Integrand size = 38, antiderivative size = 214 \[ \int \frac {4+6 x-2 x^2}{(5-3 x) \sqrt {1+2 x} \left (3-4 x^2\right )^{5/2}} \, dx=\frac {\sqrt {1+2 x} (87+242 x)}{2628 \left (3-4 x^2\right )^{3/2}}+\frac {\sqrt {1+2 x} (1713+8941 x)}{575532 \sqrt {3-4 x^2}}+\frac {8941 \sqrt {1+\sqrt {3}} E\left (\arcsin \left (\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}\right )|3-\sqrt {3}\right )}{1151064}-\frac {12367 \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}\right ),3-\sqrt {3}\right )}{1151064 \sqrt {1+\sqrt {3}}}-\frac {684 \sqrt {2 \left (53+67 \sqrt {3}\right )} \operatorname {EllipticPi}\left (-\frac {6}{73} \left (9+10 \sqrt {3}\right ),\arcsin \left (\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}\right ),3-\sqrt {3}\right )}{389017} \] Output:
1/2628*(1+2*x)^(1/2)*(87+242*x)/(-4*x^2+3)^(3/2)+1/575532*(1+2*x)^(1/2)*(1 713+8941*x)/(-4*x^2+3)^(1/2)+8941/1151064*(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))-12367/1151064*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)-684/ 389017*(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 = 22.06 (sec) , antiderivative size = 193, normalized size of antiderivative = 0.90 \[ \int \frac {4+6 x-2 x^2}{(5-3 x) \sqrt {1+2 x} \left (3-4 x^2\right )^{5/2}} \, dx=\frac {\frac {26 \sqrt {1+2 x} \left (24192+79821 x-6852 x^2-35764 x^3\right )}{\left (3-4 x^2\right )^{3/2}}-\frac {i \left (116233 \left (1+\sqrt {3}\right ) E\left (i \text {arcsinh}\left (\frac {\sqrt {1+2 x}}{\sqrt {-1+\sqrt {3}}}\right )|-2+\sqrt {3}\right )-13 \left (-3426+8941 \sqrt {3}\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {1+2 x}}{\sqrt {-1+\sqrt {3}}}\right ),-2+\sqrt {3}\right )+295488 \operatorname {EllipticPi}\left (-\frac {3}{13} \left (-1+\sqrt {3}\right ),i \text {arcsinh}\left (\frac {\sqrt {1+2 x}}{\sqrt {-1+\sqrt {3}}}\right ),-2+\sqrt {3}\right )\right )}{\sqrt {1+\sqrt {3}}}}{14963832} \] Input:
Integrate[(4 + 6*x - 2*x^2)/((5 - 3*x)*Sqrt[1 + 2*x]*(3 - 4*x^2)^(5/2)),x]
Output:
((26*Sqrt[1 + 2*x]*(24192 + 79821*x - 6852*x^2 - 35764*x^3))/(3 - 4*x^2)^( 3/2) - (I*(116233*(1 + Sqrt[3])*EllipticE[I*ArcSinh[Sqrt[1 + 2*x]/Sqrt[-1 + Sqrt[3]]], -2 + Sqrt[3]] - 13*(-3426 + 8941*Sqrt[3])*EllipticF[I*ArcSinh [Sqrt[1 + 2*x]/Sqrt[-1 + Sqrt[3]]], -2 + Sqrt[3]] + 295488*EllipticPi[(-3* (-1 + Sqrt[3]))/13, I*ArcSinh[Sqrt[1 + 2*x]/Sqrt[-1 + Sqrt[3]]], -2 + Sqrt [3]]))/Sqrt[1 + Sqrt[3]])/14963832
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.
| \(\displaystyle \int \frac {-2 x^2+6 x+4}{(5-3 x) \sqrt {2 x+1} \left (3-4 x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 2349 |
| \(\displaystyle \frac {76}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \left (3-4 x^2\right )^{5/2}}dx+\int \frac {\frac {2 x}{3}-\frac {8}{9}}{\sqrt {2 x+1} \left (3-4 x^2\right )^{5/2}}dx\) |
| \(\Big \downarrow \) 686 |
| \(\displaystyle \frac {76}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \left (3-4 x^2\right )^{5/2}}dx+\frac {1}{288} \int -\frac {8 (97-102 x)}{9 \sqrt {2 x+1} \left (3-4 x^2\right )^{3/2}}dx-\frac {\sqrt {2 x+1} (33-34 x)}{324 \left (3-4 x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {76}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \left (3-4 x^2\right )^{5/2}}dx-\frac {1}{324} \int \frac {97-102 x}{\sqrt {2 x+1} \left (3-4 x^2\right )^{3/2}}dx-\frac {\sqrt {2 x+1} (33-34 x)}{324 \left (3-4 x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 686 |
| \(\displaystyle \frac {76}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \left (3-4 x^2\right )^{5/2}}dx+\frac {1}{324} \left (-\frac {1}{96} \int \frac {32 (125 x+111)}{\sqrt {2 x+1} \sqrt {3-4 x^2}}dx-\frac {\sqrt {2 x+1} (111-125 x)}{3 \sqrt {3-4 x^2}}\right )-\frac {\sqrt {2 x+1} (33-34 x)}{324 \left (3-4 x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {76}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \left (3-4 x^2\right )^{5/2}}dx+\frac {1}{324} \left (-\frac {1}{3} \int \frac {125 x+111}{\sqrt {2 x+1} \sqrt {3-4 x^2}}dx-\frac {\sqrt {2 x+1} (111-125 x)}{3 \sqrt {3-4 x^2}}\right )-\frac {\sqrt {2 x+1} (33-34 x)}{324 \left (3-4 x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 600 |
| \(\displaystyle \frac {76}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \left (3-4 x^2\right )^{5/2}}dx+\frac {1}{324} \left (\frac {1}{3} \left (-\frac {97}{2} \int \frac {1}{\sqrt {2 x+1} \sqrt {3-4 x^2}}dx-\frac {125}{2} \int \frac {\sqrt {2 x+1}}{\sqrt {3-4 x^2}}dx\right )-\frac {(111-125 x) \sqrt {2 x+1}}{3 \sqrt {3-4 x^2}}\right )-\frac {\sqrt {2 x+1} (33-34 x)}{324 \left (3-4 x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 508 |
| \(\displaystyle \frac {76}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \left (3-4 x^2\right )^{5/2}}dx+\frac {1}{324} \left (\frac {1}{3} \left (\frac {125 \sqrt {3+\sqrt {3}} \int \frac {\sqrt {1-\frac {3-2 \sqrt {3} x}{3+\sqrt {3}}}}{\sqrt {\frac {1}{6} \left (2 \sqrt {3} x-3\right )+1}}d\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}}{2 \sqrt [4]{3}}-\frac {97}{2} \int \frac {1}{\sqrt {2 x+1} \sqrt {3-4 x^2}}dx\right )-\frac {(111-125 x) \sqrt {2 x+1}}{3 \sqrt {3-4 x^2}}\right )-\frac {\sqrt {2 x+1} (33-34 x)}{324 \left (3-4 x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {1}{324} \left (\frac {1}{3} \left (\frac {125 \sqrt {3+\sqrt {3}} E\left (\arcsin \left (\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}\right )|3-\sqrt {3}\right )}{2 \sqrt [4]{3}}-\frac {97}{2} \int \frac {1}{\sqrt {2 x+1} \sqrt {3-4 x^2}}dx\right )-\frac {(111-125 x) \sqrt {2 x+1}}{3 \sqrt {3-4 x^2}}\right )+\frac {76}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \left (3-4 x^2\right )^{5/2}}dx-\frac {\sqrt {2 x+1} (33-34 x)}{324 \left (3-4 x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 511 |
| \(\displaystyle \frac {1}{324} \left (\frac {1}{3} \left (\frac {97 \sqrt [4]{3} \int \frac {1}{\sqrt {1-\frac {3-2 \sqrt {3} x}{3+\sqrt {3}}} \sqrt {\frac {1}{6} \left (2 \sqrt {3} x-3\right )+1}}d\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}}{2 \sqrt {3+\sqrt {3}}}+\frac {125 \sqrt {3+\sqrt {3}} E\left (\arcsin \left (\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}\right )|3-\sqrt {3}\right )}{2 \sqrt [4]{3}}\right )-\frac {(111-125 x) \sqrt {2 x+1}}{3 \sqrt {3-4 x^2}}\right )+\frac {76}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \left (3-4 x^2\right )^{5/2}}dx-\frac {\sqrt {2 x+1} (33-34 x)}{324 \left (3-4 x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {76}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \left (3-4 x^2\right )^{5/2}}dx+\frac {1}{324} \left (\frac {1}{3} \left (\frac {97 \sqrt [4]{3} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}\right ),3-\sqrt {3}\right )}{2 \sqrt {3+\sqrt {3}}}+\frac {125 \sqrt {3+\sqrt {3}} E\left (\arcsin \left (\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}\right )|3-\sqrt {3}\right )}{2 \sqrt [4]{3}}\right )-\frac {(111-125 x) \sqrt {2 x+1}}{3 \sqrt {3-4 x^2}}\right )-\frac {\sqrt {2 x+1} (33-34 x)}{324 \left (3-4 x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 744 |
| \(\displaystyle \frac {76}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \left (3-4 x^2\right )^{5/2}}dx+\frac {1}{324} \left (\frac {1}{3} \left (\frac {97 \sqrt [4]{3} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}\right ),3-\sqrt {3}\right )}{2 \sqrt {3+\sqrt {3}}}+\frac {125 \sqrt {3+\sqrt {3}} E\left (\arcsin \left (\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}\right )|3-\sqrt {3}\right )}{2 \sqrt [4]{3}}\right )-\frac {(111-125 x) \sqrt {2 x+1}}{3 \sqrt {3-4 x^2}}\right )-\frac {\sqrt {2 x+1} (33-34 x)}{324 \left (3-4 x^2\right )^{3/2}}\) |
Input:
Int[(4 + 6*x - 2*x^2)/((5 - 3*x)*Sqrt[1 + 2*x]*(3 - 4*x^2)^(5/2)),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. \(445\) vs. \(2(158)=316\).
Time = 0.73 (sec) , antiderivative size = 446, normalized size of antiderivative = 2.08
| method | result | size |
| elliptic | \(\frac {\sqrt {-\left (4 x^{2}-3\right ) \left (1+2 x \right )}\, \left (\frac {\left (\frac {29}{14016}+\frac {121 x}{21024}\right ) \sqrt {-8 x^{3}-4 x^{2}+6 x +3}}{\left (x^{2}-\frac {3}{4}\right )^{2}}-\frac {2 \left (-4-8 x \right ) \left (\frac {571}{1534752}+\frac {8941 x}{4604256}\right )}{\sqrt {\left (x^{2}-\frac {3}{4}\right ) \left (-4-8 x \right )}}+\frac {571 \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 )}{287766 \sqrt {-8 x^{3}-4 x^{2}+6 x +3}}-\frac {8941 \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 )}{863298 \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 )}{5329 \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}}\) | \(446\) |
| default | \(\text {Expression too large to display}\) | \(1011\) |
Input:
int((-2*x^2+6*x+4)/(5-3*x)/(1+2*x)^(1/2)/(-4*x^2+3)^(5/2),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)*((29/14016+121/2 1024*x)*(-8*x^3-4*x^2+6*x+3)^(1/2)/(x^2-3/4)^2-2*(-4-8*x)*(571/1534752+894 1/4604256*x)/((x^2-3/4)*(-4-8*x))^(1/2)+571/287766*((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))-8941/863298*((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)))-152/5329*((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)))
\[ \int \frac {4+6 x-2 x^2}{(5-3 x) \sqrt {1+2 x} \left (3-4 x^2\right )^{5/2}} \, dx=\int { \frac {2 \, {\left (x^{2} - 3 \, x - 2\right )}}{{\left (-4 \, x^{2} + 3\right )}^{\frac {5}{2}} {\left (3 \, x - 5\right )} \sqrt {2 \, x + 1}} \,d x } \] Input:
integrate((-2*x^2+6*x+4)/(5-3*x)/(1+2*x)^(1/2)/(-4*x^2+3)^(5/2),x, algorit hm="fricas")
Output:
integral(-2*(x^2 - 3*x - 2)*sqrt(-4*x^2 + 3)*sqrt(2*x + 1)/(384*x^8 - 448* x^7 - 1184*x^6 + 1008*x^5 + 1368*x^4 - 756*x^3 - 702*x^2 + 189*x + 135), x )
Timed out. \[ \int \frac {4+6 x-2 x^2}{(5-3 x) \sqrt {1+2 x} \left (3-4 x^2\right )^{5/2}} \, dx=\text {Timed out} \] Input:
integrate((-2*x**2+6*x+4)/(5-3*x)/(1+2*x)**(1/2)/(-4*x**2+3)**(5/2),x)
Output:
Timed out
\[ \int \frac {4+6 x-2 x^2}{(5-3 x) \sqrt {1+2 x} \left (3-4 x^2\right )^{5/2}} \, dx=\int { \frac {2 \, {\left (x^{2} - 3 \, x - 2\right )}}{{\left (-4 \, x^{2} + 3\right )}^{\frac {5}{2}} {\left (3 \, x - 5\right )} \sqrt {2 \, x + 1}} \,d x } \] Input:
integrate((-2*x^2+6*x+4)/(5-3*x)/(1+2*x)^(1/2)/(-4*x^2+3)^(5/2),x, algorit hm="maxima")
Output:
2*integrate((x^2 - 3*x - 2)/((-4*x^2 + 3)^(5/2)*(3*x - 5)*sqrt(2*x + 1)), x)
\[ \int \frac {4+6 x-2 x^2}{(5-3 x) \sqrt {1+2 x} \left (3-4 x^2\right )^{5/2}} \, dx=\int { \frac {2 \, {\left (x^{2} - 3 \, x - 2\right )}}{{\left (-4 \, x^{2} + 3\right )}^{\frac {5}{2}} {\left (3 \, x - 5\right )} \sqrt {2 \, x + 1}} \,d x } \] Input:
integrate((-2*x^2+6*x+4)/(5-3*x)/(1+2*x)^(1/2)/(-4*x^2+3)^(5/2),x, algorit hm="giac")
Output:
integrate(2*(x^2 - 3*x - 2)/((-4*x^2 + 3)^(5/2)*(3*x - 5)*sqrt(2*x + 1)), x)
Timed out. \[ \int \frac {4+6 x-2 x^2}{(5-3 x) \sqrt {1+2 x} \left (3-4 x^2\right )^{5/2}} \, dx=\int -\frac {-2\,x^2+6\,x+4}{\sqrt {2\,x+1}\,\left (3\,x-5\right )\,{\left (3-4\,x^2\right )}^{5/2}} \,d x \] Input:
int(-(6*x - 2*x^2 + 4)/((2*x + 1)^(1/2)*(3*x - 5)*(3 - 4*x^2)^(5/2)),x)
Output:
int(-(6*x - 2*x^2 + 4)/((2*x + 1)^(1/2)*(3*x - 5)*(3 - 4*x^2)^(5/2)), x)
\[ \int \frac {4+6 x-2 x^2}{(5-3 x) \sqrt {1+2 x} \left (3-4 x^2\right )^{5/2}} \, dx=\int \frac {-2 x^{2}+6 x +4}{\left (5-3 x \right ) \sqrt {2 x +1}\, \left (-4 x^{2}+3\right )^{\frac {5}{2}}}d x \] Input:
int((-2*x^2+6*x+4)/(5-3*x)/(1+2*x)^(1/2)/(-4*x^2+3)^(5/2),x)
Output:
int((-2*x^2+6*x+4)/(5-3*x)/(1+2*x)^(1/2)/(-4*x^2+3)^(5/2),x)