\(\int \sqrt {2+3 x} (f+g x)^{3/2} \sqrt {4-9 x^2} \, dx\) [93]

Optimal result

Integrand size = 30, antiderivative size = 198 \[ \int \sqrt {2+3 x} (f+g x)^{3/2} \sqrt {4-9 x^2} \, dx=-\frac {(9 f-26 g) (3 f+2 g)^2 \sqrt {2-3 x} \sqrt {f+g x}}{576 g^2}+\frac {(9 f-26 g) (3 f+2 g) (2-3 x)^{3/2} \sqrt {f+g x}}{288 g}+\frac {(9 f-26 g) (2-3 x)^{3/2} (f+g x)^{3/2}}{72 g}-\frac {(2-3 x)^{3/2} (f+g x)^{5/2}}{4 g}+\frac {(9 f-26 g) (3 f+2 g)^3 \arctan \left (\frac {\sqrt {g} \sqrt {2-3 x}}{\sqrt {3} \sqrt {f+g x}}\right )}{576 \sqrt {3} g^{5/2}} \] Output:

-1/576*(9*f-26*g)*(3*f+2*g)^2*(2-3*x)^(1/2)*(g*x+f)^(1/2)/g^2+1/288*(9*f-2 
6*g)*(3*f+2*g)*(2-3*x)^(3/2)*(g*x+f)^(1/2)/g+1/72*(9*f-26*g)*(2-3*x)^(3/2) 
*(g*x+f)^(3/2)/g-1/4*(2-3*x)^(3/2)*(g*x+f)^(5/2)/g+1/1728*(9*f-26*g)*(3*f+ 
2*g)^3*arctan(1/3*g^(1/2)*(2-3*x)^(1/2)*3^(1/2)/(g*x+f)^(1/2))*3^(1/2)/g^( 
5/2)
 

Mathematica [A] (verified)

Time = 10.62 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.16 \[ \int \sqrt {2+3 x} (f+g x)^{3/2} \sqrt {4-9 x^2} \, dx=\frac {\sqrt {4-9 x^2} \left (3 \sqrt {-3 f-2 g} \sqrt {g} (-2+3 x) \left (-81 f^4+9 f^3 g (10-3 x)+2 f^2 g^2 \left (-190+321 x+351 x^2\right )+8 g^4 x \left (-13-13 x+42 x^2+54 x^3\right )+4 f g^3 \left (-26-121 x+222 x^2+270 x^3\right )\right )+(9 f-26 g) (3 f+2 g)^4 \sqrt {6-9 x} \sqrt {\frac {f+g x}{3 f+2 g}} \text {arcsinh}\left (\frac {\sqrt {g} \sqrt {2-3 x}}{\sqrt {-3 f-2 g}}\right )\right )}{1728 \sqrt {-3 f-2 g} g^{5/2} (-2+3 x) \sqrt {2+3 x} \sqrt {f+g x}} \] Input:

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

Output:

(Sqrt[4 - 9*x^2]*(3*Sqrt[-3*f - 2*g]*Sqrt[g]*(-2 + 3*x)*(-81*f^4 + 9*f^3*g 
*(10 - 3*x) + 2*f^2*g^2*(-190 + 321*x + 351*x^2) + 8*g^4*x*(-13 - 13*x + 4 
2*x^2 + 54*x^3) + 4*f*g^3*(-26 - 121*x + 222*x^2 + 270*x^3)) + (9*f - 26*g 
)*(3*f + 2*g)^4*Sqrt[6 - 9*x]*Sqrt[(f + g*x)/(3*f + 2*g)]*ArcSinh[(Sqrt[g] 
*Sqrt[2 - 3*x])/Sqrt[-3*f - 2*g]]))/(1728*Sqrt[-3*f - 2*g]*g^(5/2)*(-2 + 3 
*x)*Sqrt[2 + 3*x]*Sqrt[f + g*x])
 

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.91, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.233, Rules used = {639, 90, 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.

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

\(\Big \downarrow \) 639

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

\(\Big \downarrow \) 90

\(\displaystyle -\frac {(9 f-26 g) \int \sqrt {2-3 x} (f+g x)^{3/2}dx}{8 g}-\frac {(2-3 x)^{3/2} (f+g x)^{5/2}}{4 g}\)

\(\Big \downarrow \) 60

\(\displaystyle -\frac {(9 f-26 g) \left (\frac {1}{6} (3 f+2 g) \int \sqrt {2-3 x} \sqrt {f+g x}dx-\frac {1}{9} (2-3 x)^{3/2} (f+g x)^{3/2}\right )}{8 g}-\frac {(2-3 x)^{3/2} (f+g x)^{5/2}}{4 g}\)

\(\Big \downarrow \) 60

\(\displaystyle -\frac {(9 f-26 g) \left (\frac {1}{6} (3 f+2 g) \left (\frac {1}{12} (3 f+2 g) \int \frac {\sqrt {2-3 x}}{\sqrt {f+g x}}dx-\frac {1}{6} (2-3 x)^{3/2} \sqrt {f+g x}\right )-\frac {1}{9} (2-3 x)^{3/2} (f+g x)^{3/2}\right )}{8 g}-\frac {(2-3 x)^{3/2} (f+g x)^{5/2}}{4 g}\)

\(\Big \downarrow \) 60

\(\displaystyle -\frac {(9 f-26 g) \left (\frac {1}{6} (3 f+2 g) \left (\frac {1}{12} (3 f+2 g) \left (\frac {(3 f+2 g) \int \frac {1}{\sqrt {2-3 x} \sqrt {f+g x}}dx}{2 g}+\frac {\sqrt {2-3 x} \sqrt {f+g x}}{g}\right )-\frac {1}{6} (2-3 x)^{3/2} \sqrt {f+g x}\right )-\frac {1}{9} (2-3 x)^{3/2} (f+g x)^{3/2}\right )}{8 g}-\frac {(2-3 x)^{3/2} (f+g x)^{5/2}}{4 g}\)

\(\Big \downarrow \) 66

\(\displaystyle -\frac {(9 f-26 g) \left (\frac {1}{6} (3 f+2 g) \left (\frac {1}{12} (3 f+2 g) \left (\frac {(3 f+2 g) \int \frac {1}{-\frac {g (2-3 x)}{f+g x}-3}d\frac {\sqrt {2-3 x}}{\sqrt {f+g x}}}{g}+\frac {\sqrt {2-3 x} \sqrt {f+g x}}{g}\right )-\frac {1}{6} (2-3 x)^{3/2} \sqrt {f+g x}\right )-\frac {1}{9} (2-3 x)^{3/2} (f+g x)^{3/2}\right )}{8 g}-\frac {(2-3 x)^{3/2} (f+g x)^{5/2}}{4 g}\)

\(\Big \downarrow \) 217

\(\displaystyle -\frac {(9 f-26 g) \left (\frac {1}{6} (3 f+2 g) \left (\frac {1}{12} (3 f+2 g) \left (\frac {\sqrt {2-3 x} \sqrt {f+g x}}{g}-\frac {(3 f+2 g) \arctan \left (\frac {\sqrt {g} \sqrt {2-3 x}}{\sqrt {3} \sqrt {f+g x}}\right )}{\sqrt {3} g^{3/2}}\right )-\frac {1}{6} (2-3 x)^{3/2} \sqrt {f+g x}\right )-\frac {1}{9} (2-3 x)^{3/2} (f+g x)^{3/2}\right )}{8 g}-\frac {(2-3 x)^{3/2} (f+g x)^{5/2}}{4 g}\)

Input:

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

Output:

-1/4*((2 - 3*x)^(3/2)*(f + g*x)^(5/2))/g - ((9*f - 26*g)*(-1/9*((2 - 3*x)^ 
(3/2)*(f + g*x)^(3/2)) + ((3*f + 2*g)*(-1/6*((2 - 3*x)^(3/2)*Sqrt[f + g*x] 
) + ((3*f + 2*g)*((Sqrt[2 - 3*x]*Sqrt[f + g*x])/g - ((3*f + 2*g)*ArcTan[(S 
qrt[g]*Sqrt[2 - 3*x])/(Sqrt[3]*Sqrt[f + g*x])])/(Sqrt[3]*g^(3/2))))/12))/6 
))/(8*g)
 

Defintions of rubi rules used

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 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. \(472\) vs. \(2(161)=322\).

Time = 0.88 (sec) , antiderivative size = 473, normalized size of antiderivative = 2.39

method	result	size
default	\(-\frac {\sqrt {g x +f}\, \sqrt {-9 x^{2}+4}\, \left (-2592 g^{\frac {7}{2}} x^{3} \sqrt {-\left (g x +f \right ) \left (-2+3 x \right )}-3888 f \,g^{\frac {5}{2}} x^{2} \sqrt {-\left (g x +f \right ) \left (-2+3 x \right )}-2016 g^{\frac {7}{2}} x^{2} \sqrt {-\left (g x +f \right ) \left (-2+3 x \right )}+243 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (6 g x +3 f -2 g \right )}{6 \sqrt {g}\, \sqrt {-\left (g x +f \right ) \left (-2+3 x \right )}}\right ) f^{4}-216 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (6 g x +3 f -2 g \right )}{6 \sqrt {g}\, \sqrt {-\left (g x +f \right ) \left (-2+3 x \right )}}\right ) f^{3} g -1080 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (6 g x +3 f -2 g \right )}{6 \sqrt {g}\, \sqrt {-\left (g x +f \right ) \left (-2+3 x \right )}}\right ) f^{2} g^{2}-864 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (6 g x +3 f -2 g \right )}{6 \sqrt {g}\, \sqrt {-\left (g x +f \right ) \left (-2+3 x \right )}}\right ) f \,g^{3}-208 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (6 g x +3 f -2 g \right )}{6 \sqrt {g}\, \sqrt {-\left (g x +f \right ) \left (-2+3 x \right )}}\right ) g^{4}-324 g^{\frac {3}{2}} \sqrt {-\left (g x +f \right ) \left (-2+3 x \right )}\, f^{2} x -3312 g^{\frac {5}{2}} \sqrt {-\left (g x +f \right ) \left (-2+3 x \right )}\, f x +624 g^{\frac {7}{2}} \sqrt {-\left (g x +f \right ) \left (-2+3 x \right )}\, x +486 \sqrt {g}\, \sqrt {-\left (g x +f \right ) \left (-2+3 x \right )}\, f^{3}-540 g^{\frac {3}{2}} \sqrt {-\left (g x +f \right ) \left (-2+3 x \right )}\, f^{2}+2280 g^{\frac {5}{2}} \sqrt {-\left (g x +f \right ) \left (-2+3 x \right )}\, f +624 g^{\frac {7}{2}} \sqrt {-\left (g x +f \right ) \left (-2+3 x \right )}\right )}{3456 g^{\frac {5}{2}} \sqrt {3 x +2}\, \sqrt {-\left (g x +f \right ) \left (-2+3 x \right )}}\)	\(473\)

Input:

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

Output:

-1/3456*(g*x+f)^(1/2)*(-9*x^2+4)^(1/2)/g^(5/2)*(-2592*g^(7/2)*x^3*(-(g*x+f 
)*(-2+3*x))^(1/2)-3888*f*g^(5/2)*x^2*(-(g*x+f)*(-2+3*x))^(1/2)-2016*g^(7/2 
)*x^2*(-(g*x+f)*(-2+3*x))^(1/2)+243*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^4-216*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*g-1080*3^(1/2)*arc 
tan(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^2 
-864*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^3-208*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^4-324*g^(3/2)*(-(g*x+f)*(-2+3*x))^(1/2)*f^2*x-331 
2*g^(5/2)*(-(g*x+f)*(-2+3*x))^(1/2)*f*x+624*g^(7/2)*(-(g*x+f)*(-2+3*x))^(1 
/2)*x+486*g^(1/2)*(-(g*x+f)*(-2+3*x))^(1/2)*f^3-540*g^(3/2)*(-(g*x+f)*(-2+ 
3*x))^(1/2)*f^2+2280*g^(5/2)*(-(g*x+f)*(-2+3*x))^(1/2)*f+624*g^(7/2)*(-(g* 
x+f)*(-2+3*x))^(1/2))/(3*x+2)^(1/2)/(-(g*x+f)*(-2+3*x))^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 552, normalized size of antiderivative = 2.79 \[ \int \sqrt {2+3 x} (f+g x)^{3/2} \sqrt {4-9 x^2} \, dx=\left [\frac {\sqrt {3} {\left (486 \, f^{4} - 432 \, f^{3} g - 2160 \, f^{2} g^{2} - 1728 \, f g^{3} - 416 \, g^{4} + 3 \, {\left (243 \, f^{4} - 216 \, f^{3} g - 1080 \, f^{2} g^{2} - 864 \, f g^{3} - 208 \, g^{4}\right )} x\right )} \sqrt {-g} \log \left (-\frac {216 \, g^{2} x^{3} + 216 \, f g x^{2} - 4 \, \sqrt {3} {\left (6 \, g x + 3 \, f - 2 \, g\right )} \sqrt {g x + f} \sqrt {-9 \, x^{2} + 4} \sqrt {-g} \sqrt {3 \, x + 2} + 18 \, f^{2} - 72 \, f g + 8 \, g^{2} + 3 \, {\left (9 \, f^{2} + 12 \, f g - 28 \, g^{2}\right )} x}{3 \, x + 2}\right ) + 12 \, {\left (432 \, g^{4} x^{3} - 81 \, f^{3} g + 90 \, f^{2} g^{2} - 380 \, f g^{3} - 104 \, g^{4} + 24 \, {\left (27 \, f g^{3} + 14 \, g^{4}\right )} x^{2} + 2 \, {\left (27 \, f^{2} g^{2} + 276 \, f g^{3} - 52 \, g^{4}\right )} x\right )} \sqrt {g x + f} \sqrt {-9 \, x^{2} + 4} \sqrt {3 \, x + 2}}{6912 \, {\left (3 \, g^{3} x + 2 \, g^{3}\right )}}, \frac {\sqrt {3} {\left (486 \, f^{4} - 432 \, f^{3} g - 2160 \, f^{2} g^{2} - 1728 \, f g^{3} - 416 \, g^{4} + 3 \, {\left (243 \, f^{4} - 216 \, f^{3} g - 1080 \, f^{2} g^{2} - 864 \, f g^{3} - 208 \, g^{4}\right )} x\right )} \sqrt {g} \arctan \left (\frac {\sqrt {3} {\left (6 \, g x + 3 \, f - 2 \, g\right )} \sqrt {g x + f} \sqrt {-9 \, x^{2} + 4} \sqrt {g} \sqrt {3 \, x + 2}}{6 \, {\left (9 \, g^{2} x^{3} + 9 \, f g x^{2} - 4 \, g^{2} x - 4 \, f g\right )}}\right ) + 6 \, {\left (432 \, g^{4} x^{3} - 81 \, f^{3} g + 90 \, f^{2} g^{2} - 380 \, f g^{3} - 104 \, g^{4} + 24 \, {\left (27 \, f g^{3} + 14 \, g^{4}\right )} x^{2} + 2 \, {\left (27 \, f^{2} g^{2} + 276 \, f g^{3} - 52 \, g^{4}\right )} x\right )} \sqrt {g x + f} \sqrt {-9 \, x^{2} + 4} \sqrt {3 \, x + 2}}{3456 \, {\left (3 \, g^{3} x + 2 \, g^{3}\right )}}\right ] \] Input:

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

Output:

[1/6912*(sqrt(3)*(486*f^4 - 432*f^3*g - 2160*f^2*g^2 - 1728*f*g^3 - 416*g^ 
4 + 3*(243*f^4 - 216*f^3*g - 1080*f^2*g^2 - 864*f*g^3 - 208*g^4)*x)*sqrt(- 
g)*log(-(216*g^2*x^3 + 216*f*g*x^2 - 4*sqrt(3)*(6*g*x + 3*f - 2*g)*sqrt(g* 
x + f)*sqrt(-9*x^2 + 4)*sqrt(-g)*sqrt(3*x + 2) + 18*f^2 - 72*f*g + 8*g^2 + 
 3*(9*f^2 + 12*f*g - 28*g^2)*x)/(3*x + 2)) + 12*(432*g^4*x^3 - 81*f^3*g + 
90*f^2*g^2 - 380*f*g^3 - 104*g^4 + 24*(27*f*g^3 + 14*g^4)*x^2 + 2*(27*f^2* 
g^2 + 276*f*g^3 - 52*g^4)*x)*sqrt(g*x + f)*sqrt(-9*x^2 + 4)*sqrt(3*x + 2)) 
/(3*g^3*x + 2*g^3), 1/3456*(sqrt(3)*(486*f^4 - 432*f^3*g - 2160*f^2*g^2 - 
1728*f*g^3 - 416*g^4 + 3*(243*f^4 - 216*f^3*g - 1080*f^2*g^2 - 864*f*g^3 - 
 208*g^4)*x)*sqrt(g)*arctan(1/6*sqrt(3)*(6*g*x + 3*f - 2*g)*sqrt(g*x + f)* 
sqrt(-9*x^2 + 4)*sqrt(g)*sqrt(3*x + 2)/(9*g^2*x^3 + 9*f*g*x^2 - 4*g^2*x - 
4*f*g)) + 6*(432*g^4*x^3 - 81*f^3*g + 90*f^2*g^2 - 380*f*g^3 - 104*g^4 + 2 
4*(27*f*g^3 + 14*g^4)*x^2 + 2*(27*f^2*g^2 + 276*f*g^3 - 52*g^4)*x)*sqrt(g* 
x + f)*sqrt(-9*x^2 + 4)*sqrt(3*x + 2))/(3*g^3*x + 2*g^3)]
                                                                                    
                                                                                    
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-1)]

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

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

Output:

Timed out
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 474, normalized size of antiderivative = 2.39 \[ \int \sqrt {2+3 x} (f+g x)^{3/2} \sqrt {4-9 x^2} \, dx=\frac {729 \sqrt {g}\, \sqrt {3}\, \mathit {asin} \left (\frac {\sqrt {g}\, \sqrt {-3 x +2}}{\sqrt {3 f +2 g}}\right ) f^{5}-162 \sqrt {g}\, \sqrt {3}\, \mathit {asin} \left (\frac {\sqrt {g}\, \sqrt {-3 x +2}}{\sqrt {3 f +2 g}}\right ) f^{4} g -3672 \sqrt {g}\, \sqrt {3}\, \mathit {asin} \left (\frac {\sqrt {g}\, \sqrt {-3 x +2}}{\sqrt {3 f +2 g}}\right ) f^{3} g^{2}-4752 \sqrt {g}\, \sqrt {3}\, \mathit {asin} \left (\frac {\sqrt {g}\, \sqrt {-3 x +2}}{\sqrt {3 f +2 g}}\right ) f^{2} g^{3}-2352 \sqrt {g}\, \sqrt {3}\, \mathit {asin} \left (\frac {\sqrt {g}\, \sqrt {-3 x +2}}{\sqrt {3 f +2 g}}\right ) f \,g^{4}-416 \sqrt {g}\, \sqrt {3}\, \mathit {asin} \left (\frac {\sqrt {g}\, \sqrt {-3 x +2}}{\sqrt {3 f +2 g}}\right ) g^{5}-729 \sqrt {g x +f}\, \sqrt {-3 x +2}\, f^{4} g +486 \sqrt {g x +f}\, \sqrt {-3 x +2}\, f^{3} g^{2} x +324 \sqrt {g x +f}\, \sqrt {-3 x +2}\, f^{3} g^{2}+5832 \sqrt {g x +f}\, \sqrt {-3 x +2}\, f^{2} g^{3} x^{2}+5292 \sqrt {g x +f}\, \sqrt {-3 x +2}\, f^{2} g^{3} x -2880 \sqrt {g x +f}\, \sqrt {-3 x +2}\, f^{2} g^{3}+3888 \sqrt {g x +f}\, \sqrt {-3 x +2}\, f \,g^{4} x^{3}+6912 \sqrt {g x +f}\, \sqrt {-3 x +2}\, f \,g^{4} x^{2}+2376 \sqrt {g x +f}\, \sqrt {-3 x +2}\, f \,g^{4} x -3216 \sqrt {g x +f}\, \sqrt {-3 x +2}\, f \,g^{4}+2592 \sqrt {g x +f}\, \sqrt {-3 x +2}\, g^{5} x^{3}+2016 \sqrt {g x +f}\, \sqrt {-3 x +2}\, g^{5} x^{2}-624 \sqrt {g x +f}\, \sqrt {-3 x +2}\, g^{5} x -624 \sqrt {g x +f}\, \sqrt {-3 x +2}\, g^{5}}{1728 g^{3} \left (3 f +2 g \right )} \] Input:

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

Output:

(729*sqrt(g)*sqrt(3)*asin((sqrt(g)*sqrt( - 3*x + 2))/sqrt(3*f + 2*g))*f**5 
 - 162*sqrt(g)*sqrt(3)*asin((sqrt(g)*sqrt( - 3*x + 2))/sqrt(3*f + 2*g))*f* 
*4*g - 3672*sqrt(g)*sqrt(3)*asin((sqrt(g)*sqrt( - 3*x + 2))/sqrt(3*f + 2*g 
))*f**3*g**2 - 4752*sqrt(g)*sqrt(3)*asin((sqrt(g)*sqrt( - 3*x + 2))/sqrt(3 
*f + 2*g))*f**2*g**3 - 2352*sqrt(g)*sqrt(3)*asin((sqrt(g)*sqrt( - 3*x + 2) 
)/sqrt(3*f + 2*g))*f*g**4 - 416*sqrt(g)*sqrt(3)*asin((sqrt(g)*sqrt( - 3*x 
+ 2))/sqrt(3*f + 2*g))*g**5 - 729*sqrt(f + g*x)*sqrt( - 3*x + 2)*f**4*g + 
486*sqrt(f + g*x)*sqrt( - 3*x + 2)*f**3*g**2*x + 324*sqrt(f + g*x)*sqrt( - 
 3*x + 2)*f**3*g**2 + 5832*sqrt(f + g*x)*sqrt( - 3*x + 2)*f**2*g**3*x**2 + 
 5292*sqrt(f + g*x)*sqrt( - 3*x + 2)*f**2*g**3*x - 2880*sqrt(f + g*x)*sqrt 
( - 3*x + 2)*f**2*g**3 + 3888*sqrt(f + g*x)*sqrt( - 3*x + 2)*f*g**4*x**3 + 
 6912*sqrt(f + g*x)*sqrt( - 3*x + 2)*f*g**4*x**2 + 2376*sqrt(f + g*x)*sqrt 
( - 3*x + 2)*f*g**4*x - 3216*sqrt(f + g*x)*sqrt( - 3*x + 2)*f*g**4 + 2592* 
sqrt(f + g*x)*sqrt( - 3*x + 2)*g**5*x**3 + 2016*sqrt(f + g*x)*sqrt( - 3*x 
+ 2)*g**5*x**2 - 624*sqrt(f + g*x)*sqrt( - 3*x + 2)*g**5*x - 624*sqrt(f + 
g*x)*sqrt( - 3*x + 2)*g**5)/(1728*g**3*(3*f + 2*g))