\(\int \frac {\sqrt {2+3 x} \sqrt {4-9 x^2}}{(f+g x)^2} \, dx\) [84]

Optimal result

Integrand size = 28, antiderivative size = 94 \[ \int \frac {\sqrt {2+3 x} \sqrt {4-9 x^2}}{(f+g x)^2} \, dx=\frac {6 \sqrt {2-3 x}}{g^2}+\frac {(3 f-2 g) \sqrt {2-3 x}}{g^2 (f+g x)}-\frac {3 (9 f+2 g) \text {arctanh}\left (\frac {\sqrt {g} \sqrt {2-3 x}}{\sqrt {3 f+2 g}}\right )}{g^{5/2} \sqrt {3 f+2 g}} \] Output:

6*(2-3*x)^(1/2)/g^2+(3*f-2*g)*(2-3*x)^(1/2)/g^2/(g*x+f)-3*(9*f+2*g)*arctan 
h(g^(1/2)*(2-3*x)^(1/2)/(3*f+2*g)^(1/2))/g^(5/2)/(3*f+2*g)^(1/2)
 

Mathematica [A] (verified)

Time = 0.44 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.20 \[ \int \frac {\sqrt {2+3 x} \sqrt {4-9 x^2}}{(f+g x)^2} \, dx=\frac {(9 f-2 g+6 g x) \sqrt {4-9 x^2}}{g^2 \sqrt {2+3 x} (f+g x)}+\frac {3 (9 f+2 g) \arctan \left (\frac {\sqrt {-3 f-2 g} \sqrt {4-9 x^2}}{\sqrt {g} (-2+3 x) \sqrt {2+3 x}}\right )}{\sqrt {-3 f-2 g} g^{5/2}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.28, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {639, 87, 60, 73, 221}

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.

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

\(\Big \downarrow \) 639

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

\(\Big \downarrow \) 87

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

\(\Big \downarrow \) 60

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

((3*f - 2*g)*(2 - 3*x)^(3/2))/(g*(3*f + 2*g)*(f + g*x)) + (3*(9*f + 2*g)*( 
(2*Sqrt[2 - 3*x])/g - (2*Sqrt[3*f + 2*g]*ArcTanh[(Sqrt[g]*Sqrt[2 - 3*x])/S 
qrt[3*f + 2*g]])/g^(3/2)))/(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 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

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 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

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 [A] (verified)

Time = 0.88 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.71

method	result	size
risch	\(-\frac {6 \left (-2+3 x \right ) \sqrt {\frac {-9 x^{2}+4}{3 x +2}}\, \sqrt {3 x +2}}{g^{2} \sqrt {2-3 x}\, \sqrt {-9 x^{2}+4}}+\frac {\left (\frac {6 \left (-\frac {3 f}{2}+g \right ) \sqrt {2-3 x}}{g \left (2-3 x \right )-3 f -2 g}-\frac {3 \left (9 f +2 g \right ) \operatorname {arctanh}\left (\frac {g \sqrt {2-3 x}}{\sqrt {g \left (3 f +2 g \right )}}\right )}{\sqrt {g \left (3 f +2 g \right )}}\right ) \sqrt {\frac {-9 x^{2}+4}{3 x +2}}\, \sqrt {3 x +2}}{g^{2} \sqrt {-9 x^{2}+4}}\)	\(161\)
default	\(\frac {\sqrt {-9 x^{2}+4}\, \left (6 \sqrt {g \left (3 f +2 g \right )}\, \sqrt {2-3 x}\, g x -27 \,\operatorname {arctanh}\left (\frac {g \sqrt {2-3 x}}{\sqrt {g \left (3 f +2 g \right )}}\right ) f g x -6 \,\operatorname {arctanh}\left (\frac {g \sqrt {2-3 x}}{\sqrt {g \left (3 f +2 g \right )}}\right ) g^{2} x +9 \sqrt {g \left (3 f +2 g \right )}\, \sqrt {2-3 x}\, f -2 \sqrt {g \left (3 f +2 g \right )}\, \sqrt {2-3 x}\, g -27 \,\operatorname {arctanh}\left (\frac {g \sqrt {2-3 x}}{\sqrt {g \left (3 f +2 g \right )}}\right ) f^{2}-6 \,\operatorname {arctanh}\left (\frac {g \sqrt {2-3 x}}{\sqrt {g \left (3 f +2 g \right )}}\right ) f g \right )}{\sqrt {3 x +2}\, \sqrt {2-3 x}\, g^{2} \left (g x +f \right ) \sqrt {g \left (3 f +2 g \right )}}\)	\(215\)

Input:

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

Output:

-6*(-2+3*x)/g^2/(2-3*x)^(1/2)*((-9*x^2+4)/(3*x+2))^(1/2)*(3*x+2)^(1/2)/(-9 
*x^2+4)^(1/2)+1/g^2*(6*(-3/2*f+g)*(2-3*x)^(1/2)/(g*(2-3*x)-3*f-2*g)-3*(9*f 
+2*g)/(g*(3*f+2*g))^(1/2)*arctanh(g*(2-3*x)^(1/2)/(g*(3*f+2*g))^(1/2)))*(( 
-9*x^2+4)/(3*x+2))^(1/2)*(3*x+2)^(1/2)/(-9*x^2+4)^(1/2)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 213 vs. \(2 (80) = 160\).

Time = 0.09 (sec) , antiderivative size = 457, normalized size of antiderivative = 4.86 \[ \int \frac {\sqrt {2+3 x} \sqrt {4-9 x^2}}{(f+g x)^2} \, dx=\left [\frac {3 \, {\left (3 \, {\left (9 \, f g + 2 \, g^{2}\right )} x^{2} + 18 \, f^{2} + 4 \, f g + {\left (27 \, f^{2} + 24 \, f g + 4 \, g^{2}\right )} x\right )} \sqrt {3 \, f g + 2 \, g^{2}} \log \left (-\frac {9 \, g x^{2} - 3 \, {\left (3 \, f + 2 \, g\right )} x + 2 \, \sqrt {3 \, f g + 2 \, g^{2}} \sqrt {-9 \, x^{2} + 4} \sqrt {3 \, x + 2} - 6 \, f - 8 \, g}{3 \, g x^{2} + {\left (3 \, f + 2 \, g\right )} x + 2 \, f}\right ) + 2 \, {\left (27 \, f^{2} g + 12 \, f g^{2} - 4 \, g^{3} + 6 \, {\left (3 \, f g^{2} + 2 \, g^{3}\right )} x\right )} \sqrt {-9 \, x^{2} + 4} \sqrt {3 \, x + 2}}{2 \, {\left (6 \, f^{2} g^{3} + 4 \, f g^{4} + 3 \, {\left (3 \, f g^{4} + 2 \, g^{5}\right )} x^{2} + {\left (9 \, f^{2} g^{3} + 12 \, f g^{4} + 4 \, g^{5}\right )} x\right )}}, \frac {3 \, {\left (3 \, {\left (9 \, f g + 2 \, g^{2}\right )} x^{2} + 18 \, f^{2} + 4 \, f g + {\left (27 \, f^{2} + 24 \, f g + 4 \, g^{2}\right )} x\right )} \sqrt {-3 \, f g - 2 \, g^{2}} \arctan \left (\frac {\sqrt {-3 \, f g - 2 \, g^{2}} \sqrt {-9 \, x^{2} + 4} \sqrt {3 \, x + 2}}{3 \, {\left (3 \, f + 2 \, g\right )} x + 6 \, f + 4 \, g}\right ) + {\left (27 \, f^{2} g + 12 \, f g^{2} - 4 \, g^{3} + 6 \, {\left (3 \, f g^{2} + 2 \, g^{3}\right )} x\right )} \sqrt {-9 \, x^{2} + 4} \sqrt {3 \, x + 2}}{6 \, f^{2} g^{3} + 4 \, f g^{4} + 3 \, {\left (3 \, f g^{4} + 2 \, g^{5}\right )} x^{2} + {\left (9 \, f^{2} g^{3} + 12 \, f g^{4} + 4 \, g^{5}\right )} x}\right ] \] Input:

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

Output:

[1/2*(3*(3*(9*f*g + 2*g^2)*x^2 + 18*f^2 + 4*f*g + (27*f^2 + 24*f*g + 4*g^2 
)*x)*sqrt(3*f*g + 2*g^2)*log(-(9*g*x^2 - 3*(3*f + 2*g)*x + 2*sqrt(3*f*g + 
2*g^2)*sqrt(-9*x^2 + 4)*sqrt(3*x + 2) - 6*f - 8*g)/(3*g*x^2 + (3*f + 2*g)* 
x + 2*f)) + 2*(27*f^2*g + 12*f*g^2 - 4*g^3 + 6*(3*f*g^2 + 2*g^3)*x)*sqrt(- 
9*x^2 + 4)*sqrt(3*x + 2))/(6*f^2*g^3 + 4*f*g^4 + 3*(3*f*g^4 + 2*g^5)*x^2 + 
 (9*f^2*g^3 + 12*f*g^4 + 4*g^5)*x), (3*(3*(9*f*g + 2*g^2)*x^2 + 18*f^2 + 4 
*f*g + (27*f^2 + 24*f*g + 4*g^2)*x)*sqrt(-3*f*g - 2*g^2)*arctan(sqrt(-3*f* 
g - 2*g^2)*sqrt(-9*x^2 + 4)*sqrt(3*x + 2)/(3*(3*f + 2*g)*x + 6*f + 4*g)) + 
 (27*f^2*g + 12*f*g^2 - 4*g^3 + 6*(3*f*g^2 + 2*g^3)*x)*sqrt(-9*x^2 + 4)*sq 
rt(3*x + 2))/(6*f^2*g^3 + 4*f*g^4 + 3*(3*f*g^4 + 2*g^5)*x^2 + (9*f^2*g^3 + 
 12*f*g^4 + 4*g^5)*x)]
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.07 \[ \int \frac {\sqrt {2+3 x} \sqrt {4-9 x^2}}{(f+g x)^2} \, dx=\frac {3 \, {\left (9 \, f + 2 \, g\right )} \arctan \left (\frac {g \sqrt {-3 \, x + 2}}{\sqrt {-3 \, f g - 2 \, g^{2}}}\right )}{\sqrt {-3 \, f g - 2 \, g^{2}} g^{2}} + \frac {6 \, \sqrt {-3 \, x + 2}}{g^{2}} + \frac {3 \, {\left (3 \, f \sqrt {-3 \, x + 2} - 2 \, g \sqrt {-3 \, x + 2}\right )}}{{\left (g {\left (3 \, x - 2\right )} + 3 \, f + 2 \, g\right )} g^{2}} \] Input:

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

Output:

3*(9*f + 2*g)*arctan(g*sqrt(-3*x + 2)/sqrt(-3*f*g - 2*g^2))/(sqrt(-3*f*g - 
 2*g^2)*g^2) + 6*sqrt(-3*x + 2)/g^2 + 3*(3*f*sqrt(-3*x + 2) - 2*g*sqrt(-3* 
x + 2))/((g*(3*x - 2) + 3*f + 2*g)*g^2)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 240, normalized size of antiderivative = 2.55 \[ \int \frac {\sqrt {2+3 x} \sqrt {4-9 x^2}}{(f+g x)^2} \, dx=\frac {-27 \sqrt {g}\, \sqrt {-3 f -2 g}\, \mathit {atan} \left (\frac {\sqrt {-3 x +2}\, g}{\sqrt {g}\, \sqrt {-3 f -2 g}}\right ) f^{2}-27 \sqrt {g}\, \sqrt {-3 f -2 g}\, \mathit {atan} \left (\frac {\sqrt {-3 x +2}\, g}{\sqrt {g}\, \sqrt {-3 f -2 g}}\right ) f g x -6 \sqrt {g}\, \sqrt {-3 f -2 g}\, \mathit {atan} \left (\frac {\sqrt {-3 x +2}\, g}{\sqrt {g}\, \sqrt {-3 f -2 g}}\right ) f g -6 \sqrt {g}\, \sqrt {-3 f -2 g}\, \mathit {atan} \left (\frac {\sqrt {-3 x +2}\, g}{\sqrt {g}\, \sqrt {-3 f -2 g}}\right ) g^{2} x +27 \sqrt {-3 x +2}\, f^{2} g +18 \sqrt {-3 x +2}\, f \,g^{2} x +12 \sqrt {-3 x +2}\, f \,g^{2}+12 \sqrt {-3 x +2}\, g^{3} x -4 \sqrt {-3 x +2}\, g^{3}}{g^{3} \left (3 f g x +2 g^{2} x +3 f^{2}+2 f g \right )} \] Input:

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

Output:

( - 27*sqrt(g)*sqrt( - 3*f - 2*g)*atan((sqrt( - 3*x + 2)*g)/(sqrt(g)*sqrt( 
 - 3*f - 2*g)))*f**2 - 27*sqrt(g)*sqrt( - 3*f - 2*g)*atan((sqrt( - 3*x + 2 
)*g)/(sqrt(g)*sqrt( - 3*f - 2*g)))*f*g*x - 6*sqrt(g)*sqrt( - 3*f - 2*g)*at 
an((sqrt( - 3*x + 2)*g)/(sqrt(g)*sqrt( - 3*f - 2*g)))*f*g - 6*sqrt(g)*sqrt 
( - 3*f - 2*g)*atan((sqrt( - 3*x + 2)*g)/(sqrt(g)*sqrt( - 3*f - 2*g)))*g** 
2*x + 27*sqrt( - 3*x + 2)*f**2*g + 18*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) 
/(g**3*(3*f**2 + 3*f*g*x + 2*f*g + 2*g**2*x))