\(\int \frac {\sqrt {2+3 x} \sqrt {4-9 x^2}}{(f+g x)^{3/2}} \, dx\) [96]

Optimal result

Integrand size = 30, antiderivative size = 103 \[ \int \frac {\sqrt {2+3 x} \sqrt {4-9 x^2}}{(f+g x)^{3/2}} \, dx=\frac {2 (3 f-2 g) \sqrt {2-3 x}}{g^2 \sqrt {f+g x}}+\frac {3 \sqrt {2-3 x} \sqrt {f+g x}}{g^2}-\frac {\sqrt {3} (9 f-2 g) \arctan \left (\frac {\sqrt {g} \sqrt {2-3 x}}{\sqrt {3} \sqrt {f+g x}}\right )}{g^{5/2}} \] Output:

2*(3*f-2*g)*(2-3*x)^(1/2)/g^2/(g*x+f)^(1/2)+3*(2-3*x)^(1/2)*(g*x+f)^(1/2)/ 
g^2-3^(1/2)*(9*f-2*g)*arctan(1/3*g^(1/2)*(2-3*x)^(1/2)*3^(1/2)/(g*x+f)^(1/ 
2))/g^(5/2)
 

Mathematica [A] (verified)

Time = 1.30 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.37 \[ \int \frac {\sqrt {2+3 x} \sqrt {4-9 x^2}}{(f+g x)^{3/2}} \, dx=-\frac {\sqrt {-2+3 x} \sqrt {2+3 x} \left (\sqrt {g} \sqrt {-2+3 x} (9 f+g (-4+3 x))+2 \sqrt {3} (9 f-2 g) \sqrt {f+g x} \text {arctanh}\left (\frac {\sqrt {g} \sqrt {-2+3 x}}{\sqrt {3 f+2 g}-\sqrt {3} \sqrt {f+g x}}\right )\right )}{g^{5/2} \sqrt {f+g x} \sqrt {4-9 x^2}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.29, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {639, 87, 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.

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

\(\Big \downarrow \) 639

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

\(\Big \downarrow \) 87

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

\(\Big \downarrow \) 60

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

\(\Big \downarrow \) 66

\(\displaystyle \frac {3 (9 f-2 g) \left (\frac {(3 f+2 g) \int \frac {1}{-\frac {g (2-3 x)}{f+g x}-3}d\frac {\sqrt {2-3 x}}{\sqrt {f+g x}}}{g}+\frac {\sqrt {2-3 x} \sqrt {f+g x}}{g}\right )}{g (3 f+2 g)}+\frac {2 (2-3 x)^{3/2} (3 f-2 g)}{g (3 f+2 g) \sqrt {f+g x}}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {3 (9 f-2 g) \left (\frac {\sqrt {2-3 x} \sqrt {f+g x}}{g}-\frac {(3 f+2 g) \arctan \left (\frac {\sqrt {g} \sqrt {2-3 x}}{\sqrt {3} \sqrt {f+g x}}\right )}{\sqrt {3} g^{3/2}}\right )}{g (3 f+2 g)}+\frac {2 (2-3 x)^{3/2} (3 f-2 g)}{g (3 f+2 g) \sqrt {f+g x}}\)

Input:

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

Output:

(2*(3*f - 2*g)*(2 - 3*x)^(3/2))/(g*(3*f + 2*g)*Sqrt[f + g*x]) + (3*(9*f - 
2*g)*((Sqrt[2 - 3*x]*Sqrt[f + g*x])/g - ((3*f + 2*g)*ArcTan[(Sqrt[g]*Sqrt[ 
2 - 3*x])/(Sqrt[3]*Sqrt[f + g*x])])/(Sqrt[3]*g^(3/2))))/(g*(3*f + 2*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 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. \(270\) vs. \(2(84)=168\).

Time = 0.87 (sec) , antiderivative size = 271, normalized size of antiderivative = 2.63

method	result	size
default	\(\frac {\sqrt {-9 x^{2}+4}\, \left (9 \arctan \left (\frac {\sqrt {3}\, \left (6 g x +3 f -2 g \right )}{6 \sqrt {g}\, \sqrt {-\left (g x +f \right ) \left (-2+3 x \right )}}\right ) \sqrt {3}\, f g x -2 \arctan \left (\frac {\sqrt {3}\, \left (6 g x +3 f -2 g \right )}{6 \sqrt {g}\, \sqrt {-\left (g x +f \right ) \left (-2+3 x \right )}}\right ) \sqrt {3}\, g^{2} x +9 \arctan \left (\frac {\sqrt {3}\, \left (6 g x +3 f -2 g \right )}{6 \sqrt {g}\, \sqrt {-\left (g x +f \right ) \left (-2+3 x \right )}}\right ) \sqrt {3}\, f^{2}-2 \arctan \left (\frac {\sqrt {3}\, \left (6 g x +3 f -2 g \right )}{6 \sqrt {g}\, \sqrt {-\left (g x +f \right ) \left (-2+3 x \right )}}\right ) \sqrt {3}\, f g +6 g^{\frac {3}{2}} x \sqrt {-\left (g x +f \right ) \left (-2+3 x \right )}+18 f \sqrt {g}\, \sqrt {-\left (g x +f \right ) \left (-2+3 x \right )}-8 g^{\frac {3}{2}} \sqrt {-\left (g x +f \right ) \left (-2+3 x \right )}\right )}{2 \sqrt {-\left (g x +f \right ) \left (-2+3 x \right )}\, g^{\frac {5}{2}} \sqrt {g x +f}\, \sqrt {3 x +2}}\)	\(271\)

Input:

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

Output:

1/2*(-9*x^2+4)^(1/2)*(9*arctan(1/6*3^(1/2)/g^(1/2)*(6*g*x+3*f-2*g)/(-(g*x+ 
f)*(-2+3*x))^(1/2))*3^(1/2)*f*g*x-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))*3^(1/2)*g^2*x+9*arctan(1/6*3^(1/2)/g^(1/2) 
*(6*g*x+3*f-2*g)/(-(g*x+f)*(-2+3*x))^(1/2))*3^(1/2)*f^2-2*arctan(1/6*3^(1/ 
2)/g^(1/2)*(6*g*x+3*f-2*g)/(-(g*x+f)*(-2+3*x))^(1/2))*3^(1/2)*f*g+6*g^(3/2 
)*x*(-(g*x+f)*(-2+3*x))^(1/2)+18*f*g^(1/2)*(-(g*x+f)*(-2+3*x))^(1/2)-8*g^( 
3/2)*(-(g*x+f)*(-2+3*x))^(1/2))/(-(g*x+f)*(-2+3*x))^(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. 177 vs. \(2 (84) = 168\).

Time = 0.33 (sec) , antiderivative size = 410, normalized size of antiderivative = 3.98 \[ \int \frac {\sqrt {2+3 x} \sqrt {4-9 x^2}}{(f+g x)^{3/2}} \, dx=\left [-\frac {\sqrt {3} {\left (3 \, {\left (9 \, f g - 2 \, g^{2}\right )} x^{2} + 18 \, f^{2} - 4 \, f g + {\left (27 \, f^{2} + 12 \, f g - 4 \, g^{2}\right )} x\right )} \sqrt {-\frac {1}{g}} \log \left (-\frac {216 \, g^{2} x^{3} + 216 \, f g x^{2} - 4 \, \sqrt {3} {\left (6 \, g^{2} x + 3 \, f g - 2 \, g^{2}\right )} \sqrt {g x + f} \sqrt {-9 \, x^{2} + 4} \sqrt {3 \, x + 2} \sqrt {-\frac {1}{g}} + 18 \, f^{2} - 72 \, f g + 8 \, g^{2} + 3 \, {\left (9 \, f^{2} + 12 \, f g - 28 \, g^{2}\right )} x}{3 \, x + 2}\right ) - 4 \, {\left (3 \, g x + 9 \, f - 4 \, g\right )} \sqrt {g x + f} \sqrt {-9 \, x^{2} + 4} \sqrt {3 \, x + 2}}{4 \, {\left (3 \, g^{3} x^{2} + 2 \, f g^{2} + {\left (3 \, f g^{2} + 2 \, g^{3}\right )} x\right )}}, \frac {2 \, {\left (3 \, g x + 9 \, f - 4 \, g\right )} \sqrt {g x + f} \sqrt {-9 \, x^{2} + 4} \sqrt {3 \, x + 2} - \frac {\sqrt {3} {\left (3 \, {\left (9 \, f g - 2 \, g^{2}\right )} x^{2} + 18 \, f^{2} - 4 \, f g + {\left (27 \, f^{2} + 12 \, f g - 4 \, g^{2}\right )} x\right )} \arctan \left (\frac {2 \, \sqrt {3} \sqrt {g x + f} \sqrt {-9 \, x^{2} + 4} \sqrt {g} \sqrt {3 \, x + 2}}{18 \, g x^{2} + 3 \, {\left (3 \, f + 2 \, g\right )} x + 6 \, f - 4 \, g}\right )}{\sqrt {g}}}{2 \, {\left (3 \, g^{3} x^{2} + 2 \, f g^{2} + {\left (3 \, f g^{2} + 2 \, g^{3}\right )} x\right )}}\right ] \] Input:

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

Output:

[-1/4*(sqrt(3)*(3*(9*f*g - 2*g^2)*x^2 + 18*f^2 - 4*f*g + (27*f^2 + 12*f*g 
- 4*g^2)*x)*sqrt(-1/g)*log(-(216*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 
*(3*g*x + 9*f - 4*g)*sqrt(g*x + f)*sqrt(-9*x^2 + 4)*sqrt(3*x + 2))/(3*g^3* 
x^2 + 2*f*g^2 + (3*f*g^2 + 2*g^3)*x), 1/2*(2*(3*g*x + 9*f - 4*g)*sqrt(g*x 
+ f)*sqrt(-9*x^2 + 4)*sqrt(3*x + 2) - sqrt(3)*(3*(9*f*g - 2*g^2)*x^2 + 18* 
f^2 - 4*f*g + (27*f^2 + 12*f*g - 4*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))/(3*g^3*x^2 + 2*f*g^2 + (3*f*g^2 + 2*g^3)*x)]
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.10 \[ \int \frac {\sqrt {2+3 x} \sqrt {4-9 x^2}}{(f+g x)^{3/2}} \, dx=\frac {\sqrt {-3 \, x + 2} {\left (\frac {\sqrt {3} {\left (3 \, x - 2\right )}}{g} + \frac {9 \, \sqrt {3} f g - 2 \, \sqrt {3} g^{2}}{g^{3}}\right )}}{\sqrt {g {\left (3 \, x - 2\right )} + 3 \, f + 2 \, g}} + \frac {{\left (9 \, \sqrt {3} f - 2 \, \sqrt {3} g\right )} \log \left ({\left | -\sqrt {-g} \sqrt {-3 \, x + 2} + \sqrt {g {\left (3 \, x - 2\right )} + 3 \, f + 2 \, g} \right |}\right )}{\sqrt {-g} g^{2}} \] Input:

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

Output:

sqrt(-3*x + 2)*(sqrt(3)*(3*x - 2)/g + (9*sqrt(3)*f*g - 2*sqrt(3)*g^2)/g^3) 
/sqrt(g*(3*x - 2) + 3*f + 2*g) + (9*sqrt(3)*f - 2*sqrt(3)*g)*log(abs(-sqrt 
(-g)*sqrt(-3*x + 2) + sqrt(g*(3*x - 2) + 3*f + 2*g)))/(sqrt(-g)*g^2)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.09 \[ \int \frac {\sqrt {2+3 x} \sqrt {4-9 x^2}}{(f+g x)^{3/2}} \, dx=\frac {-9 \sqrt {g}\, \sqrt {g x +f}\, \sqrt {3}\, \mathit {asin} \left (\frac {\sqrt {g}\, \sqrt {-3 x +2}}{\sqrt {3 f +2 g}}\right ) f +2 \sqrt {g}\, \sqrt {g x +f}\, \sqrt {3}\, \mathit {asin} \left (\frac {\sqrt {g}\, \sqrt {-3 x +2}}{\sqrt {3 f +2 g}}\right ) g +9 \sqrt {-3 x +2}\, f g +3 \sqrt {-3 x +2}\, g^{2} x -4 \sqrt {-3 x +2}\, g^{2}}{\sqrt {g x +f}\, g^{3}} \] Input:

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

Output:

( - 9*sqrt(g)*sqrt(f + g*x)*sqrt(3)*asin((sqrt(g)*sqrt( - 3*x + 2))/sqrt(3 
*f + 2*g))*f + 2*sqrt(g)*sqrt(f + g*x)*sqrt(3)*asin((sqrt(g)*sqrt( - 3*x + 
 2))/sqrt(3*f + 2*g))*g + 9*sqrt( - 3*x + 2)*f*g + 3*sqrt( - 3*x + 2)*g**2 
*x - 4*sqrt( - 3*x + 2)*g**2)/(sqrt(f + g*x)*g**3)