\(\int \frac {\sqrt {2+3 x} \sqrt {4-9 x^2}}{(f+g x)^{9/2}} \, dx\) [99]

Optimal result

Integrand size = 30, antiderivative size = 124 \[ \int \frac {\sqrt {2+3 x} \sqrt {4-9 x^2}}{(f+g x)^{9/2}} \, dx=\frac {2 (3 f-2 g) (2-3 x)^{3/2}}{7 g (3 f+2 g) (f+g x)^{7/2}}-\frac {6 (9 f+22 g) (2-3 x)^{3/2}}{35 g (3 f+2 g)^2 (f+g x)^{5/2}}-\frac {12 (9 f+22 g) (2-3 x)^{3/2}}{35 g (3 f+2 g)^3 (f+g x)^{3/2}} \] Output:

2/7*(3*f-2*g)*(2-3*x)^(3/2)/g/(3*f+2*g)/(g*x+f)^(7/2)-6/35*(9*f+22*g)*(2-3 
*x)^(3/2)/g/(3*f+2*g)^2/(g*x+f)^(5/2)-12/35*(9*f+22*g)*(2-3*x)^(3/2)/g/(3* 
f+2*g)^3/(g*x+f)^(3/2)
 

Mathematica [A] (verified)

Time = 10.06 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.70 \[ \int \frac {\sqrt {2+3 x} \sqrt {4-9 x^2}}{(f+g x)^{9/2}} \, dx=\frac {2 (-2+3 x) \sqrt {4-9 x^2} \left (21 f^2 (14+9 x)+6 f g \left (32+86 x+9 x^2\right )+4 g^2 \left (10+33 x+33 x^2\right )\right )}{35 (3 f+2 g)^3 \sqrt {2+3 x} (f+g x)^{7/2}} \] Input:

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

Output:

(2*(-2 + 3*x)*Sqrt[4 - 9*x^2]*(21*f^2*(14 + 9*x) + 6*f*g*(32 + 86*x + 9*x^ 
2) + 4*g^2*(10 + 33*x + 33*x^2)))/(35*(3*f + 2*g)^3*Sqrt[2 + 3*x]*(f + g*x 
)^(7/2))
 

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.03, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {639, 87, 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.

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

\(\Big \downarrow \) 639

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

\(\Big \downarrow \) 87

\(\displaystyle \frac {3 (9 f+22 g) \int \frac {\sqrt {2-3 x}}{(f+g x)^{7/2}}dx}{7 g (3 f+2 g)}+\frac {2 (2-3 x)^{3/2} (3 f-2 g)}{7 g (3 f+2 g) (f+g x)^{7/2}}\)

\(\Big \downarrow \) 55

\(\displaystyle \frac {3 (9 f+22 g) \left (\frac {6 \int \frac {\sqrt {2-3 x}}{(f+g x)^{5/2}}dx}{5 (3 f+2 g)}-\frac {2 (2-3 x)^{3/2}}{5 (3 f+2 g) (f+g x)^{5/2}}\right )}{7 g (3 f+2 g)}+\frac {2 (2-3 x)^{3/2} (3 f-2 g)}{7 g (3 f+2 g) (f+g x)^{7/2}}\)

\(\Big \downarrow \) 48

\(\displaystyle \frac {2 (2-3 x)^{3/2} (3 f-2 g)}{7 g (3 f+2 g) (f+g x)^{7/2}}+\frac {3 (9 f+22 g) \left (-\frac {4 (2-3 x)^{3/2}}{5 (3 f+2 g)^2 (f+g x)^{3/2}}-\frac {2 (2-3 x)^{3/2}}{5 (3 f+2 g) (f+g x)^{5/2}}\right )}{7 g (3 f+2 g)}\)

Input:

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

Output:

(2*(3*f - 2*g)*(2 - 3*x)^(3/2))/(7*g*(3*f + 2*g)*(f + g*x)^(7/2)) + (3*(9* 
f + 22*g)*((-2*(2 - 3*x)^(3/2))/(5*(3*f + 2*g)*(f + g*x)^(5/2)) - (4*(2 - 
3*x)^(3/2))/(5*(3*f + 2*g)^2*(f + g*x)^(3/2))))/(7*g*(3*f + 2*g))
 

Defintions of rubi rules used

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 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 = 87, normalized size of antiderivative = 0.70

method	result	size
default	\(\frac {2 \sqrt {-9 x^{2}+4}\, \left (-2+3 x \right ) \left (54 f g \,x^{2}+132 g^{2} x^{2}+189 f^{2} x +516 f g x +132 g^{2} x +294 f^{2}+192 f g +40 g^{2}\right )}{35 \sqrt {3 x +2}\, \left (g x +f \right )^{\frac {7}{2}} \left (3 f +2 g \right )^{3}}\)	\(87\)
gosper	\(\frac {2 \left (-2+3 x \right ) \left (54 f g \,x^{2}+132 g^{2} x^{2}+189 f^{2} x +516 f g x +132 g^{2} x +294 f^{2}+192 f g +40 g^{2}\right ) \sqrt {-9 x^{2}+4}}{35 \left (g x +f \right )^{\frac {7}{2}} \left (27 f^{3}+54 f^{2} g +36 f \,g^{2}+8 g^{3}\right ) \sqrt {3 x +2}}\)	\(103\)
orering	\(\frac {2 \left (-2+3 x \right ) \left (54 f g \,x^{2}+132 g^{2} x^{2}+189 f^{2} x +516 f g x +132 g^{2} x +294 f^{2}+192 f g +40 g^{2}\right ) \sqrt {-9 x^{2}+4}}{35 \left (g x +f \right )^{\frac {7}{2}} \left (27 f^{3}+54 f^{2} g +36 f \,g^{2}+8 g^{3}\right ) \sqrt {3 x +2}}\)	\(103\)

Input:

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

Output:

2/35/(3*x+2)^(1/2)*(-9*x^2+4)^(1/2)/(g*x+f)^(7/2)*(-2+3*x)*(54*f*g*x^2+132 
*g^2*x^2+189*f^2*x+516*f*g*x+132*g^2*x+294*f^2+192*f*g+40*g^2)/(3*f+2*g)^3
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 323 vs. \(2 (106) = 212\).

Time = 0.09 (sec) , antiderivative size = 323, normalized size of antiderivative = 2.60 \[ \int \frac {\sqrt {2+3 x} \sqrt {4-9 x^2}}{(f+g x)^{9/2}} \, dx=\frac {2 \, {\left (18 \, {\left (9 \, f g + 22 \, g^{2}\right )} x^{3} + 3 \, {\left (189 \, f^{2} + 480 \, f g + 44 \, g^{2}\right )} x^{2} - 588 \, f^{2} - 384 \, f g - 80 \, g^{2} + 24 \, {\left (21 \, f^{2} - 19 \, f g - 6 \, g^{2}\right )} x\right )} \sqrt {g x + f} \sqrt {-9 \, x^{2} + 4} \sqrt {3 \, x + 2}}{35 \, {\left (54 \, f^{7} + 108 \, f^{6} g + 72 \, f^{5} g^{2} + 16 \, f^{4} g^{3} + 3 \, {\left (27 \, f^{3} g^{4} + 54 \, f^{2} g^{5} + 36 \, f g^{6} + 8 \, g^{7}\right )} x^{5} + 2 \, {\left (162 \, f^{4} g^{3} + 351 \, f^{3} g^{4} + 270 \, f^{2} g^{5} + 84 \, f g^{6} + 8 \, g^{7}\right )} x^{4} + 2 \, {\left (243 \, f^{5} g^{2} + 594 \, f^{4} g^{3} + 540 \, f^{3} g^{4} + 216 \, f^{2} g^{5} + 32 \, f g^{6}\right )} x^{3} + 12 \, {\left (27 \, f^{6} g + 81 \, f^{5} g^{2} + 90 \, f^{4} g^{3} + 44 \, f^{3} g^{4} + 8 \, f^{2} g^{5}\right )} x^{2} + {\left (81 \, f^{7} + 378 \, f^{6} g + 540 \, f^{5} g^{2} + 312 \, f^{4} g^{3} + 64 \, f^{3} g^{4}\right )} x\right )}} \] Input:

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

Output:

2/35*(18*(9*f*g + 22*g^2)*x^3 + 3*(189*f^2 + 480*f*g + 44*g^2)*x^2 - 588*f 
^2 - 384*f*g - 80*g^2 + 24*(21*f^2 - 19*f*g - 6*g^2)*x)*sqrt(g*x + f)*sqrt 
(-9*x^2 + 4)*sqrt(3*x + 2)/(54*f^7 + 108*f^6*g + 72*f^5*g^2 + 16*f^4*g^3 + 
 3*(27*f^3*g^4 + 54*f^2*g^5 + 36*f*g^6 + 8*g^7)*x^5 + 2*(162*f^4*g^3 + 351 
*f^3*g^4 + 270*f^2*g^5 + 84*f*g^6 + 8*g^7)*x^4 + 2*(243*f^5*g^2 + 594*f^4* 
g^3 + 540*f^3*g^4 + 216*f^2*g^5 + 32*f*g^6)*x^3 + 12*(27*f^6*g + 81*f^5*g^ 
2 + 90*f^4*g^3 + 44*f^3*g^4 + 8*f^2*g^5)*x^2 + (81*f^7 + 378*f^6*g + 540*f 
^5*g^2 + 312*f^4*g^3 + 64*f^3*g^4)*x)
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 215 vs. \(2 (106) = 212\).

Time = 0.22 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.73 \[ \int \frac {\sqrt {2+3 x} \sqrt {4-9 x^2}}{(f+g x)^{9/2}} \, dx=\frac {18 \, {\left ({\left (\frac {2 \, {\left (9 \, \sqrt {3} f g^{4} + 22 \, \sqrt {3} g^{5}\right )} {\left (3 \, x - 2\right )}}{27 \, f^{3} g^{3} + 54 \, f^{2} g^{4} + 36 \, f g^{5} + 8 \, g^{6}} + \frac {7 \, {\left (27 \, \sqrt {3} f^{2} g^{3} + 84 \, \sqrt {3} f g^{4} + 44 \, \sqrt {3} g^{5}\right )}}{27 \, f^{3} g^{3} + 54 \, f^{2} g^{4} + 36 \, f g^{5} + 8 \, g^{6}}\right )} {\left (3 \, x - 2\right )} + \frac {140 \, {\left (9 \, \sqrt {3} f^{2} g^{3} + 12 \, \sqrt {3} f g^{4} + 4 \, \sqrt {3} g^{5}\right )}}{27 \, f^{3} g^{3} + 54 \, f^{2} g^{4} + 36 \, f g^{5} + 8 \, g^{6}}\right )} {\left (3 \, x - 2\right )} \sqrt {-3 \, x + 2}}{35 \, {\left (g {\left (3 \, x - 2\right )} + 3 \, f + 2 \, g\right )}^{\frac {7}{2}}} \] Input:

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

Output:

18/35*((2*(9*sqrt(3)*f*g^4 + 22*sqrt(3)*g^5)*(3*x - 2)/(27*f^3*g^3 + 54*f^ 
2*g^4 + 36*f*g^5 + 8*g^6) + 7*(27*sqrt(3)*f^2*g^3 + 84*sqrt(3)*f*g^4 + 44* 
sqrt(3)*g^5)/(27*f^3*g^3 + 54*f^2*g^4 + 36*f*g^5 + 8*g^6))*(3*x - 2) + 140 
*(9*sqrt(3)*f^2*g^3 + 12*sqrt(3)*f*g^4 + 4*sqrt(3)*g^5)/(27*f^3*g^3 + 54*f 
^2*g^4 + 36*f*g^5 + 8*g^6))*(3*x - 2)*sqrt(-3*x + 2)/(g*(3*x - 2) + 3*f + 
2*g)^(7/2)
 

Mupad [B] (verification not implemented)

Time = 6.37 (sec) , antiderivative size = 262, normalized size of antiderivative = 2.11 \[ \int \frac {\sqrt {2+3 x} \sqrt {4-9 x^2}}{(f+g x)^{9/2}} \, dx=-\frac {\sqrt {f+g\,x}\,\left (\frac {\sqrt {3\,x+2}\,\sqrt {4-9\,x^2}\,\left (1176\,f^2+768\,f\,g+160\,g^2\right )}{105\,g^4\,{\left (3\,f+2\,g\right )}^3}+\frac {x\,\sqrt {3\,x+2}\,\sqrt {4-9\,x^2}\,\left (-1008\,f^2+912\,f\,g+288\,g^2\right )}{105\,g^4\,{\left (3\,f+2\,g\right )}^3}-\frac {x^2\,\sqrt {3\,x+2}\,\sqrt {4-9\,x^2}\,\left (1134\,f^2+2880\,f\,g+264\,g^2\right )}{105\,g^4\,{\left (3\,f+2\,g\right )}^3}-\frac {12\,x^3\,\sqrt {3\,x+2}\,\sqrt {4-9\,x^2}\,\left (9\,f+22\,g\right )}{35\,g^3\,{\left (3\,f+2\,g\right )}^3}\right )}{x^5+\frac {2\,f^4}{3\,g^4}+\frac {2\,x^4\,\left (6\,f+g\right )}{3\,g}+\frac {2\,f\,x^3\,\left (9\,f+4\,g\right )}{3\,g^2}+\frac {f^3\,x\,\left (3\,f+8\,g\right )}{3\,g^4}+\frac {4\,f^2\,x^2\,\left (f+g\right )}{g^3}} \] Input:

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

Output:

-((f + g*x)^(1/2)*(((3*x + 2)^(1/2)*(4 - 9*x^2)^(1/2)*(768*f*g + 1176*f^2 
+ 160*g^2))/(105*g^4*(3*f + 2*g)^3) + (x*(3*x + 2)^(1/2)*(4 - 9*x^2)^(1/2) 
*(912*f*g - 1008*f^2 + 288*g^2))/(105*g^4*(3*f + 2*g)^3) - (x^2*(3*x + 2)^ 
(1/2)*(4 - 9*x^2)^(1/2)*(2880*f*g + 1134*f^2 + 264*g^2))/(105*g^4*(3*f + 2 
*g)^3) - (12*x^3*(3*x + 2)^(1/2)*(4 - 9*x^2)^(1/2)*(9*f + 22*g))/(35*g^3*( 
3*f + 2*g)^3)))/(x^5 + (2*f^4)/(3*g^4) + (2*x^4*(6*f + g))/(3*g) + (2*f*x^ 
3*(9*f + 4*g))/(3*g^2) + (f^3*x*(3*f + 8*g))/(3*g^4) + (4*f^2*x^2*(f + g)) 
/g^3)
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.86 \[ \int \frac {\sqrt {2+3 x} \sqrt {4-9 x^2}}{(f+g x)^{9/2}} \, dx=\frac {2 \sqrt {-3 x +2}\, \left (162 f g \,x^{3}+396 g^{2} x^{3}+567 f^{2} x^{2}+1440 f g \,x^{2}+132 g^{2} x^{2}+504 f^{2} x -456 f g x -144 g^{2} x -588 f^{2}-384 f g -80 g^{2}\right )}{35 \sqrt {g x +f}\, \left (27 f^{3} g^{3} x^{3}+54 f^{2} g^{4} x^{3}+36 f \,g^{5} x^{3}+8 g^{6} x^{3}+81 f^{4} g^{2} x^{2}+162 f^{3} g^{3} x^{2}+108 f^{2} g^{4} x^{2}+24 f \,g^{5} x^{2}+81 f^{5} g x +162 f^{4} g^{2} x +108 f^{3} g^{3} x +24 f^{2} g^{4} x +27 f^{6}+54 f^{5} g +36 f^{4} g^{2}+8 f^{3} g^{3}\right )} \] Input:

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

Output:

(2*sqrt( - 3*x + 2)*(567*f**2*x**2 + 504*f**2*x - 588*f**2 + 162*f*g*x**3 
+ 1440*f*g*x**2 - 456*f*g*x - 384*f*g + 396*g**2*x**3 + 132*g**2*x**2 - 14 
4*g**2*x - 80*g**2))/(35*sqrt(f + g*x)*(27*f**6 + 81*f**5*g*x + 54*f**5*g 
+ 81*f**4*g**2*x**2 + 162*f**4*g**2*x + 36*f**4*g**2 + 27*f**3*g**3*x**3 + 
 162*f**3*g**3*x**2 + 108*f**3*g**3*x + 8*f**3*g**3 + 54*f**2*g**4*x**3 + 
108*f**2*g**4*x**2 + 24*f**2*g**4*x + 36*f*g**5*x**3 + 24*f*g**5*x**2 + 8* 
g**6*x**3))