\(\int \frac {\sqrt {2+3 x} \sqrt {4-9 x^2}}{(f+g x)^{5/2}} \, dx\) [97]

Optimal result

Integrand size = 30, antiderivative size = 107 \[ \int \frac {\sqrt {2+3 x} \sqrt {4-9 x^2}}{(f+g x)^{5/2}} \, dx=\frac {2 (3 f-2 g) (2-3 x)^{3/2}}{3 g (3 f+2 g) (f+g x)^{3/2}}-\frac {6 \sqrt {2-3 x}}{g^2 \sqrt {f+g x}}+\frac {6 \sqrt {3} \arctan \left (\frac {\sqrt {g} \sqrt {2-3 x}}{\sqrt {3} \sqrt {f+g x}}\right )}{g^{5/2}} \] Output:

2/3*(3*f-2*g)*(2-3*x)^(3/2)/g/(3*f+2*g)/(g*x+f)^(3/2)-6*(2-3*x)^(1/2)/g^2/ 
(g*x+f)^(1/2)+6*3^(1/2)*arctan(1/3*g^(1/2)*(2-3*x)^(1/2)*3^(1/2)/(g*x+f)^( 
1/2))/g^(5/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 1.55 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.45 \[ \int \frac {\sqrt {2+3 x} \sqrt {4-9 x^2}}{(f+g x)^{5/2}} \, dx=\frac {i \sqrt {-2+3 x} \sqrt {2+3 x} \left (-\frac {2 i \sqrt {-2+3 x} \left (27 f^2+4 g^2 (1+3 x)+12 f (g+3 g x)\right )}{3 g^2 (3 f+2 g) (f+g x)^{3/2}}-\frac {12 i \sqrt {3} \text {arctanh}\left (\frac {\sqrt {g} \sqrt {-2+3 x}}{\sqrt {3 f+2 g}-\sqrt {3} \sqrt {f+g x}}\right )}{g^{5/2}}\right )}{\sqrt {4-9 x^2}} \] Input:

Integrate[(Sqrt[2 + 3*x]*Sqrt[4 - 9*x^2])/(f + g*x)^(5/2),x]
 

Output:

(I*Sqrt[-2 + 3*x]*Sqrt[2 + 3*x]*((((-2*I)/3)*Sqrt[-2 + 3*x]*(27*f^2 + 4*g^ 
2*(1 + 3*x) + 12*f*(g + 3*g*x)))/(g^2*(3*f + 2*g)*(f + g*x)^(3/2)) - ((12* 
I)*Sqrt[3]*ArcTanh[(Sqrt[g]*Sqrt[-2 + 3*x])/(Sqrt[3*f + 2*g] - Sqrt[3]*Sqr 
t[f + g*x])])/g^(5/2)))/Sqrt[4 - 9*x^2]
 

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.06, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {639, 87, 57, 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.

\(\displaystyle \int \frac {\sqrt {3 x+2} \sqrt {4-9 x^2}}{(f+g x)^{5/2}} \, dx\)

\(\Big \downarrow \) 639

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

\(\Big \downarrow \) 87

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

\(\Big \downarrow \) 57

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

\(\Big \downarrow \) 66

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

\(\Big \downarrow \) 217

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

Input:

Int[(Sqrt[2 + 3*x]*Sqrt[4 - 9*x^2])/(f + g*x)^(5/2),x]
 

Output:

(2*(3*f - 2*g)*(2 - 3*x)^(3/2))/(3*g*(3*f + 2*g)*(f + g*x)^(3/2)) + (3*((- 
2*Sqrt[2 - 3*x])/(g*Sqrt[f + g*x]) + (2*Sqrt[3]*ArcTan[(Sqrt[g]*Sqrt[2 - 3 
*x])/(Sqrt[3]*Sqrt[f + g*x])])/g^(3/2)))/g
 

Defintions of rubi rules used

rule 57

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + 1))), x] - Simp[d*(n/(b*(m + 1))) 
Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] & 
& GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m 
+ n + 2, 0] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[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 639

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(418\) vs. \(2(86)=172\).

Time = 0.83 (sec) , antiderivative size = 419, normalized size of antiderivative = 3.92

method	result	size
default	\(-\frac {\left (27 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (6 g x +3 f -2 g \right )}{6 \sqrt {g}\, \sqrt {-\left (g x +f \right ) \left (-2+3 x \right )}}\right ) f \,g^{2} x^{2}+18 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (6 g x +3 f -2 g \right )}{6 \sqrt {g}\, \sqrt {-\left (g x +f \right ) \left (-2+3 x \right )}}\right ) g^{3} x^{2}+54 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (6 g x +3 f -2 g \right )}{6 \sqrt {g}\, \sqrt {-\left (g x +f \right ) \left (-2+3 x \right )}}\right ) f^{2} g x +36 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (6 g x +3 f -2 g \right )}{6 \sqrt {g}\, \sqrt {-\left (g x +f \right ) \left (-2+3 x \right )}}\right ) f \,g^{2} x +27 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (6 g x +3 f -2 g \right )}{6 \sqrt {g}\, \sqrt {-\left (g x +f \right ) \left (-2+3 x \right )}}\right ) f^{3}+18 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (6 g x +3 f -2 g \right )}{6 \sqrt {g}\, \sqrt {-\left (g x +f \right ) \left (-2+3 x \right )}}\right ) f^{2} g +72 g^{\frac {3}{2}} \sqrt {-\left (g x +f \right ) \left (-2+3 x \right )}\, f x +24 g^{\frac {5}{2}} \sqrt {-\left (g x +f \right ) \left (-2+3 x \right )}\, x +54 \sqrt {g}\, \sqrt {-\left (g x +f \right ) \left (-2+3 x \right )}\, f^{2}+24 g^{\frac {3}{2}} \sqrt {-\left (g x +f \right ) \left (-2+3 x \right )}\, f +8 g^{\frac {5}{2}} \sqrt {-\left (g x +f \right ) \left (-2+3 x \right )}\right ) \sqrt {-9 x^{2}+4}}{3 \left (3 f +2 g \right ) \sqrt {-\left (g x +f \right ) \left (-2+3 x \right )}\, g^{\frac {5}{2}} \left (g x +f \right )^{\frac {3}{2}} \sqrt {3 x +2}}\)	\(419\)

Input:

int((3*x+2)^(1/2)*(-9*x^2+4)^(1/2)/(g*x+f)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

-1/3*(27*3^(1/2)*arctan(1/6*3^(1/2)/g^(1/2)*(6*g*x+3*f-2*g)/(-(g*x+f)*(-2+ 
3*x))^(1/2))*f*g^2*x^2+18*3^(1/2)*arctan(1/6*3^(1/2)/g^(1/2)*(6*g*x+3*f-2* 
g)/(-(g*x+f)*(-2+3*x))^(1/2))*g^3*x^2+54*3^(1/2)*arctan(1/6*3^(1/2)/g^(1/2 
)*(6*g*x+3*f-2*g)/(-(g*x+f)*(-2+3*x))^(1/2))*f^2*g*x+36*3^(1/2)*arctan(1/6 
*3^(1/2)/g^(1/2)*(6*g*x+3*f-2*g)/(-(g*x+f)*(-2+3*x))^(1/2))*f*g^2*x+27*3^( 
1/2)*arctan(1/6*3^(1/2)/g^(1/2)*(6*g*x+3*f-2*g)/(-(g*x+f)*(-2+3*x))^(1/2)) 
*f^3+18*3^(1/2)*arctan(1/6*3^(1/2)/g^(1/2)*(6*g*x+3*f-2*g)/(-(g*x+f)*(-2+3 
*x))^(1/2))*f^2*g+72*g^(3/2)*(-(g*x+f)*(-2+3*x))^(1/2)*f*x+24*g^(5/2)*(-(g 
*x+f)*(-2+3*x))^(1/2)*x+54*g^(1/2)*(-(g*x+f)*(-2+3*x))^(1/2)*f^2+24*g^(3/2 
)*(-(g*x+f)*(-2+3*x))^(1/2)*f+8*g^(5/2)*(-(g*x+f)*(-2+3*x))^(1/2))*(-9*x^2 
+4)^(1/2)/(3*f+2*g)/(-(g*x+f)*(-2+3*x))^(1/2)/g^(5/2)/(g*x+f)^(3/2)/(3*x+2 
)^(1/2)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 277 vs. \(2 (86) = 172\).

Time = 0.13 (sec) , antiderivative size = 611, normalized size of antiderivative = 5.71 \[ \int \frac {\sqrt {2+3 x} \sqrt {4-9 x^2}}{(f+g x)^{5/2}} \, dx=\left [\frac {9 \, \sqrt {3} {\left (3 \, {\left (3 \, f g^{2} + 2 \, g^{3}\right )} x^{3} + 6 \, f^{3} + 4 \, f^{2} g + 2 \, {\left (9 \, f^{2} g + 9 \, f g^{2} + 2 \, g^{3}\right )} x^{2} + {\left (9 \, f^{3} + 18 \, f^{2} g + 8 \, f g^{2}\right )} x\right )} \sqrt {-\frac {1}{g}} \log \left (-\frac {216 \, g^{2} x^{3} + 216 \, f g x^{2} - 4 \, \sqrt {3} {\left (6 \, g^{2} x + 3 \, f g - 2 \, g^{2}\right )} \sqrt {g x + f} \sqrt {-9 \, x^{2} + 4} \sqrt {3 \, x + 2} \sqrt {-\frac {1}{g}} + 18 \, f^{2} - 72 \, f g + 8 \, g^{2} + 3 \, {\left (9 \, f^{2} + 12 \, f g - 28 \, g^{2}\right )} x}{3 \, x + 2}\right ) - 4 \, {\left (27 \, f^{2} + 12 \, f g + 4 \, g^{2} + 12 \, {\left (3 \, f g + g^{2}\right )} x\right )} \sqrt {g x + f} \sqrt {-9 \, x^{2} + 4} \sqrt {3 \, x + 2}}{6 \, {\left (6 \, f^{3} g^{2} + 4 \, f^{2} g^{3} + 3 \, {\left (3 \, f g^{4} + 2 \, g^{5}\right )} x^{3} + 2 \, {\left (9 \, f^{2} g^{3} + 9 \, f g^{4} + 2 \, g^{5}\right )} x^{2} + {\left (9 \, f^{3} g^{2} + 18 \, f^{2} g^{3} + 8 \, f g^{4}\right )} x\right )}}, -\frac {2 \, {\left (27 \, f^{2} + 12 \, f g + 4 \, g^{2} + 12 \, {\left (3 \, f g + g^{2}\right )} x\right )} \sqrt {g x + f} \sqrt {-9 \, x^{2} + 4} \sqrt {3 \, x + 2} - \frac {9 \, \sqrt {3} {\left (3 \, {\left (3 \, f g^{2} + 2 \, g^{3}\right )} x^{3} + 6 \, f^{3} + 4 \, f^{2} g + 2 \, {\left (9 \, f^{2} g + 9 \, f g^{2} + 2 \, g^{3}\right )} x^{2} + {\left (9 \, f^{3} + 18 \, f^{2} g + 8 \, f g^{2}\right )} x\right )} \arctan \left (\frac {2 \, \sqrt {3} \sqrt {g x + f} \sqrt {-9 \, x^{2} + 4} \sqrt {g} \sqrt {3 \, x + 2}}{18 \, g x^{2} + 3 \, {\left (3 \, f + 2 \, g\right )} x + 6 \, f - 4 \, g}\right )}{\sqrt {g}}}{3 \, {\left (6 \, f^{3} g^{2} + 4 \, f^{2} g^{3} + 3 \, {\left (3 \, f g^{4} + 2 \, g^{5}\right )} x^{3} + 2 \, {\left (9 \, f^{2} g^{3} + 9 \, f g^{4} + 2 \, g^{5}\right )} x^{2} + {\left (9 \, f^{3} g^{2} + 18 \, f^{2} g^{3} + 8 \, f g^{4}\right )} x\right )}}\right ] \] Input:

integrate((2+3*x)^(1/2)*(-9*x^2+4)^(1/2)/(g*x+f)^(5/2),x, algorithm="frica 
s")
 

Output:

[1/6*(9*sqrt(3)*(3*(3*f*g^2 + 2*g^3)*x^3 + 6*f^3 + 4*f^2*g + 2*(9*f^2*g + 
9*f*g^2 + 2*g^3)*x^2 + (9*f^3 + 18*f^2*g + 8*f*g^2)*x)*sqrt(-1/g)*log(-(21 
6*g^2*x^3 + 216*f*g*x^2 - 4*sqrt(3)*(6*g^2*x + 3*f*g - 2*g^2)*sqrt(g*x + f 
)*sqrt(-9*x^2 + 4)*sqrt(3*x + 2)*sqrt(-1/g) + 18*f^2 - 72*f*g + 8*g^2 + 3* 
(9*f^2 + 12*f*g - 28*g^2)*x)/(3*x + 2)) - 4*(27*f^2 + 12*f*g + 4*g^2 + 12* 
(3*f*g + g^2)*x)*sqrt(g*x + f)*sqrt(-9*x^2 + 4)*sqrt(3*x + 2))/(6*f^3*g^2 
+ 4*f^2*g^3 + 3*(3*f*g^4 + 2*g^5)*x^3 + 2*(9*f^2*g^3 + 9*f*g^4 + 2*g^5)*x^ 
2 + (9*f^3*g^2 + 18*f^2*g^3 + 8*f*g^4)*x), -1/3*(2*(27*f^2 + 12*f*g + 4*g^ 
2 + 12*(3*f*g + g^2)*x)*sqrt(g*x + f)*sqrt(-9*x^2 + 4)*sqrt(3*x + 2) - 9*s 
qrt(3)*(3*(3*f*g^2 + 2*g^3)*x^3 + 6*f^3 + 4*f^2*g + 2*(9*f^2*g + 9*f*g^2 + 
 2*g^3)*x^2 + (9*f^3 + 18*f^2*g + 8*f*g^2)*x)*arctan(2*sqrt(3)*sqrt(g*x + 
f)*sqrt(-9*x^2 + 4)*sqrt(g)*sqrt(3*x + 2)/(18*g*x^2 + 3*(3*f + 2*g)*x + 6* 
f - 4*g))/sqrt(g))/(6*f^3*g^2 + 4*f^2*g^3 + 3*(3*f*g^4 + 2*g^5)*x^3 + 2*(9 
*f^2*g^3 + 9*f*g^4 + 2*g^5)*x^2 + (9*f^3*g^2 + 18*f^2*g^3 + 8*f*g^4)*x)]
 

Sympy [F]

\[ \int \frac {\sqrt {2+3 x} \sqrt {4-9 x^2}}{(f+g x)^{5/2}} \, dx=\int \frac {\sqrt {- \left (3 x - 2\right ) \left (3 x + 2\right )} \sqrt {3 x + 2}}{\left (f + g x\right )^{\frac {5}{2}}}\, dx \] Input:

integrate((2+3*x)**(1/2)*(-9*x**2+4)**(1/2)/(g*x+f)**(5/2),x)
 

Output:

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

Maxima [F]

\[ \int \frac {\sqrt {2+3 x} \sqrt {4-9 x^2}}{(f+g x)^{5/2}} \, dx=\int { \frac {\sqrt {-9 \, x^{2} + 4} \sqrt {3 \, x + 2}}{{\left (g x + f\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate((2+3*x)^(1/2)*(-9*x^2+4)^(1/2)/(g*x+f)^(5/2),x, algorithm="maxim 
a")
 

Output:

integrate(sqrt(-9*x^2 + 4)*sqrt(3*x + 2)/(g*x + f)^(5/2), x)
 

Giac [A] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.44 \[ \int \frac {\sqrt {2+3 x} \sqrt {4-9 x^2}}{(f+g x)^{5/2}} \, dx=-\frac {2 \, {\left (\frac {4 \, {\left (3 \, \sqrt {3} f g^{2} + \sqrt {3} g^{3}\right )} {\left (3 \, x - 2\right )}}{3 \, f g^{3} + 2 \, g^{4}} + \frac {3 \, {\left (9 \, \sqrt {3} f^{2} g + 12 \, \sqrt {3} f g^{2} + 4 \, \sqrt {3} g^{3}\right )}}{3 \, f g^{3} + 2 \, g^{4}}\right )} \sqrt {-3 \, x + 2}}{{\left (g {\left (3 \, x - 2\right )} + 3 \, f + 2 \, g\right )}^{\frac {3}{2}}} - \frac {6 \, \sqrt {3} \log \left ({\left | -\sqrt {-g} \sqrt {-3 \, x + 2} + \sqrt {g {\left (3 \, x - 2\right )} + 3 \, f + 2 \, g} \right |}\right )}{\sqrt {-g} g^{2}} \] Input:

integrate((2+3*x)^(1/2)*(-9*x^2+4)^(1/2)/(g*x+f)^(5/2),x, algorithm="giac" 
)
 

Output:

-2*(4*(3*sqrt(3)*f*g^2 + sqrt(3)*g^3)*(3*x - 2)/(3*f*g^3 + 2*g^4) + 3*(9*s 
qrt(3)*f^2*g + 12*sqrt(3)*f*g^2 + 4*sqrt(3)*g^3)/(3*f*g^3 + 2*g^4))*sqrt(- 
3*x + 2)/(g*(3*x - 2) + 3*f + 2*g)^(3/2) - 6*sqrt(3)*log(abs(-sqrt(-g)*sqr 
t(-3*x + 2) + sqrt(g*(3*x - 2) + 3*f + 2*g)))/(sqrt(-g)*g^2)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {2+3 x} \sqrt {4-9 x^2}}{(f+g x)^{5/2}} \, dx=\int \frac {\sqrt {3\,x+2}\,\sqrt {4-9\,x^2}}{{\left (f+g\,x\right )}^{5/2}} \,d x \] Input:

int(((3*x + 2)^(1/2)*(4 - 9*x^2)^(1/2))/(f + g*x)^(5/2),x)
 

Output:

int(((3*x + 2)^(1/2)*(4 - 9*x^2)^(1/2))/(f + g*x)^(5/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 237, normalized size of antiderivative = 2.21 \[ \int \frac {\sqrt {2+3 x} \sqrt {4-9 x^2}}{(f+g x)^{5/2}} \, dx=\frac {18 \sqrt {g}\, \sqrt {g x +f}\, \sqrt {3}\, \mathit {asin} \left (\frac {\sqrt {g}\, \sqrt {-3 x +2}}{\sqrt {3 f +2 g}}\right ) f^{2}+18 \sqrt {g}\, \sqrt {g x +f}\, \sqrt {3}\, \mathit {asin} \left (\frac {\sqrt {g}\, \sqrt {-3 x +2}}{\sqrt {3 f +2 g}}\right ) f g x +12 \sqrt {g}\, \sqrt {g x +f}\, \sqrt {3}\, \mathit {asin} \left (\frac {\sqrt {g}\, \sqrt {-3 x +2}}{\sqrt {3 f +2 g}}\right ) f g +12 \sqrt {g}\, \sqrt {g x +f}\, \sqrt {3}\, \mathit {asin} \left (\frac {\sqrt {g}\, \sqrt {-3 x +2}}{\sqrt {3 f +2 g}}\right ) g^{2} x -18 \sqrt {-3 x +2}\, f^{2} g -24 \sqrt {-3 x +2}\, f \,g^{2} x -8 \sqrt {-3 x +2}\, f \,g^{2}-8 \sqrt {-3 x +2}\, g^{3} x -\frac {8 \sqrt {-3 x +2}\, g^{3}}{3}}{\sqrt {g x +f}\, g^{3} \left (3 f g x +2 g^{2} x +3 f^{2}+2 f g \right )} \] Input:

int((2+3*x)^(1/2)*(-9*x^2+4)^(1/2)/(g*x+f)^(5/2),x)
 

Output:

(2*(27*sqrt(g)*sqrt(f + g*x)*sqrt(3)*asin((sqrt(g)*sqrt( - 3*x + 2))/sqrt( 
3*f + 2*g))*f**2 + 27*sqrt(g)*sqrt(f + g*x)*sqrt(3)*asin((sqrt(g)*sqrt( - 
3*x + 2))/sqrt(3*f + 2*g))*f*g*x + 18*sqrt(g)*sqrt(f + g*x)*sqrt(3)*asin(( 
sqrt(g)*sqrt( - 3*x + 2))/sqrt(3*f + 2*g))*f*g + 18*sqrt(g)*sqrt(f + g*x)* 
sqrt(3)*asin((sqrt(g)*sqrt( - 3*x + 2))/sqrt(3*f + 2*g))*g**2*x - 27*sqrt( 
 - 3*x + 2)*f**2*g - 36*sqrt( - 3*x + 2)*f*g**2*x - 12*sqrt( - 3*x + 2)*f* 
g**2 - 12*sqrt( - 3*x + 2)*g**3*x - 4*sqrt( - 3*x + 2)*g**3))/(3*sqrt(f + 
g*x)*g**3*(3*f**2 + 3*f*g*x + 2*f*g + 2*g**2*x))