\(\int \frac {4+6 x-2 x^2}{(5-3 x) \sqrt {1+2 x} (3+4 x^2)^{3/2}} \, dx\) [12]

Optimal result

Integrand size = 38, antiderivative size = 302 \[ \int \frac {4+6 x-2 x^2}{(5-3 x) \sqrt {1+2 x} \left (3+4 x^2\right )^{3/2}} \, dx=-\frac {(111-670 x) \sqrt {1+2 x}}{3048 \sqrt {3+4 x^2}}-\frac {335 \sqrt {1+2 x} \sqrt {3+4 x^2}}{3048 (3+2 x)}+\frac {76}{127} \sqrt {\frac {3}{1651}} \text {arctanh}\left (\frac {\sqrt {\frac {127}{39}} \sqrt {1+2 x}}{\sqrt {3+4 x^2}}\right )+\frac {335 (3+2 x) \sqrt {\frac {3+4 x^2}{(3+2 x)^2}} E\left (2 \arctan \left (\frac {\sqrt {1+2 x}}{\sqrt {2}}\right )|\frac {3}{4}\right )}{1524 \sqrt {2} \sqrt {3+4 x^2}}-\frac {(3+2 x) \sqrt {\frac {3+4 x^2}{(3+2 x)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {1+2 x}}{\sqrt {2}}\right ),\frac {3}{4}\right )}{24 \sqrt {2} \sqrt {3+4 x^2}}-\frac {7 \sqrt {2} (3+2 x) \sqrt {\frac {3+4 x^2}{(3+2 x)^2}} \operatorname {EllipticPi}\left (\frac {361}{312},2 \arctan \left (\frac {\sqrt {1+2 x}}{\sqrt {2}}\right ),\frac {3}{4}\right )}{1651 \sqrt {3+4 x^2}} \] Output:

-1/3048*(111-670*x)*(1+2*x)^(1/2)/(4*x^2+3)^(1/2)-335*(1+2*x)^(1/2)*(4*x^2 
+3)^(1/2)/(9144+6096*x)+76/209677*4953^(1/2)*arctanh(1/39*4953^(1/2)*(1+2* 
x)^(1/2)/(4*x^2+3)^(1/2))+335/3048*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)-1/48*2^(1/2)*(3+2*x)*((4*x^2+3)/(3+2*x)^2)^(1/2)*InverseJacobiA 
M(2*arctan(1/2*(1+2*x)^(1/2)*2^(1/2)),1/2*3^(1/2))/(4*x^2+3)^(1/2)-7/1651* 
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 = 24.21 (sec) , antiderivative size = 659, normalized size of antiderivative = 2.18 \[ \int \frac {4+6 x-2 x^2}{(5-3 x) \sqrt {1+2 x} \left (3+4 x^2\right )^{3/2}} \, dx=\frac {\sqrt {1+2 x} \left (78 \sqrt {3} (-111+670 x)-\frac {3 (1+2 x) \left (8710 \sqrt {-\frac {i}{i+\sqrt {3}}}+\frac {34840 \sqrt {-\frac {i}{i+\sqrt {3}}}}{(1+2 x)^2}-\frac {17420 \sqrt {-\frac {i}{i+\sqrt {3}}}}{1+2 x}+\frac {4355 \left (-i+\sqrt {3}\right ) \sqrt {\frac {3 i+\sqrt {3}+2 \left (-i+\sqrt {3}\right ) x}{\left (-i+\sqrt {3}\right ) (1+2 x)}} \sqrt {\frac {-3 i+\sqrt {3}+2 \left (i+\sqrt {3}\right ) x}{\left (i+\sqrt {3}\right ) (1+2 x)}} E\left (i \text {arcsinh}\left (\frac {2 \sqrt {-\frac {i}{i+\sqrt {3}}}}{\sqrt {1+2 x}}\right )|\frac {i+\sqrt {3}}{i-\sqrt {3}}\right )}{\sqrt {1+2 x}}-\frac {5 \left (441 i+871 \sqrt {3}\right ) \sqrt {\frac {3 i+\sqrt {3}+2 \left (-i+\sqrt {3}\right ) x}{\left (-i+\sqrt {3}\right ) (1+2 x)}} \sqrt {\frac {-3 i+\sqrt {3}+2 \left (i+\sqrt {3}\right ) x}{\left (i+\sqrt {3}\right ) (1+2 x)}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {2 \sqrt {-\frac {i}{i+\sqrt {3}}}}{\sqrt {1+2 x}}\right ),\frac {i+\sqrt {3}}{i-\sqrt {3}}\right )}{\sqrt {1+2 x}}+\frac {3648 i \sqrt {\frac {3 i+\sqrt {3}+2 \left (-i+\sqrt {3}\right ) x}{\left (-i+\sqrt {3}\right ) (1+2 x)}} \sqrt {\frac {-3 i+\sqrt {3}+2 \left (i+\sqrt {3}\right ) x}{\left (i+\sqrt {3}\right ) (1+2 x)}} \operatorname {EllipticPi}\left (\frac {13}{12} \left (1-i \sqrt {3}\right ),i \text {arcsinh}\left (\frac {2 \sqrt {-\frac {i}{i+\sqrt {3}}}}{\sqrt {1+2 x}}\right ),\frac {i+\sqrt {3}}{i-\sqrt {3}}\right )}{\sqrt {1+2 x}}\right )}{\sqrt {-\frac {i}{3 \left (i+\sqrt {3}\right )}}}\right )}{237744 \sqrt {3} \sqrt {3+4 x^2}} \] Input:

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

Output:

(Sqrt[1 + 2*x]*(78*Sqrt[3]*(-111 + 670*x) - (3*(1 + 2*x)*(8710*Sqrt[(-I)/( 
I + Sqrt[3])] + (34840*Sqrt[(-I)/(I + Sqrt[3])])/(1 + 2*x)^2 - (17420*Sqrt 
[(-I)/(I + Sqrt[3])])/(1 + 2*x) + (4355*(-I + Sqrt[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))]*EllipticE[I*ArcSinh[(2*Sqrt[ 
(-I)/(I + Sqrt[3])])/Sqrt[1 + 2*x]], (I + Sqrt[3])/(I - Sqrt[3])])/Sqrt[1 
+ 2*x] - (5*(441*I + 871*Sqrt[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))]*EllipticF[I*ArcSinh[(2*Sqrt[(-I)/(I + Sqrt[3])])/ 
Sqrt[1 + 2*x]], (I + Sqrt[3])/(I - Sqrt[3])])/Sqrt[1 + 2*x] + ((3648*I)*Sq 
rt[(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))]*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]))/Sqrt[(-1/3*I)/(I + Sqr 
t[3])]))/(237744*Sqrt[3]*Sqrt[3 + 4*x^2])
 

Rubi [F]

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.

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

\(\Big \downarrow \) 2349

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

\(\Big \downarrow \) 686

\(\displaystyle \frac {76}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \left (4 x^2+3\right )^{3/2}}dx-\frac {1}{192} \int \frac {8 (2 x+33)}{9 \sqrt {2 x+1} \sqrt {4 x^2+3}}dx-\frac {\sqrt {2 x+1} (33-2 x)}{216 \sqrt {4 x^2+3}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {76}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \left (4 x^2+3\right )^{3/2}}dx-\frac {1}{216} \int \frac {2 x+33}{\sqrt {2 x+1} \sqrt {4 x^2+3}}dx-\frac {\sqrt {2 x+1} (33-2 x)}{216 \sqrt {4 x^2+3}}\)

\(\Big \downarrow \) 599

\(\displaystyle \frac {76}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \left (4 x^2+3\right )^{3/2}}dx+\frac {1}{432} \int -\frac {2 (2 x+33)}{\sqrt {(2 x+1)^2-2 (2 x+1)+4}}d\sqrt {2 x+1}-\frac {\sqrt {2 x+1} (33-2 x)}{216 \sqrt {4 x^2+3}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {76}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \left (4 x^2+3\right )^{3/2}}dx-\frac {1}{216} \int \frac {2 x+33}{\sqrt {(2 x+1)^2-2 (2 x+1)+4}}d\sqrt {2 x+1}-\frac {\sqrt {2 x+1} (33-2 x)}{216 \sqrt {4 x^2+3}}\)

\(\Big \downarrow \) 744

\(\displaystyle \frac {76}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \left (4 x^2+3\right )^{3/2}}dx-\frac {1}{216} \int \frac {2 x+33}{\sqrt {(2 x+1)^2-2 (2 x+1)+4}}d\sqrt {2 x+1}-\frac {\sqrt {2 x+1} (33-2 x)}{216 \sqrt {4 x^2+3}}\)

\(\Big \downarrow \) 1511

\(\displaystyle \frac {76}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \left (4 x^2+3\right )^{3/2}}dx+\frac {1}{216} \left (2 \int \frac {1-2 x}{2 \sqrt {(2 x+1)^2-2 (2 x+1)+4}}d\sqrt {2 x+1}-34 \int \frac {1}{\sqrt {(2 x+1)^2-2 (2 x+1)+4}}d\sqrt {2 x+1}\right )-\frac {\sqrt {2 x+1} (33-2 x)}{216 \sqrt {4 x^2+3}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {76}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \left (4 x^2+3\right )^{3/2}}dx+\frac {1}{216} \left (\int \frac {1-2 x}{\sqrt {(2 x+1)^2-2 (2 x+1)+4}}d\sqrt {2 x+1}-34 \int \frac {1}{\sqrt {(2 x+1)^2-2 (2 x+1)+4}}d\sqrt {2 x+1}\right )-\frac {\sqrt {2 x+1} (33-2 x)}{216 \sqrt {4 x^2+3}}\)

\(\Big \downarrow \) 1416

\(\displaystyle \frac {1}{216} \left (\int \frac {1-2 x}{\sqrt {(2 x+1)^2-2 (2 x+1)+4}}d\sqrt {2 x+1}-\frac {17 (2 x+3) \sqrt {\frac {(2 x+1)^2-2 (2 x+1)+4}{(2 x+3)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2 x+1}}{\sqrt {2}}\right ),\frac {3}{4}\right )}{\sqrt {2} \sqrt {(2 x+1)^2-2 (2 x+1)+4}}\right )+\frac {76}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \left (4 x^2+3\right )^{3/2}}dx-\frac {\sqrt {2 x+1} (33-2 x)}{216 \sqrt {4 x^2+3}}\)

\(\Big \downarrow \) 1509

\(\displaystyle \frac {76}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \left (4 x^2+3\right )^{3/2}}dx+\frac {1}{216} \left (-\frac {17 (2 x+3) \sqrt {\frac {(2 x+1)^2-2 (2 x+1)+4}{(2 x+3)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2 x+1}}{\sqrt {2}}\right ),\frac {3}{4}\right )}{\sqrt {2} \sqrt {(2 x+1)^2-2 (2 x+1)+4}}+\frac {\sqrt {2} (2 x+3) \sqrt {\frac {(2 x+1)^2-2 (2 x+1)+4}{(2 x+3)^2}} E\left (2 \arctan \left (\frac {\sqrt {2 x+1}}{\sqrt {2}}\right )|\frac {3}{4}\right )}{\sqrt {(2 x+1)^2-2 (2 x+1)+4}}-\frac {\sqrt {2 x+1} \sqrt {(2 x+1)^2-2 (2 x+1)+4}}{2 x+3}\right )-\frac {\sqrt {2 x+1} (33-2 x)}{216 \sqrt {4 x^2+3}}\)

Input:

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

Output:

$Aborted
 

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 686

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])
 

rule 744

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]
 

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 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 = 0.45 (sec) , antiderivative size = 490, normalized size of antiderivative = 1.62

method	result	size
elliptic	\(\frac {\sqrt {\left (4 x^{2}+3\right ) \left (1+2 x \right )}\, \left (-\frac {2 \left (4+8 x \right ) \left (\frac {37}{8128}-\frac {335 x}{12192}\right )}{\sqrt {\left (x^{2}+\frac {3}{4}\right ) \left (4+8 x \right )}}-\frac {37 \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 )}{508 \sqrt {8 x^{3}+4 x^{2}+6 x +3}}-\frac {335 \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 )}{762 \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 )}{1651 \sqrt {8 x^{3}+4 x^{2}+6 x +3}}\right )}{\sqrt {4 x^{2}+3}\, \sqrt {1+2 x}}\)	\(490\)
default	\(-\frac {\sqrt {1+2 x}\, \sqrt {4 x^{2}+3}\, \left (2912 i \sqrt {3}\, \sqrt {-\frac {1+2 x}{-1+i \sqrt {3}}}\, \sqrt {\frac {i \sqrt {3}-2 x}{1+i \sqrt {3}}}\, \sqrt {\frac {i \sqrt {3}+2 x}{-1+i \sqrt {3}}}\, \operatorname {EllipticF}\left (\sqrt {-\frac {1+2 x}{-1+i \sqrt {3}}}, \sqrt {-\frac {-1+i \sqrt {3}}{1+i \sqrt {3}}}\right )+3648 i \sqrt {3}\, \sqrt {-\frac {1+2 x}{-1+i \sqrt {3}}}\, \sqrt {\frac {i \sqrt {3}-2 x}{1+i \sqrt {3}}}\, \sqrt {\frac {i \sqrt {3}+2 x}{-1+i \sqrt {3}}}\, \operatorname {EllipticPi}\left (\sqrt {-\frac {1+2 x}{-1+i \sqrt {3}}}, \frac {3}{13}-\frac {3 i \sqrt {3}}{13}, \sqrt {-\frac {-1+i \sqrt {3}}{1+i \sqrt {3}}}\right )+14508 \sqrt {-\frac {1+2 x}{-1+i \sqrt {3}}}\, \sqrt {\frac {i \sqrt {3}-2 x}{1+i \sqrt {3}}}\, \sqrt {\frac {i \sqrt {3}+2 x}{-1+i \sqrt {3}}}\, \operatorname {EllipticF}\left (\sqrt {-\frac {1+2 x}{-1+i \sqrt {3}}}, \sqrt {-\frac {-1+i \sqrt {3}}{1+i \sqrt {3}}}\right )-3648 \sqrt {-\frac {1+2 x}{-1+i \sqrt {3}}}\, \sqrt {\frac {i \sqrt {3}-2 x}{1+i \sqrt {3}}}\, \sqrt {\frac {i \sqrt {3}+2 x}{-1+i \sqrt {3}}}\, \operatorname {EllipticPi}\left (\sqrt {-\frac {1+2 x}{-1+i \sqrt {3}}}, \frac {3}{13}-\frac {3 i \sqrt {3}}{13}, \sqrt {-\frac {-1+i \sqrt {3}}{1+i \sqrt {3}}}\right )-17420 \sqrt {-\frac {1+2 x}{-1+i \sqrt {3}}}\, \sqrt {\frac {i \sqrt {3}-2 x}{1+i \sqrt {3}}}\, \sqrt {\frac {i \sqrt {3}+2 x}{-1+i \sqrt {3}}}\, \operatorname {EllipticE}\left (\sqrt {-\frac {1+2 x}{-1+i \sqrt {3}}}, \sqrt {-\frac {-1+i \sqrt {3}}{1+i \sqrt {3}}}\right )-17420 x^{2}-5824 x +1443\right )}{39624 \left (8 x^{3}+4 x^{2}+6 x +3\right )}\)	\(615\)

Input:

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

Output:

((4*x^2+3)*(1+2*x))^(1/2)/(4*x^2+3)^(1/2)/(1+2*x)^(1/2)*(-2*(4+8*x)*(37/81 
28-335/12192*x)/((x^2+3/4)*(4+8*x))^(1/2)-37/508*(1/2-1/2*I*3^(1/2))*((x+1 
/2)/(1/2-1/2*I*3^(1/2)))^(1/2)*((x-1/2*I*3^(1/2))/(-1/2-1/2*I*3^(1/2)))^(1 
/2)*((x+1/2*I*3^(1/2))/(-1/2+1/2*I*3^(1/2)))^(1/2)/(8*x^3+4*x^2+6*x+3)^(1/ 
2)*EllipticF(((x+1/2)/(1/2-1/2*I*3^(1/2)))^(1/2),((-1/2+1/2*I*3^(1/2))/(-1 
/2-1/2*I*3^(1/2)))^(1/2))-335/762*(1/2-1/2*I*3^(1/2))*((x+1/2)/(1/2-1/2*I* 
3^(1/2)))^(1/2)*((x-1/2*I*3^(1/2))/(-1/2-1/2*I*3^(1/2)))^(1/2)*((x+1/2*I*3 
^(1/2))/(-1/2+1/2*I*3^(1/2)))^(1/2)/(8*x^3+4*x^2+6*x+3)^(1/2)*((-1/2-1/2*I 
*3^(1/2))*EllipticE(((x+1/2)/(1/2-1/2*I*3^(1/2)))^(1/2),((-1/2+1/2*I*3^(1/ 
2))/(-1/2-1/2*I*3^(1/2)))^(1/2))+1/2*I*3^(1/2)*EllipticF(((x+1/2)/(1/2-1/2 
*I*3^(1/2)))^(1/2),((-1/2+1/2*I*3^(1/2))/(-1/2-1/2*I*3^(1/2)))^(1/2)))+304 
/1651*(1/2-1/2*I*3^(1/2))*((x+1/2)/(1/2-1/2*I*3^(1/2)))^(1/2)*((x-1/2*I*3^ 
(1/2))/(-1/2-1/2*I*3^(1/2)))^(1/2)*((x+1/2*I*3^(1/2))/(-1/2+1/2*I*3^(1/2)) 
)^(1/2)/(8*x^3+4*x^2+6*x+3)^(1/2)*EllipticPi(((x+1/2)/(1/2-1/2*I*3^(1/2))) 
^(1/2),3/13-3/13*I*3^(1/2),((-1/2+1/2*I*3^(1/2))/(-1/2-1/2*I*3^(1/2)))^(1/ 
2)))
 

Fricas [F]

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

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

Output:

integral(2*sqrt(4*x^2 + 3)*(x^2 - 3*x - 2)*sqrt(2*x + 1)/(96*x^6 - 112*x^5 
 + 64*x^4 - 168*x^3 - 66*x^2 - 63*x - 45), x)
 

Sympy [F]

\[ \int \frac {4+6 x-2 x^2}{(5-3 x) \sqrt {1+2 x} \left (3+4 x^2\right )^{3/2}} \, dx=2 \left (\int \left (- \frac {3 x}{12 x^{3} \sqrt {2 x + 1} \sqrt {4 x^{2} + 3} - 20 x^{2} \sqrt {2 x + 1} \sqrt {4 x^{2} + 3} + 9 x \sqrt {2 x + 1} \sqrt {4 x^{2} + 3} - 15 \sqrt {2 x + 1} \sqrt {4 x^{2} + 3}}\right )\, dx + \int \frac {x^{2}}{12 x^{3} \sqrt {2 x + 1} \sqrt {4 x^{2} + 3} - 20 x^{2} \sqrt {2 x + 1} \sqrt {4 x^{2} + 3} + 9 x \sqrt {2 x + 1} \sqrt {4 x^{2} + 3} - 15 \sqrt {2 x + 1} \sqrt {4 x^{2} + 3}}\, dx + \int \left (- \frac {2}{12 x^{3} \sqrt {2 x + 1} \sqrt {4 x^{2} + 3} - 20 x^{2} \sqrt {2 x + 1} \sqrt {4 x^{2} + 3} + 9 x \sqrt {2 x + 1} \sqrt {4 x^{2} + 3} - 15 \sqrt {2 x + 1} \sqrt {4 x^{2} + 3}}\right )\, dx\right ) \] Input:

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

Output:

2*(Integral(-3*x/(12*x**3*sqrt(2*x + 1)*sqrt(4*x**2 + 3) - 20*x**2*sqrt(2* 
x + 1)*sqrt(4*x**2 + 3) + 9*x*sqrt(2*x + 1)*sqrt(4*x**2 + 3) - 15*sqrt(2*x 
 + 1)*sqrt(4*x**2 + 3)), x) + Integral(x**2/(12*x**3*sqrt(2*x + 1)*sqrt(4* 
x**2 + 3) - 20*x**2*sqrt(2*x + 1)*sqrt(4*x**2 + 3) + 9*x*sqrt(2*x + 1)*sqr 
t(4*x**2 + 3) - 15*sqrt(2*x + 1)*sqrt(4*x**2 + 3)), x) + Integral(-2/(12*x 
**3*sqrt(2*x + 1)*sqrt(4*x**2 + 3) - 20*x**2*sqrt(2*x + 1)*sqrt(4*x**2 + 3 
) + 9*x*sqrt(2*x + 1)*sqrt(4*x**2 + 3) - 15*sqrt(2*x + 1)*sqrt(4*x**2 + 3) 
), x))
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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