Integrand size = 35, antiderivative size = 241 \[ \int \frac {\sqrt {2+3 x} \sqrt {1+\frac {5 x}{6}-x^2}}{(f+g x)^{13/2}} \, dx=\frac {\sqrt {\frac {2}{3}} (3 f-2 g) (3-2 x)^{3/2}}{11 g (2 f+3 g) (f+g x)^{11/2}}-\frac {\sqrt {\frac {2}{3}} (18 f+131 g) (3-2 x)^{3/2}}{99 g (2 f+3 g)^2 (f+g x)^{9/2}}-\frac {4 \sqrt {\frac {2}{3}} (18 f+131 g) (3-2 x)^{3/2}}{231 g (2 f+3 g)^3 (f+g x)^{7/2}}-\frac {32 \sqrt {\frac {2}{3}} (18 f+131 g) (3-2 x)^{3/2}}{1155 g (2 f+3 g)^4 (f+g x)^{5/2}}-\frac {128 \sqrt {\frac {2}{3}} (18 f+131 g) (3-2 x)^{3/2}}{3465 g (2 f+3 g)^5 (f+g x)^{3/2}} \] Output:
1/33*6^(1/2)*(3*f-2*g)*(3-2*x)^(3/2)/g/(2*f+3*g)/(g*x+f)^(11/2)-1/297*6^(1 /2)*(18*f+131*g)*(3-2*x)^(3/2)/g/(2*f+3*g)^2/(g*x+f)^(9/2)-4/693*6^(1/2)*( 18*f+131*g)*(3-2*x)^(3/2)/g/(2*f+3*g)^3/(g*x+f)^(7/2)-32/3465*6^(1/2)*(18* f+131*g)*(3-2*x)^(3/2)/g/(2*f+3*g)^4/(g*x+f)^(5/2)-128/10395*6^(1/2)*(18*f +131*g)*(3-2*x)^(3/2)/g/(2*f+3*g)^5/(g*x+f)^(3/2)
Time = 8.34 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.63 \[ \int \frac {\sqrt {2+3 x} \sqrt {1+\frac {5 x}{6}-x^2}}{(f+g x)^{13/2}} \, dx=\frac {(-3+2 x) \sqrt {4+\frac {10 x}{3}-4 x^2} \left (3696 f^4 (19+9 x)+264 f^3 g \left (747+1160 x+108 x^2\right )+396 f^2 g^2 \left (675+1314 x+596 x^2+32 x^3\right )+2 f g^3 \left (91665+203040 x+108612 x^2+48704 x^3+1152 x^4\right )+g^4 \left (51030+123795 x+70740 x^2+37728 x^3+16768 x^4\right )\right )}{3465 (2 f+3 g)^5 \sqrt {2+3 x} (f+g x)^{11/2}} \] Input:
Integrate[(Sqrt[2 + 3*x]*Sqrt[1 + (5*x)/6 - x^2])/(f + g*x)^(13/2),x]
Output:
((-3 + 2*x)*Sqrt[4 + (10*x)/3 - 4*x^2]*(3696*f^4*(19 + 9*x) + 264*f^3*g*(7 47 + 1160*x + 108*x^2) + 396*f^2*g^2*(675 + 1314*x + 596*x^2 + 32*x^3) + 2 *f*g^3*(91665 + 203040*x + 108612*x^2 + 48704*x^3 + 1152*x^4) + g^4*(51030 + 123795*x + 70740*x^2 + 37728*x^3 + 16768*x^4)))/(3465*(2*f + 3*g)^5*Sqr t[2 + 3*x]*(f + g*x)^(11/2))
Time = 0.30 (sec) , antiderivative size = 230, normalized size of antiderivative = 0.95, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1245, 87, 27, 55, 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.
| \(\displaystyle \int \frac {\sqrt {3 x+2} \sqrt {-x^2+\frac {5 x}{6}+1}}{(f+g x)^{13/2}} \, dx\) |
| \(\Big \downarrow \) 1245 |
| \(\displaystyle \int \frac {\sqrt {\frac {1}{2}-\frac {x}{3}} (3 x+2)}{(f+g x)^{13/2}}dx\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle \frac {(18 f+131 g) \int \frac {\sqrt {3-2 x}}{\sqrt {6} (f+g x)^{11/2}}dx}{11 g (2 f+3 g)}+\frac {\sqrt {\frac {2}{3}} (3-2 x)^{3/2} (3 f-2 g)}{11 g (2 f+3 g) (f+g x)^{11/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(18 f+131 g) \int \frac {\sqrt {3-2 x}}{(f+g x)^{11/2}}dx}{11 \sqrt {6} g (2 f+3 g)}+\frac {\sqrt {\frac {2}{3}} (3-2 x)^{3/2} (3 f-2 g)}{11 g (2 f+3 g) (f+g x)^{11/2}}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle \frac {(18 f+131 g) \left (\frac {4 \int \frac {\sqrt {3-2 x}}{(f+g x)^{9/2}}dx}{3 (2 f+3 g)}-\frac {2 (3-2 x)^{3/2}}{9 (2 f+3 g) (f+g x)^{9/2}}\right )}{11 \sqrt {6} g (2 f+3 g)}+\frac {\sqrt {\frac {2}{3}} (3-2 x)^{3/2} (3 f-2 g)}{11 g (2 f+3 g) (f+g x)^{11/2}}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle \frac {(18 f+131 g) \left (\frac {4 \left (\frac {8 \int \frac {\sqrt {3-2 x}}{(f+g x)^{7/2}}dx}{7 (2 f+3 g)}-\frac {2 (3-2 x)^{3/2}}{7 (2 f+3 g) (f+g x)^{7/2}}\right )}{3 (2 f+3 g)}-\frac {2 (3-2 x)^{3/2}}{9 (2 f+3 g) (f+g x)^{9/2}}\right )}{11 \sqrt {6} g (2 f+3 g)}+\frac {\sqrt {\frac {2}{3}} (3-2 x)^{3/2} (3 f-2 g)}{11 g (2 f+3 g) (f+g x)^{11/2}}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle \frac {(18 f+131 g) \left (\frac {4 \left (\frac {8 \left (\frac {4 \int \frac {\sqrt {3-2 x}}{(f+g x)^{5/2}}dx}{5 (2 f+3 g)}-\frac {2 (3-2 x)^{3/2}}{5 (2 f+3 g) (f+g x)^{5/2}}\right )}{7 (2 f+3 g)}-\frac {2 (3-2 x)^{3/2}}{7 (2 f+3 g) (f+g x)^{7/2}}\right )}{3 (2 f+3 g)}-\frac {2 (3-2 x)^{3/2}}{9 (2 f+3 g) (f+g x)^{9/2}}\right )}{11 \sqrt {6} g (2 f+3 g)}+\frac {\sqrt {\frac {2}{3}} (3-2 x)^{3/2} (3 f-2 g)}{11 g (2 f+3 g) (f+g x)^{11/2}}\) |
| \(\Big \downarrow \) 48 |
| \(\displaystyle \frac {\sqrt {\frac {2}{3}} (3-2 x)^{3/2} (3 f-2 g)}{11 g (2 f+3 g) (f+g x)^{11/2}}+\frac {(18 f+131 g) \left (\frac {4 \left (\frac {8 \left (-\frac {8 (3-2 x)^{3/2}}{15 (2 f+3 g)^2 (f+g x)^{3/2}}-\frac {2 (3-2 x)^{3/2}}{5 (2 f+3 g) (f+g x)^{5/2}}\right )}{7 (2 f+3 g)}-\frac {2 (3-2 x)^{3/2}}{7 (2 f+3 g) (f+g x)^{7/2}}\right )}{3 (2 f+3 g)}-\frac {2 (3-2 x)^{3/2}}{9 (2 f+3 g) (f+g x)^{9/2}}\right )}{11 \sqrt {6} g (2 f+3 g)}\) |
Input:
Int[(Sqrt[2 + 3*x]*Sqrt[1 + (5*x)/6 - x^2])/(f + g*x)^(13/2),x]
Output:
(Sqrt[2/3]*(3*f - 2*g)*(3 - 2*x)^(3/2))/(11*g*(2*f + 3*g)*(f + g*x)^(11/2) ) + ((18*f + 131*g)*((-2*(3 - 2*x)^(3/2))/(9*(2*f + 3*g)*(f + g*x)^(9/2)) + (4*((-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*(2*f + 3*g))))/(11*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 = 1.45 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.79
| method | result | size |
| default | \(\frac {\sqrt {-36 x^{2}+30 x +36}\, \left (2 x -3\right ) \left (2304 g^{3} x^{4} f +16768 g^{4} x^{4}+12672 f^{2} g^{2} x^{3}+97408 f \,g^{3} x^{3}+37728 g^{4} x^{3}+28512 f^{3} g \,x^{2}+236016 f^{2} g^{2} x^{2}+217224 f \,g^{3} x^{2}+70740 g^{4} x^{2}+33264 f^{4} x +306240 f^{3} g x +520344 f^{2} g^{2} x +406080 f \,g^{3} x +123795 g^{4} x +70224 f^{4}+197208 g \,f^{3}+267300 g^{2} f^{2}+183330 f \,g^{3}+51030 g^{4}\right )}{10395 \sqrt {3 x +2}\, \left (g x +f \right )^{\frac {11}{2}} \left (2 f +3 g \right )^{5}}\) | \(191\) |
| gosper | \(\frac {\left (2 x -3\right ) \left (2304 g^{3} x^{4} f +16768 g^{4} x^{4}+12672 f^{2} g^{2} x^{3}+97408 f \,g^{3} x^{3}+37728 g^{4} x^{3}+28512 f^{3} g \,x^{2}+236016 f^{2} g^{2} x^{2}+217224 f \,g^{3} x^{2}+70740 g^{4} x^{2}+33264 f^{4} x +306240 f^{3} g x +520344 f^{2} g^{2} x +406080 f \,g^{3} x +123795 g^{4} x +70224 f^{4}+197208 g \,f^{3}+267300 g^{2} f^{2}+183330 f \,g^{3}+51030 g^{4}\right ) \sqrt {-36 x^{2}+30 x +36}}{10395 \left (g x +f \right )^{\frac {11}{2}} \left (32 f^{5}+240 f^{4} g +720 f^{3} g^{2}+1080 f^{2} g^{3}+810 f \,g^{4}+243 g^{5}\right ) \sqrt {3 x +2}}\) | \(223\) |
| orering | \(\frac {\left (2 x -3\right ) \left (2304 g^{3} x^{4} f +16768 g^{4} x^{4}+12672 f^{2} g^{2} x^{3}+97408 f \,g^{3} x^{3}+37728 g^{4} x^{3}+28512 f^{3} g \,x^{2}+236016 f^{2} g^{2} x^{2}+217224 f \,g^{3} x^{2}+70740 g^{4} x^{2}+33264 f^{4} x +306240 f^{3} g x +520344 f^{2} g^{2} x +406080 f \,g^{3} x +123795 g^{4} x +70224 f^{4}+197208 g \,f^{3}+267300 g^{2} f^{2}+183330 f \,g^{3}+51030 g^{4}\right ) \sqrt {-36 x^{2}+30 x +36}}{10395 \left (g x +f \right )^{\frac {11}{2}} \left (32 f^{5}+240 f^{4} g +720 f^{3} g^{2}+1080 f^{2} g^{3}+810 f \,g^{4}+243 g^{5}\right ) \sqrt {3 x +2}}\) | \(223\) |
Input:
int(1/6*(3*x+2)^(1/2)*(-36*x^2+30*x+36)^(1/2)/(g*x+f)^(13/2),x,method=_RET URNVERBOSE)
Output:
1/10395/(3*x+2)^(1/2)*(-36*x^2+30*x+36)^(1/2)/(g*x+f)^(11/2)*(2*x-3)*(2304 *f*g^3*x^4+16768*g^4*x^4+12672*f^2*g^2*x^3+97408*f*g^3*x^3+37728*g^4*x^3+2 8512*f^3*g*x^2+236016*f^2*g^2*x^2+217224*f*g^3*x^2+70740*g^4*x^2+33264*f^4 *x+306240*f^3*g*x+520344*f^2*g^2*x+406080*f*g^3*x+123795*g^4*x+70224*f^4+1 97208*f^3*g+267300*f^2*g^2+183330*f*g^3+51030*g^4)/(2*f+3*g)^5
Leaf count of result is larger than twice the leaf count of optimal. 653 vs. \(2 (191) = 382\).
Time = 0.11 (sec) , antiderivative size = 653, normalized size of antiderivative = 2.71 \[ \int \frac {\sqrt {2+3 x} \sqrt {1+\frac {5 x}{6}-x^2}}{(f+g x)^{13/2}} \, dx =\text {Too large to display} \] Input:
integrate(1/6*(2+3*x)^(1/2)*(-36*x^2+30*x+36)^(1/2)/(g*x+f)^(13/2),x, algo rithm="fricas")
Output:
1/10395*(256*(18*f*g^3 + 131*g^4)*x^5 + 64*(396*f^2*g^2 + 2936*f*g^3 + 393 *g^4)*x^4 - 210672*f^4 - 591624*f^3*g - 801900*f^2*g^2 - 549990*f*g^3 - 15 3090*g^4 + 24*(2376*f^3*g + 18084*f^2*g^2 + 5926*f*g^3 + 1179*g^4)*x^3 + 6 *(11088*f^4 + 87824*f^3*g + 55440*f^2*g^2 + 26748*f*g^3 + 5895*g^4)*x^2 + 3*(13552*f^4 - 174768*f^3*g - 342144*f^2*g^2 - 283860*f*g^3 - 89775*g^4)*x )*sqrt(g*x + f)*sqrt(-36*x^2 + 30*x + 36)*sqrt(3*x + 2)/(64*f^11 + 480*f^1 0*g + 1440*f^9*g^2 + 2160*f^8*g^3 + 1620*f^7*g^4 + 486*f^6*g^5 + 3*(32*f^5 *g^6 + 240*f^4*g^7 + 720*f^3*g^8 + 1080*f^2*g^9 + 810*f*g^10 + 243*g^11)*x ^7 + 2*(288*f^6*g^5 + 2192*f^5*g^6 + 6720*f^4*g^7 + 10440*f^3*g^8 + 8370*f ^2*g^9 + 2997*f*g^10 + 243*g^11)*x^6 + 3*(480*f^7*g^4 + 3728*f^6*g^5 + 117 60*f^5*g^6 + 19080*f^4*g^7 + 16470*f^3*g^8 + 6885*f^2*g^9 + 972*f*g^10)*x^ 5 + 30*(64*f^8*g^3 + 512*f^7*g^4 + 1680*f^6*g^5 + 2880*f^5*g^6 + 2700*f^4* g^7 + 1296*f^3*g^8 + 243*f^2*g^9)*x^4 + 5*(288*f^9*g^2 + 2416*f^8*g^3 + 84 00*f^7*g^4 + 15480*f^6*g^5 + 15930*f^5*g^6 + 8667*f^4*g^7 + 1944*f^3*g^8)* x^3 + 6*(96*f^10*g + 880*f^9*g^2 + 3360*f^8*g^3 + 6840*f^7*g^4 + 7830*f^6* g^5 + 4779*f^5*g^6 + 1215*f^4*g^7)*x^2 + 3*(32*f^11 + 368*f^10*g + 1680*f^ 9*g^2 + 3960*f^8*g^3 + 5130*f^7*g^4 + 3483*f^6*g^5 + 972*f^5*g^6)*x)
Timed out. \[ \int \frac {\sqrt {2+3 x} \sqrt {1+\frac {5 x}{6}-x^2}}{(f+g x)^{13/2}} \, dx=\text {Timed out} \] Input:
integrate(1/6*(2+3*x)**(1/2)*(-36*x**2+30*x+36)**(1/2)/(g*x+f)**(13/2),x)
Output:
Timed out
\[ \int \frac {\sqrt {2+3 x} \sqrt {1+\frac {5 x}{6}-x^2}}{(f+g x)^{13/2}} \, dx=\int { \frac {\sqrt {-36 \, x^{2} + 30 \, x + 36} \sqrt {3 \, x + 2}}{6 \, {\left (g x + f\right )}^{\frac {13}{2}}} \,d x } \] Input:
integrate(1/6*(2+3*x)^(1/2)*(-36*x^2+30*x+36)^(1/2)/(g*x+f)^(13/2),x, algo rithm="maxima")
Output:
1/6*integrate(sqrt(-36*x^2 + 30*x + 36)*sqrt(3*x + 2)/(g*x + f)^(13/2), x)
Leaf count of result is larger than twice the leaf count of optimal. 491 vs. \(2 (191) = 382\).
Time = 0.41 (sec) , antiderivative size = 491, normalized size of antiderivative = 2.04 \[ \int \frac {\sqrt {2+3 x} \sqrt {1+\frac {5 x}{6}-x^2}}{(f+g x)^{13/2}} \, dx=\frac {16 \, \sqrt {6} {\left ({\left (2 \, {\left (4 \, {\left (\frac {2 \, {\left (18 \, \sqrt {2} f g^{8} + 131 \, \sqrt {2} g^{9}\right )} {\left (2 \, x - 3\right )}}{32 \, f^{5} g^{5} + 240 \, f^{4} g^{6} + 720 \, f^{3} g^{7} + 1080 \, f^{2} g^{8} + 810 \, f g^{9} + 243 \, g^{10}} + \frac {11 \, {\left (36 \, \sqrt {2} f^{2} g^{7} + 316 \, \sqrt {2} f g^{8} + 393 \, \sqrt {2} g^{9}\right )}}{32 \, f^{5} g^{5} + 240 \, f^{4} g^{6} + 720 \, f^{3} g^{7} + 1080 \, f^{2} g^{8} + 810 \, f g^{9} + 243 \, g^{10}}\right )} {\left (2 \, x - 3\right )} + \frac {99 \, {\left (72 \, \sqrt {2} f^{3} g^{6} + 740 \, \sqrt {2} f^{2} g^{7} + 1734 \, \sqrt {2} f g^{8} + 1179 \, \sqrt {2} g^{9}\right )}}{32 \, f^{5} g^{5} + 240 \, f^{4} g^{6} + 720 \, f^{3} g^{7} + 1080 \, f^{2} g^{8} + 810 \, f g^{9} + 243 \, g^{10}}\right )} {\left (2 \, x - 3\right )} + \frac {231 \, {\left (144 \, \sqrt {2} f^{4} g^{5} + 1696 \, \sqrt {2} f^{3} g^{6} + 5688 \, \sqrt {2} f^{2} g^{7} + 7560 \, \sqrt {2} f g^{8} + 3537 \, \sqrt {2} g^{9}\right )}}{32 \, f^{5} g^{5} + 240 \, f^{4} g^{6} + 720 \, f^{3} g^{7} + 1080 \, f^{2} g^{8} + 810 \, f g^{9} + 243 \, g^{10}}\right )} {\left (2 \, x - 3\right )} + \frac {15015 \, {\left (16 \, \sqrt {2} f^{4} g^{5} + 96 \, \sqrt {2} f^{3} g^{6} + 216 \, \sqrt {2} f^{2} g^{7} + 216 \, \sqrt {2} f g^{8} + 81 \, \sqrt {2} g^{9}\right )}}{32 \, f^{5} g^{5} + 240 \, f^{4} g^{6} + 720 \, f^{3} g^{7} + 1080 \, f^{2} g^{8} + 810 \, f g^{9} + 243 \, g^{10}}\right )} {\left (2 \, x - 3\right )} \sqrt {-2 \, x + 3}}{10395 \, {\left (g {\left (2 \, x - 3\right )} + 2 \, f + 3 \, g\right )}^{\frac {11}{2}}} \] Input:
integrate(1/6*(2+3*x)^(1/2)*(-36*x^2+30*x+36)^(1/2)/(g*x+f)^(13/2),x, algo rithm="giac")
Output:
16/10395*sqrt(6)*((2*(4*(2*(18*sqrt(2)*f*g^8 + 131*sqrt(2)*g^9)*(2*x - 3)/ (32*f^5*g^5 + 240*f^4*g^6 + 720*f^3*g^7 + 1080*f^2*g^8 + 810*f*g^9 + 243*g ^10) + 11*(36*sqrt(2)*f^2*g^7 + 316*sqrt(2)*f*g^8 + 393*sqrt(2)*g^9)/(32*f ^5*g^5 + 240*f^4*g^6 + 720*f^3*g^7 + 1080*f^2*g^8 + 810*f*g^9 + 243*g^10)) *(2*x - 3) + 99*(72*sqrt(2)*f^3*g^6 + 740*sqrt(2)*f^2*g^7 + 1734*sqrt(2)*f *g^8 + 1179*sqrt(2)*g^9)/(32*f^5*g^5 + 240*f^4*g^6 + 720*f^3*g^7 + 1080*f^ 2*g^8 + 810*f*g^9 + 243*g^10))*(2*x - 3) + 231*(144*sqrt(2)*f^4*g^5 + 1696 *sqrt(2)*f^3*g^6 + 5688*sqrt(2)*f^2*g^7 + 7560*sqrt(2)*f*g^8 + 3537*sqrt(2 )*g^9)/(32*f^5*g^5 + 240*f^4*g^6 + 720*f^3*g^7 + 1080*f^2*g^8 + 810*f*g^9 + 243*g^10))*(2*x - 3) + 15015*(16*sqrt(2)*f^4*g^5 + 96*sqrt(2)*f^3*g^6 + 216*sqrt(2)*f^2*g^7 + 216*sqrt(2)*f*g^8 + 81*sqrt(2)*g^9)/(32*f^5*g^5 + 24 0*f^4*g^6 + 720*f^3*g^7 + 1080*f^2*g^8 + 810*f*g^9 + 243*g^10))*(2*x - 3)* sqrt(-2*x + 3)/(g*(2*x - 3) + 2*f + 3*g)^(11/2)
Time = 12.91 (sec) , antiderivative size = 436, normalized size of antiderivative = 1.81 \[ \int \frac {\sqrt {2+3 x} \sqrt {1+\frac {5 x}{6}-x^2}}{(f+g x)^{13/2}} \, dx=\frac {\sqrt {-36\,x^2+30\,x+36}\,\left (\frac {x^2\,\sqrt {3\,x+2}\,\left (66528\,f^4+526944\,f^3\,g+332640\,f^2\,g^2+160488\,f\,g^3+35370\,g^4\right )}{31185\,g^5\,{\left (2\,f+3\,g\right )}^5}-\frac {x\,\sqrt {3\,x+2}\,\left (-40656\,f^4+524304\,f^3\,g+1026432\,f^2\,g^2+851580\,f\,g^3+269325\,g^4\right )}{31185\,g^5\,{\left (2\,f+3\,g\right )}^5}-\frac {\sqrt {3\,x+2}\,\left (210672\,f^4+591624\,f^3\,g+801900\,f^2\,g^2+549990\,f\,g^3+153090\,g^4\right )}{31185\,g^5\,{\left (2\,f+3\,g\right )}^5}+\frac {8\,x^3\,\sqrt {3\,x+2}\,\left (2376\,f^3+18084\,f^2\,g+5926\,f\,g^2+1179\,g^3\right )}{10395\,g^4\,{\left (2\,f+3\,g\right )}^5}+\frac {64\,x^4\,\sqrt {3\,x+2}\,\left (396\,f^2+2936\,f\,g+393\,g^2\right )}{31185\,g^3\,{\left (2\,f+3\,g\right )}^5}+\frac {256\,x^5\,\sqrt {3\,x+2}\,\left (18\,f+131\,g\right )}{31185\,g^2\,{\left (2\,f+3\,g\right )}^5}\right )}{x^6\,\sqrt {f+g\,x}+\frac {2\,f^5\,\sqrt {f+g\,x}}{3\,g^5}+\frac {x^5\,\sqrt {f+g\,x}\,\left (15\,f+2\,g\right )}{3\,g}+\frac {10\,f^2\,x^3\,\sqrt {f+g\,x}\,\left (3\,f+2\,g\right )}{3\,g^3}+\frac {5\,f^3\,x^2\,\sqrt {f+g\,x}\,\left (3\,f+4\,g\right )}{3\,g^4}+\frac {10\,f\,x^4\,\sqrt {f+g\,x}\,\left (3\,f+g\right )}{3\,g^2}+\frac {f^4\,x\,\sqrt {f+g\,x}\,\left (3\,f+10\,g\right )}{3\,g^5}} \] Input:
int(((3*x + 2)^(1/2)*(30*x - 36*x^2 + 36)^(1/2))/(6*(f + g*x)^(13/2)),x)
Output:
((30*x - 36*x^2 + 36)^(1/2)*((x^2*(3*x + 2)^(1/2)*(160488*f*g^3 + 526944*f ^3*g + 66528*f^4 + 35370*g^4 + 332640*f^2*g^2))/(31185*g^5*(2*f + 3*g)^5) - (x*(3*x + 2)^(1/2)*(851580*f*g^3 + 524304*f^3*g - 40656*f^4 + 269325*g^4 + 1026432*f^2*g^2))/(31185*g^5*(2*f + 3*g)^5) - ((3*x + 2)^(1/2)*(549990* f*g^3 + 591624*f^3*g + 210672*f^4 + 153090*g^4 + 801900*f^2*g^2))/(31185*g ^5*(2*f + 3*g)^5) + (8*x^3*(3*x + 2)^(1/2)*(5926*f*g^2 + 18084*f^2*g + 237 6*f^3 + 1179*g^3))/(10395*g^4*(2*f + 3*g)^5) + (64*x^4*(3*x + 2)^(1/2)*(29 36*f*g + 396*f^2 + 393*g^2))/(31185*g^3*(2*f + 3*g)^5) + (256*x^5*(3*x + 2 )^(1/2)*(18*f + 131*g))/(31185*g^2*(2*f + 3*g)^5)))/(x^6*(f + g*x)^(1/2) + (2*f^5*(f + g*x)^(1/2))/(3*g^5) + (x^5*(f + g*x)^(1/2)*(15*f + 2*g))/(3*g ) + (10*f^2*x^3*(f + g*x)^(1/2)*(3*f + 2*g))/(3*g^3) + (5*f^3*x^2*(f + g*x )^(1/2)*(3*f + 4*g))/(3*g^4) + (10*f*x^4*(f + g*x)^(1/2)*(3*f + g))/(3*g^2 ) + (f^4*x*(f + g*x)^(1/2)*(3*f + 10*g))/(3*g^5))
Time = 0.22 (sec) , antiderivative size = 566, normalized size of antiderivative = 2.35 \[ \int \frac {\sqrt {2+3 x} \sqrt {1+\frac {5 x}{6}-x^2}}{(f+g x)^{13/2}} \, dx=\frac {\sqrt {-2 x +3}\, \sqrt {6}\, \left (4608 f \,g^{3} x^{5}+33536 g^{4} x^{5}+25344 f^{2} g^{2} x^{4}+187904 f \,g^{3} x^{4}+25152 g^{4} x^{4}+57024 f^{3} g \,x^{3}+434016 f^{2} g^{2} x^{3}+142224 f \,g^{3} x^{3}+28296 g^{4} x^{3}+66528 f^{4} x^{2}+526944 f^{3} g \,x^{2}+332640 f^{2} g^{2} x^{2}+160488 f \,g^{3} x^{2}+35370 g^{4} x^{2}+40656 f^{4} x -524304 f^{3} g x -1026432 f^{2} g^{2} x -851580 f \,g^{3} x -269325 g^{4} x -210672 f^{4}-591624 f^{3} g -801900 f^{2} g^{2}-549990 f \,g^{3}-153090 g^{4}\right )}{10395 \sqrt {g x +f}\, \left (32 f^{5} g^{5} x^{5}+240 f^{4} g^{6} x^{5}+720 f^{3} g^{7} x^{5}+1080 f^{2} g^{8} x^{5}+810 f \,g^{9} x^{5}+243 g^{10} x^{5}+160 f^{6} g^{4} x^{4}+1200 f^{5} g^{5} x^{4}+3600 f^{4} g^{6} x^{4}+5400 f^{3} g^{7} x^{4}+4050 f^{2} g^{8} x^{4}+1215 f \,g^{9} x^{4}+320 f^{7} g^{3} x^{3}+2400 f^{6} g^{4} x^{3}+7200 f^{5} g^{5} x^{3}+10800 f^{4} g^{6} x^{3}+8100 f^{3} g^{7} x^{3}+2430 f^{2} g^{8} x^{3}+320 f^{8} g^{2} x^{2}+2400 f^{7} g^{3} x^{2}+7200 f^{6} g^{4} x^{2}+10800 f^{5} g^{5} x^{2}+8100 f^{4} g^{6} x^{2}+2430 f^{3} g^{7} x^{2}+160 f^{9} g x +1200 f^{8} g^{2} x +3600 f^{7} g^{3} x +5400 f^{6} g^{4} x +4050 f^{5} g^{5} x +1215 f^{4} g^{6} x +32 f^{10}+240 f^{9} g +720 f^{8} g^{2}+1080 f^{7} g^{3}+810 f^{6} g^{4}+243 f^{5} g^{5}\right )} \] Input:
int(1/6*(2+3*x)^(1/2)*(-36*x^2+30*x+36)^(1/2)/(g*x+f)^(13/2),x)
Output:
(sqrt( - 2*x + 3)*sqrt(6)*(66528*f**4*x**2 + 40656*f**4*x - 210672*f**4 + 57024*f**3*g*x**3 + 526944*f**3*g*x**2 - 524304*f**3*g*x - 591624*f**3*g + 25344*f**2*g**2*x**4 + 434016*f**2*g**2*x**3 + 332640*f**2*g**2*x**2 - 10 26432*f**2*g**2*x - 801900*f**2*g**2 + 4608*f*g**3*x**5 + 187904*f*g**3*x* *4 + 142224*f*g**3*x**3 + 160488*f*g**3*x**2 - 851580*f*g**3*x - 549990*f* g**3 + 33536*g**4*x**5 + 25152*g**4*x**4 + 28296*g**4*x**3 + 35370*g**4*x* *2 - 269325*g**4*x - 153090*g**4))/(10395*sqrt(f + g*x)*(32*f**10 + 160*f* *9*g*x + 240*f**9*g + 320*f**8*g**2*x**2 + 1200*f**8*g**2*x + 720*f**8*g** 2 + 320*f**7*g**3*x**3 + 2400*f**7*g**3*x**2 + 3600*f**7*g**3*x + 1080*f** 7*g**3 + 160*f**6*g**4*x**4 + 2400*f**6*g**4*x**3 + 7200*f**6*g**4*x**2 + 5400*f**6*g**4*x + 810*f**6*g**4 + 32*f**5*g**5*x**5 + 1200*f**5*g**5*x**4 + 7200*f**5*g**5*x**3 + 10800*f**5*g**5*x**2 + 4050*f**5*g**5*x + 243*f** 5*g**5 + 240*f**4*g**6*x**5 + 3600*f**4*g**6*x**4 + 10800*f**4*g**6*x**3 + 8100*f**4*g**6*x**2 + 1215*f**4*g**6*x + 720*f**3*g**7*x**5 + 5400*f**3*g **7*x**4 + 8100*f**3*g**7*x**3 + 2430*f**3*g**7*x**2 + 1080*f**2*g**8*x**5 + 4050*f**2*g**8*x**4 + 2430*f**2*g**8*x**3 + 810*f*g**9*x**5 + 1215*f*g* *9*x**4 + 243*g**10*x**5))