Integrand size = 37, antiderivative size = 60 \[ \int \frac {\sqrt {2-3 x}}{\sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^{3/2}} \, dx=\frac {2 \sqrt {\frac {11}{39}} \sqrt {5-2 x} E\left (\arcsin \left (\frac {\sqrt {\frac {39}{22}} \sqrt {1+4 x}}{\sqrt {7+5 x}}\right )|\frac {62}{39}\right )}{23 \sqrt {-5+2 x}} \]
2/897*EllipticE(1/22*858^(1/2)*(1+4*x)^(1/2)/(7+5*x)^(1/2),1/39*2418^(1/2) )*429^(1/2)*(5-2*x)^(1/2)/(-5+2*x)^(1/2)
Leaf count is larger than twice the leaf count of optimal. \(237\) vs. \(2(60)=120\).
Time = 28.52 (sec) , antiderivative size = 237, normalized size of antiderivative = 3.95 \[ \int \frac {\sqrt {2-3 x}}{\sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^{3/2}} \, dx=\frac {\sqrt {-5+2 x} \sqrt {1+4 x} \left (-1922 \sqrt {\frac {7+5 x}{-2+3 x}} \left (-5-18 x+8 x^2\right )+62 \sqrt {682} \sqrt {\frac {-5-18 x+8 x^2}{(2-3 x)^2}} \left (-14+11 x+15 x^2\right ) E\left (\arcsin \left (\sqrt {\frac {31}{39}} \sqrt {\frac {-5+2 x}{-2+3 x}}\right )|\frac {39}{62}\right )-23 \sqrt {682} \sqrt {\frac {-5-18 x+8 x^2}{(2-3 x)^2}} \left (-14+11 x+15 x^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {31}{39}} \sqrt {\frac {-5+2 x}{-2+3 x}}\right ),\frac {39}{62}\right )\right )}{27807 \sqrt {2-3 x} \sqrt {7+5 x} \sqrt {\frac {7+5 x}{-2+3 x}} \left (-5-18 x+8 x^2\right )} \]
(Sqrt[-5 + 2*x]*Sqrt[1 + 4*x]*(-1922*Sqrt[(7 + 5*x)/(-2 + 3*x)]*(-5 - 18*x + 8*x^2) + 62*Sqrt[682]*Sqrt[(-5 - 18*x + 8*x^2)/(2 - 3*x)^2]*(-14 + 11*x + 15*x^2)*EllipticE[ArcSin[Sqrt[31/39]*Sqrt[(-5 + 2*x)/(-2 + 3*x)]], 39/6 2] - 23*Sqrt[682]*Sqrt[(-5 - 18*x + 8*x^2)/(2 - 3*x)^2]*(-14 + 11*x + 15*x ^2)*EllipticF[ArcSin[Sqrt[31/39]*Sqrt[(-5 + 2*x)/(-2 + 3*x)]], 39/62]))/(2 7807*Sqrt[2 - 3*x]*Sqrt[7 + 5*x]*Sqrt[(7 + 5*x)/(-2 + 3*x)]*(-5 - 18*x + 8 *x^2))
Leaf count is larger than twice the leaf count of optimal. \(362\) vs. \(2(60)=120\).
Time = 0.32 (sec) , antiderivative size = 362, normalized size of antiderivative = 6.03, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.162, Rules used = {194, 27, 324, 320, 388, 313}
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.
| \(\displaystyle \int \frac {\sqrt {2-3 x}}{\sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 194 |
| \(\displaystyle \frac {\sqrt {2} \sqrt {2-3 x} \sqrt {\frac {4 x+1}{5 x+7}} \int \frac {\sqrt {2} \sqrt {\frac {31 (2 x-5)}{5 x+7}+11}}{\sqrt {\frac {23 (2 x-5)}{5 x+7}+22}}d\frac {\sqrt {2 x-5}}{\sqrt {5 x+7}}}{39 \sqrt {4 x+1} \sqrt {-\frac {2-3 x}{5 x+7}}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \sqrt {2-3 x} \sqrt {\frac {4 x+1}{5 x+7}} \int \frac {\sqrt {\frac {31 (2 x-5)}{5 x+7}+11}}{\sqrt {\frac {23 (2 x-5)}{5 x+7}+22}}d\frac {\sqrt {2 x-5}}{\sqrt {5 x+7}}}{39 \sqrt {4 x+1} \sqrt {-\frac {2-3 x}{5 x+7}}}\) |
| \(\Big \downarrow \) 324 |
| \(\displaystyle \frac {2 \sqrt {2-3 x} \sqrt {\frac {4 x+1}{5 x+7}} \left (11 \int \frac {1}{\sqrt {\frac {23 (2 x-5)}{5 x+7}+22} \sqrt {\frac {31 (2 x-5)}{5 x+7}+11}}d\frac {\sqrt {2 x-5}}{\sqrt {5 x+7}}+31 \int \frac {2 x-5}{(5 x+7) \sqrt {\frac {23 (2 x-5)}{5 x+7}+22} \sqrt {\frac {31 (2 x-5)}{5 x+7}+11}}d\frac {\sqrt {2 x-5}}{\sqrt {5 x+7}}\right )}{39 \sqrt {4 x+1} \sqrt {-\frac {2-3 x}{5 x+7}}}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {2 \sqrt {2-3 x} \sqrt {\frac {4 x+1}{5 x+7}} \left (31 \int \frac {2 x-5}{(5 x+7) \sqrt {\frac {23 (2 x-5)}{5 x+7}+22} \sqrt {\frac {31 (2 x-5)}{5 x+7}+11}}d\frac {\sqrt {2 x-5}}{\sqrt {5 x+7}}+\frac {\sqrt {\frac {11}{62}} \sqrt {\frac {23 (2 x-5)}{5 x+7}+22} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {\frac {31}{11}} \sqrt {2 x-5}}{\sqrt {5 x+7}}\right ),\frac {39}{62}\right )}{\sqrt {\frac {\frac {23 (2 x-5)}{5 x+7}+22}{\frac {31 (2 x-5)}{5 x+7}+11}} \sqrt {\frac {31 (2 x-5)}{5 x+7}+11}}\right )}{39 \sqrt {4 x+1} \sqrt {-\frac {2-3 x}{5 x+7}}}\) |
| \(\Big \downarrow \) 388 |
| \(\displaystyle \frac {2 \sqrt {2-3 x} \sqrt {\frac {4 x+1}{5 x+7}} \left (31 \left (\frac {\sqrt {2 x-5} \sqrt {\frac {23 (2 x-5)}{5 x+7}+22}}{23 \sqrt {5 x+7} \sqrt {\frac {31 (2 x-5)}{5 x+7}+11}}-\frac {11}{23} \int \frac {\sqrt {\frac {23 (2 x-5)}{5 x+7}+22}}{\left (\frac {31 (2 x-5)}{5 x+7}+11\right )^{3/2}}d\frac {\sqrt {2 x-5}}{\sqrt {5 x+7}}\right )+\frac {\sqrt {\frac {11}{62}} \sqrt {\frac {23 (2 x-5)}{5 x+7}+22} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {\frac {31}{11}} \sqrt {2 x-5}}{\sqrt {5 x+7}}\right ),\frac {39}{62}\right )}{\sqrt {\frac {\frac {23 (2 x-5)}{5 x+7}+22}{\frac {31 (2 x-5)}{5 x+7}+11}} \sqrt {\frac {31 (2 x-5)}{5 x+7}+11}}\right )}{39 \sqrt {4 x+1} \sqrt {-\frac {2-3 x}{5 x+7}}}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {2 \sqrt {2-3 x} \sqrt {\frac {4 x+1}{5 x+7}} \left (\frac {\sqrt {\frac {11}{62}} \sqrt {\frac {23 (2 x-5)}{5 x+7}+22} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {\frac {31}{11}} \sqrt {2 x-5}}{\sqrt {5 x+7}}\right ),\frac {39}{62}\right )}{\sqrt {\frac {\frac {23 (2 x-5)}{5 x+7}+22}{\frac {31 (2 x-5)}{5 x+7}+11}} \sqrt {\frac {31 (2 x-5)}{5 x+7}+11}}+31 \left (\frac {\sqrt {2 x-5} \sqrt {\frac {23 (2 x-5)}{5 x+7}+22}}{23 \sqrt {5 x+7} \sqrt {\frac {31 (2 x-5)}{5 x+7}+11}}-\frac {\sqrt {\frac {22}{31}} \sqrt {\frac {23 (2 x-5)}{5 x+7}+22} E\left (\arctan \left (\frac {\sqrt {\frac {31}{11}} \sqrt {2 x-5}}{\sqrt {5 x+7}}\right )|\frac {39}{62}\right )}{23 \sqrt {\frac {\frac {23 (2 x-5)}{5 x+7}+22}{\frac {31 (2 x-5)}{5 x+7}+11}} \sqrt {\frac {31 (2 x-5)}{5 x+7}+11}}\right )\right )}{39 \sqrt {4 x+1} \sqrt {-\frac {2-3 x}{5 x+7}}}\) |
(2*Sqrt[2 - 3*x]*Sqrt[(1 + 4*x)/(7 + 5*x)]*(31*((Sqrt[-5 + 2*x]*Sqrt[22 + (23*(-5 + 2*x))/(7 + 5*x)])/(23*Sqrt[7 + 5*x]*Sqrt[11 + (31*(-5 + 2*x))/(7 + 5*x)]) - (Sqrt[22/31]*Sqrt[22 + (23*(-5 + 2*x))/(7 + 5*x)]*EllipticE[Ar cTan[(Sqrt[31/11]*Sqrt[-5 + 2*x])/Sqrt[7 + 5*x]], 39/62])/(23*Sqrt[(22 + ( 23*(-5 + 2*x))/(7 + 5*x))/(11 + (31*(-5 + 2*x))/(7 + 5*x))]*Sqrt[11 + (31* (-5 + 2*x))/(7 + 5*x)])) + (Sqrt[11/62]*Sqrt[22 + (23*(-5 + 2*x))/(7 + 5*x )]*EllipticF[ArcTan[(Sqrt[31/11]*Sqrt[-5 + 2*x])/Sqrt[7 + 5*x]], 39/62])/( Sqrt[(22 + (23*(-5 + 2*x))/(7 + 5*x))/(11 + (31*(-5 + 2*x))/(7 + 5*x))]*Sq rt[11 + (31*(-5 + 2*x))/(7 + 5*x)])))/(39*Sqrt[1 + 4*x]*Sqrt[-((2 - 3*x)/( 7 + 5*x))])
3.1.97.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[Sqrt[(c_.) + (d_.)*(x_)]/(((a_.) + (b_.)*(x_))^(3/2)*Sqrt[(e_.) + (f_.) *(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x_] :> Simp[-2*Sqrt[c + d*x]*(Sqrt[(-(b*e - a*f))*((g + h*x)/((f*g - e*h)*(a + b*x)))]/((b*e - a*f)*Sqrt[g + h*x]*Sq rt[(b*e - a*f)*((c + d*x)/((d*e - c*f)*(a + b*x)))])) Subst[Int[Sqrt[1 + (b*c - a*d)*(x^2/(d*e - c*f))]/Sqrt[1 - (b*g - a*h)*(x^2/(f*g - e*h))], x], x, Sqrt[e + f*x]/Sqrt[a + b*x]], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]
Int[Sqrt[(a_) + (b_.)*(x_)^2]/((c_) + (d_.)*(x_)^2)^(3/2), x_Symbol] :> Sim p[(Sqrt[a + b*x^2]/(c*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*(c + d*x^2)))]))*EllipticE[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /; FreeQ [{a, b, c, d}, x] && PosQ[b/a] && PosQ[d/c]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[(Sqrt[a + b*x^2]/(a*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*( c + d*x^2)))]))*EllipticF[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /; Fre eQ[{a, b, c, d}, x] && PosQ[d/c] && PosQ[b/a] && !SimplerSqrtQ[b/a, d/c]
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ a Int[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x], x] + Simp[b Int[x^2/(Sqr t[a + b*x^2]*Sqrt[c + d*x^2]), x], x] /; FreeQ[{a, b, c, d}, x] && PosQ[d/c ] && PosQ[b/a]
Int[(x_)^2/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[x*(Sqrt[a + b*x^2]/(b*Sqrt[c + d*x^2])), x] - Simp[c/b Int[Sqrt[ a + b*x^2]/(c + d*x^2)^(3/2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && PosQ[b/a] && PosQ[d/c] && !SimplerSqrtQ[b/a, d/c]
Result contains complex when optimal does not.
Time = 1.62 (sec) , antiderivative size = 435, normalized size of antiderivative = 7.25
| method | result | size |
| elliptic | \(\frac {\sqrt {-\left (7+5 x \right ) \left (-2+3 x \right ) \left (-5+2 x \right ) \left (1+4 x \right )}\, \left (-\frac {2 \left (-120 x^{3}+350 x^{2}-105 x -50\right )}{897 \sqrt {\left (x +\frac {7}{5}\right ) \left (-120 x^{3}+350 x^{2}-105 x -50\right )}}+\frac {34 \sqrt {-\frac {3795 \left (x +\frac {7}{5}\right )}{-\frac {2}{3}+x}}\, \left (-\frac {2}{3}+x \right )^{2} \sqrt {806}\, \sqrt {\frac {x -\frac {5}{2}}{-\frac {2}{3}+x}}\, \sqrt {2139}\, \sqrt {\frac {x +\frac {1}{4}}{-\frac {2}{3}+x}}\, F\left (\frac {\sqrt {-\frac {3795 \left (x +\frac {7}{5}\right )}{-\frac {2}{3}+x}}}{69}, \frac {i \sqrt {897}}{39}\right )}{24942879 \sqrt {-30 \left (x +\frac {7}{5}\right ) \left (-\frac {2}{3}+x \right ) \left (x -\frac {5}{2}\right ) \left (x +\frac {1}{4}\right )}}+\frac {28 \sqrt {-\frac {3795 \left (x +\frac {7}{5}\right )}{-\frac {2}{3}+x}}\, \left (-\frac {2}{3}+x \right )^{2} \sqrt {806}\, \sqrt {\frac {x -\frac {5}{2}}{-\frac {2}{3}+x}}\, \sqrt {2139}\, \sqrt {\frac {x +\frac {1}{4}}{-\frac {2}{3}+x}}\, \left (\frac {2 F\left (\frac {\sqrt {-\frac {3795 \left (x +\frac {7}{5}\right )}{-\frac {2}{3}+x}}}{69}, \frac {i \sqrt {897}}{39}\right )}{3}-\frac {31 \Pi \left (\frac {\sqrt {-\frac {3795 \left (x +\frac {7}{5}\right )}{-\frac {2}{3}+x}}}{69}, -\frac {69}{55}, \frac {i \sqrt {897}}{39}\right )}{15}\right )}{21105513 \sqrt {-30 \left (x +\frac {7}{5}\right ) \left (-\frac {2}{3}+x \right ) \left (x -\frac {5}{2}\right ) \left (x +\frac {1}{4}\right )}}-\frac {40 \left (\left (x +\frac {7}{5}\right ) \left (x -\frac {5}{2}\right ) \left (x +\frac {1}{4}\right )-\frac {\sqrt {-\frac {3795 \left (x +\frac {7}{5}\right )}{-\frac {2}{3}+x}}\, \left (-\frac {2}{3}+x \right )^{2} \sqrt {806}\, \sqrt {\frac {x -\frac {5}{2}}{-\frac {2}{3}+x}}\, \sqrt {2139}\, \sqrt {\frac {x +\frac {1}{4}}{-\frac {2}{3}+x}}\, \left (\frac {181 F\left (\frac {\sqrt {-\frac {3795 \left (x +\frac {7}{5}\right )}{-\frac {2}{3}+x}}}{69}, \frac {i \sqrt {897}}{39}\right )}{341}-\frac {117 E\left (\frac {\sqrt {-\frac {3795 \left (x +\frac {7}{5}\right )}{-\frac {2}{3}+x}}}{69}, \frac {i \sqrt {897}}{39}\right )}{62}+\frac {91 \Pi \left (\frac {\sqrt {-\frac {3795 \left (x +\frac {7}{5}\right )}{-\frac {2}{3}+x}}}{69}, -\frac {69}{55}, \frac {i \sqrt {897}}{39}\right )}{55}\right )}{80730}\right )}{299 \sqrt {-30 \left (x +\frac {7}{5}\right ) \left (-\frac {2}{3}+x \right ) \left (x -\frac {5}{2}\right ) \left (x +\frac {1}{4}\right )}}\right )}{\sqrt {2-3 x}\, \sqrt {-5+2 x}\, \sqrt {1+4 x}\, \sqrt {7+5 x}}\) | \(435\) |
| default | \(-\frac {2 \sqrt {2-3 x}\, \sqrt {7+5 x}\, \sqrt {-5+2 x}\, \sqrt {1+4 x}\, \left (9 \sqrt {-\frac {253 \left (7+5 x \right )}{-2+3 x}}\, \sqrt {13}\, \sqrt {3}\, \sqrt {\frac {-5+2 x}{-2+3 x}}\, \sqrt {23}\, \sqrt {\frac {1+4 x}{-2+3 x}}\, x^{2} F\left (\frac {\sqrt {-\frac {253 \left (7+5 x \right )}{-2+3 x}}}{23}, \frac {i \sqrt {897}}{39}\right )-9 \sqrt {-\frac {253 \left (7+5 x \right )}{-2+3 x}}\, \sqrt {13}\, \sqrt {3}\, \sqrt {\frac {-5+2 x}{-2+3 x}}\, \sqrt {23}\, \sqrt {\frac {1+4 x}{-2+3 x}}\, x^{2} E\left (\frac {\sqrt {-\frac {253 \left (7+5 x \right )}{-2+3 x}}}{23}, \frac {i \sqrt {897}}{39}\right )-12 \sqrt {-\frac {253 \left (7+5 x \right )}{-2+3 x}}\, \sqrt {13}\, \sqrt {3}\, \sqrt {\frac {-5+2 x}{-2+3 x}}\, \sqrt {23}\, \sqrt {\frac {1+4 x}{-2+3 x}}\, x F\left (\frac {\sqrt {-\frac {253 \left (7+5 x \right )}{-2+3 x}}}{23}, \frac {i \sqrt {897}}{39}\right )+12 \sqrt {-\frac {253 \left (7+5 x \right )}{-2+3 x}}\, \sqrt {13}\, \sqrt {3}\, \sqrt {\frac {-5+2 x}{-2+3 x}}\, \sqrt {23}\, \sqrt {\frac {1+4 x}{-2+3 x}}\, x E\left (\frac {\sqrt {-\frac {253 \left (7+5 x \right )}{-2+3 x}}}{23}, \frac {i \sqrt {897}}{39}\right )+4 \sqrt {-\frac {253 \left (7+5 x \right )}{-2+3 x}}\, \sqrt {13}\, \sqrt {3}\, \sqrt {\frac {-5+2 x}{-2+3 x}}\, \sqrt {23}\, \sqrt {\frac {1+4 x}{-2+3 x}}\, F\left (\frac {\sqrt {-\frac {253 \left (7+5 x \right )}{-2+3 x}}}{23}, \frac {i \sqrt {897}}{39}\right )-4 \sqrt {-\frac {253 \left (7+5 x \right )}{-2+3 x}}\, \sqrt {13}\, \sqrt {3}\, \sqrt {\frac {-5+2 x}{-2+3 x}}\, \sqrt {23}\, \sqrt {\frac {1+4 x}{-2+3 x}}\, E\left (\frac {\sqrt {-\frac {253 \left (7+5 x \right )}{-2+3 x}}}{23}, \frac {i \sqrt {897}}{39}\right )-5704 x^{2}+12834 x +3565\right )}{20631 \left (120 x^{4}-182 x^{3}-385 x^{2}+197 x +70\right )}\) | \(563\) |
(-(7+5*x)*(-2+3*x)*(-5+2*x)*(1+4*x))^(1/2)/(2-3*x)^(1/2)/(-5+2*x)^(1/2)/(1 +4*x)^(1/2)/(7+5*x)^(1/2)*(-2/897*(-120*x^3+350*x^2-105*x-50)/((x+7/5)*(-1 20*x^3+350*x^2-105*x-50))^(1/2)+34/24942879*(-3795*(x+7/5)/(-2/3+x))^(1/2) *(-2/3+x)^2*806^(1/2)*((x-5/2)/(-2/3+x))^(1/2)*2139^(1/2)*((x+1/4)/(-2/3+x ))^(1/2)/(-30*(x+7/5)*(-2/3+x)*(x-5/2)*(x+1/4))^(1/2)*EllipticF(1/69*(-379 5*(x+7/5)/(-2/3+x))^(1/2),1/39*I*897^(1/2))+28/21105513*(-3795*(x+7/5)/(-2 /3+x))^(1/2)*(-2/3+x)^2*806^(1/2)*((x-5/2)/(-2/3+x))^(1/2)*2139^(1/2)*((x+ 1/4)/(-2/3+x))^(1/2)/(-30*(x+7/5)*(-2/3+x)*(x-5/2)*(x+1/4))^(1/2)*(2/3*Ell ipticF(1/69*(-3795*(x+7/5)/(-2/3+x))^(1/2),1/39*I*897^(1/2))-31/15*Ellipti cPi(1/69*(-3795*(x+7/5)/(-2/3+x))^(1/2),-69/55,1/39*I*897^(1/2)))-40/299*( (x+7/5)*(x-5/2)*(x+1/4)-1/80730*(-3795*(x+7/5)/(-2/3+x))^(1/2)*(-2/3+x)^2* 806^(1/2)*((x-5/2)/(-2/3+x))^(1/2)*2139^(1/2)*((x+1/4)/(-2/3+x))^(1/2)*(18 1/341*EllipticF(1/69*(-3795*(x+7/5)/(-2/3+x))^(1/2),1/39*I*897^(1/2))-117/ 62*EllipticE(1/69*(-3795*(x+7/5)/(-2/3+x))^(1/2),1/39*I*897^(1/2))+91/55*E llipticPi(1/69*(-3795*(x+7/5)/(-2/3+x))^(1/2),-69/55,1/39*I*897^(1/2))))/( -30*(x+7/5)*(-2/3+x)*(x-5/2)*(x+1/4))^(1/2))
\[ \int \frac {\sqrt {2-3 x}}{\sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^{3/2}} \, dx=\int { \frac {\sqrt {-3 \, x + 2}}{{\left (5 \, x + 7\right )}^{\frac {3}{2}} \sqrt {4 \, x + 1} \sqrt {2 \, x - 5}} \,d x } \]
integral(sqrt(5*x + 7)*sqrt(4*x + 1)*sqrt(2*x - 5)*sqrt(-3*x + 2)/(200*x^4 + 110*x^3 - 993*x^2 - 1232*x - 245), x)
\[ \int \frac {\sqrt {2-3 x}}{\sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^{3/2}} \, dx=\int \frac {\sqrt {2 - 3 x}}{\sqrt {2 x - 5} \sqrt {4 x + 1} \left (5 x + 7\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {\sqrt {2-3 x}}{\sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^{3/2}} \, dx=\int { \frac {\sqrt {-3 \, x + 2}}{{\left (5 \, x + 7\right )}^{\frac {3}{2}} \sqrt {4 \, x + 1} \sqrt {2 \, x - 5}} \,d x } \]
\[ \int \frac {\sqrt {2-3 x}}{\sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^{3/2}} \, dx=\int { \frac {\sqrt {-3 \, x + 2}}{{\left (5 \, x + 7\right )}^{\frac {3}{2}} \sqrt {4 \, x + 1} \sqrt {2 \, x - 5}} \,d x } \]
Timed out. \[ \int \frac {\sqrt {2-3 x}}{\sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^{3/2}} \, dx=\int \frac {\sqrt {2-3\,x}}{\sqrt {4\,x+1}\,\sqrt {2\,x-5}\,{\left (5\,x+7\right )}^{3/2}} \,d x \]