Integrand size = 25, antiderivative size = 129 \[ \int \frac {2-5 x}{\sqrt {x} \sqrt {2+5 x+3 x^2}} \, dx=-\frac {10 \sqrt {x} (2+3 x)}{3 \sqrt {2+5 x+3 x^2}}+\frac {10 \sqrt {2} (1+x) \sqrt {\frac {2+3 x}{1+x}} E\left (\arctan \left (\sqrt {x}\right )|-\frac {1}{2}\right )}{3 \sqrt {2+5 x+3 x^2}}+\frac {2 \sqrt {2} (1+x) \sqrt {\frac {2+3 x}{1+x}} \operatorname {EllipticF}\left (\arctan \left (\sqrt {x}\right ),-\frac {1}{2}\right )}{\sqrt {2+5 x+3 x^2}} \]
-10/3*(2+3*x)*x^(1/2)/(3*x^2+5*x+2)^(1/2)+10/3*(1+x)^(3/2)*(1/(1+x))^(1/2) *EllipticE(x^(1/2)/(1+x)^(1/2),1/2*I*2^(1/2))*2^(1/2)*((2+3*x)/(1+x))^(1/2 )/(3*x^2+5*x+2)^(1/2)+2*(1+x)^(3/2)*(1/(1+x))^(1/2)*EllipticF(x^(1/2)/(1+x )^(1/2),1/2*I*2^(1/2))*2^(1/2)*((2+3*x)/(1+x))^(1/2)/(3*x^2+5*x+2)^(1/2)
Result contains complex when optimal does not.
Time = 21.14 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.16 \[ \int \frac {2-5 x}{\sqrt {x} \sqrt {2+5 x+3 x^2}} \, dx=-\frac {2 x^{3/2} \left (5 \left (3+\frac {2}{x^2}+\frac {5}{x}\right )+\frac {5 i \sqrt {2} \sqrt {1+\frac {1}{x}} \sqrt {3+\frac {2}{x}} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {2}{3}}}{\sqrt {x}}\right )|\frac {3}{2}\right )}{\sqrt {x}}-\frac {8 i \sqrt {2} \sqrt {1+\frac {1}{x}} \sqrt {3+\frac {2}{x}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {2}{3}}}{\sqrt {x}}\right ),\frac {3}{2}\right )}{\sqrt {x}}\right )}{3 \sqrt {2+5 x+3 x^2}} \]
(-2*x^(3/2)*(5*(3 + 2/x^2 + 5/x) + ((5*I)*Sqrt[2]*Sqrt[1 + x^(-1)]*Sqrt[3 + 2/x]*EllipticE[I*ArcSinh[Sqrt[2/3]/Sqrt[x]], 3/2])/Sqrt[x] - ((8*I)*Sqrt [2]*Sqrt[1 + x^(-1)]*Sqrt[3 + 2/x]*EllipticF[I*ArcSinh[Sqrt[2/3]/Sqrt[x]], 3/2])/Sqrt[x]))/(3*Sqrt[2 + 5*x + 3*x^2])
Time = 0.28 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.03, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {1240, 1503, 1413, 1456}
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.
\(\displaystyle \int \frac {2-5 x}{\sqrt {x} \sqrt {3 x^2+5 x+2}} \, dx\) |
\(\Big \downarrow \) 1240 |
\(\displaystyle 2 \int \frac {2-5 x}{\sqrt {3 x^2+5 x+2}}d\sqrt {x}\) |
\(\Big \downarrow \) 1503 |
\(\displaystyle 2 \left (2 \int \frac {1}{\sqrt {3 x^2+5 x+2}}d\sqrt {x}-5 \int \frac {x}{\sqrt {3 x^2+5 x+2}}d\sqrt {x}\right )\) |
\(\Big \downarrow \) 1413 |
\(\displaystyle 2 \left (\frac {\sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} \operatorname {EllipticF}\left (\arctan \left (\sqrt {x}\right ),-\frac {1}{2}\right )}{\sqrt {3 x^2+5 x+2}}-5 \int \frac {x}{\sqrt {3 x^2+5 x+2}}d\sqrt {x}\right )\) |
\(\Big \downarrow \) 1456 |
\(\displaystyle 2 \left (\frac {\sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} \operatorname {EllipticF}\left (\arctan \left (\sqrt {x}\right ),-\frac {1}{2}\right )}{\sqrt {3 x^2+5 x+2}}-5 \left (\frac {\sqrt {x} (3 x+2)}{3 \sqrt {3 x^2+5 x+2}}-\frac {\sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} E\left (\arctan \left (\sqrt {x}\right )|-\frac {1}{2}\right )}{3 \sqrt {3 x^2+5 x+2}}\right )\right )\) |
2*(-5*((Sqrt[x]*(2 + 3*x))/(3*Sqrt[2 + 5*x + 3*x^2]) - (Sqrt[2]*(1 + x)*Sq rt[(2 + 3*x)/(1 + x)]*EllipticE[ArcTan[Sqrt[x]], -1/2])/(3*Sqrt[2 + 5*x + 3*x^2])) + (Sqrt[2]*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*EllipticF[ArcTan[Sqrt[ x]], -1/2])/Sqrt[2 + 5*x + 3*x^2])
3.11.60.3.1 Defintions of rubi rules used
Int[((f_) + (g_.)*(x_))/(Sqrt[x_]*Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Simp[2 Subst[Int[(f + g*x^2)/Sqrt[a + b*x^2 + c*x^4], x], x, Sqrt[x]], x] /; FreeQ[{a, b, c, f, g}, x]
Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b ^2 - 4*a*c, 2]}, Simp[(2*a + (b - q)*x^2)*(Sqrt[(2*a + (b + q)*x^2)/(2*a + (b - q)*x^2)]/(2*a*Rt[(b - q)/(2*a), 2]*Sqrt[a + b*x^2 + c*x^4]))*EllipticF [ArcTan[Rt[(b - q)/(2*a), 2]*x], -2*(q/(b - q))], x] /; PosQ[(b - q)/a]] /; FreeQ[{a, b, c}, x] && GtQ[b^2 - 4*a*c, 0]
Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[x*((b - q + 2*c*x^2)/(2*c*Sqrt[a + b*x^2 + c*x^4 ])), x] - Simp[Rt[(b - q)/(2*a), 2]*(2*a + (b - q)*x^2)*(Sqrt[(2*a + (b + q )*x^2)/(2*a + (b - q)*x^2)]/(2*c*Sqrt[a + b*x^2 + c*x^4]))*EllipticE[ArcTan [Rt[(b - q)/(2*a), 2]*x], -2*(q/(b - q))], x] /; PosQ[(b - q)/a]] /; FreeQ[ {a, b, c}, x] && GtQ[b^2 - 4*a*c, 0]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo l] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[d Int[1/Sqrt[a + b*x^2 + c*x^4] , x], x] + Simp[e Int[x^2/Sqrt[a + b*x^2 + c*x^4], x], x] /; PosQ[(b + q) /a] || PosQ[(b - q)/a]] /; FreeQ[{a, b, c, d, e}, x] && GtQ[b^2 - 4*a*c, 0]
Time = 0.19 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.60
method | result | size |
default | \(\frac {\sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {6}\, \sqrt {-x}\, \left (21 F\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )-5 E\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )\right )}{9 \sqrt {x}\, \sqrt {3 x^{2}+5 x +2}}\) | \(77\) |
elliptic | \(\frac {\sqrt {x \left (3 x^{2}+5 x +2\right )}\, \left (\frac {2 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {-6 x}\, F\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{3 \sqrt {3 x^{3}+5 x^{2}+2 x}}-\frac {5 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {-6 x}\, \left (\frac {E\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{3}-F\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )\right )}{3 \sqrt {3 x^{3}+5 x^{2}+2 x}}\right )}{\sqrt {x}\, \sqrt {3 x^{2}+5 x +2}}\) | \(159\) |
1/9/x^(1/2)/(3*x^2+5*x+2)^(1/2)*(6*x+4)^(1/2)*(3+3*x)^(1/2)*6^(1/2)*(-x)^( 1/2)*(21*EllipticF(1/2*(6*x+4)^(1/2),I*2^(1/2))-5*EllipticE(1/2*(6*x+4)^(1 /2),I*2^(1/2)))
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.07 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.20 \[ \int \frac {2-5 x}{\sqrt {x} \sqrt {2+5 x+3 x^2}} \, dx=\frac {86}{27} \, \sqrt {3} {\rm weierstrassPInverse}\left (\frac {28}{27}, \frac {80}{729}, x + \frac {5}{9}\right ) + \frac {10}{3} \, \sqrt {3} {\rm weierstrassZeta}\left (\frac {28}{27}, \frac {80}{729}, {\rm weierstrassPInverse}\left (\frac {28}{27}, \frac {80}{729}, x + \frac {5}{9}\right )\right ) \]
86/27*sqrt(3)*weierstrassPInverse(28/27, 80/729, x + 5/9) + 10/3*sqrt(3)*w eierstrassZeta(28/27, 80/729, weierstrassPInverse(28/27, 80/729, x + 5/9))
\[ \int \frac {2-5 x}{\sqrt {x} \sqrt {2+5 x+3 x^2}} \, dx=- \int \left (- \frac {2}{\sqrt {x} \sqrt {3 x^{2} + 5 x + 2}}\right )\, dx - \int \frac {5 \sqrt {x}}{\sqrt {3 x^{2} + 5 x + 2}}\, dx \]
-Integral(-2/(sqrt(x)*sqrt(3*x**2 + 5*x + 2)), x) - Integral(5*sqrt(x)/sqr t(3*x**2 + 5*x + 2), x)
\[ \int \frac {2-5 x}{\sqrt {x} \sqrt {2+5 x+3 x^2}} \, dx=\int { -\frac {5 \, x - 2}{\sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {x}} \,d x } \]
\[ \int \frac {2-5 x}{\sqrt {x} \sqrt {2+5 x+3 x^2}} \, dx=\int { -\frac {5 \, x - 2}{\sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {x}} \,d x } \]
Timed out. \[ \int \frac {2-5 x}{\sqrt {x} \sqrt {2+5 x+3 x^2}} \, dx=-\int \frac {5\,x-2}{\sqrt {x}\,\sqrt {3\,x^2+5\,x+2}} \,d x \]