - 3)^3*sqrt(-2*x + 3) + 3402*(2*x - 3)^2*sqrt(-2*x + 3) - 11340 
*(-2*x + 3)^(3/2) + 25515*sqrt(-2*x + 3))*f*g^2 - 2970*(5*(2*x - 3)^3*sqrt 
(-2*x + 3) + 63*(2*x - 3)^2*sqrt(-2*x + 3) - 315*(-2*x + 3)^(3/2) + 945*sq 
rt(-2*x + 3))*f*g^2 - 49896*((2*x - 3)^2*sqrt(-2*x + 3) - 10*(-2*x + 3)^(3 
/2) + 45*sqrt(-2*x + 3))*f*g^2 + 45*(21*(2*x - 3)^5*sqrt(-2*x + 3) + 385*( 
2*x - 3)^4*sqrt(-2*x + 3) + 2970*(2*x - 3)^3*sqrt(-2*x + 3) + 12474*(2*x - 
 3)^2*sqrt(-2*x + 3) - 31185*(-2*x + 3)^(3/2) + 56133*sqrt(-2*x + 3))*g^3 
- 55*(35*(2*x - 3)^4*sqrt(-2*x + 3) + 540*(2*x - 3)^3*sqrt(-2*x + 3) + 340 
2*(2*x - 3)^2*sqrt(-2*x + 3) - 11340*(-2*x + 3)^(3/2) + 25515*sqrt(-2*x + 
3))*g^3 - 1188*(5*(2*x - 3)^3*sqrt(-2*x + 3) + 63*(2*x - 3)^2*sqrt(-2*x + 
3) - 315*(-2*x + 3)^(3/2) + 945*sqrt(-2*x + 3))*g^3 - 332640*f^3*sqrt(-2*x 
 + 3))
 

Mupad [B] (verification not implemented)

Time = 0.44 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.25 \[ \int \sqrt {2+3 x} (f+g x)^3 \sqrt {1+\frac {5 x}{6}-x^2} \, dx=-\frac {\sqrt {6}\,\sqrt {-6\,x^2+5\,x+6}\,\left (\sqrt {3\,x+2}\,\left (\frac {19\,f^3}{90}+\frac {16\,f^2\,g}{35}+\frac {3\,f\,g^2}{7}+\frac {58\,g^3}{385}\right )+x\,\sqrt {3\,x+2}\,\left (-\frac {11\,f^3}{270}+\frac {16\,f^2\,g}{105}+\frac {f\,g^2}{7}+\frac {58\,g^3}{1155}\right )+x^2\,\sqrt {3\,x+2}\,\left (-\frac {f^3}{15}-\frac {19\,f^2\,g}{210}+\frac {f\,g^2}{14}+\frac {29\,g^3}{1155}\right )-\frac {g^3\,x^5\,\sqrt {3\,x+2}}{33}-\frac {g\,x^3\,\sqrt {3\,x+2}\,\left (594\,f^2+297\,f\,g-58\,g^2\right )}{4158}-g^2\,x^4\,\sqrt {3\,x+2}\,\left (\frac {f}{9}+\frac {35\,g}{1782}\right )\right )}{x+\frac {2}{3}} \] Input:

int(((f + g*x)^3*(3*x + 2)^(1/2)*(30*x - 36*x^2 + 36)^(1/2))/6,x)
 

Output:

-(6^(1/2)*(5*x - 6*x^2 + 6)^(1/2)*((3*x + 2)^(1/2)*((3*f*g^2)/7 + (16*f^2* 
g)/35 + (19*f^3)/90 + (58*g^3)/385) + x*(3*x + 2)^(1/2)*((f*g^2)/7 + (16*f 
^2*g)/105 - (11*f^3)/270 + (58*g^3)/1155) + x^2*(3*x + 2)^(1/2)*((f*g^2)/1 
4 - (19*f^2*g)/210 - f^3/15 + (29*g^3)/1155) - (g^3*x^5*(3*x + 2)^(1/2))/3 
3 - (g*x^3*(3*x + 2)^(1/2)*(297*f*g + 594*f^2 - 58*g^2))/4158 - g^2*x^4*(3 
*x + 2)^(1/2)*(f/9 + (35*g)/1782)))/(x + 2/3)
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.97 \[ \int \sqrt {2+3 x} (f+g x)^3 \sqrt {1+\frac {5 x}{6}-x^2} \, dx=\frac {\sqrt {-2 x +3}\, \sqrt {6}\, \left (1890 g^{3} x^{5}+6930 f \,g^{2} x^{4}+1225 g^{3} x^{4}+8910 f^{2} g \,x^{3}+4455 f \,g^{2} x^{3}-870 g^{3} x^{3}+4158 f^{3} x^{2}+5643 f^{2} g \,x^{2}-4455 f \,g^{2} x^{2}-1566 g^{3} x^{2}+2541 f^{3} x -9504 f^{2} g x -8910 f \,g^{2} x -3132 g^{3} x -13167 f^{3}-28512 f^{2} g -26730 f \,g^{2}-9396 g^{3}\right )}{20790} \] Input:

int(1/6*(2+3*x)^(1/2)*(g*x+f)^3*(-36*x^2+30*x+36)^(1/2),x)
 

Output:

(sqrt( - 2*x + 3)*sqrt(6)*(4158*f**3*x**2 + 2541*f**3*x - 13167*f**3 + 891 
0*f**2*g*x**3 + 5643*f**2*g*x**2 - 9504*f**2*g*x - 28512*f**2*g + 6930*f*g 
**2*x**4 + 4455*f*g**2*x**3 - 4455*f*g**2*x**2 - 8910*f*g**2*x - 26730*f*g 
**2 + 1890*g**3*x**5 + 1225*g**3*x**4 - 870*g**3*x**3 - 1566*g**3*x**2 - 3 
132*g**3*x - 9396*g**3))/20790