Integrand size = 38, antiderivative size = 156 \[ \int \frac {4+6 x-2 x^2}{(5-3 x) \sqrt {1+2 x} \sqrt {3-4 x^2}} \, dx=-\frac {1}{3} \sqrt {1+\sqrt {3}} E\left (\arcsin \left (\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}\right )|3-\sqrt {3}\right )+\frac {11 \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}\right ),3-\sqrt {3}\right )}{9 \sqrt {1+\sqrt {3}}}-\frac {76}{657} \sqrt {2 \left (53+67 \sqrt {3}\right )} \operatorname {EllipticPi}\left (-\frac {6}{73} \left (9+10 \sqrt {3}\right ),\arcsin \left (\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}\right ),3-\sqrt {3}\right ) \] Output:
-1/3*(1+3^(1/2))^(1/2)*EllipticE(1/6*(3-2*x*3^(1/2))^(1/2)*6^(1/2),(3-3^(1 /2))^(1/2))+11/9*EllipticF(1/6*(3-2*x*3^(1/2))^(1/2)*6^(1/2),(3-3^(1/2))^( 1/2))/(1+3^(1/2))^(1/2)-76/657*(106+134*3^(1/2))^(1/2)*EllipticPi(1/6*(3-2 *x*3^(1/2))^(1/2)*6^(1/2),-54/73-60/73*3^(1/2),(3-3^(1/2))^(1/2))
Result contains complex when optimal does not.
Time = 22.61 (sec) , antiderivative size = 301, normalized size of antiderivative = 1.93 \[ \int \frac {4+6 x-2 x^2}{(5-3 x) \sqrt {1+2 x} \sqrt {3-4 x^2}} \, dx=-\frac {(3+6 x)^{3/2} \left (-78+\frac {156}{(1+2 x)^2}+\frac {156}{1+2 x}-\frac {78 i \sqrt {-1+\sqrt {3}} \sqrt {\frac {-3+4 x^2}{(1+2 x)^2}} E\left (i \text {arcsinh}\left (\frac {\sqrt {-1+\sqrt {3}}}{\sqrt {1+2 x}}\right )|-2-\sqrt {3}\right )}{\sqrt {1+2 x}}+\frac {6 i \left (-16+13 \sqrt {3}\right ) \sqrt {\frac {-3+4 x^2}{(1+2 x)^2}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-1+\sqrt {3}}}{\sqrt {1+2 x}}\right ),-2-\sqrt {3}\right )}{\sqrt {-1+\sqrt {3}} \sqrt {1+2 x}}+\frac {304 i \sqrt {\frac {-3+4 x^2}{(1+2 x)^2}} \operatorname {EllipticPi}\left (-\frac {13}{6} \left (1+\sqrt {3}\right ),i \text {arcsinh}\left (\frac {\sqrt {-1+\sqrt {3}}}{\sqrt {1+2 x}}\right ),-2-\sqrt {3}\right )}{\sqrt {-1+\sqrt {3}} \sqrt {1+2 x}}\right )}{702 \sqrt {9-12 x^2}} \] Input:
Integrate[(4 + 6*x - 2*x^2)/((5 - 3*x)*Sqrt[1 + 2*x]*Sqrt[3 - 4*x^2]),x]
Output:
-1/702*((3 + 6*x)^(3/2)*(-78 + 156/(1 + 2*x)^2 + 156/(1 + 2*x) - ((78*I)*S qrt[-1 + Sqrt[3]]*Sqrt[(-3 + 4*x^2)/(1 + 2*x)^2]*EllipticE[I*ArcSinh[Sqrt[ -1 + Sqrt[3]]/Sqrt[1 + 2*x]], -2 - Sqrt[3]])/Sqrt[1 + 2*x] + ((6*I)*(-16 + 13*Sqrt[3])*Sqrt[(-3 + 4*x^2)/(1 + 2*x)^2]*EllipticF[I*ArcSinh[Sqrt[-1 + Sqrt[3]]/Sqrt[1 + 2*x]], -2 - Sqrt[3]])/(Sqrt[-1 + Sqrt[3]]*Sqrt[1 + 2*x]) + ((304*I)*Sqrt[(-3 + 4*x^2)/(1 + 2*x)^2]*EllipticPi[(-13*(1 + Sqrt[3]))/ 6, I*ArcSinh[Sqrt[-1 + Sqrt[3]]/Sqrt[1 + 2*x]], -2 - Sqrt[3]])/(Sqrt[-1 + Sqrt[3]]*Sqrt[1 + 2*x])))/Sqrt[9 - 12*x^2]
Time = 0.96 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.06, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.289, Rules used = {2349, 600, 508, 327, 511, 321, 730, 27, 186, 25, 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.
| \(\displaystyle \int \frac {-2 x^2+6 x+4}{(5-3 x) \sqrt {2 x+1} \sqrt {3-4 x^2}} \, dx\) |
| \(\Big \downarrow \) 2349 |
| \(\displaystyle \frac {76}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \sqrt {3-4 x^2}}dx+\int \frac {\frac {2 x}{3}-\frac {8}{9}}{\sqrt {2 x+1} \sqrt {3-4 x^2}}dx\) |
| \(\Big \downarrow \) 600 |
| \(\displaystyle -\frac {11}{9} \int \frac {1}{\sqrt {2 x+1} \sqrt {3-4 x^2}}dx+\frac {76}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \sqrt {3-4 x^2}}dx+\frac {1}{3} \int \frac {\sqrt {2 x+1}}{\sqrt {3-4 x^2}}dx\) |
| \(\Big \downarrow \) 508 |
| \(\displaystyle -\frac {11}{9} \int \frac {1}{\sqrt {2 x+1} \sqrt {3-4 x^2}}dx+\frac {76}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \sqrt {3-4 x^2}}dx-\frac {\sqrt {3+\sqrt {3}} \int \frac {\sqrt {1-\frac {3-2 \sqrt {3} x}{3+\sqrt {3}}}}{\sqrt {\frac {1}{6} \left (2 \sqrt {3} x-3\right )+1}}d\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}}{3 \sqrt [4]{3}}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle -\frac {11}{9} \int \frac {1}{\sqrt {2 x+1} \sqrt {3-4 x^2}}dx+\frac {76}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \sqrt {3-4 x^2}}dx-\frac {\sqrt {3+\sqrt {3}} E\left (\arcsin \left (\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}\right )|3-\sqrt {3}\right )}{3 \sqrt [4]{3}}\) |
| \(\Big \downarrow \) 511 |
| \(\displaystyle \frac {76}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \sqrt {3-4 x^2}}dx+\frac {11 \int \frac {1}{\sqrt {1-\frac {3-2 \sqrt {3} x}{3+\sqrt {3}}} \sqrt {\frac {1}{6} \left (2 \sqrt {3} x-3\right )+1}}d\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}}{3\ 3^{3/4} \sqrt {3+\sqrt {3}}}-\frac {\sqrt {3+\sqrt {3}} E\left (\arcsin \left (\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}\right )|3-\sqrt {3}\right )}{3 \sqrt [4]{3}}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {76}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \sqrt {3-4 x^2}}dx+\frac {11 \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}\right ),3-\sqrt {3}\right )}{3\ 3^{3/4} \sqrt {3+\sqrt {3}}}-\frac {\sqrt {3+\sqrt {3}} E\left (\arcsin \left (\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}\right )|3-\sqrt {3}\right )}{3 \sqrt [4]{3}}\) |
| \(\Big \downarrow \) 730 |
| \(\displaystyle \frac {76 \int \frac {3}{(5-3 x) \sqrt {2 x+1} \sqrt {3-2 \sqrt {3} x} \sqrt {2 \sqrt {3} x+3}}dx}{9 \sqrt {3}}+\frac {11 \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}\right ),3-\sqrt {3}\right )}{3\ 3^{3/4} \sqrt {3+\sqrt {3}}}-\frac {\sqrt {3+\sqrt {3}} E\left (\arcsin \left (\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}\right )|3-\sqrt {3}\right )}{3 \sqrt [4]{3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {76 \int \frac {1}{(5-3 x) \sqrt {2 x+1} \sqrt {3-2 \sqrt {3} x} \sqrt {2 \sqrt {3} x+3}}dx}{3 \sqrt {3}}+\frac {11 \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}\right ),3-\sqrt {3}\right )}{3\ 3^{3/4} \sqrt {3+\sqrt {3}}}-\frac {\sqrt {3+\sqrt {3}} E\left (\arcsin \left (\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}\right )|3-\sqrt {3}\right )}{3 \sqrt [4]{3}}\) |
| \(\Big \downarrow \) 186 |
| \(\displaystyle -\frac {152 \int -\frac {1}{(13-3 (2 x+1)) \sqrt {-\sqrt {3} (2 x+1)+\sqrt {3}+3} \sqrt {\sqrt {3} (2 x+1)-\sqrt {3}+3}}d\sqrt {2 x+1}}{3 \sqrt {3}}+\frac {11 \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}\right ),3-\sqrt {3}\right )}{3\ 3^{3/4} \sqrt {3+\sqrt {3}}}-\frac {\sqrt {3+\sqrt {3}} E\left (\arcsin \left (\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}\right )|3-\sqrt {3}\right )}{3 \sqrt [4]{3}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {152 \int \frac {1}{(13-3 (2 x+1)) \sqrt {-\sqrt {3} (2 x+1)+\sqrt {3}+3} \sqrt {\sqrt {3} (2 x+1)-\sqrt {3}+3}}d\sqrt {2 x+1}}{3 \sqrt {3}}+\frac {11 \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}\right ),3-\sqrt {3}\right )}{3\ 3^{3/4} \sqrt {3+\sqrt {3}}}-\frac {\sqrt {3+\sqrt {3}} E\left (\arcsin \left (\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}\right )|3-\sqrt {3}\right )}{3 \sqrt [4]{3}}\) |
| \(\Big \downarrow \) 412 |
| \(\displaystyle \frac {11 \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}\right ),3-\sqrt {3}\right )}{3\ 3^{3/4} \sqrt {3+\sqrt {3}}}-\frac {\sqrt {3+\sqrt {3}} E\left (\arcsin \left (\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}\right )|3-\sqrt {3}\right )}{3 \sqrt [4]{3}}+\frac {152 \operatorname {EllipticPi}\left (\frac {3}{13} \left (1+\sqrt {3}\right ),\arcsin \left (\sqrt {\frac {1}{2} \left (-1+\sqrt {3}\right )} \sqrt {2 x+1}\right ),-2-\sqrt {3}\right )}{117 \sqrt {\sqrt {3}-1}}\) |
Input:
Int[(4 + 6*x - 2*x^2)/((5 - 3*x)*Sqrt[1 + 2*x]*Sqrt[3 - 4*x^2]),x]
Output:
-1/3*(Sqrt[3 + Sqrt[3]]*EllipticE[ArcSin[Sqrt[3 - 2*Sqrt[3]*x]/Sqrt[6]], 3 - Sqrt[3]])/3^(1/4) + (11*EllipticF[ArcSin[Sqrt[3 - 2*Sqrt[3]*x]/Sqrt[6]] , 3 - Sqrt[3]])/(3*3^(3/4)*Sqrt[3 + Sqrt[3]]) + (152*EllipticPi[(3*(1 + Sq rt[3]))/13, ArcSin[Sqrt[(-1 + Sqrt[3])/2]*Sqrt[1 + 2*x]], -2 - Sqrt[3]])/( 117*Sqrt[-1 + Sqrt[3]])
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/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c /(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0] && !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ (Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) )], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 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[Sqrt[(c_) + (d_.)*(x_)]/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> With[{q = Rt[-b/a, 2]}, Simp[-2*(Sqrt[c + d*x]/(Sqrt[a]*q*Sqrt[q*((c + d*x)/(d + c *q))])) Subst[Int[Sqrt[1 - 2*d*(x^2/(d + c*q))]/Sqrt[1 - x^2], x], x, Sqr t[(1 - q*x)/2]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && GtQ[a, 0]
Int[1/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Wit h[{q = Rt[-b/a, 2]}, Simp[-2*(Sqrt[q*((c + d*x)/(d + c*q))]/(Sqrt[a]*q*Sqrt [c + d*x])) Subst[Int[1/(Sqrt[1 - 2*d*(x^2/(d + c*q))]*Sqrt[1 - x^2]), x] , x, Sqrt[(1 - q*x)/2]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && GtQ[ a, 0]
Int[((A_.) + (B_.)*(x_))/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2] ), x_Symbol] :> Simp[B/d Int[Sqrt[c + d*x]/Sqrt[a + b*x^2], x], x] - Simp [(B*c - A*d)/d Int[1/(Sqrt[c + d*x]*Sqrt[a + b*x^2]), x], x] /; FreeQ[{a, b, c, d, A, B}, x] && NegQ[b/a]
Int[1/(Sqrt[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))*Sqrt[(a_) + (b_.)*(x_) ^2]), x_Symbol] :> With[{q = Rt[-b/a, 2]}, Simp[1/Sqrt[a] Int[1/((e + f*x )*Sqrt[c + d*x]*Sqrt[1 - q*x]*Sqrt[1 + q*x]), x], x]] /; FreeQ[{a, b, c, d, e, f}, x] && NegQ[b/a] && GtQ[a, 0]
Int[(Px_)*((c_) + (d_.)*(x_))^(m_.)*((e_) + (f_.)*(x_))^(n_.)*((a_.) + (b_. )*(x_)^2)^(p_.), x_Symbol] :> Int[PolynomialQuotient[Px, c + d*x, x]*(c + d *x)^(m + 1)*(e + f*x)^n*(a + b*x^2)^p, x] + Simp[PolynomialRemainder[Px, c + d*x, x] Int[(c + d*x)^m*(e + f*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && PolynomialQ[Px, x] && LtQ[m, 0] && !IntegerQ[n ] && IntegersQ[2*m, 2*n, 2*p]
Leaf count of result is larger than twice the leaf count of optimal. \(345\) vs. \(2(112)=224\).
Time = 0.36 (sec) , antiderivative size = 346, normalized size of antiderivative = 2.22
| method | result | size |
| default | \(\frac {\sqrt {1+2 x}\, \sqrt {-4 x^{2}+3}\, \sqrt {\left (2 x +\sqrt {3}\right ) \sqrt {3}}\, \sqrt {-\left (1+2 x \right ) \left (\sqrt {3}-1\right )}\, \sqrt {3}\, \sqrt {\left (-2 x +\sqrt {3}\right ) \sqrt {3}}\, \left (-3 \operatorname {EllipticE}\left (\frac {\sqrt {3}\, \sqrt {2}\, \sqrt {\left (2 x +\sqrt {3}\right ) \sqrt {3}}}{6}, \frac {\sqrt {2}\, \sqrt {\sqrt {3}\, \left (\sqrt {3}-1\right )}}{\sqrt {3}-1}\right )+33 \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {2}\, \sqrt {\left (2 x +\sqrt {3}\right ) \sqrt {3}}}{6}, \frac {\sqrt {2}\, \sqrt {\sqrt {3}\, \left (\sqrt {3}-1\right )}}{\sqrt {3}-1}\right ) \sqrt {3}+21 \operatorname {EllipticE}\left (\frac {\sqrt {3}\, \sqrt {2}\, \sqrt {\left (2 x +\sqrt {3}\right ) \sqrt {3}}}{6}, \frac {\sqrt {2}\, \sqrt {\sqrt {3}\, \left (\sqrt {3}-1\right )}}{\sqrt {3}-1}\right ) \sqrt {3}+110 \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {2}\, \sqrt {\left (2 x +\sqrt {3}\right ) \sqrt {3}}}{6}, \frac {\sqrt {2}\, \sqrt {\sqrt {3}\, \left (\sqrt {3}-1\right )}}{\sqrt {3}-1}\right )-152 \operatorname {EllipticPi}\left (\frac {\sqrt {3}\, \sqrt {2}\, \sqrt {\left (2 x +\sqrt {3}\right ) \sqrt {3}}}{6}, \frac {6 \sqrt {3}}{10+3 \sqrt {3}}, \frac {\sqrt {2}\, \sqrt {\sqrt {3}\, \left (\sqrt {3}-1\right )}}{\sqrt {3}-1}\right )\right )}{27 \left (\sqrt {3}-1\right ) \left (8 x^{3}+4 x^{2}-6 x -3\right ) \left (10+3 \sqrt {3}\right )}\) | \(346\) |
| elliptic | \(\frac {\sqrt {-\left (4 x^{2}-3\right ) \left (1+2 x \right )}\, \left (-\frac {16 \sqrt {\left (x +\frac {\sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {-\frac {x +\frac {1}{2}}{\frac {\sqrt {3}}{2}-\frac {1}{2}}}\, \sqrt {-3 \left (x -\frac {\sqrt {3}}{2}\right ) \sqrt {3}}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {\left (x +\frac {\sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {\sqrt {3}}{\frac {\sqrt {3}}{2}-\frac {1}{2}}}\right )}{27 \sqrt {-8 x^{3}-4 x^{2}+6 x +3}}+\frac {4 \sqrt {\left (x +\frac {\sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {-\frac {x +\frac {1}{2}}{\frac {\sqrt {3}}{2}-\frac {1}{2}}}\, \sqrt {-3 \left (x -\frac {\sqrt {3}}{2}\right ) \sqrt {3}}\, \left (\left (-\frac {\sqrt {3}}{2}+\frac {1}{2}\right ) \operatorname {EllipticE}\left (\frac {\sqrt {3}\, \sqrt {\left (x +\frac {\sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {\sqrt {3}}{\frac {\sqrt {3}}{2}-\frac {1}{2}}}\right )-\frac {\operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {\left (x +\frac {\sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {\sqrt {3}}{\frac {\sqrt {3}}{2}-\frac {1}{2}}}\right )}{2}\right )}{9 \sqrt {-8 x^{3}-4 x^{2}+6 x +3}}-\frac {152 \sqrt {\left (x +\frac {\sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {-\frac {x +\frac {1}{2}}{\frac {\sqrt {3}}{2}-\frac {1}{2}}}\, \sqrt {-3 \left (x -\frac {\sqrt {3}}{2}\right ) \sqrt {3}}\, \operatorname {EllipticPi}\left (\frac {\sqrt {3}\, \sqrt {\left (x +\frac {\sqrt {3}}{2}\right ) \sqrt {3}}}{3}, -\frac {\sqrt {3}}{-\frac {\sqrt {3}}{2}-\frac {5}{3}}, \sqrt {\frac {\sqrt {3}}{\frac {\sqrt {3}}{2}-\frac {1}{2}}}\right )}{81 \sqrt {-8 x^{3}-4 x^{2}+6 x +3}\, \left (-\frac {\sqrt {3}}{2}-\frac {5}{3}\right )}\right )}{\sqrt {-4 x^{2}+3}\, \sqrt {1+2 x}}\) | \(391\) |
Input:
int((-2*x^2+6*x+4)/(5-3*x)/(1+2*x)^(1/2)/(-4*x^2+3)^(1/2),x,method=_RETURN VERBOSE)
Output:
1/27*(1+2*x)^(1/2)*(-4*x^2+3)^(1/2)*((2*x+3^(1/2))*3^(1/2))^(1/2)/(3^(1/2) -1)*(-(1+2*x)*(3^(1/2)-1))^(1/2)*3^(1/2)*((-2*x+3^(1/2))*3^(1/2))^(1/2)*(- 3*EllipticE(1/6*3^(1/2)*2^(1/2)*((2*x+3^(1/2))*3^(1/2))^(1/2),2^(1/2)/(3^( 1/2)-1)*(3^(1/2)*(3^(1/2)-1))^(1/2))+33*EllipticF(1/6*3^(1/2)*2^(1/2)*((2* x+3^(1/2))*3^(1/2))^(1/2),2^(1/2)/(3^(1/2)-1)*(3^(1/2)*(3^(1/2)-1))^(1/2)) *3^(1/2)+21*EllipticE(1/6*3^(1/2)*2^(1/2)*((2*x+3^(1/2))*3^(1/2))^(1/2),2^ (1/2)/(3^(1/2)-1)*(3^(1/2)*(3^(1/2)-1))^(1/2))*3^(1/2)+110*EllipticF(1/6*3 ^(1/2)*2^(1/2)*((2*x+3^(1/2))*3^(1/2))^(1/2),2^(1/2)/(3^(1/2)-1)*(3^(1/2)* (3^(1/2)-1))^(1/2))-152*EllipticPi(1/6*3^(1/2)*2^(1/2)*((2*x+3^(1/2))*3^(1 /2))^(1/2),6*3^(1/2)/(10+3*3^(1/2)),2^(1/2)/(3^(1/2)-1)*(3^(1/2)*(3^(1/2)- 1))^(1/2)))/(8*x^3+4*x^2-6*x-3)/(10+3*3^(1/2))
\[ \int \frac {4+6 x-2 x^2}{(5-3 x) \sqrt {1+2 x} \sqrt {3-4 x^2}} \, dx=\int { \frac {2 \, {\left (x^{2} - 3 \, x - 2\right )}}{\sqrt {-4 \, x^{2} + 3} {\left (3 \, x - 5\right )} \sqrt {2 \, x + 1}} \,d x } \] Input:
integrate((-2*x^2+6*x+4)/(5-3*x)/(1+2*x)^(1/2)/(-4*x^2+3)^(1/2),x, algorit hm="fricas")
Output:
integral(-2*(x^2 - 3*x - 2)*sqrt(-4*x^2 + 3)*sqrt(2*x + 1)/(24*x^4 - 28*x^ 3 - 38*x^2 + 21*x + 15), x)
\[ \int \frac {4+6 x-2 x^2}{(5-3 x) \sqrt {1+2 x} \sqrt {3-4 x^2}} \, dx=2 \left (\int \left (- \frac {3 x}{3 x \sqrt {3 - 4 x^{2}} \sqrt {2 x + 1} - 5 \sqrt {3 - 4 x^{2}} \sqrt {2 x + 1}}\right )\, dx + \int \frac {x^{2}}{3 x \sqrt {3 - 4 x^{2}} \sqrt {2 x + 1} - 5 \sqrt {3 - 4 x^{2}} \sqrt {2 x + 1}}\, dx + \int \left (- \frac {2}{3 x \sqrt {3 - 4 x^{2}} \sqrt {2 x + 1} - 5 \sqrt {3 - 4 x^{2}} \sqrt {2 x + 1}}\right )\, dx\right ) \] Input:
integrate((-2*x**2+6*x+4)/(5-3*x)/(1+2*x)**(1/2)/(-4*x**2+3)**(1/2),x)
Output:
2*(Integral(-3*x/(3*x*sqrt(3 - 4*x**2)*sqrt(2*x + 1) - 5*sqrt(3 - 4*x**2)* sqrt(2*x + 1)), x) + Integral(x**2/(3*x*sqrt(3 - 4*x**2)*sqrt(2*x + 1) - 5 *sqrt(3 - 4*x**2)*sqrt(2*x + 1)), x) + Integral(-2/(3*x*sqrt(3 - 4*x**2)*s qrt(2*x + 1) - 5*sqrt(3 - 4*x**2)*sqrt(2*x + 1)), x))
\[ \int \frac {4+6 x-2 x^2}{(5-3 x) \sqrt {1+2 x} \sqrt {3-4 x^2}} \, dx=\int { \frac {2 \, {\left (x^{2} - 3 \, x - 2\right )}}{\sqrt {-4 \, x^{2} + 3} {\left (3 \, x - 5\right )} \sqrt {2 \, x + 1}} \,d x } \] Input:
integrate((-2*x^2+6*x+4)/(5-3*x)/(1+2*x)^(1/2)/(-4*x^2+3)^(1/2),x, algorit hm="maxima")
Output:
2*integrate((x^2 - 3*x - 2)/(sqrt(-4*x^2 + 3)*(3*x - 5)*sqrt(2*x + 1)), x)
\[ \int \frac {4+6 x-2 x^2}{(5-3 x) \sqrt {1+2 x} \sqrt {3-4 x^2}} \, dx=\int { \frac {2 \, {\left (x^{2} - 3 \, x - 2\right )}}{\sqrt {-4 \, x^{2} + 3} {\left (3 \, x - 5\right )} \sqrt {2 \, x + 1}} \,d x } \] Input:
integrate((-2*x^2+6*x+4)/(5-3*x)/(1+2*x)^(1/2)/(-4*x^2+3)^(1/2),x, algorit hm="giac")
Output:
integrate(2*(x^2 - 3*x - 2)/(sqrt(-4*x^2 + 3)*(3*x - 5)*sqrt(2*x + 1)), x)
Timed out. \[ \int \frac {4+6 x-2 x^2}{(5-3 x) \sqrt {1+2 x} \sqrt {3-4 x^2}} \, dx=\int -\frac {-2\,x^2+6\,x+4}{\sqrt {2\,x+1}\,\left (3\,x-5\right )\,\sqrt {3-4\,x^2}} \,d x \] Input:
int(-(6*x - 2*x^2 + 4)/((2*x + 1)^(1/2)*(3*x - 5)*(3 - 4*x^2)^(1/2)),x)
Output:
int(-(6*x - 2*x^2 + 4)/((2*x + 1)^(1/2)*(3*x - 5)*(3 - 4*x^2)^(1/2)), x)
\[ \int \frac {4+6 x-2 x^2}{(5-3 x) \sqrt {1+2 x} \sqrt {3-4 x^2}} \, dx=-2 \left (\int \frac {\sqrt {2 x +1}\, \sqrt {-4 x^{2}+3}\, x^{2}}{24 x^{4}-28 x^{3}-38 x^{2}+21 x +15}d x \right )+6 \left (\int \frac {\sqrt {2 x +1}\, \sqrt {-4 x^{2}+3}\, x}{24 x^{4}-28 x^{3}-38 x^{2}+21 x +15}d x \right )+4 \left (\int \frac {\sqrt {2 x +1}\, \sqrt {-4 x^{2}+3}}{24 x^{4}-28 x^{3}-38 x^{2}+21 x +15}d x \right ) \] Input:
int((-2*x^2+6*x+4)/(5-3*x)/(1+2*x)^(1/2)/(-4*x^2+3)^(1/2),x)
Output:
2*( - int((sqrt(2*x + 1)*sqrt( - 4*x**2 + 3)*x**2)/(24*x**4 - 28*x**3 - 38 *x**2 + 21*x + 15),x) + 3*int((sqrt(2*x + 1)*sqrt( - 4*x**2 + 3)*x)/(24*x* *4 - 28*x**3 - 38*x**2 + 21*x + 15),x) + 2*int((sqrt(2*x + 1)*sqrt( - 4*x* *2 + 3))/(24*x**4 - 28*x**3 - 38*x**2 + 21*x + 15),x))