\(\int \frac {4+6 x-2 x^2}{(5-3 x) \sqrt {1+2 x} (2+7 x+4 x^2)^{7/2}} \, dx\) [56]

Optimal result

Integrand size = 41, antiderivative size = 418 \[ \int \frac {4+6 x-2 x^2}{(5-3 x) \sqrt {1+2 x} \left (2+7 x+4 x^2\right )^{7/2}} \, dx=-\frac {8 \sqrt {1+2 x} (328+319 x)}{18955 \left (2+7 x+4 x^2\right )^{5/2}}+\frac {4 \sqrt {1+2 x} (10115413+9883660 x)}{215575215 \left (2+7 x+4 x^2\right )^{3/2}}-\frac {8 \sqrt {1+2 x} (7809126509+7301723844 x)}{163449128013 \sqrt {2+7 x+4 x^2}}+\frac {4867815896 \sqrt {3+\sqrt {17}} \sqrt {1+2 x} \sqrt {-2-7 x-4 x^2} E\left (\arcsin \left (\frac {\sqrt {17+\sqrt {17} (7+8 x)}}{\sqrt {34}}\right )|\frac {1}{4} \left (17-3 \sqrt {17}\right )\right )}{54483042671 \sqrt {-1-2 x} \sqrt {2+7 x+4 x^2}}+\frac {21088452736 \sqrt {-1-2 x} \sqrt {-2-7 x-4 x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {17+\sqrt {17} (7+8 x)}}{\sqrt {34}}\right ),\frac {1}{4} \left (17-3 \sqrt {17}\right )\right )}{163449128013 \sqrt {3+\sqrt {17}} \sqrt {1+2 x} \sqrt {2+7 x+4 x^2}}+\frac {98496 \sqrt {-1-2 x} \sqrt {-2-7 x-4 x^2} \operatorname {EllipticPi}\left (-\frac {3 \left (51-61 \sqrt {17}\right )}{1784},\arcsin \left (\frac {\sqrt {17+\sqrt {17} (7+8 x)}}{\sqrt {34}}\right ),\frac {1}{4} \left (17-3 \sqrt {17}\right )\right )}{11089567 \sqrt {3+\sqrt {17}} \left (61+3 \sqrt {17}\right ) \sqrt {1+2 x} \sqrt {2+7 x+4 x^2}} \] Output:

-8/18955*(1+2*x)^(1/2)*(328+319*x)/(4*x^2+7*x+2)^(5/2)+4/215575215*(1+2*x) 
^(1/2)*(10115413+9883660*x)/(4*x^2+7*x+2)^(3/2)-8/163449128013*(1+2*x)^(1/ 
2)*(7809126509+7301723844*x)/(4*x^2+7*x+2)^(1/2)[phone]/54483042671*(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)[phone]/163449128013*(-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+17^(1/2))^(1/2)/(1+2*x)^(1/2)/(4*x^2+7*x+2)^(1/2)+98496/11089 
567*(-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),-153/1784+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 = 24.13 (sec) , antiderivative size = 373, normalized size of antiderivative = 0.89 \[ \int \frac {4+6 x-2 x^2}{(5-3 x) \sqrt {1+2 x} \left (2+7 x+4 x^2\right )^{7/2}} \, dx=\frac {4 \left (-13 (1+2 x) \left (263953467202+2162761148813 x+6704742099018 x^2+8969355523400 x^3+5338425594080 x^4+1168275815040 x^5\right )+\frac {10 \left (2+7 x+4 x^2\right )^2 \left (47461204986 \sqrt {3+\sqrt {17}} \left (2+7 x+4 x^2\right )[phone] i \left (3+\sqrt {17}\right ) (1+2 x)^{3/2} \sqrt {\frac {2+7 x+4 x^2}{(1+2 x)^2}} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {2}{-3+\sqrt {17}}}}{\sqrt {1+2 x}}\right )|-\frac {13}{4}+\frac {3 \sqrt {17}}{4}\right )-i \left (36877705141+23730602493 \sqrt {17}\right ) (1+2 x)^{3/2} \sqrt {\frac {2+7 x+4 x^2}{(1+2 x)^2}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {2}{-3+\sqrt {17}}}}{\sqrt {1+2 x}}\right ),-\frac {13}{4}+\frac {3 \sqrt {17}}{4}\right )-45366642 i (1+2 x)^{3/2} \sqrt {\frac {2+7 x+4 x^2}{(1+2 x)^2}} \operatorname {EllipticPi}\left (-\frac {13}{6} \left (-3+\sqrt {17}\right ),i \text {arcsinh}\left (\frac {\sqrt {\frac {2}{-3+\sqrt {17}}}}{\sqrt {1+2 x}}\right ),\frac {1}{4} \left (-13+3 \sqrt {17}\right )\right )\right )}{\sqrt {3+\sqrt {17}}}\right )}{10624193320845 \sqrt {1+2 x} \left (2+7 x+4 x^2\right )^{5/2}} \] Input:

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

Output:

(4*(-13*(1 + 2*x)*(263953467202 + 2162761148813*x + 6704742099018*x^2 + 89 
69355523400*x^3 + 5338425594080*x^4 + 1168275815040*x^5) + (10*(2 + 7*x + 
4*x^2)^2*(47461204986*Sqrt[3 + Sqrt[17]]*(2 + 7*x + 4*x^2) + (23730602493* 
I)*(3 + Sqrt[17])*(1 + 2*x)^(3/2)*Sqrt[(2 + 7*x + 4*x^2)/(1 + 2*x)^2]*Elli 
pticE[I*ArcSinh[Sqrt[2/(-3 + Sqrt[17])]/Sqrt[1 + 2*x]], -13/4 + (3*Sqrt[17 
])/4] - I*(36877705141 + 23730602493*Sqrt[17])*(1 + 2*x)^(3/2)*Sqrt[(2 + 7 
*x + 4*x^2)/(1 + 2*x)^2]*EllipticF[I*ArcSinh[Sqrt[2/(-3 + Sqrt[17])]/Sqrt[ 
1 + 2*x]], -13/4 + (3*Sqrt[17])/4] - (45366642*I)*(1 + 2*x)^(3/2)*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 + S 
qrt[17]]))/(10624193320845*Sqrt[1 + 2*x]*(2 + 7*x + 4*x^2)^(5/2))
 

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.

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

\(\Big \downarrow \) 2154

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

\(\Big \downarrow \) 1235

\(\displaystyle \frac {76}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \left (4 x^2+7 x+2\right )^{7/2}}dx+\frac {1}{85} \int \frac {2 (84 x+131)}{\sqrt {2 x+1} \left (4 x^2+7 x+2\right )^{5/2}}dx+\frac {8 \sqrt {2 x+1} (27 x+47)}{765 \left (4 x^2+7 x+2\right )^{5/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {76}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \left (4 x^2+7 x+2\right )^{7/2}}dx+\frac {2}{85} \int \frac {84 x+131}{\sqrt {2 x+1} \left (4 x^2+7 x+2\right )^{5/2}}dx+\frac {8 \sqrt {2 x+1} (27 x+47)}{765 \left (4 x^2+7 x+2\right )^{5/2}}\)

\(\Big \downarrow \) 1235

\(\displaystyle \frac {76}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \left (4 x^2+7 x+2\right )^{7/2}}dx+\frac {2}{85} \left (\frac {1}{51} \int -\frac {4 (2106 x+2681)}{\sqrt {2 x+1} \left (4 x^2+7 x+2\right )^{3/2}}dx-\frac {2 \sqrt {2 x+1} (1404 x+1985)}{51 \left (4 x^2+7 x+2\right )^{3/2}}\right )+\frac {8 \sqrt {2 x+1} (27 x+47)}{765 \left (4 x^2+7 x+2\right )^{5/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {76}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \left (4 x^2+7 x+2\right )^{7/2}}dx+\frac {2}{85} \left (-\frac {4}{51} \int \frac {2106 x+2681}{\sqrt {2 x+1} \left (4 x^2+7 x+2\right )^{3/2}}dx-\frac {2 \sqrt {2 x+1} (1404 x+1985)}{51 \left (4 x^2+7 x+2\right )^{3/2}}\right )+\frac {8 \sqrt {2 x+1} (27 x+47)}{765 \left (4 x^2+7 x+2\right )^{5/2}}\)

\(\Big \downarrow \) 1235

\(\displaystyle \frac {76}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \left (4 x^2+7 x+2\right )^{7/2}}dx+\frac {2}{85} \left (-\frac {4}{51} \left (\frac {1}{17} \int \frac {2 (27960 x+10627)}{\sqrt {2 x+1} \sqrt {4 x^2+7 x+2}}dx-\frac {2 \sqrt {2 x+1} (27960 x+38303)}{17 \sqrt {4 x^2+7 x+2}}\right )-\frac {2 \sqrt {2 x+1} (1404 x+1985)}{51 \left (4 x^2+7 x+2\right )^{3/2}}\right )+\frac {8 \sqrt {2 x+1} (27 x+47)}{765 \left (4 x^2+7 x+2\right )^{5/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {76}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \left (4 x^2+7 x+2\right )^{7/2}}dx+\frac {2}{85} \left (-\frac {4}{51} \left (\frac {2}{17} \int \frac {27960 x+10627}{\sqrt {2 x+1} \sqrt {4 x^2+7 x+2}}dx-\frac {2 \sqrt {2 x+1} (27960 x+38303)}{17 \sqrt {4 x^2+7 x+2}}\right )-\frac {2 \sqrt {2 x+1} (1404 x+1985)}{51 \left (4 x^2+7 x+2\right )^{3/2}}\right )+\frac {8 \sqrt {2 x+1} (27 x+47)}{765 \left (4 x^2+7 x+2\right )^{5/2}}\)

\(\Big \downarrow \) 1269

\(\displaystyle \frac {76}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \left (4 x^2+7 x+2\right )^{7/2}}dx+\frac {2}{85} \left (-\frac {4}{51} \left (\frac {2}{17} \left (13980 \int \frac {\sqrt {2 x+1}}{\sqrt {4 x^2+7 x+2}}dx-3353 \int \frac {1}{\sqrt {2 x+1} \sqrt {4 x^2+7 x+2}}dx\right )-\frac {2 \sqrt {2 x+1} (27960 x+38303)}{17 \sqrt {4 x^2+7 x+2}}\right )-\frac {2 \sqrt {2 x+1} (1404 x+1985)}{51 \left (4 x^2+7 x+2\right )^{3/2}}\right )+\frac {8 \sqrt {2 x+1} (27 x+47)}{765 \left (4 x^2+7 x+2\right )^{5/2}}\)

\(\Big \downarrow \) 1172

\(\displaystyle \frac {76}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \left (4 x^2+7 x+2\right )^{7/2}}dx+\frac {2}{85} \left (-\frac {4}{51} \left (\frac {2}{17} \left (\frac {6990 \sqrt {3+\sqrt {17}} \sqrt {2 x+1} \sqrt {-4 x^2-7 x-2} \int \frac {\sqrt {1-\frac {8 x+\sqrt {17}+7}{3+\sqrt {17}}}}{\sqrt {1-\frac {8 x+\sqrt {17}+7}{2 \sqrt {17}}}}d\frac {\sqrt {8 x+\sqrt {17}+7}}{\sqrt {2} \sqrt [4]{17}}}{\sqrt {-2 x-1} \sqrt {4 x^2+7 x+2}}-\frac {6706 \sqrt {-2 x-1} \sqrt {-4 x^2-7 x-2} \int \frac {1}{\sqrt {1-\frac {8 x+\sqrt {17}+7}{2 \sqrt {17}}} \sqrt {1-\frac {8 x+\sqrt {17}+7}{3+\sqrt {17}}}}d\frac {\sqrt {8 x+\sqrt {17}+7}}{\sqrt {2} \sqrt [4]{17}}}{\sqrt {3+\sqrt {17}} \sqrt {2 x+1} \sqrt {4 x^2+7 x+2}}\right )-\frac {2 \sqrt {2 x+1} (27960 x+38303)}{17 \sqrt {4 x^2+7 x+2}}\right )-\frac {2 \sqrt {2 x+1} (1404 x+1985)}{51 \left (4 x^2+7 x+2\right )^{3/2}}\right )+\frac {8 \sqrt {2 x+1} (27 x+47)}{765 \left (4 x^2+7 x+2\right )^{5/2}}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {2}{85} \left (-\frac {4}{51} \left (\frac {2}{17} \left (\frac {6990 \sqrt {3+\sqrt {17}} \sqrt {2 x+1} \sqrt {-4 x^2-7 x-2} \int \frac {\sqrt {1-\frac {8 x+\sqrt {17}+7}{3+\sqrt {17}}}}{\sqrt {1-\frac {8 x+\sqrt {17}+7}{2 \sqrt {17}}}}d\frac {\sqrt {8 x+\sqrt {17}+7}}{\sqrt {2} \sqrt [4]{17}}}{\sqrt {-2 x-1} \sqrt {4 x^2+7 x+2}}-\frac {6706 \sqrt {-2 x-1} \sqrt {-4 x^2-7 x-2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {8 x+\sqrt {17}+7}}{\sqrt {2} \sqrt [4]{17}}\right ),\frac {1}{4} \left (17-3 \sqrt {17}\right )\right )}{\sqrt {3+\sqrt {17}} \sqrt {2 x+1} \sqrt {4 x^2+7 x+2}}\right )-\frac {2 \sqrt {2 x+1} (27960 x+38303)}{17 \sqrt {4 x^2+7 x+2}}\right )-\frac {2 \sqrt {2 x+1} (1404 x+1985)}{51 \left (4 x^2+7 x+2\right )^{3/2}}\right )+\frac {76}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \left (4 x^2+7 x+2\right )^{7/2}}dx+\frac {8 \sqrt {2 x+1} (27 x+47)}{765 \left (4 x^2+7 x+2\right )^{5/2}}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {76}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \left (4 x^2+7 x+2\right )^{7/2}}dx+\frac {2}{85} \left (-\frac {4}{51} \left (\frac {2}{17} \left (\frac {6990 \sqrt {3+\sqrt {17}} \sqrt {2 x+1} \sqrt {-4 x^2-7 x-2} E\left (\arcsin \left (\frac {\sqrt {8 x+\sqrt {17}+7}}{\sqrt {2} \sqrt [4]{17}}\right )|\frac {1}{4} \left (17-3 \sqrt {17}\right )\right )}{\sqrt {-2 x-1} \sqrt {4 x^2+7 x+2}}-\frac {6706 \sqrt {-2 x-1} \sqrt {-4 x^2-7 x-2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {8 x+\sqrt {17}+7}}{\sqrt {2} \sqrt [4]{17}}\right ),\frac {1}{4} \left (17-3 \sqrt {17}\right )\right )}{\sqrt {3+\sqrt {17}} \sqrt {2 x+1} \sqrt {4 x^2+7 x+2}}\right )-\frac {2 \sqrt {2 x+1} (27960 x+38303)}{17 \sqrt {4 x^2+7 x+2}}\right )-\frac {2 \sqrt {2 x+1} (1404 x+1985)}{51 \left (4 x^2+7 x+2\right )^{3/2}}\right )+\frac {8 \sqrt {2 x+1} (27 x+47)}{765 \left (4 x^2+7 x+2\right )^{5/2}}\)

\(\Big \downarrow \) 1292

\(\displaystyle \frac {76}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \left (4 x^2+7 x+2\right )^{7/2}}dx+\frac {2}{85} \left (-\frac {4}{51} \left (\frac {2}{17} \left (\frac {6990 \sqrt {3+\sqrt {17}} \sqrt {2 x+1} \sqrt {-4 x^2-7 x-2} E\left (\arcsin \left (\frac {\sqrt {8 x+\sqrt {17}+7}}{\sqrt {2} \sqrt [4]{17}}\right )|\frac {1}{4} \left (17-3 \sqrt {17}\right )\right )}{\sqrt {-2 x-1} \sqrt {4 x^2+7 x+2}}-\frac {6706 \sqrt {-2 x-1} \sqrt {-4 x^2-7 x-2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {8 x+\sqrt {17}+7}}{\sqrt {2} \sqrt [4]{17}}\right ),\frac {1}{4} \left (17-3 \sqrt {17}\right )\right )}{\sqrt {3+\sqrt {17}} \sqrt {2 x+1} \sqrt {4 x^2+7 x+2}}\right )-\frac {2 \sqrt {2 x+1} (27960 x+38303)}{17 \sqrt {4 x^2+7 x+2}}\right )-\frac {2 \sqrt {2 x+1} (1404 x+1985)}{51 \left (4 x^2+7 x+2\right )^{3/2}}\right )+\frac {8 \sqrt {2 x+1} (27 x+47)}{765 \left (4 x^2+7 x+2\right )^{5/2}}\)

Input:

Int[(4 + 6*x - 2*x^2)/((5 - 3*x)*Sqrt[1 + 2*x]*(2 + 7*x + 4*x^2)^(7/2)),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 = 1.00 (sec) , antiderivative size = 563, normalized size of antiderivative = 1.35

method	result	size
elliptic	\(\frac {\sqrt {\left (4 x^{2}+7 x +2\right ) \left (1+2 x \right )}\, \left (\frac {\left (-\frac {41}{18955}-\frac {319 x}{151640}\right ) \sqrt {8 x^{3}+18 x^{2}+11 x +2}}{\left (x^{2}+\frac {7}{4} x +\frac {1}{2}\right )^{3}}+\frac {\left (\frac {10115413}{862300860}+\frac {494183 x}{43115043}\right ) \sqrt {8 x^{3}+18 x^{2}+11 x +2}}{\left (x^{2}+\frac {7}{4} x +\frac {1}{2}\right )^{2}}-\frac {2 \left (4+8 x \right ) \left (\frac {7809126509}{163449128013}+\frac {2433907948 x}{54483042671}\right )}{\sqrt {\left (x^{2}+\frac {7}{4} x +\frac {1}{2}\right ) \left (4+8 x \right )}}+\frac {79502243488 \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 )}{163449128013 \sqrt {8 x^{3}+18 x^{2}+11 x +2}}+\frac {38942527168 \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 )}{54483042671 \sqrt {8 x^{3}+18 x^{2}+11 x +2}}+\frac {24624 \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 )}{144164371 \sqrt {8 x^{3}+18 x^{2}+11 x +2}}\right )}{\sqrt {4 x^{2}+7 x +2}\, \sqrt {1+2 x}}\)	\(563\)
default	\(\text {Expression too large to display}\)	\(2642\)

Input:

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

Output:

((4*x^2+7*x+2)*(1+2*x))^(1/2)/(4*x^2+7*x+2)^(1/2)/(1+2*x)^(1/2)*((-41/1895 
5-319/151640*x)*(8*x^3+18*x^2+11*x+2)^(1/2)/(x^2+7/4*x+1/2)^3+(10115413/86 
2300860+494183/43115043*x)*(8*x^3+18*x^2+11*x+2)^(1/2)/(x^2+7/4*x+1/2)^2-2 
*(4+8*x)*(7809126509/163449128013+2433907948/54483042671*x)/((x^2+7/4*x+1/ 
2)*(4+8*x))^(1/2)[phone]/163449128013*(-3/8-1/8*17^(1/2))*(-(x+1/2)/( 
3/8+1/8*17^(1/2)))^(1/2)*(-(x+7/8-1/8*17^(1/2))/(-3/8+1/8*17^(1/2)))^(1/2) 
*((x+7/8+1/8*17^(1/2))/(3/8+1/8*17^(1/2)))^(1/2)/(8*x^3+18*x^2+11*x+2)^(1/ 
2)*EllipticF((-(x+1/2)/(3/8+1/8*17^(1/2)))^(1/2),I*((3/8+1/8*17^(1/2))/(-3 
/8+1/8*17^(1/2)))^(1/2))[phone]/54483042671*(-3/8-1/8*17^(1/2))*(-(x+ 
1/2)/(3/8+1/8*17^(1/2)))^(1/2)*(-(x+7/8-1/8*17^(1/2))/(-3/8+1/8*17^(1/2))) 
^(1/2)*((x+7/8+1/8*17^(1/2))/(3/8+1/8*17^(1/2)))^(1/2)/(8*x^3+18*x^2+11*x+ 
2)^(1/2)*((3/8-1/8*17^(1/2))*EllipticE((-(x+1/2)/(3/8+1/8*17^(1/2)))^(1/2) 
,I*((3/8+1/8*17^(1/2))/(-3/8+1/8*17^(1/2)))^(1/2))+(-7/8+1/8*17^(1/2))*Ell 
ipticF((-(x+1/2)/(3/8+1/8*17^(1/2)))^(1/2),I*((3/8+1/8*17^(1/2))/(-3/8+1/8 
*17^(1/2)))^(1/2)))+24624/144164371*(-3/8-1/8*17^(1/2))*(-(x+1/2)/(3/8+1/8 
*17^(1/2)))^(1/2)*(-(x+7/8-1/8*17^(1/2))/(-3/8+1/8*17^(1/2)))^(1/2)*((x+7/ 
8+1/8*17^(1/2))/(3/8+1/8*17^(1/2)))^(1/2)/(8*x^3+18*x^2+11*x+2)^(1/2)*Elli 
pticPi((-(x+1/2)/(3/8+1/8*17^(1/2)))^(1/2),-9/52-3/52*17^(1/2),I*((3/8+1/8 
*17^(1/2))/(-3/8+1/8*17^(1/2)))^(1/2)))
 

Fricas [F]

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

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

Output:

integral(2*sqrt(4*x^2 + 7*x + 2)*(x^2 - 3*x - 2)*sqrt(2*x + 1)/(1536*x^10 
+ 8960*x^9 + 17472*x^8 + 3584*x^7 - 38378*x^6 - 68775*x^5 - 58237*x^4 - 28 
224*x^3 - 7992*x^2 - 1232*x - 80), x)
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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