Integrand size = 38, antiderivative size = 243 \[ \int \frac {4+6 x-2 x^2}{(5-3 x) \sqrt {1+2 x} \left (3-4 x^2\right )^{7/2}} \, dx=\frac {\sqrt {1+2 x} (87+242 x)}{4380 \left (3-4 x^2\right )^{5/2}}+\frac {\sqrt {1+2 x} (1614+33395 x)}{2877660 \left (3-4 x^2\right )^{3/2}}+\frac {\sqrt {1+2 x} (8374881+3813406 x)}{1008332064 \sqrt {3-4 x^2}}+\frac {1906703 \sqrt {1+\sqrt {3}} E\left (\arcsin \left (\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}\right )|3-\sqrt {3}\right )}{1008332064}-\frac {642599 \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}\right ),3-\sqrt {3}\right )}{63020754 \sqrt {1+\sqrt {3}}}+\frac {6156 \sqrt {2 \left (53+67 \sqrt {3}\right )} \operatorname {EllipticPi}\left (-\frac {6}{73} \left (9+10 \sqrt {3}\right ),\arcsin \left (\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}\right ),3-\sqrt {3}\right )}{28398241} \] Output:
1/4380*(1+2*x)^(1/2)*(87+242*x)/(-4*x^2+3)^(5/2)+1/2877660*(1+2*x)^(1/2)*( 1614+33395*x)/(-4*x^2+3)^(3/2)+1/1008332064*(1+2*x)^(1/2)*(8374881+3813406 *x)/(-4*x^2+3)^(1/2)+1906703/1008332064*(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))-642599/63020754*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)+6156 /28398241*(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 = 22.04 (sec) , antiderivative size = 203, normalized size of antiderivative = 0.84 \[ \int \frac {4+6 x-2 x^2}{(5-3 x) \sqrt {1+2 x} \left (3-4 x^2\right )^{7/2}} \, dx=\frac {\frac {13 \sqrt {1+2 x} \left (485495397+625684878 x-1016296632 x^2-691640880 x^3+669990480 x^4+305072480 x^5\right )}{\left (3-4 x^2\right )^{5/2}}-\frac {5 i \left (24787139 \left (1+\sqrt {3}\right ) E\left (i \text {arcsinh}\left (\frac {\sqrt {1+2 x}}{\sqrt {-1+\sqrt {3}}}\right )|-2+\sqrt {3}\right )-13 \left (-8374881+1906703 \sqrt {3}\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {1+2 x}}{\sqrt {-1+\sqrt {3}}}\right ),-2+\sqrt {3}\right )-31912704 \operatorname {EllipticPi}\left (-\frac {3}{13} \left (-1+\sqrt {3}\right ),i \text {arcsinh}\left (\frac {\sqrt {1+2 x}}{\sqrt {-1+\sqrt {3}}}\right ),-2+\sqrt {3}\right )\right )}{\sqrt {1+\sqrt {3}}}}{65541584160} \] Input:
Integrate[(4 + 6*x - 2*x^2)/((5 - 3*x)*Sqrt[1 + 2*x]*(3 - 4*x^2)^(7/2)),x]
Output:
((13*Sqrt[1 + 2*x]*(485495397 + 625684878*x - 1016296632*x^2 - 691640880*x ^3 + 669990480*x^4 + 305072480*x^5))/(3 - 4*x^2)^(5/2) - ((5*I)*(24787139* (1 + Sqrt[3])*EllipticE[I*ArcSinh[Sqrt[1 + 2*x]/Sqrt[-1 + Sqrt[3]]], -2 + Sqrt[3]] - 13*(-8374881 + 1906703*Sqrt[3])*EllipticF[I*ArcSinh[Sqrt[1 + 2* x]/Sqrt[-1 + Sqrt[3]]], -2 + Sqrt[3]] - 31912704*EllipticPi[(-3*(-1 + Sqrt [3]))/13, I*ArcSinh[Sqrt[1 + 2*x]/Sqrt[-1 + Sqrt[3]]], -2 + Sqrt[3]]))/Sqr t[1 + Sqrt[3]])/65541584160
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.
\(\displaystyle \int \frac {-2 x^2+6 x+4}{(5-3 x) \sqrt {2 x+1} \left (3-4 x^2\right )^{7/2}} \, dx\) |
\(\Big \downarrow \) 2349 |
\(\displaystyle \frac {76}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \left (3-4 x^2\right )^{7/2}}dx+\int \frac {\frac {2 x}{3}-\frac {8}{9}}{\sqrt {2 x+1} \left (3-4 x^2\right )^{7/2}}dx\) |
\(\Big \downarrow \) 686 |
\(\displaystyle \frac {76}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \left (3-4 x^2\right )^{7/2}}dx+\frac {1}{480} \int -\frac {56 (23-34 x)}{9 \sqrt {2 x+1} \left (3-4 x^2\right )^{5/2}}dx-\frac {\sqrt {2 x+1} (33-34 x)}{540 \left (3-4 x^2\right )^{5/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {76}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \left (3-4 x^2\right )^{7/2}}dx-\frac {7}{540} \int \frac {23-34 x}{\sqrt {2 x+1} \left (3-4 x^2\right )^{5/2}}dx-\frac {\sqrt {2 x+1} (33-34 x)}{540 \left (3-4 x^2\right )^{5/2}}\) |
\(\Big \downarrow \) 686 |
\(\displaystyle \frac {76}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \left (3-4 x^2\right )^{7/2}}dx-\frac {7}{540} \left (\frac {1}{288} \int \frac {32 (76-111 x)}{\sqrt {2 x+1} \left (3-4 x^2\right )^{3/2}}dx+\frac {\sqrt {2 x+1} (30-37 x)}{9 \left (3-4 x^2\right )^{3/2}}\right )-\frac {\sqrt {2 x+1} (33-34 x)}{540 \left (3-4 x^2\right )^{5/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {76}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \left (3-4 x^2\right )^{7/2}}dx-\frac {7}{540} \left (\frac {1}{9} \int \frac {76-111 x}{\sqrt {2 x+1} \left (3-4 x^2\right )^{3/2}}dx+\frac {\sqrt {2 x+1} (30-37 x)}{9 \left (3-4 x^2\right )^{3/2}}\right )-\frac {\sqrt {2 x+1} (33-34 x)}{540 \left (3-4 x^2\right )^{5/2}}\) |
\(\Big \downarrow \) 686 |
\(\displaystyle \frac {76}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \left (3-4 x^2\right )^{7/2}}dx-\frac {7}{540} \left (\frac {1}{9} \left (\frac {1}{96} \int \frac {4 (970 x+789)}{\sqrt {2 x+1} \sqrt {3-4 x^2}}dx+\frac {\sqrt {2 x+1} (789-970 x)}{24 \sqrt {3-4 x^2}}\right )+\frac {\sqrt {2 x+1} (30-37 x)}{9 \left (3-4 x^2\right )^{3/2}}\right )-\frac {\sqrt {2 x+1} (33-34 x)}{540 \left (3-4 x^2\right )^{5/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {76}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \left (3-4 x^2\right )^{7/2}}dx-\frac {7}{540} \left (\frac {1}{9} \left (\frac {1}{24} \int \frac {970 x+789}{\sqrt {2 x+1} \sqrt {3-4 x^2}}dx+\frac {\sqrt {2 x+1} (789-970 x)}{24 \sqrt {3-4 x^2}}\right )+\frac {\sqrt {2 x+1} (30-37 x)}{9 \left (3-4 x^2\right )^{3/2}}\right )-\frac {\sqrt {2 x+1} (33-34 x)}{540 \left (3-4 x^2\right )^{5/2}}\) |
\(\Big \downarrow \) 600 |
\(\displaystyle \frac {76}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \left (3-4 x^2\right )^{7/2}}dx-\frac {7}{540} \left (\frac {1}{9} \left (\frac {1}{24} \left (304 \int \frac {1}{\sqrt {2 x+1} \sqrt {3-4 x^2}}dx+485 \int \frac {\sqrt {2 x+1}}{\sqrt {3-4 x^2}}dx\right )+\frac {\sqrt {2 x+1} (789-970 x)}{24 \sqrt {3-4 x^2}}\right )+\frac {\sqrt {2 x+1} (30-37 x)}{9 \left (3-4 x^2\right )^{3/2}}\right )-\frac {\sqrt {2 x+1} (33-34 x)}{540 \left (3-4 x^2\right )^{5/2}}\) |
\(\Big \downarrow \) 508 |
\(\displaystyle \frac {76}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \left (3-4 x^2\right )^{7/2}}dx-\frac {7}{540} \left (\frac {1}{9} \left (\frac {1}{24} \left (304 \int \frac {1}{\sqrt {2 x+1} \sqrt {3-4 x^2}}dx-\frac {485 \sqrt {3+\sqrt {3}} \int \frac {\sqrt {1-\frac {3-2 \sqrt {3} x}{3+\sqrt {3}}}}{\sqrt {\frac {1}{6} \left (2 \sqrt {3} x-3\right )+1}}d\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}}{\sqrt [4]{3}}\right )+\frac {\sqrt {2 x+1} (789-970 x)}{24 \sqrt {3-4 x^2}}\right )+\frac {\sqrt {2 x+1} (30-37 x)}{9 \left (3-4 x^2\right )^{3/2}}\right )-\frac {\sqrt {2 x+1} (33-34 x)}{540 \left (3-4 x^2\right )^{5/2}}\) |
\(\Big \downarrow \) 327 |
\(\displaystyle -\frac {7}{540} \left (\frac {1}{9} \left (\frac {1}{24} \left (304 \int \frac {1}{\sqrt {2 x+1} \sqrt {3-4 x^2}}dx-\frac {485 \sqrt {3+\sqrt {3}} E\left (\arcsin \left (\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}\right )|3-\sqrt {3}\right )}{\sqrt [4]{3}}\right )+\frac {\sqrt {2 x+1} (789-970 x)}{24 \sqrt {3-4 x^2}}\right )+\frac {\sqrt {2 x+1} (30-37 x)}{9 \left (3-4 x^2\right )^{3/2}}\right )+\frac {76}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \left (3-4 x^2\right )^{7/2}}dx-\frac {\sqrt {2 x+1} (33-34 x)}{540 \left (3-4 x^2\right )^{5/2}}\) |
\(\Big \downarrow \) 511 |
\(\displaystyle -\frac {7}{540} \left (\frac {1}{9} \left (\frac {1}{24} \left (-\frac {304 \sqrt [4]{3} \int \frac {1}{\sqrt {1-\frac {3-2 \sqrt {3} x}{3+\sqrt {3}}} \sqrt {\frac {1}{6} \left (2 \sqrt {3} x-3\right )+1}}d\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}}{\sqrt {3+\sqrt {3}}}-\frac {485 \sqrt {3+\sqrt {3}} E\left (\arcsin \left (\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}\right )|3-\sqrt {3}\right )}{\sqrt [4]{3}}\right )+\frac {\sqrt {2 x+1} (789-970 x)}{24 \sqrt {3-4 x^2}}\right )+\frac {\sqrt {2 x+1} (30-37 x)}{9 \left (3-4 x^2\right )^{3/2}}\right )+\frac {76}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \left (3-4 x^2\right )^{7/2}}dx-\frac {\sqrt {2 x+1} (33-34 x)}{540 \left (3-4 x^2\right )^{5/2}}\) |
\(\Big \downarrow \) 321 |
\(\displaystyle \frac {76}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \left (3-4 x^2\right )^{7/2}}dx-\frac {7}{540} \left (\frac {1}{9} \left (\frac {1}{24} \left (-\frac {304 \sqrt [4]{3} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}\right ),3-\sqrt {3}\right )}{\sqrt {3+\sqrt {3}}}-\frac {485 \sqrt {3+\sqrt {3}} E\left (\arcsin \left (\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}\right )|3-\sqrt {3}\right )}{\sqrt [4]{3}}\right )+\frac {\sqrt {2 x+1} (789-970 x)}{24 \sqrt {3-4 x^2}}\right )+\frac {\sqrt {2 x+1} (30-37 x)}{9 \left (3-4 x^2\right )^{3/2}}\right )-\frac {\sqrt {2 x+1} (33-34 x)}{540 \left (3-4 x^2\right )^{5/2}}\) |
\(\Big \downarrow \) 744 |
\(\displaystyle \frac {76}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \left (3-4 x^2\right )^{7/2}}dx-\frac {7}{540} \left (\frac {1}{9} \left (\frac {1}{24} \left (-\frac {304 \sqrt [4]{3} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}\right ),3-\sqrt {3}\right )}{\sqrt {3+\sqrt {3}}}-\frac {485 \sqrt {3+\sqrt {3}} E\left (\arcsin \left (\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}\right )|3-\sqrt {3}\right )}{\sqrt [4]{3}}\right )+\frac {\sqrt {2 x+1} (789-970 x)}{24 \sqrt {3-4 x^2}}\right )+\frac {\sqrt {2 x+1} (30-37 x)}{9 \left (3-4 x^2\right )^{3/2}}\right )-\frac {\sqrt {2 x+1} (33-34 x)}{540 \left (3-4 x^2\right )^{5/2}}\) |
Input:
Int[(4 + 6*x - 2*x^2)/((5 - 3*x)*Sqrt[1 + 2*x]*(3 - 4*x^2)^(7/2)),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. \(475\) vs. \(2(181)=362\).
Time = 0.89 (sec) , antiderivative size = 476, normalized size of antiderivative = 1.96
method | result | size |
elliptic | \(\frac {\sqrt {-\left (4 x^{2}-3\right ) \left (1+2 x \right )}\, \left (\frac {\left (-\frac {29}{93440}-\frac {121 x}{140160}\right ) \sqrt {-8 x^{3}-4 x^{2}+6 x +3}}{\left (x^{2}-\frac {3}{4}\right )^{3}}+\frac {\left (\frac {269}{7673760}+\frac {6679 x}{9208512}\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 {2791627}{2688885504}+\frac {1906703 x}{4033328256}\right )}{\sqrt {\left (x^{2}-\frac {3}{4}\right ) \left (-4-8 x \right )}}+\frac {2791627 \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 )}{504166032 \sqrt {-8 x^{3}-4 x^{2}+6 x +3}}-\frac {1906703 \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 )}{756249048 \sqrt {-8 x^{3}-4 x^{2}+6 x +3}}+\frac {1368 \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 )}{389017 \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}}\) | \(476\) |
default | \(\text {Expression too large to display}\) | \(1505\) |
Input:
int((-2*x^2+6*x+4)/(5-3*x)/(1+2*x)^(1/2)/(-4*x^2+3)^(7/2),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)*((-29/93440-121/ 140160*x)*(-8*x^3-4*x^2+6*x+3)^(1/2)/(x^2-3/4)^3+(269/7673760+6679/9208512 *x)*(-8*x^3-4*x^2+6*x+3)^(1/2)/(x^2-3/4)^2-2*(-4-8*x)*(2791627/2688885504+ 1906703/4033328256*x)/((x^2-3/4)*(-4-8*x))^(1/2)+2791627/504166032*((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))-1906703/756 249048*((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)))+1368/389017*((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)))
\[ \int \frac {4+6 x-2 x^2}{(5-3 x) \sqrt {1+2 x} \left (3-4 x^2\right )^{7/2}} \, dx=\int { \frac {2 \, {\left (x^{2} - 3 \, x - 2\right )}}{{\left (-4 \, x^{2} + 3\right )}^{\frac {7}{2}} {\left (3 \, x - 5\right )} \sqrt {2 \, x + 1}} \,d x } \] Input:
integrate((-2*x^2+6*x+4)/(5-3*x)/(1+2*x)^(1/2)/(-4*x^2+3)^(7/2),x, algorit hm="fricas")
Output:
integral(2*(x^2 - 3*x - 2)*sqrt(-4*x^2 + 3)*sqrt(2*x + 1)/(1536*x^10 - 179 2*x^9 - 5888*x^8 + 5376*x^7 + 9024*x^6 - 6048*x^5 - 6912*x^4 + 3024*x^3 + 2646*x^2 - 567*x - 405), x)
Timed out. \[ \int \frac {4+6 x-2 x^2}{(5-3 x) \sqrt {1+2 x} \left (3-4 x^2\right )^{7/2}} \, dx=\text {Timed out} \] Input:
integrate((-2*x**2+6*x+4)/(5-3*x)/(1+2*x)**(1/2)/(-4*x**2+3)**(7/2),x)
Output:
Timed out
\[ \int \frac {4+6 x-2 x^2}{(5-3 x) \sqrt {1+2 x} \left (3-4 x^2\right )^{7/2}} \, dx=\int { \frac {2 \, {\left (x^{2} - 3 \, x - 2\right )}}{{\left (-4 \, x^{2} + 3\right )}^{\frac {7}{2}} {\left (3 \, x - 5\right )} \sqrt {2 \, x + 1}} \,d x } \] Input:
integrate((-2*x^2+6*x+4)/(5-3*x)/(1+2*x)^(1/2)/(-4*x^2+3)^(7/2),x, algorit hm="maxima")
Output:
2*integrate((x^2 - 3*x - 2)/((-4*x^2 + 3)^(7/2)*(3*x - 5)*sqrt(2*x + 1)), x)
\[ \int \frac {4+6 x-2 x^2}{(5-3 x) \sqrt {1+2 x} \left (3-4 x^2\right )^{7/2}} \, dx=\int { \frac {2 \, {\left (x^{2} - 3 \, x - 2\right )}}{{\left (-4 \, x^{2} + 3\right )}^{\frac {7}{2}} {\left (3 \, x - 5\right )} \sqrt {2 \, x + 1}} \,d x } \] Input:
integrate((-2*x^2+6*x+4)/(5-3*x)/(1+2*x)^(1/2)/(-4*x^2+3)^(7/2),x, algorit hm="giac")
Output:
integrate(2*(x^2 - 3*x - 2)/((-4*x^2 + 3)^(7/2)*(3*x - 5)*sqrt(2*x + 1)), x)
Timed out. \[ \int \frac {4+6 x-2 x^2}{(5-3 x) \sqrt {1+2 x} \left (3-4 x^2\right )^{7/2}} \, dx=\int -\frac {-2\,x^2+6\,x+4}{\sqrt {2\,x+1}\,\left (3\,x-5\right )\,{\left (3-4\,x^2\right )}^{7/2}} \,d x \] Input:
int(-(6*x - 2*x^2 + 4)/((2*x + 1)^(1/2)*(3*x - 5)*(3 - 4*x^2)^(7/2)),x)
Output:
int(-(6*x - 2*x^2 + 4)/((2*x + 1)^(1/2)*(3*x - 5)*(3 - 4*x^2)^(7/2)), x)
\[ \int \frac {4+6 x-2 x^2}{(5-3 x) \sqrt {1+2 x} \left (3-4 x^2\right )^{7/2}} \, dx=-12 \left (\int \frac {\sqrt {2 x +1}\, x^{4}}{2304 \sqrt {-4 x^{2}+3}\, x^{10}-12160 \sqrt {-4 x^{2}+3}\, x^{8}+21184 \sqrt {-4 x^{2}+3}\, x^{6}-16344 \sqrt {-4 x^{2}+3}\, x^{4}+5643 \sqrt {-4 x^{2}+3}\, x^{2}-675 \sqrt {-4 x^{2}+3}}d x \right )-14 \left (\int \frac {\sqrt {2 x +1}\, x^{3}}{2304 \sqrt {-4 x^{2}+3}\, x^{10}-12160 \sqrt {-4 x^{2}+3}\, x^{8}+21184 \sqrt {-4 x^{2}+3}\, x^{6}-16344 \sqrt {-4 x^{2}+3}\, x^{4}+5643 \sqrt {-4 x^{2}+3}\, x^{2}-675 \sqrt {-4 x^{2}+3}}d x \right )+10 \left (\int \frac {\sqrt {2 x +1}\, x^{2}}{2304 \sqrt {-4 x^{2}+3}\, x^{10}-12160 \sqrt {-4 x^{2}+3}\, x^{8}+21184 \sqrt {-4 x^{2}+3}\, x^{6}-16344 \sqrt {-4 x^{2}+3}\, x^{4}+5643 \sqrt {-4 x^{2}+3}\, x^{2}-675 \sqrt {-4 x^{2}+3}}d x \right )-6 \left (\int \frac {\sqrt {2 x +1}\, \sqrt {-4 x^{2}+3}\, x}{1536 x^{10}-1792 x^{9}-5888 x^{8}+5376 x^{7}+9024 x^{6}-6048 x^{5}-6912 x^{4}+3024 x^{3}+2646 x^{2}-567 x -405}d x \right )-4 \left (\int \frac {\sqrt {2 x +1}\, \sqrt {-4 x^{2}+3}}{1536 x^{10}-1792 x^{9}-5888 x^{8}+5376 x^{7}+9024 x^{6}-6048 x^{5}-6912 x^{4}+3024 x^{3}+2646 x^{2}-567 x -405}d x \right ) \] Input:
int((-2*x^2+6*x+4)/(5-3*x)/(1+2*x)^(1/2)/(-4*x^2+3)^(7/2),x)
Output:
2*( - 6*int((sqrt(2*x + 1)*x**4)/(2304*sqrt( - 4*x**2 + 3)*x**10 - 12160*s qrt( - 4*x**2 + 3)*x**8 + 21184*sqrt( - 4*x**2 + 3)*x**6 - 16344*sqrt( - 4 *x**2 + 3)*x**4 + 5643*sqrt( - 4*x**2 + 3)*x**2 - 675*sqrt( - 4*x**2 + 3)) ,x) - 7*int((sqrt(2*x + 1)*x**3)/(2304*sqrt( - 4*x**2 + 3)*x**10 - 12160*s qrt( - 4*x**2 + 3)*x**8 + 21184*sqrt( - 4*x**2 + 3)*x**6 - 16344*sqrt( - 4 *x**2 + 3)*x**4 + 5643*sqrt( - 4*x**2 + 3)*x**2 - 675*sqrt( - 4*x**2 + 3)) ,x) + 5*int((sqrt(2*x + 1)*x**2)/(2304*sqrt( - 4*x**2 + 3)*x**10 - 12160*s qrt( - 4*x**2 + 3)*x**8 + 21184*sqrt( - 4*x**2 + 3)*x**6 - 16344*sqrt( - 4 *x**2 + 3)*x**4 + 5643*sqrt( - 4*x**2 + 3)*x**2 - 675*sqrt( - 4*x**2 + 3)) ,x) - 3*int((sqrt(2*x + 1)*sqrt( - 4*x**2 + 3)*x)/(1536*x**10 - 1792*x**9 - 5888*x**8 + 5376*x**7 + 9024*x**6 - 6048*x**5 - 6912*x**4 + 3024*x**3 + 2646*x**2 - 567*x - 405),x) - 2*int((sqrt(2*x + 1)*sqrt( - 4*x**2 + 3))/(1 536*x**10 - 1792*x**9 - 5888*x**8 + 5376*x**7 + 9024*x**6 - 6048*x**5 - 69 12*x**4 + 3024*x**3 + 2646*x**2 - 567*x - 405),x))