Integrand size = 25, antiderivative size = 185 \[ \int \frac {2-5 x}{\sqrt {x} \left (2+5 x+3 x^2\right )^{5/2}} \, dx=\frac {2 \sqrt {x} (38+45 x)}{3 \left (2+5 x+3 x^2\right )^{3/2}}+\frac {715 \sqrt {x} (2+3 x)}{3 \sqrt {2+5 x+3 x^2}}-\frac {5 \sqrt {x} (361+429 x)}{3 \sqrt {2+5 x+3 x^2}}-\frac {715 \sqrt {2} (1+x) \sqrt {\frac {2+3 x}{1+x}} E\left (\arctan \left (\sqrt {x}\right )|-\frac {1}{2}\right )}{3 \sqrt {2+5 x+3 x^2}}+\frac {295 \sqrt {2} (1+x) \sqrt {\frac {2+3 x}{1+x}} \operatorname {EllipticF}\left (\arctan \left (\sqrt {x}\right ),-\frac {1}{2}\right )}{\sqrt {2+5 x+3 x^2}} \]
2/3*(38+45*x)*x^(1/2)/(3*x^2+5*x+2)^(3/2)+715/3*(2+3*x)*x^(1/2)/(3*x^2+5*x +2)^(1/2)-5/3*(361+429*x)*x^(1/2)/(3*x^2+5*x+2)^(1/2)-715/3*(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)+295*(1+x)^(3/2)*(1/(1+x))^(1/2)*Ellipti cF(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 = 21.20 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.90 \[ \int \frac {2-5 x}{\sqrt {x} \left (2+5 x+3 x^2\right )^{5/2}} \, dx=\frac {2 \left (1430+5383 x+6615 x^2+2655 x^3\right )+715 i \sqrt {2+\frac {2}{x}} \sqrt {3+\frac {2}{x}} x^{3/2} \left (2+5 x+3 x^2\right ) E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {2}{3}}}{\sqrt {x}}\right )|\frac {3}{2}\right )+170 i \sqrt {2+\frac {2}{x}} \sqrt {3+\frac {2}{x}} x^{3/2} \left (2+5 x+3 x^2\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {2}{3}}}{\sqrt {x}}\right ),\frac {3}{2}\right )}{3 \sqrt {x} \left (2+5 x+3 x^2\right )^{3/2}} \]
(2*(1430 + 5383*x + 6615*x^2 + 2655*x^3) + (715*I)*Sqrt[2 + 2/x]*Sqrt[3 + 2/x]*x^(3/2)*(2 + 5*x + 3*x^2)*EllipticE[I*ArcSinh[Sqrt[2/3]/Sqrt[x]], 3/2 ] + (170*I)*Sqrt[2 + 2/x]*Sqrt[3 + 2/x]*x^(3/2)*(2 + 5*x + 3*x^2)*Elliptic F[I*ArcSinh[Sqrt[2/3]/Sqrt[x]], 3/2])/(3*Sqrt[x]*(2 + 5*x + 3*x^2)^(3/2))
Time = 0.37 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.04, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {1235, 27, 1235, 27, 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.
| \(\displaystyle \int \frac {2-5 x}{\sqrt {x} \left (3 x^2+5 x+2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 1235 |
| \(\displaystyle \frac {2 \sqrt {x} (45 x+38)}{3 \left (3 x^2+5 x+2\right )^{3/2}}-\frac {1}{3} \int \frac {5 (7-27 x)}{\sqrt {x} \left (3 x^2+5 x+2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \sqrt {x} (45 x+38)}{3 \left (3 x^2+5 x+2\right )^{3/2}}-\frac {5}{3} \int \frac {7-27 x}{\sqrt {x} \left (3 x^2+5 x+2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 1235 |
| \(\displaystyle \frac {2 \sqrt {x} (45 x+38)}{3 \left (3 x^2+5 x+2\right )^{3/2}}-\frac {5}{3} \left (\frac {\sqrt {x} (429 x+361)}{\sqrt {3 x^2+5 x+2}}-\int \frac {3 (143 x+118)}{2 \sqrt {x} \sqrt {3 x^2+5 x+2}}dx\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \sqrt {x} (45 x+38)}{3 \left (3 x^2+5 x+2\right )^{3/2}}-\frac {5}{3} \left (\frac {\sqrt {x} (429 x+361)}{\sqrt {3 x^2+5 x+2}}-\frac {3}{2} \int \frac {143 x+118}{\sqrt {x} \sqrt {3 x^2+5 x+2}}dx\right )\) |
| \(\Big \downarrow \) 1240 |
| \(\displaystyle \frac {2 \sqrt {x} (45 x+38)}{3 \left (3 x^2+5 x+2\right )^{3/2}}-\frac {5}{3} \left (\frac {\sqrt {x} (429 x+361)}{\sqrt {3 x^2+5 x+2}}-3 \int \frac {143 x+118}{\sqrt {3 x^2+5 x+2}}d\sqrt {x}\right )\) |
| \(\Big \downarrow \) 1503 |
| \(\displaystyle \frac {2 \sqrt {x} (45 x+38)}{3 \left (3 x^2+5 x+2\right )^{3/2}}-\frac {5}{3} \left (\frac {\sqrt {x} (429 x+361)}{\sqrt {3 x^2+5 x+2}}-3 \left (118 \int \frac {1}{\sqrt {3 x^2+5 x+2}}d\sqrt {x}+143 \int \frac {x}{\sqrt {3 x^2+5 x+2}}d\sqrt {x}\right )\right )\) |
| \(\Big \downarrow \) 1413 |
| \(\displaystyle \frac {2 \sqrt {x} (45 x+38)}{3 \left (3 x^2+5 x+2\right )^{3/2}}-\frac {5}{3} \left (\frac {\sqrt {x} (429 x+361)}{\sqrt {3 x^2+5 x+2}}-3 \left (143 \int \frac {x}{\sqrt {3 x^2+5 x+2}}d\sqrt {x}+\frac {59 \sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} \operatorname {EllipticF}\left (\arctan \left (\sqrt {x}\right ),-\frac {1}{2}\right )}{\sqrt {3 x^2+5 x+2}}\right )\right )\) |
| \(\Big \downarrow \) 1456 |
| \(\displaystyle \frac {2 \sqrt {x} (45 x+38)}{3 \left (3 x^2+5 x+2\right )^{3/2}}-\frac {5}{3} \left (\frac {\sqrt {x} (429 x+361)}{\sqrt {3 x^2+5 x+2}}-3 \left (\frac {59 \sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} \operatorname {EllipticF}\left (\arctan \left (\sqrt {x}\right ),-\frac {1}{2}\right )}{\sqrt {3 x^2+5 x+2}}+143 \left (\frac {\sqrt {x} (3 x+2)}{3 \sqrt {3 x^2+5 x+2}}-\frac {\sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} E\left (\arctan \left (\sqrt {x}\right )|-\frac {1}{2}\right )}{3 \sqrt {3 x^2+5 x+2}}\right )\right )\right )\) |
(2*Sqrt[x]*(38 + 45*x))/(3*(2 + 5*x + 3*x^2)^(3/2)) - (5*((Sqrt[x]*(361 + 429*x))/Sqrt[2 + 5*x + 3*x^2] - 3*(143*((Sqrt[x]*(2 + 3*x))/(3*Sqrt[2 + 5* x + 3*x^2]) - (Sqrt[2]*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*EllipticE[ArcTan[Sq rt[x]], -1/2])/(3*Sqrt[2 + 5*x + 3*x^2])) + (59*Sqrt[2]*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*EllipticF[ArcTan[Sqrt[x]], -1/2])/Sqrt[2 + 5*x + 3*x^2])))/3
3.11.79.3.1 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[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(f*(b*c*d - b^2*e + 2 *a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*e))*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))), x] + Simp[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^m *(a + b*x + c*x^2)^(p + 1)*Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2*(p + m + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d* m + b*e*m) - b*d*(3*c*d - b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - f*(2*c*d - b*e))*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && LtQ[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p] )
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 = 0.21 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.16
| method | result | size |
| elliptic | \(\frac {\sqrt {x \left (3 x^{2}+5 x +2\right )}\, \left (\frac {\left (\frac {76}{27}+\frac {10 x}{3}\right ) \sqrt {3 x^{3}+5 x^{2}+2 x}}{\left (x^{2}+\frac {5}{3} x +\frac {2}{3}\right )^{2}}-\frac {2 x \left (\frac {1805}{18}+\frac {715 x}{6}\right ) \sqrt {3}}{\sqrt {x \left (x^{2}+\frac {5}{3} x +\frac {2}{3}\right )}}+\frac {295 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {-6 x}\, F\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{3 \sqrt {3 x^{3}+5 x^{2}+2 x}}+\frac {715 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {-6 x}\, \left (\frac {E\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{3}-F\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )\right )}{6 \sqrt {3 x^{3}+5 x^{2}+2 x}}\right )}{\sqrt {x}\, \sqrt {3 x^{2}+5 x +2}}\) | \(214\) |
| default | \(-\frac {\left (1125 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {6}\, \sqrt {-x}\, F\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right ) x^{2}-2145 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {6}\, \sqrt {-x}\, E\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right ) x^{2}+1875 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {6}\, \sqrt {-x}\, F\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right ) x -3575 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {6}\, \sqrt {-x}\, E\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right ) x +750 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {6}\, \sqrt {-x}\, F\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )-1430 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {6}\, \sqrt {-x}\, E\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )+38610 x^{4}+96840 x^{3}+79350 x^{2}+21204 x \right ) \sqrt {3 x^{2}+5 x +2}}{18 \sqrt {x}\, \left (2+3 x \right )^{2} \left (1+x \right )^{2}}\) | \(297\) |
(x*(3*x^2+5*x+2))^(1/2)/x^(1/2)/(3*x^2+5*x+2)^(1/2)*((76/27+10/3*x)*(3*x^3 +5*x^2+2*x)^(1/2)/(x^2+5/3*x+2/3)^2-2*x*(1805/18+715/6*x)*3^(1/2)/(x*(x^2+ 5/3*x+2/3))^(1/2)+295/3*(6*x+4)^(1/2)*(3+3*x)^(1/2)*(-6*x)^(1/2)/(3*x^3+5* x^2+2*x)^(1/2)*EllipticF(1/2*(6*x+4)^(1/2),I*2^(1/2))+715/6*(6*x+4)^(1/2)* (3+3*x)^(1/2)*(-6*x)^(1/2)/(3*x^3+5*x^2+2*x)^(1/2)*(1/3*EllipticE(1/2*(6*x +4)^(1/2),I*2^(1/2))-EllipticF(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 = 0.08 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.66 \[ \int \frac {2-5 x}{\sqrt {x} \left (2+5 x+3 x^2\right )^{5/2}} \, dx=\frac {1735 \, \sqrt {3} {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} {\rm weierstrassPInverse}\left (\frac {28}{27}, \frac {80}{729}, x + \frac {5}{9}\right ) - 6435 \, \sqrt {3} {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} {\rm weierstrassZeta}\left (\frac {28}{27}, \frac {80}{729}, {\rm weierstrassPInverse}\left (\frac {28}{27}, \frac {80}{729}, x + \frac {5}{9}\right )\right ) - 9 \, {\left (6435 \, x^{3} + 16140 \, x^{2} + 13225 \, x + 3534\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {x}}{27 \, {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )}} \]
1/27*(1735*sqrt(3)*(9*x^4 + 30*x^3 + 37*x^2 + 20*x + 4)*weierstrassPInvers e(28/27, 80/729, x + 5/9) - 6435*sqrt(3)*(9*x^4 + 30*x^3 + 37*x^2 + 20*x + 4)*weierstrassZeta(28/27, 80/729, weierstrassPInverse(28/27, 80/729, x + 5/9)) - 9*(6435*x^3 + 16140*x^2 + 13225*x + 3534)*sqrt(3*x^2 + 5*x + 2)*sq rt(x))/(9*x^4 + 30*x^3 + 37*x^2 + 20*x + 4)
\[ \int \frac {2-5 x}{\sqrt {x} \left (2+5 x+3 x^2\right )^{5/2}} \, dx=- \int \frac {5 \sqrt {x}}{9 x^{4} \sqrt {3 x^{2} + 5 x + 2} + 30 x^{3} \sqrt {3 x^{2} + 5 x + 2} + 37 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 20 x \sqrt {3 x^{2} + 5 x + 2} + 4 \sqrt {3 x^{2} + 5 x + 2}}\, dx - \int \left (- \frac {2}{9 x^{\frac {9}{2}} \sqrt {3 x^{2} + 5 x + 2} + 30 x^{\frac {7}{2}} \sqrt {3 x^{2} + 5 x + 2} + 37 x^{\frac {5}{2}} \sqrt {3 x^{2} + 5 x + 2} + 20 x^{\frac {3}{2}} \sqrt {3 x^{2} + 5 x + 2} + 4 \sqrt {x} \sqrt {3 x^{2} + 5 x + 2}}\right )\, dx \]
-Integral(5*sqrt(x)/(9*x**4*sqrt(3*x**2 + 5*x + 2) + 30*x**3*sqrt(3*x**2 + 5*x + 2) + 37*x**2*sqrt(3*x**2 + 5*x + 2) + 20*x*sqrt(3*x**2 + 5*x + 2) + 4*sqrt(3*x**2 + 5*x + 2)), x) - Integral(-2/(9*x**(9/2)*sqrt(3*x**2 + 5*x + 2) + 30*x**(7/2)*sqrt(3*x**2 + 5*x + 2) + 37*x**(5/2)*sqrt(3*x**2 + 5*x + 2) + 20*x**(3/2)*sqrt(3*x**2 + 5*x + 2) + 4*sqrt(x)*sqrt(3*x**2 + 5*x + 2)), x)
\[ \int \frac {2-5 x}{\sqrt {x} \left (2+5 x+3 x^2\right )^{5/2}} \, dx=\int { -\frac {5 \, x - 2}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} \sqrt {x}} \,d x } \]
\[ \int \frac {2-5 x}{\sqrt {x} \left (2+5 x+3 x^2\right )^{5/2}} \, dx=\int { -\frac {5 \, x - 2}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} \sqrt {x}} \,d x } \]
Timed out. \[ \int \frac {2-5 x}{\sqrt {x} \left (2+5 x+3 x^2\right )^{5/2}} \, dx=-\int \frac {5\,x-2}{\sqrt {x}\,{\left (3\,x^2+5\,x+2\right )}^{5/2}} \,d x \]