\(\int \frac {1}{(1+x)^{3/2} (1-x+x^2)^{3/2}} \, dx\) [513]

Optimal result

Integrand size = 20, antiderivative size = 137 \[ \int \frac {1}{(1+x)^{3/2} \left (1-x+x^2\right )^{3/2}} \, dx=\frac {2 x}{3 \sqrt {1+x} \sqrt {1-x+x^2}}+\frac {2 \sqrt {2+\sqrt {3}} \sqrt {1+x} \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1-\sqrt {3}+x}{1+\sqrt {3}+x}\right ),-7-4 \sqrt {3}\right )}{3 \sqrt [4]{3} \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1-x+x^2}} \]

[Out]

2/3*x/(1+x)^(1/2)/(x^2-x+1)^(1/2)+2/9*EllipticF((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)*3^(3/4)/(x^2-x+1)^(1/2)/((1+x)/(1+x+3^(1/2))^2)^(1/2)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {727, 205, 224} \[ \int \frac {1}{(1+x)^{3/2} \left (1-x+x^2\right )^{3/2}} \, dx=\frac {2 \sqrt {2+\sqrt {3}} \sqrt {x+1} \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{3 \sqrt [4]{3} \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^2-x+1}}+\frac {2 x}{3 \sqrt {x+1} \sqrt {x^2-x+1}} \]

[In]

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

[Out]

(2*x)/(3*Sqrt[1 + x]*Sqrt[1 - x + x^2]) + (2*Sqrt[2 + Sqrt[3]]*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]])/(3*3^(1/4)*Sqrt[(1 + x)/(1 + Sqrt
[3] + x)^2]*Sqrt[1 - x + x^2])

Rule 205

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^n)^(p + 1)/(a*n*(p + 1))), x] + Dist[(n*(p
+ 1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (
IntegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[
p])

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 727

Int[((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[(d + e*x)^(m - p)*(a*d + c*e*x^3)^p, x], x]
/; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[b*d + a*e, 0] && EqQ[c*d + b*e, 0] && IGtQ[m - p + 1, 0] &&  !Intege
rQ[p]

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {1+x^3} \int \frac {1}{\left (1+x^3\right )^{3/2}} \, dx}{\sqrt {1+x} \sqrt {1-x+x^2}} \\ & = \frac {2 x}{3 \sqrt {1+x} \sqrt {1-x+x^2}}+\frac {\sqrt {1+x^3} \int \frac {1}{\sqrt {1+x^3}} \, dx}{3 \sqrt {1+x} \sqrt {1-x+x^2}} \\ & = \frac {2 x}{3 \sqrt {1+x} \sqrt {1-x+x^2}}+\frac {2 \sqrt {2+\sqrt {3}} \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 )}{3 \sqrt [4]{3} \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1-x+x^2}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 20.27 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.58 \[ \int \frac {1}{(1+x)^{3/2} \left (1-x+x^2\right )^{3/2}} \, dx=\sqrt {3-3 (1+x)+(1+x)^2} \left (-\frac {2}{9 \sqrt {1+x}}+\frac {2 (1+x)^{3/2}}{9 \left (3-3 (1+x)+(1+x)^2\right )}\right )+\frac {i \sqrt {\frac {2}{3}} (1+x) \sqrt {1-\frac {6}{\left (3-i \sqrt {3}\right ) (1+x)}} \sqrt {1-\frac {6}{\left (3+i \sqrt {3}\right ) (1+x)}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-\frac {6}{3-i \sqrt {3}}}}{\sqrt {1+x}}\right ),\frac {3-i \sqrt {3}}{3+i \sqrt {3}}\right )}{3 \sqrt {-\frac {1}{3-i \sqrt {3}}} \sqrt {3-3 (1+x)+(1+x)^2}} \]

[In]

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

[Out]

Sqrt[3 - 3*(1 + x) + (1 + x)^2]*(-2/(9*Sqrt[1 + x]) + (2*(1 + x)^(3/2))/(9*(3 - 3*(1 + x) + (1 + x)^2))) + ((I
/3)*Sqrt[2/3]*(1 + x)*Sqrt[1 - 6/((3 - I*Sqrt[3])*(1 + x))]*Sqrt[1 - 6/((3 + I*Sqrt[3])*(1 + x))]*EllipticF[I*
ArcSinh[Sqrt[-6/(3 - I*Sqrt[3])]/Sqrt[1 + x]], (3 - I*Sqrt[3])/(3 + I*Sqrt[3])])/(Sqrt[-(3 - I*Sqrt[3])^(-1)]*
Sqrt[3 - 3*(1 + x) + (1 + x)^2])

Maple [A] (verified)

Time = 0.72 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.15

method	result	size
elliptic	\(\frac {\sqrt {\left (1+x \right ) \left (x^{2}-x +1\right )}\, \left (\frac {2 x}{3 \sqrt {x^{3}+1}}+\frac {2 \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, F\left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {x^{3}+1}}\right )}{\sqrt {1+x}\, \sqrt {x^{2}-x +1}}\)	\(157\)
risch	\(\frac {2 x}{3 \sqrt {1+x}\, \sqrt {x^{2}-x +1}}+\frac {2 \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, F\left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right ) \sqrt {\left (1+x \right ) \left (x^{2}-x +1\right )}}{3 \sqrt {x^{3}+1}\, \sqrt {1+x}\, \sqrt {x^{2}-x +1}}\)	\(164\)
default	\(-\frac {\sqrt {1+x}\, \sqrt {x^{2}-x +1}\, \left (i \sqrt {3}\, \sqrt {-\frac {2 \left (1+x \right )}{-3+i \sqrt {3}}}\, \sqrt {\frac {i \sqrt {3}-2 x +1}{i \sqrt {3}+3}}\, \sqrt {\frac {i \sqrt {3}+2 x -1}{-3+i \sqrt {3}}}\, F\left (\sqrt {-\frac {2 \left (1+x \right )}{-3+i \sqrt {3}}}, \sqrt {-\frac {-3+i \sqrt {3}}{i \sqrt {3}+3}}\right )-3 \sqrt {-\frac {2 \left (1+x \right )}{-3+i \sqrt {3}}}\, \sqrt {\frac {i \sqrt {3}-2 x +1}{i \sqrt {3}+3}}\, \sqrt {\frac {i \sqrt {3}+2 x -1}{-3+i \sqrt {3}}}\, F\left (\sqrt {-\frac {2 \left (1+x \right )}{-3+i \sqrt {3}}}, \sqrt {-\frac {-3+i \sqrt {3}}{i \sqrt {3}+3}}\right )-2 x \right )}{3 \left (x^{3}+1\right )}\)	\(247\)

[In]

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

[Out]

((1+x)*(x^2-x+1))^(1/2)/(1+x)^(1/2)/(x^2-x+1)^(1/2)*(2/3*x/(x^3+1)^(1/2)+2/3*(3/2-1/2*I*3^(1/2))*((1+x)/(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(((1+x)/(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 [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.12 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.27 \[ \int \frac {1}{(1+x)^{3/2} \left (1-x+x^2\right )^{3/2}} \, dx=\frac {2 \, {\left (\sqrt {x^{2} - x + 1} \sqrt {x + 1} x + {\left (x^{3} + 1\right )} {\rm weierstrassPInverse}\left (0, -4, x\right )\right )}}{3 \, {\left (x^{3} + 1\right )}} \]

[In]

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

[Out]

2/3*(sqrt(x^2 - x + 1)*sqrt(x + 1)*x + (x^3 + 1)*weierstrassPInverse(0, -4, x))/(x^3 + 1)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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