\(\int \frac {4+6 x-2 x^2}{(5-3 x) \sqrt {1+2 x} (2+5 x+4 x^2)^{5/2}} \, dx\) [67]

Optimal result

Integrand size = 41, antiderivative size = 480 \[ \int \frac {4+6 x-2 x^2}{(5-3 x) \sqrt {1+2 x} \left (2+5 x+4 x^2\right )^{5/2}} \, dx=\frac {8 \sqrt {1+2 x} (146+221 x)}{4053 \left (2+5 x+4 x^2\right )^{3/2}}+\frac {4 \sqrt {1+2 x} (1335743+1956796 x)}{5475603 \sqrt {2+5 x+4 x^2}}-\frac {3913592 \sqrt {2} \sqrt {1+2 x} \sqrt {2+5 x+4 x^2}}{5475603 \left (1+\sqrt {2} (1+2 x)\right )}+\frac {684 \sqrt {\frac {3}{2509}} \text {arctanh}\left (\frac {\sqrt {\frac {193}{39}} \sqrt {1+2 x}}{\sqrt {2+5 x+4 x^2}}\right )}{37249}+\frac {1956796\ 2^{3/4} \sqrt {\frac {2+5 x+4 x^2}{\left (1+\sqrt {2} (1+2 x)\right )^2}} \left (1+\sqrt {2} (1+2 x)\right ) E\left (2 \arctan \left (\sqrt [4]{2} \sqrt {1+2 x}\right )|\frac {1}{8} \left (4-\sqrt {2}\right )\right )}{5475603 \sqrt {2+5 x+4 x^2}}-\frac {2\ 2^{3/4} \left (119243+15571 \sqrt {2}\right ) \sqrt {\frac {2+5 x+4 x^2}{\left (1+\sqrt {2} (1+2 x)\right )^2}} \left (1+\sqrt {2} (1+2 x)\right ) \operatorname {EllipticF}\left (2 \arctan \left (\sqrt [4]{2} \sqrt {1+2 x}\right ),\frac {1}{8} \left (4-\sqrt {2}\right )\right )}{1333437 \sqrt {2+5 x+4 x^2}}-\frac {342 \sqrt [4]{2} \left (347-78 \sqrt {2}\right ) \sqrt {\frac {2+5 x+4 x^2}{\left (1+\sqrt {2} (1+2 x)\right )^2}} \left (1+\sqrt {2} (1+2 x)\right ) \operatorname {EllipticPi}\left (\frac {1}{312} \left (156+347 \sqrt {2}\right ),2 \arctan \left (\sqrt [4]{2} \sqrt {1+2 x}\right ),\frac {1}{8} \left (4-\sqrt {2}\right )\right )}{159313973 \sqrt {2+5 x+4 x^2}} \] Output:

8/4053*(1+2*x)^(1/2)*(146+221*x)/(4*x^2+5*x+2)^(3/2)+4/5475603*(1+2*x)^(1/ 
2)*(1335743+1956796*x)/(4*x^2+5*x+2)^(1/2)-3913592*2^(1/2)*(1+2*x)^(1/2)*( 
4*x^2+5*x+2)^(1/2)/(5475603+5475603*2^(1/2)*(1+2*x))+684/93457741*7527^(1/ 
2)*arctanh(1/39*7527^(1/2)*(1+2*x)^(1/2)/(4*x^2+5*x+2)^(1/2))+1956796/5475 
603*((4*x^2+5*x+2)/(1+2^(1/2)*(1+2*x))^2)^(1/2)*(1+2^(1/2)*(1+2*x))*Ellipt 
icE(sin(2*arctan(2^(1/4)*(1+2*x)^(1/2))),1/4*(8-2*2^(1/2))^(1/2))*2^(3/4)/ 
(4*x^2+5*x+2)^(1/2)-2/1333437*2^(3/4)*(119243+15571*2^(1/2))*((4*x^2+5*x+2 
)/(1+2^(1/2)*(1+2*x))^2)^(1/2)*(1+2^(1/2)*(1+2*x))*InverseJacobiAM(2*arcta 
n(2^(1/4)*(1+2*x)^(1/2)),1/4*(8-2*2^(1/2))^(1/2))/(4*x^2+5*x+2)^(1/2)-342/ 
159313973*2^(1/4)*(347-78*2^(1/2))*((4*x^2+5*x+2)/(1+2^(1/2)*(1+2*x))^2)^( 
1/2)*(1+2^(1/2)*(1+2*x))*EllipticPi(sin(2*arctan(2^(1/4)*(1+2*x)^(1/2))),1 
/2+347/312*2^(1/2),1/4*(8-2*2^(1/2))^(1/2))/(4*x^2+5*x+2)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 26.91 (sec) , antiderivative size = 638, normalized size of antiderivative = 1.33 \[ \int \frac {4+6 x-2 x^2}{(5-3 x) \sqrt {1+2 x} \left (2+5 x+4 x^2\right )^{5/2}} \, dx=2 \left (\frac {2 \sqrt {1+2 x} \left (3065978+11189449 x+15126952 x^2+7827184 x^3\right )}{5475603 \left (2+5 x+4 x^2\right )^{3/2}}-\frac {(1+2 x)^{3/2} \left (\frac {50876696 \sqrt {-\frac {i}{-i+\sqrt {7}}} \left (2+5 x+4 x^2\right )}{(1+2 x)^2}+\frac {6359587 \left (i+\sqrt {7}\right ) \sqrt {\frac {-3 i+\sqrt {7}+2 \left (-i+\sqrt {7}\right ) x}{\left (-i+\sqrt {7}\right ) (1+2 x)}} \sqrt {\frac {3 i+\sqrt {7}+2 \left (i+\sqrt {7}\right ) x}{\left (i+\sqrt {7}\right ) (1+2 x)}} E\left (i \text {arcsinh}\left (\frac {\sqrt {-\frac {2 i}{-i+\sqrt {7}}}}{\sqrt {1+2 x}}\right )|\frac {i-\sqrt {7}}{i+\sqrt {7}}\right )}{\sqrt {\frac {1}{2}+x}}-\frac {\left (3031931 i+6359587 \sqrt {7}\right ) \sqrt {\frac {-3 i+\sqrt {7}+2 \left (-i+\sqrt {7}\right ) x}{\left (-i+\sqrt {7}\right ) (1+2 x)}} \sqrt {\frac {3 i+\sqrt {7}+2 \left (i+\sqrt {7}\right ) x}{\left (i+\sqrt {7}\right ) (1+2 x)}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-\frac {2 i}{-i+\sqrt {7}}}}{\sqrt {1+2 x}}\right ),\frac {i-\sqrt {7}}{i+\sqrt {7}}\right )}{\sqrt {\frac {1}{2}+x}}+\frac {100548 i \sqrt {\frac {-3 i+\sqrt {7}+2 \left (-i+\sqrt {7}\right ) x}{\left (-i+\sqrt {7}\right ) (1+2 x)}} \sqrt {\frac {3 i+\sqrt {7}+2 \left (i+\sqrt {7}\right ) x}{\left (i+\sqrt {7}\right ) (1+2 x)}} \operatorname {EllipticPi}\left (-\frac {13}{6} \left (1+i \sqrt {7}\right ),i \text {arcsinh}\left (\frac {\sqrt {-\frac {2 i}{-i+\sqrt {7}}}}{\sqrt {1+2 x}}\right ),\frac {i-\sqrt {7}}{i+\sqrt {7}}\right )}{\sqrt {\frac {1}{2}+x}}\right )}{142365678 \sqrt {-\frac {i}{-i+\sqrt {7}}} \sqrt {2+5 x+4 x^2}}\right ) \] Input:

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

Output:

2*((2*Sqrt[1 + 2*x]*(3065978 + 11189449*x + 15126952*x^2 + 7827184*x^3))/( 
5475603*(2 + 5*x + 4*x^2)^(3/2)) - ((1 + 2*x)^(3/2)*((50876696*Sqrt[(-I)/( 
-I + Sqrt[7])]*(2 + 5*x + 4*x^2))/(1 + 2*x)^2 + (6359587*(I + Sqrt[7])*Sqr 
t[(-3*I + Sqrt[7] + 2*(-I + Sqrt[7])*x)/((-I + Sqrt[7])*(1 + 2*x))]*Sqrt[( 
3*I + Sqrt[7] + 2*(I + Sqrt[7])*x)/((I + Sqrt[7])*(1 + 2*x))]*EllipticE[I* 
ArcSinh[Sqrt[(-2*I)/(-I + Sqrt[7])]/Sqrt[1 + 2*x]], (I - Sqrt[7])/(I + Sqr 
t[7])])/Sqrt[1/2 + x] - ((3031931*I + 6359587*Sqrt[7])*Sqrt[(-3*I + Sqrt[7 
] + 2*(-I + Sqrt[7])*x)/((-I + Sqrt[7])*(1 + 2*x))]*Sqrt[(3*I + Sqrt[7] + 
2*(I + Sqrt[7])*x)/((I + Sqrt[7])*(1 + 2*x))]*EllipticF[I*ArcSinh[Sqrt[(-2 
*I)/(-I + Sqrt[7])]/Sqrt[1 + 2*x]], (I - Sqrt[7])/(I + Sqrt[7])])/Sqrt[1/2 
 + x] + ((100548*I)*Sqrt[(-3*I + Sqrt[7] + 2*(-I + Sqrt[7])*x)/((-I + Sqrt 
[7])*(1 + 2*x))]*Sqrt[(3*I + Sqrt[7] + 2*(I + Sqrt[7])*x)/((I + Sqrt[7])*( 
1 + 2*x))]*EllipticPi[(-13*(1 + I*Sqrt[7]))/6, I*ArcSinh[Sqrt[(-2*I)/(-I + 
 Sqrt[7])]/Sqrt[1 + 2*x]], (I - Sqrt[7])/(I + Sqrt[7])])/Sqrt[1/2 + x]))/( 
142365678*Sqrt[(-I)/(-I + Sqrt[7])]*Sqrt[2 + 5*x + 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.

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

\(\Big \downarrow \) 2154

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

\(\Big \downarrow \) 1235

\(\displaystyle \frac {76}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \left (4 x^2+5 x+2\right )^{5/2}}dx+\frac {1}{21} \int -\frac {2 (33-68 x)}{3 \sqrt {2 x+1} \left (4 x^2+5 x+2\right )^{3/2}}dx+\frac {8 \sqrt {2 x+1} (17 x+1)}{189 \left (4 x^2+5 x+2\right )^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {76}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \left (4 x^2+5 x+2\right )^{5/2}}dx-\frac {2}{63} \int \frac {33-68 x}{\sqrt {2 x+1} \left (4 x^2+5 x+2\right )^{3/2}}dx+\frac {8 \sqrt {2 x+1} (17 x+1)}{189 \left (4 x^2+5 x+2\right )^{3/2}}\)

\(\Big \downarrow \) 1235

\(\displaystyle \frac {76}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \left (4 x^2+5 x+2\right )^{5/2}}dx-\frac {2}{63} \left (\frac {1}{7} \int \frac {8 (135 x+143)}{\sqrt {2 x+1} \sqrt {4 x^2+5 x+2}}dx-\frac {2 \sqrt {2 x+1} (540 x+103)}{7 \sqrt {4 x^2+5 x+2}}\right )+\frac {8 \sqrt {2 x+1} (17 x+1)}{189 \left (4 x^2+5 x+2\right )^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {76}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \left (4 x^2+5 x+2\right )^{5/2}}dx-\frac {2}{63} \left (\frac {8}{7} \int \frac {135 x+143}{\sqrt {2 x+1} \sqrt {4 x^2+5 x+2}}dx-\frac {2 \sqrt {2 x+1} (540 x+103)}{7 \sqrt {4 x^2+5 x+2}}\right )+\frac {8 \sqrt {2 x+1} (17 x+1)}{189 \left (4 x^2+5 x+2\right )^{3/2}}\)

\(\Big \downarrow \) 1269

\(\displaystyle \frac {76}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \left (4 x^2+5 x+2\right )^{5/2}}dx-\frac {2}{63} \left (\frac {8}{7} \left (\frac {151}{2} \int \frac {1}{\sqrt {2 x+1} \sqrt {4 x^2+5 x+2}}dx+\frac {135}{2} \int \frac {\sqrt {2 x+1}}{\sqrt {4 x^2+5 x+2}}dx\right )-\frac {2 \sqrt {2 x+1} (540 x+103)}{7 \sqrt {4 x^2+5 x+2}}\right )+\frac {8 \sqrt {2 x+1} (17 x+1)}{189 \left (4 x^2+5 x+2\right )^{3/2}}\)

\(\Big \downarrow \) 1172

\(\displaystyle \frac {76}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \left (4 x^2+5 x+2\right )^{5/2}}dx-\frac {2}{63} \left (-\frac {2 \sqrt {2 x+1} (540 x+103)}{7 \sqrt {4 x^2+5 x+2}}+\frac {8}{7} \left (\frac {151 i \sqrt {-\frac {2 x+1}{1+i \sqrt {7}}} \int \frac {1}{\sqrt {\frac {i \left (8 x+i \sqrt {7}+5\right )}{2 \sqrt {7}}+1} \sqrt {1-\frac {i \left (8 x+i \sqrt {7}+5\right )}{i-\sqrt {7}}}}d\frac {\sqrt {-i \left (8 x+i \sqrt {7}+5\right )}}{\sqrt {2} \sqrt [4]{7}}}{\sqrt {2 x+1}}+\frac {135 i \sqrt {2 x+1} \int \frac {\sqrt {1-\frac {i \left (8 x+i \sqrt {7}+5\right )}{i-\sqrt {7}}}}{\sqrt {\frac {i \left (8 x+i \sqrt {7}+5\right )}{2 \sqrt {7}}+1}}d\frac {\sqrt {-i \left (8 x+i \sqrt {7}+5\right )}}{\sqrt {2} \sqrt [4]{7}}}{4 \sqrt {-\frac {2 x+1}{1+i \sqrt {7}}}}\right )\right )+\frac {8 \sqrt {2 x+1} (17 x+1)}{189 \left (4 x^2+5 x+2\right )^{3/2}}\)

\(\Big \downarrow \) 321

\(\displaystyle -\frac {2}{63} \left (-\frac {2 \sqrt {2 x+1} (540 x+103)}{7 \sqrt {4 x^2+5 x+2}}+\frac {8}{7} \left (\frac {135 i \sqrt {2 x+1} \int \frac {\sqrt {1-\frac {i \left (8 x+i \sqrt {7}+5\right )}{i-\sqrt {7}}}}{\sqrt {\frac {i \left (8 x+i \sqrt {7}+5\right )}{2 \sqrt {7}}+1}}d\frac {\sqrt {-i \left (8 x+i \sqrt {7}+5\right )}}{\sqrt {2} \sqrt [4]{7}}}{4 \sqrt {-\frac {2 x+1}{1+i \sqrt {7}}}}+\frac {151 i \sqrt {-\frac {2 x+1}{1+i \sqrt {7}}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {-i \left (8 x+i \sqrt {7}+5\right )}}{\sqrt {2} \sqrt [4]{7}}\right ),-\frac {2 \sqrt {7}}{i-\sqrt {7}}\right )}{\sqrt {2 x+1}}\right )\right )+\frac {76}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \left (4 x^2+5 x+2\right )^{5/2}}dx+\frac {8 \sqrt {2 x+1} (17 x+1)}{189 \left (4 x^2+5 x+2\right )^{3/2}}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {76}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \left (4 x^2+5 x+2\right )^{5/2}}dx-\frac {2}{63} \left (-\frac {2 \sqrt {2 x+1} (540 x+103)}{7 \sqrt {4 x^2+5 x+2}}+\frac {8}{7} \left (\frac {151 i \sqrt {-\frac {2 x+1}{1+i \sqrt {7}}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {-i \left (8 x+i \sqrt {7}+5\right )}}{\sqrt {2} \sqrt [4]{7}}\right ),-\frac {2 \sqrt {7}}{i-\sqrt {7}}\right )}{\sqrt {2 x+1}}+\frac {135 i \sqrt {2 x+1} E\left (\arcsin \left (\frac {\sqrt {-i \left (8 x+i \sqrt {7}+5\right )}}{\sqrt {2} \sqrt [4]{7}}\right )|-\frac {2 \sqrt {7}}{i-\sqrt {7}}\right )}{4 \sqrt {-\frac {2 x+1}{1+i \sqrt {7}}}}\right )\right )+\frac {8 \sqrt {2 x+1} (17 x+1)}{189 \left (4 x^2+5 x+2\right )^{3/2}}\)

\(\Big \downarrow \) 1292

\(\displaystyle \frac {76}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \left (4 x^2+5 x+2\right )^{5/2}}dx-\frac {2}{63} \left (-\frac {2 \sqrt {2 x+1} (540 x+103)}{7 \sqrt {4 x^2+5 x+2}}+\frac {8}{7} \left (\frac {151 i \sqrt {-\frac {2 x+1}{1+i \sqrt {7}}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {-i \left (8 x+i \sqrt {7}+5\right )}}{\sqrt {2} \sqrt [4]{7}}\right ),-\frac {2 \sqrt {7}}{i-\sqrt {7}}\right )}{\sqrt {2 x+1}}+\frac {135 i \sqrt {2 x+1} E\left (\arcsin \left (\frac {\sqrt {-i \left (8 x+i \sqrt {7}+5\right )}}{\sqrt {2} \sqrt [4]{7}}\right )|-\frac {2 \sqrt {7}}{i-\sqrt {7}}\right )}{4 \sqrt {-\frac {2 x+1}{1+i \sqrt {7}}}}\right )\right )+\frac {8 \sqrt {2 x+1} (17 x+1)}{189 \left (4 x^2+5 x+2\right )^{3/2}}\)

Input:

Int[(4 + 6*x - 2*x^2)/((5 - 3*x)*Sqrt[1 + 2*x]*(2 + 5*x + 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 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 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 1235

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(f*(b*c*d - b^2*e + 2 
*a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*e))*x)*((a 
+ b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))), x] 
 + Simp[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))   Int[(d + e*x)^m 
*(a + b*x + c*x^2)^(p + 1)*Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2*(p + m + 
 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d* 
m + b*e*m) - b*d*(3*c*d - b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - 
f*(2*c*d - b*e))*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, 
 m}, x] && LtQ[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p] 
)
 

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 1292

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x 
_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Unintegrable[(d + e*x)^m*(f + g*x)^n* 
(a + b*x + c*x^2)^p, x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, 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 = 0.88 (sec) , antiderivative size = 541, normalized size of antiderivative = 1.13

method	result	size
elliptic	\(\frac {\sqrt {\left (4 x^{2}+5 x +2\right ) \left (1+2 x \right )}\, \left (\frac {\left (\frac {73}{4053}+\frac {221 x}{8106}\right ) \sqrt {8 x^{3}+14 x^{2}+9 x +2}}{\left (x^{2}+\frac {5}{4} x +\frac {1}{2}\right )^{2}}-\frac {2 \left (4+8 x \right ) \left (-\frac {1335743}{10951206}-\frac {978398 x}{5475603}\right )}{\sqrt {\left (x^{2}+\frac {5}{4} x +\frac {1}{2}\right ) \left (4+8 x \right )}}-\frac {2960672 \left (-\frac {1}{8}-\frac {i \sqrt {7}}{8}\right ) \sqrt {\frac {x +\frac {1}{2}}{-\frac {1}{8}-\frac {i \sqrt {7}}{8}}}\, \sqrt {\frac {x +\frac {5}{8}-\frac {i \sqrt {7}}{8}}{\frac {1}{8}-\frac {i \sqrt {7}}{8}}}\, \sqrt {\frac {x +\frac {5}{8}+\frac {i \sqrt {7}}{8}}{\frac {1}{8}+\frac {i \sqrt {7}}{8}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {1}{2}}{-\frac {1}{8}-\frac {i \sqrt {7}}{8}}}, \sqrt {\frac {\frac {1}{8}+\frac {i \sqrt {7}}{8}}{\frac {1}{8}-\frac {i \sqrt {7}}{8}}}\right )}{1825201 \sqrt {8 x^{3}+14 x^{2}+9 x +2}}-\frac {15654368 \left (-\frac {1}{8}-\frac {i \sqrt {7}}{8}\right ) \sqrt {\frac {x +\frac {1}{2}}{-\frac {1}{8}-\frac {i \sqrt {7}}{8}}}\, \sqrt {\frac {x +\frac {5}{8}-\frac {i \sqrt {7}}{8}}{\frac {1}{8}-\frac {i \sqrt {7}}{8}}}\, \sqrt {\frac {x +\frac {5}{8}+\frac {i \sqrt {7}}{8}}{\frac {1}{8}+\frac {i \sqrt {7}}{8}}}\, \left (\left (\frac {1}{8}-\frac {i \sqrt {7}}{8}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +\frac {1}{2}}{-\frac {1}{8}-\frac {i \sqrt {7}}{8}}}, \sqrt {\frac {\frac {1}{8}+\frac {i \sqrt {7}}{8}}{\frac {1}{8}-\frac {i \sqrt {7}}{8}}}\right )+\left (-\frac {5}{8}+\frac {i \sqrt {7}}{8}\right ) \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {1}{2}}{-\frac {1}{8}-\frac {i \sqrt {7}}{8}}}, \sqrt {\frac {\frac {1}{8}+\frac {i \sqrt {7}}{8}}{\frac {1}{8}-\frac {i \sqrt {7}}{8}}}\right )\right )}{5475603 \sqrt {8 x^{3}+14 x^{2}+9 x +2}}+\frac {2736 \left (-\frac {1}{8}-\frac {i \sqrt {7}}{8}\right ) \sqrt {\frac {x +\frac {1}{2}}{-\frac {1}{8}-\frac {i \sqrt {7}}{8}}}\, \sqrt {\frac {x +\frac {5}{8}-\frac {i \sqrt {7}}{8}}{\frac {1}{8}-\frac {i \sqrt {7}}{8}}}\, \sqrt {\frac {x +\frac {5}{8}+\frac {i \sqrt {7}}{8}}{\frac {1}{8}+\frac {i \sqrt {7}}{8}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {x +\frac {1}{2}}{-\frac {1}{8}-\frac {i \sqrt {7}}{8}}}, -\frac {3}{52}-\frac {3 i \sqrt {7}}{52}, \sqrt {\frac {\frac {1}{8}+\frac {i \sqrt {7}}{8}}{\frac {1}{8}-\frac {i \sqrt {7}}{8}}}\right )}{484237 \sqrt {8 x^{3}+14 x^{2}+9 x +2}}\right )}{\sqrt {4 x^{2}+5 x +2}\, \sqrt {1+2 x}}\)	\(541\)
default	\(\text {Expression too large to display}\)	\(1829\)

Input:

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

Output:

((4*x^2+5*x+2)*(1+2*x))^(1/2)/(4*x^2+5*x+2)^(1/2)/(1+2*x)^(1/2)*((73/4053+ 
221/8106*x)*(8*x^3+14*x^2+9*x+2)^(1/2)/(x^2+5/4*x+1/2)^2-2*(4+8*x)*(-13357 
43/10951206-978398/5475603*x)/((x^2+5/4*x+1/2)*(4+8*x))^(1/2)-2960672/1825 
201*(-1/8-1/8*I*7^(1/2))*((x+1/2)/(-1/8-1/8*I*7^(1/2)))^(1/2)*((x+5/8-1/8* 
I*7^(1/2))/(1/8-1/8*I*7^(1/2)))^(1/2)*((x+5/8+1/8*I*7^(1/2))/(1/8+1/8*I*7^ 
(1/2)))^(1/2)/(8*x^3+14*x^2+9*x+2)^(1/2)*EllipticF(((x+1/2)/(-1/8-1/8*I*7^ 
(1/2)))^(1/2),((1/8+1/8*I*7^(1/2))/(1/8-1/8*I*7^(1/2)))^(1/2))-15654368/54 
75603*(-1/8-1/8*I*7^(1/2))*((x+1/2)/(-1/8-1/8*I*7^(1/2)))^(1/2)*((x+5/8-1/ 
8*I*7^(1/2))/(1/8-1/8*I*7^(1/2)))^(1/2)*((x+5/8+1/8*I*7^(1/2))/(1/8+1/8*I* 
7^(1/2)))^(1/2)/(8*x^3+14*x^2+9*x+2)^(1/2)*((1/8-1/8*I*7^(1/2))*EllipticE( 
((x+1/2)/(-1/8-1/8*I*7^(1/2)))^(1/2),((1/8+1/8*I*7^(1/2))/(1/8-1/8*I*7^(1/ 
2)))^(1/2))+(-5/8+1/8*I*7^(1/2))*EllipticF(((x+1/2)/(-1/8-1/8*I*7^(1/2)))^ 
(1/2),((1/8+1/8*I*7^(1/2))/(1/8-1/8*I*7^(1/2)))^(1/2)))+2736/484237*(-1/8- 
1/8*I*7^(1/2))*((x+1/2)/(-1/8-1/8*I*7^(1/2)))^(1/2)*((x+5/8-1/8*I*7^(1/2)) 
/(1/8-1/8*I*7^(1/2)))^(1/2)*((x+5/8+1/8*I*7^(1/2))/(1/8+1/8*I*7^(1/2)))^(1 
/2)/(8*x^3+14*x^2+9*x+2)^(1/2)*EllipticPi(((x+1/2)/(-1/8-1/8*I*7^(1/2)))^( 
1/2),-3/52-3/52*I*7^(1/2),((1/8+1/8*I*7^(1/2))/(1/8-1/8*I*7^(1/2)))^(1/2)) 
)
 

Fricas [F]

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

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

Output:

integral(2*sqrt(4*x^2 + 5*x + 2)*(x^2 - 3*x - 2)*sqrt(2*x + 1)/(384*x^8 + 
992*x^7 + 376*x^6 - 1782*x^5 - 3347*x^4 - 2851*x^3 - 1362*x^2 - 356*x - 40 
), x)
 

Sympy [F]

\[ \int \frac {4+6 x-2 x^2}{(5-3 x) \sqrt {1+2 x} \left (2+5 x+4 x^2\right )^{5/2}} \, dx=2 \left (\int \left (- \frac {3 x}{48 x^{5} \sqrt {2 x + 1} \sqrt {4 x^{2} + 5 x + 2} + 40 x^{4} \sqrt {2 x + 1} \sqrt {4 x^{2} + 5 x + 2} - 77 x^{3} \sqrt {2 x + 1} \sqrt {4 x^{2} + 5 x + 2} - 145 x^{2} \sqrt {2 x + 1} \sqrt {4 x^{2} + 5 x + 2} - 88 x \sqrt {2 x + 1} \sqrt {4 x^{2} + 5 x + 2} - 20 \sqrt {2 x + 1} \sqrt {4 x^{2} + 5 x + 2}}\right )\, dx + \int \frac {x^{2}}{48 x^{5} \sqrt {2 x + 1} \sqrt {4 x^{2} + 5 x + 2} + 40 x^{4} \sqrt {2 x + 1} \sqrt {4 x^{2} + 5 x + 2} - 77 x^{3} \sqrt {2 x + 1} \sqrt {4 x^{2} + 5 x + 2} - 145 x^{2} \sqrt {2 x + 1} \sqrt {4 x^{2} + 5 x + 2} - 88 x \sqrt {2 x + 1} \sqrt {4 x^{2} + 5 x + 2} - 20 \sqrt {2 x + 1} \sqrt {4 x^{2} + 5 x + 2}}\, dx + \int \left (- \frac {2}{48 x^{5} \sqrt {2 x + 1} \sqrt {4 x^{2} + 5 x + 2} + 40 x^{4} \sqrt {2 x + 1} \sqrt {4 x^{2} + 5 x + 2} - 77 x^{3} \sqrt {2 x + 1} \sqrt {4 x^{2} + 5 x + 2} - 145 x^{2} \sqrt {2 x + 1} \sqrt {4 x^{2} + 5 x + 2} - 88 x \sqrt {2 x + 1} \sqrt {4 x^{2} + 5 x + 2} - 20 \sqrt {2 x + 1} \sqrt {4 x^{2} + 5 x + 2}}\right )\, dx\right ) \] Input:

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

Output:

2*(Integral(-3*x/(48*x**5*sqrt(2*x + 1)*sqrt(4*x**2 + 5*x + 2) + 40*x**4*s 
qrt(2*x + 1)*sqrt(4*x**2 + 5*x + 2) - 77*x**3*sqrt(2*x + 1)*sqrt(4*x**2 + 
5*x + 2) - 145*x**2*sqrt(2*x + 1)*sqrt(4*x**2 + 5*x + 2) - 88*x*sqrt(2*x + 
 1)*sqrt(4*x**2 + 5*x + 2) - 20*sqrt(2*x + 1)*sqrt(4*x**2 + 5*x + 2)), x) 
+ Integral(x**2/(48*x**5*sqrt(2*x + 1)*sqrt(4*x**2 + 5*x + 2) + 40*x**4*sq 
rt(2*x + 1)*sqrt(4*x**2 + 5*x + 2) - 77*x**3*sqrt(2*x + 1)*sqrt(4*x**2 + 5 
*x + 2) - 145*x**2*sqrt(2*x + 1)*sqrt(4*x**2 + 5*x + 2) - 88*x*sqrt(2*x + 
1)*sqrt(4*x**2 + 5*x + 2) - 20*sqrt(2*x + 1)*sqrt(4*x**2 + 5*x + 2)), x) + 
 Integral(-2/(48*x**5*sqrt(2*x + 1)*sqrt(4*x**2 + 5*x + 2) + 40*x**4*sqrt( 
2*x + 1)*sqrt(4*x**2 + 5*x + 2) - 77*x**3*sqrt(2*x + 1)*sqrt(4*x**2 + 5*x 
+ 2) - 145*x**2*sqrt(2*x + 1)*sqrt(4*x**2 + 5*x + 2) - 88*x*sqrt(2*x + 1)* 
sqrt(4*x**2 + 5*x + 2) - 20*sqrt(2*x + 1)*sqrt(4*x**2 + 5*x + 2)), x))
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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