Integrand size = 35, antiderivative size = 189 \[ \int \frac {\sqrt {2-3 x}}{\sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^2} \, dx=-\frac {5 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{897 (7+5 x)}+\frac {2 \sqrt {11} \sqrt {-5+2 x} E\left (\arcsin \left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|-\frac {1}{2}\right )}{897 \sqrt {5-2 x}}-\frac {2 \sqrt {\frac {6}{11}} \sqrt {5-2 x} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {1+4 x}\right ),\frac {1}{3}\right )}{115 \sqrt {-5+2 x}}-\frac {3571 \sqrt {5-2 x} \operatorname {EllipticPi}\left (\frac {55}{124},\arcsin \left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right ),-\frac {1}{2}\right )}{92690 \sqrt {11} \sqrt {-5+2 x}} \]
-2/1265*EllipticF(1/11*33^(1/2)*(1+4*x)^(1/2),1/3*3^(1/2))*66^(1/2)*(5-2*x )^(1/2)/(-5+2*x)^(1/2)-3571/1019590*EllipticPi(2/11*(2-3*x)^(1/2)*11^(1/2) ,55/124,1/2*I*2^(1/2))*(5-2*x)^(1/2)*11^(1/2)/(-5+2*x)^(1/2)+2/897*Ellipti cE(2/11*(2-3*x)^(1/2)*11^(1/2),1/2*I*2^(1/2))*11^(1/2)*(-5+2*x)^(1/2)/(5-2 *x)^(1/2)-5/897*(2-3*x)^(1/2)*(-5+2*x)^(1/2)*(1+4*x)^(1/2)/(7+5*x)
Time = 5.68 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.69 \[ \int \frac {\sqrt {2-3 x}}{\sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^2} \, dx=\frac {-\frac {51150 \sqrt {2-3 x} (-5+2 x) \sqrt {1+4 x}}{7+5 x}-3 \sqrt {55-22 x} \left (6820 E\left (\arcsin \left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|-\frac {1}{2}\right )-14508 \operatorname {EllipticF}\left (\arcsin \left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right ),-\frac {1}{2}\right )+10713 \operatorname {EllipticPi}\left (\frac {55}{124},\arcsin \left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right ),-\frac {1}{2}\right )\right )}{9176310 \sqrt {-5+2 x}} \]
((-51150*Sqrt[2 - 3*x]*(-5 + 2*x)*Sqrt[1 + 4*x])/(7 + 5*x) - 3*Sqrt[55 - 2 2*x]*(6820*EllipticE[ArcSin[(2*Sqrt[2 - 3*x])/Sqrt[11]], -1/2] - 14508*Ell ipticF[ArcSin[(2*Sqrt[2 - 3*x])/Sqrt[11]], -1/2] + 10713*EllipticPi[55/124 , ArcSin[(2*Sqrt[2 - 3*x])/Sqrt[11]], -1/2]))/(9176310*Sqrt[-5 + 2*x])
Time = 0.45 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.06, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {195, 25, 2110, 176, 124, 123, 131, 27, 129, 186, 27, 413, 27, 412}
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.
| \(\displaystyle \int \frac {\sqrt {2-3 x}}{\sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^2} \, dx\) |
| \(\Big \downarrow \) 195 |
| \(\displaystyle -\frac {\int -\frac {-120 x^2-336 x+479}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)}dx}{1794}-\frac {5 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{897 (5 x+7)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {-120 x^2-336 x+479}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)}dx}{1794}-\frac {5 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{897 (5 x+7)}\) |
| \(\Big \downarrow \) 2110 |
| \(\displaystyle \frac {\int \frac {-24 x-\frac {168}{5}}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}dx+\frac {3571}{5} \int \frac {1}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)}dx}{1794}-\frac {5 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{897 (5 x+7)}\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle \frac {-\frac {468}{5} \int \frac {1}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}dx-12 \int \frac {\sqrt {2 x-5}}{\sqrt {2-3 x} \sqrt {4 x+1}}dx+\frac {3571}{5} \int \frac {1}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)}dx}{1794}-\frac {5 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{897 (5 x+7)}\) |
| \(\Big \downarrow \) 124 |
| \(\displaystyle \frac {-\frac {12 \sqrt {2 x-5} \int \frac {\sqrt {5-2 x}}{\sqrt {2-3 x} \sqrt {4 x+1}}dx}{\sqrt {5-2 x}}-\frac {468}{5} \int \frac {1}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}dx+\frac {3571}{5} \int \frac {1}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)}dx}{1794}-\frac {5 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{897 (5 x+7)}\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle \frac {-\frac {468}{5} \int \frac {1}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}dx+\frac {3571}{5} \int \frac {1}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)}dx-\frac {2 \sqrt {66} \sqrt {2 x-5} E\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right )|\frac {1}{3}\right )}{\sqrt {5-2 x}}}{1794}-\frac {5 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{897 (5 x+7)}\) |
| \(\Big \downarrow \) 131 |
| \(\displaystyle \frac {-\frac {468 \sqrt {\frac {2}{11}} \sqrt {5-2 x} \int \frac {\sqrt {\frac {11}{2}}}{\sqrt {2-3 x} \sqrt {5-2 x} \sqrt {4 x+1}}dx}{5 \sqrt {2 x-5}}+\frac {3571}{5} \int \frac {1}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)}dx-\frac {2 \sqrt {66} \sqrt {2 x-5} E\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right )|\frac {1}{3}\right )}{\sqrt {5-2 x}}}{1794}-\frac {5 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{897 (5 x+7)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {468 \sqrt {5-2 x} \int \frac {1}{\sqrt {2-3 x} \sqrt {5-2 x} \sqrt {4 x+1}}dx}{5 \sqrt {2 x-5}}+\frac {3571}{5} \int \frac {1}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)}dx-\frac {2 \sqrt {66} \sqrt {2 x-5} E\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right )|\frac {1}{3}\right )}{\sqrt {5-2 x}}}{1794}-\frac {5 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{897 (5 x+7)}\) |
| \(\Big \downarrow \) 129 |
| \(\displaystyle \frac {\frac {3571}{5} \int \frac {1}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)}dx-\frac {156 \sqrt {\frac {6}{11}} \sqrt {5-2 x} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right ),\frac {1}{3}\right )}{5 \sqrt {2 x-5}}-\frac {2 \sqrt {66} \sqrt {2 x-5} E\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right )|\frac {1}{3}\right )}{\sqrt {5-2 x}}}{1794}-\frac {5 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{897 (5 x+7)}\) |
| \(\Big \downarrow \) 186 |
| \(\displaystyle \frac {-\frac {7142}{5} \int \frac {3}{(31-5 (2-3 x)) \sqrt {11-4 (2-3 x)} \sqrt {-2 (2-3 x)-11}}d\sqrt {2-3 x}-\frac {156 \sqrt {\frac {6}{11}} \sqrt {5-2 x} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right ),\frac {1}{3}\right )}{5 \sqrt {2 x-5}}-\frac {2 \sqrt {66} \sqrt {2 x-5} E\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right )|\frac {1}{3}\right )}{\sqrt {5-2 x}}}{1794}-\frac {5 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{897 (5 x+7)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {21426}{5} \int \frac {1}{(31-5 (2-3 x)) \sqrt {11-4 (2-3 x)} \sqrt {-2 (2-3 x)-11}}d\sqrt {2-3 x}-\frac {156 \sqrt {\frac {6}{11}} \sqrt {5-2 x} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right ),\frac {1}{3}\right )}{5 \sqrt {2 x-5}}-\frac {2 \sqrt {66} \sqrt {2 x-5} E\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right )|\frac {1}{3}\right )}{\sqrt {5-2 x}}}{1794}-\frac {5 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{897 (5 x+7)}\) |
| \(\Big \downarrow \) 413 |
| \(\displaystyle \frac {-\frac {21426 \sqrt {2 (2-3 x)+11} \int \frac {\sqrt {11}}{(31-5 (2-3 x)) \sqrt {11-4 (2-3 x)} \sqrt {2 (2-3 x)+11}}d\sqrt {2-3 x}}{5 \sqrt {11} \sqrt {-2 (2-3 x)-11}}-\frac {156 \sqrt {\frac {6}{11}} \sqrt {5-2 x} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right ),\frac {1}{3}\right )}{5 \sqrt {2 x-5}}-\frac {2 \sqrt {66} \sqrt {2 x-5} E\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right )|\frac {1}{3}\right )}{\sqrt {5-2 x}}}{1794}-\frac {5 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{897 (5 x+7)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {21426 \sqrt {2 (2-3 x)+11} \int \frac {1}{(31-5 (2-3 x)) \sqrt {11-4 (2-3 x)} \sqrt {2 (2-3 x)+11}}d\sqrt {2-3 x}}{5 \sqrt {-2 (2-3 x)-11}}-\frac {156 \sqrt {\frac {6}{11}} \sqrt {5-2 x} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right ),\frac {1}{3}\right )}{5 \sqrt {2 x-5}}-\frac {2 \sqrt {66} \sqrt {2 x-5} E\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right )|\frac {1}{3}\right )}{\sqrt {5-2 x}}}{1794}-\frac {5 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{897 (5 x+7)}\) |
| \(\Big \downarrow \) 412 |
| \(\displaystyle \frac {-\frac {156 \sqrt {\frac {6}{11}} \sqrt {5-2 x} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right ),\frac {1}{3}\right )}{5 \sqrt {2 x-5}}-\frac {2 \sqrt {66} \sqrt {2 x-5} E\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right )|\frac {1}{3}\right )}{\sqrt {5-2 x}}-\frac {10713 \sqrt {2 (2-3 x)+11} \operatorname {EllipticPi}\left (\frac {55}{124},\arcsin \left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right ),-\frac {1}{2}\right )}{155 \sqrt {11} \sqrt {-2 (2-3 x)-11}}}{1794}-\frac {5 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{897 (5 x+7)}\) |
(-5*Sqrt[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x])/(897*(7 + 5*x)) + ((-2*Sqr t[66]*Sqrt[-5 + 2*x]*EllipticE[ArcSin[Sqrt[3/11]*Sqrt[1 + 4*x]], 1/3])/Sqr t[5 - 2*x] - (156*Sqrt[6/11]*Sqrt[5 - 2*x]*EllipticF[ArcSin[Sqrt[3/11]*Sqr t[1 + 4*x]], 1/3])/(5*Sqrt[-5 + 2*x]) - (10713*Sqrt[11 + 2*(2 - 3*x)]*Elli pticPi[55/124, ArcSin[(2*Sqrt[2 - 3*x])/Sqrt[11]], -1/2])/(155*Sqrt[11]*Sq rt[-11 - 2*(2 - 3*x)]))/1794
3.1.56.3.1 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[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_ )]), x_] :> Simp[(2/b)*Rt[-(b*e - a*f)/d, 2]*EllipticE[ArcSin[Sqrt[a + b*x] /Rt[-(b*c - a*d)/d, 2]], f*((b*c - a*d)/(d*(b*e - a*f)))], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[b/(b*c - a*d), 0] && GtQ[b/(b*e - a*f), 0] && !L tQ[-(b*c - a*d)/d, 0] && !(SimplerQ[c + d*x, a + b*x] && GtQ[-d/(b*c - a*d ), 0] && GtQ[d/(d*e - c*f), 0] && !LtQ[(b*c - a*d)/b, 0])
Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_ )]), x_] :> Simp[Sqrt[e + f*x]*(Sqrt[b*((c + d*x)/(b*c - a*d))]/(Sqrt[c + d *x]*Sqrt[b*((e + f*x)/(b*e - a*f))])) Int[Sqrt[b*(e/(b*e - a*f)) + b*f*(x /(b*e - a*f))]/(Sqrt[a + b*x]*Sqrt[b*(c/(b*c - a*d)) + b*d*(x/(b*c - a*d))] ), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && !(GtQ[b/(b*c - a*d), 0] && Gt Q[b/(b*e - a*f), 0]) && !LtQ[-(b*c - a*d)/d, 0]
Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x _)]), x_] :> Simp[2*(Rt[-b/d, 2]/(b*Sqrt[(b*e - a*f)/b]))*EllipticF[ArcSin[ Sqrt[a + b*x]/(Rt[-b/d, 2]*Sqrt[(b*c - a*d)/b])], f*((b*c - a*d)/(d*(b*e - a*f)))], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[(b*c - a*d)/b, 0] && GtQ [(b*e - a*f)/b, 0] && PosQ[-b/d] && !(SimplerQ[c + d*x, a + b*x] && GtQ[(d *e - c*f)/d, 0] && GtQ[-d/b, 0]) && !(SimplerQ[c + d*x, a + b*x] && GtQ[(( -b)*e + a*f)/f, 0] && GtQ[-f/b, 0]) && !(SimplerQ[e + f*x, a + b*x] && GtQ [((-d)*e + c*f)/f, 0] && GtQ[((-b)*e + a*f)/f, 0] && (PosQ[-f/d] || PosQ[-f /b]))
Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x _)]), x_] :> Simp[Sqrt[b*((c + d*x)/(b*c - a*d))]/Sqrt[c + d*x] Int[1/(Sq rt[a + b*x]*Sqrt[b*(c/(b*c - a*d)) + b*d*(x/(b*c - a*d))]*Sqrt[e + f*x]), x ], x] /; FreeQ[{a, b, c, d, e, f}, x] && !GtQ[(b*c - a*d)/b, 0] && Simpler Q[a + b*x, c + d*x] && SimplerQ[a + b*x, e + f*x]
Int[((g_.) + (h_.)*(x_))/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]* Sqrt[(e_) + (f_.)*(x_)]), x_] :> Simp[h/f Int[Sqrt[e + f*x]/(Sqrt[a + b*x ]*Sqrt[c + d*x]), x], x] + Simp[(f*g - e*h)/f Int[1/(Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && Sim plerQ[a + b*x, e + f*x] && SimplerQ[c + d*x, e + f*x]
Int[1/(((a_.) + (b_.)*(x_))*Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_ )]*Sqrt[(g_.) + (h_.)*(x_)]), x_] :> Simp[-2 Subst[Int[1/(Simp[b*c - a*d - b*x^2, x]*Sqrt[Simp[(d*e - c*f)/d + f*(x^2/d), x]]*Sqrt[Simp[(d*g - c*h)/ d + h*(x^2/d), x]]), x], x, Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && GtQ[(d*e - c*f)/d, 0]
Int[(((a_.) + (b_.)*(x_))^(m_)*Sqrt[(c_.) + (d_.)*(x_)])/(Sqrt[(e_.) + (f_. )*(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x_] :> Simp[b*(a + b*x)^(m + 1)*Sqrt[c + d*x]*Sqrt[e + f*x]*(Sqrt[g + h*x]/((m + 1)*(b*e - a*f)*(b*g - a*h))), x] + Simp[1/(2*(m + 1)*(b*e - a*f)*(b*g - a*h)) Int[((a + b*x)^(m + 1)/(Sqrt[ c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]))*Simp[2*a*c*f*h*(m + 1) - b*(d*e*g + c*(2*m + 3)*(f*g + e*h)) + 2*(a*d*f*h*(m + 1) - b*(m + 2)*(d*f*g + d*e*h + c*f*h))*x - b*d*f*h*(2*m + 5)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g , h, m}, x] && IntegerQ[2*m] && LeQ[m, -2]
Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x _)^2]), x_Symbol] :> Simp[(1/(a*Sqrt[c]*Sqrt[e]*Rt[-d/c, 2]))*EllipticPi[b* (c/(a*d)), ArcSin[Rt[-d/c, 2]*x], c*(f/(d*e))], x] /; FreeQ[{a, b, c, d, e, f}, x] && !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] && !( !GtQ[f/e, 0] && S implerSqrtQ[-f/e, -d/c])
Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x _)^2]), x_Symbol] :> Simp[Sqrt[1 + (d/c)*x^2]/Sqrt[c + d*x^2] Int[1/((a + b*x^2)*Sqrt[1 + (d/c)*x^2]*Sqrt[e + f*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && !GtQ[c, 0]
Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f _.)*(x_))^(p_.)*((g_.) + (h_.)*(x_))^(q_.), x_Symbol] :> Simp[PolynomialRem ainder[Px, a + b*x, x] Int[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p*(g + h*x)^ q, x], x] + Int[PolynomialQuotient[Px, a + b*x, x]*(a + b*x)^(m + 1)*(c + d *x)^n*(e + f*x)^p*(g + h*x)^q, x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n, p , q}, x] && PolyQ[Px, x] && EqQ[m, -1]
Time = 1.65 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.31
| method | result | size |
| elliptic | \(\frac {\sqrt {-\left (-2+3 x \right ) \left (-5+2 x \right ) \left (1+4 x \right )}\, \left (-\frac {5 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}{897 \left (7+5 x \right )}-\frac {28 \sqrt {11+44 x}\, \sqrt {22-33 x}\, \sqrt {110-44 x}\, F\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right )}{180895 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}-\frac {4 \sqrt {11+44 x}\, \sqrt {22-33 x}\, \sqrt {110-44 x}\, \left (-\frac {11 E\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right )}{12}+\frac {2 F\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right )}{3}\right )}{36179 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}+\frac {7142 \sqrt {11+44 x}\, \sqrt {22-33 x}\, \sqrt {110-44 x}\, \Pi \left (\frac {\sqrt {11+44 x}}{11}, -\frac {55}{23}, \sqrt {3}\right )}{12481755 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}\right )}{\sqrt {2-3 x}\, \sqrt {-5+2 x}\, \sqrt {1+4 x}}\) | \(247\) |
| default | \(\frac {\sqrt {2-3 x}\, \sqrt {-5+2 x}\, \sqrt {1+4 x}\, \left (14260 \sqrt {1+4 x}\, \sqrt {2-3 x}\, \sqrt {22}\, \sqrt {5-2 x}\, F\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right ) x -6325 \sqrt {1+4 x}\, \sqrt {2-3 x}\, \sqrt {22}\, \sqrt {5-2 x}\, E\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right ) x -35710 \sqrt {1+4 x}\, \sqrt {2-3 x}\, \sqrt {22}\, \sqrt {5-2 x}\, \Pi \left (\frac {\sqrt {11+44 x}}{11}, -\frac {55}{23}, \sqrt {3}\right ) x +19964 \sqrt {1+4 x}\, \sqrt {2-3 x}\, \sqrt {22}\, \sqrt {5-2 x}\, F\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right )-8855 \sqrt {1+4 x}\, \sqrt {2-3 x}\, \sqrt {22}\, \sqrt {5-2 x}\, E\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right )-49994 \sqrt {1+4 x}\, \sqrt {2-3 x}\, \sqrt {22}\, \sqrt {5-2 x}\, \Pi \left (\frac {\sqrt {11+44 x}}{11}, -\frac {55}{23}, \sqrt {3}\right )-151800 x^{3}+442750 x^{2}-132825 x -63250\right )}{1134705 \left (24 x^{3}-70 x^{2}+21 x +10\right ) \left (7+5 x \right )}\) | \(302\) |
| risch | \(\frac {5 \left (-2+3 x \right ) \sqrt {-5+2 x}\, \sqrt {1+4 x}\, \sqrt {\left (2-3 x \right ) \left (-5+2 x \right ) \left (1+4 x \right )}}{897 \left (7+5 x \right ) \sqrt {-\left (-2+3 x \right ) \left (-5+2 x \right ) \left (1+4 x \right )}\, \sqrt {2-3 x}}-\frac {\left (-\frac {4 \sqrt {22-33 x}\, \sqrt {-66 x +165}\, \sqrt {33+132 x}\, \left (-\frac {11 E\left (\frac {2 \sqrt {22-33 x}}{11}, \frac {i \sqrt {2}}{2}\right )}{6}+\frac {5 F\left (\frac {2 \sqrt {22-33 x}}{11}, \frac {i \sqrt {2}}{2}\right )}{2}\right )}{108537 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}-\frac {28 \sqrt {22-33 x}\, \sqrt {-66 x +165}\, \sqrt {33+132 x}\, F\left (\frac {2 \sqrt {22-33 x}}{11}, \frac {i \sqrt {2}}{2}\right )}{542685 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}+\frac {3571 \sqrt {22-33 x}\, \sqrt {-66 x +165}\, \sqrt {33+132 x}\, \Pi \left (\frac {2 \sqrt {22-33 x}}{11}, \frac {55}{124}, \frac {i \sqrt {2}}{2}\right )}{33646470 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}\right ) \sqrt {\left (2-3 x \right ) \left (-5+2 x \right ) \left (1+4 x \right )}}{\sqrt {2-3 x}\, \sqrt {-5+2 x}\, \sqrt {1+4 x}}\) | \(306\) |
(-(-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)*(-5/897/(7+5*x)*(-24*x^3+70*x^2-21*x-10)^(1/2)-28/180895*(11+44*x)^(1/ 2)*(22-33*x)^(1/2)*(110-44*x)^(1/2)/(-24*x^3+70*x^2-21*x-10)^(1/2)*Ellipti cF(1/11*(11+44*x)^(1/2),3^(1/2))-4/36179*(11+44*x)^(1/2)*(22-33*x)^(1/2)*( 110-44*x)^(1/2)/(-24*x^3+70*x^2-21*x-10)^(1/2)*(-11/12*EllipticE(1/11*(11+ 44*x)^(1/2),3^(1/2))+2/3*EllipticF(1/11*(11+44*x)^(1/2),3^(1/2)))+7142/124 81755*(11+44*x)^(1/2)*(22-33*x)^(1/2)*(110-44*x)^(1/2)/(-24*x^3+70*x^2-21* x-10)^(1/2)*EllipticPi(1/11*(11+44*x)^(1/2),-55/23,3^(1/2)))
\[ \int \frac {\sqrt {2-3 x}}{\sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^2} \, dx=\int { \frac {\sqrt {-3 \, x + 2}}{{\left (5 \, x + 7\right )}^{2} \sqrt {4 \, x + 1} \sqrt {2 \, x - 5}} \,d x } \]
integral(sqrt(4*x + 1)*sqrt(2*x - 5)*sqrt(-3*x + 2)/(200*x^4 + 110*x^3 - 9 93*x^2 - 1232*x - 245), x)
\[ \int \frac {\sqrt {2-3 x}}{\sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^2} \, dx=\int \frac {\sqrt {2 - 3 x}}{\sqrt {2 x - 5} \sqrt {4 x + 1} \left (5 x + 7\right )^{2}}\, dx \]
\[ \int \frac {\sqrt {2-3 x}}{\sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^2} \, dx=\int { \frac {\sqrt {-3 \, x + 2}}{{\left (5 \, x + 7\right )}^{2} \sqrt {4 \, x + 1} \sqrt {2 \, x - 5}} \,d x } \]
\[ \int \frac {\sqrt {2-3 x}}{\sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^2} \, dx=\int { \frac {\sqrt {-3 \, x + 2}}{{\left (5 \, x + 7\right )}^{2} \sqrt {4 \, x + 1} \sqrt {2 \, x - 5}} \,d x } \]
Timed out. \[ \int \frac {\sqrt {2-3 x}}{\sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^2} \, dx=\int \frac {\sqrt {2-3\,x}}{\sqrt {4\,x+1}\,\sqrt {2\,x-5}\,{\left (5\,x+7\right )}^2} \,d x \]