\(\int \frac {4+6 x-2 x^2}{(5-3 x) \sqrt {1+2 x} (2+7 x+4 x^2)^{3/2}} \, dx\) [54]

Optimal result

Integrand size = 41, antiderivative size = 354 \[ \int \frac {4+6 x-2 x^2}{(5-3 x) \sqrt {1+2 x} \left (2+7 x+4 x^2\right )^{3/2}} \, dx=-\frac {8 \sqrt {1+2 x} (328+319 x)}{3791 \sqrt {2+7 x+4 x^2}}+\frac {638 \sqrt {3+\sqrt {17}} \sqrt {1+2 x} \sqrt {-2-7 x-4 x^2} E\left (\arcsin \left (\frac {\sqrt {17+\sqrt {17} (7+8 x)}}{\sqrt {34}}\right )|\frac {1}{4} \left (17-3 \sqrt {17}\right )\right )}{3791 \sqrt {-1-2 x} \sqrt {2+7 x+4 x^2}}+\frac {1132 \sqrt {-1-2 x} \sqrt {-2-7 x-4 x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {17+\sqrt {17} (7+8 x)}}{\sqrt {34}}\right ),\frac {1}{4} \left (17-3 \sqrt {17}\right )\right )}{3791 \sqrt {3+\sqrt {17}} \sqrt {1+2 x} \sqrt {2+7 x+4 x^2}}+\frac {1216 \sqrt {-1-2 x} \sqrt {-2-7 x-4 x^2} \operatorname {EllipticPi}\left (-\frac {3 \left (51-61 \sqrt {17}\right )}{1784},\arcsin \left (\frac {\sqrt {17+\sqrt {17} (7+8 x)}}{\sqrt {34}}\right ),\frac {1}{4} \left (17-3 \sqrt {17}\right )\right )}{223 \sqrt {3+\sqrt {17}} \left (61+3 \sqrt {17}\right ) \sqrt {1+2 x} \sqrt {2+7 x+4 x^2}} \] Output:

-8/3791*(1+2*x)^(1/2)*(328+319*x)/(4*x^2+7*x+2)^(1/2)+638/3791*(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)+1132/3791*(-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)+1216/223*(-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+1 
83/1784*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 = 24.21 (sec) , antiderivative size = 421, normalized size of antiderivative = 1.19 \[ \int \frac {4+6 x-2 x^2}{(5-3 x) \sqrt {1+2 x} \left (2+7 x+4 x^2\right )^{3/2}} \, dx=\frac {2 \left (-468 \sqrt {3+\sqrt {17}}+7358 \sqrt {3+\sqrt {17}} x+4147 i \left (3+\sqrt {17}\right ) (1+2 x)^{3/2} \sqrt {\frac {2+7 x+4 x^2}{(1+2 x)^2}} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {2}{-3+\sqrt {17}}}}{\sqrt {1+2 x}}\right )|-\frac {13}{4}+\frac {3 \sqrt {17}}{4}\right )-i \left (2499+4147 \sqrt {17}\right ) (1+2 x)^{3/2} \sqrt {\frac {2+7 x+4 x^2}{(1+2 x)^2}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {2}{-3+\sqrt {17}}}}{\sqrt {1+2 x}}\right ),-\frac {13}{4}+\frac {3 \sqrt {17}}{4}\right )-2584 i \sqrt {1+2 x} \sqrt {\frac {2+7 x+4 x^2}{(1+2 x)^2}} \operatorname {EllipticPi}\left (-\frac {13}{6} \left (-3+\sqrt {17}\right ),i \text {arcsinh}\left (\frac {\sqrt {\frac {2}{-3+\sqrt {17}}}}{\sqrt {1+2 x}}\right ),\frac {1}{4} \left (-13+3 \sqrt {17}\right )\right )-5168 i x \sqrt {1+2 x} \sqrt {\frac {2+7 x+4 x^2}{(1+2 x)^2}} \operatorname {EllipticPi}\left (-\frac {13}{6} \left (-3+\sqrt {17}\right ),i \text {arcsinh}\left (\frac {\sqrt {\frac {2}{-3+\sqrt {17}}}}{\sqrt {1+2 x}}\right ),\frac {1}{4} \left (-13+3 \sqrt {17}\right )\right )\right )}{49283 \sqrt {3+\sqrt {17}} \sqrt {1+2 x} \sqrt {2+7 x+4 x^2}} \] Input:

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

Output:

(2*(-468*Sqrt[3 + Sqrt[17]] + 7358*Sqrt[3 + Sqrt[17]]*x + (4147*I)*(3 + Sq 
rt[17])*(1 + 2*x)^(3/2)*Sqrt[(2 + 7*x + 4*x^2)/(1 + 2*x)^2]*EllipticE[I*Ar 
cSinh[Sqrt[2/(-3 + Sqrt[17])]/Sqrt[1 + 2*x]], -13/4 + (3*Sqrt[17])/4] - I* 
(2499 + 4147*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*Sq 
rt[17])/4] - (2584*I)*Sqrt[1 + 2*x]*Sqrt[(2 + 7*x + 4*x^2)/(1 + 2*x)^2]*El 
lipticPi[(-13*(-3 + Sqrt[17]))/6, I*ArcSinh[Sqrt[2/(-3 + Sqrt[17])]/Sqrt[1 
 + 2*x]], (-13 + 3*Sqrt[17])/4] - (5168*I)*x*Sqrt[1 + 2*x]*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]))/(49283*Sqrt[3 + S 
qrt[17]]*Sqrt[1 + 2*x]*Sqrt[2 + 7*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.

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

\(\Big \downarrow \) 2154

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

\(\Big \downarrow \) 1235

\(\displaystyle \frac {76}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \left (4 x^2+7 x+2\right )^{3/2}}dx+\frac {1}{17} \int -\frac {2 (108 x+1)}{9 \sqrt {2 x+1} \sqrt {4 x^2+7 x+2}}dx+\frac {8 \sqrt {2 x+1} (27 x+47)}{153 \sqrt {4 x^2+7 x+2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {76}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \left (4 x^2+7 x+2\right )^{3/2}}dx-\frac {2}{153} \int \frac {108 x+1}{\sqrt {2 x+1} \sqrt {4 x^2+7 x+2}}dx+\frac {8 \sqrt {2 x+1} (27 x+47)}{153 \sqrt {4 x^2+7 x+2}}\)

\(\Big \downarrow \) 1269

\(\displaystyle \frac {76}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \left (4 x^2+7 x+2\right )^{3/2}}dx-\frac {2}{153} \left (54 \int \frac {\sqrt {2 x+1}}{\sqrt {4 x^2+7 x+2}}dx-53 \int \frac {1}{\sqrt {2 x+1} \sqrt {4 x^2+7 x+2}}dx\right )+\frac {8 \sqrt {2 x+1} (27 x+47)}{153 \sqrt {4 x^2+7 x+2}}\)

\(\Big \downarrow \) 1172

\(\displaystyle \frac {76}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \left (4 x^2+7 x+2\right )^{3/2}}dx-\frac {2}{153} \left (\frac {27 \sqrt {3+\sqrt {17}} \sqrt {2 x+1} \sqrt {-4 x^2-7 x-2} \int \frac {\sqrt {1-\frac {8 x+\sqrt {17}+7}{3+\sqrt {17}}}}{\sqrt {1-\frac {8 x+\sqrt {17}+7}{2 \sqrt {17}}}}d\frac {\sqrt {8 x+\sqrt {17}+7}}{\sqrt {2} \sqrt [4]{17}}}{\sqrt {-2 x-1} \sqrt {4 x^2+7 x+2}}-\frac {106 \sqrt {-2 x-1} \sqrt {-4 x^2-7 x-2} \int \frac {1}{\sqrt {1-\frac {8 x+\sqrt {17}+7}{2 \sqrt {17}}} \sqrt {1-\frac {8 x+\sqrt {17}+7}{3+\sqrt {17}}}}d\frac {\sqrt {8 x+\sqrt {17}+7}}{\sqrt {2} \sqrt [4]{17}}}{\sqrt {3+\sqrt {17}} \sqrt {2 x+1} \sqrt {4 x^2+7 x+2}}\right )+\frac {8 \sqrt {2 x+1} (27 x+47)}{153 \sqrt {4 x^2+7 x+2}}\)

\(\Big \downarrow \) 321

\(\displaystyle -\frac {2}{153} \left (\frac {27 \sqrt {3+\sqrt {17}} \sqrt {2 x+1} \sqrt {-4 x^2-7 x-2} \int \frac {\sqrt {1-\frac {8 x+\sqrt {17}+7}{3+\sqrt {17}}}}{\sqrt {1-\frac {8 x+\sqrt {17}+7}{2 \sqrt {17}}}}d\frac {\sqrt {8 x+\sqrt {17}+7}}{\sqrt {2} \sqrt [4]{17}}}{\sqrt {-2 x-1} \sqrt {4 x^2+7 x+2}}-\frac {106 \sqrt {-2 x-1} \sqrt {-4 x^2-7 x-2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {8 x+\sqrt {17}+7}}{\sqrt {2} \sqrt [4]{17}}\right ),\frac {1}{4} \left (17-3 \sqrt {17}\right )\right )}{\sqrt {3+\sqrt {17}} \sqrt {2 x+1} \sqrt {4 x^2+7 x+2}}\right )+\frac {76}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \left (4 x^2+7 x+2\right )^{3/2}}dx+\frac {8 \sqrt {2 x+1} (27 x+47)}{153 \sqrt {4 x^2+7 x+2}}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {76}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \left (4 x^2+7 x+2\right )^{3/2}}dx-\frac {2}{153} \left (\frac {27 \sqrt {3+\sqrt {17}} \sqrt {2 x+1} \sqrt {-4 x^2-7 x-2} E\left (\arcsin \left (\frac {\sqrt {8 x+\sqrt {17}+7}}{\sqrt {2} \sqrt [4]{17}}\right )|\frac {1}{4} \left (17-3 \sqrt {17}\right )\right )}{\sqrt {-2 x-1} \sqrt {4 x^2+7 x+2}}-\frac {106 \sqrt {-2 x-1} \sqrt {-4 x^2-7 x-2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {8 x+\sqrt {17}+7}}{\sqrt {2} \sqrt [4]{17}}\right ),\frac {1}{4} \left (17-3 \sqrt {17}\right )\right )}{\sqrt {3+\sqrt {17}} \sqrt {2 x+1} \sqrt {4 x^2+7 x+2}}\right )+\frac {8 \sqrt {2 x+1} (27 x+47)}{153 \sqrt {4 x^2+7 x+2}}\)

\(\Big \downarrow \) 1292

\(\displaystyle \frac {76}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \left (4 x^2+7 x+2\right )^{3/2}}dx-\frac {2}{153} \left (\frac {27 \sqrt {3+\sqrt {17}} \sqrt {2 x+1} \sqrt {-4 x^2-7 x-2} E\left (\arcsin \left (\frac {\sqrt {8 x+\sqrt {17}+7}}{\sqrt {2} \sqrt [4]{17}}\right )|\frac {1}{4} \left (17-3 \sqrt {17}\right )\right )}{\sqrt {-2 x-1} \sqrt {4 x^2+7 x+2}}-\frac {106 \sqrt {-2 x-1} \sqrt {-4 x^2-7 x-2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {8 x+\sqrt {17}+7}}{\sqrt {2} \sqrt [4]{17}}\right ),\frac {1}{4} \left (17-3 \sqrt {17}\right )\right )}{\sqrt {3+\sqrt {17}} \sqrt {2 x+1} \sqrt {4 x^2+7 x+2}}\right )+\frac {8 \sqrt {2 x+1} (27 x+47)}{153 \sqrt {4 x^2+7 x+2}}\)

Input:

Int[(4 + 6*x - 2*x^2)/((5 - 3*x)*Sqrt[1 + 2*x]*(2 + 7*x + 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 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 = 0.58 (sec) , antiderivative size = 497, normalized size of antiderivative = 1.40

method	result	size
elliptic	\(\frac {\sqrt {\left (4 x^{2}+7 x +2\right ) \left (1+2 x \right )}\, \left (-\frac {2 \left (4+8 x \right ) \left (\frac {328}{3791}+\frac {319 x}{3791}\right )}{\sqrt {\left (x^{2}+\frac {7}{4} x +\frac {1}{2}\right ) \left (4+8 x \right )}}+\frac {3684 \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 )}{3791 \sqrt {8 x^{3}+18 x^{2}+11 x +2}}+\frac {5104 \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 )}{3791 \sqrt {8 x^{3}+18 x^{2}+11 x +2}}+\frac {304 \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 )}{2899 \sqrt {8 x^{3}+18 x^{2}+11 x +2}}\right )}{\sqrt {4 x^{2}+7 x +2}\, \sqrt {1+2 x}}\)	\(497\)
default	\(-\frac {\sqrt {1+2 x}\, \sqrt {4 x^{2}+7 x +2}\, \left (3679 \sqrt {-\left (1+2 x \right ) \left (3+\sqrt {17}\right )}\, \sqrt {\left (-8 x -7+\sqrt {17}\right ) \left (-3+\sqrt {17}\right )}\, \sqrt {\left (8 x +7+\sqrt {17}\right ) \left (3+\sqrt {17}\right )}\, \operatorname {EllipticF}\left (\frac {2 \sqrt {-\left (1+2 x \right ) \left (3+\sqrt {17}\right )}}{3+\sqrt {17}}, \frac {i \sqrt {\left (3+\sqrt {17}\right ) \left (-3+\sqrt {17}\right )}}{-3+\sqrt {17}}\right ) \sqrt {17}+1292 \sqrt {-\left (1+2 x \right ) \left (3+\sqrt {17}\right )}\, \sqrt {\left (-8 x -7+\sqrt {17}\right ) \left (-3+\sqrt {17}\right )}\, \sqrt {\left (8 x +7+\sqrt {17}\right ) \left (3+\sqrt {17}\right )}\, \operatorname {EllipticPi}\left (\frac {2 \sqrt {-\left (1+2 x \right ) \left (3+\sqrt {17}\right )}}{3+\sqrt {17}}, -\frac {9}{52}-\frac {3 \sqrt {17}}{52}, \frac {i \sqrt {\left (3+\sqrt {17}\right ) \left (-3+\sqrt {17}\right )}}{-3+\sqrt {17}}\right ) \sqrt {17}+530816 \sqrt {17}\, x^{2}+27625 \sqrt {-\left (1+2 x \right ) \left (3+\sqrt {17}\right )}\, \sqrt {\left (-8 x -7+\sqrt {17}\right ) \left (-3+\sqrt {17}\right )}\, \sqrt {\left (8 x +7+\sqrt {17}\right ) \left (3+\sqrt {17}\right )}\, \operatorname {EllipticF}\left (\frac {2 \sqrt {-\left (1+2 x \right ) \left (3+\sqrt {17}\right )}}{3+\sqrt {17}}, \frac {i \sqrt {\left (3+\sqrt {17}\right ) \left (-3+\sqrt {17}\right )}}{-3+\sqrt {17}}\right )+3876 \sqrt {-\left (1+2 x \right ) \left (3+\sqrt {17}\right )}\, \sqrt {\left (-8 x -7+\sqrt {17}\right ) \left (-3+\sqrt {17}\right )}\, \sqrt {\left (8 x +7+\sqrt {17}\right ) \left (3+\sqrt {17}\right )}\, \operatorname {EllipticPi}\left (\frac {2 \sqrt {-\left (1+2 x \right ) \left (3+\sqrt {17}\right )}}{3+\sqrt {17}}, -\frac {9}{52}-\frac {3 \sqrt {17}}{52}, \frac {i \sqrt {\left (3+\sqrt {17}\right ) \left (-3+\sqrt {17}\right )}}{-3+\sqrt {17}}\right )-16588 \sqrt {-\left (1+2 x \right ) \left (3+\sqrt {17}\right )}\, \sqrt {\left (-8 x -7+\sqrt {17}\right ) \left (-3+\sqrt {17}\right )}\, \sqrt {\left (8 x +7+\sqrt {17}\right ) \left (3+\sqrt {17}\right )}\, \operatorname {EllipticE}\left (\frac {2 \sqrt {-\left (1+2 x \right ) \left (3+\sqrt {17}\right )}}{3+\sqrt {17}}, \frac {i \sqrt {\left (3+\sqrt {17}\right ) \left (-3+\sqrt {17}\right )}}{-3+\sqrt {17}}\right )+811200 \sqrt {17}\, x +1592448 x^{2}+272896 \sqrt {17}+2433600 x +818688\right )}{49283 \left (3+\sqrt {17}\right )^{2} \left (-3+\sqrt {17}\right ) \left (8 x^{3}+18 x^{2}+11 x +2\right )}\)	\(577\)

Input:

int((-2*x^2+6*x+4)/(5-3*x)/(1+2*x)^(1/2)/(4*x^2+7*x+2)^(3/2),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)*(-2*(4+8*x 
)*(328/3791+319/3791*x)/((x^2+7/4*x+1/2)*(4+8*x))^(1/2)+3684/3791*(-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))+5104/3791*(-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*1 
7^(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))*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)))+304/2899*(-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)*El 
lipticPi((-(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]

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

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

Output:

integral(2*sqrt(4*x^2 + 7*x + 2)*(x^2 - 3*x - 2)*sqrt(2*x + 1)/(96*x^6 + 2 
24*x^5 - 82*x^4 - 567*x^3 - 497*x^2 - 168*x - 20), x)
 

Sympy [F]

\[ \int \frac {4+6 x-2 x^2}{(5-3 x) \sqrt {1+2 x} \left (2+7 x+4 x^2\right )^{3/2}} \, dx=2 \left (\int \left (- \frac {3 x}{12 x^{3} \sqrt {2 x + 1} \sqrt {4 x^{2} + 7 x + 2} + x^{2} \sqrt {2 x + 1} \sqrt {4 x^{2} + 7 x + 2} - 29 x \sqrt {2 x + 1} \sqrt {4 x^{2} + 7 x + 2} - 10 \sqrt {2 x + 1} \sqrt {4 x^{2} + 7 x + 2}}\right )\, dx + \int \frac {x^{2}}{12 x^{3} \sqrt {2 x + 1} \sqrt {4 x^{2} + 7 x + 2} + x^{2} \sqrt {2 x + 1} \sqrt {4 x^{2} + 7 x + 2} - 29 x \sqrt {2 x + 1} \sqrt {4 x^{2} + 7 x + 2} - 10 \sqrt {2 x + 1} \sqrt {4 x^{2} + 7 x + 2}}\, dx + \int \left (- \frac {2}{12 x^{3} \sqrt {2 x + 1} \sqrt {4 x^{2} + 7 x + 2} + x^{2} \sqrt {2 x + 1} \sqrt {4 x^{2} + 7 x + 2} - 29 x \sqrt {2 x + 1} \sqrt {4 x^{2} + 7 x + 2} - 10 \sqrt {2 x + 1} \sqrt {4 x^{2} + 7 x + 2}}\right )\, dx\right ) \] Input:

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

Output:

2*(Integral(-3*x/(12*x**3*sqrt(2*x + 1)*sqrt(4*x**2 + 7*x + 2) + x**2*sqrt 
(2*x + 1)*sqrt(4*x**2 + 7*x + 2) - 29*x*sqrt(2*x + 1)*sqrt(4*x**2 + 7*x + 
2) - 10*sqrt(2*x + 1)*sqrt(4*x**2 + 7*x + 2)), x) + Integral(x**2/(12*x**3 
*sqrt(2*x + 1)*sqrt(4*x**2 + 7*x + 2) + x**2*sqrt(2*x + 1)*sqrt(4*x**2 + 7 
*x + 2) - 29*x*sqrt(2*x + 1)*sqrt(4*x**2 + 7*x + 2) - 10*sqrt(2*x + 1)*sqr 
t(4*x**2 + 7*x + 2)), x) + Integral(-2/(12*x**3*sqrt(2*x + 1)*sqrt(4*x**2 
+ 7*x + 2) + x**2*sqrt(2*x + 1)*sqrt(4*x**2 + 7*x + 2) - 29*x*sqrt(2*x + 1 
)*sqrt(4*x**2 + 7*x + 2) - 10*sqrt(2*x + 1)*sqrt(4*x**2 + 7*x + 2)), x))
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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