Integrand size = 38, antiderivative size = 185 \[ \int \frac {4+6 x-2 x^2}{(5-3 x) \sqrt {1+2 x} \left (3-4 x^2\right )^{3/2}} \, dx=\frac {\sqrt {1+2 x} (87+242 x)}{876 \sqrt {3-4 x^2}}+\frac {121}{876} \sqrt {1+\sqrt {3}} E\left (\arcsin \left (\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}\right )|3-\sqrt {3}\right )-\frac {52 \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}\right ),3-\sqrt {3}\right )}{219 \sqrt {1+\sqrt {3}}}+\frac {76 \sqrt {2 \left (53+67 \sqrt {3}\right )} \operatorname {EllipticPi}\left (-\frac {6}{73} \left (9+10 \sqrt {3}\right ),\arcsin \left (\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}\right ),3-\sqrt {3}\right )}{5329} \] Output:
1/876*(1+2*x)^(1/2)*(87+242*x)/(-4*x^2+3)^(1/2)+121/876*(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))-52/219*Elli pticF(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)+76/5329*(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 = 21.95 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.99 \[ \int \frac {4+6 x-2 x^2}{(5-3 x) \sqrt {1+2 x} \left (3-4 x^2\right )^{3/2}} \, dx=\frac {\frac {13 \sqrt {1+2 x} (87+242 x)}{\sqrt {3-4 x^2}}-\frac {i \left (1573 \left (1+\sqrt {3}\right ) E\left (i \text {arcsinh}\left (\frac {\sqrt {1+2 x}}{\sqrt {-1+\sqrt {3}}}\right )|-2+\sqrt {3}\right )-13 \left (-87+121 \sqrt {3}\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {1+2 x}}{\sqrt {-1+\sqrt {3}}}\right ),-2+\sqrt {3}\right )-1824 \operatorname {EllipticPi}\left (-\frac {3}{13} \left (-1+\sqrt {3}\right ),i \text {arcsinh}\left (\frac {\sqrt {1+2 x}}{\sqrt {-1+\sqrt {3}}}\right ),-2+\sqrt {3}\right )\right )}{\sqrt {1+\sqrt {3}}}}{11388} \] Input:
Integrate[(4 + 6*x - 2*x^2)/((5 - 3*x)*Sqrt[1 + 2*x]*(3 - 4*x^2)^(3/2)),x]
Output:
((13*Sqrt[1 + 2*x]*(87 + 242*x))/Sqrt[3 - 4*x^2] - (I*(1573*(1 + Sqrt[3])* EllipticE[I*ArcSinh[Sqrt[1 + 2*x]/Sqrt[-1 + Sqrt[3]]], -2 + Sqrt[3]] - 13* (-87 + 121*Sqrt[3])*EllipticF[I*ArcSinh[Sqrt[1 + 2*x]/Sqrt[-1 + Sqrt[3]]], -2 + Sqrt[3]] - 1824*EllipticPi[(-3*(-1 + Sqrt[3]))/13, I*ArcSinh[Sqrt[1 + 2*x]/Sqrt[-1 + Sqrt[3]]], -2 + Sqrt[3]]))/Sqrt[1 + Sqrt[3]])/11388
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.
| \(\displaystyle \int \frac {-2 x^2+6 x+4}{(5-3 x) \sqrt {2 x+1} \left (3-4 x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 2349 |
| \(\displaystyle \frac {76}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \left (3-4 x^2\right )^{3/2}}dx+\int \frac {\frac {2 x}{3}-\frac {8}{9}}{\sqrt {2 x+1} \left (3-4 x^2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 686 |
| \(\displaystyle \frac {76}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \left (3-4 x^2\right )^{3/2}}dx+\frac {1}{96} \int -\frac {8 (34 x+33)}{9 \sqrt {2 x+1} \sqrt {3-4 x^2}}dx-\frac {\sqrt {2 x+1} (33-34 x)}{108 \sqrt {3-4 x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {76}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \left (3-4 x^2\right )^{3/2}}dx-\frac {1}{108} \int \frac {34 x+33}{\sqrt {2 x+1} \sqrt {3-4 x^2}}dx-\frac {\sqrt {2 x+1} (33-34 x)}{108 \sqrt {3-4 x^2}}\) |
| \(\Big \downarrow \) 600 |
| \(\displaystyle \frac {76}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \left (3-4 x^2\right )^{3/2}}dx+\frac {1}{108} \left (-16 \int \frac {1}{\sqrt {2 x+1} \sqrt {3-4 x^2}}dx-17 \int \frac {\sqrt {2 x+1}}{\sqrt {3-4 x^2}}dx\right )-\frac {\sqrt {2 x+1} (33-34 x)}{108 \sqrt {3-4 x^2}}\) |
| \(\Big \downarrow \) 508 |
| \(\displaystyle \frac {76}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \left (3-4 x^2\right )^{3/2}}dx+\frac {1}{108} \left (\frac {17 \sqrt {3+\sqrt {3}} \int \frac {\sqrt {1-\frac {3-2 \sqrt {3} x}{3+\sqrt {3}}}}{\sqrt {\frac {1}{6} \left (2 \sqrt {3} x-3\right )+1}}d\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}}{\sqrt [4]{3}}-16 \int \frac {1}{\sqrt {2 x+1} \sqrt {3-4 x^2}}dx\right )-\frac {\sqrt {2 x+1} (33-34 x)}{108 \sqrt {3-4 x^2}}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {1}{108} \left (\frac {17 \sqrt {3+\sqrt {3}} E\left (\arcsin \left (\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}\right )|3-\sqrt {3}\right )}{\sqrt [4]{3}}-16 \int \frac {1}{\sqrt {2 x+1} \sqrt {3-4 x^2}}dx\right )+\frac {76}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \left (3-4 x^2\right )^{3/2}}dx-\frac {\sqrt {2 x+1} (33-34 x)}{108 \sqrt {3-4 x^2}}\) |
| \(\Big \downarrow \) 511 |
| \(\displaystyle \frac {1}{108} \left (\frac {16 \sqrt [4]{3} \int \frac {1}{\sqrt {1-\frac {3-2 \sqrt {3} x}{3+\sqrt {3}}} \sqrt {\frac {1}{6} \left (2 \sqrt {3} x-3\right )+1}}d\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}}{\sqrt {3+\sqrt {3}}}+\frac {17 \sqrt {3+\sqrt {3}} E\left (\arcsin \left (\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}\right )|3-\sqrt {3}\right )}{\sqrt [4]{3}}\right )+\frac {76}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \left (3-4 x^2\right )^{3/2}}dx-\frac {\sqrt {2 x+1} (33-34 x)}{108 \sqrt {3-4 x^2}}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {76}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \left (3-4 x^2\right )^{3/2}}dx+\frac {1}{108} \left (\frac {16 \sqrt [4]{3} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}\right ),3-\sqrt {3}\right )}{\sqrt {3+\sqrt {3}}}+\frac {17 \sqrt {3+\sqrt {3}} E\left (\arcsin \left (\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}\right )|3-\sqrt {3}\right )}{\sqrt [4]{3}}\right )-\frac {\sqrt {2 x+1} (33-34 x)}{108 \sqrt {3-4 x^2}}\) |
| \(\Big \downarrow \) 744 |
| \(\displaystyle \frac {76}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \left (3-4 x^2\right )^{3/2}}dx+\frac {1}{108} \left (\frac {16 \sqrt [4]{3} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}\right ),3-\sqrt {3}\right )}{\sqrt {3+\sqrt {3}}}+\frac {17 \sqrt {3+\sqrt {3}} E\left (\arcsin \left (\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}\right )|3-\sqrt {3}\right )}{\sqrt [4]{3}}\right )-\frac {\sqrt {2 x+1} (33-34 x)}{108 \sqrt {3-4 x^2}}\) |
Input:
Int[(4 + 6*x - 2*x^2)/((5 - 3*x)*Sqrt[1 + 2*x]*(3 - 4*x^2)^(3/2)),x]
Output:
$Aborted
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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[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[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _), x_Symbol] :> Simp[(-(d + e*x)^(m + 1))*(f*a*c*e - a*g*c*d + c*(c*d*f + a*e*g)*x)*((a + c*x^2)^(p + 1)/(2*a*c*(p + 1)*(c*d^2 + a*e^2))), x] + Simp[ 1/(2*a*c*(p + 1)*(c*d^2 + a*e^2)) Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*Sim p[f*(c^2*d^2*(2*p + 3) + a*c*e^2*(m + 2*p + 3)) - a*c*d*e*g*m + c*e*(c*d*f + a*e*g)*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x] && LtQ [p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_) + (c_.)*(x_ )^2)^(p_), x_Symbol] :> Unintegrable[(d + e*x)^m*(f + g*x)^n*(a + c*x^2)^p, x] /; FreeQ[{a, c, d, e, f, g, m, n, p}, x]
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. \(415\) vs. \(2(135)=270\).
Time = 0.47 (sec) , antiderivative size = 416, normalized size of antiderivative = 2.25
| method | result | size |
| elliptic | \(\frac {\sqrt {-\left (4 x^{2}-3\right ) \left (1+2 x \right )}\, \left (-\frac {2 \left (-4-8 x \right ) \left (\frac {29}{2336}+\frac {121 x}{3504}\right )}{\sqrt {\left (x^{2}-\frac {3}{4}\right ) \left (-4-8 x \right )}}+\frac {29 \sqrt {\left (x +\frac {\sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {-\frac {x +\frac {1}{2}}{\frac {\sqrt {3}}{2}-\frac {1}{2}}}\, \sqrt {-3 \left (x -\frac {\sqrt {3}}{2}\right ) \sqrt {3}}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {\left (x +\frac {\sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {\sqrt {3}}{\frac {\sqrt {3}}{2}-\frac {1}{2}}}\right )}{438 \sqrt {-8 x^{3}-4 x^{2}+6 x +3}}-\frac {121 \sqrt {\left (x +\frac {\sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {-\frac {x +\frac {1}{2}}{\frac {\sqrt {3}}{2}-\frac {1}{2}}}\, \sqrt {-3 \left (x -\frac {\sqrt {3}}{2}\right ) \sqrt {3}}\, \left (\left (-\frac {\sqrt {3}}{2}+\frac {1}{2}\right ) \operatorname {EllipticE}\left (\frac {\sqrt {3}\, \sqrt {\left (x +\frac {\sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {\sqrt {3}}{\frac {\sqrt {3}}{2}-\frac {1}{2}}}\right )-\frac {\operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {\left (x +\frac {\sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {\sqrt {3}}{\frac {\sqrt {3}}{2}-\frac {1}{2}}}\right )}{2}\right )}{657 \sqrt {-8 x^{3}-4 x^{2}+6 x +3}}+\frac {152 \sqrt {\left (x +\frac {\sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {-\frac {x +\frac {1}{2}}{\frac {\sqrt {3}}{2}-\frac {1}{2}}}\, \sqrt {-3 \left (x -\frac {\sqrt {3}}{2}\right ) \sqrt {3}}\, \operatorname {EllipticPi}\left (\frac {\sqrt {3}\, \sqrt {\left (x +\frac {\sqrt {3}}{2}\right ) \sqrt {3}}}{3}, -\frac {\sqrt {3}}{-\frac {\sqrt {3}}{2}-\frac {5}{3}}, \sqrt {\frac {\sqrt {3}}{\frac {\sqrt {3}}{2}-\frac {1}{2}}}\right )}{657 \sqrt {-8 x^{3}-4 x^{2}+6 x +3}\, \left (-\frac {\sqrt {3}}{2}-\frac {5}{3}\right )}\right )}{\sqrt {-4 x^{2}+3}\, \sqrt {1+2 x}}\) | \(416\) |
| default | \(-\frac {\sqrt {1+2 x}\, \sqrt {-4 x^{2}+3}\, \left (2080 \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}}\, \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 )-1824 \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}}\, \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 )-121 \sqrt {-\left (1+2 x \right ) \left (\sqrt {3}-1\right )}\, \sqrt {3}\, \sqrt {\left (-2 x +\sqrt {3}\right ) \sqrt {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 ) \sqrt {\left (2 x +\sqrt {3}\right ) \sqrt {3}}-1452 x^{2}+1872 \sqrt {\left (2 x +\sqrt {3}\right ) \sqrt {3}}\, \sqrt {-\left (1+2 x \right ) \left (\sqrt {3}-1\right )}\, \sqrt {\left (-2 x +\sqrt {3}\right ) \sqrt {3}}\, \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 )+2541 \sqrt {-\left (1+2 x \right ) \left (\sqrt {3}-1\right )}\, \sqrt {\left (-2 x +\sqrt {3}\right ) \sqrt {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 ) \sqrt {\left (2 x +\sqrt {3}\right ) \sqrt {3}}-1248 x +10164 x^{2} \sqrt {3}-261+8736 \sqrt {3}\, x +1827 \sqrt {3}\right )}{2628 \left (\sqrt {3}-1\right ) \left (8 x^{3}+4 x^{2}-6 x -3\right ) \left (10+3 \sqrt {3}\right )}\) | \(534\) |
Input:
int((-2*x^2+6*x+4)/(5-3*x)/(1+2*x)^(1/2)/(-4*x^2+3)^(3/2),x,method=_RETURN VERBOSE)
Output:
(-(4*x^2-3)*(1+2*x))^(1/2)/(-4*x^2+3)^(1/2)/(1+2*x)^(1/2)*(-2*(-4-8*x)*(29 /2336+121/3504*x)/((x^2-3/4)*(-4-8*x))^(1/2)+29/438*((x+1/2*3^(1/2))*3^(1/ 2))^(1/2)*(-(x+1/2)/(1/2*3^(1/2)-1/2))^(1/2)*(-3*(x-1/2*3^(1/2))*3^(1/2))^ (1/2)/(-8*x^3-4*x^2+6*x+3)^(1/2)*EllipticF(1/3*3^(1/2)*((x+1/2*3^(1/2))*3^ (1/2))^(1/2),(3^(1/2)/(1/2*3^(1/2)-1/2))^(1/2))-121/657*((x+1/2*3^(1/2))*3 ^(1/2))^(1/2)*(-(x+1/2)/(1/2*3^(1/2)-1/2))^(1/2)*(-3*(x-1/2*3^(1/2))*3^(1/ 2))^(1/2)/(-8*x^3-4*x^2+6*x+3)^(1/2)*((-1/2*3^(1/2)+1/2)*EllipticE(1/3*3^( 1/2)*((x+1/2*3^(1/2))*3^(1/2))^(1/2),(3^(1/2)/(1/2*3^(1/2)-1/2))^(1/2))-1/ 2*EllipticF(1/3*3^(1/2)*((x+1/2*3^(1/2))*3^(1/2))^(1/2),(3^(1/2)/(1/2*3^(1 /2)-1/2))^(1/2)))+152/657*((x+1/2*3^(1/2))*3^(1/2))^(1/2)*(-(x+1/2)/(1/2*3 ^(1/2)-1/2))^(1/2)*(-3*(x-1/2*3^(1/2))*3^(1/2))^(1/2)/(-8*x^3-4*x^2+6*x+3) ^(1/2)/(-1/2*3^(1/2)-5/3)*EllipticPi(1/3*3^(1/2)*((x+1/2*3^(1/2))*3^(1/2)) ^(1/2),-3^(1/2)/(-1/2*3^(1/2)-5/3),(3^(1/2)/(1/2*3^(1/2)-1/2))^(1/2)))
\[ \int \frac {4+6 x-2 x^2}{(5-3 x) \sqrt {1+2 x} \left (3-4 x^2\right )^{3/2}} \, dx=\int { \frac {2 \, {\left (x^{2} - 3 \, x - 2\right )}}{{\left (-4 \, x^{2} + 3\right )}^{\frac {3}{2}} {\left (3 \, x - 5\right )} \sqrt {2 \, x + 1}} \,d x } \] Input:
integrate((-2*x^2+6*x+4)/(5-3*x)/(1+2*x)^(1/2)/(-4*x^2+3)^(3/2),x, algorit hm="fricas")
Output:
integral(2*(x^2 - 3*x - 2)*sqrt(-4*x^2 + 3)*sqrt(2*x + 1)/(96*x^6 - 112*x^ 5 - 224*x^4 + 168*x^3 + 174*x^2 - 63*x - 45), x)
\[ \int \frac {4+6 x-2 x^2}{(5-3 x) \sqrt {1+2 x} \left (3-4 x^2\right )^{3/2}} \, dx=2 \left (\int \left (- \frac {3 x}{- 12 x^{3} \sqrt {3 - 4 x^{2}} \sqrt {2 x + 1} + 20 x^{2} \sqrt {3 - 4 x^{2}} \sqrt {2 x + 1} + 9 x \sqrt {3 - 4 x^{2}} \sqrt {2 x + 1} - 15 \sqrt {3 - 4 x^{2}} \sqrt {2 x + 1}}\right )\, dx + \int \frac {x^{2}}{- 12 x^{3} \sqrt {3 - 4 x^{2}} \sqrt {2 x + 1} + 20 x^{2} \sqrt {3 - 4 x^{2}} \sqrt {2 x + 1} + 9 x \sqrt {3 - 4 x^{2}} \sqrt {2 x + 1} - 15 \sqrt {3 - 4 x^{2}} \sqrt {2 x + 1}}\, dx + \int \left (- \frac {2}{- 12 x^{3} \sqrt {3 - 4 x^{2}} \sqrt {2 x + 1} + 20 x^{2} \sqrt {3 - 4 x^{2}} \sqrt {2 x + 1} + 9 x \sqrt {3 - 4 x^{2}} \sqrt {2 x + 1} - 15 \sqrt {3 - 4 x^{2}} \sqrt {2 x + 1}}\right )\, dx\right ) \] Input:
integrate((-2*x**2+6*x+4)/(5-3*x)/(1+2*x)**(1/2)/(-4*x**2+3)**(3/2),x)
Output:
2*(Integral(-3*x/(-12*x**3*sqrt(3 - 4*x**2)*sqrt(2*x + 1) + 20*x**2*sqrt(3 - 4*x**2)*sqrt(2*x + 1) + 9*x*sqrt(3 - 4*x**2)*sqrt(2*x + 1) - 15*sqrt(3 - 4*x**2)*sqrt(2*x + 1)), x) + Integral(x**2/(-12*x**3*sqrt(3 - 4*x**2)*sq rt(2*x + 1) + 20*x**2*sqrt(3 - 4*x**2)*sqrt(2*x + 1) + 9*x*sqrt(3 - 4*x**2 )*sqrt(2*x + 1) - 15*sqrt(3 - 4*x**2)*sqrt(2*x + 1)), x) + Integral(-2/(-1 2*x**3*sqrt(3 - 4*x**2)*sqrt(2*x + 1) + 20*x**2*sqrt(3 - 4*x**2)*sqrt(2*x + 1) + 9*x*sqrt(3 - 4*x**2)*sqrt(2*x + 1) - 15*sqrt(3 - 4*x**2)*sqrt(2*x + 1)), x))
\[ \int \frac {4+6 x-2 x^2}{(5-3 x) \sqrt {1+2 x} \left (3-4 x^2\right )^{3/2}} \, dx=\int { \frac {2 \, {\left (x^{2} - 3 \, x - 2\right )}}{{\left (-4 \, x^{2} + 3\right )}^{\frac {3}{2}} {\left (3 \, x - 5\right )} \sqrt {2 \, x + 1}} \,d x } \] Input:
integrate((-2*x^2+6*x+4)/(5-3*x)/(1+2*x)^(1/2)/(-4*x^2+3)^(3/2),x, algorit hm="maxima")
Output:
2*integrate((x^2 - 3*x - 2)/((-4*x^2 + 3)^(3/2)*(3*x - 5)*sqrt(2*x + 1)), x)
\[ \int \frac {4+6 x-2 x^2}{(5-3 x) \sqrt {1+2 x} \left (3-4 x^2\right )^{3/2}} \, dx=\int { \frac {2 \, {\left (x^{2} - 3 \, x - 2\right )}}{{\left (-4 \, x^{2} + 3\right )}^{\frac {3}{2}} {\left (3 \, x - 5\right )} \sqrt {2 \, x + 1}} \,d x } \] Input:
integrate((-2*x^2+6*x+4)/(5-3*x)/(1+2*x)^(1/2)/(-4*x^2+3)^(3/2),x, algorit hm="giac")
Output:
integrate(2*(x^2 - 3*x - 2)/((-4*x^2 + 3)^(3/2)*(3*x - 5)*sqrt(2*x + 1)), x)
Timed out. \[ \int \frac {4+6 x-2 x^2}{(5-3 x) \sqrt {1+2 x} \left (3-4 x^2\right )^{3/2}} \, dx=\int -\frac {-2\,x^2+6\,x+4}{\sqrt {2\,x+1}\,\left (3\,x-5\right )\,{\left (3-4\,x^2\right )}^{3/2}} \,d x \] Input:
int(-(6*x - 2*x^2 + 4)/((2*x + 1)^(1/2)*(3*x - 5)*(3 - 4*x^2)^(3/2)),x)
Output:
int(-(6*x - 2*x^2 + 4)/((2*x + 1)^(1/2)*(3*x - 5)*(3 - 4*x^2)^(3/2)), x)
\[ \int \frac {4+6 x-2 x^2}{(5-3 x) \sqrt {1+2 x} \left (3-4 x^2\right )^{3/2}} \, dx=2 \left (\int \frac {\sqrt {2 x +1}\, \sqrt {-4 x^{2}+3}\, x^{2}}{96 x^{6}-112 x^{5}-224 x^{4}+168 x^{3}+174 x^{2}-63 x -45}d x \right )-6 \left (\int \frac {\sqrt {2 x +1}\, \sqrt {-4 x^{2}+3}\, x}{96 x^{6}-112 x^{5}-224 x^{4}+168 x^{3}+174 x^{2}-63 x -45}d x \right )-4 \left (\int \frac {\sqrt {2 x +1}\, \sqrt {-4 x^{2}+3}}{96 x^{6}-112 x^{5}-224 x^{4}+168 x^{3}+174 x^{2}-63 x -45}d x \right ) \] Input:
int((-2*x^2+6*x+4)/(5-3*x)/(1+2*x)^(1/2)/(-4*x^2+3)^(3/2),x)
Output:
2*(int((sqrt(2*x + 1)*sqrt( - 4*x**2 + 3)*x**2)/(96*x**6 - 112*x**5 - 224* x**4 + 168*x**3 + 174*x**2 - 63*x - 45),x) - 3*int((sqrt(2*x + 1)*sqrt( - 4*x**2 + 3)*x)/(96*x**6 - 112*x**5 - 224*x**4 + 168*x**3 + 174*x**2 - 63*x - 45),x) - 2*int((sqrt(2*x + 1)*sqrt( - 4*x**2 + 3))/(96*x**6 - 112*x**5 - 224*x**4 + 168*x**3 + 174*x**2 - 63*x - 45),x))