\(\int \frac {4+6 x-2 x^2}{(5-3 x) \sqrt {1+2 x} \sqrt {3+4 x^2}} \, dx\) [11]

Optimal result

Integrand size = 38, antiderivative size = 271 \[ \int \frac {4+6 x-2 x^2}{(5-3 x) \sqrt {1+2 x} \sqrt {3+4 x^2}} \, dx=\frac {\sqrt {1+2 x} \sqrt {3+4 x^2}}{3 (3+2 x)}+\frac {76 \text {arctanh}\left (\frac {\sqrt {\frac {127}{39}} \sqrt {1+2 x}}{\sqrt {3+4 x^2}}\right )}{3 \sqrt {4953}}-\frac {\sqrt {2} (3+2 x) \sqrt {\frac {3+4 x^2}{(3+2 x)^2}} E\left (2 \arctan \left (\frac {\sqrt {1+2 x}}{\sqrt {2}}\right )|\frac {3}{4}\right )}{3 \sqrt {3+4 x^2}}+\frac {(3+2 x) \sqrt {\frac {3+4 x^2}{(3+2 x)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {1+2 x}}{\sqrt {2}}\right ),\frac {3}{4}\right )}{6 \sqrt {2} \sqrt {3+4 x^2}}-\frac {7 \sqrt {2} (3+2 x) \sqrt {\frac {3+4 x^2}{(3+2 x)^2}} \operatorname {EllipticPi}\left (\frac {361}{312},2 \arctan \left (\frac {\sqrt {1+2 x}}{\sqrt {2}}\right ),\frac {3}{4}\right )}{117 \sqrt {3+4 x^2}} \] Output:

(1+2*x)^(1/2)*(4*x^2+3)^(1/2)/(9+6*x)+76/14859*4953^(1/2)*arctanh(1/39*495 
3^(1/2)*(1+2*x)^(1/2)/(4*x^2+3)^(1/2))-1/3*2^(1/2)*(3+2*x)*((4*x^2+3)/(3+2 
*x)^2)^(1/2)*EllipticE(sin(2*arctan(1/2*(1+2*x)^(1/2)*2^(1/2))),1/2*3^(1/2 
))/(4*x^2+3)^(1/2)+1/12*2^(1/2)*(3+2*x)*((4*x^2+3)/(3+2*x)^2)^(1/2)*Invers 
eJacobiAM(2*arctan(1/2*(1+2*x)^(1/2)*2^(1/2)),1/2*3^(1/2))/(4*x^2+3)^(1/2) 
-7/117*2^(1/2)*(3+2*x)*((4*x^2+3)/(3+2*x)^2)^(1/2)*EllipticPi(sin(2*arctan 
(1/2*(1+2*x)^(1/2)*2^(1/2))),361/312,1/2*3^(1/2))/(4*x^2+3)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 23.87 (sec) , antiderivative size = 632, normalized size of antiderivative = 2.33 \[ \int \frac {4+6 x-2 x^2}{(5-3 x) \sqrt {1+2 x} \sqrt {3+4 x^2}} \, dx=\frac {(3+6 x)^{3/2} \left (78 \sqrt {-\frac {i}{i+\sqrt {3}}}+\frac {312 \sqrt {-\frac {i}{i+\sqrt {3}}}}{(1+2 x)^2}-\frac {156 \sqrt {-\frac {i}{i+\sqrt {3}}}}{1+2 x}+\frac {39 \left (-i+\sqrt {3}\right ) \sqrt {\frac {3 i+\sqrt {3}+2 \left (-i+\sqrt {3}\right ) x}{\left (-i+\sqrt {3}\right ) (1+2 x)}} \sqrt {\frac {-3 i+\sqrt {3}+2 \left (i+\sqrt {3}\right ) x}{\left (i+\sqrt {3}\right ) (1+2 x)}} E\left (i \text {arcsinh}\left (\frac {2 \sqrt {-\frac {i}{i+\sqrt {3}}}}{\sqrt {1+2 x}}\right )|\frac {i+\sqrt {3}}{i-\sqrt {3}}\right )}{\sqrt {1+2 x}}-\frac {3 \left (-16 i+13 \sqrt {3}\right ) \sqrt {\frac {3 i+\sqrt {3}+2 \left (-i+\sqrt {3}\right ) x}{\left (-i+\sqrt {3}\right ) (1+2 x)}} \sqrt {\frac {-3 i+\sqrt {3}+2 \left (i+\sqrt {3}\right ) x}{\left (i+\sqrt {3}\right ) (1+2 x)}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {2 \sqrt {-\frac {i}{i+\sqrt {3}}}}{\sqrt {1+2 x}}\right ),\frac {i+\sqrt {3}}{i-\sqrt {3}}\right )}{\sqrt {1+2 x}}-\frac {152 i \sqrt {\frac {3 i+\sqrt {3}+2 \left (-i+\sqrt {3}\right ) x}{\left (-i+\sqrt {3}\right ) (1+2 x)}} \sqrt {\frac {-3 i+\sqrt {3}+2 \left (i+\sqrt {3}\right ) x}{\left (i+\sqrt {3}\right ) (1+2 x)}} \operatorname {EllipticPi}\left (\frac {13}{12} \left (1-i \sqrt {3}\right ),i \text {arcsinh}\left (\frac {2 \sqrt {-\frac {i}{i+\sqrt {3}}}}{\sqrt {1+2 x}}\right ),\frac {i+\sqrt {3}}{i-\sqrt {3}}\right )}{\sqrt {1+2 x}}\right )}{702 \sqrt {-\frac {3 i}{i+\sqrt {3}}} \sqrt {3+4 x^2}} \] Input:

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

Output:

((3 + 6*x)^(3/2)*(78*Sqrt[(-I)/(I + Sqrt[3])] + (312*Sqrt[(-I)/(I + Sqrt[3 
])])/(1 + 2*x)^2 - (156*Sqrt[(-I)/(I + Sqrt[3])])/(1 + 2*x) + (39*(-I + Sq 
rt[3])*Sqrt[(3*I + Sqrt[3] + 2*(-I + Sqrt[3])*x)/((-I + Sqrt[3])*(1 + 2*x) 
)]*Sqrt[(-3*I + Sqrt[3] + 2*(I + Sqrt[3])*x)/((I + Sqrt[3])*(1 + 2*x))]*El 
lipticE[I*ArcSinh[(2*Sqrt[(-I)/(I + Sqrt[3])])/Sqrt[1 + 2*x]], (I + Sqrt[3 
])/(I - Sqrt[3])])/Sqrt[1 + 2*x] - (3*(-16*I + 13*Sqrt[3])*Sqrt[(3*I + Sqr 
t[3] + 2*(-I + Sqrt[3])*x)/((-I + Sqrt[3])*(1 + 2*x))]*Sqrt[(-3*I + Sqrt[3 
] + 2*(I + Sqrt[3])*x)/((I + Sqrt[3])*(1 + 2*x))]*EllipticF[I*ArcSinh[(2*S 
qrt[(-I)/(I + Sqrt[3])])/Sqrt[1 + 2*x]], (I + Sqrt[3])/(I - Sqrt[3])])/Sqr 
t[1 + 2*x] - ((152*I)*Sqrt[(3*I + Sqrt[3] + 2*(-I + Sqrt[3])*x)/((-I + Sqr 
t[3])*(1 + 2*x))]*Sqrt[(-3*I + Sqrt[3] + 2*(I + Sqrt[3])*x)/((I + Sqrt[3]) 
*(1 + 2*x))]*EllipticPi[(13*(1 - I*Sqrt[3]))/12, I*ArcSinh[(2*Sqrt[(-I)/(I 
 + Sqrt[3])])/Sqrt[1 + 2*x]], (I + Sqrt[3])/(I - Sqrt[3])])/Sqrt[1 + 2*x]) 
)/(702*Sqrt[(-3*I)/(I + Sqrt[3])]*Sqrt[3 + 4*x^2])
 

Rubi [A] (verified)

Time = 0.75 (sec) , antiderivative size = 440, normalized size of antiderivative = 1.62, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {2349, 599, 27, 729, 1511, 27, 1416, 1509, 1540, 27, 1416, 2222}

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.

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

\(\Big \downarrow \) 2349

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

\(\Big \downarrow \) 599

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 729

\(\displaystyle \frac {152}{9} \int \frac {1}{(13-3 (2 x+1)) \sqrt {(2 x+1)^2-2 (2 x+1)+4}}d\sqrt {2 x+1}-\frac {1}{9} \int \frac {11-3 (2 x+1)}{\sqrt {(2 x+1)^2-2 (2 x+1)+4}}d\sqrt {2 x+1}\)

\(\Big \downarrow \) 1511

\(\displaystyle \frac {1}{9} \left (-5 \int \frac {1}{\sqrt {(2 x+1)^2-2 (2 x+1)+4}}d\sqrt {2 x+1}-6 \int \frac {1-2 x}{2 \sqrt {(2 x+1)^2-2 (2 x+1)+4}}d\sqrt {2 x+1}\right )+\frac {152}{9} \int \frac {1}{(13-3 (2 x+1)) \sqrt {(2 x+1)^2-2 (2 x+1)+4}}d\sqrt {2 x+1}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{9} \left (-5 \int \frac {1}{\sqrt {(2 x+1)^2-2 (2 x+1)+4}}d\sqrt {2 x+1}-3 \int \frac {1-2 x}{\sqrt {(2 x+1)^2-2 (2 x+1)+4}}d\sqrt {2 x+1}\right )+\frac {152}{9} \int \frac {1}{(13-3 (2 x+1)) \sqrt {(2 x+1)^2-2 (2 x+1)+4}}d\sqrt {2 x+1}\)

\(\Big \downarrow \) 1416

\(\displaystyle \frac {1}{9} \left (-3 \int \frac {1-2 x}{\sqrt {(2 x+1)^2-2 (2 x+1)+4}}d\sqrt {2 x+1}-\frac {5 (2 x+3) \sqrt {\frac {(2 x+1)^2-2 (2 x+1)+4}{(2 x+3)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2 x+1}}{\sqrt {2}}\right ),\frac {3}{4}\right )}{2 \sqrt {2} \sqrt {(2 x+1)^2-2 (2 x+1)+4}}\right )+\frac {152}{9} \int \frac {1}{(13-3 (2 x+1)) \sqrt {(2 x+1)^2-2 (2 x+1)+4}}d\sqrt {2 x+1}\)

\(\Big \downarrow \) 1509

\(\displaystyle \frac {152}{9} \int \frac {1}{(13-3 (2 x+1)) \sqrt {(2 x+1)^2-2 (2 x+1)+4}}d\sqrt {2 x+1}+\frac {1}{9} \left (-\frac {5 (2 x+3) \sqrt {\frac {(2 x+1)^2-2 (2 x+1)+4}{(2 x+3)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2 x+1}}{\sqrt {2}}\right ),\frac {3}{4}\right )}{2 \sqrt {2} \sqrt {(2 x+1)^2-2 (2 x+1)+4}}-3 \left (\frac {\sqrt {2} (2 x+3) \sqrt {\frac {(2 x+1)^2-2 (2 x+1)+4}{(2 x+3)^2}} E\left (2 \arctan \left (\frac {\sqrt {2 x+1}}{\sqrt {2}}\right )|\frac {3}{4}\right )}{\sqrt {(2 x+1)^2-2 (2 x+1)+4}}-\frac {\sqrt {2 x+1} \sqrt {(2 x+1)^2-2 (2 x+1)+4}}{2 x+3}\right )\right )\)

\(\Big \downarrow \) 1540

\(\displaystyle \frac {152}{9} \left (\frac {1}{19} \int \frac {1}{\sqrt {(2 x+1)^2-2 (2 x+1)+4}}d\sqrt {2 x+1}+\frac {6}{19} \int \frac {2 x+3}{2 (13-3 (2 x+1)) \sqrt {(2 x+1)^2-2 (2 x+1)+4}}d\sqrt {2 x+1}\right )+\frac {1}{9} \left (-\frac {5 (2 x+3) \sqrt {\frac {(2 x+1)^2-2 (2 x+1)+4}{(2 x+3)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2 x+1}}{\sqrt {2}}\right ),\frac {3}{4}\right )}{2 \sqrt {2} \sqrt {(2 x+1)^2-2 (2 x+1)+4}}-3 \left (\frac {\sqrt {2} (2 x+3) \sqrt {\frac {(2 x+1)^2-2 (2 x+1)+4}{(2 x+3)^2}} E\left (2 \arctan \left (\frac {\sqrt {2 x+1}}{\sqrt {2}}\right )|\frac {3}{4}\right )}{\sqrt {(2 x+1)^2-2 (2 x+1)+4}}-\frac {\sqrt {2 x+1} \sqrt {(2 x+1)^2-2 (2 x+1)+4}}{2 x+3}\right )\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {152}{9} \left (\frac {1}{19} \int \frac {1}{\sqrt {(2 x+1)^2-2 (2 x+1)+4}}d\sqrt {2 x+1}+\frac {3}{19} \int \frac {2 x+3}{(13-3 (2 x+1)) \sqrt {(2 x+1)^2-2 (2 x+1)+4}}d\sqrt {2 x+1}\right )+\frac {1}{9} \left (-\frac {5 (2 x+3) \sqrt {\frac {(2 x+1)^2-2 (2 x+1)+4}{(2 x+3)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2 x+1}}{\sqrt {2}}\right ),\frac {3}{4}\right )}{2 \sqrt {2} \sqrt {(2 x+1)^2-2 (2 x+1)+4}}-3 \left (\frac {\sqrt {2} (2 x+3) \sqrt {\frac {(2 x+1)^2-2 (2 x+1)+4}{(2 x+3)^2}} E\left (2 \arctan \left (\frac {\sqrt {2 x+1}}{\sqrt {2}}\right )|\frac {3}{4}\right )}{\sqrt {(2 x+1)^2-2 (2 x+1)+4}}-\frac {\sqrt {2 x+1} \sqrt {(2 x+1)^2-2 (2 x+1)+4}}{2 x+3}\right )\right )\)

\(\Big \downarrow \) 1416

\(\displaystyle \frac {152}{9} \left (\frac {3}{19} \int \frac {2 x+3}{(13-3 (2 x+1)) \sqrt {(2 x+1)^2-2 (2 x+1)+4}}d\sqrt {2 x+1}+\frac {(2 x+3) \sqrt {\frac {(2 x+1)^2-2 (2 x+1)+4}{(2 x+3)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2 x+1}}{\sqrt {2}}\right ),\frac {3}{4}\right )}{38 \sqrt {2} \sqrt {(2 x+1)^2-2 (2 x+1)+4}}\right )+\frac {1}{9} \left (-\frac {5 (2 x+3) \sqrt {\frac {(2 x+1)^2-2 (2 x+1)+4}{(2 x+3)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2 x+1}}{\sqrt {2}}\right ),\frac {3}{4}\right )}{2 \sqrt {2} \sqrt {(2 x+1)^2-2 (2 x+1)+4}}-3 \left (\frac {\sqrt {2} (2 x+3) \sqrt {\frac {(2 x+1)^2-2 (2 x+1)+4}{(2 x+3)^2}} E\left (2 \arctan \left (\frac {\sqrt {2 x+1}}{\sqrt {2}}\right )|\frac {3}{4}\right )}{\sqrt {(2 x+1)^2-2 (2 x+1)+4}}-\frac {\sqrt {2 x+1} \sqrt {(2 x+1)^2-2 (2 x+1)+4}}{2 x+3}\right )\right )\)

\(\Big \downarrow \) 2222

\(\displaystyle \frac {152}{9} \left (\frac {3}{19} \left (\frac {19 \text {arctanh}\left (\frac {\sqrt {\frac {127}{39}} \sqrt {2 x+1}}{\sqrt {(2 x+1)^2-2 (2 x+1)+4}}\right )}{2 \sqrt {4953}}-\frac {7 (2 x+3) \sqrt {\frac {(2 x+1)^2-2 (2 x+1)+4}{(2 x+3)^2}} \operatorname {EllipticPi}\left (\frac {361}{312},2 \arctan \left (\frac {\sqrt {2 x+1}}{\sqrt {2}}\right ),\frac {3}{4}\right )}{156 \sqrt {2} \sqrt {(2 x+1)^2-2 (2 x+1)+4}}\right )+\frac {(2 x+3) \sqrt {\frac {(2 x+1)^2-2 (2 x+1)+4}{(2 x+3)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2 x+1}}{\sqrt {2}}\right ),\frac {3}{4}\right )}{38 \sqrt {2} \sqrt {(2 x+1)^2-2 (2 x+1)+4}}\right )+\frac {1}{9} \left (-\frac {5 (2 x+3) \sqrt {\frac {(2 x+1)^2-2 (2 x+1)+4}{(2 x+3)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2 x+1}}{\sqrt {2}}\right ),\frac {3}{4}\right )}{2 \sqrt {2} \sqrt {(2 x+1)^2-2 (2 x+1)+4}}-3 \left (\frac {\sqrt {2} (2 x+3) \sqrt {\frac {(2 x+1)^2-2 (2 x+1)+4}{(2 x+3)^2}} E\left (2 \arctan \left (\frac {\sqrt {2 x+1}}{\sqrt {2}}\right )|\frac {3}{4}\right )}{\sqrt {(2 x+1)^2-2 (2 x+1)+4}}-\frac {\sqrt {2 x+1} \sqrt {(2 x+1)^2-2 (2 x+1)+4}}{2 x+3}\right )\right )\)

Input:

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

Output:

(-3*(-((Sqrt[1 + 2*x]*Sqrt[4 - 2*(1 + 2*x) + (1 + 2*x)^2])/(3 + 2*x)) + (S 
qrt[2]*(3 + 2*x)*Sqrt[(4 - 2*(1 + 2*x) + (1 + 2*x)^2)/(3 + 2*x)^2]*Ellipti 
cE[2*ArcTan[Sqrt[1 + 2*x]/Sqrt[2]], 3/4])/Sqrt[4 - 2*(1 + 2*x) + (1 + 2*x) 
^2]) - (5*(3 + 2*x)*Sqrt[(4 - 2*(1 + 2*x) + (1 + 2*x)^2)/(3 + 2*x)^2]*Elli 
pticF[2*ArcTan[Sqrt[1 + 2*x]/Sqrt[2]], 3/4])/(2*Sqrt[2]*Sqrt[4 - 2*(1 + 2* 
x) + (1 + 2*x)^2]))/9 + (152*(((3 + 2*x)*Sqrt[(4 - 2*(1 + 2*x) + (1 + 2*x) 
^2)/(3 + 2*x)^2]*EllipticF[2*ArcTan[Sqrt[1 + 2*x]/Sqrt[2]], 3/4])/(38*Sqrt 
[2]*Sqrt[4 - 2*(1 + 2*x) + (1 + 2*x)^2]) + (3*((19*ArcTanh[(Sqrt[127/39]*S 
qrt[1 + 2*x])/Sqrt[4 - 2*(1 + 2*x) + (1 + 2*x)^2]])/(2*Sqrt[4953]) - (7*(3 
 + 2*x)*Sqrt[(4 - 2*(1 + 2*x) + (1 + 2*x)^2)/(3 + 2*x)^2]*EllipticPi[361/3 
12, 2*ArcTan[Sqrt[1 + 2*x]/Sqrt[2]], 3/4])/(156*Sqrt[2]*Sqrt[4 - 2*(1 + 2* 
x) + (1 + 2*x)^2])))/19))/9
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 599

Int[((A_.) + (B_.)*(x_))/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2] 
), x_Symbol] :> Simp[-2/d^2   Subst[Int[(B*c - A*d - B*x^2)/Sqrt[(b*c^2 + a 
*d^2)/d^2 - 2*b*c*(x^2/d^2) + b*(x^4/d^2)], x], x, Sqrt[c + d*x]], x] /; Fr 
eeQ[{a, b, c, d, A, B}, x] && PosQ[b/a]
 

rule 729

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

rule 1416

Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c 
/a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/ 
(2*q*Sqrt[a + b*x^2 + c*x^4]))*EllipticF[2*ArcTan[q*x], 1/2 - b*(q^2/(4*c)) 
], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
 

rule 1509

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo 
l] :> With[{q = Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a + b*x^2 + c*x^4]/(a*(1 + q 
^2*x^2))), x] + Simp[d*(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2* 
x^2)^2)]/(q*Sqrt[a + b*x^2 + c*x^4]))*EllipticE[2*ArcTan[q*x], 1/2 - b*(q^2 
/(4*c))], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 
- 4*a*c, 0] && PosQ[c/a]
 

rule 1511

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo 
l] :> With[{q = Rt[c/a, 2]}, Simp[(e + d*q)/q   Int[1/Sqrt[a + b*x^2 + c*x^ 
4], x], x] - Simp[e/q   Int[(1 - q*x^2)/Sqrt[a + b*x^2 + c*x^4], x], x] /; 
NeQ[e + d*q, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && Pos 
Q[c/a]
 

rule 1540

Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4]), x_S 
ymbol] :> With[{q = Rt[c/a, 2]}, Simp[(c*d + a*e*q)/(c*d^2 - a*e^2)   Int[1 
/Sqrt[a + b*x^2 + c*x^4], x], x] - Simp[(a*e*(e + d*q))/(c*d^2 - a*e^2)   I 
nt[(1 + q*x^2)/((d + e*x^2)*Sqrt[a + b*x^2 + c*x^4]), x], x]] /; FreeQ[{a, 
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && 
NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a]
 

rule 2222

Int[((A_) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^2 + 
 (c_.)*(x_)^4]), x_Symbol] :> With[{q = Rt[B/A, 2]}, Simp[(-(B*d - A*e))*(A 
rcTanh[Rt[b - c*(d/e) - a*(e/d), 2]*(x/Sqrt[a + b*x^2 + c*x^4])]/(2*d*e*Rt[ 
b - c*(d/e) - a*(e/d), 2])), x] + Simp[(B*d + A*e)*(1 + q^2*x^2)*(Sqrt[(a + 
 b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/(4*d*e*q*Sqrt[a + b*x^2 + c*x^4]))*Ell 
ipticPi[-(e - d*q^2)^2/(4*d*e*q^2), 2*ArcTan[q*x], 1/2 - b/(4*a*q^2)], x]] 
/; FreeQ[{a, b, c, d, e, A, B}, x] && NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a] && 
 EqQ[c*A^2 - a*B^2, 0] && PosQ[B/A] && NegQ[-b + c*(d/e) + a*(e/d)]
 

rule 2349

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]
 

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.29 (sec) , antiderivative size = 346, normalized size of antiderivative = 1.28

method	result	size
default	\(-\frac {\sqrt {1+2 x}\, \sqrt {4 x^{2}+3}\, \left (-1+i \sqrt {3}\right ) \sqrt {-\frac {1+2 x}{-1+i \sqrt {3}}}\, \sqrt {\frac {i \sqrt {3}-2 x}{1+i \sqrt {3}}}\, \sqrt {\frac {i \sqrt {3}+2 x}{-1+i \sqrt {3}}}\, \left (39 i \sqrt {3}\, \operatorname {EllipticF}\left (\sqrt {-\frac {1+2 x}{-1+i \sqrt {3}}}, \sqrt {-\frac {-1+i \sqrt {3}}{1+i \sqrt {3}}}\right )-39 i \operatorname {EllipticE}\left (\sqrt {-\frac {1+2 x}{-1+i \sqrt {3}}}, \sqrt {-\frac {-1+i \sqrt {3}}{1+i \sqrt {3}}}\right ) \sqrt {3}-104 \operatorname {EllipticF}\left (\sqrt {-\frac {1+2 x}{-1+i \sqrt {3}}}, \sqrt {-\frac {-1+i \sqrt {3}}{1+i \sqrt {3}}}\right )-39 \operatorname {EllipticE}\left (\sqrt {-\frac {1+2 x}{-1+i \sqrt {3}}}, \sqrt {-\frac {-1+i \sqrt {3}}{1+i \sqrt {3}}}\right )+152 \operatorname {EllipticPi}\left (\sqrt {-\frac {1+2 x}{-1+i \sqrt {3}}}, \frac {3}{13}-\frac {3 i \sqrt {3}}{13}, \sqrt {-\frac {-1+i \sqrt {3}}{1+i \sqrt {3}}}\right )\right )}{117 \left (8 x^{3}+4 x^{2}+6 x +3\right )}\)	\(346\)
elliptic	\(\frac {\sqrt {\left (4 x^{2}+3\right ) \left (1+2 x \right )}\, \left (-\frac {16 \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x +\frac {1}{2}}{\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {i \sqrt {3}}{2}}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {i \sqrt {3}}{2}}{-\frac {1}{2}+\frac {i \sqrt {3}}{2}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {1}{2}}{\frac {1}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\right )}{9 \sqrt {8 x^{3}+4 x^{2}+6 x +3}}+\frac {4 \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x +\frac {1}{2}}{\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {i \sqrt {3}}{2}}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {i \sqrt {3}}{2}}{-\frac {1}{2}+\frac {i \sqrt {3}}{2}}}\, \left (\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +\frac {1}{2}}{\frac {1}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\right )+\frac {i \sqrt {3}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {1}{2}}{\frac {1}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\right )}{2}\right )}{3 \sqrt {8 x^{3}+4 x^{2}+6 x +3}}+\frac {304 \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x +\frac {1}{2}}{\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {i \sqrt {3}}{2}}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {i \sqrt {3}}{2}}{-\frac {1}{2}+\frac {i \sqrt {3}}{2}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {x +\frac {1}{2}}{\frac {1}{2}-\frac {i \sqrt {3}}{2}}}, \frac {3}{13}-\frac {3 i \sqrt {3}}{13}, \sqrt {\frac {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\right )}{117 \sqrt {8 x^{3}+4 x^{2}+6 x +3}}\right )}{\sqrt {4 x^{2}+3}\, \sqrt {1+2 x}}\)	\(465\)

Input:

int((-2*x^2+6*x+4)/(5-3*x)/(1+2*x)^(1/2)/(4*x^2+3)^(1/2),x,method=_RETURNV 
ERBOSE)
 

Output:

-1/117*(1+2*x)^(1/2)*(4*x^2+3)^(1/2)*(-1+I*3^(1/2))*(-(1+2*x)/(-1+I*3^(1/2 
)))^(1/2)*((I*3^(1/2)-2*x)/(1+I*3^(1/2)))^(1/2)*((I*3^(1/2)+2*x)/(-1+I*3^( 
1/2)))^(1/2)*(39*I*3^(1/2)*EllipticF((-(1+2*x)/(-1+I*3^(1/2)))^(1/2),(-(-1 
+I*3^(1/2))/(1+I*3^(1/2)))^(1/2))-39*I*EllipticE((-(1+2*x)/(-1+I*3^(1/2))) 
^(1/2),(-(-1+I*3^(1/2))/(1+I*3^(1/2)))^(1/2))*3^(1/2)-104*EllipticF((-(1+2 
*x)/(-1+I*3^(1/2)))^(1/2),(-(-1+I*3^(1/2))/(1+I*3^(1/2)))^(1/2))-39*Ellipt 
icE((-(1+2*x)/(-1+I*3^(1/2)))^(1/2),(-(-1+I*3^(1/2))/(1+I*3^(1/2)))^(1/2)) 
+152*EllipticPi((-(1+2*x)/(-1+I*3^(1/2)))^(1/2),3/13-3/13*I*3^(1/2),(-(-1+ 
I*3^(1/2))/(1+I*3^(1/2)))^(1/2)))/(8*x^3+4*x^2+6*x+3)
 

Fricas [F]

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

integrate((-2*x^2+6*x+4)/(5-3*x)/(1+2*x)^(1/2)/(4*x^2+3)^(1/2),x, algorith 
m="fricas")
 

Output:

integral(2*sqrt(4*x^2 + 3)*(x^2 - 3*x - 2)*sqrt(2*x + 1)/(24*x^4 - 28*x^3 
- 2*x^2 - 21*x - 15), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

integrate((-2*x^2+6*x+4)/(5-3*x)/(1+2*x)^(1/2)/(4*x^2+3)^(1/2),x, algorith 
m="maxima")
 

Output:

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

Giac [F]

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

integrate((-2*x^2+6*x+4)/(5-3*x)/(1+2*x)^(1/2)/(4*x^2+3)^(1/2),x, algorith 
m="giac")
 

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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