Integrand size = 35, antiderivative size = 193 \[ \int \frac {\sqrt {2+3 x} \sqrt {1+\frac {5 x}{6}-x^2}}{(f+g x)^{11/2}} \, dx=\frac {\sqrt {\frac {2}{3}} (3 f-2 g) (3-2 x)^{3/2}}{9 g (2 f+3 g) (f+g x)^{9/2}}-\frac {\sqrt {\frac {2}{3}} (6 f+35 g) (3-2 x)^{3/2}}{21 g (2 f+3 g)^2 (f+g x)^{7/2}}-\frac {8 \sqrt {\frac {2}{3}} (6 f+35 g) (3-2 x)^{3/2}}{105 g (2 f+3 g)^3 (f+g x)^{5/2}}-\frac {32 \sqrt {\frac {2}{3}} (6 f+35 g) (3-2 x)^{3/2}}{315 g (2 f+3 g)^4 (f+g x)^{3/2}} \] Output:
1/27*6^(1/2)*(3*f-2*g)*(3-2*x)^(3/2)/g/(2*f+3*g)/(g*x+f)^(9/2)-1/63*6^(1/2 )*(6*f+35*g)*(3-2*x)^(3/2)/g/(2*f+3*g)^2/(g*x+f)^(7/2)-8/315*6^(1/2)*(6*f+ 35*g)*(3-2*x)^(3/2)/g/(2*f+3*g)^3/(g*x+f)^(5/2)-32/945*6^(1/2)*(6*f+35*g)* (3-2*x)^(3/2)/g/(2*f+3*g)^4/(g*x+f)^(3/2)
Time = 7.93 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.62 \[ \int \frac {\sqrt {2+3 x} \sqrt {1+\frac {5 x}{6}-x^2}}{(f+g x)^{11/2}} \, dx=\frac {(-3+2 x) \sqrt {4+\frac {10 x}{3}-4 x^2} \left (168 f^3 (19+9 x)+36 f^2 g \left (180+299 x+24 x^2\right )+35 g^3 \left (54+135 x+72 x^2+32 x^3\right )+6 f g^2 \left (945+2025 x+912 x^2+32 x^3\right )\right )}{315 (2 f+3 g)^4 \sqrt {2+3 x} (f+g x)^{9/2}} \] Input:
Integrate[(Sqrt[2 + 3*x]*Sqrt[1 + (5*x)/6 - x^2])/(f + g*x)^(11/2),x]
Output:
((-3 + 2*x)*Sqrt[4 + (10*x)/3 - 4*x^2]*(168*f^3*(19 + 9*x) + 36*f^2*g*(180 + 299*x + 24*x^2) + 35*g^3*(54 + 135*x + 72*x^2 + 32*x^3) + 6*f*g^2*(945 + 2025*x + 912*x^2 + 32*x^3)))/(315*(2*f + 3*g)^4*Sqrt[2 + 3*x]*(f + g*x)^ (9/2))
Time = 0.29 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.96, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {1245, 87, 27, 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.
| \(\displaystyle \int \frac {\sqrt {3 x+2} \sqrt {-x^2+\frac {5 x}{6}+1}}{(f+g x)^{11/2}} \, dx\) |
| \(\Big \downarrow \) 1245 |
| \(\displaystyle \int \frac {\sqrt {\frac {1}{2}-\frac {x}{3}} (3 x+2)}{(f+g x)^{11/2}}dx\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle \frac {(6 f+35 g) \int \frac {\sqrt {3-2 x}}{\sqrt {6} (f+g x)^{9/2}}dx}{3 g (2 f+3 g)}+\frac {\sqrt {\frac {2}{3}} (3-2 x)^{3/2} (3 f-2 g)}{9 g (2 f+3 g) (f+g x)^{9/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(6 f+35 g) \int \frac {\sqrt {3-2 x}}{(f+g x)^{9/2}}dx}{3 \sqrt {6} g (2 f+3 g)}+\frac {\sqrt {\frac {2}{3}} (3-2 x)^{3/2} (3 f-2 g)}{9 g (2 f+3 g) (f+g x)^{9/2}}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle \frac {(6 f+35 g) \left (\frac {8 \int \frac {\sqrt {3-2 x}}{(f+g x)^{7/2}}dx}{7 (2 f+3 g)}-\frac {2 (3-2 x)^{3/2}}{7 (2 f+3 g) (f+g x)^{7/2}}\right )}{3 \sqrt {6} g (2 f+3 g)}+\frac {\sqrt {\frac {2}{3}} (3-2 x)^{3/2} (3 f-2 g)}{9 g (2 f+3 g) (f+g x)^{9/2}}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle \frac {(6 f+35 g) \left (\frac {8 \left (\frac {4 \int \frac {\sqrt {3-2 x}}{(f+g x)^{5/2}}dx}{5 (2 f+3 g)}-\frac {2 (3-2 x)^{3/2}}{5 (2 f+3 g) (f+g x)^{5/2}}\right )}{7 (2 f+3 g)}-\frac {2 (3-2 x)^{3/2}}{7 (2 f+3 g) (f+g x)^{7/2}}\right )}{3 \sqrt {6} g (2 f+3 g)}+\frac {\sqrt {\frac {2}{3}} (3-2 x)^{3/2} (3 f-2 g)}{9 g (2 f+3 g) (f+g x)^{9/2}}\) |
| \(\Big \downarrow \) 48 |
| \(\displaystyle \frac {\sqrt {\frac {2}{3}} (3-2 x)^{3/2} (3 f-2 g)}{9 g (2 f+3 g) (f+g x)^{9/2}}+\frac {(6 f+35 g) \left (\frac {8 \left (-\frac {8 (3-2 x)^{3/2}}{15 (2 f+3 g)^2 (f+g x)^{3/2}}-\frac {2 (3-2 x)^{3/2}}{5 (2 f+3 g) (f+g x)^{5/2}}\right )}{7 (2 f+3 g)}-\frac {2 (3-2 x)^{3/2}}{7 (2 f+3 g) (f+g x)^{7/2}}\right )}{3 \sqrt {6} g (2 f+3 g)}\) |
Input:
Int[(Sqrt[2 + 3*x]*Sqrt[1 + (5*x)/6 - x^2])/(f + g*x)^(11/2),x]
Output:
(Sqrt[2/3]*(3*f - 2*g)*(3 - 2*x)^(3/2))/(9*g*(2*f + 3*g)*(f + g*x)^(9/2)) + ((6*f + 35*g)*((-2*(3 - 2*x)^(3/2))/(7*(2*f + 3*g)*(f + g*x)^(7/2)) + (8 *((-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*(2*f + 3*g))))/(3*Sqrt[6]*g*(2* f + 3*g))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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[((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]
Time = 1.58 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.70
| method | result | size |
| default | \(\frac {\sqrt {-36 x^{2}+30 x +36}\, \left (2 x -3\right ) \left (192 x^{3} f \,g^{2}+1120 x^{3} g^{3}+864 x^{2} f^{2} g +5472 f \,g^{2} x^{2}+2520 x^{2} g^{3}+1512 x \,f^{3}+10764 x \,f^{2} g +12150 x f \,g^{2}+4725 x \,g^{3}+3192 f^{3}+6480 f^{2} g +5670 f \,g^{2}+1890 g^{3}\right )}{945 \sqrt {3 x +2}\, \left (g x +f \right )^{\frac {9}{2}} \left (2 f +3 g \right )^{4}}\) | \(135\) |
| gosper | \(\frac {\left (2 x -3\right ) \left (192 x^{3} f \,g^{2}+1120 x^{3} g^{3}+864 x^{2} f^{2} g +5472 f \,g^{2} x^{2}+2520 x^{2} g^{3}+1512 x \,f^{3}+10764 x \,f^{2} g +12150 x f \,g^{2}+4725 x \,g^{3}+3192 f^{3}+6480 f^{2} g +5670 f \,g^{2}+1890 g^{3}\right ) \sqrt {-36 x^{2}+30 x +36}}{945 \left (g x +f \right )^{\frac {9}{2}} \left (16 f^{4}+96 g \,f^{3}+216 g^{2} f^{2}+216 f \,g^{3}+81 g^{4}\right ) \sqrt {3 x +2}}\) | \(159\) |
| orering | \(\frac {\left (2 x -3\right ) \left (192 x^{3} f \,g^{2}+1120 x^{3} g^{3}+864 x^{2} f^{2} g +5472 f \,g^{2} x^{2}+2520 x^{2} g^{3}+1512 x \,f^{3}+10764 x \,f^{2} g +12150 x f \,g^{2}+4725 x \,g^{3}+3192 f^{3}+6480 f^{2} g +5670 f \,g^{2}+1890 g^{3}\right ) \sqrt {-36 x^{2}+30 x +36}}{945 \left (g x +f \right )^{\frac {9}{2}} \left (16 f^{4}+96 g \,f^{3}+216 g^{2} f^{2}+216 f \,g^{3}+81 g^{4}\right ) \sqrt {3 x +2}}\) | \(159\) |
Input:
int(1/6*(3*x+2)^(1/2)*(-36*x^2+30*x+36)^(1/2)/(g*x+f)^(11/2),x,method=_RET URNVERBOSE)
Output:
1/945/(3*x+2)^(1/2)*(-36*x^2+30*x+36)^(1/2)/(g*x+f)^(9/2)*(2*x-3)*(192*f*g ^2*x^3+1120*g^3*x^3+864*f^2*g*x^2+5472*f*g^2*x^2+2520*g^3*x^2+1512*f^3*x+1 0764*f^2*g*x+12150*f*g^2*x+4725*g^3*x+3192*f^3+6480*f^2*g+5670*f*g^2+1890* g^3)/(2*f+3*g)^4
Leaf count of result is larger than twice the leaf count of optimal. 476 vs. \(2 (153) = 306\).
Time = 0.10 (sec) , antiderivative size = 476, normalized size of antiderivative = 2.47 \[ \int \frac {\sqrt {2+3 x} \sqrt {1+\frac {5 x}{6}-x^2}}{(f+g x)^{11/2}} \, dx=\frac {{\left (64 \, {\left (6 \, f g^{2} + 35 \, g^{3}\right )} x^{4} + 48 \, {\left (36 \, f^{2} g + 216 \, f g^{2} + 35 \, g^{3}\right )} x^{3} - 9576 \, f^{3} - 19440 \, f^{2} g - 17010 \, f g^{2} - 5670 \, g^{3} + 18 \, {\left (168 \, f^{3} + 1052 \, f^{2} g + 438 \, f g^{2} + 105 \, g^{3}\right )} x^{2} + 3 \, {\left (616 \, f^{3} - 6444 \, f^{2} g - 8370 \, f g^{2} - 3465 \, g^{3}\right )} x\right )} \sqrt {g x + f} \sqrt {-36 \, x^{2} + 30 \, x + 36} \sqrt {3 \, x + 2}}{945 \, {\left (32 \, f^{9} + 192 \, f^{8} g + 432 \, f^{7} g^{2} + 432 \, f^{6} g^{3} + 162 \, f^{5} g^{4} + 3 \, {\left (16 \, f^{4} g^{5} + 96 \, f^{3} g^{6} + 216 \, f^{2} g^{7} + 216 \, f g^{8} + 81 \, g^{9}\right )} x^{6} + {\left (240 \, f^{5} g^{4} + 1472 \, f^{4} g^{5} + 3432 \, f^{3} g^{6} + 3672 \, f^{2} g^{7} + 1647 \, f g^{8} + 162 \, g^{9}\right )} x^{5} + 10 \, {\left (48 \, f^{6} g^{3} + 304 \, f^{5} g^{4} + 744 \, f^{4} g^{5} + 864 \, f^{3} g^{6} + 459 \, f^{2} g^{7} + 81 \, f g^{8}\right )} x^{4} + 10 \, {\left (48 \, f^{7} g^{2} + 320 \, f^{6} g^{3} + 840 \, f^{5} g^{4} + 1080 \, f^{4} g^{5} + 675 \, f^{3} g^{6} + 162 \, f^{2} g^{7}\right )} x^{3} + 5 \, {\left (48 \, f^{8} g + 352 \, f^{7} g^{2} + 1032 \, f^{6} g^{3} + 1512 \, f^{5} g^{4} + 1107 \, f^{4} g^{5} + 324 \, f^{3} g^{6}\right )} x^{2} + {\left (48 \, f^{9} + 448 \, f^{8} g + 1608 \, f^{7} g^{2} + 2808 \, f^{6} g^{3} + 2403 \, f^{5} g^{4} + 810 \, f^{4} g^{5}\right )} x\right )}} \] Input:
integrate(1/6*(2+3*x)^(1/2)*(-36*x^2+30*x+36)^(1/2)/(g*x+f)^(11/2),x, algo rithm="fricas")
Output:
1/945*(64*(6*f*g^2 + 35*g^3)*x^4 + 48*(36*f^2*g + 216*f*g^2 + 35*g^3)*x^3 - 9576*f^3 - 19440*f^2*g - 17010*f*g^2 - 5670*g^3 + 18*(168*f^3 + 1052*f^2 *g + 438*f*g^2 + 105*g^3)*x^2 + 3*(616*f^3 - 6444*f^2*g - 8370*f*g^2 - 346 5*g^3)*x)*sqrt(g*x + f)*sqrt(-36*x^2 + 30*x + 36)*sqrt(3*x + 2)/(32*f^9 + 192*f^8*g + 432*f^7*g^2 + 432*f^6*g^3 + 162*f^5*g^4 + 3*(16*f^4*g^5 + 96*f ^3*g^6 + 216*f^2*g^7 + 216*f*g^8 + 81*g^9)*x^6 + (240*f^5*g^4 + 1472*f^4*g ^5 + 3432*f^3*g^6 + 3672*f^2*g^7 + 1647*f*g^8 + 162*g^9)*x^5 + 10*(48*f^6* g^3 + 304*f^5*g^4 + 744*f^4*g^5 + 864*f^3*g^6 + 459*f^2*g^7 + 81*f*g^8)*x^ 4 + 10*(48*f^7*g^2 + 320*f^6*g^3 + 840*f^5*g^4 + 1080*f^4*g^5 + 675*f^3*g^ 6 + 162*f^2*g^7)*x^3 + 5*(48*f^8*g + 352*f^7*g^2 + 1032*f^6*g^3 + 1512*f^5 *g^4 + 1107*f^4*g^5 + 324*f^3*g^6)*x^2 + (48*f^9 + 448*f^8*g + 1608*f^7*g^ 2 + 2808*f^6*g^3 + 2403*f^5*g^4 + 810*f^4*g^5)*x)
Timed out. \[ \int \frac {\sqrt {2+3 x} \sqrt {1+\frac {5 x}{6}-x^2}}{(f+g x)^{11/2}} \, dx=\text {Timed out} \] Input:
integrate(1/6*(2+3*x)**(1/2)*(-36*x**2+30*x+36)**(1/2)/(g*x+f)**(11/2),x)
Output:
Timed out
\[ \int \frac {\sqrt {2+3 x} \sqrt {1+\frac {5 x}{6}-x^2}}{(f+g x)^{11/2}} \, dx=\int { \frac {\sqrt {-36 \, x^{2} + 30 \, x + 36} \sqrt {3 \, x + 2}}{6 \, {\left (g x + f\right )}^{\frac {11}{2}}} \,d x } \] Input:
integrate(1/6*(2+3*x)^(1/2)*(-36*x^2+30*x+36)^(1/2)/(g*x+f)^(11/2),x, algo rithm="maxima")
Output:
1/6*integrate(sqrt(-36*x^2 + 30*x + 36)*sqrt(3*x + 2)/(g*x + f)^(11/2), x)
Leaf count of result is larger than twice the leaf count of optimal. 341 vs. \(2 (153) = 306\).
Time = 0.77 (sec) , antiderivative size = 341, normalized size of antiderivative = 1.77 \[ \int \frac {\sqrt {2+3 x} \sqrt {1+\frac {5 x}{6}-x^2}}{(f+g x)^{11/2}} \, dx=\frac {8 \, \sqrt {6} {\left ({\left (4 \, {\left (\frac {2 \, {\left (6 \, \sqrt {2} f g^{6} + 35 \, \sqrt {2} g^{7}\right )} {\left (2 \, x - 3\right )}}{16 \, f^{4} g^{4} + 96 \, f^{3} g^{5} + 216 \, f^{2} g^{6} + 216 \, f g^{7} + 81 \, g^{8}} + \frac {9 \, {\left (12 \, \sqrt {2} f^{2} g^{5} + 88 \, \sqrt {2} f g^{6} + 105 \, \sqrt {2} g^{7}\right )}}{16 \, f^{4} g^{4} + 96 \, f^{3} g^{5} + 216 \, f^{2} g^{6} + 216 \, f g^{7} + 81 \, g^{8}}\right )} {\left (2 \, x - 3\right )} + \frac {63 \, {\left (24 \, \sqrt {2} f^{3} g^{4} + 212 \, \sqrt {2} f^{2} g^{5} + 474 \, \sqrt {2} f g^{6} + 315 \, \sqrt {2} g^{7}\right )}}{16 \, f^{4} g^{4} + 96 \, f^{3} g^{5} + 216 \, f^{2} g^{6} + 216 \, f g^{7} + 81 \, g^{8}}\right )} {\left (2 \, x - 3\right )} + \frac {1365 \, {\left (8 \, \sqrt {2} f^{3} g^{4} + 36 \, \sqrt {2} f^{2} g^{5} + 54 \, \sqrt {2} f g^{6} + 27 \, \sqrt {2} g^{7}\right )}}{16 \, f^{4} g^{4} + 96 \, f^{3} g^{5} + 216 \, f^{2} g^{6} + 216 \, f g^{7} + 81 \, g^{8}}\right )} {\left (2 \, x - 3\right )} \sqrt {-2 \, x + 3}}{945 \, {\left (g {\left (2 \, x - 3\right )} + 2 \, f + 3 \, g\right )}^{\frac {9}{2}}} \] Input:
integrate(1/6*(2+3*x)^(1/2)*(-36*x^2+30*x+36)^(1/2)/(g*x+f)^(11/2),x, algo rithm="giac")
Output:
8/945*sqrt(6)*((4*(2*(6*sqrt(2)*f*g^6 + 35*sqrt(2)*g^7)*(2*x - 3)/(16*f^4* g^4 + 96*f^3*g^5 + 216*f^2*g^6 + 216*f*g^7 + 81*g^8) + 9*(12*sqrt(2)*f^2*g ^5 + 88*sqrt(2)*f*g^6 + 105*sqrt(2)*g^7)/(16*f^4*g^4 + 96*f^3*g^5 + 216*f^ 2*g^6 + 216*f*g^7 + 81*g^8))*(2*x - 3) + 63*(24*sqrt(2)*f^3*g^4 + 212*sqrt (2)*f^2*g^5 + 474*sqrt(2)*f*g^6 + 315*sqrt(2)*g^7)/(16*f^4*g^4 + 96*f^3*g^ 5 + 216*f^2*g^6 + 216*f*g^7 + 81*g^8))*(2*x - 3) + 1365*(8*sqrt(2)*f^3*g^4 + 36*sqrt(2)*f^2*g^5 + 54*sqrt(2)*f*g^6 + 27*sqrt(2)*g^7)/(16*f^4*g^4 + 9 6*f^3*g^5 + 216*f^2*g^6 + 216*f*g^7 + 81*g^8))*(2*x - 3)*sqrt(-2*x + 3)/(g *(2*x - 3) + 2*f + 3*g)^(9/2)
Time = 11.91 (sec) , antiderivative size = 336, normalized size of antiderivative = 1.74 \[ \int \frac {\sqrt {2+3 x} \sqrt {1+\frac {5 x}{6}-x^2}}{(f+g x)^{11/2}} \, dx=\frac {\sqrt {-36\,x^2+30\,x+36}\,\left (\frac {x^2\,\sqrt {3\,x+2}\,\left (3024\,f^3+18936\,f^2\,g+7884\,f\,g^2+1890\,g^3\right )}{2835\,g^4\,{\left (2\,f+3\,g\right )}^4}-\frac {x\,\sqrt {3\,x+2}\,\left (-1848\,f^3+19332\,f^2\,g+25110\,f\,g^2+10395\,g^3\right )}{2835\,g^4\,{\left (2\,f+3\,g\right )}^4}-\frac {\sqrt {3\,x+2}\,\left (9576\,f^3+19440\,f^2\,g+17010\,f\,g^2+5670\,g^3\right )}{2835\,g^4\,{\left (2\,f+3\,g\right )}^4}+\frac {16\,x^3\,\sqrt {3\,x+2}\,\left (36\,f^2+216\,f\,g+35\,g^2\right )}{945\,g^3\,{\left (2\,f+3\,g\right )}^4}+\frac {64\,x^4\,\sqrt {3\,x+2}\,\left (6\,f+35\,g\right )}{2835\,g^2\,{\left (2\,f+3\,g\right )}^4}\right )}{x^5\,\sqrt {f+g\,x}+\frac {2\,f^4\,\sqrt {f+g\,x}}{3\,g^4}+\frac {2\,x^4\,\sqrt {f+g\,x}\,\left (6\,f+g\right )}{3\,g}+\frac {4\,f^2\,x^2\,\sqrt {f+g\,x}\,\left (f+g\right )}{g^3}+\frac {2\,f\,x^3\,\sqrt {f+g\,x}\,\left (9\,f+4\,g\right )}{3\,g^2}+\frac {f^3\,x\,\sqrt {f+g\,x}\,\left (3\,f+8\,g\right )}{3\,g^4}} \] Input:
int(((3*x + 2)^(1/2)*(30*x - 36*x^2 + 36)^(1/2))/(6*(f + g*x)^(11/2)),x)
Output:
((30*x - 36*x^2 + 36)^(1/2)*((x^2*(3*x + 2)^(1/2)*(7884*f*g^2 + 18936*f^2* g + 3024*f^3 + 1890*g^3))/(2835*g^4*(2*f + 3*g)^4) - (x*(3*x + 2)^(1/2)*(2 5110*f*g^2 + 19332*f^2*g - 1848*f^3 + 10395*g^3))/(2835*g^4*(2*f + 3*g)^4) - ((3*x + 2)^(1/2)*(17010*f*g^2 + 19440*f^2*g + 9576*f^3 + 5670*g^3))/(28 35*g^4*(2*f + 3*g)^4) + (16*x^3*(3*x + 2)^(1/2)*(216*f*g + 36*f^2 + 35*g^2 ))/(945*g^3*(2*f + 3*g)^4) + (64*x^4*(3*x + 2)^(1/2)*(6*f + 35*g))/(2835*g ^2*(2*f + 3*g)^4)))/(x^5*(f + g*x)^(1/2) + (2*f^4*(f + g*x)^(1/2))/(3*g^4) + (2*x^4*(f + g*x)^(1/2)*(6*f + g))/(3*g) + (4*f^2*x^2*(f + g*x)^(1/2)*(f + g))/g^3 + (2*f*x^3*(f + g*x)^(1/2)*(9*f + 4*g))/(3*g^2) + (f^3*x*(f + g *x)^(1/2)*(3*f + 8*g))/(3*g^4))
Time = 0.21 (sec) , antiderivative size = 383, normalized size of antiderivative = 1.98 \[ \int \frac {\sqrt {2+3 x} \sqrt {1+\frac {5 x}{6}-x^2}}{(f+g x)^{11/2}} \, dx=\frac {\sqrt {-2 x +3}\, \sqrt {6}\, \left (384 f \,g^{2} x^{4}+2240 g^{3} x^{4}+1728 f^{2} g \,x^{3}+10368 f \,g^{2} x^{3}+1680 g^{3} x^{3}+3024 f^{3} x^{2}+18936 f^{2} g \,x^{2}+7884 f \,g^{2} x^{2}+1890 g^{3} x^{2}+1848 f^{3} x -19332 f^{2} g x -25110 f \,g^{2} x -10395 g^{3} x -9576 f^{3}-19440 f^{2} g -17010 f \,g^{2}-5670 g^{3}\right )}{945 \sqrt {g x +f}\, \left (16 f^{4} g^{4} x^{4}+96 f^{3} g^{5} x^{4}+216 f^{2} g^{6} x^{4}+216 f \,g^{7} x^{4}+81 g^{8} x^{4}+64 f^{5} g^{3} x^{3}+384 f^{4} g^{4} x^{3}+864 f^{3} g^{5} x^{3}+864 f^{2} g^{6} x^{3}+324 f \,g^{7} x^{3}+96 f^{6} g^{2} x^{2}+576 f^{5} g^{3} x^{2}+1296 f^{4} g^{4} x^{2}+1296 f^{3} g^{5} x^{2}+486 f^{2} g^{6} x^{2}+64 f^{7} g x +384 f^{6} g^{2} x +864 f^{5} g^{3} x +864 f^{4} g^{4} x +324 f^{3} g^{5} x +16 f^{8}+96 f^{7} g +216 f^{6} g^{2}+216 f^{5} g^{3}+81 f^{4} g^{4}\right )} \] Input:
int(1/6*(2+3*x)^(1/2)*(-36*x^2+30*x+36)^(1/2)/(g*x+f)^(11/2),x)
Output:
(sqrt( - 2*x + 3)*sqrt(6)*(3024*f**3*x**2 + 1848*f**3*x - 9576*f**3 + 1728 *f**2*g*x**3 + 18936*f**2*g*x**2 - 19332*f**2*g*x - 19440*f**2*g + 384*f*g **2*x**4 + 10368*f*g**2*x**3 + 7884*f*g**2*x**2 - 25110*f*g**2*x - 17010*f *g**2 + 2240*g**3*x**4 + 1680*g**3*x**3 + 1890*g**3*x**2 - 10395*g**3*x - 5670*g**3))/(945*sqrt(f + g*x)*(16*f**8 + 64*f**7*g*x + 96*f**7*g + 96*f** 6*g**2*x**2 + 384*f**6*g**2*x + 216*f**6*g**2 + 64*f**5*g**3*x**3 + 576*f* *5*g**3*x**2 + 864*f**5*g**3*x + 216*f**5*g**3 + 16*f**4*g**4*x**4 + 384*f **4*g**4*x**3 + 1296*f**4*g**4*x**2 + 864*f**4*g**4*x + 81*f**4*g**4 + 96* f**3*g**5*x**4 + 864*f**3*g**5*x**3 + 1296*f**3*g**5*x**2 + 324*f**3*g**5* x + 216*f**2*g**6*x**4 + 864*f**2*g**6*x**3 + 486*f**2*g**6*x**2 + 216*f*g **7*x**4 + 324*f*g**7*x**3 + 81*g**8*x**4))