\(\int \frac {\sqrt {2+3 x} \sqrt {1+\frac {5 x}{6}-x^2}}{(f+g x)^{9/2}} \, dx\) [308]

Optimal result

Integrand size = 35, antiderivative size = 145 \[ \int \frac {\sqrt {2+3 x} \sqrt {1+\frac {5 x}{6}-x^2}}{(f+g x)^{9/2}} \, dx=\frac {\sqrt {\frac {2}{3}} (3 f-2 g) (3-2 x)^{3/2}}{7 g (2 f+3 g) (f+g x)^{7/2}}-\frac {\sqrt {\frac {2}{3}} (18 f+79 g) (3-2 x)^{3/2}}{35 g (2 f+3 g)^2 (f+g x)^{5/2}}-\frac {4 \sqrt {\frac {2}{3}} (18 f+79 g) (3-2 x)^{3/2}}{105 g (2 f+3 g)^3 (f+g x)^{3/2}} \] Output:

1/21*6^(1/2)*(3*f-2*g)*(3-2*x)^(3/2)/g/(2*f+3*g)/(g*x+f)^(7/2)-1/105*6^(1/ 
2)*(18*f+79*g)*(3-2*x)^(3/2)/g/(2*f+3*g)^2/(g*x+f)^(5/2)-4/315*6^(1/2)*(18 
*f+79*g)*(3-2*x)^(3/2)/g/(2*f+3*g)^3/(g*x+f)^(3/2)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 10.10 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.63 \[ \int \frac {\sqrt {2+3 x} \sqrt {1+\frac {5 x}{6}-x^2}}{(f+g x)^{9/2}} \, dx=\frac {(-3+2 x) \sqrt {4+\frac {10 x}{3}-4 x^2} \left (28 f^2 (19+9 x)+2 f g \left (333+634 x+36 x^2\right )+g^2 \left (270+711 x+316 x^2\right )\right )}{105 (2 f+3 g)^3 \sqrt {2+3 x} (f+g x)^{7/2}} \] Input:

Integrate[(Sqrt[2 + 3*x]*Sqrt[1 + (5*x)/6 - x^2])/(f + g*x)^(9/2),x]
 

Output:

((-3 + 2*x)*Sqrt[4 + (10*x)/3 - 4*x^2]*(28*f^2*(19 + 9*x) + 2*f*g*(333 + 6 
34*x + 36*x^2) + g^2*(270 + 711*x + 316*x^2)))/(105*(2*f + 3*g)^3*Sqrt[2 + 
 3*x]*(f + g*x)^(7/2))
 

Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.97, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {1245, 87, 27, 55, 48}

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.

\(\displaystyle \int \frac {\sqrt {3 x+2} \sqrt {-x^2+\frac {5 x}{6}+1}}{(f+g x)^{9/2}} \, dx\)

\(\Big \downarrow \) 1245

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

\(\Big \downarrow \) 87

\(\displaystyle \frac {(18 f+79 g) \int \frac {\sqrt {3-2 x}}{\sqrt {6} (f+g x)^{7/2}}dx}{7 g (2 f+3 g)}+\frac {\sqrt {\frac {2}{3}} (3-2 x)^{3/2} (3 f-2 g)}{7 g (2 f+3 g) (f+g x)^{7/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {(18 f+79 g) \int \frac {\sqrt {3-2 x}}{(f+g x)^{7/2}}dx}{7 \sqrt {6} g (2 f+3 g)}+\frac {\sqrt {\frac {2}{3}} (3-2 x)^{3/2} (3 f-2 g)}{7 g (2 f+3 g) (f+g x)^{7/2}}\)

\(\Big \downarrow \) 55

\(\displaystyle \frac {(18 f+79 g) \left (\frac {4 \int \frac {\sqrt {3-2 x}}{(f+g x)^{5/2}}dx}{5 (2 f+3 g)}-\frac {2 (3-2 x)^{3/2}}{5 (2 f+3 g) (f+g x)^{5/2}}\right )}{7 \sqrt {6} g (2 f+3 g)}+\frac {\sqrt {\frac {2}{3}} (3-2 x)^{3/2} (3 f-2 g)}{7 g (2 f+3 g) (f+g x)^{7/2}}\)

\(\Big \downarrow \) 48

\(\displaystyle \frac {\sqrt {\frac {2}{3}} (3-2 x)^{3/2} (3 f-2 g)}{7 g (2 f+3 g) (f+g x)^{7/2}}+\frac {(18 f+79 g) \left (-\frac {8 (3-2 x)^{3/2}}{15 (2 f+3 g)^2 (f+g x)^{3/2}}-\frac {2 (3-2 x)^{3/2}}{5 (2 f+3 g) (f+g x)^{5/2}}\right )}{7 \sqrt {6} g (2 f+3 g)}\)

Input:

Int[(Sqrt[2 + 3*x]*Sqrt[1 + (5*x)/6 - x^2])/(f + g*x)^(9/2),x]
 

Output:

(Sqrt[2/3]*(3*f - 2*g)*(3 - 2*x)^(3/2))/(7*g*(2*f + 3*g)*(f + g*x)^(7/2)) 
+ ((18*f + 79*g)*((-2*(3 - 2*x)^(3/2))/(5*(2*f + 3*g)*(f + g*x)^(5/2)) - ( 
8*(3 - 2*x)^(3/2))/(15*(2*f + 3*g)^2*(f + g*x)^(3/2))))/(7*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 48

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp 
[(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{ 
a, b, c, d, m, n}, x] && EqQ[m + n + 2, 0] && NeQ[m, -1]
 

rule 55

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(S 
implify[m + n + 2]/((b*c - a*d)*(m + 1)))   Int[(a + b*x)^Simplify[m + 1]*( 
c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && ILtQ[Simplify[m + n + 
 2], 0] && NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[ 
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (SumSimplerQ[m, 1] ||  !SumSimp 
lerQ[n, 1])
 

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 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 [A] (verified)

Time = 1.52 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.62

method	result	size
default	\(\frac {\sqrt {-36 x^{2}+30 x +36}\, \left (2 x -3\right ) \left (72 f g \,x^{2}+316 g^{2} x^{2}+252 f^{2} x +1268 f g x +711 g^{2} x +532 f^{2}+666 f g +270 g^{2}\right )}{315 \sqrt {3 x +2}\, \left (g x +f \right )^{\frac {7}{2}} \left (2 f +3 g \right )^{3}}\)	\(90\)
gosper	\(\frac {\left (2 x -3\right ) \left (72 f g \,x^{2}+316 g^{2} x^{2}+252 f^{2} x +1268 f g x +711 g^{2} x +532 f^{2}+666 f g +270 g^{2}\right ) \sqrt {-36 x^{2}+30 x +36}}{315 \left (g x +f \right )^{\frac {7}{2}} \left (8 f^{3}+36 f^{2} g +54 f \,g^{2}+27 g^{3}\right ) \sqrt {3 x +2}}\)	\(106\)
orering	\(\frac {\left (2 x -3\right ) \left (72 f g \,x^{2}+316 g^{2} x^{2}+252 f^{2} x +1268 f g x +711 g^{2} x +532 f^{2}+666 f g +270 g^{2}\right ) \sqrt {-36 x^{2}+30 x +36}}{315 \left (g x +f \right )^{\frac {7}{2}} \left (8 f^{3}+36 f^{2} g +54 f \,g^{2}+27 g^{3}\right ) \sqrt {3 x +2}}\)	\(106\)

Input:

int(1/6*(3*x+2)^(1/2)*(-36*x^2+30*x+36)^(1/2)/(g*x+f)^(9/2),x,method=_RETU 
RNVERBOSE)
 

Output:

1/315/(3*x+2)^(1/2)*(-36*x^2+30*x+36)^(1/2)/(g*x+f)^(7/2)*(2*x-3)*(72*f*g* 
x^2+316*g^2*x^2+252*f^2*x+1268*f*g*x+711*g^2*x+532*f^2+666*f*g+270*g^2)/(2 
*f+3*g)^3
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 325 vs. \(2 (115) = 230\).

Time = 0.09 (sec) , antiderivative size = 325, normalized size of antiderivative = 2.24 \[ \int \frac {\sqrt {2+3 x} \sqrt {1+\frac {5 x}{6}-x^2}}{(f+g x)^{9/2}} \, dx=\frac {{\left (8 \, {\left (18 \, f g + 79 \, g^{2}\right )} x^{3} + 2 \, {\left (252 \, f^{2} + 1160 \, f g + 237 \, g^{2}\right )} x^{2} - 1596 \, f^{2} - 1998 \, f g - 810 \, g^{2} + {\left (308 \, f^{2} - 2472 \, f g - 1593 \, g^{2}\right )} x\right )} \sqrt {g x + f} \sqrt {-36 \, x^{2} + 30 \, x + 36} \sqrt {3 \, x + 2}}{315 \, {\left (16 \, f^{7} + 72 \, f^{6} g + 108 \, f^{5} g^{2} + 54 \, f^{4} g^{3} + 3 \, {\left (8 \, f^{3} g^{4} + 36 \, f^{2} g^{5} + 54 \, f g^{6} + 27 \, g^{7}\right )} x^{5} + 2 \, {\left (48 \, f^{4} g^{3} + 224 \, f^{3} g^{4} + 360 \, f^{2} g^{5} + 216 \, f g^{6} + 27 \, g^{7}\right )} x^{4} + 2 \, {\left (72 \, f^{5} g^{2} + 356 \, f^{4} g^{3} + 630 \, f^{3} g^{4} + 459 \, f^{2} g^{5} + 108 \, f g^{6}\right )} x^{3} + 12 \, {\left (8 \, f^{6} g + 44 \, f^{5} g^{2} + 90 \, f^{4} g^{3} + 81 \, f^{3} g^{4} + 27 \, f^{2} g^{5}\right )} x^{2} + {\left (24 \, f^{7} + 172 \, f^{6} g + 450 \, f^{5} g^{2} + 513 \, f^{4} g^{3} + 216 \, f^{3} g^{4}\right )} x\right )}} \] Input:

integrate(1/6*(2+3*x)^(1/2)*(-36*x^2+30*x+36)^(1/2)/(g*x+f)^(9/2),x, algor 
ithm="fricas")
 

Output:

1/315*(8*(18*f*g + 79*g^2)*x^3 + 2*(252*f^2 + 1160*f*g + 237*g^2)*x^2 - 15 
96*f^2 - 1998*f*g - 810*g^2 + (308*f^2 - 2472*f*g - 1593*g^2)*x)*sqrt(g*x 
+ f)*sqrt(-36*x^2 + 30*x + 36)*sqrt(3*x + 2)/(16*f^7 + 72*f^6*g + 108*f^5* 
g^2 + 54*f^4*g^3 + 3*(8*f^3*g^4 + 36*f^2*g^5 + 54*f*g^6 + 27*g^7)*x^5 + 2* 
(48*f^4*g^3 + 224*f^3*g^4 + 360*f^2*g^5 + 216*f*g^6 + 27*g^7)*x^4 + 2*(72* 
f^5*g^2 + 356*f^4*g^3 + 630*f^3*g^4 + 459*f^2*g^5 + 108*f*g^6)*x^3 + 12*(8 
*f^6*g + 44*f^5*g^2 + 90*f^4*g^3 + 81*f^3*g^4 + 27*f^2*g^5)*x^2 + (24*f^7 
+ 172*f^6*g + 450*f^5*g^2 + 513*f^4*g^3 + 216*f^3*g^4)*x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\sqrt {2+3 x} \sqrt {1+\frac {5 x}{6}-x^2}}{(f+g x)^{9/2}} \, dx=\text {Timed out} \] Input:

integrate(1/6*(2+3*x)**(1/2)*(-36*x**2+30*x+36)**(1/2)/(g*x+f)**(9/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\sqrt {2+3 x} \sqrt {1+\frac {5 x}{6}-x^2}}{(f+g x)^{9/2}} \, dx=\int { \frac {\sqrt {-36 \, x^{2} + 30 \, x + 36} \sqrt {3 \, x + 2}}{6 \, {\left (g x + f\right )}^{\frac {9}{2}}} \,d x } \] Input:

integrate(1/6*(2+3*x)^(1/2)*(-36*x^2+30*x+36)^(1/2)/(g*x+f)^(9/2),x, algor 
ithm="maxima")
 

Output:

1/6*integrate(sqrt(-36*x^2 + 30*x + 36)*sqrt(3*x + 2)/(g*x + f)^(9/2), x)
 

Giac [A] (verification not implemented)

Time = 0.73 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.50 \[ \int \frac {\sqrt {2+3 x} \sqrt {1+\frac {5 x}{6}-x^2}}{(f+g x)^{9/2}} \, dx=\frac {4 \, \sqrt {6} {\left ({\left (\frac {2 \, {\left (18 \, \sqrt {2} f g^{4} + 79 \, \sqrt {2} g^{5}\right )} {\left (2 \, x - 3\right )}}{8 \, f^{3} g^{3} + 36 \, f^{2} g^{4} + 54 \, f g^{5} + 27 \, g^{6}} + \frac {7 \, {\left (36 \, \sqrt {2} f^{2} g^{3} + 212 \, \sqrt {2} f g^{4} + 237 \, \sqrt {2} g^{5}\right )}}{8 \, f^{3} g^{3} + 36 \, f^{2} g^{4} + 54 \, f g^{5} + 27 \, g^{6}}\right )} {\left (2 \, x - 3\right )} + \frac {455 \, {\left (4 \, \sqrt {2} f^{2} g^{3} + 12 \, \sqrt {2} f g^{4} + 9 \, \sqrt {2} g^{5}\right )}}{8 \, f^{3} g^{3} + 36 \, f^{2} g^{4} + 54 \, f g^{5} + 27 \, g^{6}}\right )} {\left (2 \, x - 3\right )} \sqrt {-2 \, x + 3}}{315 \, {\left (g {\left (2 \, x - 3\right )} + 2 \, f + 3 \, g\right )}^{\frac {7}{2}}} \] Input:

integrate(1/6*(2+3*x)^(1/2)*(-36*x^2+30*x+36)^(1/2)/(g*x+f)^(9/2),x, algor 
ithm="giac")
 

Output:

4/315*sqrt(6)*((2*(18*sqrt(2)*f*g^4 + 79*sqrt(2)*g^5)*(2*x - 3)/(8*f^3*g^3 
 + 36*f^2*g^4 + 54*f*g^5 + 27*g^6) + 7*(36*sqrt(2)*f^2*g^3 + 212*sqrt(2)*f 
*g^4 + 237*sqrt(2)*g^5)/(8*f^3*g^3 + 36*f^2*g^4 + 54*f*g^5 + 27*g^6))*(2*x 
 - 3) + 455*(4*sqrt(2)*f^2*g^3 + 12*sqrt(2)*f*g^4 + 9*sqrt(2)*g^5)/(8*f^3* 
g^3 + 36*f^2*g^4 + 54*f*g^5 + 27*g^6))*(2*x - 3)*sqrt(-2*x + 3)/(g*(2*x - 
3) + 2*f + 3*g)^(7/2)
 

Mupad [B] (verification not implemented)

Time = 11.87 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.73 \[ \int \frac {\sqrt {2+3 x} \sqrt {1+\frac {5 x}{6}-x^2}}{(f+g x)^{9/2}} \, dx=-\frac {\sqrt {-36\,x^2+30\,x+36}\,\left (\frac {\sqrt {3\,x+2}\,\left (1596\,f^2+1998\,f\,g+810\,g^2\right )}{945\,g^3\,{\left (2\,f+3\,g\right )}^3}+\frac {x\,\sqrt {3\,x+2}\,\left (-308\,f^2+2472\,f\,g+1593\,g^2\right )}{945\,g^3\,{\left (2\,f+3\,g\right )}^3}-\frac {x^2\,\sqrt {3\,x+2}\,\left (504\,f^2+2320\,f\,g+474\,g^2\right )}{945\,g^3\,{\left (2\,f+3\,g\right )}^3}-\frac {8\,x^3\,\sqrt {3\,x+2}\,\left (18\,f+79\,g\right )}{945\,g^2\,{\left (2\,f+3\,g\right )}^3}\right )}{x^4\,\sqrt {f+g\,x}+\frac {2\,f^3\,\sqrt {f+g\,x}}{3\,g^3}+\frac {x^3\,\sqrt {f+g\,x}\,\left (9\,f+2\,g\right )}{3\,g}+\frac {f^2\,x\,\sqrt {f+g\,x}\,\left (f+2\,g\right )}{g^3}+\frac {f\,x^2\,\sqrt {f+g\,x}\,\left (3\,f+2\,g\right )}{g^2}} \] Input:

int(((3*x + 2)^(1/2)*(30*x - 36*x^2 + 36)^(1/2))/(6*(f + g*x)^(9/2)),x)
 

Output:

-((30*x - 36*x^2 + 36)^(1/2)*(((3*x + 2)^(1/2)*(1998*f*g + 1596*f^2 + 810* 
g^2))/(945*g^3*(2*f + 3*g)^3) + (x*(3*x + 2)^(1/2)*(2472*f*g - 308*f^2 + 1 
593*g^2))/(945*g^3*(2*f + 3*g)^3) - (x^2*(3*x + 2)^(1/2)*(2320*f*g + 504*f 
^2 + 474*g^2))/(945*g^3*(2*f + 3*g)^3) - (8*x^3*(3*x + 2)^(1/2)*(18*f + 79 
*g))/(945*g^2*(2*f + 3*g)^3)))/(x^4*(f + g*x)^(1/2) + (2*f^3*(f + g*x)^(1/ 
2))/(3*g^3) + (x^3*(f + g*x)^(1/2)*(9*f + 2*g))/(3*g) + (f^2*x*(f + g*x)^( 
1/2)*(f + 2*g))/g^3 + (f*x^2*(f + g*x)^(1/2)*(3*f + 2*g))/g^2)
 

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.61 \[ \int \frac {\sqrt {2+3 x} \sqrt {1+\frac {5 x}{6}-x^2}}{(f+g x)^{9/2}} \, dx=\frac {\sqrt {-2 x +3}\, \sqrt {6}\, \left (144 f g \,x^{3}+632 g^{2} x^{3}+504 f^{2} x^{2}+2320 f g \,x^{2}+474 g^{2} x^{2}+308 f^{2} x -2472 f g x -1593 g^{2} x -1596 f^{2}-1998 f g -810 g^{2}\right )}{315 \sqrt {g x +f}\, \left (8 f^{3} g^{3} x^{3}+36 f^{2} g^{4} x^{3}+54 f \,g^{5} x^{3}+27 g^{6} x^{3}+24 f^{4} g^{2} x^{2}+108 f^{3} g^{3} x^{2}+162 f^{2} g^{4} x^{2}+81 f \,g^{5} x^{2}+24 f^{5} g x +108 f^{4} g^{2} x +162 f^{3} g^{3} x +81 f^{2} g^{4} x +8 f^{6}+36 f^{5} g +54 f^{4} g^{2}+27 f^{3} g^{3}\right )} \] Input:

int(1/6*(2+3*x)^(1/2)*(-36*x^2+30*x+36)^(1/2)/(g*x+f)^(9/2),x)
 

Output:

(sqrt( - 2*x + 3)*sqrt(6)*(504*f**2*x**2 + 308*f**2*x - 1596*f**2 + 144*f* 
g*x**3 + 2320*f*g*x**2 - 2472*f*g*x - 1998*f*g + 632*g**2*x**3 + 474*g**2* 
x**2 - 1593*g**2*x - 810*g**2))/(315*sqrt(f + g*x)*(8*f**6 + 24*f**5*g*x + 
 36*f**5*g + 24*f**4*g**2*x**2 + 108*f**4*g**2*x + 54*f**4*g**2 + 8*f**3*g 
**3*x**3 + 108*f**3*g**3*x**2 + 162*f**3*g**3*x + 27*f**3*g**3 + 36*f**2*g 
**4*x**3 + 162*f**2*g**4*x**2 + 81*f**2*g**4*x + 54*f*g**5*x**3 + 81*f*g** 
5*x**2 + 27*g**6*x**3))