\(\int \sqrt {2+3 x} (f+g x)^{3/2} \sqrt {1+\frac {5 x}{6}-x^2} \, dx\) [302]

Optimal result

Integrand size = 35, antiderivative size = 220 \[ \int \sqrt {2+3 x} (f+g x)^{3/2} \sqrt {1+\frac {5 x}{6}-x^2} \, dx=-\frac {(18 f-77 g) (2 f+3 g)^2 \sqrt {3-2 x} \sqrt {f+g x}}{512 \sqrt {6} g^2}+\frac {(18 f-77 g) (2 f+3 g) (3-2 x)^{3/2} \sqrt {f+g x}}{256 \sqrt {6} g}+\frac {(18 f-77 g) (3-2 x)^{3/2} (f+g x)^{3/2}}{96 \sqrt {6} g}-\frac {\sqrt {\frac {3}{2}} (3-2 x)^{3/2} (f+g x)^{5/2}}{8 g}+\frac {(18 f-77 g) (2 f+3 g)^3 \arctan \left (\frac {\sqrt {g} \sqrt {3-2 x}}{\sqrt {2} \sqrt {f+g x}}\right )}{1024 \sqrt {3} g^{5/2}} \] Output:

-1/3072*(18*f-77*g)*(2*f+3*g)^2*(3-2*x)^(1/2)*(g*x+f)^(1/2)*6^(1/2)/g^2+1/ 
1536*(18*f-77*g)*(2*f+3*g)*(3-2*x)^(3/2)*(g*x+f)^(1/2)*6^(1/2)/g+1/576*(18 
*f-77*g)*(3-2*x)^(3/2)*(g*x+f)^(3/2)*6^(1/2)/g-1/16*6^(1/2)*(3-2*x)^(3/2)* 
(g*x+f)^(5/2)/g+1/3072*(18*f-77*g)*(2*f+3*g)^3*arctan(1/2*g^(1/2)*(3-2*x)^ 
(1/2)*2^(1/2)/(g*x+f)^(1/2))*3^(1/2)/g^(5/2)
 

Mathematica [A] (verified)

Time = 1.95 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.15 \[ \int \sqrt {2+3 x} (f+g x)^{3/2} \sqrt {1+\frac {5 x}{6}-x^2} \, dx=-\frac {\sqrt {6+5 x-6 x^2} \left (-\sqrt {3} \sqrt {g} \sqrt {-3+2 x}+\sqrt {6 f+6 g x}\right ) \left (27 \sqrt {2} \sqrt {g} \sqrt {-3+2 x} \sqrt {f+g x} \left (-216 f^3+12 f^2 g (5+12 x)+2 f g^2 \left (-1605+536 x+864 x^2\right )+g^3 \left (-2079-924 x+736 x^2+1152 x^3\right )\right )+81 (2 f+3 g)^3 (-18 f+77 g) \log \left (-\sqrt {3} \sqrt {g} \sqrt {-3+2 x}+\sqrt {6} \sqrt {f+g x}\right )\right )}{41472 g^{5/2} \sqrt {-9+6 x} \sqrt {4+6 x} \left (\sqrt {6} \sqrt {g} \sqrt {-3+2 x}-2 \sqrt {3} \sqrt {f+g x}\right )} \] Input:

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

Output:

-1/41472*(Sqrt[6 + 5*x - 6*x^2]*(-(Sqrt[3]*Sqrt[g]*Sqrt[-3 + 2*x]) + Sqrt[ 
6*f + 6*g*x])*(27*Sqrt[2]*Sqrt[g]*Sqrt[-3 + 2*x]*Sqrt[f + g*x]*(-216*f^3 + 
 12*f^2*g*(5 + 12*x) + 2*f*g^2*(-1605 + 536*x + 864*x^2) + g^3*(-2079 - 92 
4*x + 736*x^2 + 1152*x^3)) + 81*(2*f + 3*g)^3*(-18*f + 77*g)*Log[-(Sqrt[3] 
*Sqrt[g]*Sqrt[-3 + 2*x]) + Sqrt[6]*Sqrt[f + g*x]]))/(g^(5/2)*Sqrt[-9 + 6*x 
]*Sqrt[4 + 6*x]*(Sqrt[6]*Sqrt[g]*Sqrt[-3 + 2*x] - 2*Sqrt[3]*Sqrt[f + g*x]) 
)
 

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.87, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.229, Rules used = {1245, 90, 27, 60, 60, 60, 66, 217}

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.

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

\(\Big \downarrow \) 1245

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

\(\Big \downarrow \) 90

\(\displaystyle -\frac {(18 f-77 g) \int \frac {\sqrt {3-2 x} (f+g x)^{3/2}}{\sqrt {6}}dx}{16 g}-\frac {\sqrt {\frac {3}{2}} (3-2 x)^{3/2} (f+g x)^{5/2}}{8 g}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {(18 f-77 g) \int \sqrt {3-2 x} (f+g x)^{3/2}dx}{16 \sqrt {6} g}-\frac {\sqrt {\frac {3}{2}} (3-2 x)^{3/2} (f+g x)^{5/2}}{8 g}\)

\(\Big \downarrow \) 60

\(\displaystyle -\frac {(18 f-77 g) \left (\frac {1}{4} (2 f+3 g) \int \sqrt {3-2 x} \sqrt {f+g x}dx-\frac {1}{6} (3-2 x)^{3/2} (f+g x)^{3/2}\right )}{16 \sqrt {6} g}-\frac {\sqrt {\frac {3}{2}} (3-2 x)^{3/2} (f+g x)^{5/2}}{8 g}\)

\(\Big \downarrow \) 60

\(\displaystyle -\frac {(18 f-77 g) \left (\frac {1}{4} (2 f+3 g) \left (\frac {1}{8} (2 f+3 g) \int \frac {\sqrt {3-2 x}}{\sqrt {f+g x}}dx-\frac {1}{4} (3-2 x)^{3/2} \sqrt {f+g x}\right )-\frac {1}{6} (3-2 x)^{3/2} (f+g x)^{3/2}\right )}{16 \sqrt {6} g}-\frac {\sqrt {\frac {3}{2}} (3-2 x)^{3/2} (f+g x)^{5/2}}{8 g}\)

\(\Big \downarrow \) 60

\(\displaystyle -\frac {(18 f-77 g) \left (\frac {1}{4} (2 f+3 g) \left (\frac {1}{8} (2 f+3 g) \left (\frac {(2 f+3 g) \int \frac {1}{\sqrt {3-2 x} \sqrt {f+g x}}dx}{2 g}+\frac {\sqrt {3-2 x} \sqrt {f+g x}}{g}\right )-\frac {1}{4} (3-2 x)^{3/2} \sqrt {f+g x}\right )-\frac {1}{6} (3-2 x)^{3/2} (f+g x)^{3/2}\right )}{16 \sqrt {6} g}-\frac {\sqrt {\frac {3}{2}} (3-2 x)^{3/2} (f+g x)^{5/2}}{8 g}\)

\(\Big \downarrow \) 66

\(\displaystyle -\frac {(18 f-77 g) \left (\frac {1}{4} (2 f+3 g) \left (\frac {1}{8} (2 f+3 g) \left (\frac {(2 f+3 g) \int \frac {1}{-\frac {g (3-2 x)}{f+g x}-2}d\frac {\sqrt {3-2 x}}{\sqrt {f+g x}}}{g}+\frac {\sqrt {3-2 x} \sqrt {f+g x}}{g}\right )-\frac {1}{4} (3-2 x)^{3/2} \sqrt {f+g x}\right )-\frac {1}{6} (3-2 x)^{3/2} (f+g x)^{3/2}\right )}{16 \sqrt {6} g}-\frac {\sqrt {\frac {3}{2}} (3-2 x)^{3/2} (f+g x)^{5/2}}{8 g}\)

\(\Big \downarrow \) 217

\(\displaystyle -\frac {(18 f-77 g) \left (\frac {1}{4} (2 f+3 g) \left (\frac {1}{8} (2 f+3 g) \left (\frac {\sqrt {3-2 x} \sqrt {f+g x}}{g}-\frac {(2 f+3 g) \arctan \left (\frac {\sqrt {g} \sqrt {3-2 x}}{\sqrt {2} \sqrt {f+g x}}\right )}{\sqrt {2} g^{3/2}}\right )-\frac {1}{4} (3-2 x)^{3/2} \sqrt {f+g x}\right )-\frac {1}{6} (3-2 x)^{3/2} (f+g x)^{3/2}\right )}{16 \sqrt {6} g}-\frac {\sqrt {\frac {3}{2}} (3-2 x)^{3/2} (f+g x)^{5/2}}{8 g}\)

Input:

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

Output:

-1/8*(Sqrt[3/2]*(3 - 2*x)^(3/2)*(f + g*x)^(5/2))/g - ((18*f - 77*g)*(-1/6* 
((3 - 2*x)^(3/2)*(f + g*x)^(3/2)) + ((2*f + 3*g)*(-1/4*((3 - 2*x)^(3/2)*Sq 
rt[f + g*x]) + ((2*f + 3*g)*((Sqrt[3 - 2*x]*Sqrt[f + g*x])/g - ((2*f + 3*g 
)*ArcTan[(Sqrt[g]*Sqrt[3 - 2*x])/(Sqrt[2]*Sqrt[f + g*x])])/(Sqrt[2]*g^(3/2 
))))/8))/4))/(16*Sqrt[6]*g)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 60

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( 
b*(m + n + 1)))   Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, 
 c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !Integer 
Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinear 
Q[a, b, c, d, m, n, x]
 

rule 66

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ 
2   Subst[Int[1/(b - d*x^2), x], x, Sqrt[a + b*x]/Sqrt[c + d*x]], x] /; Fre 
eQ[{a, b, c, d}, x] &&  !GtQ[c - a*(d/b), 0]
 

rule 90

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), 
 x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p 
+ 2))   Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, 
p}, x] && NeQ[n + p + 2, 0]
 

rule 217

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( 
-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & 
& (LtQ[a, 0] || LtQ[b, 0])
 

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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(493\) vs. \(2(173)=346\).

Time = 1.36 (sec) , antiderivative size = 494, normalized size of antiderivative = 2.25

method	result	size
default	\(-\frac {\sqrt {g x +f}\, \sqrt {-6 x^{2}+5 x +6}\, \sqrt {3}\, \left (-2304 \sqrt {2}\, g^{\frac {7}{2}} x^{3} \sqrt {-\left (2 x -3\right ) \left (g x +f \right )}-3456 \sqrt {2}\, f \,g^{\frac {5}{2}} x^{2} \sqrt {-\left (2 x -3\right ) \left (g x +f \right )}-1472 \sqrt {2}\, g^{\frac {7}{2}} x^{2} \sqrt {-\left (2 x -3\right ) \left (g x +f \right )}-288 g^{\frac {3}{2}} \sqrt {2}\, \sqrt {-\left (2 x -3\right ) \left (g x +f \right )}\, f^{2} x -2144 g^{\frac {5}{2}} \sqrt {2}\, \sqrt {-\left (2 x -3\right ) \left (g x +f \right )}\, f x +1848 g^{\frac {7}{2}} \sqrt {2}\, \sqrt {-\left (2 x -3\right ) \left (g x +f \right )}\, x +432 \sqrt {g}\, \sqrt {2}\, \sqrt {-\left (2 x -3\right ) \left (g x +f \right )}\, f^{3}-120 g^{\frac {3}{2}} \sqrt {2}\, \sqrt {-\left (2 x -3\right ) \left (g x +f \right )}\, f^{2}+6420 g^{\frac {5}{2}} \sqrt {2}\, \sqrt {-\left (2 x -3\right ) \left (g x +f \right )}\, f +4158 g^{\frac {7}{2}} \sqrt {2}\, \sqrt {-\left (2 x -3\right ) \left (g x +f \right )}+432 \arctan \left (\frac {\left (4 g x +2 f -3 g \right ) \sqrt {2}}{4 \sqrt {g}\, \sqrt {-\left (2 x -3\right ) \left (g x +f \right )}}\right ) f^{4}+96 \arctan \left (\frac {\left (4 g x +2 f -3 g \right ) \sqrt {2}}{4 \sqrt {g}\, \sqrt {-\left (2 x -3\right ) \left (g x +f \right )}}\right ) f^{3} g -5400 \arctan \left (\frac {\left (4 g x +2 f -3 g \right ) \sqrt {2}}{4 \sqrt {g}\, \sqrt {-\left (2 x -3\right ) \left (g x +f \right )}}\right ) f^{2} g^{2}-11016 \arctan \left (\frac {\left (4 g x +2 f -3 g \right ) \sqrt {2}}{4 \sqrt {g}\, \sqrt {-\left (2 x -3\right ) \left (g x +f \right )}}\right ) f \,g^{3}-6237 \arctan \left (\frac {\left (4 g x +2 f -3 g \right ) \sqrt {2}}{4 \sqrt {g}\, \sqrt {-\left (2 x -3\right ) \left (g x +f \right )}}\right ) g^{4}\right )}{18432 g^{\frac {5}{2}} \sqrt {3 x +2}\, \sqrt {-\left (2 x -3\right ) \left (g x +f \right )}}\)	\(494\)

Input:

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

Output:

-1/18432*(g*x+f)^(1/2)*(-6*x^2+5*x+6)^(1/2)*3^(1/2)/g^(5/2)*(-2304*2^(1/2) 
*g^(7/2)*x^3*(-(2*x-3)*(g*x+f))^(1/2)-3456*2^(1/2)*f*g^(5/2)*x^2*(-(2*x-3) 
*(g*x+f))^(1/2)-1472*2^(1/2)*g^(7/2)*x^2*(-(2*x-3)*(g*x+f))^(1/2)-288*g^(3 
/2)*2^(1/2)*(-(2*x-3)*(g*x+f))^(1/2)*f^2*x-2144*g^(5/2)*2^(1/2)*(-(2*x-3)* 
(g*x+f))^(1/2)*f*x+1848*g^(7/2)*2^(1/2)*(-(2*x-3)*(g*x+f))^(1/2)*x+432*g^( 
1/2)*2^(1/2)*(-(2*x-3)*(g*x+f))^(1/2)*f^3-120*g^(3/2)*2^(1/2)*(-(2*x-3)*(g 
*x+f))^(1/2)*f^2+6420*g^(5/2)*2^(1/2)*(-(2*x-3)*(g*x+f))^(1/2)*f+4158*g^(7 
/2)*2^(1/2)*(-(2*x-3)*(g*x+f))^(1/2)+432*arctan(1/4/g^(1/2)*(4*g*x+2*f-3*g 
)*2^(1/2)/(-(2*x-3)*(g*x+f))^(1/2))*f^4+96*arctan(1/4/g^(1/2)*(4*g*x+2*f-3 
*g)*2^(1/2)/(-(2*x-3)*(g*x+f))^(1/2))*f^3*g-5400*arctan(1/4/g^(1/2)*(4*g*x 
+2*f-3*g)*2^(1/2)/(-(2*x-3)*(g*x+f))^(1/2))*f^2*g^2-11016*arctan(1/4/g^(1/ 
2)*(4*g*x+2*f-3*g)*2^(1/2)/(-(2*x-3)*(g*x+f))^(1/2))*f*g^3-6237*arctan(1/4 
/g^(1/2)*(4*g*x+2*f-3*g)*2^(1/2)/(-(2*x-3)*(g*x+f))^(1/2))*g^4)/(3*x+2)^(1 
/2)/(-(2*x-3)*(g*x+f))^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 588, normalized size of antiderivative = 2.67 \[ \int \sqrt {2+3 x} (f+g x)^{3/2} \sqrt {1+\frac {5 x}{6}-x^2} \, dx=\left [\frac {3 \, \sqrt {3} {\left (288 \, f^{4} + 64 \, f^{3} g - 3600 \, f^{2} g^{2} - 7344 \, f g^{3} - 4158 \, g^{4} + 3 \, {\left (144 \, f^{4} + 32 \, f^{3} g - 1800 \, f^{2} g^{2} - 3672 \, f g^{3} - 2079 \, g^{4}\right )} x\right )} \sqrt {-g} \log \left (-\frac {288 \, g^{2} x^{3} - 4 \, \sqrt {3} {\left (4 \, g x + 2 \, f - 3 \, g\right )} \sqrt {g x + f} \sqrt {-36 \, x^{2} + 30 \, x + 36} \sqrt {-g} \sqrt {3 \, x + 2} + 48 \, {\left (6 \, f g - 5 \, g^{2}\right )} x^{2} + 24 \, f^{2} - 216 \, f g + 54 \, g^{2} + 3 \, {\left (12 \, f^{2} - 44 \, f g - 69 \, g^{2}\right )} x}{3 \, x + 2}\right ) + 4 \, {\left (1152 \, g^{4} x^{3} - 216 \, f^{3} g + 60 \, f^{2} g^{2} - 3210 \, f g^{3} - 2079 \, g^{4} + 32 \, {\left (54 \, f g^{3} + 23 \, g^{4}\right )} x^{2} + 4 \, {\left (36 \, f^{2} g^{2} + 268 \, f g^{3} - 231 \, g^{4}\right )} x\right )} \sqrt {g x + f} \sqrt {-36 \, x^{2} + 30 \, x + 36} \sqrt {3 \, x + 2}}{36864 \, {\left (3 \, g^{3} x + 2 \, g^{3}\right )}}, \frac {3 \, \sqrt {3} {\left (288 \, f^{4} + 64 \, f^{3} g - 3600 \, f^{2} g^{2} - 7344 \, f g^{3} - 4158 \, g^{4} + 3 \, {\left (144 \, f^{4} + 32 \, f^{3} g - 1800 \, f^{2} g^{2} - 3672 \, f g^{3} - 2079 \, g^{4}\right )} x\right )} \sqrt {g} \arctan \left (\frac {\sqrt {3} {\left (4 \, g x + 2 \, f - 3 \, g\right )} \sqrt {g x + f} \sqrt {-36 \, x^{2} + 30 \, x + 36} \sqrt {g} \sqrt {3 \, x + 2}}{12 \, {\left (6 \, g^{2} x^{3} + {\left (6 \, f g - 5 \, g^{2}\right )} x^{2} - 6 \, f g - {\left (5 \, f g + 6 \, g^{2}\right )} x\right )}}\right ) + 2 \, {\left (1152 \, g^{4} x^{3} - 216 \, f^{3} g + 60 \, f^{2} g^{2} - 3210 \, f g^{3} - 2079 \, g^{4} + 32 \, {\left (54 \, f g^{3} + 23 \, g^{4}\right )} x^{2} + 4 \, {\left (36 \, f^{2} g^{2} + 268 \, f g^{3} - 231 \, g^{4}\right )} x\right )} \sqrt {g x + f} \sqrt {-36 \, x^{2} + 30 \, x + 36} \sqrt {3 \, x + 2}}{18432 \, {\left (3 \, g^{3} x + 2 \, g^{3}\right )}}\right ] \] Input:

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

Output:

[1/36864*(3*sqrt(3)*(288*f^4 + 64*f^3*g - 3600*f^2*g^2 - 7344*f*g^3 - 4158 
*g^4 + 3*(144*f^4 + 32*f^3*g - 1800*f^2*g^2 - 3672*f*g^3 - 2079*g^4)*x)*sq 
rt(-g)*log(-(288*g^2*x^3 - 4*sqrt(3)*(4*g*x + 2*f - 3*g)*sqrt(g*x + f)*sqr 
t(-36*x^2 + 30*x + 36)*sqrt(-g)*sqrt(3*x + 2) + 48*(6*f*g - 5*g^2)*x^2 + 2 
4*f^2 - 216*f*g + 54*g^2 + 3*(12*f^2 - 44*f*g - 69*g^2)*x)/(3*x + 2)) + 4* 
(1152*g^4*x^3 - 216*f^3*g + 60*f^2*g^2 - 3210*f*g^3 - 2079*g^4 + 32*(54*f* 
g^3 + 23*g^4)*x^2 + 4*(36*f^2*g^2 + 268*f*g^3 - 231*g^4)*x)*sqrt(g*x + f)* 
sqrt(-36*x^2 + 30*x + 36)*sqrt(3*x + 2))/(3*g^3*x + 2*g^3), 1/18432*(3*sqr 
t(3)*(288*f^4 + 64*f^3*g - 3600*f^2*g^2 - 7344*f*g^3 - 4158*g^4 + 3*(144*f 
^4 + 32*f^3*g - 1800*f^2*g^2 - 3672*f*g^3 - 2079*g^4)*x)*sqrt(g)*arctan(1/ 
12*sqrt(3)*(4*g*x + 2*f - 3*g)*sqrt(g*x + f)*sqrt(-36*x^2 + 30*x + 36)*sqr 
t(g)*sqrt(3*x + 2)/(6*g^2*x^3 + (6*f*g - 5*g^2)*x^2 - 6*f*g - (5*f*g + 6*g 
^2)*x)) + 2*(1152*g^4*x^3 - 216*f^3*g + 60*f^2*g^2 - 3210*f*g^3 - 2079*g^4 
 + 32*(54*f*g^3 + 23*g^4)*x^2 + 4*(36*f^2*g^2 + 268*f*g^3 - 231*g^4)*x)*sq 
rt(g*x + f)*sqrt(-36*x^2 + 30*x + 36)*sqrt(3*x + 2))/(3*g^3*x + 2*g^3)]
 

Sympy [F(-1)]

Timed out. \[ \int \sqrt {2+3 x} (f+g x)^{3/2} \sqrt {1+\frac {5 x}{6}-x^2} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Maxima [F]

\[ \int \sqrt {2+3 x} (f+g x)^{3/2} \sqrt {1+\frac {5 x}{6}-x^2} \, dx=\int { \frac {1}{6} \, {\left (g x + f\right )}^{\frac {3}{2}} \sqrt {-36 \, x^{2} + 30 \, x + 36} \sqrt {3 \, x + 2} \,d x } \] Input:

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

Output:

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

Giac [F(-1)]

Timed out. \[ \int \sqrt {2+3 x} (f+g x)^{3/2} \sqrt {1+\frac {5 x}{6}-x^2} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Mupad [F(-1)]

Timed out. \[ \int \sqrt {2+3 x} (f+g x)^{3/2} \sqrt {1+\frac {5 x}{6}-x^2} \, dx=\int \frac {{\left (f+g\,x\right )}^{3/2}\,\sqrt {3\,x+2}\,\sqrt {-36\,x^2+30\,x+36}}{6} \,d x \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 492, normalized size of antiderivative = 2.24 \[ \int \sqrt {2+3 x} (f+g x)^{3/2} \sqrt {1+\frac {5 x}{6}-x^2} \, dx=\frac {\sqrt {3}\, \left (864 \sqrt {g}\, \mathit {asin} \left (\frac {\sqrt {g}\, \sqrt {-2 x +3}}{\sqrt {2 f +3 g}}\right ) f^{5}+1488 \sqrt {g}\, \mathit {asin} \left (\frac {\sqrt {g}\, \sqrt {-2 x +3}}{\sqrt {2 f +3 g}}\right ) f^{4} g -10512 \sqrt {g}\, \mathit {asin} \left (\frac {\sqrt {g}\, \sqrt {-2 x +3}}{\sqrt {2 f +3 g}}\right ) f^{3} g^{2}-38232 \sqrt {g}\, \mathit {asin} \left (\frac {\sqrt {g}\, \sqrt {-2 x +3}}{\sqrt {2 f +3 g}}\right ) f^{2} g^{3}-45522 \sqrt {g}\, \mathit {asin} \left (\frac {\sqrt {g}\, \sqrt {-2 x +3}}{\sqrt {2 f +3 g}}\right ) f \,g^{4}-18711 \sqrt {g}\, \mathit {asin} \left (\frac {\sqrt {g}\, \sqrt {-2 x +3}}{\sqrt {2 f +3 g}}\right ) g^{5}-432 \sqrt {g x +f}\, \sqrt {-2 x +3}\, \sqrt {2}\, f^{4} g +288 \sqrt {g x +f}\, \sqrt {-2 x +3}\, \sqrt {2}\, f^{3} g^{2} x -528 \sqrt {g x +f}\, \sqrt {-2 x +3}\, \sqrt {2}\, f^{3} g^{2}+3456 \sqrt {g x +f}\, \sqrt {-2 x +3}\, \sqrt {2}\, f^{2} g^{3} x^{2}+2576 \sqrt {g x +f}\, \sqrt {-2 x +3}\, \sqrt {2}\, f^{2} g^{3} x -6240 \sqrt {g x +f}\, \sqrt {-2 x +3}\, \sqrt {2}\, f^{2} g^{3}+2304 \sqrt {g x +f}\, \sqrt {-2 x +3}\, \sqrt {2}\, f \,g^{4} x^{3}+6656 \sqrt {g x +f}\, \sqrt {-2 x +3}\, \sqrt {2}\, f \,g^{4} x^{2}+1368 \sqrt {g x +f}\, \sqrt {-2 x +3}\, \sqrt {2}\, f \,g^{4} x -13788 \sqrt {g x +f}\, \sqrt {-2 x +3}\, \sqrt {2}\, f \,g^{4}+3456 \sqrt {g x +f}\, \sqrt {-2 x +3}\, \sqrt {2}\, g^{5} x^{3}+2208 \sqrt {g x +f}\, \sqrt {-2 x +3}\, \sqrt {2}\, g^{5} x^{2}-2772 \sqrt {g x +f}\, \sqrt {-2 x +3}\, \sqrt {2}\, g^{5} x -6237 \sqrt {g x +f}\, \sqrt {-2 x +3}\, \sqrt {2}\, g^{5}\right )}{9216 g^{3} \left (2 f +3 g \right )} \] Input:

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

Output:

(sqrt(3)*(864*sqrt(g)*asin((sqrt(g)*sqrt( - 2*x + 3))/sqrt(2*f + 3*g))*f** 
5 + 1488*sqrt(g)*asin((sqrt(g)*sqrt( - 2*x + 3))/sqrt(2*f + 3*g))*f**4*g - 
 10512*sqrt(g)*asin((sqrt(g)*sqrt( - 2*x + 3))/sqrt(2*f + 3*g))*f**3*g**2 
- 38232*sqrt(g)*asin((sqrt(g)*sqrt( - 2*x + 3))/sqrt(2*f + 3*g))*f**2*g**3 
 - 45522*sqrt(g)*asin((sqrt(g)*sqrt( - 2*x + 3))/sqrt(2*f + 3*g))*f*g**4 - 
 18711*sqrt(g)*asin((sqrt(g)*sqrt( - 2*x + 3))/sqrt(2*f + 3*g))*g**5 - 432 
*sqrt(f + g*x)*sqrt( - 2*x + 3)*sqrt(2)*f**4*g + 288*sqrt(f + g*x)*sqrt( - 
 2*x + 3)*sqrt(2)*f**3*g**2*x - 528*sqrt(f + g*x)*sqrt( - 2*x + 3)*sqrt(2) 
*f**3*g**2 + 3456*sqrt(f + g*x)*sqrt( - 2*x + 3)*sqrt(2)*f**2*g**3*x**2 + 
2576*sqrt(f + g*x)*sqrt( - 2*x + 3)*sqrt(2)*f**2*g**3*x - 6240*sqrt(f + g* 
x)*sqrt( - 2*x + 3)*sqrt(2)*f**2*g**3 + 2304*sqrt(f + g*x)*sqrt( - 2*x + 3 
)*sqrt(2)*f*g**4*x**3 + 6656*sqrt(f + g*x)*sqrt( - 2*x + 3)*sqrt(2)*f*g**4 
*x**2 + 1368*sqrt(f + g*x)*sqrt( - 2*x + 3)*sqrt(2)*f*g**4*x - 13788*sqrt( 
f + g*x)*sqrt( - 2*x + 3)*sqrt(2)*f*g**4 + 3456*sqrt(f + g*x)*sqrt( - 2*x 
+ 3)*sqrt(2)*g**5*x**3 + 2208*sqrt(f + g*x)*sqrt( - 2*x + 3)*sqrt(2)*g**5* 
x**2 - 2772*sqrt(f + g*x)*sqrt( - 2*x + 3)*sqrt(2)*g**5*x - 6237*sqrt(f + 
g*x)*sqrt( - 2*x + 3)*sqrt(2)*g**5))/(9216*g**3*(2*f + 3*g))