\(\int \frac {4+6 x-2 x^2}{(5-3 x) \sqrt {1+2 x} (3+4 x^2)^{5/2}} \, dx\) [13]

Optimal result

Integrand size = 38, antiderivative size = 331 \[ \int \frac {4+6 x-2 x^2}{(5-3 x) \sqrt {1+2 x} \left (3+4 x^2\right )^{5/2}} \, dx=-\frac {(111-670 x) \sqrt {1+2 x}}{9144 \left (3+4 x^2\right )^{3/2}}+\frac {\sqrt {1+2 x} (36501+345268 x)}{6967728 \sqrt {3+4 x^2}}-\frac {86317 \sqrt {1+2 x} \sqrt {3+4 x^2}}{3483864 (3+2 x)}+\frac {684 \sqrt {\frac {3}{1651}} \text {arctanh}\left (\frac {\sqrt {\frac {127}{39}} \sqrt {1+2 x}}{\sqrt {3+4 x^2}}\right )}{16129}+\frac {86317 (3+2 x) \sqrt {\frac {3+4 x^2}{(3+2 x)^2}} E\left (2 \arctan \left (\frac {\sqrt {1+2 x}}{\sqrt {2}}\right )|\frac {3}{4}\right )}{1741932 \sqrt {2} \sqrt {3+4 x^2}}-\frac {827 (3+2 x) \sqrt {\frac {3+4 x^2}{(3+2 x)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {1+2 x}}{\sqrt {2}}\right ),\frac {3}{4}\right )}{109728 \sqrt {2} \sqrt {3+4 x^2}}-\frac {63 \sqrt {2} (3+2 x) \sqrt {\frac {3+4 x^2}{(3+2 x)^2}} \operatorname {EllipticPi}\left (\frac {361}{312},2 \arctan \left (\frac {\sqrt {1+2 x}}{\sqrt {2}}\right ),\frac {3}{4}\right )}{209677 \sqrt {3+4 x^2}} \] Output:

-1/9144*(111-670*x)*(1+2*x)^(1/2)/(4*x^2+3)^(3/2)+1/6967728*(1+2*x)^(1/2)* 
(36501+345268*x)/(4*x^2+3)^(1/2)-86317*(1+2*x)^(1/2)*(4*x^2+3)^(1/2)/(1045 
1592+6967728*x)+684/26628979*4953^(1/2)*arctanh(1/39*4953^(1/2)*(1+2*x)^(1 
/2)/(4*x^2+3)^(1/2))+86317/3483864*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)-827/219456*2^(1/2)*(3+2*x)*((4*x^2+3)/(3+2*x)^2)^(1/2)*InverseJ 
acobiAM(2*arctan(1/2*(1+2*x)^(1/2)*2^(1/2)),1/2*3^(1/2))/(4*x^2+3)^(1/2)-6 
3/209677*2^(1/2)*(3+2*x)*((4*x^2+3)/(3+2*x)^2)^(1/2)*EllipticPi(sin(2*arct 
an(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 = 26.10 (sec) , antiderivative size = 623, normalized size of antiderivative = 1.88 \[ \int \frac {4+6 x-2 x^2}{(5-3 x) \sqrt {1+2 x} \left (3+4 x^2\right )^{5/2}} \, dx=\frac {\sqrt {1+2 x} \left (26 \left (24921+1546344 x+146004 x^2+1381072 x^3\right )+\frac {(1+2 x) \left (3+4 x^2\right ) \left (-\frac {2244242 \left (-i+\sqrt {3}\right ) \sqrt {\frac {3 i+\sqrt {3}+2 \left (-i+\sqrt {3}\right ) x}{\left (-i+\sqrt {3}\right ) (1+2 x)}} \sqrt {\frac {-3 i+\sqrt {3}+2 \left (i+\sqrt {3}\right ) x}{\left (i+\sqrt {3}\right ) (1+2 x)}} E\left (i \text {arcsinh}\left (\frac {2 \sqrt {-\frac {i}{i+\sqrt {3}}}}{\sqrt {1+2 x}}\right )|\frac {i+\sqrt {3}}{i-\sqrt {3}}\right )}{\sqrt {1+2 x}}+\frac {\left (1065489 i+2244242 \sqrt {3}\right ) \sqrt {\frac {3 i+\sqrt {3}+2 \left (-i+\sqrt {3}\right ) x}{\left (-i+\sqrt {3}\right ) (1+2 x)}} \sqrt {\frac {-3 i+\sqrt {3}+2 \left (i+\sqrt {3}\right ) x}{\left (i+\sqrt {3}\right ) (1+2 x)}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {2 \sqrt {-\frac {i}{i+\sqrt {3}}}}{\sqrt {1+2 x}}\right ),\frac {i+\sqrt {3}}{i-\sqrt {3}}\right )}{\sqrt {1+2 x}}-76 \left (\frac {59059 \sqrt {-\frac {i}{i+\sqrt {3}}} \left (3+4 x^2\right )}{(1+2 x)^2}+\frac {7776 i \sqrt {\frac {3 i+\sqrt {3}+2 \left (-i+\sqrt {3}\right ) x}{\left (-i+\sqrt {3}\right ) (1+2 x)}} \sqrt {\frac {-3 i+\sqrt {3}+2 \left (i+\sqrt {3}\right ) x}{\left (i+\sqrt {3}\right ) (1+2 x)}} \operatorname {EllipticPi}\left (\frac {13}{12} \left (1-i \sqrt {3}\right ),i \text {arcsinh}\left (\frac {2 \sqrt {-\frac {i}{i+\sqrt {3}}}}{\sqrt {1+2 x}}\right ),\frac {i+\sqrt {3}}{i-\sqrt {3}}\right )}{\sqrt {1+2 x}}\right )\right )}{\sqrt {-\frac {i}{i+\sqrt {3}}}}\right )}{181160928 \left (3+4 x^2\right )^{3/2}} \] Input:

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

Output:

(Sqrt[1 + 2*x]*(26*(24921 + 1546344*x + 146004*x^2 + 1381072*x^3) + ((1 + 
2*x)*(3 + 4*x^2)*((-2244242*(-I + Sqrt[3])*Sqrt[(3*I + Sqrt[3] + 2*(-I + S 
qrt[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 + Sq 
rt[3])])/Sqrt[1 + 2*x]], (I + Sqrt[3])/(I - Sqrt[3])])/Sqrt[1 + 2*x] + ((1 
065489*I + 2244242*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 + Sq 
rt[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] - 76*((59059*Sqrt[(- 
I)/(I + Sqrt[3])]*(3 + 4*x^2))/(1 + 2*x)^2 + ((7776*I)*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))]*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[(-I)/(I + Sqrt[3])]))/(181160928* 
(3 + 4*x^2)^(3/2))
 

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.

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

\(\Big \downarrow \) 2349

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

\(\Big \downarrow \) 686

\(\displaystyle \frac {76}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \left (4 x^2+3\right )^{5/2}}dx-\frac {1}{576} \int \frac {8 (161-6 x)}{9 \sqrt {2 x+1} \left (4 x^2+3\right )^{3/2}}dx-\frac {\sqrt {2 x+1} (33-2 x)}{648 \left (4 x^2+3\right )^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {76}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \left (4 x^2+3\right )^{5/2}}dx-\frac {1}{648} \int \frac {161-6 x}{\sqrt {2 x+1} \left (4 x^2+3\right )^{3/2}}dx-\frac {\sqrt {2 x+1} (33-2 x)}{648 \left (4 x^2+3\right )^{3/2}}\)

\(\Big \downarrow \) 686

\(\displaystyle \frac {76}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \left (4 x^2+3\right )^{5/2}}dx+\frac {1}{648} \left (\frac {1}{192} \int -\frac {32 (123-76 x)}{\sqrt {2 x+1} \sqrt {4 x^2+3}}dx-\frac {\sqrt {2 x+1} (76 x+123)}{6 \sqrt {4 x^2+3}}\right )-\frac {\sqrt {2 x+1} (33-2 x)}{648 \left (4 x^2+3\right )^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {76}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \left (4 x^2+3\right )^{5/2}}dx+\frac {1}{648} \left (-\frac {1}{6} \int \frac {123-76 x}{\sqrt {2 x+1} \sqrt {4 x^2+3}}dx-\frac {\sqrt {2 x+1} (76 x+123)}{6 \sqrt {4 x^2+3}}\right )-\frac {\sqrt {2 x+1} (33-2 x)}{648 \left (4 x^2+3\right )^{3/2}}\)

\(\Big \downarrow \) 599

\(\displaystyle \frac {76}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \left (4 x^2+3\right )^{5/2}}dx+\frac {1}{648} \left (\frac {1}{12} \int -\frac {2 (161-38 (2 x+1))}{\sqrt {(2 x+1)^2-2 (2 x+1)+4}}d\sqrt {2 x+1}-\frac {\sqrt {2 x+1} (76 x+123)}{6 \sqrt {4 x^2+3}}\right )-\frac {\sqrt {2 x+1} (33-2 x)}{648 \left (4 x^2+3\right )^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {76}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \left (4 x^2+3\right )^{5/2}}dx+\frac {1}{648} \left (-\frac {1}{6} \int \frac {161-38 (2 x+1)}{\sqrt {(2 x+1)^2-2 (2 x+1)+4}}d\sqrt {2 x+1}-\frac {\sqrt {2 x+1} (76 x+123)}{6 \sqrt {4 x^2+3}}\right )-\frac {\sqrt {2 x+1} (33-2 x)}{648 \left (4 x^2+3\right )^{3/2}}\)

\(\Big \downarrow \) 744

\(\displaystyle \frac {76}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \left (4 x^2+3\right )^{5/2}}dx+\frac {1}{648} \left (-\frac {1}{6} \int \frac {161-38 (2 x+1)}{\sqrt {(2 x+1)^2-2 (2 x+1)+4}}d\sqrt {2 x+1}-\frac {\sqrt {2 x+1} (76 x+123)}{6 \sqrt {4 x^2+3}}\right )-\frac {\sqrt {2 x+1} (33-2 x)}{648 \left (4 x^2+3\right )^{3/2}}\)

\(\Big \downarrow \) 1511

\(\displaystyle \frac {76}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \left (4 x^2+3\right )^{5/2}}dx+\frac {1}{648} \left (\frac {1}{6} \left (-85 \int \frac {1}{\sqrt {(2 x+1)^2-2 (2 x+1)+4}}d\sqrt {2 x+1}-76 \int \frac {1-2 x}{2 \sqrt {(2 x+1)^2-2 (2 x+1)+4}}d\sqrt {2 x+1}\right )-\frac {\sqrt {2 x+1} (76 x+123)}{6 \sqrt {4 x^2+3}}\right )-\frac {\sqrt {2 x+1} (33-2 x)}{648 \left (4 x^2+3\right )^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {76}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \left (4 x^2+3\right )^{5/2}}dx+\frac {1}{648} \left (\frac {1}{6} \left (-85 \int \frac {1}{\sqrt {(2 x+1)^2-2 (2 x+1)+4}}d\sqrt {2 x+1}-38 \int \frac {1-2 x}{\sqrt {(2 x+1)^2-2 (2 x+1)+4}}d\sqrt {2 x+1}\right )-\frac {\sqrt {2 x+1} (76 x+123)}{6 \sqrt {4 x^2+3}}\right )-\frac {\sqrt {2 x+1} (33-2 x)}{648 \left (4 x^2+3\right )^{3/2}}\)

\(\Big \downarrow \) 1416

\(\displaystyle \frac {1}{648} \left (\frac {1}{6} \left (-38 \int \frac {1-2 x}{\sqrt {(2 x+1)^2-2 (2 x+1)+4}}d\sqrt {2 x+1}-\frac {85 (2 x+3) \sqrt {\frac {(2 x+1)^2-2 (2 x+1)+4}{(2 x+3)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2 x+1}}{\sqrt {2}}\right ),\frac {3}{4}\right )}{2 \sqrt {2} \sqrt {(2 x+1)^2-2 (2 x+1)+4}}\right )-\frac {\sqrt {2 x+1} (76 x+123)}{6 \sqrt {4 x^2+3}}\right )+\frac {76}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \left (4 x^2+3\right )^{5/2}}dx-\frac {\sqrt {2 x+1} (33-2 x)}{648 \left (4 x^2+3\right )^{3/2}}\)

\(\Big \downarrow \) 1509

\(\displaystyle \frac {76}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \left (4 x^2+3\right )^{5/2}}dx+\frac {1}{648} \left (\frac {1}{6} \left (-\frac {85 (2 x+3) \sqrt {\frac {(2 x+1)^2-2 (2 x+1)+4}{(2 x+3)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2 x+1}}{\sqrt {2}}\right ),\frac {3}{4}\right )}{2 \sqrt {2} \sqrt {(2 x+1)^2-2 (2 x+1)+4}}-38 \left (\frac {\sqrt {2} (2 x+3) \sqrt {\frac {(2 x+1)^2-2 (2 x+1)+4}{(2 x+3)^2}} E\left (2 \arctan \left (\frac {\sqrt {2 x+1}}{\sqrt {2}}\right )|\frac {3}{4}\right )}{\sqrt {(2 x+1)^2-2 (2 x+1)+4}}-\frac {\sqrt {2 x+1} \sqrt {(2 x+1)^2-2 (2 x+1)+4}}{2 x+3}\right )\right )-\frac {\sqrt {2 x+1} (76 x+123)}{6 \sqrt {4 x^2+3}}\right )-\frac {\sqrt {2 x+1} (33-2 x)}{648 \left (4 x^2+3\right )^{3/2}}\)

Input:

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

Output:

$Aborted
 

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 = 0.62 (sec) , antiderivative size = 520, normalized size of antiderivative = 1.57

method	result	size
elliptic	\(\frac {\sqrt {\left (4 x^{2}+3\right ) \left (1+2 x \right )}\, \left (\frac {\left (-\frac {37}{48768}+\frac {335 x}{73152}\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 {12167}{18580608}-\frac {86317 x}{13935456}\right )}{\sqrt {\left (x^{2}+\frac {3}{4}\right ) \left (4+8 x \right )}}+\frac {12167 \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 )}{1161288 \sqrt {8 x^{3}+4 x^{2}+6 x +3}}-\frac {86317 \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 )}{870966 \sqrt {8 x^{3}+4 x^{2}+6 x +3}}+\frac {2736 \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 )}{209677 \sqrt {8 x^{3}+4 x^{2}+6 x +3}}\right )}{\sqrt {4 x^{2}+3}\, \sqrt {1+2 x}}\)	\(520\)
default	\(\text {Expression too large to display}\)	\(1192\)

Input:

int((-2*x^2+6*x+4)/(5-3*x)/(1+2*x)^(1/2)/(4*x^2+3)^(5/2),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)*((-37/48768+335/73 
152*x)*(8*x^3+4*x^2+6*x+3)^(1/2)/(x^2+3/4)^2-2*(4+8*x)*(-12167/18580608-86 
317/13935456*x)/((x^2+3/4)*(4+8*x))^(1/2)+12167/1161288*(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))-86317/870966*(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)))+2736/209677*(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]

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

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

Output:

integral(2*sqrt(4*x^2 + 3)*(x^2 - 3*x - 2)*sqrt(2*x + 1)/(384*x^8 - 448*x^ 
7 + 544*x^6 - 1008*x^5 - 72*x^4 - 756*x^3 - 378*x^2 - 189*x - 135), x)
 

Sympy [F(-1)]

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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