Integrand size = 30, antiderivative size = 165 \[ \int \frac {\sqrt {2+3 x} \sqrt {4-9 x^2}}{(f+g x)^{11/2}} \, dx=\frac {2 (3 f-2 g) (2-3 x)^{3/2}}{9 g (3 f+2 g) (f+g x)^{9/2}}-\frac {2 (3 f+10 g) (2-3 x)^{3/2}}{7 g (3 f+2 g)^2 (f+g x)^{7/2}}-\frac {24 (3 f+10 g) (2-3 x)^{3/2}}{35 g (3 f+2 g)^3 (f+g x)^{5/2}}-\frac {48 (3 f+10 g) (2-3 x)^{3/2}}{35 g (3 f+2 g)^4 (f+g x)^{3/2}} \] Output:
2/9*(3*f-2*g)*(2-3*x)^(3/2)/g/(3*f+2*g)/(g*x+f)^(9/2)-2/7*(3*f+10*g)*(2-3* x)^(3/2)/g/(3*f+2*g)^2/(g*x+f)^(7/2)-24/35*(3*f+10*g)*(2-3*x)^(3/2)/g/(3*f +2*g)^3/(g*x+f)^(5/2)-48/35*(3*f+10*g)*(2-3*x)^(3/2)/g/(3*f+2*g)^4/(g*x+f) ^(3/2)
Time = 3.63 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.71 \[ \int \frac {\sqrt {2+3 x} \sqrt {4-9 x^2}}{(f+g x)^{11/2}} \, dx=-\frac {2 (2-3 x)^2 \sqrt {2+3 x} \left (567 f^3 (14+9 x)+162 f^2 g \left (50+123 x+18 x^2\right )+40 g^3 \left (14+45 x+54 x^2+54 x^3\right )+12 f g^2 \left (290+855 x+864 x^2+54 x^3\right )\right )}{315 (3 f+2 g)^4 (f+g x)^{9/2} \sqrt {4-9 x^2}} \] Input:
Integrate[(Sqrt[2 + 3*x]*Sqrt[4 - 9*x^2])/(f + g*x)^(11/2),x]
Output:
(-2*(2 - 3*x)^2*Sqrt[2 + 3*x]*(567*f^3*(14 + 9*x) + 162*f^2*g*(50 + 123*x + 18*x^2) + 40*g^3*(14 + 45*x + 54*x^2 + 54*x^3) + 12*f*g^2*(290 + 855*x + 864*x^2 + 54*x^3)))/(315*(3*f + 2*g)^4*(f + g*x)^(9/2)*Sqrt[4 - 9*x^2])
Time = 0.25 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.03, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {639, 87, 55, 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.
| \(\displaystyle \int \frac {\sqrt {3 x+2} \sqrt {4-9 x^2}}{(f+g x)^{11/2}} \, dx\) |
| \(\Big \downarrow \) 639 |
| \(\displaystyle \int \frac {\sqrt {2-3 x} (3 x+2)}{(f+g x)^{11/2}}dx\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle \frac {(3 f+10 g) \int \frac {\sqrt {2-3 x}}{(f+g x)^{9/2}}dx}{g (3 f+2 g)}+\frac {2 (2-3 x)^{3/2} (3 f-2 g)}{9 g (3 f+2 g) (f+g x)^{9/2}}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle \frac {(3 f+10 g) \left (\frac {12 \int \frac {\sqrt {2-3 x}}{(f+g x)^{7/2}}dx}{7 (3 f+2 g)}-\frac {2 (2-3 x)^{3/2}}{7 (3 f+2 g) (f+g x)^{7/2}}\right )}{g (3 f+2 g)}+\frac {2 (2-3 x)^{3/2} (3 f-2 g)}{9 g (3 f+2 g) (f+g x)^{9/2}}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle \frac {(3 f+10 g) \left (\frac {12 \left (\frac {6 \int \frac {\sqrt {2-3 x}}{(f+g x)^{5/2}}dx}{5 (3 f+2 g)}-\frac {2 (2-3 x)^{3/2}}{5 (3 f+2 g) (f+g x)^{5/2}}\right )}{7 (3 f+2 g)}-\frac {2 (2-3 x)^{3/2}}{7 (3 f+2 g) (f+g x)^{7/2}}\right )}{g (3 f+2 g)}+\frac {2 (2-3 x)^{3/2} (3 f-2 g)}{9 g (3 f+2 g) (f+g x)^{9/2}}\) |
| \(\Big \downarrow \) 48 |
| \(\displaystyle \frac {2 (2-3 x)^{3/2} (3 f-2 g)}{9 g (3 f+2 g) (f+g x)^{9/2}}+\frac {(3 f+10 g) \left (\frac {12 \left (-\frac {4 (2-3 x)^{3/2}}{5 (3 f+2 g)^2 (f+g x)^{3/2}}-\frac {2 (2-3 x)^{3/2}}{5 (3 f+2 g) (f+g x)^{5/2}}\right )}{7 (3 f+2 g)}-\frac {2 (2-3 x)^{3/2}}{7 (3 f+2 g) (f+g x)^{7/2}}\right )}{g (3 f+2 g)}\) |
Input:
Int[(Sqrt[2 + 3*x]*Sqrt[4 - 9*x^2])/(f + g*x)^(11/2),x]
Output:
(2*(3*f - 2*g)*(2 - 3*x)^(3/2))/(9*g*(3*f + 2*g)*(f + g*x)^(9/2)) + ((3*f + 10*g)*((-2*(2 - 3*x)^(3/2))/(7*(3*f + 2*g)*(f + g*x)^(7/2)) + (12*((-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*(3*f + 2*g))))/(g*(3*f + 2*g))
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]
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])
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]))))
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]))
Time = 0.84 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.80
| method | result | size |
| default | \(\frac {2 \sqrt {-9 x^{2}+4}\, \left (-2+3 x \right ) \left (648 x^{3} f \,g^{2}+2160 x^{3} g^{3}+2916 x^{2} f^{2} g +10368 f \,g^{2} x^{2}+2160 x^{2} g^{3}+5103 x \,f^{3}+19926 x \,f^{2} g +10260 x f \,g^{2}+1800 x \,g^{3}+7938 f^{3}+8100 f^{2} g +3480 f \,g^{2}+560 g^{3}\right )}{315 \sqrt {3 x +2}\, \left (g x +f \right )^{\frac {9}{2}} \left (3 f +2 g \right )^{4}}\) | \(132\) |
| gosper | \(\frac {2 \left (-2+3 x \right ) \left (648 x^{3} f \,g^{2}+2160 x^{3} g^{3}+2916 x^{2} f^{2} g +10368 f \,g^{2} x^{2}+2160 x^{2} g^{3}+5103 x \,f^{3}+19926 x \,f^{2} g +10260 x f \,g^{2}+1800 x \,g^{3}+7938 f^{3}+8100 f^{2} g +3480 f \,g^{2}+560 g^{3}\right ) \sqrt {-9 x^{2}+4}}{315 \left (g x +f \right )^{\frac {9}{2}} \left (81 f^{4}+216 g \,f^{3}+216 g^{2} f^{2}+96 f \,g^{3}+16 g^{4}\right ) \sqrt {3 x +2}}\) | \(156\) |
| orering | \(\frac {2 \left (-2+3 x \right ) \left (648 x^{3} f \,g^{2}+2160 x^{3} g^{3}+2916 x^{2} f^{2} g +10368 f \,g^{2} x^{2}+2160 x^{2} g^{3}+5103 x \,f^{3}+19926 x \,f^{2} g +10260 x f \,g^{2}+1800 x \,g^{3}+7938 f^{3}+8100 f^{2} g +3480 f \,g^{2}+560 g^{3}\right ) \sqrt {-9 x^{2}+4}}{315 \left (g x +f \right )^{\frac {9}{2}} \left (81 f^{4}+216 g \,f^{3}+216 g^{2} f^{2}+96 f \,g^{3}+16 g^{4}\right ) \sqrt {3 x +2}}\) | \(156\) |
Input:
int((3*x+2)^(1/2)*(-9*x^2+4)^(1/2)/(g*x+f)^(11/2),x,method=_RETURNVERBOSE)
Output:
2/315/(3*x+2)^(1/2)*(-9*x^2+4)^(1/2)/(g*x+f)^(9/2)*(-2+3*x)*(648*f*g^2*x^3 +2160*g^3*x^3+2916*f^2*g*x^2+10368*f*g^2*x^2+2160*g^3*x^2+5103*f^3*x+19926 *f^2*g*x+10260*f*g^2*x+1800*g^3*x+7938*f^3+8100*f^2*g+3480*f*g^2+560*g^3)/ (3*f+2*g)^4
Leaf count of result is larger than twice the leaf count of optimal. 473 vs. \(2 (141) = 282\).
Time = 0.10 (sec) , antiderivative size = 473, normalized size of antiderivative = 2.87 \[ \int \frac {\sqrt {2+3 x} \sqrt {4-9 x^2}}{(f+g x)^{11/2}} \, dx=\frac {2 \, {\left (648 \, {\left (3 \, f g^{2} + 10 \, g^{3}\right )} x^{4} + 108 \, {\left (81 \, f^{2} g + 276 \, f g^{2} + 20 \, g^{3}\right )} x^{3} - 15876 \, f^{3} - 16200 \, f^{2} g - 6960 \, f g^{2} - 1120 \, g^{3} + 27 \, {\left (567 \, f^{3} + 1998 \, f^{2} g + 372 \, f g^{2} + 40 \, g^{3}\right )} x^{2} + 24 \, {\left (567 \, f^{3} - 648 \, f^{2} g - 420 \, f g^{2} - 80 \, g^{3}\right )} x\right )} \sqrt {g x + f} \sqrt {-9 \, x^{2} + 4} \sqrt {3 \, x + 2}}{315 \, {\left (162 \, f^{9} + 432 \, f^{8} g + 432 \, f^{7} g^{2} + 192 \, f^{6} g^{3} + 32 \, f^{5} g^{4} + 3 \, {\left (81 \, f^{4} g^{5} + 216 \, f^{3} g^{6} + 216 \, f^{2} g^{7} + 96 \, f g^{8} + 16 \, g^{9}\right )} x^{6} + {\left (1215 \, f^{5} g^{4} + 3402 \, f^{4} g^{5} + 3672 \, f^{3} g^{6} + 1872 \, f^{2} g^{7} + 432 \, f g^{8} + 32 \, g^{9}\right )} x^{5} + 10 \, {\left (243 \, f^{6} g^{3} + 729 \, f^{5} g^{4} + 864 \, f^{4} g^{5} + 504 \, f^{3} g^{6} + 144 \, f^{2} g^{7} + 16 \, f g^{8}\right )} x^{4} + 10 \, {\left (243 \, f^{7} g^{2} + 810 \, f^{6} g^{3} + 1080 \, f^{5} g^{4} + 720 \, f^{4} g^{5} + 240 \, f^{3} g^{6} + 32 \, f^{2} g^{7}\right )} x^{3} + 5 \, {\left (243 \, f^{8} g + 972 \, f^{7} g^{2} + 1512 \, f^{6} g^{3} + 1152 \, f^{5} g^{4} + 432 \, f^{4} g^{5} + 64 \, f^{3} g^{6}\right )} x^{2} + {\left (243 \, f^{9} + 1458 \, f^{8} g + 2808 \, f^{7} g^{2} + 2448 \, f^{6} g^{3} + 1008 \, f^{5} g^{4} + 160 \, f^{4} g^{5}\right )} x\right )}} \] Input:
integrate((2+3*x)^(1/2)*(-9*x^2+4)^(1/2)/(g*x+f)^(11/2),x, algorithm="fric as")
Output:
2/315*(648*(3*f*g^2 + 10*g^3)*x^4 + 108*(81*f^2*g + 276*f*g^2 + 20*g^3)*x^ 3 - 15876*f^3 - 16200*f^2*g - 6960*f*g^2 - 1120*g^3 + 27*(567*f^3 + 1998*f ^2*g + 372*f*g^2 + 40*g^3)*x^2 + 24*(567*f^3 - 648*f^2*g - 420*f*g^2 - 80* g^3)*x)*sqrt(g*x + f)*sqrt(-9*x^2 + 4)*sqrt(3*x + 2)/(162*f^9 + 432*f^8*g + 432*f^7*g^2 + 192*f^6*g^3 + 32*f^5*g^4 + 3*(81*f^4*g^5 + 216*f^3*g^6 + 2 16*f^2*g^7 + 96*f*g^8 + 16*g^9)*x^6 + (1215*f^5*g^4 + 3402*f^4*g^5 + 3672* f^3*g^6 + 1872*f^2*g^7 + 432*f*g^8 + 32*g^9)*x^5 + 10*(243*f^6*g^3 + 729*f ^5*g^4 + 864*f^4*g^5 + 504*f^3*g^6 + 144*f^2*g^7 + 16*f*g^8)*x^4 + 10*(243 *f^7*g^2 + 810*f^6*g^3 + 1080*f^5*g^4 + 720*f^4*g^5 + 240*f^3*g^6 + 32*f^2 *g^7)*x^3 + 5*(243*f^8*g + 972*f^7*g^2 + 1512*f^6*g^3 + 1152*f^5*g^4 + 432 *f^4*g^5 + 64*f^3*g^6)*x^2 + (243*f^9 + 1458*f^8*g + 2808*f^7*g^2 + 2448*f ^6*g^3 + 1008*f^5*g^4 + 160*f^4*g^5)*x)
Timed out. \[ \int \frac {\sqrt {2+3 x} \sqrt {4-9 x^2}}{(f+g x)^{11/2}} \, dx=\text {Timed out} \] Input:
integrate((2+3*x)**(1/2)*(-9*x**2+4)**(1/2)/(g*x+f)**(11/2),x)
Output:
Timed out
\[ \int \frac {\sqrt {2+3 x} \sqrt {4-9 x^2}}{(f+g x)^{11/2}} \, dx=\int { \frac {\sqrt {-9 \, x^{2} + 4} \sqrt {3 \, x + 2}}{{\left (g x + f\right )}^{\frac {11}{2}}} \,d x } \] Input:
integrate((2+3*x)^(1/2)*(-9*x^2+4)^(1/2)/(g*x+f)^(11/2),x, algorithm="maxi ma")
Output:
integrate(sqrt(-9*x^2 + 4)*sqrt(3*x + 2)/(g*x + f)^(11/2), x)
Leaf count of result is larger than twice the leaf count of optimal. 338 vs. \(2 (141) = 282\).
Time = 0.25 (sec) , antiderivative size = 338, normalized size of antiderivative = 2.05 \[ \int \frac {\sqrt {2+3 x} \sqrt {4-9 x^2}}{(f+g x)^{11/2}} \, dx=\frac {18 \, {\left ({\left (4 \, {\left (\frac {2 \, {\left (3 \, \sqrt {3} f g^{6} + 10 \, \sqrt {3} g^{7}\right )} {\left (3 \, x - 2\right )}}{81 \, f^{4} g^{4} + 216 \, f^{3} g^{5} + 216 \, f^{2} g^{6} + 96 \, f g^{7} + 16 \, g^{8}} + \frac {9 \, {\left (9 \, \sqrt {3} f^{2} g^{5} + 36 \, \sqrt {3} f g^{6} + 20 \, \sqrt {3} g^{7}\right )}}{81 \, f^{4} g^{4} + 216 \, f^{3} g^{5} + 216 \, f^{2} g^{6} + 96 \, f g^{7} + 16 \, g^{8}}\right )} {\left (3 \, x - 2\right )} + \frac {63 \, {\left (27 \, \sqrt {3} f^{3} g^{4} + 126 \, \sqrt {3} f^{2} g^{5} + 132 \, \sqrt {3} f g^{6} + 40 \, \sqrt {3} g^{7}\right )}}{81 \, f^{4} g^{4} + 216 \, f^{3} g^{5} + 216 \, f^{2} g^{6} + 96 \, f g^{7} + 16 \, g^{8}}\right )} {\left (3 \, x - 2\right )} + \frac {420 \, {\left (27 \, \sqrt {3} f^{3} g^{4} + 54 \, \sqrt {3} f^{2} g^{5} + 36 \, \sqrt {3} f g^{6} + 8 \, \sqrt {3} g^{7}\right )}}{81 \, f^{4} g^{4} + 216 \, f^{3} g^{5} + 216 \, f^{2} g^{6} + 96 \, f g^{7} + 16 \, g^{8}}\right )} {\left (3 \, x - 2\right )} \sqrt {-3 \, x + 2}}{35 \, {\left (g {\left (3 \, x - 2\right )} + 3 \, f + 2 \, g\right )}^{\frac {9}{2}}} \] Input:
integrate((2+3*x)^(1/2)*(-9*x^2+4)^(1/2)/(g*x+f)^(11/2),x, algorithm="giac ")
Output:
18/35*((4*(2*(3*sqrt(3)*f*g^6 + 10*sqrt(3)*g^7)*(3*x - 2)/(81*f^4*g^4 + 21 6*f^3*g^5 + 216*f^2*g^6 + 96*f*g^7 + 16*g^8) + 9*(9*sqrt(3)*f^2*g^5 + 36*s qrt(3)*f*g^6 + 20*sqrt(3)*g^7)/(81*f^4*g^4 + 216*f^3*g^5 + 216*f^2*g^6 + 9 6*f*g^7 + 16*g^8))*(3*x - 2) + 63*(27*sqrt(3)*f^3*g^4 + 126*sqrt(3)*f^2*g^ 5 + 132*sqrt(3)*f*g^6 + 40*sqrt(3)*g^7)/(81*f^4*g^4 + 216*f^3*g^5 + 216*f^ 2*g^6 + 96*f*g^7 + 16*g^8))*(3*x - 2) + 420*(27*sqrt(3)*f^3*g^4 + 54*sqrt( 3)*f^2*g^5 + 36*sqrt(3)*f*g^6 + 8*sqrt(3)*g^7)/(81*f^4*g^4 + 216*f^3*g^5 + 216*f^2*g^6 + 96*f*g^7 + 16*g^8))*(3*x - 2)*sqrt(-3*x + 2)/(g*(3*x - 2) + 3*f + 2*g)^(9/2)
Time = 6.77 (sec) , antiderivative size = 355, normalized size of antiderivative = 2.15 \[ \int \frac {\sqrt {2+3 x} \sqrt {4-9 x^2}}{(f+g x)^{11/2}} \, dx=\frac {\sqrt {f+g\,x}\,\left (\frac {x^2\,\sqrt {3\,x+2}\,\sqrt {4-9\,x^2}\,\left (30618\,f^3+107892\,f^2\,g+20088\,f\,g^2+2160\,g^3\right )}{945\,g^5\,{\left (3\,f+2\,g\right )}^4}-\frac {x\,\sqrt {3\,x+2}\,\sqrt {4-9\,x^2}\,\left (-27216\,f^3+31104\,f^2\,g+20160\,f\,g^2+3840\,g^3\right )}{945\,g^5\,{\left (3\,f+2\,g\right )}^4}-\frac {\sqrt {3\,x+2}\,\sqrt {4-9\,x^2}\,\left (31752\,f^3+32400\,f^2\,g+13920\,f\,g^2+2240\,g^3\right )}{945\,g^5\,{\left (3\,f+2\,g\right )}^4}+\frac {8\,x^3\,\sqrt {3\,x+2}\,\sqrt {4-9\,x^2}\,\left (81\,f^2+276\,f\,g+20\,g^2\right )}{35\,g^4\,{\left (3\,f+2\,g\right )}^4}+\frac {48\,x^4\,\sqrt {3\,x+2}\,\sqrt {4-9\,x^2}\,\left (3\,f+10\,g\right )}{35\,g^3\,{\left (3\,f+2\,g\right )}^4}\right )}{x^6+\frac {2\,f^5}{3\,g^5}+\frac {x^5\,\left (15\,f+2\,g\right )}{3\,g}+\frac {f^4\,x\,\left (3\,f+10\,g\right )}{3\,g^5}+\frac {10\,f^2\,x^3\,\left (3\,f+2\,g\right )}{3\,g^3}+\frac {5\,f^3\,x^2\,\left (3\,f+4\,g\right )}{3\,g^4}+\frac {10\,f\,x^4\,\left (3\,f+g\right )}{3\,g^2}} \] Input:
int(((3*x + 2)^(1/2)*(4 - 9*x^2)^(1/2))/(f + g*x)^(11/2),x)
Output:
((f + g*x)^(1/2)*((x^2*(3*x + 2)^(1/2)*(4 - 9*x^2)^(1/2)*(20088*f*g^2 + 10 7892*f^2*g + 30618*f^3 + 2160*g^3))/(945*g^5*(3*f + 2*g)^4) - (x*(3*x + 2) ^(1/2)*(4 - 9*x^2)^(1/2)*(20160*f*g^2 + 31104*f^2*g - 27216*f^3 + 3840*g^3 ))/(945*g^5*(3*f + 2*g)^4) - ((3*x + 2)^(1/2)*(4 - 9*x^2)^(1/2)*(13920*f*g ^2 + 32400*f^2*g + 31752*f^3 + 2240*g^3))/(945*g^5*(3*f + 2*g)^4) + (8*x^3 *(3*x + 2)^(1/2)*(4 - 9*x^2)^(1/2)*(276*f*g + 81*f^2 + 20*g^2))/(35*g^4*(3 *f + 2*g)^4) + (48*x^4*(3*x + 2)^(1/2)*(4 - 9*x^2)^(1/2)*(3*f + 10*g))/(35 *g^3*(3*f + 2*g)^4)))/(x^6 + (2*f^5)/(3*g^5) + (x^5*(15*f + 2*g))/(3*g) + (f^4*x*(3*f + 10*g))/(3*g^5) + (10*f^2*x^3*(3*f + 2*g))/(3*g^3) + (5*f^3*x ^2*(3*f + 4*g))/(3*g^4) + (10*f*x^4*(3*f + g))/(3*g^2))
Time = 0.21 (sec) , antiderivative size = 381, normalized size of antiderivative = 2.31 \[ \int \frac {\sqrt {2+3 x} \sqrt {4-9 x^2}}{(f+g x)^{11/2}} \, dx=\frac {2 \sqrt {-3 x +2}\, \left (1944 f \,g^{2} x^{4}+6480 g^{3} x^{4}+8748 f^{2} g \,x^{3}+29808 f \,g^{2} x^{3}+2160 g^{3} x^{3}+15309 f^{3} x^{2}+53946 f^{2} g \,x^{2}+10044 f \,g^{2} x^{2}+1080 g^{3} x^{2}+13608 f^{3} x -15552 f^{2} g x -10080 f \,g^{2} x -1920 g^{3} x -15876 f^{3}-16200 f^{2} g -6960 f \,g^{2}-1120 g^{3}\right )}{315 \sqrt {g x +f}\, \left (81 f^{4} g^{4} x^{4}+216 f^{3} g^{5} x^{4}+216 f^{2} g^{6} x^{4}+96 f \,g^{7} x^{4}+16 g^{8} x^{4}+324 f^{5} g^{3} x^{3}+864 f^{4} g^{4} x^{3}+864 f^{3} g^{5} x^{3}+384 f^{2} g^{6} x^{3}+64 f \,g^{7} x^{3}+486 f^{6} g^{2} x^{2}+1296 f^{5} g^{3} x^{2}+1296 f^{4} g^{4} x^{2}+576 f^{3} g^{5} x^{2}+96 f^{2} g^{6} x^{2}+324 f^{7} g x +864 f^{6} g^{2} x +864 f^{5} g^{3} x +384 f^{4} g^{4} x +64 f^{3} g^{5} x +81 f^{8}+216 f^{7} g +216 f^{6} g^{2}+96 f^{5} g^{3}+16 f^{4} g^{4}\right )} \] Input:
int((2+3*x)^(1/2)*(-9*x^2+4)^(1/2)/(g*x+f)^(11/2),x)
Output:
(2*sqrt( - 3*x + 2)*(15309*f**3*x**2 + 13608*f**3*x - 15876*f**3 + 8748*f* *2*g*x**3 + 53946*f**2*g*x**2 - 15552*f**2*g*x - 16200*f**2*g + 1944*f*g** 2*x**4 + 29808*f*g**2*x**3 + 10044*f*g**2*x**2 - 10080*f*g**2*x - 6960*f*g **2 + 6480*g**3*x**4 + 2160*g**3*x**3 + 1080*g**3*x**2 - 1920*g**3*x - 112 0*g**3))/(315*sqrt(f + g*x)*(81*f**8 + 324*f**7*g*x + 216*f**7*g + 486*f** 6*g**2*x**2 + 864*f**6*g**2*x + 216*f**6*g**2 + 324*f**5*g**3*x**3 + 1296* f**5*g**3*x**2 + 864*f**5*g**3*x + 96*f**5*g**3 + 81*f**4*g**4*x**4 + 864* f**4*g**4*x**3 + 1296*f**4*g**4*x**2 + 384*f**4*g**4*x + 16*f**4*g**4 + 21 6*f**3*g**5*x**4 + 864*f**3*g**5*x**3 + 576*f**3*g**5*x**2 + 64*f**3*g**5* x + 216*f**2*g**6*x**4 + 384*f**2*g**6*x**3 + 96*f**2*g**6*x**2 + 96*f*g** 7*x**4 + 64*f*g**7*x**3 + 16*g**8*x**4))