\(\int \frac {4+6 x-2 x^2}{(5-3 x) \sqrt {1+2 x} \sqrt {2+7 x+4 x^2}} \, dx\) [53]

Optimal result

Integrand size = 41, antiderivative size = 322 \[ \int \frac {4+6 x-2 x^2}{(5-3 x) \sqrt {1+2 x} \sqrt {2+7 x+4 x^2}} \, dx=\frac {\sqrt {3+\sqrt {17}} \sqrt {1+2 x} \sqrt {-2-7 x-4 x^2} E\left (\arcsin \left (\frac {\sqrt {17+\sqrt {17} (7+8 x)}}{\sqrt {34}}\right )|\frac {1}{4} \left (17-3 \sqrt {17}\right )\right )}{6 \sqrt {-1-2 x} \sqrt {2+7 x+4 x^2}}-\frac {22 \sqrt {-1-2 x} \sqrt {-2-7 x-4 x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {17+\sqrt {17} (7+8 x)}}{\sqrt {34}}\right ),\frac {1}{4} \left (17-3 \sqrt {17}\right )\right )}{9 \sqrt {3+\sqrt {17}} \sqrt {1+2 x} \sqrt {2+7 x+4 x^2}}+\frac {1216 \sqrt {-1-2 x} \sqrt {-2-7 x-4 x^2} \operatorname {EllipticPi}\left (-\frac {3 \left (51-61 \sqrt {17}\right )}{1784},\arcsin \left (\frac {\sqrt {17+\sqrt {17} (7+8 x)}}{\sqrt {34}}\right ),\frac {1}{4} \left (17-3 \sqrt {17}\right )\right )}{9 \sqrt {3+\sqrt {17}} \left (61+3 \sqrt {17}\right ) \sqrt {1+2 x} \sqrt {2+7 x+4 x^2}} \] Output:

1/6*(3+17^(1/2))^(1/2)*(1+2*x)^(1/2)*(-4*x^2-7*x-2)^(1/2)*EllipticE(1/34*( 
17+17^(1/2)*(7+8*x))^(1/2)*34^(1/2),1/2*(17-3*17^(1/2))^(1/2))/(-1-2*x)^(1 
/2)/(4*x^2+7*x+2)^(1/2)-22/9*(-1-2*x)^(1/2)*(-4*x^2-7*x-2)^(1/2)*EllipticF 
(1/34*(17+17^(1/2)*(7+8*x))^(1/2)*34^(1/2),1/2*(17-3*17^(1/2))^(1/2))/(3+1 
7^(1/2))^(1/2)/(1+2*x)^(1/2)/(4*x^2+7*x+2)^(1/2)+1216/9*(-1-2*x)^(1/2)*(-4 
*x^2-7*x-2)^(1/2)*EllipticPi(1/34*(17+17^(1/2)*(7+8*x))^(1/2)*34^(1/2),-15 
3/1784+183/1784*17^(1/2),1/2*(17-3*17^(1/2))^(1/2))/(3+17^(1/2))^(1/2)/(61 
+3*17^(1/2))/(1+2*x)^(1/2)/(4*x^2+7*x+2)^(1/2)
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 24.16 (sec) , antiderivative size = 342, normalized size of antiderivative = 1.06 \[ \int \frac {4+6 x-2 x^2}{(5-3 x) \sqrt {1+2 x} \sqrt {2+7 x+4 x^2}} \, dx=\frac {(3+6 x)^{3/2} \left (78-\frac {39}{(1+2 x)^2}+\frac {117}{1+2 x}+\frac {39 i \sqrt {3+\sqrt {17}} \sqrt {\frac {2+7 x+4 x^2}{(1+2 x)^2}} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {2}{-3+\sqrt {17}}}}{\sqrt {1+2 x}}\right )|-\frac {13}{4}+\frac {3 \sqrt {17}}{4}\right )}{\sqrt {1+2 x}}-\frac {3 i \left (27+13 \sqrt {17}\right ) \sqrt {\frac {2+7 x+4 x^2}{(1+2 x)^2}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {2}{-3+\sqrt {17}}}}{\sqrt {1+2 x}}\right ),-\frac {13}{4}+\frac {3 \sqrt {17}}{4}\right )}{\sqrt {3+\sqrt {17}} \sqrt {1+2 x}}-\frac {608 i \sqrt {\frac {2+7 x+4 x^2}{(1+2 x)^2}} \operatorname {EllipticPi}\left (-\frac {13}{6} \left (-3+\sqrt {17}\right ),i \text {arcsinh}\left (\frac {\sqrt {\frac {2}{-3+\sqrt {17}}}}{\sqrt {1+2 x}}\right ),\frac {1}{4} \left (-13+3 \sqrt {17}\right )\right )}{\sqrt {3+\sqrt {17}} \sqrt {1+2 x}}\right )}{702 \sqrt {3} \sqrt {2+7 x+4 x^2}} \] Input:

Integrate[(4 + 6*x - 2*x^2)/((5 - 3*x)*Sqrt[1 + 2*x]*Sqrt[2 + 7*x + 4*x^2] 
),x]
 

Output:

((3 + 6*x)^(3/2)*(78 - 39/(1 + 2*x)^2 + 117/(1 + 2*x) + ((39*I)*Sqrt[3 + S 
qrt[17]]*Sqrt[(2 + 7*x + 4*x^2)/(1 + 2*x)^2]*EllipticE[I*ArcSinh[Sqrt[2/(- 
3 + Sqrt[17])]/Sqrt[1 + 2*x]], -13/4 + (3*Sqrt[17])/4])/Sqrt[1 + 2*x] - (( 
3*I)*(27 + 13*Sqrt[17])*Sqrt[(2 + 7*x + 4*x^2)/(1 + 2*x)^2]*EllipticF[I*Ar 
cSinh[Sqrt[2/(-3 + Sqrt[17])]/Sqrt[1 + 2*x]], -13/4 + (3*Sqrt[17])/4])/(Sq 
rt[3 + Sqrt[17]]*Sqrt[1 + 2*x]) - ((608*I)*Sqrt[(2 + 7*x + 4*x^2)/(1 + 2*x 
)^2]*EllipticPi[(-13*(-3 + Sqrt[17]))/6, I*ArcSinh[Sqrt[2/(-3 + Sqrt[17])] 
/Sqrt[1 + 2*x]], (-13 + 3*Sqrt[17])/4])/(Sqrt[3 + Sqrt[17]]*Sqrt[1 + 2*x]) 
))/(702*Sqrt[3]*Sqrt[2 + 7*x + 4*x^2])
 

Rubi [A] (warning: unable to verify)

Time = 1.24 (sec) , antiderivative size = 358, normalized size of antiderivative = 1.11, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.244, Rules used = {2154, 1269, 1172, 321, 327, 1279, 186, 25, 413, 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.

\(\displaystyle \int \frac {-2 x^2+6 x+4}{(5-3 x) \sqrt {2 x+1} \sqrt {4 x^2+7 x+2}} \, dx\)

\(\Big \downarrow \) 2154

\(\displaystyle \frac {76}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \sqrt {4 x^2+7 x+2}}dx+\int \frac {\frac {2 x}{3}-\frac {8}{9}}{\sqrt {2 x+1} \sqrt {4 x^2+7 x+2}}dx\)

\(\Big \downarrow \) 1269

\(\displaystyle -\frac {11}{9} \int \frac {1}{\sqrt {2 x+1} \sqrt {4 x^2+7 x+2}}dx+\frac {76}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \sqrt {4 x^2+7 x+2}}dx+\frac {1}{3} \int \frac {\sqrt {2 x+1}}{\sqrt {4 x^2+7 x+2}}dx\)

\(\Big \downarrow \) 1172

\(\displaystyle \frac {76}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \sqrt {4 x^2+7 x+2}}dx-\frac {22 \sqrt {-2 x-1} \sqrt {-4 x^2-7 x-2} \int \frac {1}{\sqrt {1-\frac {8 x+\sqrt {17}+7}{2 \sqrt {17}}} \sqrt {1-\frac {8 x+\sqrt {17}+7}{3+\sqrt {17}}}}d\frac {\sqrt {8 x+\sqrt {17}+7}}{\sqrt {2} \sqrt [4]{17}}}{9 \sqrt {3+\sqrt {17}} \sqrt {2 x+1} \sqrt {4 x^2+7 x+2}}+\frac {\sqrt {3+\sqrt {17}} \sqrt {2 x+1} \sqrt {-4 x^2-7 x-2} \int \frac {\sqrt {1-\frac {8 x+\sqrt {17}+7}{3+\sqrt {17}}}}{\sqrt {1-\frac {8 x+\sqrt {17}+7}{2 \sqrt {17}}}}d\frac {\sqrt {8 x+\sqrt {17}+7}}{\sqrt {2} \sqrt [4]{17}}}{6 \sqrt {-2 x-1} \sqrt {4 x^2+7 x+2}}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {76}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \sqrt {4 x^2+7 x+2}}dx+\frac {\sqrt {3+\sqrt {17}} \sqrt {2 x+1} \sqrt {-4 x^2-7 x-2} \int \frac {\sqrt {1-\frac {8 x+\sqrt {17}+7}{3+\sqrt {17}}}}{\sqrt {1-\frac {8 x+\sqrt {17}+7}{2 \sqrt {17}}}}d\frac {\sqrt {8 x+\sqrt {17}+7}}{\sqrt {2} \sqrt [4]{17}}}{6 \sqrt {-2 x-1} \sqrt {4 x^2+7 x+2}}-\frac {22 \sqrt {-2 x-1} \sqrt {-4 x^2-7 x-2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {8 x+\sqrt {17}+7}}{\sqrt {2} \sqrt [4]{17}}\right ),\frac {1}{4} \left (17-3 \sqrt {17}\right )\right )}{9 \sqrt {3+\sqrt {17}} \sqrt {2 x+1} \sqrt {4 x^2+7 x+2}}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {76}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \sqrt {4 x^2+7 x+2}}dx-\frac {22 \sqrt {-2 x-1} \sqrt {-4 x^2-7 x-2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {8 x+\sqrt {17}+7}}{\sqrt {2} \sqrt [4]{17}}\right ),\frac {1}{4} \left (17-3 \sqrt {17}\right )\right )}{9 \sqrt {3+\sqrt {17}} \sqrt {2 x+1} \sqrt {4 x^2+7 x+2}}+\frac {\sqrt {3+\sqrt {17}} \sqrt {2 x+1} \sqrt {-4 x^2-7 x-2} E\left (\arcsin \left (\frac {\sqrt {8 x+\sqrt {17}+7}}{\sqrt {2} \sqrt [4]{17}}\right )|\frac {1}{4} \left (17-3 \sqrt {17}\right )\right )}{6 \sqrt {-2 x-1} \sqrt {4 x^2+7 x+2}}\)

\(\Big \downarrow \) 1279

\(\displaystyle \frac {76 \sqrt {8 x-\sqrt {17}+7} \sqrt {8 x+\sqrt {17}+7} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \sqrt {8 x-\sqrt {17}+7} \sqrt {8 x+\sqrt {17}+7}}dx}{9 \sqrt {4 x^2+7 x+2}}-\frac {22 \sqrt {-2 x-1} \sqrt {-4 x^2-7 x-2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {8 x+\sqrt {17}+7}}{\sqrt {2} \sqrt [4]{17}}\right ),\frac {1}{4} \left (17-3 \sqrt {17}\right )\right )}{9 \sqrt {3+\sqrt {17}} \sqrt {2 x+1} \sqrt {4 x^2+7 x+2}}+\frac {\sqrt {3+\sqrt {17}} \sqrt {2 x+1} \sqrt {-4 x^2-7 x-2} E\left (\arcsin \left (\frac {\sqrt {8 x+\sqrt {17}+7}}{\sqrt {2} \sqrt [4]{17}}\right )|\frac {1}{4} \left (17-3 \sqrt {17}\right )\right )}{6 \sqrt {-2 x-1} \sqrt {4 x^2+7 x+2}}\)

\(\Big \downarrow \) 186

\(\displaystyle -\frac {152 \sqrt {8 x-\sqrt {17}+7} \sqrt {8 x+\sqrt {17}+7} \int -\frac {1}{(13-3 (2 x+1)) \sqrt {4 (2 x+1)-\sqrt {17}+3} \sqrt {4 (2 x+1)+\sqrt {17}+3}}d\sqrt {2 x+1}}{9 \sqrt {4 x^2+7 x+2}}-\frac {22 \sqrt {-2 x-1} \sqrt {-4 x^2-7 x-2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {8 x+\sqrt {17}+7}}{\sqrt {2} \sqrt [4]{17}}\right ),\frac {1}{4} \left (17-3 \sqrt {17}\right )\right )}{9 \sqrt {3+\sqrt {17}} \sqrt {2 x+1} \sqrt {4 x^2+7 x+2}}+\frac {\sqrt {3+\sqrt {17}} \sqrt {2 x+1} \sqrt {-4 x^2-7 x-2} E\left (\arcsin \left (\frac {\sqrt {8 x+\sqrt {17}+7}}{\sqrt {2} \sqrt [4]{17}}\right )|\frac {1}{4} \left (17-3 \sqrt {17}\right )\right )}{6 \sqrt {-2 x-1} \sqrt {4 x^2+7 x+2}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {152 \sqrt {8 x-\sqrt {17}+7} \sqrt {8 x+\sqrt {17}+7} \int \frac {1}{(13-3 (2 x+1)) \sqrt {4 (2 x+1)-\sqrt {17}+3} \sqrt {4 (2 x+1)+\sqrt {17}+3}}d\sqrt {2 x+1}}{9 \sqrt {4 x^2+7 x+2}}-\frac {22 \sqrt {-2 x-1} \sqrt {-4 x^2-7 x-2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {8 x+\sqrt {17}+7}}{\sqrt {2} \sqrt [4]{17}}\right ),\frac {1}{4} \left (17-3 \sqrt {17}\right )\right )}{9 \sqrt {3+\sqrt {17}} \sqrt {2 x+1} \sqrt {4 x^2+7 x+2}}+\frac {\sqrt {3+\sqrt {17}} \sqrt {2 x+1} \sqrt {-4 x^2-7 x-2} E\left (\arcsin \left (\frac {\sqrt {8 x+\sqrt {17}+7}}{\sqrt {2} \sqrt [4]{17}}\right )|\frac {1}{4} \left (17-3 \sqrt {17}\right )\right )}{6 \sqrt {-2 x-1} \sqrt {4 x^2+7 x+2}}\)

\(\Big \downarrow \) 413

\(\displaystyle \frac {152 \sqrt {8 x-\sqrt {17}+7} \sqrt {8 x+\sqrt {17}+7} \sqrt {\frac {4 (2 x+1)}{3-\sqrt {17}}+1} \int \frac {1}{(13-3 (2 x+1)) \sqrt {4 (2 x+1)+\sqrt {17}+3} \sqrt {\frac {4 (2 x+1)}{3-\sqrt {17}}+1}}d\sqrt {2 x+1}}{9 \sqrt {4 x^2+7 x+2} \sqrt {4 (2 x+1)-\sqrt {17}+3}}-\frac {22 \sqrt {-2 x-1} \sqrt {-4 x^2-7 x-2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {8 x+\sqrt {17}+7}}{\sqrt {2} \sqrt [4]{17}}\right ),\frac {1}{4} \left (17-3 \sqrt {17}\right )\right )}{9 \sqrt {3+\sqrt {17}} \sqrt {2 x+1} \sqrt {4 x^2+7 x+2}}+\frac {\sqrt {3+\sqrt {17}} \sqrt {2 x+1} \sqrt {-4 x^2-7 x-2} E\left (\arcsin \left (\frac {\sqrt {8 x+\sqrt {17}+7}}{\sqrt {2} \sqrt [4]{17}}\right )|\frac {1}{4} \left (17-3 \sqrt {17}\right )\right )}{6 \sqrt {-2 x-1} \sqrt {4 x^2+7 x+2}}\)

\(\Big \downarrow \) 412

\(\displaystyle -\frac {22 \sqrt {-2 x-1} \sqrt {-4 x^2-7 x-2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {8 x+\sqrt {17}+7}}{\sqrt {2} \sqrt [4]{17}}\right ),\frac {1}{4} \left (17-3 \sqrt {17}\right )\right )}{9 \sqrt {3+\sqrt {17}} \sqrt {2 x+1} \sqrt {4 x^2+7 x+2}}+\frac {\sqrt {3+\sqrt {17}} \sqrt {2 x+1} \sqrt {-4 x^2-7 x-2} E\left (\arcsin \left (\frac {\sqrt {8 x+\sqrt {17}+7}}{\sqrt {2} \sqrt [4]{17}}\right )|\frac {1}{4} \left (17-3 \sqrt {17}\right )\right )}{6 \sqrt {-2 x-1} \sqrt {4 x^2+7 x+2}}+\frac {38 \sqrt {13-3 \sqrt {17}} \sqrt {8 x-\sqrt {17}+7} \sqrt {8 x+\sqrt {17}+7} \sqrt {\frac {4 (2 x+1)}{3-\sqrt {17}}+1} \operatorname {EllipticPi}\left (-\frac {3}{52} \left (3-\sqrt {17}\right ),\arcsin \left (\frac {2 \sqrt {2 x+1}}{\sqrt {-3+\sqrt {17}}}\right ),\frac {1}{4} \left (-13+3 \sqrt {17}\right )\right )}{117 \sqrt {4 x^2+7 x+2} \sqrt {4 (2 x+1)-\sqrt {17}+3}}\)

Input:

Int[(4 + 6*x - 2*x^2)/((5 - 3*x)*Sqrt[1 + 2*x]*Sqrt[2 + 7*x + 4*x^2]),x]
 

Output:

(Sqrt[3 + Sqrt[17]]*Sqrt[1 + 2*x]*Sqrt[-2 - 7*x - 4*x^2]*EllipticE[ArcSin[ 
Sqrt[7 + Sqrt[17] + 8*x]/(Sqrt[2]*17^(1/4))], (17 - 3*Sqrt[17])/4])/(6*Sqr 
t[-1 - 2*x]*Sqrt[2 + 7*x + 4*x^2]) - (22*Sqrt[-1 - 2*x]*Sqrt[-2 - 7*x - 4* 
x^2]*EllipticF[ArcSin[Sqrt[7 + Sqrt[17] + 8*x]/(Sqrt[2]*17^(1/4))], (17 - 
3*Sqrt[17])/4])/(9*Sqrt[3 + Sqrt[17]]*Sqrt[1 + 2*x]*Sqrt[2 + 7*x + 4*x^2]) 
 + (38*Sqrt[13 - 3*Sqrt[17]]*Sqrt[7 - Sqrt[17] + 8*x]*Sqrt[7 + Sqrt[17] + 
8*x]*Sqrt[1 + (4*(1 + 2*x))/(3 - Sqrt[17])]*EllipticPi[(-3*(3 - Sqrt[17])) 
/52, ArcSin[(2*Sqrt[1 + 2*x])/Sqrt[-3 + Sqrt[17]]], (-13 + 3*Sqrt[17])/4]) 
/(117*Sqrt[2 + 7*x + 4*x^2]*Sqrt[3 - Sqrt[17] + 4*(1 + 2*x)])
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 186

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]
 

rule 321

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])
 

rule 327

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]
 

rule 412

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])
 

rule 413

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x 
_)^2]), x_Symbol] :> Simp[Sqrt[1 + (d/c)*x^2]/Sqrt[c + d*x^2]   Int[1/((a + 
 b*x^2)*Sqrt[1 + (d/c)*x^2]*Sqrt[e + f*x^2]), x], x] /; FreeQ[{a, b, c, d, 
e, f}, x] &&  !GtQ[c, 0]
 

rule 1172

Int[((d_.) + (e_.)*(x_))^(m_)/Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Sy 
mbol] :> Simp[2*Rt[b^2 - 4*a*c, 2]*(d + e*x)^m*(Sqrt[(-c)*((a + b*x + c*x^2 
)/(b^2 - 4*a*c))]/(c*Sqrt[a + b*x + c*x^2]*(2*c*((d + e*x)/(2*c*d - b*e - e 
*Rt[b^2 - 4*a*c, 2])))^m))   Subst[Int[(1 + 2*e*Rt[b^2 - 4*a*c, 2]*(x^2/(2* 
c*d - b*e - e*Rt[b^2 - 4*a*c, 2])))^m/Sqrt[1 - x^2], x], x, Sqrt[(b + Rt[b^ 
2 - 4*a*c, 2] + 2*c*x)/(2*Rt[b^2 - 4*a*c, 2])]], x] /; FreeQ[{a, b, c, d, e 
}, x] && EqQ[m^2, 1/4]
 

rule 1269

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g/e   Int[(d + e*x)^(m + 1)*(a + b*x + 
 c*x^2)^p, x], x] + Simp[(e*f - d*g)/e   Int[(d + e*x)^m*(a + b*x + c*x^2)^ 
p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] &&  !IGtQ[m, 0]
 

rule 1279

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(f_.) + (g_.)*(x_)]*Sqrt[(a_.) + (b_.)*(x_ 
) + (c_.)*(x_)^2]), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[Sqrt[b 
 - q + 2*c*x]*(Sqrt[b + q + 2*c*x]/Sqrt[a + b*x + c*x^2])   Int[1/((d + e*x 
)*Sqrt[f + g*x]*Sqrt[b - q + 2*c*x]*Sqrt[b + q + 2*c*x]), x], x]] /; FreeQ[ 
{a, b, c, d, e, f, g}, x]
 

rule 2154

Int[(Px_)*((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b 
_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[PolynomialQuotient[Px, d + 
 e*x, x]*(d + e*x)^(m + 1)*(f + g*x)^n*(a + b*x + c*x^2)^p, x] + Simp[Polyn 
omialRemainder[Px, d + e*x, x]   Int[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x 
^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && PolynomialQ[Px, x 
] && LtQ[m, 0] &&  !IntegerQ[n] && IntegersQ[2*m, 2*n, 2*p]
 

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.42 (sec) , antiderivative size = 358, normalized size of antiderivative = 1.11

method	result	size
default	\(-\frac {\sqrt {1+2 x}\, \sqrt {4 x^{2}+7 x +2}\, \sqrt {-\left (1+2 x \right ) \left (3+\sqrt {17}\right )}\, \sqrt {\left (-8 x -7+\sqrt {17}\right ) \left (-3+\sqrt {17}\right )}\, \sqrt {\left (8 x +7+\sqrt {17}\right ) \left (3+\sqrt {17}\right )}\, \left (39 \operatorname {EllipticF}\left (\frac {2 \sqrt {-\left (1+2 x \right ) \left (3+\sqrt {17}\right )}}{3+\sqrt {17}}, \frac {i \sqrt {\left (3+\sqrt {17}\right ) \left (-3+\sqrt {17}\right )}}{-3+\sqrt {17}}\right ) \sqrt {17}-39 \operatorname {EllipticE}\left (\frac {2 \sqrt {-\left (1+2 x \right ) \left (3+\sqrt {17}\right )}}{3+\sqrt {17}}, \frac {i \sqrt {\left (3+\sqrt {17}\right ) \left (-3+\sqrt {17}\right )}}{-3+\sqrt {17}}\right ) \sqrt {17}-689 \operatorname {EllipticF}\left (\frac {2 \sqrt {-\left (1+2 x \right ) \left (3+\sqrt {17}\right )}}{3+\sqrt {17}}, \frac {i \sqrt {\left (3+\sqrt {17}\right ) \left (-3+\sqrt {17}\right )}}{-3+\sqrt {17}}\right )+117 \operatorname {EllipticE}\left (\frac {2 \sqrt {-\left (1+2 x \right ) \left (3+\sqrt {17}\right )}}{3+\sqrt {17}}, \frac {i \sqrt {\left (3+\sqrt {17}\right ) \left (-3+\sqrt {17}\right )}}{-3+\sqrt {17}}\right )+608 \operatorname {EllipticPi}\left (\frac {2 \sqrt {-\left (1+2 x \right ) \left (3+\sqrt {17}\right )}}{3+\sqrt {17}}, -\frac {9}{52}-\frac {3 \sqrt {17}}{52}, \frac {i \sqrt {\left (3+\sqrt {17}\right ) \left (-3+\sqrt {17}\right )}}{-3+\sqrt {17}}\right )\right )}{936 \left (3+\sqrt {17}\right ) \left (-3+\sqrt {17}\right ) \left (8 x^{3}+18 x^{2}+11 x +2\right )}\)	\(358\)
elliptic	\(\frac {\sqrt {\left (4 x^{2}+7 x +2\right ) \left (1+2 x \right )}\, \left (-\frac {16 \left (-\frac {3}{8}-\frac {\sqrt {17}}{8}\right ) \sqrt {-\frac {x +\frac {1}{2}}{\frac {3}{8}+\frac {\sqrt {17}}{8}}}\, \sqrt {-\frac {x +\frac {7}{8}-\frac {\sqrt {17}}{8}}{-\frac {3}{8}+\frac {\sqrt {17}}{8}}}\, \sqrt {\frac {x +\frac {7}{8}+\frac {\sqrt {17}}{8}}{\frac {3}{8}+\frac {\sqrt {17}}{8}}}\, \operatorname {EllipticF}\left (\sqrt {-\frac {x +\frac {1}{2}}{\frac {3}{8}+\frac {\sqrt {17}}{8}}}, i \sqrt {\frac {\frac {3}{8}+\frac {\sqrt {17}}{8}}{-\frac {3}{8}+\frac {\sqrt {17}}{8}}}\right )}{9 \sqrt {8 x^{3}+18 x^{2}+11 x +2}}+\frac {4 \left (-\frac {3}{8}-\frac {\sqrt {17}}{8}\right ) \sqrt {-\frac {x +\frac {1}{2}}{\frac {3}{8}+\frac {\sqrt {17}}{8}}}\, \sqrt {-\frac {x +\frac {7}{8}-\frac {\sqrt {17}}{8}}{-\frac {3}{8}+\frac {\sqrt {17}}{8}}}\, \sqrt {\frac {x +\frac {7}{8}+\frac {\sqrt {17}}{8}}{\frac {3}{8}+\frac {\sqrt {17}}{8}}}\, \left (\left (\frac {3}{8}-\frac {\sqrt {17}}{8}\right ) \operatorname {EllipticE}\left (\sqrt {-\frac {x +\frac {1}{2}}{\frac {3}{8}+\frac {\sqrt {17}}{8}}}, i \sqrt {\frac {\frac {3}{8}+\frac {\sqrt {17}}{8}}{-\frac {3}{8}+\frac {\sqrt {17}}{8}}}\right )+\left (-\frac {7}{8}+\frac {\sqrt {17}}{8}\right ) \operatorname {EllipticF}\left (\sqrt {-\frac {x +\frac {1}{2}}{\frac {3}{8}+\frac {\sqrt {17}}{8}}}, i \sqrt {\frac {\frac {3}{8}+\frac {\sqrt {17}}{8}}{-\frac {3}{8}+\frac {\sqrt {17}}{8}}}\right )\right )}{3 \sqrt {8 x^{3}+18 x^{2}+11 x +2}}+\frac {304 \left (-\frac {3}{8}-\frac {\sqrt {17}}{8}\right ) \sqrt {-\frac {x +\frac {1}{2}}{\frac {3}{8}+\frac {\sqrt {17}}{8}}}\, \sqrt {-\frac {x +\frac {7}{8}-\frac {\sqrt {17}}{8}}{-\frac {3}{8}+\frac {\sqrt {17}}{8}}}\, \sqrt {\frac {x +\frac {7}{8}+\frac {\sqrt {17}}{8}}{\frac {3}{8}+\frac {\sqrt {17}}{8}}}\, \operatorname {EllipticPi}\left (\sqrt {-\frac {x +\frac {1}{2}}{\frac {3}{8}+\frac {\sqrt {17}}{8}}}, -\frac {9}{52}-\frac {3 \sqrt {17}}{52}, i \sqrt {\frac {\frac {3}{8}+\frac {\sqrt {17}}{8}}{-\frac {3}{8}+\frac {\sqrt {17}}{8}}}\right )}{117 \sqrt {8 x^{3}+18 x^{2}+11 x +2}}\right )}{\sqrt {4 x^{2}+7 x +2}\, \sqrt {1+2 x}}\)	\(469\)

Input:

int((-2*x^2+6*x+4)/(5-3*x)/(1+2*x)^(1/2)/(4*x^2+7*x+2)^(1/2),x,method=_RET 
URNVERBOSE)
 

Output:

-1/936*(1+2*x)^(1/2)*(4*x^2+7*x+2)^(1/2)/(3+17^(1/2))*(-(1+2*x)*(3+17^(1/2 
)))^(1/2)/(-3+17^(1/2))*((-8*x-7+17^(1/2))*(-3+17^(1/2)))^(1/2)*((8*x+7+17 
^(1/2))*(3+17^(1/2)))^(1/2)*(39*EllipticF(2/(3+17^(1/2))*(-(1+2*x)*(3+17^( 
1/2)))^(1/2),I/(-3+17^(1/2))*((3+17^(1/2))*(-3+17^(1/2)))^(1/2))*17^(1/2)- 
39*EllipticE(2/(3+17^(1/2))*(-(1+2*x)*(3+17^(1/2)))^(1/2),I/(-3+17^(1/2))* 
((3+17^(1/2))*(-3+17^(1/2)))^(1/2))*17^(1/2)-689*EllipticF(2/(3+17^(1/2))* 
(-(1+2*x)*(3+17^(1/2)))^(1/2),I/(-3+17^(1/2))*((3+17^(1/2))*(-3+17^(1/2))) 
^(1/2))+117*EllipticE(2/(3+17^(1/2))*(-(1+2*x)*(3+17^(1/2)))^(1/2),I/(-3+1 
7^(1/2))*((3+17^(1/2))*(-3+17^(1/2)))^(1/2))+608*EllipticPi(2/(3+17^(1/2)) 
*(-(1+2*x)*(3+17^(1/2)))^(1/2),-9/52-3/52*17^(1/2),I/(-3+17^(1/2))*((3+17^ 
(1/2))*(-3+17^(1/2)))^(1/2)))/(8*x^3+18*x^2+11*x+2)
 

Fricas [F]

\[ \int \frac {4+6 x-2 x^2}{(5-3 x) \sqrt {1+2 x} \sqrt {2+7 x+4 x^2}} \, dx=\int { \frac {2 \, {\left (x^{2} - 3 \, x - 2\right )}}{\sqrt {4 \, x^{2} + 7 \, x + 2} {\left (3 \, x - 5\right )} \sqrt {2 \, x + 1}} \,d x } \] Input:

integrate((-2*x^2+6*x+4)/(5-3*x)/(1+2*x)^(1/2)/(4*x^2+7*x+2)^(1/2),x, algo 
rithm="fricas")
 

Output:

integral(2*sqrt(4*x^2 + 7*x + 2)*(x^2 - 3*x - 2)*sqrt(2*x + 1)/(24*x^4 + 1 
4*x^3 - 57*x^2 - 49*x - 10), x)
 

Sympy [F]

\[ \int \frac {4+6 x-2 x^2}{(5-3 x) \sqrt {1+2 x} \sqrt {2+7 x+4 x^2}} \, dx=2 \left (\int \left (- \frac {3 x}{3 x \sqrt {2 x + 1} \sqrt {4 x^{2} + 7 x + 2} - 5 \sqrt {2 x + 1} \sqrt {4 x^{2} + 7 x + 2}}\right )\, dx + \int \frac {x^{2}}{3 x \sqrt {2 x + 1} \sqrt {4 x^{2} + 7 x + 2} - 5 \sqrt {2 x + 1} \sqrt {4 x^{2} + 7 x + 2}}\, dx + \int \left (- \frac {2}{3 x \sqrt {2 x + 1} \sqrt {4 x^{2} + 7 x + 2} - 5 \sqrt {2 x + 1} \sqrt {4 x^{2} + 7 x + 2}}\right )\, dx\right ) \] Input:

integrate((-2*x**2+6*x+4)/(5-3*x)/(1+2*x)**(1/2)/(4*x**2+7*x+2)**(1/2),x)
 

Output:

2*(Integral(-3*x/(3*x*sqrt(2*x + 1)*sqrt(4*x**2 + 7*x + 2) - 5*sqrt(2*x + 
1)*sqrt(4*x**2 + 7*x + 2)), x) + Integral(x**2/(3*x*sqrt(2*x + 1)*sqrt(4*x 
**2 + 7*x + 2) - 5*sqrt(2*x + 1)*sqrt(4*x**2 + 7*x + 2)), x) + Integral(-2 
/(3*x*sqrt(2*x + 1)*sqrt(4*x**2 + 7*x + 2) - 5*sqrt(2*x + 1)*sqrt(4*x**2 + 
 7*x + 2)), x))
 

Maxima [F]

\[ \int \frac {4+6 x-2 x^2}{(5-3 x) \sqrt {1+2 x} \sqrt {2+7 x+4 x^2}} \, dx=\int { \frac {2 \, {\left (x^{2} - 3 \, x - 2\right )}}{\sqrt {4 \, x^{2} + 7 \, x + 2} {\left (3 \, x - 5\right )} \sqrt {2 \, x + 1}} \,d x } \] Input:

integrate((-2*x^2+6*x+4)/(5-3*x)/(1+2*x)^(1/2)/(4*x^2+7*x+2)^(1/2),x, algo 
rithm="maxima")
 

Output:

2*integrate((x^2 - 3*x - 2)/(sqrt(4*x^2 + 7*x + 2)*(3*x - 5)*sqrt(2*x + 1) 
), x)
 

Giac [F]

\[ \int \frac {4+6 x-2 x^2}{(5-3 x) \sqrt {1+2 x} \sqrt {2+7 x+4 x^2}} \, dx=\int { \frac {2 \, {\left (x^{2} - 3 \, x - 2\right )}}{\sqrt {4 \, x^{2} + 7 \, x + 2} {\left (3 \, x - 5\right )} \sqrt {2 \, x + 1}} \,d x } \] Input:

integrate((-2*x^2+6*x+4)/(5-3*x)/(1+2*x)^(1/2)/(4*x^2+7*x+2)^(1/2),x, algo 
rithm="giac")
 

Output:

integrate(2*(x^2 - 3*x - 2)/(sqrt(4*x^2 + 7*x + 2)*(3*x - 5)*sqrt(2*x + 1) 
), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {4+6 x-2 x^2}{(5-3 x) \sqrt {1+2 x} \sqrt {2+7 x+4 x^2}} \, dx=\int -\frac {-2\,x^2+6\,x+4}{\sqrt {2\,x+1}\,\left (3\,x-5\right )\,\sqrt {4\,x^2+7\,x+2}} \,d x \] Input:

int(-(6*x - 2*x^2 + 4)/((2*x + 1)^(1/2)*(3*x - 5)*(7*x + 4*x^2 + 2)^(1/2)) 
,x)
 

Output:

int(-(6*x - 2*x^2 + 4)/((2*x + 1)^(1/2)*(3*x - 5)*(7*x + 4*x^2 + 2)^(1/2)) 
, x)
 

Reduce [F]

\[ \int \frac {4+6 x-2 x^2}{(5-3 x) \sqrt {1+2 x} \sqrt {2+7 x+4 x^2}} \, dx=\int \frac {-2 x^{2}+6 x +4}{\left (5-3 x \right ) \sqrt {2 x +1}\, \sqrt {4 x^{2}+7 x +2}}d x \] Input:

int((-2*x^2+6*x+4)/(5-3*x)/(1+2*x)^(1/2)/(4*x^2+7*x+2)^(1/2),x)
 

Output:

int((-2*x^2+6*x+4)/(5-3*x)/(1+2*x)^(1/2)/(4*x^2+7*x+2)^(1/2),x)