Integrand size = 35, antiderivative size = 119 \[ \int \frac {\sqrt {2+3 x} \sqrt {1+\frac {5 x}{6}-x^2}}{(f+g x)^{5/2}} \, dx=\frac {\sqrt {\frac {2}{3}} (3 f-2 g) (3-2 x)^{3/2}}{3 g (2 f+3 g) (f+g x)^{3/2}}-\frac {\sqrt {6} \sqrt {3-2 x}}{g^2 \sqrt {f+g x}}+\frac {2 \sqrt {3} \arctan \left (\frac {\sqrt {g} \sqrt {3-2 x}}{\sqrt {2} \sqrt {f+g x}}\right )}{g^{5/2}} \] Output:
1/9*6^(1/2)*(3*f-2*g)*(3-2*x)^(3/2)/g/(2*f+3*g)/(g*x+f)^(3/2)-6^(1/2)*(3-2 *x)^(1/2)/g^2/(g*x+f)^(1/2)+2*3^(1/2)*arctan(1/2*g^(1/2)*(3-2*x)^(1/2)*2^( 1/2)/(g*x+f)^(1/2))/g^(5/2)
Time = 10.93 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.39 \[ \int \frac {\sqrt {2+3 x} \sqrt {1+\frac {5 x}{6}-x^2}}{(f+g x)^{5/2}} \, dx=\frac {\sqrt {6+5 x-6 x^2} \left (\frac {\sqrt {g} \left (18 f^2+6 f g (3+4 x)+g^2 (6+23 x)\right )}{-2 f-3 g}+\frac {9 \sqrt {6-4 x} (f+g x)^2 \text {arcsinh}\left (\frac {\sqrt {g} \sqrt {3-2 x}}{\sqrt {-2 f-3 g}}\right )}{\sqrt {-2 f-3 g} (-3+2 x) \sqrt {\frac {f+g x}{2 f+3 g}}}\right )}{3 g^{5/2} \sqrt {3+\frac {9 x}{2}} (f+g x)^{3/2}} \] Input:
Integrate[(Sqrt[2 + 3*x]*Sqrt[1 + (5*x)/6 - x^2])/(f + g*x)^(5/2),x]
Output:
(Sqrt[6 + 5*x - 6*x^2]*((Sqrt[g]*(18*f^2 + 6*f*g*(3 + 4*x) + g^2*(6 + 23*x )))/(-2*f - 3*g) + (9*Sqrt[6 - 4*x]*(f + g*x)^2*ArcSinh[(Sqrt[g]*Sqrt[3 - 2*x])/Sqrt[-2*f - 3*g]])/(Sqrt[-2*f - 3*g]*(-3 + 2*x)*Sqrt[(f + g*x)/(2*f + 3*g)])))/(3*g^(5/2)*Sqrt[3 + (9*x)/2]*(f + g*x)^(3/2))
Time = 0.25 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.06, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {1245, 87, 27, 57, 66, 217}
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.
| \(\displaystyle \int \frac {\sqrt {3 x+2} \sqrt {-x^2+\frac {5 x}{6}+1}}{(f+g x)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 1245 |
| \(\displaystyle \int \frac {\sqrt {\frac {1}{2}-\frac {x}{3}} (3 x+2)}{(f+g x)^{5/2}}dx\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle \frac {3 \int \frac {\sqrt {3-2 x}}{\sqrt {6} (f+g x)^{3/2}}dx}{g}+\frac {\sqrt {\frac {2}{3}} (3-2 x)^{3/2} (3 f-2 g)}{3 g (2 f+3 g) (f+g x)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {\frac {3}{2}} \int \frac {\sqrt {3-2 x}}{(f+g x)^{3/2}}dx}{g}+\frac {\sqrt {\frac {2}{3}} (3-2 x)^{3/2} (3 f-2 g)}{3 g (2 f+3 g) (f+g x)^{3/2}}\) |
| \(\Big \downarrow \) 57 |
| \(\displaystyle \frac {\sqrt {\frac {3}{2}} \left (-\frac {2 \int \frac {1}{\sqrt {3-2 x} \sqrt {f+g x}}dx}{g}-\frac {2 \sqrt {3-2 x}}{g \sqrt {f+g x}}\right )}{g}+\frac {\sqrt {\frac {2}{3}} (3-2 x)^{3/2} (3 f-2 g)}{3 g (2 f+3 g) (f+g x)^{3/2}}\) |
| \(\Big \downarrow \) 66 |
| \(\displaystyle \frac {\sqrt {\frac {3}{2}} \left (-\frac {4 \int \frac {1}{-\frac {g (3-2 x)}{f+g x}-2}d\frac {\sqrt {3-2 x}}{\sqrt {f+g x}}}{g}-\frac {2 \sqrt {3-2 x}}{g \sqrt {f+g x}}\right )}{g}+\frac {\sqrt {\frac {2}{3}} (3-2 x)^{3/2} (3 f-2 g)}{3 g (2 f+3 g) (f+g x)^{3/2}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\sqrt {\frac {3}{2}} \left (\frac {2 \sqrt {2} \arctan \left (\frac {\sqrt {g} \sqrt {3-2 x}}{\sqrt {2} \sqrt {f+g x}}\right )}{g^{3/2}}-\frac {2 \sqrt {3-2 x}}{g \sqrt {f+g x}}\right )}{g}+\frac {\sqrt {\frac {2}{3}} (3-2 x)^{3/2} (3 f-2 g)}{3 g (2 f+3 g) (f+g x)^{3/2}}\) |
Input:
Int[(Sqrt[2 + 3*x]*Sqrt[1 + (5*x)/6 - x^2])/(f + g*x)^(5/2),x]
Output:
(Sqrt[2/3]*(3*f - 2*g)*(3 - 2*x)^(3/2))/(3*g*(2*f + 3*g)*(f + g*x)^(3/2)) + (Sqrt[3/2]*((-2*Sqrt[3 - 2*x])/(g*Sqrt[f + g*x]) + (2*Sqrt[2]*ArcTan[(Sq rt[g]*Sqrt[3 - 2*x])/(Sqrt[2]*Sqrt[f + g*x])])/g^(3/2)))/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/(b*(m + 1))), x] - Simp[d*(n/(b*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] & & GtQ[n, 0] && LtQ[m, -1] && !(IntegerQ[n] && !IntegerQ[m]) && !(ILeQ[m + n + 2, 0] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c , d, m, n, x]
Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ 2 Subst[Int[1/(b - d*x^2), x], x, Sqrt[a + b*x]/Sqrt[c + d*x]], x] /; Fre eQ[{a, b, c, d}, x] && !GtQ[c - a*(d/b), 0]
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[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
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]
Leaf count of result is larger than twice the leaf count of optimal. \(421\) vs. \(2(92)=184\).
Time = 1.50 (sec) , antiderivative size = 422, normalized size of antiderivative = 3.55
| method | result | size |
| default | \(-\frac {\left (24 \sqrt {2}\, \sqrt {-\left (2 x -3\right ) \left (g x +f \right )}\, g^{\frac {3}{2}} f x +23 \sqrt {2}\, \sqrt {-\left (2 x -3\right ) \left (g x +f \right )}\, g^{\frac {5}{2}} x +18 \arctan \left (\frac {\left (4 g x +2 f -3 g \right ) \sqrt {2}}{4 \sqrt {g}\, \sqrt {-\left (2 x -3\right ) \left (g x +f \right )}}\right ) f \,g^{2} x^{2}+27 \arctan \left (\frac {\left (4 g x +2 f -3 g \right ) \sqrt {2}}{4 \sqrt {g}\, \sqrt {-\left (2 x -3\right ) \left (g x +f \right )}}\right ) g^{3} x^{2}+18 \sqrt {2}\, \sqrt {-\left (2 x -3\right ) \left (g x +f \right )}\, \sqrt {g}\, f^{2}+18 \sqrt {2}\, \sqrt {-\left (2 x -3\right ) \left (g x +f \right )}\, g^{\frac {3}{2}} f +6 \sqrt {2}\, \sqrt {-\left (2 x -3\right ) \left (g x +f \right )}\, g^{\frac {5}{2}}+36 \arctan \left (\frac {\left (4 g x +2 f -3 g \right ) \sqrt {2}}{4 \sqrt {g}\, \sqrt {-\left (2 x -3\right ) \left (g x +f \right )}}\right ) f^{2} g x +54 \arctan \left (\frac {\left (4 g x +2 f -3 g \right ) \sqrt {2}}{4 \sqrt {g}\, \sqrt {-\left (2 x -3\right ) \left (g x +f \right )}}\right ) f \,g^{2} x +18 \arctan \left (\frac {\left (4 g x +2 f -3 g \right ) \sqrt {2}}{4 \sqrt {g}\, \sqrt {-\left (2 x -3\right ) \left (g x +f \right )}}\right ) f^{3}+27 \arctan \left (\frac {\left (4 g x +2 f -3 g \right ) \sqrt {2}}{4 \sqrt {g}\, \sqrt {-\left (2 x -3\right ) \left (g x +f \right )}}\right ) f^{2} g \right ) \sqrt {3}\, \sqrt {-6 x^{2}+5 x +6}}{9 \left (2 f +3 g \right ) \sqrt {-\left (2 x -3\right ) \left (g x +f \right )}\, g^{\frac {5}{2}} \left (g x +f \right )^{\frac {3}{2}} \sqrt {3 x +2}}\) | \(422\) |
Input:
int(1/6*(3*x+2)^(1/2)*(-36*x^2+30*x+36)^(1/2)/(g*x+f)^(5/2),x,method=_RETU RNVERBOSE)
Output:
-1/9*(24*2^(1/2)*(-(2*x-3)*(g*x+f))^(1/2)*g^(3/2)*f*x+23*2^(1/2)*(-(2*x-3) *(g*x+f))^(1/2)*g^(5/2)*x+18*arctan(1/4/g^(1/2)*(4*g*x+2*f-3*g)*2^(1/2)/(- (2*x-3)*(g*x+f))^(1/2))*f*g^2*x^2+27*arctan(1/4/g^(1/2)*(4*g*x+2*f-3*g)*2^ (1/2)/(-(2*x-3)*(g*x+f))^(1/2))*g^3*x^2+18*2^(1/2)*(-(2*x-3)*(g*x+f))^(1/2 )*g^(1/2)*f^2+18*2^(1/2)*(-(2*x-3)*(g*x+f))^(1/2)*g^(3/2)*f+6*2^(1/2)*(-(2 *x-3)*(g*x+f))^(1/2)*g^(5/2)+36*arctan(1/4/g^(1/2)*(4*g*x+2*f-3*g)*2^(1/2) /(-(2*x-3)*(g*x+f))^(1/2))*f^2*g*x+54*arctan(1/4/g^(1/2)*(4*g*x+2*f-3*g)*2 ^(1/2)/(-(2*x-3)*(g*x+f))^(1/2))*f*g^2*x+18*arctan(1/4/g^(1/2)*(4*g*x+2*f- 3*g)*2^(1/2)/(-(2*x-3)*(g*x+f))^(1/2))*f^3+27*arctan(1/4/g^(1/2)*(4*g*x+2* f-3*g)*2^(1/2)/(-(2*x-3)*(g*x+f))^(1/2))*f^2*g)*3^(1/2)*(-6*x^2+5*x+6)^(1/ 2)/(2*f+3*g)/(-(2*x-3)*(g*x+f))^(1/2)/g^(5/2)/(g*x+f)^(3/2)/(3*x+2)^(1/2)
Leaf count of result is larger than twice the leaf count of optimal. 282 vs. \(2 (92) = 184\).
Time = 0.11 (sec) , antiderivative size = 631, normalized size of antiderivative = 5.30 \[ \int \frac {\sqrt {2+3 x} \sqrt {1+\frac {5 x}{6}-x^2}}{(f+g x)^{5/2}} \, dx=\left [\frac {9 \, \sqrt {3} {\left (3 \, {\left (2 \, f g^{2} + 3 \, g^{3}\right )} x^{3} + 4 \, f^{3} + 6 \, f^{2} g + 2 \, {\left (6 \, f^{2} g + 11 \, f g^{2} + 3 \, g^{3}\right )} x^{2} + {\left (6 \, f^{3} + 17 \, f^{2} g + 12 \, f g^{2}\right )} x\right )} \sqrt {-\frac {1}{g}} \log \left (-\frac {288 \, g^{2} x^{3} - 4 \, \sqrt {3} {\left (4 \, g^{2} x + 2 \, f g - 3 \, g^{2}\right )} \sqrt {g x + f} \sqrt {-36 \, x^{2} + 30 \, x + 36} \sqrt {3 \, x + 2} \sqrt {-\frac {1}{g}} + 48 \, {\left (6 \, f g - 5 \, g^{2}\right )} x^{2} + 24 \, f^{2} - 216 \, f g + 54 \, g^{2} + 3 \, {\left (12 \, f^{2} - 44 \, f g - 69 \, g^{2}\right )} x}{3 \, x + 2}\right ) - 2 \, {\left (18 \, f^{2} + 18 \, f g + 6 \, g^{2} + {\left (24 \, f g + 23 \, g^{2}\right )} x\right )} \sqrt {g x + f} \sqrt {-36 \, x^{2} + 30 \, x + 36} \sqrt {3 \, x + 2}}{18 \, {\left (4 \, f^{3} g^{2} + 6 \, f^{2} g^{3} + 3 \, {\left (2 \, f g^{4} + 3 \, g^{5}\right )} x^{3} + 2 \, {\left (6 \, f^{2} g^{3} + 11 \, f g^{4} + 3 \, g^{5}\right )} x^{2} + {\left (6 \, f^{3} g^{2} + 17 \, f^{2} g^{3} + 12 \, f g^{4}\right )} x\right )}}, -\frac {{\left (18 \, f^{2} + 18 \, f g + 6 \, g^{2} + {\left (24 \, f g + 23 \, g^{2}\right )} x\right )} \sqrt {g x + f} \sqrt {-36 \, x^{2} + 30 \, x + 36} \sqrt {3 \, x + 2} - \frac {9 \, \sqrt {3} {\left (3 \, {\left (2 \, f g^{2} + 3 \, g^{3}\right )} x^{3} + 4 \, f^{3} + 6 \, f^{2} g + 2 \, {\left (6 \, f^{2} g + 11 \, f g^{2} + 3 \, g^{3}\right )} x^{2} + {\left (6 \, f^{3} + 17 \, f^{2} g + 12 \, f g^{2}\right )} x\right )} \arctan \left (\frac {2 \, \sqrt {3} \sqrt {g x + f} \sqrt {-36 \, x^{2} + 30 \, x + 36} \sqrt {g} \sqrt {3 \, x + 2}}{3 \, {\left (12 \, g x^{2} + {\left (6 \, f - g\right )} x + 4 \, f - 6 \, g\right )}}\right )}{\sqrt {g}}}{9 \, {\left (4 \, f^{3} g^{2} + 6 \, f^{2} g^{3} + 3 \, {\left (2 \, f g^{4} + 3 \, g^{5}\right )} x^{3} + 2 \, {\left (6 \, f^{2} g^{3} + 11 \, f g^{4} + 3 \, g^{5}\right )} x^{2} + {\left (6 \, f^{3} g^{2} + 17 \, f^{2} g^{3} + 12 \, f g^{4}\right )} x\right )}}\right ] \] Input:
integrate(1/6*(2+3*x)^(1/2)*(-36*x^2+30*x+36)^(1/2)/(g*x+f)^(5/2),x, algor ithm="fricas")
Output:
[1/18*(9*sqrt(3)*(3*(2*f*g^2 + 3*g^3)*x^3 + 4*f^3 + 6*f^2*g + 2*(6*f^2*g + 11*f*g^2 + 3*g^3)*x^2 + (6*f^3 + 17*f^2*g + 12*f*g^2)*x)*sqrt(-1/g)*log(- (288*g^2*x^3 - 4*sqrt(3)*(4*g^2*x + 2*f*g - 3*g^2)*sqrt(g*x + f)*sqrt(-36* x^2 + 30*x + 36)*sqrt(3*x + 2)*sqrt(-1/g) + 48*(6*f*g - 5*g^2)*x^2 + 24*f^ 2 - 216*f*g + 54*g^2 + 3*(12*f^2 - 44*f*g - 69*g^2)*x)/(3*x + 2)) - 2*(18* f^2 + 18*f*g + 6*g^2 + (24*f*g + 23*g^2)*x)*sqrt(g*x + f)*sqrt(-36*x^2 + 3 0*x + 36)*sqrt(3*x + 2))/(4*f^3*g^2 + 6*f^2*g^3 + 3*(2*f*g^4 + 3*g^5)*x^3 + 2*(6*f^2*g^3 + 11*f*g^4 + 3*g^5)*x^2 + (6*f^3*g^2 + 17*f^2*g^3 + 12*f*g^ 4)*x), -1/9*((18*f^2 + 18*f*g + 6*g^2 + (24*f*g + 23*g^2)*x)*sqrt(g*x + f) *sqrt(-36*x^2 + 30*x + 36)*sqrt(3*x + 2) - 9*sqrt(3)*(3*(2*f*g^2 + 3*g^3)* x^3 + 4*f^3 + 6*f^2*g + 2*(6*f^2*g + 11*f*g^2 + 3*g^3)*x^2 + (6*f^3 + 17*f ^2*g + 12*f*g^2)*x)*arctan(2/3*sqrt(3)*sqrt(g*x + f)*sqrt(-36*x^2 + 30*x + 36)*sqrt(g)*sqrt(3*x + 2)/(12*g*x^2 + (6*f - g)*x + 4*f - 6*g))/sqrt(g))/ (4*f^3*g^2 + 6*f^2*g^3 + 3*(2*f*g^4 + 3*g^5)*x^3 + 2*(6*f^2*g^3 + 11*f*g^4 + 3*g^5)*x^2 + (6*f^3*g^2 + 17*f^2*g^3 + 12*f*g^4)*x)]
Timed out. \[ \int \frac {\sqrt {2+3 x} \sqrt {1+\frac {5 x}{6}-x^2}}{(f+g x)^{5/2}} \, dx=\text {Timed out} \] Input:
integrate(1/6*(2+3*x)**(1/2)*(-36*x**2+30*x+36)**(1/2)/(g*x+f)**(5/2),x)
Output:
Timed out
\[ \int \frac {\sqrt {2+3 x} \sqrt {1+\frac {5 x}{6}-x^2}}{(f+g x)^{5/2}} \, dx=\int { \frac {\sqrt {-36 \, x^{2} + 30 \, x + 36} \sqrt {3 \, x + 2}}{6 \, {\left (g x + f\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate(1/6*(2+3*x)^(1/2)*(-36*x^2+30*x+36)^(1/2)/(g*x+f)^(5/2),x, algor ithm="maxima")
Output:
1/6*integrate(sqrt(-36*x^2 + 30*x + 36)*sqrt(3*x + 2)/(g*x + f)^(5/2), x)
Time = 0.44 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.33 \[ \int \frac {\sqrt {2+3 x} \sqrt {1+\frac {5 x}{6}-x^2}}{(f+g x)^{5/2}} \, dx=-\frac {1}{9} \, \sqrt {6} {\left (\frac {{\left (\frac {{\left (24 \, \sqrt {2} f g^{2} + 23 \, \sqrt {2} g^{3}\right )} {\left (2 \, x - 3\right )}}{2 \, f g^{3} + 3 \, g^{4}} + \frac {9 \, {\left (4 \, \sqrt {2} f^{2} g + 12 \, \sqrt {2} f g^{2} + 9 \, \sqrt {2} g^{3}\right )}}{2 \, f g^{3} + 3 \, g^{4}}\right )} \sqrt {-2 \, x + 3}}{{\left (g {\left (2 \, x - 3\right )} + 2 \, f + 3 \, g\right )}^{\frac {3}{2}}} + \frac {9 \, \sqrt {2} \log \left ({\left | -\sqrt {-g} \sqrt {-2 \, x + 3} + \sqrt {g {\left (2 \, x - 3\right )} + 2 \, f + 3 \, g} \right |}\right )}{\sqrt {-g} g^{2}}\right )} \] Input:
integrate(1/6*(2+3*x)^(1/2)*(-36*x^2+30*x+36)^(1/2)/(g*x+f)^(5/2),x, algor ithm="giac")
Output:
-1/9*sqrt(6)*(((24*sqrt(2)*f*g^2 + 23*sqrt(2)*g^3)*(2*x - 3)/(2*f*g^3 + 3* g^4) + 9*(4*sqrt(2)*f^2*g + 12*sqrt(2)*f*g^2 + 9*sqrt(2)*g^3)/(2*f*g^3 + 3 *g^4))*sqrt(-2*x + 3)/(g*(2*x - 3) + 2*f + 3*g)^(3/2) + 9*sqrt(2)*log(abs( -sqrt(-g)*sqrt(-2*x + 3) + sqrt(g*(2*x - 3) + 2*f + 3*g)))/(sqrt(-g)*g^2))
Timed out. \[ \int \frac {\sqrt {2+3 x} \sqrt {1+\frac {5 x}{6}-x^2}}{(f+g x)^{5/2}} \, dx=\int \frac {\sqrt {3\,x+2}\,\sqrt {-36\,x^2+30\,x+36}}{6\,{\left (f+g\,x\right )}^{5/2}} \,d x \] Input:
int(((3*x + 2)^(1/2)*(30*x - 36*x^2 + 36)^(1/2))/(6*(f + g*x)^(5/2)),x)
Output:
int(((3*x + 2)^(1/2)*(30*x - 36*x^2 + 36)^(1/2))/(6*(f + g*x)^(5/2)), x)
Time = 0.21 (sec) , antiderivative size = 241, normalized size of antiderivative = 2.03 \[ \int \frac {\sqrt {2+3 x} \sqrt {1+\frac {5 x}{6}-x^2}}{(f+g x)^{5/2}} \, dx=\frac {\sqrt {3}\, \left (36 \sqrt {g}\, \sqrt {g x +f}\, \mathit {asin} \left (\frac {\sqrt {g}\, \sqrt {-2 x +3}}{\sqrt {2 f +3 g}}\right ) f^{2}+36 \sqrt {g}\, \sqrt {g x +f}\, \mathit {asin} \left (\frac {\sqrt {g}\, \sqrt {-2 x +3}}{\sqrt {2 f +3 g}}\right ) f g x +54 \sqrt {g}\, \sqrt {g x +f}\, \mathit {asin} \left (\frac {\sqrt {g}\, \sqrt {-2 x +3}}{\sqrt {2 f +3 g}}\right ) f g +54 \sqrt {g}\, \sqrt {g x +f}\, \mathit {asin} \left (\frac {\sqrt {g}\, \sqrt {-2 x +3}}{\sqrt {2 f +3 g}}\right ) g^{2} x -18 \sqrt {-2 x +3}\, \sqrt {2}\, f^{2} g -24 \sqrt {-2 x +3}\, \sqrt {2}\, f \,g^{2} x -18 \sqrt {-2 x +3}\, \sqrt {2}\, f \,g^{2}-23 \sqrt {-2 x +3}\, \sqrt {2}\, g^{3} x -6 \sqrt {-2 x +3}\, \sqrt {2}\, g^{3}\right )}{9 \sqrt {g x +f}\, g^{3} \left (2 f g x +3 g^{2} x +2 f^{2}+3 f g \right )} \] Input:
int(1/6*(2+3*x)^(1/2)*(-36*x^2+30*x+36)^(1/2)/(g*x+f)^(5/2),x)
Output:
(sqrt(3)*(36*sqrt(g)*sqrt(f + g*x)*asin((sqrt(g)*sqrt( - 2*x + 3))/sqrt(2* f + 3*g))*f**2 + 36*sqrt(g)*sqrt(f + g*x)*asin((sqrt(g)*sqrt( - 2*x + 3))/ sqrt(2*f + 3*g))*f*g*x + 54*sqrt(g)*sqrt(f + g*x)*asin((sqrt(g)*sqrt( - 2* x + 3))/sqrt(2*f + 3*g))*f*g + 54*sqrt(g)*sqrt(f + g*x)*asin((sqrt(g)*sqrt ( - 2*x + 3))/sqrt(2*f + 3*g))*g**2*x - 18*sqrt( - 2*x + 3)*sqrt(2)*f**2*g - 24*sqrt( - 2*x + 3)*sqrt(2)*f*g**2*x - 18*sqrt( - 2*x + 3)*sqrt(2)*f*g* *2 - 23*sqrt( - 2*x + 3)*sqrt(2)*g**3*x - 6*sqrt( - 2*x + 3)*sqrt(2)*g**3) )/(9*sqrt(f + g*x)*g**3*(2*f**2 + 2*f*g*x + 3*f*g + 3*g**2*x))