\(\int \frac {\sqrt {2+3 x} \sqrt {1+\frac {5 x}{6}-x^2}}{\sqrt {f+g x}} \, dx\) [304]

Optimal result

Integrand size = 35, antiderivative size = 128 \[ \int \frac {\sqrt {2+3 x} \sqrt {1+\frac {5 x}{6}-x^2}}{\sqrt {f+g x}} \, dx=-\frac {(18 f-25 g) \sqrt {3-2 x} \sqrt {f+g x}}{8 \sqrt {6} g^2}-\frac {\sqrt {\frac {3}{2}} (3-2 x)^{3/2} \sqrt {f+g x}}{4 g}+\frac {(18 f-25 g) (2 f+3 g) \arctan \left (\frac {\sqrt {g} \sqrt {3-2 x}}{\sqrt {2} \sqrt {f+g x}}\right )}{16 \sqrt {3} g^{5/2}} \] Output:

-1/48*(18*f-25*g)*(3-2*x)^(1/2)*(g*x+f)^(1/2)*6^(1/2)/g^2-1/8*6^(1/2)*(3-2 
*x)^(3/2)*(g*x+f)^(1/2)/g+1/48*(18*f-25*g)*(2*f+3*g)*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 = 5.09 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.45 \[ \int \frac {\sqrt {2+3 x} \sqrt {1+\frac {5 x}{6}-x^2}}{\sqrt {f+g x}} \, dx=\frac {\sqrt {6+5 x-6 x^2} \left (-2 \sqrt {-2 f-3 g} \sqrt {g} (-3+2 x) \left (18 f^2+f g (-7+6 x)-g^2 x (7+12 x)\right )+(18 f-25 g) (2 f+3 g)^2 \sqrt {6-4 x} \sqrt {\frac {f+g x}{2 f+3 g}} \text {arcsinh}\left (\frac {\sqrt {g} \sqrt {3-2 x}}{\sqrt {-2 f-3 g}}\right )\right )}{16 \sqrt {6} \sqrt {-2 f-3 g} g^{5/2} (-3+2 x) \sqrt {2+3 x} \sqrt {f+g x}} \] Input:

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

Output:

(Sqrt[6 + 5*x - 6*x^2]*(-2*Sqrt[-2*f - 3*g]*Sqrt[g]*(-3 + 2*x)*(18*f^2 + f 
*g*(-7 + 6*x) - g^2*x*(7 + 12*x)) + (18*f - 25*g)*(2*f + 3*g)^2*Sqrt[6 - 4 
*x]*Sqrt[(f + g*x)/(2*f + 3*g)]*ArcSinh[(Sqrt[g]*Sqrt[3 - 2*x])/Sqrt[-2*f 
- 3*g]]))/(16*Sqrt[6]*Sqrt[-2*f - 3*g]*g^(5/2)*(-3 + 2*x)*Sqrt[2 + 3*x]*Sq 
rt[f + g*x])
 

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.97, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {1245, 90, 27, 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.

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

\(\Big \downarrow \) 1245

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

\(\Big \downarrow \) 90

\(\displaystyle -\frac {(18 f-25 g) \int \frac {\sqrt {3-2 x}}{\sqrt {6} \sqrt {f+g x}}dx}{8 g}-\frac {\sqrt {\frac {3}{2}} (3-2 x)^{3/2} \sqrt {f+g x}}{4 g}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {(18 f-25 g) \int \frac {\sqrt {3-2 x}}{\sqrt {f+g x}}dx}{8 \sqrt {6} g}-\frac {\sqrt {\frac {3}{2}} (3-2 x)^{3/2} \sqrt {f+g x}}{4 g}\)

\(\Big \downarrow \) 60

\(\displaystyle -\frac {(18 f-25 g) \left (\frac {(2 f+3 g) \int \frac {1}{\sqrt {3-2 x} \sqrt {f+g x}}dx}{2 g}+\frac {\sqrt {3-2 x} \sqrt {f+g x}}{g}\right )}{8 \sqrt {6} g}-\frac {\sqrt {\frac {3}{2}} (3-2 x)^{3/2} \sqrt {f+g x}}{4 g}\)

\(\Big \downarrow \) 66

\(\displaystyle -\frac {(18 f-25 g) \left (\frac {(2 f+3 g) \int \frac {1}{-\frac {g (3-2 x)}{f+g x}-2}d\frac {\sqrt {3-2 x}}{\sqrt {f+g x}}}{g}+\frac {\sqrt {3-2 x} \sqrt {f+g x}}{g}\right )}{8 \sqrt {6} g}-\frac {\sqrt {\frac {3}{2}} (3-2 x)^{3/2} \sqrt {f+g x}}{4 g}\)

\(\Big \downarrow \) 217

\(\displaystyle -\frac {(18 f-25 g) \left (\frac {\sqrt {3-2 x} \sqrt {f+g x}}{g}-\frac {(2 f+3 g) \arctan \left (\frac {\sqrt {g} \sqrt {3-2 x}}{\sqrt {2} \sqrt {f+g x}}\right )}{\sqrt {2} g^{3/2}}\right )}{8 \sqrt {6} g}-\frac {\sqrt {\frac {3}{2}} (3-2 x)^{3/2} \sqrt {f+g x}}{4 g}\)

Input:

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

Output:

-1/4*(Sqrt[3/2]*(3 - 2*x)^(3/2)*Sqrt[f + g*x])/g - ((18*f - 25*g)*((Sqrt[3 
 - 2*x]*Sqrt[f + g*x])/g - ((2*f + 3*g)*ArcTan[(Sqrt[g]*Sqrt[3 - 2*x])/(Sq 
rt[2]*Sqrt[f + g*x])])/(Sqrt[2]*g^(3/2))))/(8*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. \(233\) vs. \(2(97)=194\).

Time = 1.31 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.83

method	result	size
default	\(-\frac {\sqrt {-6 x^{2}+5 x +6}\, \sqrt {g x +f}\, \sqrt {3}\, \left (-24 \sqrt {2}\, g^{\frac {3}{2}} x \sqrt {-\left (2 x -3\right ) \left (g x +f \right )}+36 \sqrt {2}\, f \sqrt {-\left (2 x -3\right ) \left (g x +f \right )}\, \sqrt {g}-14 \sqrt {2}\, g^{\frac {3}{2}} \sqrt {-\left (2 x -3\right ) \left (g x +f \right )}+36 \arctan \left (\frac {\left (4 g x +2 f -3 g \right ) \sqrt {2}}{4 \sqrt {g}\, \sqrt {-\left (2 x -3\right ) \left (g x +f \right )}}\right ) f^{2}+4 \arctan \left (\frac {\left (4 g x +2 f -3 g \right ) \sqrt {2}}{4 \sqrt {g}\, \sqrt {-\left (2 x -3\right ) \left (g x +f \right )}}\right ) f g -75 \arctan \left (\frac {\left (4 g x +2 f -3 g \right ) \sqrt {2}}{4 \sqrt {g}\, \sqrt {-\left (2 x -3\right ) \left (g x +f \right )}}\right ) g^{2}\right )}{96 g^{\frac {5}{2}} \sqrt {3 x +2}\, \sqrt {-\left (2 x -3\right ) \left (g x +f \right )}}\)	\(234\)

Input:

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

Output:

-1/96*(-6*x^2+5*x+6)^(1/2)*(g*x+f)^(1/2)*3^(1/2)/g^(5/2)*(-24*2^(1/2)*g^(3 
/2)*x*(-(2*x-3)*(g*x+f))^(1/2)+36*2^(1/2)*f*(-(2*x-3)*(g*x+f))^(1/2)*g^(1/ 
2)-14*2^(1/2)*g^(3/2)*(-(2*x-3)*(g*x+f))^(1/2)+36*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+4*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-75*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^2)/(3*x+2)^(1/2)/(-(2*x- 
3)*(g*x+f))^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 406, normalized size of antiderivative = 3.17 \[ \int \frac {\sqrt {2+3 x} \sqrt {1+\frac {5 x}{6}-x^2}}{\sqrt {f+g x}} \, dx=\left [\frac {\sqrt {3} {\left (72 \, f^{2} + 8 \, f g - 150 \, g^{2} + 3 \, {\left (36 \, f^{2} + 4 \, f g - 75 \, g^{2}\right )} x\right )} \sqrt {-g} \log \left (-\frac {288 \, g^{2} x^{3} - 4 \, \sqrt {3} {\left (4 \, g x + 2 \, f - 3 \, g\right )} \sqrt {g x + f} \sqrt {-36 \, x^{2} + 30 \, x + 36} \sqrt {-g} \sqrt {3 \, x + 2} + 48 \, {\left (6 \, f g - 5 \, g^{2}\right )} x^{2} + 24 \, f^{2} - 216 \, f g + 54 \, g^{2} + 3 \, {\left (12 \, f^{2} - 44 \, f g - 69 \, g^{2}\right )} x}{3 \, x + 2}\right ) + 4 \, {\left (12 \, g^{2} x - 18 \, f g + 7 \, g^{2}\right )} \sqrt {g x + f} \sqrt {-36 \, x^{2} + 30 \, x + 36} \sqrt {3 \, x + 2}}{192 \, {\left (3 \, g^{3} x + 2 \, g^{3}\right )}}, \frac {\sqrt {3} {\left (72 \, f^{2} + 8 \, f g - 150 \, g^{2} + 3 \, {\left (36 \, f^{2} + 4 \, f g - 75 \, g^{2}\right )} x\right )} \sqrt {g} \arctan \left (\frac {\sqrt {3} {\left (4 \, g x + 2 \, f - 3 \, g\right )} \sqrt {g x + f} \sqrt {-36 \, x^{2} + 30 \, x + 36} \sqrt {g} \sqrt {3 \, x + 2}}{12 \, {\left (6 \, g^{2} x^{3} + {\left (6 \, f g - 5 \, g^{2}\right )} x^{2} - 6 \, f g - {\left (5 \, f g + 6 \, g^{2}\right )} x\right )}}\right ) + 2 \, {\left (12 \, g^{2} x - 18 \, f g + 7 \, g^{2}\right )} \sqrt {g x + f} \sqrt {-36 \, x^{2} + 30 \, x + 36} \sqrt {3 \, x + 2}}{96 \, {\left (3 \, g^{3} x + 2 \, g^{3}\right )}}\right ] \] Input:

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

Output:

[1/192*(sqrt(3)*(72*f^2 + 8*f*g - 150*g^2 + 3*(36*f^2 + 4*f*g - 75*g^2)*x) 
*sqrt(-g)*log(-(288*g^2*x^3 - 4*sqrt(3)*(4*g*x + 2*f - 3*g)*sqrt(g*x + f)* 
sqrt(-36*x^2 + 30*x + 36)*sqrt(-g)*sqrt(3*x + 2) + 48*(6*f*g - 5*g^2)*x^2 
+ 24*f^2 - 216*f*g + 54*g^2 + 3*(12*f^2 - 44*f*g - 69*g^2)*x)/(3*x + 2)) + 
 4*(12*g^2*x - 18*f*g + 7*g^2)*sqrt(g*x + f)*sqrt(-36*x^2 + 30*x + 36)*sqr 
t(3*x + 2))/(3*g^3*x + 2*g^3), 1/96*(sqrt(3)*(72*f^2 + 8*f*g - 150*g^2 + 3 
*(36*f^2 + 4*f*g - 75*g^2)*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)*sqrt(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*(12*g^2*x - 18* 
f*g + 7*g^2)*sqrt(g*x + f)*sqrt(-36*x^2 + 30*x + 36)*sqrt(3*x + 2))/(3*g^3 
*x + 2*g^3)]
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.42 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.91 \[ \int \frac {\sqrt {2+3 x} \sqrt {1+\frac {5 x}{6}-x^2}}{\sqrt {f+g x}} \, dx=\frac {1}{96} \, \sqrt {6} \sqrt {2} {\left (\sqrt {g {\left (2 \, x - 3\right )} + 2 \, f + 3 \, g} \sqrt {-2 \, x + 3} {\left (\frac {6 \, {\left (2 \, x - 3\right )}}{g} - \frac {18 \, f g - 25 \, g^{2}}{g^{3}}\right )} - \frac {{\left (36 \, f^{2} + 4 \, f g - 75 \, g^{2}\right )} \log \left ({\left | -\sqrt {-g} \sqrt {-2 \, x + 3} + \sqrt {g {\left (2 \, x - 3\right )} + 2 \, f + 3 \, g} \right |}\right )}{\sqrt {-g} g^{2}}\right )} \] Input:

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

Output:

1/96*sqrt(6)*sqrt(2)*(sqrt(g*(2*x - 3) + 2*f + 3*g)*sqrt(-2*x + 3)*(6*(2*x 
 - 3)/g - (18*f*g - 25*g^2)/g^3) - (36*f^2 + 4*f*g - 75*g^2)*log(abs(-sqrt 
(-g)*sqrt(-2*x + 3) + sqrt(g*(2*x - 3) + 2*f + 3*g)))/(sqrt(-g)*g^2))
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.77 \[ \int \frac {\sqrt {2+3 x} \sqrt {1+\frac {5 x}{6}-x^2}}{\sqrt {f+g x}} \, dx=\frac {\sqrt {3}\, \left (72 \sqrt {g}\, \mathit {asin} \left (\frac {\sqrt {g}\, \sqrt {-2 x +3}}{\sqrt {2 f +3 g}}\right ) f^{3}+116 \sqrt {g}\, \mathit {asin} \left (\frac {\sqrt {g}\, \sqrt {-2 x +3}}{\sqrt {2 f +3 g}}\right ) f^{2} g -138 \sqrt {g}\, \mathit {asin} \left (\frac {\sqrt {g}\, \sqrt {-2 x +3}}{\sqrt {2 f +3 g}}\right ) f \,g^{2}-225 \sqrt {g}\, \mathit {asin} \left (\frac {\sqrt {g}\, \sqrt {-2 x +3}}{\sqrt {2 f +3 g}}\right ) g^{3}-36 \sqrt {g x +f}\, \sqrt {-2 x +3}\, \sqrt {2}\, f^{2} g +24 \sqrt {g x +f}\, \sqrt {-2 x +3}\, \sqrt {2}\, f \,g^{2} x -40 \sqrt {g x +f}\, \sqrt {-2 x +3}\, \sqrt {2}\, f \,g^{2}+36 \sqrt {g x +f}\, \sqrt {-2 x +3}\, \sqrt {2}\, g^{3} x +21 \sqrt {g x +f}\, \sqrt {-2 x +3}\, \sqrt {2}\, g^{3}\right )}{48 g^{3} \left (2 f +3 g \right )} \] Input:

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

Output:

(sqrt(3)*(72*sqrt(g)*asin((sqrt(g)*sqrt( - 2*x + 3))/sqrt(2*f + 3*g))*f**3 
 + 116*sqrt(g)*asin((sqrt(g)*sqrt( - 2*x + 3))/sqrt(2*f + 3*g))*f**2*g - 1 
38*sqrt(g)*asin((sqrt(g)*sqrt( - 2*x + 3))/sqrt(2*f + 3*g))*f*g**2 - 225*s 
qrt(g)*asin((sqrt(g)*sqrt( - 2*x + 3))/sqrt(2*f + 3*g))*g**3 - 36*sqrt(f + 
 g*x)*sqrt( - 2*x + 3)*sqrt(2)*f**2*g + 24*sqrt(f + g*x)*sqrt( - 2*x + 3)* 
sqrt(2)*f*g**2*x - 40*sqrt(f + g*x)*sqrt( - 2*x + 3)*sqrt(2)*f*g**2 + 36*s 
qrt(f + g*x)*sqrt( - 2*x + 3)*sqrt(2)*g**3*x + 21*sqrt(f + g*x)*sqrt( - 2* 
x + 3)*sqrt(2)*g**3))/(48*g**3*(2*f + 3*g))