\(\int \frac {\sqrt {2+3 x} \sqrt {1+\frac {5 x}{6}-x^2}}{(f+g x)^2} \, dx\) [300]

Optimal result

Integrand size = 33, antiderivative size = 110 \[ \int \frac {\sqrt {2+3 x} \sqrt {1+\frac {5 x}{6}-x^2}}{(f+g x)^2} \, dx=\frac {\sqrt {6} \sqrt {3-2 x}}{g^2}+\frac {(3 f-2 g) \sqrt {3-2 x}}{\sqrt {6} g^2 (f+g x)}-\frac {\sqrt {\frac {2}{3}} (9 f+7 g) \text {arctanh}\left (\frac {\sqrt {g} \sqrt {3-2 x}}{\sqrt {2 f+3 g}}\right )}{g^{5/2} \sqrt {2 f+3 g}} \] Output:

6^(1/2)*(3-2*x)^(1/2)/g^2+1/6*(3*f-2*g)*(3-2*x)^(1/2)*6^(1/2)/g^2/(g*x+f)- 
1/3*6^(1/2)*(9*f+7*g)*arctanh(g^(1/2)*(3-2*x)^(1/2)/(2*f+3*g)^(1/2))/g^(5/ 
2)/(2*f+3*g)^(1/2)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 1.24 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.15 \[ \int \frac {\sqrt {2+3 x} \sqrt {1+\frac {5 x}{6}-x^2}}{(f+g x)^2} \, dx=\frac {\frac {\sqrt {g} (9 f-2 g+6 g x) \sqrt {6+5 x-6 x^2}}{\sqrt {2+3 x} (f+g x)}+\frac {2 (9 f+7 g) \text {arctanh}\left (\frac {\sqrt {2 f+3 g} \sqrt {6+5 x-6 x^2}}{\sqrt {g} (-3+2 x) \sqrt {2+3 x}}\right )}{\sqrt {2 f+3 g}}}{\sqrt {6} g^{5/2}} \] Input:

Integrate[(Sqrt[2 + 3*x]*Sqrt[1 + (5*x)/6 - x^2])/(f + g*x)^2,x]
 

Output:

((Sqrt[g]*(9*f - 2*g + 6*g*x)*Sqrt[6 + 5*x - 6*x^2])/(Sqrt[2 + 3*x]*(f + g 
*x)) + (2*(9*f + 7*g)*ArcTanh[(Sqrt[2*f + 3*g]*Sqrt[6 + 5*x - 6*x^2])/(Sqr 
t[g]*(-3 + 2*x)*Sqrt[2 + 3*x])])/Sqrt[2*f + 3*g])/(Sqrt[6]*g^(5/2))
 

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.15, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {1245, 87, 27, 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.

\(\displaystyle \int \frac {\sqrt {3 x+2} \sqrt {-x^2+\frac {5 x}{6}+1}}{(f+g x)^2} \, dx\)

\(\Big \downarrow \) 1245

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

\(\Big \downarrow \) 87

\(\displaystyle \frac {(9 f+7 g) \int \frac {\sqrt {3-2 x}}{\sqrt {6} (f+g x)}dx}{g (2 f+3 g)}+\frac {(3-2 x)^{3/2} (3 f-2 g)}{\sqrt {6} g (2 f+3 g) (f+g x)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {(9 f+7 g) \int \frac {\sqrt {3-2 x}}{f+g x}dx}{\sqrt {6} g (2 f+3 g)}+\frac {(3-2 x)^{3/2} (3 f-2 g)}{\sqrt {6} g (2 f+3 g) (f+g x)}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {(9 f+7 g) \left (\frac {(2 f+3 g) \int \frac {1}{\sqrt {3-2 x} (f+g x)}dx}{g}+\frac {2 \sqrt {3-2 x}}{g}\right )}{\sqrt {6} g (2 f+3 g)}+\frac {(3-2 x)^{3/2} (3 f-2 g)}{\sqrt {6} g (2 f+3 g) (f+g x)}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {(9 f+7 g) \left (\frac {2 \sqrt {3-2 x}}{g}-\frac {(2 f+3 g) \int \frac {1}{\frac {1}{2} (2 f+3 g)-\frac {1}{2} g (3-2 x)}d\sqrt {3-2 x}}{g}\right )}{\sqrt {6} g (2 f+3 g)}+\frac {(3-2 x)^{3/2} (3 f-2 g)}{\sqrt {6} g (2 f+3 g) (f+g x)}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {(9 f+7 g) \left (\frac {2 \sqrt {3-2 x}}{g}-\frac {2 \sqrt {2 f+3 g} \text {arctanh}\left (\frac {\sqrt {g} \sqrt {3-2 x}}{\sqrt {2 f+3 g}}\right )}{g^{3/2}}\right )}{\sqrt {6} g (2 f+3 g)}+\frac {(3-2 x)^{3/2} (3 f-2 g)}{\sqrt {6} g (2 f+3 g) (f+g x)}\)

Input:

Int[(Sqrt[2 + 3*x]*Sqrt[1 + (5*x)/6 - x^2])/(f + g*x)^2,x]
 

Output:

((3*f - 2*g)*(3 - 2*x)^(3/2))/(Sqrt[6]*g*(2*f + 3*g)*(f + g*x)) + ((9*f + 
7*g)*((2*Sqrt[3 - 2*x])/g - (2*Sqrt[2*f + 3*g]*ArcTanh[(Sqrt[g]*Sqrt[3 - 2 
*x])/Sqrt[2*f + 3*g]])/g^(3/2)))/(Sqrt[6]*g*(2*f + 3*g))
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

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 1245

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(181\) vs. \(2(89)=178\).

Time = 1.33 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.65

method	result	size
risch	\(-\frac {6 \left (2 x -3\right ) \sqrt {\frac {-36 x^{2}+30 x +36}{3 x +2}}\, \sqrt {3 x +2}}{g^{2} \sqrt {-12 x +18}\, \sqrt {-36 x^{2}+30 x +36}}+\frac {\left (\frac {2 \left (-3 f +2 g \right ) \sqrt {-12 x +18}}{g \left (-12 x +18\right )-12 f -18 g}-\frac {\left (9 f +7 g \right ) \sqrt {6}\, \operatorname {arctanh}\left (\frac {g \sqrt {-12 x +18}\, \sqrt {6}}{6 \sqrt {g \left (2 f +3 g \right )}}\right )}{3 \sqrt {g \left (2 f +3 g \right )}}\right ) \sqrt {\frac {-36 x^{2}+30 x +36}{3 x +2}}\, \sqrt {3 x +2}}{g^{2} \sqrt {-36 x^{2}+30 x +36}}\)	\(182\)
default	\(\frac {\left (-18 \,\operatorname {arctanh}\left (\frac {g \sqrt {-12 x +18}\, \sqrt {6}}{6 \sqrt {g \left (2 f +3 g \right )}}\right ) f g x -14 \,\operatorname {arctanh}\left (\frac {g \sqrt {-12 x +18}\, \sqrt {6}}{6 \sqrt {g \left (2 f +3 g \right )}}\right ) g^{2} x +6 \sqrt {g \left (2 f +3 g \right )}\, \sqrt {3-2 x}\, g x -18 \,\operatorname {arctanh}\left (\frac {g \sqrt {-12 x +18}\, \sqrt {6}}{6 \sqrt {g \left (2 f +3 g \right )}}\right ) f^{2}-14 \,\operatorname {arctanh}\left (\frac {g \sqrt {-12 x +18}\, \sqrt {6}}{6 \sqrt {g \left (2 f +3 g \right )}}\right ) f g +9 \sqrt {g \left (2 f +3 g \right )}\, \sqrt {3-2 x}\, f -2 \sqrt {g \left (2 f +3 g \right )}\, \sqrt {3-2 x}\, g \right ) \sqrt {-6 x^{2}+5 x +6}\, \sqrt {6}}{6 \sqrt {3 x +2}\, \sqrt {3-2 x}\, \sqrt {g \left (2 f +3 g \right )}\, g^{2} \left (g x +f \right )}\)	\(238\)

Input:

int(1/6*(3*x+2)^(1/2)*(-36*x^2+30*x+36)^(1/2)/(g*x+f)^2,x,method=_RETURNVE 
RBOSE)
 

Output:

-6*(2*x-3)/g^2/(-12*x+18)^(1/2)*((-36*x^2+30*x+36)/(3*x+2))^(1/2)*(3*x+2)^ 
(1/2)/(-36*x^2+30*x+36)^(1/2)+1/g^2*(2*(-3*f+2*g)*(-12*x+18)^(1/2)/(g*(-12 
*x+18)-12*f-18*g)-1/3*(9*f+7*g)*6^(1/2)/(g*(2*f+3*g))^(1/2)*arctanh(1/6*g* 
(-12*x+18)^(1/2)*6^(1/2)/(g*(2*f+3*g))^(1/2)))*((-36*x^2+30*x+36)/(3*x+2)) 
^(1/2)*(3*x+2)^(1/2)/(-36*x^2+30*x+36)^(1/2)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 189 vs. \(2 (89) = 178\).

Time = 0.08 (sec) , antiderivative size = 396, normalized size of antiderivative = 3.60 \[ \int \frac {\sqrt {2+3 x} \sqrt {1+\frac {5 x}{6}-x^2}}{(f+g x)^2} \, dx=\left [\frac {{\left (6 \, g x + 9 \, f - 2 \, g\right )} \sqrt {-36 \, x^{2} + 30 \, x + 36} \sqrt {3 \, x + 2} + \frac {3 \, \sqrt {\frac {2}{3}} {\left (3 \, {\left (9 \, f g + 7 \, g^{2}\right )} x^{2} + 18 \, f^{2} + 14 \, f g + {\left (27 \, f^{2} + 39 \, f g + 14 \, g^{2}\right )} x\right )} \log \left (-\frac {6 \, g x^{2} + \sqrt {\frac {2}{3}} \sqrt {2 \, f g + 3 \, g^{2}} \sqrt {-36 \, x^{2} + 30 \, x + 36} \sqrt {3 \, x + 2} - 2 \, {\left (3 \, f + 7 \, g\right )} x - 4 \, f - 12 \, g}{3 \, g x^{2} + {\left (3 \, f + 2 \, g\right )} x + 2 \, f}\right )}{\sqrt {2 \, f g + 3 \, g^{2}}}}{6 \, {\left (3 \, g^{3} x^{2} + 2 \, f g^{2} + {\left (3 \, f g^{2} + 2 \, g^{3}\right )} x\right )}}, -\frac {6 \, \sqrt {\frac {2}{3}} {\left (3 \, {\left (9 \, f g + 7 \, g^{2}\right )} x^{2} + 18 \, f^{2} + 14 \, f g + {\left (27 \, f^{2} + 39 \, f g + 14 \, g^{2}\right )} x\right )} \sqrt {-\frac {1}{2 \, f g + 3 \, g^{2}}} \arctan \left (\frac {\sqrt {\frac {2}{3}} \sqrt {-36 \, x^{2} + 30 \, x + 36} {\left (2 \, f + 3 \, g\right )} \sqrt {3 \, x + 2} \sqrt {-\frac {1}{2 \, f g + 3 \, g^{2}}}}{2 \, {\left (6 \, x^{2} - 5 \, x - 6\right )}}\right ) - {\left (6 \, g x + 9 \, f - 2 \, g\right )} \sqrt {-36 \, x^{2} + 30 \, x + 36} \sqrt {3 \, x + 2}}{6 \, {\left (3 \, g^{3} x^{2} + 2 \, f g^{2} + {\left (3 \, f g^{2} + 2 \, g^{3}\right )} x\right )}}\right ] \] Input:

integrate(1/6*(2+3*x)^(1/2)*(-36*x^2+30*x+36)^(1/2)/(g*x+f)^2,x, algorithm 
="fricas")
 

Output:

[1/6*((6*g*x + 9*f - 2*g)*sqrt(-36*x^2 + 30*x + 36)*sqrt(3*x + 2) + 3*sqrt 
(2/3)*(3*(9*f*g + 7*g^2)*x^2 + 18*f^2 + 14*f*g + (27*f^2 + 39*f*g + 14*g^2 
)*x)*log(-(6*g*x^2 + sqrt(2/3)*sqrt(2*f*g + 3*g^2)*sqrt(-36*x^2 + 30*x + 3 
6)*sqrt(3*x + 2) - 2*(3*f + 7*g)*x - 4*f - 12*g)/(3*g*x^2 + (3*f + 2*g)*x 
+ 2*f))/sqrt(2*f*g + 3*g^2))/(3*g^3*x^2 + 2*f*g^2 + (3*f*g^2 + 2*g^3)*x), 
-1/6*(6*sqrt(2/3)*(3*(9*f*g + 7*g^2)*x^2 + 18*f^2 + 14*f*g + (27*f^2 + 39* 
f*g + 14*g^2)*x)*sqrt(-1/(2*f*g + 3*g^2))*arctan(1/2*sqrt(2/3)*sqrt(-36*x^ 
2 + 30*x + 36)*(2*f + 3*g)*sqrt(3*x + 2)*sqrt(-1/(2*f*g + 3*g^2))/(6*x^2 - 
 5*x - 6)) - (6*g*x + 9*f - 2*g)*sqrt(-36*x^2 + 30*x + 36)*sqrt(3*x + 2))/ 
(3*g^3*x^2 + 2*f*g^2 + (3*f*g^2 + 2*g^3)*x)]
 

Sympy [F]

\[ \int \frac {\sqrt {2+3 x} \sqrt {1+\frac {5 x}{6}-x^2}}{(f+g x)^2} \, dx=\frac {\sqrt {6} \int \frac {\sqrt {3 x + 2} \sqrt {- 6 x^{2} + 5 x + 6}}{f^{2} + 2 f g x + g^{2} x^{2}}\, dx}{6} \] Input:

integrate(1/6*(2+3*x)**(1/2)*(-36*x**2+30*x+36)**(1/2)/(g*x+f)**2,x)
 

Output:

sqrt(6)*Integral(sqrt(3*x + 2)*sqrt(-6*x**2 + 5*x + 6)/(f**2 + 2*f*g*x + g 
**2*x**2), x)/6
 

Maxima [F]

\[ \int \frac {\sqrt {2+3 x} \sqrt {1+\frac {5 x}{6}-x^2}}{(f+g x)^2} \, dx=\int { \frac {\sqrt {-36 \, x^{2} + 30 \, x + 36} \sqrt {3 \, x + 2}}{6 \, {\left (g x + f\right )}^{2}} \,d x } \] Input:

integrate(1/6*(2+3*x)^(1/2)*(-36*x^2+30*x+36)^(1/2)/(g*x+f)^2,x, algorithm 
="maxima")
                                                                                    
                                                                                    
 

Output:

1/6*integrate(sqrt(-36*x^2 + 30*x + 36)*sqrt(3*x + 2)/(g*x + f)^2, x)
 

Giac [A] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.95 \[ \int \frac {\sqrt {2+3 x} \sqrt {1+\frac {5 x}{6}-x^2}}{(f+g x)^2} \, dx=\frac {1}{3} \, \sqrt {6} {\left (\frac {{\left (9 \, f + 7 \, g\right )} \arctan \left (\frac {g \sqrt {-2 \, x + 3}}{\sqrt {-2 \, f g - 3 \, g^{2}}}\right )}{\sqrt {-2 \, f g - 3 \, g^{2}} g^{2}} + \frac {3 \, \sqrt {-2 \, x + 3}}{g^{2}} + \frac {3 \, f \sqrt {-2 \, x + 3} - 2 \, g \sqrt {-2 \, x + 3}}{{\left (g {\left (2 \, x - 3\right )} + 2 \, f + 3 \, g\right )} g^{2}}\right )} \] Input:

integrate(1/6*(2+3*x)^(1/2)*(-36*x^2+30*x+36)^(1/2)/(g*x+f)^2,x, algorithm 
="giac")
 

Output:

1/3*sqrt(6)*((9*f + 7*g)*arctan(g*sqrt(-2*x + 3)/sqrt(-2*f*g - 3*g^2))/(sq 
rt(-2*f*g - 3*g^2)*g^2) + 3*sqrt(-2*x + 3)/g^2 + (3*f*sqrt(-2*x + 3) - 2*g 
*sqrt(-2*x + 3))/((g*(2*x - 3) + 2*f + 3*g)*g^2))
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {2+3 x} \sqrt {1+\frac {5 x}{6}-x^2}}{(f+g x)^2} \, dx=\int \frac {\sqrt {3\,x+2}\,\sqrt {-36\,x^2+30\,x+36}}{6\,{\left (f+g\,x\right )}^2} \,d x \] Input:

int(((3*x + 2)^(1/2)*(30*x - 36*x^2 + 36)^(1/2))/(6*(f + g*x)^2),x)
 

Output:

int(((3*x + 2)^(1/2)*(30*x - 36*x^2 + 36)^(1/2))/(6*(f + g*x)^2), x)
 

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 243, normalized size of antiderivative = 2.21 \[ \int \frac {\sqrt {2+3 x} \sqrt {1+\frac {5 x}{6}-x^2}}{(f+g x)^2} \, dx=\frac {\sqrt {6}\, \left (-18 \sqrt {g}\, \sqrt {-2 f -3 g}\, \mathit {atan} \left (\frac {\sqrt {-2 x +3}\, g}{\sqrt {g}\, \sqrt {-2 f -3 g}}\right ) f^{2}-18 \sqrt {g}\, \sqrt {-2 f -3 g}\, \mathit {atan} \left (\frac {\sqrt {-2 x +3}\, g}{\sqrt {g}\, \sqrt {-2 f -3 g}}\right ) f g x -14 \sqrt {g}\, \sqrt {-2 f -3 g}\, \mathit {atan} \left (\frac {\sqrt {-2 x +3}\, g}{\sqrt {g}\, \sqrt {-2 f -3 g}}\right ) f g -14 \sqrt {g}\, \sqrt {-2 f -3 g}\, \mathit {atan} \left (\frac {\sqrt {-2 x +3}\, g}{\sqrt {g}\, \sqrt {-2 f -3 g}}\right ) g^{2} x +18 \sqrt {-2 x +3}\, f^{2} g +12 \sqrt {-2 x +3}\, f \,g^{2} x +23 \sqrt {-2 x +3}\, f \,g^{2}+18 \sqrt {-2 x +3}\, g^{3} x -6 \sqrt {-2 x +3}\, g^{3}\right )}{6 g^{3} \left (2 f g x +3 g^{2} x +2 f^{2}+3 f g \right )} \] Input:

int(1/6*(2+3*x)^(1/2)*(-36*x^2+30*x+36)^(1/2)/(g*x+f)^2,x)
 

Output:

(sqrt(6)*( - 18*sqrt(g)*sqrt( - 2*f - 3*g)*atan((sqrt( - 2*x + 3)*g)/(sqrt 
(g)*sqrt( - 2*f - 3*g)))*f**2 - 18*sqrt(g)*sqrt( - 2*f - 3*g)*atan((sqrt( 
- 2*x + 3)*g)/(sqrt(g)*sqrt( - 2*f - 3*g)))*f*g*x - 14*sqrt(g)*sqrt( - 2*f 
 - 3*g)*atan((sqrt( - 2*x + 3)*g)/(sqrt(g)*sqrt( - 2*f - 3*g)))*f*g - 14*s 
qrt(g)*sqrt( - 2*f - 3*g)*atan((sqrt( - 2*x + 3)*g)/(sqrt(g)*sqrt( - 2*f - 
 3*g)))*g**2*x + 18*sqrt( - 2*x + 3)*f**2*g + 12*sqrt( - 2*x + 3)*f*g**2*x 
 + 23*sqrt( - 2*x + 3)*f*g**2 + 18*sqrt( - 2*x + 3)*g**3*x - 6*sqrt( - 2*x 
 + 3)*g**3))/(6*g**3*(2*f**2 + 2*f*g*x + 3*f*g + 3*g**2*x))