\(\int \frac {\sqrt {2+3 x} \sqrt {4-9 x^2}}{f+g x} \, dx\) [83]

Optimal result

Integrand size = 28, antiderivative size = 90 \[ \int \frac {\sqrt {2+3 x} \sqrt {4-9 x^2}}{f+g x} \, dx=-\frac {2 (3 f-2 g) \sqrt {2-3 x}}{g^2}-\frac {2 (2-3 x)^{3/2}}{3 g}+\frac {2 (3 f-2 g) \sqrt {3 f+2 g} \text {arctanh}\left (\frac {\sqrt {g} \sqrt {2-3 x}}{\sqrt {3 f+2 g}}\right )}{g^{5/2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.44 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.26 \[ \int \frac {\sqrt {2+3 x} \sqrt {4-9 x^2}}{f+g x} \, dx=\frac {2 \sqrt {4-9 x^2} (-9 f+g (4+3 x))}{3 g^2 \sqrt {2+3 x}}+\frac {2 \left (9 f^2-4 g^2\right ) \arctan \left (\frac {\sqrt {-3 f-2 g} \sqrt {4-9 x^2}}{\sqrt {g} (2-3 x) \sqrt {2+3 x}}\right )}{\sqrt {-3 f-2 g} g^{5/2}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.99, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {639, 90, 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.

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

\(\Big \downarrow \) 639

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

\(\Big \downarrow \) 90

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

\(\Big \downarrow \) 60

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

(-2*(2 - 3*x)^(3/2))/(3*g) - ((3*f - 2*g)*((2*Sqrt[2 - 3*x])/g - (2*Sqrt[3 
*f + 2*g]*ArcTanh[(Sqrt[g]*Sqrt[2 - 3*x])/Sqrt[3*f + 2*g]])/g^(3/2)))/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 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 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 = 0.86 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.60

method	result	size
risch	\(\frac {2 \left (-3 g x +9 f -4 g \right ) \left (-2+3 x \right ) \sqrt {\frac {-9 x^{2}+4}{3 x +2}}\, \sqrt {3 x +2}}{3 g^{2} \sqrt {2-3 x}\, \sqrt {-9 x^{2}+4}}+\frac {2 \left (9 f^{2}-4 g^{2}\right ) \operatorname {arctanh}\left (\frac {g \sqrt {2-3 x}}{\sqrt {g \left (3 f +2 g \right )}}\right ) \sqrt {\frac {-9 x^{2}+4}{3 x +2}}\, \sqrt {3 x +2}}{g^{2} \sqrt {g \left (3 f +2 g \right )}\, \sqrt {-9 x^{2}+4}}\)	\(144\)
default	\(\frac {2 \sqrt {-9 x^{2}+4}\, \left (3 \sqrt {g \left (3 f +2 g \right )}\, \sqrt {2-3 x}\, g x +27 \,\operatorname {arctanh}\left (\frac {g \sqrt {2-3 x}}{\sqrt {g \left (3 f +2 g \right )}}\right ) f^{2}-12 \,\operatorname {arctanh}\left (\frac {g \sqrt {2-3 x}}{\sqrt {g \left (3 f +2 g \right )}}\right ) g^{2}-9 \sqrt {g \left (3 f +2 g \right )}\, \sqrt {2-3 x}\, f +4 \sqrt {g \left (3 f +2 g \right )}\, \sqrt {2-3 x}\, g \right )}{3 \sqrt {3 x +2}\, \sqrt {2-3 x}\, g^{2} \sqrt {g \left (3 f +2 g \right )}}\)	\(157\)

Input:

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

Output:

2/3*(-3*g*x+9*f-4*g)*(-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)+2*(9*f^2-4*g^2)/g^2/(g*(3*f+2*g))^(1/2)*ar 
ctanh(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 [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 289, normalized size of antiderivative = 3.21 \[ \int \frac {\sqrt {2+3 x} \sqrt {4-9 x^2}}{f+g x} \, dx=\left [-\frac {3 \, {\left (3 \, {\left (3 \, f - 2 \, g\right )} x + 6 \, f - 4 \, g\right )} \sqrt {\frac {3 \, f + 2 \, g}{g}} \log \left (-\frac {9 \, g x^{2} + 2 \, \sqrt {-9 \, x^{2} + 4} g \sqrt {3 \, x + 2} \sqrt {\frac {3 \, f + 2 \, g}{g}} - 3 \, {\left (3 \, f + 2 \, g\right )} x - 6 \, f - 8 \, g}{3 \, g x^{2} + {\left (3 \, f + 2 \, g\right )} x + 2 \, f}\right ) - 2 \, {\left (3 \, g x - 9 \, f + 4 \, g\right )} \sqrt {-9 \, x^{2} + 4} \sqrt {3 \, x + 2}}{3 \, {\left (3 \, g^{2} x + 2 \, g^{2}\right )}}, -\frac {2 \, {\left (3 \, {\left (3 \, {\left (3 \, f - 2 \, g\right )} x + 6 \, f - 4 \, g\right )} \sqrt {-\frac {3 \, f + 2 \, g}{g}} \arctan \left (\frac {\sqrt {-9 \, x^{2} + 4} g \sqrt {3 \, x + 2} \sqrt {-\frac {3 \, f + 2 \, g}{g}}}{3 \, {\left (3 \, f + 2 \, g\right )} x + 6 \, f + 4 \, g}\right ) - {\left (3 \, g x - 9 \, f + 4 \, g\right )} \sqrt {-9 \, x^{2} + 4} \sqrt {3 \, x + 2}\right )}}{3 \, {\left (3 \, g^{2} x + 2 \, g^{2}\right )}}\right ] \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.01 \[ \int \frac {\sqrt {2+3 x} \sqrt {4-9 x^2}}{f+g x} \, dx=-\frac {2 \, {\left (9 \, f^{2} - 4 \, g^{2}\right )} \arctan \left (\frac {g \sqrt {-3 \, x + 2}}{\sqrt {-3 \, f g - 2 \, g^{2}}}\right )}{\sqrt {-3 \, f g - 2 \, g^{2}} g^{2}} - \frac {2 \, {\left (g^{2} {\left (-3 \, x + 2\right )}^{\frac {3}{2}} + 9 \, f g \sqrt {-3 \, x + 2} - 6 \, g^{2} \sqrt {-3 \, x + 2}\right )}}{3 \, g^{3}} \] Input:

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

Output:

-2*(9*f^2 - 4*g^2)*arctan(g*sqrt(-3*x + 2)/sqrt(-3*f*g - 2*g^2))/(sqrt(-3* 
f*g - 2*g^2)*g^2) - 2/3*(g^2*(-3*x + 2)^(3/2) + 9*f*g*sqrt(-3*x + 2) - 6*g 
^2*sqrt(-3*x + 2))/g^3
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.23 \[ \int \frac {\sqrt {2+3 x} \sqrt {4-9 x^2}}{f+g x} \, dx=\frac {6 \sqrt {g}\, \sqrt {-3 f -2 g}\, \mathit {atan} \left (\frac {\sqrt {-3 x +2}\, g}{\sqrt {g}\, \sqrt {-3 f -2 g}}\right ) f -4 \sqrt {g}\, \sqrt {-3 f -2 g}\, \mathit {atan} \left (\frac {\sqrt {-3 x +2}\, g}{\sqrt {g}\, \sqrt {-3 f -2 g}}\right ) g -6 \sqrt {-3 x +2}\, f g +2 \sqrt {-3 x +2}\, g^{2} x +\frac {8 \sqrt {-3 x +2}\, g^{2}}{3}}{g^{3}} \] Input:

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

Output:

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