3.6.22 \(\int \frac {1}{x^2 (1+x)^{5/2} (1-x+x^2)^{5/2}} \, dx\) [522]

Optimal. Leaf size=349 \[ \frac {22}{27 x \sqrt {1+x} \sqrt {1-x+x^2}}+\frac {2}{9 x \sqrt {1+x} \sqrt {1-x+x^2} \left (1+x^3\right )}-\frac {55 \left (1+x^3\right )}{27 x \sqrt {1+x} \sqrt {1-x+x^2}}+\frac {55 \left (1+x^3\right )}{27 \sqrt {1+x} \left (1+\sqrt {3}+x\right ) \sqrt {1-x+x^2}}-\frac {55 \sqrt {2-\sqrt {3}} \sqrt {1+x} \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} E\left (\sin ^{-1}\left (\frac {1-\sqrt {3}+x}{1+\sqrt {3}+x}\right )|-7-4 \sqrt {3}\right )}{18\ 3^{3/4} \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1-x+x^2}}+\frac {55 \sqrt {2} \sqrt {1+x} \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} F\left (\sin ^{-1}\left (\frac {1-\sqrt {3}+x}{1+\sqrt {3}+x}\right )|-7-4 \sqrt {3}\right )}{27 \sqrt [4]{3} \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1-x+x^2}} \]

[Out]

22/27/x/(1+x)^(1/2)/(x^2-x+1)^(1/2)+2/9/x/(x^3+1)/(1+x)^(1/2)/(x^2-x+1)^(1/2)-55/27*(x^3+1)/x/(1+x)^(1/2)/(x^2
-x+1)^(1/2)+55/27*(x^3+1)/(1+x+3^(1/2))/(1+x)^(1/2)/(x^2-x+1)^(1/2)+55/81*EllipticF((1+x-3^(1/2))/(1+x+3^(1/2)
),I*3^(1/2)+2*I)*2^(1/2)*(1+x)^(1/2)*((x^2-x+1)/(1+x+3^(1/2))^2)^(1/2)*3^(3/4)/(x^2-x+1)^(1/2)/((1+x)/(1+x+3^(
1/2))^2)^(1/2)-55/54*3^(1/4)*EllipticE((1+x-3^(1/2))/(1+x+3^(1/2)),I*3^(1/2)+2*I)*(1+x)^(1/2)*(1/2*6^(1/2)-1/2
*2^(1/2))*((x^2-x+1)/(1+x+3^(1/2))^2)^(1/2)/(x^2-x+1)^(1/2)/((1+x)/(1+x+3^(1/2))^2)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.09, antiderivative size = 349, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {929, 296, 331, 309, 224, 1891} \begin {gather*} \frac {55 \sqrt {2} \sqrt {x+1} \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} F\left (\text {ArcSin}\left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right )|-7-4 \sqrt {3}\right )}{27 \sqrt [4]{3} \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^2-x+1}}-\frac {55 \sqrt {2-\sqrt {3}} \sqrt {x+1} \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} E\left (\text {ArcSin}\left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right )|-7-4 \sqrt {3}\right )}{18\ 3^{3/4} \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^2-x+1}}+\frac {22}{27 x \sqrt {x+1} \sqrt {x^2-x+1}}-\frac {55 \left (x^3+1\right )}{27 x \sqrt {x+1} \sqrt {x^2-x+1}}+\frac {55 \left (x^3+1\right )}{27 \sqrt {x+1} \left (x+\sqrt {3}+1\right ) \sqrt {x^2-x+1}}+\frac {2}{9 x \sqrt {x+1} \sqrt {x^2-x+1} \left (x^3+1\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

22/(27*x*Sqrt[1 + x]*Sqrt[1 - x + x^2]) + 2/(9*x*Sqrt[1 + x]*Sqrt[1 - x + x^2]*(1 + x^3)) - (55*(1 + x^3))/(27
*x*Sqrt[1 + x]*Sqrt[1 - x + x^2]) + (55*(1 + x^3))/(27*Sqrt[1 + x]*(1 + Sqrt[3] + x)*Sqrt[1 - x + x^2]) - (55*
Sqrt[2 - Sqrt[3]]*Sqrt[1 + x]*Sqrt[(1 - x + x^2)/(1 + Sqrt[3] + x)^2]*EllipticE[ArcSin[(1 - Sqrt[3] + x)/(1 +
Sqrt[3] + x)], -7 - 4*Sqrt[3]])/(18*3^(3/4)*Sqrt[(1 + x)/(1 + Sqrt[3] + x)^2]*Sqrt[1 - x + x^2]) + (55*Sqrt[2]
*Sqrt[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]])/(27*3^(1/4)*Sqrt[(1 + x)/(1 + Sqrt[3] + x)^2]*Sqrt[1 - x + x^2])

Rule 224

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]*Sq
rt[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 296

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] + Dist[(m + n*(p + 1) + 1)/(a*n*(p + 1)), Int[(c*x)^m*(a + b*x^n)^(p + 1), x], x] /; Free
Q[{a, b, c, m}, x] && IGtQ[n, 0] && LtQ[p, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 309

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

Rule 331

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] - Dist[b*((m + n*(p + 1) + 1)/(a*c^n*(m + 1))), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 929

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

Rule 1891

Int[((c_) + (d_.)*(x_))/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Simplify[(1 - Sqrt[3])*(d/c)]]
, s = Denom[Simplify[(1 - Sqrt[3])*(d/c)]]}, Simp[2*d*s^3*(Sqrt[a + b*x^3]/(a*r^2*((1 + Sqrt[3])*s + r*x))), x
] - Simp[3^(1/4)*Sqrt[2 - Sqrt[3]]*d*s*(s + r*x)*(Sqrt[(s^2 - r*s*x + r^2*x^2)/((1 + Sqrt[3])*s + r*x)^2]/(r^2
*Sqrt[a + b*x^3]*Sqrt[s*((s + r*x)/((1 + Sqrt[3])*s + r*x)^2)]))*EllipticE[ArcSin[((1 - Sqrt[3])*s + r*x)/((1
+ Sqrt[3])*s + r*x)], -7 - 4*Sqrt[3]], x]] /; FreeQ[{a, b, c, d}, x] && PosQ[a] && EqQ[b*c^3 - 2*(5 - 3*Sqrt[3
])*a*d^3, 0]

Rubi steps

\begin {align*} \int \frac {1}{x^2 (1+x)^{5/2} \left (1-x+x^2\right )^{5/2}} \, dx &=\frac {\sqrt {1+x^3} \int \frac {1}{x^2 \left (1+x^3\right )^{5/2}} \, dx}{\sqrt {1+x} \sqrt {1-x+x^2}}\\ &=\frac {2}{9 x \sqrt {1+x} \sqrt {1-x+x^2} \left (1+x^3\right )}+\frac {\left (11 \sqrt {1+x^3}\right ) \int \frac {1}{x^2 \left (1+x^3\right )^{3/2}} \, dx}{9 \sqrt {1+x} \sqrt {1-x+x^2}}\\ &=\frac {22}{27 x \sqrt {1+x} \sqrt {1-x+x^2}}+\frac {2}{9 x \sqrt {1+x} \sqrt {1-x+x^2} \left (1+x^3\right )}+\frac {\left (55 \sqrt {1+x^3}\right ) \int \frac {1}{x^2 \sqrt {1+x^3}} \, dx}{27 \sqrt {1+x} \sqrt {1-x+x^2}}\\ &=\frac {22}{27 x \sqrt {1+x} \sqrt {1-x+x^2}}+\frac {2}{9 x \sqrt {1+x} \sqrt {1-x+x^2} \left (1+x^3\right )}-\frac {55 \left (1+x^3\right )}{27 x \sqrt {1+x} \sqrt {1-x+x^2}}+\frac {\left (55 \sqrt {1+x^3}\right ) \int \frac {x}{\sqrt {1+x^3}} \, dx}{54 \sqrt {1+x} \sqrt {1-x+x^2}}\\ &=\frac {22}{27 x \sqrt {1+x} \sqrt {1-x+x^2}}+\frac {2}{9 x \sqrt {1+x} \sqrt {1-x+x^2} \left (1+x^3\right )}-\frac {55 \left (1+x^3\right )}{27 x \sqrt {1+x} \sqrt {1-x+x^2}}+\frac {\left (55 \sqrt {1+x^3}\right ) \int \frac {1-\sqrt {3}+x}{\sqrt {1+x^3}} \, dx}{54 \sqrt {1+x} \sqrt {1-x+x^2}}+\frac {\left (55 \sqrt {\frac {1}{2} \left (2-\sqrt {3}\right )} \sqrt {1+x^3}\right ) \int \frac {1}{\sqrt {1+x^3}} \, dx}{27 \sqrt {1+x} \sqrt {1-x+x^2}}\\ &=\frac {22}{27 x \sqrt {1+x} \sqrt {1-x+x^2}}+\frac {2}{9 x \sqrt {1+x} \sqrt {1-x+x^2} \left (1+x^3\right )}-\frac {55 \left (1+x^3\right )}{27 x \sqrt {1+x} \sqrt {1-x+x^2}}+\frac {55 \left (1+x^3\right )}{27 \sqrt {1+x} \left (1+\sqrt {3}+x\right ) \sqrt {1-x+x^2}}-\frac {55 \sqrt {2-\sqrt {3}} \sqrt {1+x} \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} E\left (\sin ^{-1}\left (\frac {1-\sqrt {3}+x}{1+\sqrt {3}+x}\right )|-7-4 \sqrt {3}\right )}{18\ 3^{3/4} \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1-x+x^2}}+\frac {55 \sqrt {2} \sqrt {1+x} \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} F\left (\sin ^{-1}\left (\frac {1-\sqrt {3}+x}{1+\sqrt {3}+x}\right )|-7-4 \sqrt {3}\right )}{27 \sqrt [4]{3} \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1-x+x^2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C] Result contains complex when optimal does not.
time = 10.50, size = 414, normalized size = 1.19 \begin {gather*} -\frac {27+88 x^3+55 x^6}{27 x (1+x)^{3/2} \left (1-x+x^2\right )^{3/2}}+\frac {55 (1+x)^{3/2} \left (\frac {12 \sqrt {-\frac {i}{3 i+\sqrt {3}}} \left (1-x+x^2\right )}{(1+x)^2}+\frac {3 \sqrt {2} \left (1-i \sqrt {3}\right ) \sqrt {\frac {3 i+\sqrt {3}-\frac {6 i}{1+x}}{3 i+\sqrt {3}}} \sqrt {\frac {-3 i+\sqrt {3}+\frac {6 i}{1+x}}{-3 i+\sqrt {3}}} E\left (i \sinh ^{-1}\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 {1+x}}+\frac {i \sqrt {2} \left (3 i+\sqrt {3}\right ) \sqrt {\frac {3 i+\sqrt {3}-\frac {6 i}{1+x}}{3 i+\sqrt {3}}} \sqrt {\frac {-3 i+\sqrt {3}+\frac {6 i}{1+x}}{-3 i+\sqrt {3}}} F\left (i \sinh ^{-1}\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 {1+x}}\right )}{324 \sqrt {-\frac {i}{3 i+\sqrt {3}}} \sqrt {1-x+x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/27*(27 + 88*x^3 + 55*x^6)/(x*(1 + x)^(3/2)*(1 - x + x^2)^(3/2)) + (55*(1 + x)^(3/2)*((12*Sqrt[(-I)/(3*I + S
qrt[3])]*(1 - x + x^2))/(1 + x)^2 + (3*Sqrt[2]*(1 - I*Sqrt[3])*Sqrt[(3*I + Sqrt[3] - (6*I)/(1 + x))/(3*I + Sqr
t[3])]*Sqrt[(-3*I + Sqrt[3] + (6*I)/(1 + x))/(-3*I + Sqrt[3])]*EllipticE[I*ArcSinh[Sqrt[(-6*I)/(3*I + Sqrt[3])
]/Sqrt[1 + x]], (3*I + Sqrt[3])/(3*I - Sqrt[3])])/Sqrt[1 + x] + (I*Sqrt[2]*(3*I + Sqrt[3])*Sqrt[(3*I + Sqrt[3]
- (6*I)/(1 + x))/(3*I + Sqrt[3])]*Sqrt[(-3*I + Sqrt[3] + (6*I)/(1 + x))/(-3*I + Sqrt[3])]*EllipticF[I*ArcSinh
[Sqrt[(-6*I)/(3*I + Sqrt[3])]/Sqrt[1 + x]], (3*I + Sqrt[3])/(3*I - Sqrt[3])])/Sqrt[1 + x]))/(324*Sqrt[(-I)/(3*
I + Sqrt[3])]*Sqrt[1 - x + x^2])

________________________________________________________________________________________

Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 694 vs. \(2 (277 ) = 554\).
time = 0.12, size = 695, normalized size = 1.99

method	result	size
elliptic	\(\frac {\sqrt {\left (1+x \right ) \left (x^{2}-x +1\right )}\, \left (-\frac {\sqrt {x^{3}+1}}{x}-\frac {2 x^{2}}{9 \left (x^{3}+1\right )^{\frac {3}{2}}}-\frac {28 x^{2}}{27 \sqrt {x^{3}+1}}+\frac {55 \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \left (\left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \EllipticE \left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )+\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \EllipticF \left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )\right )}{27 \sqrt {x^{3}+1}}\right )}{\sqrt {1+x}\, \sqrt {x^{2}-x +1}}\)	\(240\)
default	\(\frac {55 i \EllipticF \left (\sqrt {-\frac {2 \left (1+x \right )}{-3+i \sqrt {3}}}, \sqrt {-\frac {-3+i \sqrt {3}}{i \sqrt {3}+3}}\right ) \sqrt {3}\, x^{4} \sqrt {-\frac {2 \left (1+x \right )}{-3+i \sqrt {3}}}\, \sqrt {\frac {i \sqrt {3}-2 x +1}{i \sqrt {3}+3}}\, \sqrt {\frac {i \sqrt {3}+2 x -1}{-3+i \sqrt {3}}}+165 \EllipticF \left (\sqrt {-\frac {2 \left (1+x \right )}{-3+i \sqrt {3}}}, \sqrt {-\frac {-3+i \sqrt {3}}{i \sqrt {3}+3}}\right ) x^{4} \sqrt {-\frac {2 \left (1+x \right )}{-3+i \sqrt {3}}}\, \sqrt {\frac {i \sqrt {3}-2 x +1}{i \sqrt {3}+3}}\, \sqrt {\frac {i \sqrt {3}+2 x -1}{-3+i \sqrt {3}}}-330 \EllipticE \left (\sqrt {-\frac {2 \left (1+x \right )}{-3+i \sqrt {3}}}, \sqrt {-\frac {-3+i \sqrt {3}}{i \sqrt {3}+3}}\right ) x^{4} \sqrt {-\frac {2 \left (1+x \right )}{-3+i \sqrt {3}}}\, \sqrt {\frac {i \sqrt {3}-2 x +1}{i \sqrt {3}+3}}\, \sqrt {\frac {i \sqrt {3}+2 x -1}{-3+i \sqrt {3}}}+55 i \sqrt {3}\, \sqrt {-\frac {2 \left (1+x \right )}{-3+i \sqrt {3}}}\, \sqrt {\frac {i \sqrt {3}-2 x +1}{i \sqrt {3}+3}}\, \sqrt {\frac {i \sqrt {3}+2 x -1}{-3+i \sqrt {3}}}\, \EllipticF \left (\sqrt {-\frac {2 \left (1+x \right )}{-3+i \sqrt {3}}}, \sqrt {-\frac {-3+i \sqrt {3}}{i \sqrt {3}+3}}\right ) x -110 x^{6}+165 \sqrt {-\frac {2 \left (1+x \right )}{-3+i \sqrt {3}}}\, \sqrt {\frac {i \sqrt {3}-2 x +1}{i \sqrt {3}+3}}\, \sqrt {\frac {i \sqrt {3}+2 x -1}{-3+i \sqrt {3}}}\, \EllipticF \left (\sqrt {-\frac {2 \left (1+x \right )}{-3+i \sqrt {3}}}, \sqrt {-\frac {-3+i \sqrt {3}}{i \sqrt {3}+3}}\right ) x -330 \sqrt {-\frac {2 \left (1+x \right )}{-3+i \sqrt {3}}}\, \sqrt {\frac {i \sqrt {3}-2 x +1}{i \sqrt {3}+3}}\, \sqrt {\frac {i \sqrt {3}+2 x -1}{-3+i \sqrt {3}}}\, \EllipticE \left (\sqrt {-\frac {2 \left (1+x \right )}{-3+i \sqrt {3}}}, \sqrt {-\frac {-3+i \sqrt {3}}{i \sqrt {3}+3}}\right ) x -176 x^{3}-54}{54 x \left (x^{2}-x +1\right )^{\frac {3}{2}} \left (1+x \right )^{\frac {3}{2}}}\)	\(695\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/54*(55*I*EllipticF((-2*(1+x)/(-3+I*3^(1/2)))^(1/2),(-(-3+I*3^(1/2))/(I*3^(1/2)+3))^(1/2))*3^(1/2)*x^4*(-2*(1
+x)/(-3+I*3^(1/2)))^(1/2)*((I*3^(1/2)-2*x+1)/(I*3^(1/2)+3))^(1/2)*((I*3^(1/2)+2*x-1)/(-3+I*3^(1/2)))^(1/2)+165
*EllipticF((-2*(1+x)/(-3+I*3^(1/2)))^(1/2),(-(-3+I*3^(1/2))/(I*3^(1/2)+3))^(1/2))*x^4*(-2*(1+x)/(-3+I*3^(1/2))
)^(1/2)*((I*3^(1/2)-2*x+1)/(I*3^(1/2)+3))^(1/2)*((I*3^(1/2)+2*x-1)/(-3+I*3^(1/2)))^(1/2)-330*EllipticE((-2*(1+
x)/(-3+I*3^(1/2)))^(1/2),(-(-3+I*3^(1/2))/(I*3^(1/2)+3))^(1/2))*x^4*(-2*(1+x)/(-3+I*3^(1/2)))^(1/2)*((I*3^(1/2
)-2*x+1)/(I*3^(1/2)+3))^(1/2)*((I*3^(1/2)+2*x-1)/(-3+I*3^(1/2)))^(1/2)+55*I*EllipticF((-2*(1+x)/(-3+I*3^(1/2))
)^(1/2),(-(-3+I*3^(1/2))/(I*3^(1/2)+3))^(1/2))*3^(1/2)*x*(-2*(1+x)/(-3+I*3^(1/2)))^(1/2)*((I*3^(1/2)-2*x+1)/(I
*3^(1/2)+3))^(1/2)*((I*3^(1/2)+2*x-1)/(-3+I*3^(1/2)))^(1/2)-110*x^6+165*(-2*(1+x)/(-3+I*3^(1/2)))^(1/2)*((I*3^
(1/2)-2*x+1)/(I*3^(1/2)+3))^(1/2)*((I*3^(1/2)+2*x-1)/(-3+I*3^(1/2)))^(1/2)*EllipticF((-2*(1+x)/(-3+I*3^(1/2)))
^(1/2),(-(-3+I*3^(1/2))/(I*3^(1/2)+3))^(1/2))*x-330*(-2*(1+x)/(-3+I*3^(1/2)))^(1/2)*((I*3^(1/2)-2*x+1)/(I*3^(1
/2)+3))^(1/2)*((I*3^(1/2)+2*x-1)/(-3+I*3^(1/2)))^(1/2)*EllipticE((-2*(1+x)/(-3+I*3^(1/2)))^(1/2),(-(-3+I*3^(1/
2))/(I*3^(1/2)+3))^(1/2))*x-176*x^3-54)/x/(x^2-x+1)^(3/2)/(1+x)^(3/2)

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.30, size = 62, normalized size = 0.18 \begin {gather*} -\frac {{\left (55 \, x^{6} + 88 \, x^{3} + 27\right )} \sqrt {x^{2} - x + 1} \sqrt {x + 1} + 55 \, {\left (x^{7} + 2 \, x^{4} + x\right )} {\rm weierstrassZeta}\left (0, -4, {\rm weierstrassPInverse}\left (0, -4, x\right )\right )}{27 \, {\left (x^{7} + 2 \, x^{4} + x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/27*((55*x^6 + 88*x^3 + 27)*sqrt(x^2 - x + 1)*sqrt(x + 1) + 55*(x^7 + 2*x^4 + x)*weierstrassZeta(0, -4, weie
rstrassPInverse(0, -4, x)))/(x^7 + 2*x^4 + x)

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^{2} \left (x + 1\right )^{\frac {5}{2}} \left (x^{2} - x + 1\right )^{\frac {5}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{x^2\,{\left (x+1\right )}^{5/2}\,{\left (x^2-x+1\right )}^{5/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________