\(\int \frac {\sqrt {2+3 x} \sqrt {1+\frac {5 x}{6}-x^2}}{(f+g x)^{3/2}} \, dx\) [305]

Optimal result

Integrand size = 35, antiderivative size = 115 \[ \int \frac {\sqrt {2+3 x} \sqrt {1+\frac {5 x}{6}-x^2}}{(f+g x)^{3/2}} \, dx=\frac {\sqrt {\frac {2}{3}} (3 f-2 g) \sqrt {3-2 x}}{g^2 \sqrt {f+g x}}+\frac {\sqrt {\frac {3}{2}} \sqrt {3-2 x} \sqrt {f+g x}}{g^2}-\frac {(18 f+g) \arctan \left (\frac {\sqrt {g} \sqrt {3-2 x}}{\sqrt {2} \sqrt {f+g x}}\right )}{2 \sqrt {3} g^{5/2}} \] Output:

1/3*6^(1/2)*(3*f-2*g)*(3-2*x)^(1/2)/g^2/(g*x+f)^(1/2)+1/2*6^(1/2)*(3-2*x)^ 
(1/2)*(g*x+f)^(1/2)/g^2-1/6*(18*f+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 = 10.32 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.49 \[ \int \frac {\sqrt {2+3 x} \sqrt {1+\frac {5 x}{6}-x^2}}{(f+g x)^{3/2}} \, dx=\frac {\sqrt {6+5 x-6 x^2} \left (2 \sqrt {-2 f-3 g} \sqrt {g} (-3+2 x) (9 f+g (-4+3 x))-\left (36 f^2+56 f g+3 g^2\right ) \sqrt {6-4 x} \sqrt {\frac {f+g x}{2 f+3 g}} \text {arcsinh}\left (\frac {\sqrt {g} \sqrt {3-2 x}}{\sqrt {-2 f-3 g}}\right )\right )}{2 \sqrt {6} \sqrt {-2 f-3 g} g^{5/2} (-3+2 x) \sqrt {2+3 x} \sqrt {f+g x}} \] Input:

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

Output:

(Sqrt[6 + 5*x - 6*x^2]*(2*Sqrt[-2*f - 3*g]*Sqrt[g]*(-3 + 2*x)*(9*f + g*(-4 
 + 3*x)) - (36*f^2 + 56*f*g + 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]]))/(2*Sqrt[6]*Sqrt[- 
2*f - 3*g]*g^(5/2)*(-3 + 2*x)*Sqrt[2 + 3*x]*Sqrt[f + g*x])
 

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.23, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {1245, 87, 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.

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

\(\Big \downarrow \) 1245

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

\(\Big \downarrow \) 87

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 60

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

\(\Big \downarrow \) 66

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

\(\Big \downarrow \) 217

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

Input:

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

Output:

(Sqrt[2/3]*(3*f - 2*g)*(3 - 2*x)^(3/2))/(g*(2*f + 3*g)*Sqrt[f + g*x]) + (( 
18*f + 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))))/(Sqrt[6]*g*(2 
*f + 3*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 87

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

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. \(271\) vs. \(2(88)=176\).

Time = 1.38 (sec) , antiderivative size = 272, normalized size of antiderivative = 2.37

method	result	size
default	\(\frac {\sqrt {-6 x^{2}+5 x +6}\, \sqrt {3}\, \left (6 \sqrt {2}\, g^{\frac {3}{2}} x \sqrt {-\left (2 x -3\right ) \left (g x +f \right )}+18 \sqrt {2}\, f \sqrt {-\left (2 x -3\right ) \left (g x +f \right )}\, \sqrt {g}-8 \sqrt {2}\, g^{\frac {3}{2}} \sqrt {-\left (2 x -3\right ) \left (g x +f \right )}+18 \arctan \left (\frac {\left (4 g x +2 f -3 g \right ) \sqrt {2}}{4 \sqrt {g}\, \sqrt {-\left (2 x -3\right ) \left (g x +f \right )}}\right ) f g x +\arctan \left (\frac {\left (4 g x +2 f -3 g \right ) \sqrt {2}}{4 \sqrt {g}\, \sqrt {-\left (2 x -3\right ) \left (g x +f \right )}}\right ) g^{2} x +18 \arctan \left (\frac {\left (4 g x +2 f -3 g \right ) \sqrt {2}}{4 \sqrt {g}\, \sqrt {-\left (2 x -3\right ) \left (g x +f \right )}}\right ) f^{2}+\arctan \left (\frac {\left (4 g x +2 f -3 g \right ) \sqrt {2}}{4 \sqrt {g}\, \sqrt {-\left (2 x -3\right ) \left (g x +f \right )}}\right ) f g \right )}{12 \sqrt {-\left (2 x -3\right ) \left (g x +f \right )}\, g^{\frac {5}{2}} \sqrt {g x +f}\, \sqrt {3 x +2}}\)	\(272\)

Input:

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

Output:

1/12*(-6*x^2+5*x+6)^(1/2)*3^(1/2)*(6*2^(1/2)*g^(3/2)*x*(-(2*x-3)*(g*x+f))^ 
(1/2)+18*2^(1/2)*f*(-(2*x-3)*(g*x+f))^(1/2)*g^(1/2)-8*2^(1/2)*g^(3/2)*(-(2 
*x-3)*(g*x+f))^(1/2)+18*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*x+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*x+18*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+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)/(-(2*x-3)*(g*x+f))^(1/2)/g^(5/2)/(g*x+f)^(1/2) 
/(3*x+2)^(1/2)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 213 vs. \(2 (88) = 176\).

Time = 0.11 (sec) , antiderivative size = 454, normalized size of antiderivative = 3.95 \[ \int \frac {\sqrt {2+3 x} \sqrt {1+\frac {5 x}{6}-x^2}}{(f+g x)^{3/2}} \, dx=\left [-\frac {\sqrt {3} {\left (3 \, {\left (18 \, f g + g^{2}\right )} x^{2} + 36 \, f^{2} + 2 \, f g + {\left (54 \, f^{2} + 39 \, f g + 2 \, g^{2}\right )} x\right )} \sqrt {-g} \log \left (-\frac {288 \, g^{2} x^{3} - 4 \, \sqrt {3} {\left (4 \, g x + 2 \, f - 3 \, g\right )} \sqrt {g x + f} \sqrt {-36 \, x^{2} + 30 \, x + 36} \sqrt {-g} \sqrt {3 \, x + 2} + 48 \, {\left (6 \, f g - 5 \, g^{2}\right )} x^{2} + 24 \, f^{2} - 216 \, f g + 54 \, g^{2} + 3 \, {\left (12 \, f^{2} - 44 \, f g - 69 \, g^{2}\right )} x}{3 \, x + 2}\right ) - 4 \, {\left (3 \, g^{2} x + 9 \, f g - 4 \, g^{2}\right )} \sqrt {g x + f} \sqrt {-36 \, x^{2} + 30 \, x + 36} \sqrt {3 \, x + 2}}{24 \, {\left (3 \, g^{4} x^{2} + 2 \, f g^{3} + {\left (3 \, f g^{3} + 2 \, g^{4}\right )} x\right )}}, -\frac {\sqrt {3} {\left (3 \, {\left (18 \, f g + g^{2}\right )} x^{2} + 36 \, f^{2} + 2 \, f g + {\left (54 \, f^{2} + 39 \, f g + 2 \, g^{2}\right )} x\right )} \sqrt {g} \arctan \left (\frac {\sqrt {3} {\left (4 \, g x + 2 \, f - 3 \, g\right )} \sqrt {g x + f} \sqrt {-36 \, x^{2} + 30 \, x + 36} \sqrt {g} \sqrt {3 \, x + 2}}{12 \, {\left (6 \, g^{2} x^{3} + {\left (6 \, f g - 5 \, g^{2}\right )} x^{2} - 6 \, f g - {\left (5 \, f g + 6 \, g^{2}\right )} x\right )}}\right ) - 2 \, {\left (3 \, g^{2} x + 9 \, f g - 4 \, g^{2}\right )} \sqrt {g x + f} \sqrt {-36 \, x^{2} + 30 \, x + 36} \sqrt {3 \, x + 2}}{12 \, {\left (3 \, g^{4} x^{2} + 2 \, f g^{3} + {\left (3 \, f g^{3} + 2 \, g^{4}\right )} x\right )}}\right ] \] Input:

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

Output:

[-1/24*(sqrt(3)*(3*(18*f*g + g^2)*x^2 + 36*f^2 + 2*f*g + (54*f^2 + 39*f*g 
+ 2*g^2)*x)*sqrt(-g)*log(-(288*g^2*x^3 - 4*sqrt(3)*(4*g*x + 2*f - 3*g)*sqr 
t(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*(3*g^2*x + 9*f*g - 4*g^2)*sqrt(g*x + f)*sqrt(-36*x^2 + 30*x 
+ 36)*sqrt(3*x + 2))/(3*g^4*x^2 + 2*f*g^3 + (3*f*g^3 + 2*g^4)*x), -1/12*(s 
qrt(3)*(3*(18*f*g + g^2)*x^2 + 36*f^2 + 2*f*g + (54*f^2 + 39*f*g + 2*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*(3*g^2*x + 9*f*g - 4*g^2)*sqrt(g*x + f)*sq 
rt(-36*x^2 + 30*x + 36)*sqrt(3*x + 2))/(3*g^4*x^2 + 2*f*g^3 + (3*f*g^3 + 2 
*g^4)*x)]
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.45 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.02 \[ \int \frac {\sqrt {2+3 x} \sqrt {1+\frac {5 x}{6}-x^2}}{(f+g x)^{3/2}} \, dx=\frac {1}{12} \, \sqrt {6} {\left (\frac {\sqrt {-2 \, x + 3} {\left (\frac {3 \, \sqrt {2} {\left (2 \, x - 3\right )}}{g} + \frac {18 \, \sqrt {2} f g + \sqrt {2} g^{2}}{g^{3}}\right )}}{\sqrt {g {\left (2 \, x - 3\right )} + 2 \, f + 3 \, g}} + \frac {{\left (18 \, \sqrt {2} f + \sqrt {2} g\right )} \log \left ({\left | -\sqrt {-g} \sqrt {-2 \, x + 3} + \sqrt {g {\left (2 \, x - 3\right )} + 2 \, f + 3 \, g} \right |}\right )}{\sqrt {-g} g^{2}}\right )} \] Input:

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

Output:

1/12*sqrt(6)*(sqrt(-2*x + 3)*(3*sqrt(2)*(2*x - 3)/g + (18*sqrt(2)*f*g + sq 
rt(2)*g^2)/g^3)/sqrt(g*(2*x - 3) + 2*f + 3*g) + (18*sqrt(2)*f + sqrt(2)*g) 
*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. \[ \int \frac {\sqrt {2+3 x} \sqrt {1+\frac {5 x}{6}-x^2}}{(f+g x)^{3/2}} \, dx=\int \frac {\sqrt {3\,x+2}\,\sqrt {-36\,x^2+30\,x+36}}{6\,{\left (f+g\,x\right )}^{3/2}} \,d x \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.02 \[ \int \frac {\sqrt {2+3 x} \sqrt {1+\frac {5 x}{6}-x^2}}{(f+g x)^{3/2}} \, dx=\frac {\sqrt {3}\, \left (-18 \sqrt {g}\, \sqrt {g x +f}\, \mathit {asin} \left (\frac {\sqrt {g}\, \sqrt {-2 x +3}}{\sqrt {2 f +3 g}}\right ) f -\sqrt {g}\, \sqrt {g x +f}\, \mathit {asin} \left (\frac {\sqrt {g}\, \sqrt {-2 x +3}}{\sqrt {2 f +3 g}}\right ) g +9 \sqrt {-2 x +3}\, \sqrt {2}\, f g +3 \sqrt {-2 x +3}\, \sqrt {2}\, g^{2} x -4 \sqrt {-2 x +3}\, \sqrt {2}\, g^{2}\right )}{6 \sqrt {g x +f}\, g^{3}} \] Input:

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

Output:

(sqrt(3)*( - 18*sqrt(g)*sqrt(f + g*x)*asin((sqrt(g)*sqrt( - 2*x + 3))/sqrt 
(2*f + 3*g))*f - sqrt(g)*sqrt(f + g*x)*asin((sqrt(g)*sqrt( - 2*x + 3))/sqr 
t(2*f + 3*g))*g + 9*sqrt( - 2*x + 3)*sqrt(2)*f*g + 3*sqrt( - 2*x + 3)*sqrt 
(2)*g**2*x - 4*sqrt( - 2*x + 3)*sqrt(2)*g**2))/(6*sqrt(f + g*x)*g**3)