Integrand size = 38, antiderivative size = 214 \[ \int \frac {\left (3-4 x^2\right )^{3/2} \left (4+6 x-2 x^2\right )}{(5-3 x) \sqrt {1+2 x}} \, dx=\frac {2 \sqrt {1+2 x} (17237+8982 x) \sqrt {3-4 x^2}}{8505}-\frac {2}{189} (16-7 x) \sqrt {1+2 x} \left (3-4 x^2\right )^{3/2}+\frac {53104 \sqrt {1+\sqrt {3}} E\left (\arcsin \left (\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}\right )|3-\sqrt {3}\right )}{1701}+\frac {2304832 \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}\right ),3-\sqrt {3}\right )}{25515 \sqrt {1+\sqrt {3}}}-\frac {5548}{729} \sqrt {2 \left (53+67 \sqrt {3}\right )} \operatorname {EllipticPi}\left (-\frac {6}{73} \left (9+10 \sqrt {3}\right ),\arcsin \left (\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}\right ),3-\sqrt {3}\right ) \] Output:
2/8505*(1+2*x)^(1/2)*(17237+8982*x)*(-4*x^2+3)^(1/2)-2/189*(16-7*x)*(1+2*x )^(1/2)*(-4*x^2+3)^(3/2)+53104/1701*(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))+2304832/25515*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)-5548/729*( 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 = 21.87 (sec) , antiderivative size = 193, normalized size of antiderivative = 0.90 \[ \int \frac {\left (3-4 x^2\right )^{3/2} \left (4+6 x-2 x^2\right )}{(5-3 x) \sqrt {1+2 x}} \, dx=\frac {2 \left (-117 \sqrt {1+2 x} \sqrt {3-4 x^2} \left (-15077-9927 x-2880 x^2+1260 x^3\right )-\frac {12 i \left (1294410 \left (1+\sqrt {3}\right ) E\left (i \text {arcsinh}\left (\frac {\sqrt {1+2 x}}{\sqrt {-1+\sqrt {3}}}\right )|-2+\sqrt {3}\right )-26 \left (193837+49785 \sqrt {3}\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {1+2 x}}{\sqrt {-1+\sqrt {3}}}\right ),-2+\sqrt {3}\right )+3543785 \operatorname {EllipticPi}\left (-\frac {3}{13} \left (-1+\sqrt {3}\right ),i \text {arcsinh}\left (\frac {\sqrt {1+2 x}}{\sqrt {-1+\sqrt {3}}}\right ),-2+\sqrt {3}\right )\right )}{\sqrt {1+\sqrt {3}}}\right )}{995085} \] Input:
Integrate[((3 - 4*x^2)^(3/2)*(4 + 6*x - 2*x^2))/((5 - 3*x)*Sqrt[1 + 2*x]), x]
Output:
(2*(-117*Sqrt[1 + 2*x]*Sqrt[3 - 4*x^2]*(-15077 - 9927*x - 2880*x^2 + 1260* x^3) - ((12*I)*(1294410*(1 + Sqrt[3])*EllipticE[I*ArcSinh[Sqrt[1 + 2*x]/Sq rt[-1 + Sqrt[3]]], -2 + Sqrt[3]] - 26*(193837 + 49785*Sqrt[3])*EllipticF[I *ArcSinh[Sqrt[1 + 2*x]/Sqrt[-1 + Sqrt[3]]], -2 + Sqrt[3]] + 3543785*Ellipt icPi[(-3*(-1 + Sqrt[3]))/13, I*ArcSinh[Sqrt[1 + 2*x]/Sqrt[-1 + Sqrt[3]]], -2 + Sqrt[3]]))/Sqrt[1 + Sqrt[3]]))/995085
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.
| \(\displaystyle \int \frac {\left (3-4 x^2\right )^{3/2} \left (-2 x^2+6 x+4\right )}{(5-3 x) \sqrt {2 x+1}} \, dx\) |
| \(\Big \downarrow \) 2349 |
| \(\displaystyle \frac {76}{9} \int \frac {\left (3-4 x^2\right )^{3/2}}{(5-3 x) \sqrt {2 x+1}}dx+\int \frac {\left (\frac {2 x}{3}-\frac {8}{9}\right ) \left (3-4 x^2\right )^{3/2}}{\sqrt {2 x+1}}dx\) |
| \(\Big \downarrow \) 682 |
| \(\displaystyle -\frac {1}{84} \int \frac {8 (22 x+75) \sqrt {3-4 x^2}}{3 \sqrt {2 x+1}}dx+\frac {76}{9} \int \frac {\left (3-4 x^2\right )^{3/2}}{(5-3 x) \sqrt {2 x+1}}dx-\frac {2}{189} (16-7 x) \sqrt {2 x+1} \left (3-4 x^2\right )^{3/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2}{63} \int \frac {(22 x+75) \sqrt {3-4 x^2}}{\sqrt {2 x+1}}dx+\frac {76}{9} \int \frac {\left (3-4 x^2\right )^{3/2}}{(5-3 x) \sqrt {2 x+1}}dx-\frac {2}{189} (16-7 x) \sqrt {2 x+1} \left (3-4 x^2\right )^{3/2}\) |
| \(\Big \downarrow \) 682 |
| \(\displaystyle -\frac {2}{63} \left (\frac {1}{15} \sqrt {2 x+1} (66 x+331) \sqrt {3-4 x^2}-\frac {1}{60} \int -\frac {32 (215 x+273)}{\sqrt {2 x+1} \sqrt {3-4 x^2}}dx\right )+\frac {76}{9} \int \frac {\left (3-4 x^2\right )^{3/2}}{(5-3 x) \sqrt {2 x+1}}dx-\frac {2}{189} (16-7 x) \sqrt {2 x+1} \left (3-4 x^2\right )^{3/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2}{63} \left (\frac {8}{15} \int \frac {215 x+273}{\sqrt {2 x+1} \sqrt {3-4 x^2}}dx+\frac {1}{15} \sqrt {2 x+1} \sqrt {3-4 x^2} (66 x+331)\right )+\frac {76}{9} \int \frac {\left (3-4 x^2\right )^{3/2}}{(5-3 x) \sqrt {2 x+1}}dx-\frac {2}{189} (16-7 x) \sqrt {2 x+1} \left (3-4 x^2\right )^{3/2}\) |
| \(\Big \downarrow \) 600 |
| \(\displaystyle -\frac {2}{63} \left (\frac {8}{15} \left (\frac {331}{2} \int \frac {1}{\sqrt {2 x+1} \sqrt {3-4 x^2}}dx+\frac {215}{2} \int \frac {\sqrt {2 x+1}}{\sqrt {3-4 x^2}}dx\right )+\frac {1}{15} \sqrt {2 x+1} \sqrt {3-4 x^2} (66 x+331)\right )+\frac {76}{9} \int \frac {\left (3-4 x^2\right )^{3/2}}{(5-3 x) \sqrt {2 x+1}}dx-\frac {2}{189} (16-7 x) \sqrt {2 x+1} \left (3-4 x^2\right )^{3/2}\) |
| \(\Big \downarrow \) 508 |
| \(\displaystyle \frac {76}{9} \int \frac {\left (3-4 x^2\right )^{3/2}}{(5-3 x) \sqrt {2 x+1}}dx-\frac {2}{63} \left (\frac {8}{15} \left (\frac {331}{2} \int \frac {1}{\sqrt {2 x+1} \sqrt {3-4 x^2}}dx-\frac {215 \sqrt {3+\sqrt {3}} \int \frac {\sqrt {1-\frac {3-2 \sqrt {3} x}{3+\sqrt {3}}}}{\sqrt {\frac {1}{6} \left (2 \sqrt {3} x-3\right )+1}}d\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}}{2 \sqrt [4]{3}}\right )+\frac {1}{15} \sqrt {2 x+1} \sqrt {3-4 x^2} (66 x+331)\right )-\frac {2}{189} (16-7 x) \sqrt {2 x+1} \left (3-4 x^2\right )^{3/2}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle -\frac {2}{63} \left (\frac {8}{15} \left (\frac {331}{2} \int \frac {1}{\sqrt {2 x+1} \sqrt {3-4 x^2}}dx-\frac {215 \sqrt {3+\sqrt {3}} E\left (\arcsin \left (\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}\right )|3-\sqrt {3}\right )}{2 \sqrt [4]{3}}\right )+\frac {1}{15} \sqrt {2 x+1} \sqrt {3-4 x^2} (66 x+331)\right )+\frac {76}{9} \int \frac {\left (3-4 x^2\right )^{3/2}}{(5-3 x) \sqrt {2 x+1}}dx-\frac {2}{189} (16-7 x) \sqrt {2 x+1} \left (3-4 x^2\right )^{3/2}\) |
| \(\Big \downarrow \) 511 |
| \(\displaystyle -\frac {2}{63} \left (\frac {8}{15} \left (-\frac {331 \sqrt [4]{3} \int \frac {1}{\sqrt {1-\frac {3-2 \sqrt {3} x}{3+\sqrt {3}}} \sqrt {\frac {1}{6} \left (2 \sqrt {3} x-3\right )+1}}d\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}}{2 \sqrt {3+\sqrt {3}}}-\frac {215 \sqrt {3+\sqrt {3}} E\left (\arcsin \left (\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}\right )|3-\sqrt {3}\right )}{2 \sqrt [4]{3}}\right )+\frac {1}{15} \sqrt {2 x+1} \sqrt {3-4 x^2} (66 x+331)\right )+\frac {76}{9} \int \frac {\left (3-4 x^2\right )^{3/2}}{(5-3 x) \sqrt {2 x+1}}dx-\frac {2}{189} (16-7 x) \sqrt {2 x+1} \left (3-4 x^2\right )^{3/2}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {76}{9} \int \frac {\left (3-4 x^2\right )^{3/2}}{(5-3 x) \sqrt {2 x+1}}dx-\frac {2}{63} \left (\frac {8}{15} \left (-\frac {331 \sqrt [4]{3} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}\right ),3-\sqrt {3}\right )}{2 \sqrt {3+\sqrt {3}}}-\frac {215 \sqrt {3+\sqrt {3}} E\left (\arcsin \left (\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}\right )|3-\sqrt {3}\right )}{2 \sqrt [4]{3}}\right )+\frac {1}{15} \sqrt {2 x+1} \sqrt {3-4 x^2} (66 x+331)\right )-\frac {2}{189} (16-7 x) \sqrt {2 x+1} \left (3-4 x^2\right )^{3/2}\) |
| \(\Big \downarrow \) 744 |
| \(\displaystyle \frac {76}{9} \int \frac {\left (3-4 x^2\right )^{3/2}}{(5-3 x) \sqrt {2 x+1}}dx-\frac {2}{63} \left (\frac {8}{15} \left (-\frac {331 \sqrt [4]{3} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}\right ),3-\sqrt {3}\right )}{2 \sqrt {3+\sqrt {3}}}-\frac {215 \sqrt {3+\sqrt {3}} E\left (\arcsin \left (\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}\right )|3-\sqrt {3}\right )}{2 \sqrt [4]{3}}\right )+\frac {1}{15} \sqrt {2 x+1} \sqrt {3-4 x^2} (66 x+331)\right )-\frac {2}{189} (16-7 x) \sqrt {2 x+1} \left (3-4 x^2\right )^{3/2}\) |
Input:
Int[((3 - 4*x^2)^(3/2)*(4 + 6*x - 2*x^2))/((5 - 3*x)*Sqrt[1 + 2*x]),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)*(c*e*f*(m + 2*p + 2) - g*c*d*(2*p + 1) + g*c*e*(m + 2*p + 1)*x)*((a + c*x^2)^p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2))), x] + Simp[2*(p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2))) Int[(d + e*x) ^m*(a + c*x^2)^(p - 1)*Simp[f*a*c*e^2*(m + 2*p + 2) + a*c*d*e*g*m - (c^2*f* d*e*(m + 2*p + 2) - g*(c^2*d^2*(2*p + 1) + a*c*e^2*(m + 2*p + 1)))*x, x], x ], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && GtQ[p, 0] && (IntegerQ[p] || ! RationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) && !ILtQ[m + 2*p, 0] && (Intege rQ[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. \(462\) vs. \(2(158)=316\).
Time = 0.46 (sec) , antiderivative size = 463, normalized size of antiderivative = 2.16
| method | result | size |
| risch | \(\frac {2 \left (1260 x^{3}-2880 x^{2}-9927 x -15077\right ) \left (4 x^{2}-3\right ) \sqrt {1+2 x}\, \sqrt {\left (-4 x^{2}+3\right ) \left (1+2 x \right )}}{8505 \sqrt {-\left (4 x^{2}-3\right ) \left (1+2 x \right )}\, \sqrt {-4 x^{2}+3}}+\frac {2 \left (-\frac {443056 \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 )}{10935 \sqrt {-8 x^{3}-4 x^{2}+6 x +3}}-\frac {106208 \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 )}{5103 \sqrt {-8 x^{3}-4 x^{2}+6 x +3}}-\frac {405004 \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 )}{6561 \sqrt {-8 x^{3}-4 x^{2}+6 x +3}\, \left (-\frac {\sqrt {3}}{2}-\frac {5}{3}\right )}\right ) \sqrt {\left (-4 x^{2}+3\right ) \left (1+2 x \right )}}{\sqrt {-4 x^{2}+3}\, \sqrt {1+2 x}}\) | \(463\) |
| elliptic | \(\frac {\sqrt {-\left (4 x^{2}-3\right ) \left (1+2 x \right )}\, \left (-\frac {8 x^{3} \sqrt {-8 x^{3}-4 x^{2}+6 x +3}}{27}+\frac {128 x^{2} \sqrt {-8 x^{3}-4 x^{2}+6 x +3}}{189}+\frac {2206 x \sqrt {-8 x^{3}-4 x^{2}+6 x +3}}{945}+\frac {30154 \sqrt {-8 x^{3}-4 x^{2}+6 x +3}}{8505}-\frac {886112 \sqrt {\left (x +\frac {\sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {-\frac {x +\frac {1}{2}}{\frac {\sqrt {3}}{2}-\frac {1}{2}}}\, \sqrt {-3 \left (x -\frac {\sqrt {3}}{2}\right ) \sqrt {3}}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {\left (x +\frac {\sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {\sqrt {3}}{\frac {\sqrt {3}}{2}-\frac {1}{2}}}\right )}{10935 \sqrt {-8 x^{3}-4 x^{2}+6 x +3}}-\frac {212416 \sqrt {\left (x +\frac {\sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {-\frac {x +\frac {1}{2}}{\frac {\sqrt {3}}{2}-\frac {1}{2}}}\, \sqrt {-3 \left (x -\frac {\sqrt {3}}{2}\right ) \sqrt {3}}\, \left (\left (-\frac {\sqrt {3}}{2}+\frac {1}{2}\right ) \operatorname {EllipticE}\left (\frac {\sqrt {3}\, \sqrt {\left (x +\frac {\sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {\sqrt {3}}{\frac {\sqrt {3}}{2}-\frac {1}{2}}}\right )-\frac {\operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {\left (x +\frac {\sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {\sqrt {3}}{\frac {\sqrt {3}}{2}-\frac {1}{2}}}\right )}{2}\right )}{5103 \sqrt {-8 x^{3}-4 x^{2}+6 x +3}}-\frac {810008 \sqrt {\left (x +\frac {\sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {-\frac {x +\frac {1}{2}}{\frac {\sqrt {3}}{2}-\frac {1}{2}}}\, \sqrt {-3 \left (x -\frac {\sqrt {3}}{2}\right ) \sqrt {3}}\, \operatorname {EllipticPi}\left (\frac {\sqrt {3}\, \sqrt {\left (x +\frac {\sqrt {3}}{2}\right ) \sqrt {3}}}{3}, -\frac {\sqrt {3}}{-\frac {\sqrt {3}}{2}-\frac {5}{3}}, \sqrt {\frac {\sqrt {3}}{\frac {\sqrt {3}}{2}-\frac {1}{2}}}\right )}{6561 \sqrt {-8 x^{3}-4 x^{2}+6 x +3}\, \left (-\frac {\sqrt {3}}{2}-\frac {5}{3}\right )}\right )}{\sqrt {-4 x^{2}+3}\, \sqrt {1+2 x}}\) | \(474\) |
| default | \(\frac {2 \sqrt {-4 x^{2}+3}\, \sqrt {1+2 x}\, \left (407079+1082187 x -162000 x^{5}-2849553 \sqrt {3}-886464 x^{4}+71046 x^{2}-1321416 x^{3}+90720 x^{6}-7575309 \sqrt {3}\, x -8363880 \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}}-635040 \sqrt {3}\, x^{6}+1134000 x^{5} \sqrt {3}+6205248 x^{4} \sqrt {3}+9249912 x^{3} \sqrt {3}-497322 x^{2} \sqrt {3}+10371744 \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 )+11524160 \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 )+398280 \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}}-14175140 \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 )\right )}{76545 \left (\sqrt {3}-1\right ) \left (8 x^{3}+4 x^{2}-6 x -3\right ) \left (10+3 \sqrt {3}\right )}\) | \(586\) |
Input:
int((-4*x^2+3)^(3/2)*(-2*x^2+6*x+4)/(5-3*x)/(1+2*x)^(1/2),x,method=_RETURN VERBOSE)
Output:
2/8505*(1260*x^3-2880*x^2-9927*x-15077)*(4*x^2-3)*(1+2*x)^(1/2)/(-(4*x^2-3 )*(1+2*x))^(1/2)*((-4*x^2+3)*(1+2*x))^(1/2)/(-4*x^2+3)^(1/2)+2*(-443056/10 935*((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))- 106208/5103*((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)))-405004/6561*((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)))*((-4*x^2+3)*(1+2*x))^(1/2)/(-4*x^2+3)^(1/2) /(1+2*x)^(1/2)
\[ \int \frac {\left (3-4 x^2\right )^{3/2} \left (4+6 x-2 x^2\right )}{(5-3 x) \sqrt {1+2 x}} \, dx=\int { \frac {2 \, {\left (x^{2} - 3 \, x - 2\right )} {\left (-4 \, x^{2} + 3\right )}^{\frac {3}{2}}}{{\left (3 \, x - 5\right )} \sqrt {2 \, x + 1}} \,d x } \] Input:
integrate((-4*x^2+3)^(3/2)*(-2*x^2+6*x+4)/(5-3*x)/(1+2*x)^(1/2),x, algorit hm="fricas")
Output:
integral(-2*(4*x^4 - 12*x^3 - 11*x^2 + 9*x + 6)*sqrt(-4*x^2 + 3)*sqrt(2*x + 1)/(6*x^2 - 7*x - 5), x)
\[ \int \frac {\left (3-4 x^2\right )^{3/2} \left (4+6 x-2 x^2\right )}{(5-3 x) \sqrt {1+2 x}} \, dx=2 \left (\int \left (- \frac {6 \sqrt {3 - 4 x^{2}}}{3 x \sqrt {2 x + 1} - 5 \sqrt {2 x + 1}}\right )\, dx + \int \left (- \frac {9 x \sqrt {3 - 4 x^{2}}}{3 x \sqrt {2 x + 1} - 5 \sqrt {2 x + 1}}\right )\, dx + \int \frac {11 x^{2} \sqrt {3 - 4 x^{2}}}{3 x \sqrt {2 x + 1} - 5 \sqrt {2 x + 1}}\, dx + \int \frac {12 x^{3} \sqrt {3 - 4 x^{2}}}{3 x \sqrt {2 x + 1} - 5 \sqrt {2 x + 1}}\, dx + \int \left (- \frac {4 x^{4} \sqrt {3 - 4 x^{2}}}{3 x \sqrt {2 x + 1} - 5 \sqrt {2 x + 1}}\right )\, dx\right ) \] Input:
integrate((-4*x**2+3)**(3/2)*(-2*x**2+6*x+4)/(5-3*x)/(1+2*x)**(1/2),x)
Output:
2*(Integral(-6*sqrt(3 - 4*x**2)/(3*x*sqrt(2*x + 1) - 5*sqrt(2*x + 1)), x) + Integral(-9*x*sqrt(3 - 4*x**2)/(3*x*sqrt(2*x + 1) - 5*sqrt(2*x + 1)), x) + Integral(11*x**2*sqrt(3 - 4*x**2)/(3*x*sqrt(2*x + 1) - 5*sqrt(2*x + 1)) , x) + Integral(12*x**3*sqrt(3 - 4*x**2)/(3*x*sqrt(2*x + 1) - 5*sqrt(2*x + 1)), x) + Integral(-4*x**4*sqrt(3 - 4*x**2)/(3*x*sqrt(2*x + 1) - 5*sqrt(2 *x + 1)), x))
\[ \int \frac {\left (3-4 x^2\right )^{3/2} \left (4+6 x-2 x^2\right )}{(5-3 x) \sqrt {1+2 x}} \, dx=\int { \frac {2 \, {\left (x^{2} - 3 \, x - 2\right )} {\left (-4 \, x^{2} + 3\right )}^{\frac {3}{2}}}{{\left (3 \, x - 5\right )} \sqrt {2 \, x + 1}} \,d x } \] Input:
integrate((-4*x^2+3)^(3/2)*(-2*x^2+6*x+4)/(5-3*x)/(1+2*x)^(1/2),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)
\[ \int \frac {\left (3-4 x^2\right )^{3/2} \left (4+6 x-2 x^2\right )}{(5-3 x) \sqrt {1+2 x}} \, dx=\int { \frac {2 \, {\left (x^{2} - 3 \, x - 2\right )} {\left (-4 \, x^{2} + 3\right )}^{\frac {3}{2}}}{{\left (3 \, x - 5\right )} \sqrt {2 \, x + 1}} \,d x } \] Input:
integrate((-4*x^2+3)^(3/2)*(-2*x^2+6*x+4)/(5-3*x)/(1+2*x)^(1/2),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. \[ \int \frac {\left (3-4 x^2\right )^{3/2} \left (4+6 x-2 x^2\right )}{(5-3 x) \sqrt {1+2 x}} \, dx=\int -\frac {{\left (3-4\,x^2\right )}^{3/2}\,\left (-2\,x^2+6\,x+4\right )}{\sqrt {2\,x+1}\,\left (3\,x-5\right )} \,d x \] Input:
int(-((3 - 4*x^2)^(3/2)*(6*x - 2*x^2 + 4))/((2*x + 1)^(1/2)*(3*x - 5)),x)
Output:
int(-((3 - 4*x^2)^(3/2)*(6*x - 2*x^2 + 4))/((2*x + 1)^(1/2)*(3*x - 5)), x)
\[ \int \frac {\left (3-4 x^2\right )^{3/2} \left (4+6 x-2 x^2\right )}{(5-3 x) \sqrt {1+2 x}} \, dx=-\frac {8 \sqrt {2 x +1}\, \sqrt {-4 x^{2}+3}\, x^{3}}{27}+\frac {128 \sqrt {2 x +1}\, \sqrt {-4 x^{2}+3}\, x^{2}}{189}+\frac {2206 \sqrt {2 x +1}\, \sqrt {-4 x^{2}+3}\, x}{945}-\frac {1012 \sqrt {2 x +1}\, \sqrt {-4 x^{2}+3}}{2835}+\frac {26552 \left (\int \frac {\sqrt {2 x +1}\, \sqrt {-4 x^{2}+3}\, x^{3}}{24 x^{4}-28 x^{3}-38 x^{2}+21 x +15}d x \right )}{189}-\frac {172106 \left (\int \frac {\sqrt {2 x +1}\, \sqrt {-4 x^{2}+3}\, x}{24 x^{4}-28 x^{3}-38 x^{2}+21 x +15}d x \right )}{2835}+\frac {1198 \left (\int \frac {\sqrt {2 x +1}\, \sqrt {-4 x^{2}+3}}{24 x^{4}-28 x^{3}-38 x^{2}+21 x +15}d x \right )}{189} \] Input:
int((-4*x^2+3)^(3/2)*(-2*x^2+6*x+4)/(5-3*x)/(1+2*x)^(1/2),x)
Output:
(2*( - 420*sqrt(2*x + 1)*sqrt( - 4*x**2 + 3)*x**3 + 960*sqrt(2*x + 1)*sqrt ( - 4*x**2 + 3)*x**2 + 3309*sqrt(2*x + 1)*sqrt( - 4*x**2 + 3)*x - 506*sqrt (2*x + 1)*sqrt( - 4*x**2 + 3) + 199140*int((sqrt(2*x + 1)*sqrt( - 4*x**2 + 3)*x**3)/(24*x**4 - 28*x**3 - 38*x**2 + 21*x + 15),x) - 86053*int((sqrt(2 *x + 1)*sqrt( - 4*x**2 + 3)*x)/(24*x**4 - 28*x**3 - 38*x**2 + 21*x + 15),x ) + 8985*int((sqrt(2*x + 1)*sqrt( - 4*x**2 + 3))/(24*x**4 - 28*x**3 - 38*x **2 + 21*x + 15),x)))/2835