Integrand size = 35, antiderivative size = 51 \[ \int \frac {1}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)} \, dx=-\frac {3 \sqrt {5-2 x} \operatorname {EllipticPi}\left (\frac {55}{124},\arcsin \left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right ),-\frac {1}{2}\right )}{31 \sqrt {11} \sqrt {-5+2 x}} \]
-3/341*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)
Result contains complex when optimal does not.
Time = 3.57 (sec) , antiderivative size = 109, normalized size of antiderivative = 2.14 \[ \int \frac {1}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)} \, dx=\frac {3 i (-2+3 x) \sqrt {\frac {-5-18 x+8 x^2}{(2-3 x)^2}} \left (\operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {11}{2}}}{\sqrt {2-3 x}}\right ),-\frac {1}{2}\right )-\operatorname {EllipticPi}\left (-\frac {62}{55},i \text {arcsinh}\left (\frac {\sqrt {\frac {11}{2}}}{\sqrt {2-3 x}}\right ),-\frac {1}{2}\right )\right )}{31 \sqrt {1+4 x} \sqrt {-55+22 x}} \]
(((3*I)/31)*(-2 + 3*x)*Sqrt[(-5 - 18*x + 8*x^2)/(2 - 3*x)^2]*(EllipticF[I* ArcSinh[Sqrt[11/2]/Sqrt[2 - 3*x]], -1/2] - EllipticPi[-62/55, I*ArcSinh[Sq rt[11/2]/Sqrt[2 - 3*x]], -1/2]))/(Sqrt[1 + 4*x]*Sqrt[-55 + 22*x])
Time = 0.19 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.16, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {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.
\(\displaystyle \int \frac {1}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)} \, dx\) |
\(\Big \downarrow \) 186 |
\(\displaystyle -2 \int \frac {3}{(31-5 (2-3 x)) \sqrt {11-4 (2-3 x)} \sqrt {-2 (2-3 x)-11}}d\sqrt {2-3 x}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -6 \int \frac {1}{(31-5 (2-3 x)) \sqrt {11-4 (2-3 x)} \sqrt {-2 (2-3 x)-11}}d\sqrt {2-3 x}\) |
\(\Big \downarrow \) 413 |
\(\displaystyle -\frac {6 \sqrt {2 (2-3 x)+11} \int \frac {\sqrt {11}}{(31-5 (2-3 x)) \sqrt {11-4 (2-3 x)} \sqrt {2 (2-3 x)+11}}d\sqrt {2-3 x}}{\sqrt {11} \sqrt {-2 (2-3 x)-11}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {6 \sqrt {2 (2-3 x)+11} \int \frac {1}{(31-5 (2-3 x)) \sqrt {11-4 (2-3 x)} \sqrt {2 (2-3 x)+11}}d\sqrt {2-3 x}}{\sqrt {-2 (2-3 x)-11}}\) |
\(\Big \downarrow \) 412 |
\(\displaystyle -\frac {3 \sqrt {2 (2-3 x)+11} \operatorname {EllipticPi}\left (\frac {55}{124},\arcsin \left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right ),-\frac {1}{2}\right )}{31 \sqrt {11} \sqrt {-2 (2-3 x)-11}}\) |
(-3*Sqrt[11 + 2*(2 - 3*x)]*EllipticPi[55/124, ArcSin[(2*Sqrt[2 - 3*x])/Sqr t[11]], -1/2])/(31*Sqrt[11]*Sqrt[-11 - 2*(2 - 3*x)])
3.1.65.3.1 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[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[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]
Time = 1.61 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.67
method | result | size |
default | \(\frac {4 \Pi \left (\frac {\sqrt {11+44 x}}{11}, -\frac {55}{23}, \sqrt {3}\right ) \sqrt {5-2 x}\, \sqrt {22}}{253 \sqrt {-5+2 x}}\) | \(34\) |
elliptic | \(\frac {4 \sqrt {-\left (-2+3 x \right ) \left (-5+2 x \right ) \left (1+4 x \right )}\, \sqrt {11+44 x}\, \sqrt {22-33 x}\, \sqrt {110-44 x}\, \Pi \left (\frac {\sqrt {11+44 x}}{11}, -\frac {55}{23}, \sqrt {3}\right )}{2783 \sqrt {2-3 x}\, \sqrt {-5+2 x}\, \sqrt {1+4 x}\, \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}\) | \(95\) |
\[ \int \frac {1}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)} \, dx=\int { \frac {1}{{\left (5 \, x + 7\right )} \sqrt {4 \, x + 1} \sqrt {2 \, x - 5} \sqrt {-3 \, x + 2}} \,d x } \]
integral(-sqrt(4*x + 1)*sqrt(2*x - 5)*sqrt(-3*x + 2)/(120*x^4 - 182*x^3 - 385*x^2 + 197*x + 70), x)
\[ \int \frac {1}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)} \, dx=\int \frac {1}{\sqrt {2 - 3 x} \sqrt {2 x - 5} \sqrt {4 x + 1} \cdot \left (5 x + 7\right )}\, dx \]
\[ \int \frac {1}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)} \, dx=\int { \frac {1}{{\left (5 \, x + 7\right )} \sqrt {4 \, x + 1} \sqrt {2 \, x - 5} \sqrt {-3 \, x + 2}} \,d x } \]
\[ \int \frac {1}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)} \, dx=\int { \frac {1}{{\left (5 \, x + 7\right )} \sqrt {4 \, x + 1} \sqrt {2 \, x - 5} \sqrt {-3 \, x + 2}} \,d x } \]
Timed out. \[ \int \frac {1}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)} \, dx=\int \frac {1}{\sqrt {2-3\,x}\,\sqrt {4\,x+1}\,\sqrt {2\,x-5}\,\left (5\,x+7\right )} \,d x \]