\(\int \frac {1}{x^3 (1+x)^{3/2} (1-x+x^2)^{3/2}} \, dx\) [499]

Optimal result

Integrand size = 23, antiderivative size = 170 \[ \int \frac {1}{x^3 (1+x)^{3/2} \left (1-x+x^2\right )^{3/2}} \, dx=\frac {2}{3 x^2 \sqrt {1+x} \sqrt {1-x+x^2}}-\frac {7 \left (1+x^3\right )}{6 x^2 \sqrt {1+x} \sqrt {1-x+x^2}}-\frac {7 \sqrt {2+\sqrt {3}} \sqrt {1+x} \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1-\sqrt {3}+x}{1+\sqrt {3}+x}\right ),-7-4 \sqrt {3}\right )}{6 \sqrt [4]{3} \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1-x+x^2}} \] Output:

2/3/x^2/(1+x)^(1/2)/(x^2-x+1)^(1/2)-7/6*(x^3+1)/x^2/(1+x)^(1/2)/(x^2-x+1)^ 
(1/2)-7/18*(1/2*6^(1/2)+1/2*2^(1/2))*(1+x)^(1/2)*((x^2-x+1)/(1+x+3^(1/2))^ 
2)^(1/2)*EllipticF((1+x-3^(1/2))/(1+x+3^(1/2)),I*3^(1/2)+2*I)*3^(3/4)/((1+ 
x)/(1+x+3^(1/2))^2)^(1/2)/(x^2-x+1)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 10.58 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^3 (1+x)^{3/2} \left (1-x+x^2\right )^{3/2}} \, dx=\frac {-\frac {6 \left (3+7 x^3\right )}{x^2 \sqrt {1+x}}-\frac {7 i (1+x) \sqrt {1+\frac {6 i}{\left (-3 i+\sqrt {3}\right ) (1+x)}} \sqrt {6-\frac {36 i}{\left (3 i+\sqrt {3}\right ) (1+x)}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-\frac {6 i}{3 i+\sqrt {3}}}}{\sqrt {1+x}}\right ),\frac {3 i+\sqrt {3}}{3 i-\sqrt {3}}\right )}{\sqrt {-\frac {i}{3 i+\sqrt {3}}}}}{36 \sqrt {1-x+x^2}} \] Input:

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

Output:

((-6*(3 + 7*x^3))/(x^2*Sqrt[1 + x]) - ((7*I)*(1 + x)*Sqrt[1 + (6*I)/((-3*I 
 + Sqrt[3])*(1 + x))]*Sqrt[6 - (36*I)/((3*I + Sqrt[3])*(1 + x))]*EllipticF 
[I*ArcSinh[Sqrt[(-6*I)/(3*I + Sqrt[3])]/Sqrt[1 + x]], (3*I + Sqrt[3])/(3*I 
 - Sqrt[3])])/Sqrt[(-I)/(3*I + Sqrt[3])])/(36*Sqrt[1 - x + x^2])
 

Rubi [A] (verified)

Time = 0.44 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.01, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {1210, 819, 847, 759}

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.

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

\(\Big \downarrow \) 1210

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

\(\Big \downarrow \) 819

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

\(\Big \downarrow \) 847

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

\(\Big \downarrow \) 759

\(\displaystyle \frac {\sqrt {x^3+1} \left (\frac {7}{3} \left (-\frac {\sqrt {2+\sqrt {3}} (x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{2 \sqrt [4]{3} \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^3+1}}-\frac {\sqrt {x^3+1}}{2 x^2}\right )+\frac {2}{3 x^2 \sqrt {x^3+1}}\right )}{\sqrt {x+1} \sqrt {x^2-x+1}}\)

Input:

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

Output:

(Sqrt[1 + x^3]*(2/(3*x^2*Sqrt[1 + x^3]) + (7*(-1/2*Sqrt[1 + x^3]/x^2 - (Sq 
rt[2 + Sqrt[3]]*(1 + x)*Sqrt[(1 - x + x^2)/(1 + Sqrt[3] + x)^2]*EllipticF[ 
ArcSin[(1 - Sqrt[3] + x)/(1 + Sqrt[3] + x)], -7 - 4*Sqrt[3]])/(2*3^(1/4)*S 
qrt[(1 + x)/(1 + Sqrt[3] + x)^2]*Sqrt[1 + x^3])))/3))/(Sqrt[1 + x]*Sqrt[1 
- x + x^2])
 

Defintions of rubi rules used

rule 759

Int[1/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], 
s = Denom[Rt[b/a, 3]]}, Simp[2*Sqrt[2 + Sqrt[3]]*(s + r*x)*(Sqrt[(s^2 - r*s 
*x + r^2*x^2)/((1 + Sqrt[3])*s + r*x)^2]/(3^(1/4)*r*Sqrt[a + b*x^3]*Sqrt[s* 
((s + r*x)/((1 + Sqrt[3])*s + r*x)^2)]))*EllipticF[ArcSin[((1 - Sqrt[3])*s 
+ r*x)/((1 + Sqrt[3])*s + r*x)], -7 - 4*Sqrt[3]], x]] /; FreeQ[{a, b}, x] & 
& PosQ[a]
 

rule 819

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-( 
c*x)^(m + 1))*((a + b*x^n)^(p + 1)/(a*c*n*(p + 1))), x] + Simp[(m + n*(p + 
1) + 1)/(a*n*(p + 1))   Int[(c*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a 
, b, c, m}, x] && IGtQ[n, 0] && LtQ[p, -1] && IntBinomialQ[a, b, c, n, m, p 
, x]
 

rule 847

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x 
)^(m + 1)*((a + b*x^n)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + n*(p + 1) 
+ 1)/(a*c^n*(m + 1)))   Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a 
, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p 
, x]
 

rule 1210

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_.)*((a_) + (b_.)*(x_) 
 + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^FracPart[p]*((a + b*x + 
c*x^2)^FracPart[p]/(a*d + c*e*x^3)^FracPart[p])   Int[(f + g*x)^n*(a*d + c* 
e*x^3)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && EqQ[b*d + a*e, 
 0] && EqQ[c*d + b*e, 0] && EqQ[m, p]
 

Maple [A] (verified)

Time = 2.87 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.99

method	result	size
elliptic	\(\frac {\sqrt {\left (x +1\right ) \left (x^{2}-x +1\right )}\, \left (-\frac {2 x}{3 \sqrt {x^{3}+1}}-\frac {\sqrt {x^{3}+1}}{2 x^{2}}-\frac {7 \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x +1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{6 \sqrt {x^{3}+1}}\right )}{\sqrt {x +1}\, \sqrt {x^{2}-x +1}}\)	\(169\)
risch	\(-\frac {7 x^{3}+3}{6 x^{2} \sqrt {x +1}\, \sqrt {x^{2}-x +1}}-\frac {7 \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x +1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right ) \sqrt {\left (x +1\right ) \left (x^{2}-x +1\right )}}{6 \sqrt {x^{3}+1}\, \sqrt {x +1}\, \sqrt {x^{2}-x +1}}\)	\(173\)
default	\(\frac {\sqrt {x +1}\, \sqrt {x^{2}-x +1}\, \left (7 i \sqrt {3}\, \sqrt {-\frac {2 \left (x +1\right )}{i \sqrt {3}-3}}\, \sqrt {\frac {i \sqrt {3}-2 x +1}{i \sqrt {3}+3}}\, \sqrt {\frac {i \sqrt {3}+2 x -1}{i \sqrt {3}-3}}\, \operatorname {EllipticF}\left (\sqrt {-\frac {2 \left (x +1\right )}{i \sqrt {3}-3}}, \sqrt {-\frac {i \sqrt {3}-3}{i \sqrt {3}+3}}\right ) x^{2}-21 \sqrt {-\frac {2 \left (x +1\right )}{i \sqrt {3}-3}}\, \sqrt {\frac {i \sqrt {3}-2 x +1}{i \sqrt {3}+3}}\, \sqrt {\frac {i \sqrt {3}+2 x -1}{i \sqrt {3}-3}}\, \operatorname {EllipticF}\left (\sqrt {-\frac {2 \left (x +1\right )}{i \sqrt {3}-3}}, \sqrt {-\frac {i \sqrt {3}-3}{i \sqrt {3}+3}}\right ) x^{2}-14 x^{3}-6\right )}{12 \left (x^{3}+1\right ) x^{2}}\)	\(259\)

Input:

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

Output:

((x+1)*(x^2-x+1))^(1/2)/(x+1)^(1/2)/(x^2-x+1)^(1/2)*(-2/3*x/(x^3+1)^(1/2)- 
1/2/x^2*(x^3+1)^(1/2)-7/6*(3/2-1/2*I*3^(1/2))*((x+1)/(3/2-1/2*I*3^(1/2)))^ 
(1/2)*((x-1/2-1/2*I*3^(1/2))/(-3/2-1/2*I*3^(1/2)))^(1/2)*((x-1/2+1/2*I*3^( 
1/2))/(-3/2+1/2*I*3^(1/2)))^(1/2)/(x^3+1)^(1/2)*EllipticF(((x+1)/(3/2-1/2* 
I*3^(1/2)))^(1/2),((-3/2+1/2*I*3^(1/2))/(-3/2-1/2*I*3^(1/2)))^(1/2)))
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.28 \[ \int \frac {1}{x^3 (1+x)^{3/2} \left (1-x+x^2\right )^{3/2}} \, dx=-\frac {{\left (7 \, x^{3} + 3\right )} \sqrt {x^{2} - x + 1} \sqrt {x + 1} + 7 \, {\left (x^{5} + x^{2}\right )} {\rm weierstrassPInverse}\left (0, -4, x\right )}{6 \, {\left (x^{5} + x^{2}\right )}} \] Input:

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

Output:

-1/6*((7*x^3 + 3)*sqrt(x^2 - x + 1)*sqrt(x + 1) + 7*(x^5 + x^2)*weierstras 
sPInverse(0, -4, x))/(x^5 + x^2)
 

Sympy [F]

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

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

Output:

Integral(1/(x**3*(x + 1)**(3/2)*(x**2 - x + 1)**(3/2)), x)
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {1}{x^3 (1+x)^{3/2} \left (1-x+x^2\right )^{3/2}} \, dx=\int \frac {\sqrt {x +1}\, \sqrt {x^{2}-x +1}}{x^{9}+2 x^{6}+x^{3}}d x \] Input:

int(1/x^3/(1+x)^(3/2)/(x^2-x+1)^(3/2),x)
                                                                                    
                                                                                    
 

Output:

int((sqrt(x + 1)*sqrt(x**2 - x + 1))/(x**9 + 2*x**6 + x**3),x)