\(\int \frac {1}{(5+x)^2 \sqrt {2 x+3 x^2+x^3}} \, dx\) [8]

Optimal result

Integrand size = 22, antiderivative size = 221 \[ \int \frac {1}{(5+x)^2 \sqrt {2 x+3 x^2+x^3}} \, dx=-\frac {x (2+x)}{60 \sqrt {2 x+3 x^2+x^3}}+\frac {\sqrt {2 x+3 x^2+x^3}}{60 (5+x)}+\frac {\sqrt {x} (1+x) \sqrt {\frac {2+x}{1+x}} E\left (\arctan \left (\sqrt {x}\right )|\frac {1}{2}\right )}{30 \sqrt {2} \sqrt {2 x+3 x^2+x^3}}+\frac {9 \sqrt {x} (1+x) \sqrt {\frac {2+x}{1+x}} \operatorname {EllipticF}\left (\arctan \left (\sqrt {x}\right ),\frac {1}{2}\right )}{80 \sqrt {2} \sqrt {2 x+3 x^2+x^3}}-\frac {47 \sqrt {x} (2+x) \operatorname {EllipticPi}\left (\frac {4}{5},\arctan \left (\sqrt {x}\right ),\frac {1}{2}\right )}{1200 \sqrt {2} \sqrt {\frac {2+x}{1+x}} \sqrt {2 x+3 x^2+x^3}} \] Output:

-1/60*x*(2+x)/(x^3+3*x^2+2*x)^(1/2)+(x^3+3*x^2+2*x)^(1/2)/(300+60*x)+1/60* 
2^(1/2)*x^(1/2)*(1+x)*((2+x)/(1+x))^(1/2)*EllipticE(x^(1/2)/(1+x)^(1/2),1/ 
2*2^(1/2))/(x^3+3*x^2+2*x)^(1/2)+9/160*2^(1/2)*x^(1/2)*(1+x)*((2+x)/(1+x)) 
^(1/2)*InverseJacobiAM(arctan(x^(1/2)),1/2*2^(1/2))/(x^3+3*x^2+2*x)^(1/2)- 
47/2400*x^(1/2)*(2+x)*EllipticPi(x^(1/2)/(1+x)^(1/2),4/5,1/2*2^(1/2))*2^(1 
/2)/((2+x)/(1+x))^(1/2)/(x^3+3*x^2+2*x)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 23.10 (sec) , antiderivative size = 209, normalized size of antiderivative = 0.95 \[ \int \frac {1}{(5+x)^2 \sqrt {2 x+3 x^2+x^3}} \, dx=\frac {x \left (-30-\frac {20}{x}-10 x+\frac {10 \left (2+3 x+x^2\right )}{5+x}-10 i \sqrt {2} \sqrt {1+\frac {1}{x}} \sqrt {x} \sqrt {\frac {2+x}{x}} E\left (i \text {arcsinh}\left (\frac {\sqrt {2}}{\sqrt {x}}\right )|\frac {1}{2}\right )+32 i \sqrt {2} \sqrt {1+\frac {1}{x}} \sqrt {x} \sqrt {\frac {2+x}{x}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {2}}{\sqrt {x}}\right ),\frac {1}{2}\right )-47 i \sqrt {2} \sqrt {1+\frac {1}{x}} \sqrt {x} \sqrt {\frac {2+x}{x}} \operatorname {EllipticPi}\left (\frac {5}{2},i \text {arcsinh}\left (\frac {\sqrt {2}}{\sqrt {x}}\right ),\frac {1}{2}\right )\right )}{600 \sqrt {x \left (2+3 x+x^2\right )}} \] Input:

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

Output:

(x*(-30 - 20/x - 10*x + (10*(2 + 3*x + x^2))/(5 + x) - (10*I)*Sqrt[2]*Sqrt 
[1 + x^(-1)]*Sqrt[x]*Sqrt[(2 + x)/x]*EllipticE[I*ArcSinh[Sqrt[2]/Sqrt[x]], 
 1/2] + (32*I)*Sqrt[2]*Sqrt[1 + x^(-1)]*Sqrt[x]*Sqrt[(2 + x)/x]*EllipticF[ 
I*ArcSinh[Sqrt[2]/Sqrt[x]], 1/2] - (47*I)*Sqrt[2]*Sqrt[1 + x^(-1)]*Sqrt[x] 
*Sqrt[(2 + x)/x]*EllipticPi[5/2, I*ArcSinh[Sqrt[2]/Sqrt[x]], 1/2]))/(600*S 
qrt[x*(2 + 3*x + x^2)])
 

Rubi [A] (verified)

Time = 0.78 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.25, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.545, Rules used = {2467, 1282, 2035, 2234, 1503, 1412, 1455, 1538, 27, 1412, 1786, 414}

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.

\(\displaystyle \int \frac {1}{(x+5)^2 \sqrt {x^3+3 x^2+2 x}} \, dx\)

\(\Big \downarrow \) 2467

\(\displaystyle \frac {\sqrt {x} \sqrt {x^2+3 x+2} \int \frac {1}{\sqrt {x} (x+5)^2 \sqrt {x^2+3 x+2}}dx}{\sqrt {x^3+3 x^2+2 x}}\)

\(\Big \downarrow \) 1282

\(\displaystyle \frac {\sqrt {x} \sqrt {x^2+3 x+2} \left (\frac {1}{120} \int \frac {-x^2-10 x+22}{\sqrt {x} (x+5) \sqrt {x^2+3 x+2}}dx+\frac {\sqrt {x} \sqrt {x^2+3 x+2}}{60 (x+5)}\right )}{\sqrt {x^3+3 x^2+2 x}}\)

\(\Big \downarrow \) 2035

\(\displaystyle \frac {\sqrt {x} \sqrt {x^2+3 x+2} \left (\frac {1}{60} \int \frac {-x^2-10 x+22}{(x+5) \sqrt {x^2+3 x+2}}d\sqrt {x}+\frac {\sqrt {x} \sqrt {x^2+3 x+2}}{60 (x+5)}\right )}{\sqrt {x^3+3 x^2+2 x}}\)

\(\Big \downarrow \) 2234

\(\displaystyle \frac {\sqrt {x} \sqrt {x^2+3 x+2} \left (\frac {1}{60} \left (47 \int \frac {1}{(x+5) \sqrt {x^2+3 x+2}}d\sqrt {x}-\int \frac {x+5}{\sqrt {x^2+3 x+2}}d\sqrt {x}\right )+\frac {\sqrt {x} \sqrt {x^2+3 x+2}}{60 (x+5)}\right )}{\sqrt {x^3+3 x^2+2 x}}\)

\(\Big \downarrow \) 1503

\(\displaystyle \frac {\sqrt {x} \sqrt {x^2+3 x+2} \left (\frac {1}{60} \left (-5 \int \frac {1}{\sqrt {x^2+3 x+2}}d\sqrt {x}-\int \frac {x}{\sqrt {x^2+3 x+2}}d\sqrt {x}+47 \int \frac {1}{(x+5) \sqrt {x^2+3 x+2}}d\sqrt {x}\right )+\frac {\sqrt {x} \sqrt {x^2+3 x+2}}{60 (x+5)}\right )}{\sqrt {x^3+3 x^2+2 x}}\)

\(\Big \downarrow \) 1412

\(\displaystyle \frac {\sqrt {x} \sqrt {x^2+3 x+2} \left (\frac {1}{60} \left (-\int \frac {x}{\sqrt {x^2+3 x+2}}d\sqrt {x}+47 \int \frac {1}{(x+5) \sqrt {x^2+3 x+2}}d\sqrt {x}-\frac {5 (x+1) \sqrt {\frac {x+2}{x+1}} \operatorname {EllipticF}\left (\arctan \left (\sqrt {x}\right ),\frac {1}{2}\right )}{\sqrt {2} \sqrt {x^2+3 x+2}}\right )+\frac {\sqrt {x} \sqrt {x^2+3 x+2}}{60 (x+5)}\right )}{\sqrt {x^3+3 x^2+2 x}}\)

\(\Big \downarrow \) 1455

\(\displaystyle \frac {\sqrt {x} \sqrt {x^2+3 x+2} \left (\frac {1}{60} \left (47 \int \frac {1}{(x+5) \sqrt {x^2+3 x+2}}d\sqrt {x}-\frac {5 (x+1) \sqrt {\frac {x+2}{x+1}} \operatorname {EllipticF}\left (\arctan \left (\sqrt {x}\right ),\frac {1}{2}\right )}{\sqrt {2} \sqrt {x^2+3 x+2}}+\frac {\sqrt {2} (x+1) \sqrt {\frac {x+2}{x+1}} E\left (\arctan \left (\sqrt {x}\right )|\frac {1}{2}\right )}{\sqrt {x^2+3 x+2}}-\frac {\sqrt {x} (x+2)}{\sqrt {x^2+3 x+2}}\right )+\frac {\sqrt {x} \sqrt {x^2+3 x+2}}{60 (x+5)}\right )}{\sqrt {x^3+3 x^2+2 x}}\)

\(\Big \downarrow \) 1538

\(\displaystyle \frac {\sqrt {x} \sqrt {x^2+3 x+2} \left (\frac {1}{60} \left (47 \left (\frac {1}{4} \int \frac {1}{\sqrt {x^2+3 x+2}}d\sqrt {x}-\frac {1}{8} \int \frac {2 (x+1)}{(x+5) \sqrt {x^2+3 x+2}}d\sqrt {x}\right )-\frac {5 (x+1) \sqrt {\frac {x+2}{x+1}} \operatorname {EllipticF}\left (\arctan \left (\sqrt {x}\right ),\frac {1}{2}\right )}{\sqrt {2} \sqrt {x^2+3 x+2}}+\frac {\sqrt {2} (x+1) \sqrt {\frac {x+2}{x+1}} E\left (\arctan \left (\sqrt {x}\right )|\frac {1}{2}\right )}{\sqrt {x^2+3 x+2}}-\frac {\sqrt {x} (x+2)}{\sqrt {x^2+3 x+2}}\right )+\frac {\sqrt {x} \sqrt {x^2+3 x+2}}{60 (x+5)}\right )}{\sqrt {x^3+3 x^2+2 x}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {x} \sqrt {x^2+3 x+2} \left (\frac {1}{60} \left (47 \left (\frac {1}{4} \int \frac {1}{\sqrt {x^2+3 x+2}}d\sqrt {x}-\frac {1}{4} \int \frac {x+1}{(x+5) \sqrt {x^2+3 x+2}}d\sqrt {x}\right )-\frac {5 (x+1) \sqrt {\frac {x+2}{x+1}} \operatorname {EllipticF}\left (\arctan \left (\sqrt {x}\right ),\frac {1}{2}\right )}{\sqrt {2} \sqrt {x^2+3 x+2}}+\frac {\sqrt {2} (x+1) \sqrt {\frac {x+2}{x+1}} E\left (\arctan \left (\sqrt {x}\right )|\frac {1}{2}\right )}{\sqrt {x^2+3 x+2}}-\frac {\sqrt {x} (x+2)}{\sqrt {x^2+3 x+2}}\right )+\frac {\sqrt {x} \sqrt {x^2+3 x+2}}{60 (x+5)}\right )}{\sqrt {x^3+3 x^2+2 x}}\)

\(\Big \downarrow \) 1412

\(\displaystyle \frac {\sqrt {x} \sqrt {x^2+3 x+2} \left (\frac {1}{60} \left (47 \left (\frac {(x+1) \sqrt {\frac {x+2}{x+1}} \operatorname {EllipticF}\left (\arctan \left (\sqrt {x}\right ),\frac {1}{2}\right )}{4 \sqrt {2} \sqrt {x^2+3 x+2}}-\frac {1}{4} \int \frac {x+1}{(x+5) \sqrt {x^2+3 x+2}}d\sqrt {x}\right )-\frac {5 (x+1) \sqrt {\frac {x+2}{x+1}} \operatorname {EllipticF}\left (\arctan \left (\sqrt {x}\right ),\frac {1}{2}\right )}{\sqrt {2} \sqrt {x^2+3 x+2}}+\frac {\sqrt {2} (x+1) \sqrt {\frac {x+2}{x+1}} E\left (\arctan \left (\sqrt {x}\right )|\frac {1}{2}\right )}{\sqrt {x^2+3 x+2}}-\frac {\sqrt {x} (x+2)}{\sqrt {x^2+3 x+2}}\right )+\frac {\sqrt {x} \sqrt {x^2+3 x+2}}{60 (x+5)}\right )}{\sqrt {x^3+3 x^2+2 x}}\)

\(\Big \downarrow \) 1786

\(\displaystyle \frac {\sqrt {x} \sqrt {x^2+3 x+2} \left (\frac {1}{60} \left (47 \left (\frac {(x+1) \sqrt {\frac {x+2}{x+1}} \operatorname {EllipticF}\left (\arctan \left (\sqrt {x}\right ),\frac {1}{2}\right )}{4 \sqrt {2} \sqrt {x^2+3 x+2}}-\frac {\sqrt {x+1} \sqrt {x+2} \int \frac {\sqrt {x+1}}{\sqrt {x+2} (x+5)}d\sqrt {x}}{4 \sqrt {x^2+3 x+2}}\right )-\frac {5 (x+1) \sqrt {\frac {x+2}{x+1}} \operatorname {EllipticF}\left (\arctan \left (\sqrt {x}\right ),\frac {1}{2}\right )}{\sqrt {2} \sqrt {x^2+3 x+2}}+\frac {\sqrt {2} (x+1) \sqrt {\frac {x+2}{x+1}} E\left (\arctan \left (\sqrt {x}\right )|\frac {1}{2}\right )}{\sqrt {x^2+3 x+2}}-\frac {\sqrt {x} (x+2)}{\sqrt {x^2+3 x+2}}\right )+\frac {\sqrt {x} \sqrt {x^2+3 x+2}}{60 (x+5)}\right )}{\sqrt {x^3+3 x^2+2 x}}\)

\(\Big \downarrow \) 414

\(\displaystyle \frac {\sqrt {x} \sqrt {x^2+3 x+2} \left (\frac {1}{60} \left (-\frac {5 (x+1) \sqrt {\frac {x+2}{x+1}} \operatorname {EllipticF}\left (\arctan \left (\sqrt {x}\right ),\frac {1}{2}\right )}{\sqrt {2} \sqrt {x^2+3 x+2}}+\frac {\sqrt {2} (x+1) \sqrt {\frac {x+2}{x+1}} E\left (\arctan \left (\sqrt {x}\right )|\frac {1}{2}\right )}{\sqrt {x^2+3 x+2}}+47 \left (\frac {(x+1) \sqrt {\frac {x+2}{x+1}} \operatorname {EllipticF}\left (\arctan \left (\sqrt {x}\right ),\frac {1}{2}\right )}{4 \sqrt {2} \sqrt {x^2+3 x+2}}-\frac {(x+2) \operatorname {EllipticPi}\left (\frac {4}{5},\arctan \left (\sqrt {x}\right ),\frac {1}{2}\right )}{20 \sqrt {2} \sqrt {\frac {x+2}{x+1}} \sqrt {x^2+3 x+2}}\right )-\frac {\sqrt {x} (x+2)}{\sqrt {x^2+3 x+2}}\right )+\frac {\sqrt {x} \sqrt {x^2+3 x+2}}{60 (x+5)}\right )}{\sqrt {x^3+3 x^2+2 x}}\)

Input:

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

Output:

(Sqrt[x]*Sqrt[2 + 3*x + x^2]*((Sqrt[x]*Sqrt[2 + 3*x + x^2])/(60*(5 + x)) + 
 (-((Sqrt[x]*(2 + x))/Sqrt[2 + 3*x + x^2]) + (Sqrt[2]*(1 + x)*Sqrt[(2 + x) 
/(1 + x)]*EllipticE[ArcTan[Sqrt[x]], 1/2])/Sqrt[2 + 3*x + x^2] - (5*(1 + x 
)*Sqrt[(2 + x)/(1 + x)]*EllipticF[ArcTan[Sqrt[x]], 1/2])/(Sqrt[2]*Sqrt[2 + 
 3*x + x^2]) + 47*(((1 + x)*Sqrt[(2 + x)/(1 + x)]*EllipticF[ArcTan[Sqrt[x] 
], 1/2])/(4*Sqrt[2]*Sqrt[2 + 3*x + x^2]) - ((2 + x)*EllipticPi[4/5, ArcTan 
[Sqrt[x]], 1/2])/(20*Sqrt[2]*Sqrt[(2 + x)/(1 + x)]*Sqrt[2 + 3*x + x^2])))/ 
60))/Sqrt[2*x + 3*x^2 + x^3]
 

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 414

Int[Sqrt[(c_) + (d_.)*(x_)^2]/(((a_) + (b_.)*(x_)^2)*Sqrt[(e_) + (f_.)*(x_) 
^2]), x_Symbol] :> Simp[c*(Sqrt[e + f*x^2]/(a*e*Rt[d/c, 2]*Sqrt[c + d*x^2]* 
Sqrt[c*((e + f*x^2)/(e*(c + d*x^2)))]))*EllipticPi[1 - b*(c/(a*d)), ArcTan[ 
Rt[d/c, 2]*x], 1 - c*(f/(d*e))], x] /; FreeQ[{a, b, c, d, e, f}, x] && PosQ 
[d/c]
 

rule 1282

Int[((d_.) + (e_.)*(x_))^(m_)/(Sqrt[(f_.) + (g_.)*(x_)]*Sqrt[(a_.) + (b_.)* 
(x_) + (c_.)*(x_)^2]), x_Symbol] :> Simp[e^2*(d + e*x)^(m + 1)*Sqrt[f + g*x 
]*(Sqrt[a + b*x + c*x^2]/((m + 1)*(e*f - d*g)*(c*d^2 - b*d*e + a*e^2))), x] 
 + Simp[1/(2*(m + 1)*(e*f - d*g)*(c*d^2 - b*d*e + a*e^2))   Int[((d + e*x)^ 
(m + 1)/(Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2]))*Simp[2*d*(c*e*f - c*d*g + b* 
e*g)*(m + 1) - e^2*(b*f + a*g)*(2*m + 3) + 2*e*(c*d*g*(m + 1) - e*(c*f + b* 
g)*(m + 2))*x - c*e^2*g*(2*m + 5)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, 
f, g}, x] && IntegerQ[2*m] && LeQ[m, -2]
 

rule 1412

Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b 
^2 - 4*a*c, 2]}, Simp[(2*a + (b + q)*x^2)*(Sqrt[(2*a + (b - q)*x^2)/(2*a + 
(b + q)*x^2)]/(2*a*Rt[(b + q)/(2*a), 2]*Sqrt[a + b*x^2 + c*x^4]))*EllipticF 
[ArcTan[Rt[(b + q)/(2*a), 2]*x], 2*(q/(b + q))], x] /; PosQ[(b + q)/a] && 
!(PosQ[(b - q)/a] && SimplerSqrtQ[(b - q)/(2*a), (b + q)/(2*a)])] /; FreeQ[ 
{a, b, c}, x] && GtQ[b^2 - 4*a*c, 0]
 

rule 1455

Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = 
 Rt[b^2 - 4*a*c, 2]}, Simp[x*((b + q + 2*c*x^2)/(2*c*Sqrt[a + b*x^2 + c*x^4 
])), x] - Simp[Rt[(b + q)/(2*a), 2]*(2*a + (b + q)*x^2)*(Sqrt[(2*a + (b - q 
)*x^2)/(2*a + (b + q)*x^2)]/(2*c*Sqrt[a + b*x^2 + c*x^4]))*EllipticE[ArcTan 
[Rt[(b + q)/(2*a), 2]*x], 2*(q/(b + q))], x] /; PosQ[(b + q)/a] &&  !(PosQ[ 
(b - q)/a] && SimplerSqrtQ[(b - q)/(2*a), (b + q)/(2*a)])] /; FreeQ[{a, b, 
c}, x] && GtQ[b^2 - 4*a*c, 0]
 

rule 1503

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo 
l] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[d   Int[1/Sqrt[a + b*x^2 + c*x^4] 
, x], x] + Simp[e   Int[x^2/Sqrt[a + b*x^2 + c*x^4], x], x] /; PosQ[(b + q) 
/a] || PosQ[(b - q)/a]] /; FreeQ[{a, b, c, d, e}, x] && GtQ[b^2 - 4*a*c, 0]
 

rule 1538

Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4]), x_S 
ymbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[2*(c/(2*c*d - e*(b - q)))   I 
nt[1/Sqrt[a + b*x^2 + c*x^4], x], x] - Simp[e/(2*c*d - e*(b - q))   Int[(b 
- q + 2*c*x^2)/((d + e*x^2)*Sqrt[a + b*x^2 + c*x^4]), x], x]] /; FreeQ[{a, 
b, c, d, e}, x] && GtQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && 
 !LtQ[c, 0]
 

rule 1786

Int[((d_) + (e_.)*(x_)^(n_))^(q_.)*((f_) + (g_.)*(x_)^(n_))^(r_.)*((a_) + ( 
b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_))^(p_), x_Symbol] :> Simp[(a + b*x^n + c*x 
^(2*n))^FracPart[p]/((d + e*x^n)^FracPart[p]*(a/d + (c*x^n)/e)^FracPart[p]) 
   Int[(d + e*x^n)^(p + q)*(f + g*x^n)^r*(a/d + (c/e)*x^n)^p, x], x] /; Fre 
eQ[{a, b, c, d, e, f, g, n, p, q, r}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c 
, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p]
 

rule 2035

Int[(Fx_)*(x_)^(m_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k   Subst 
[Int[x^(k*(m + 1) - 1)*SubstPower[Fx, x, k], x], x, x^(1/k)], x]] /; Fracti 
onQ[m] && AlgebraicFunctionQ[Fx, x]
 

rule 2234

Int[(P4x_)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4]) 
, x_Symbol] :> With[{A = Coeff[P4x, x, 0], B = Coeff[P4x, x, 2], C = Coeff[ 
P4x, x, 4]}, Simp[-(e^2)^(-1)   Int[(C*d - B*e - C*e*x^2)/Sqrt[a + b*x^2 + 
c*x^4], x], x] + Simp[(C*d^2 - B*d*e + A*e^2)/e^2   Int[1/((d + e*x^2)*Sqrt 
[a + b*x^2 + c*x^4]), x], x]] /; FreeQ[{a, b, c, d, e}, x] && PolyQ[P4x, x^ 
2, 2] && NeQ[c*d^2 - a*e^2, 0]
 

rule 2467

Int[(Fx_.)*(Px_)^(p_), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[Px^F 
racPart[p]/(x^(r*FracPart[p])*ExpandToSum[Px/x^r, x]^FracPart[p])   Int[x^( 
p*r)*ExpandToSum[Px/x^r, x]^p*Fx, x], x] /; IGtQ[r, 0]] /; FreeQ[p, x] && P 
olyQ[Px, x] &&  !IntegerQ[p] &&  !MonomialQ[Px, x] &&  !PolyQ[Fx, x]
 

Maple [A] (verified)

Time = 0.47 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.84

method	result	size
default	\(\frac {\sqrt {x^{3}+3 x^{2}+2 x}}{300+60 x}-\frac {\sqrt {2 x +4}\, \sqrt {-1-x}\, \sqrt {-2 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {2 x +4}}{2}, \sqrt {2}\right )}{24 \sqrt {x^{3}+3 x^{2}+2 x}}-\frac {\sqrt {2 x +4}\, \sqrt {-1-x}\, \sqrt {-2 x}\, \left (-\operatorname {EllipticE}\left (\frac {\sqrt {2 x +4}}{2}, \sqrt {2}\right )-\operatorname {EllipticF}\left (\frac {\sqrt {2 x +4}}{2}, \sqrt {2}\right )\right )}{120 \sqrt {x^{3}+3 x^{2}+2 x}}+\frac {47 \sqrt {2 x +4}\, \sqrt {-1-x}\, \sqrt {-2 x}\, \operatorname {EllipticPi}\left (\frac {\sqrt {2 x +4}}{2}, -\frac {2}{3}, \sqrt {2}\right )}{360 \sqrt {x^{3}+3 x^{2}+2 x}}\)	\(186\)
elliptic	\(\frac {\sqrt {x^{3}+3 x^{2}+2 x}}{300+60 x}-\frac {\sqrt {2 x +4}\, \sqrt {-1-x}\, \sqrt {-2 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {2 x +4}}{2}, \sqrt {2}\right )}{24 \sqrt {x^{3}+3 x^{2}+2 x}}-\frac {\sqrt {2 x +4}\, \sqrt {-1-x}\, \sqrt {-2 x}\, \left (-\operatorname {EllipticE}\left (\frac {\sqrt {2 x +4}}{2}, \sqrt {2}\right )-\operatorname {EllipticF}\left (\frac {\sqrt {2 x +4}}{2}, \sqrt {2}\right )\right )}{120 \sqrt {x^{3}+3 x^{2}+2 x}}+\frac {47 \sqrt {2 x +4}\, \sqrt {-1-x}\, \sqrt {-2 x}\, \operatorname {EllipticPi}\left (\frac {\sqrt {2 x +4}}{2}, -\frac {2}{3}, \sqrt {2}\right )}{360 \sqrt {x^{3}+3 x^{2}+2 x}}\)	\(186\)
risch	\(\frac {x \left (x^{2}+3 x +2\right )}{60 \left (5+x \right ) \sqrt {x \left (x^{2}+3 x +2\right )}}-\frac {\sqrt {2 x +4}\, \sqrt {-1-x}\, \sqrt {-2 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {2 x +4}}{2}, \sqrt {2}\right )}{24 \sqrt {x^{3}+3 x^{2}+2 x}}-\frac {\sqrt {2 x +4}\, \sqrt {-1-x}\, \sqrt {-2 x}\, \left (-\operatorname {EllipticE}\left (\frac {\sqrt {2 x +4}}{2}, \sqrt {2}\right )-\operatorname {EllipticF}\left (\frac {\sqrt {2 x +4}}{2}, \sqrt {2}\right )\right )}{120 \sqrt {x^{3}+3 x^{2}+2 x}}+\frac {47 \sqrt {2 x +4}\, \sqrt {-1-x}\, \sqrt {-2 x}\, \operatorname {EllipticPi}\left (\frac {\sqrt {2 x +4}}{2}, -\frac {2}{3}, \sqrt {2}\right )}{360 \sqrt {x^{3}+3 x^{2}+2 x}}\)	\(193\)

Input:

int(1/(5+x)^2/(x^3+3*x^2+2*x)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

1/60*(x^3+3*x^2+2*x)^(1/2)/(5+x)-1/24*(2*x+4)^(1/2)*(-1-x)^(1/2)*(-2*x)^(1 
/2)/(x^3+3*x^2+2*x)^(1/2)*EllipticF(1/2*(2*x+4)^(1/2),2^(1/2))-1/120*(2*x+ 
4)^(1/2)*(-1-x)^(1/2)*(-2*x)^(1/2)/(x^3+3*x^2+2*x)^(1/2)*(-EllipticE(1/2*( 
2*x+4)^(1/2),2^(1/2))-EllipticF(1/2*(2*x+4)^(1/2),2^(1/2)))+47/360*(2*x+4) 
^(1/2)*(-1-x)^(1/2)*(-2*x)^(1/2)/(x^3+3*x^2+2*x)^(1/2)*EllipticPi(1/2*(2*x 
+4)^(1/2),-2/3,2^(1/2))
 

Fricas [F]

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

integrate(1/(5+x)^2/(x^3+3*x^2+2*x)^(1/2),x, algorithm="fricas")
 

Output:

integral(sqrt(x^3 + 3*x^2 + 2*x)/(x^5 + 13*x^4 + 57*x^3 + 95*x^2 + 50*x), 
x)
 

Sympy [F]

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

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

Output:

Integral(1/(sqrt(x*(x + 1)*(x + 2))*(x + 5)**2), x)
 

Maxima [F]

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

integrate(1/(5+x)^2/(x^3+3*x^2+2*x)^(1/2),x, algorithm="maxima")
 

Output:

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

Giac [F]

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

integrate(1/(5+x)^2/(x^3+3*x^2+2*x)^(1/2),x, algorithm="giac")
 

Output:

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

Mupad [B] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.69 \[ \int \frac {1}{(5+x)^2 \sqrt {2 x+3 x^2+x^3}} \, dx=\frac {\sqrt {x^3+3\,x^2+2\,x}}{60\,\left (x+5\right )}-\frac {\sqrt {-x}\,\mathrm {E}\left (\mathrm {asin}\left (\sqrt {-x-1}\right )\middle |-1\right )\,\sqrt {-x-1}\,\sqrt {x+2}}{60\,\sqrt {x^3+3\,x^2+2\,x}}+\frac {\sqrt {-x}\,\mathrm {F}\left (\mathrm {asin}\left (\sqrt {-x-1}\right )\middle |-1\right )\,\sqrt {-x-1}\,\sqrt {x+2}}{12\,\sqrt {x^3+3\,x^2+2\,x}}-\frac {47\,\sqrt {-x}\,\sqrt {-x-1}\,\sqrt {x+2}\,\Pi \left (\frac {1}{4};\mathrm {asin}\left (\sqrt {-x-1}\right )\middle |-1\right )}{240\,\sqrt {x^3+3\,x^2+2\,x}} \] Input:

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

Output:

(2*x + 3*x^2 + x^3)^(1/2)/(60*(x + 5)) - ((-x)^(1/2)*ellipticE(asin((- x - 
 1)^(1/2)), -1)*(- x - 1)^(1/2)*(x + 2)^(1/2))/(60*(2*x + 3*x^2 + x^3)^(1/ 
2)) + ((-x)^(1/2)*ellipticF(asin((- x - 1)^(1/2)), -1)*(- x - 1)^(1/2)*(x 
+ 2)^(1/2))/(12*(2*x + 3*x^2 + x^3)^(1/2)) - (47*(-x)^(1/2)*(- x - 1)^(1/2 
)*(x + 2)^(1/2)*ellipticPi(1/4, asin((- x - 1)^(1/2)), -1))/(240*(2*x + 3* 
x^2 + x^3)^(1/2))
 

Reduce [F]

\[ \int \frac {1}{(5+x)^2 \sqrt {2 x+3 x^2+x^3}} \, dx=\int \frac {\sqrt {x^{2}+3 x +2}}{\sqrt {x}\, x^{4}+13 \sqrt {x}\, x^{3}+57 \sqrt {x}\, x^{2}+95 \sqrt {x}\, x +50 \sqrt {x}}d x \] Input:

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

Output:

int(sqrt(x**2 + 3*x + 2)/(sqrt(x)*x**4 + 13*sqrt(x)*x**3 + 57*sqrt(x)*x**2 
 + 95*sqrt(x)*x + 50*sqrt(x)),x)