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

Optimal. Leaf size=101 \[ -\frac {1}{2} \sqrt {-\frac {1}{5}-\frac {2 i}{5}} \tan ^{-1}\left (\frac {\sqrt {1-2 i} \sqrt {x^3-x^2-x}}{x^2-x-1}\right )-\frac {1}{2} \sqrt {-\frac {1}{5}+\frac {2 i}{5}} \tan ^{-1}\left (\frac {\sqrt {1+2 i} \sqrt {x^3-x^2-x}}{x^2-x-1}\right ) \]

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Rubi [C]  time = 0.97, antiderivative size = 239, normalized size of antiderivative = 2.37, number of steps used = 17, number of rules used = 9, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {2056, 6725, 943, 716, 1098, 934, 168, 538, 537} \begin {gather*} \frac {i \sqrt {3+\sqrt {5}} \sqrt {x} \sqrt {2 x+\sqrt {5}-1} \sqrt {1-\frac {2 x}{1+\sqrt {5}}} \Pi \left (-\frac {1}{2} i \left (1+\sqrt {5}\right );\sin ^{-1}\left (\sqrt {\frac {2}{1+\sqrt {5}}} \sqrt {x}\right )|\frac {1}{2} \left (-3-\sqrt {5}\right )\right )}{2 \sqrt {x^3-x^2-x}}-\frac {i \sqrt {3+\sqrt {5}} \sqrt {x} \sqrt {2 x+\sqrt {5}-1} \sqrt {1-\frac {2 x}{1+\sqrt {5}}} \Pi \left (\frac {1}{2} i \left (1+\sqrt {5}\right );\sin ^{-1}\left (\sqrt {\frac {2}{1+\sqrt {5}}} \sqrt {x}\right )|\frac {1}{2} \left (-3-\sqrt {5}\right )\right )}{2 \sqrt {x^3-x^2-x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((I/2)*Sqrt[3 + Sqrt[5]]*Sqrt[x]*Sqrt[-1 + Sqrt[5] + 2*x]*Sqrt[1 - (2*x)/(1 + Sqrt[5])]*EllipticPi[(-1/2*I)*(1
+ Sqrt[5]), ArcSin[Sqrt[2/(1 + Sqrt[5])]*Sqrt[x]], (-3 - Sqrt[5])/2])/Sqrt[-x - x^2 + x^3] - ((I/2)*Sqrt[3 +
Sqrt[5]]*Sqrt[x]*Sqrt[-1 + Sqrt[5] + 2*x]*Sqrt[1 - (2*x)/(1 + Sqrt[5])]*EllipticPi[(I/2)*(1 + Sqrt[5]), ArcSin
[Sqrt[2/(1 + Sqrt[5])]*Sqrt[x]], (-3 - Sqrt[5])/2])/Sqrt[-x - x^2 + x^3]

Rule 168

Int[1/(((a_.) + (b_.)*(x_))*Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x_Sym
bol] :> Dist[-2, Subst[Int[1/(Simp[b*c - a*d - b*x^2, x]*Sqrt[Simp[(d*e - c*f)/d + (f*x^2)/d, x]]*Sqrt[Simp[(d
*g - c*h)/d + (h*x^2)/d, x]]), x], x, Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && GtQ[(d*e - c
*f)/d, 0]

Rule 537

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[(1*Ellipt
icPi[(b*c)/(a*d), ArcSin[Rt[-(d/c), 2]*x], (c*f)/(d*e)])/(a*Sqrt[c]*Sqrt[e]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b,
c, d, e, f}, x] &&  !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] &&  !( !GtQ[f/e, 0] && SimplerSqrtQ[-(f/e), -(d/c)
])

Rule 538

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Dist[Sqrt[1 +
(d*x^2)/c]/Sqrt[c + d*x^2], Int[1/((a + b*x^2)*Sqrt[1 + (d*x^2)/c]*Sqrt[e + f*x^2]), x], x] /; FreeQ[{a, b, c,
d, e, f}, x] &&  !GtQ[c, 0]

Rule 716

Int[(x_)^(m_)/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[x^(2*m + 1)/Sqrt[a + b*x^
2 + c*x^4], x], x, Sqrt[x]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[m^2, 1/4]

Rule 934

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(f_.) + (g_.)*(x_)]*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Wi
th[{q = Rt[b^2 - 4*a*c, 2]}, Dist[(Sqrt[b - q + 2*c*x]*Sqrt[b + q + 2*c*x])/Sqrt[a + b*x + c*x^2], Int[1/((d +
e*x)*Sqrt[f + g*x]*Sqrt[b - q + 2*c*x]*Sqrt[b + q + 2*c*x]), x], x]] /; FreeQ[{a, b, c, d, e, f, g}, x] && Ne
Q[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]

Rule 943

Int[Sqrt[(f_.) + (g_.)*(x_)]/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist
[g/e, Int[1/(Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2]), x], x] + Dist[(e*f - d*g)/e, Int[1/((d + e*x)*Sqrt[f + g*x]
*Sqrt[a + b*x + c*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0]
&& NeQ[c*d^2 - b*d*e + a*e^2, 0]

Rule 1098

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

Rule 2056

Int[(u_.)*(P_)^(p_.), x_Symbol] :> With[{m = MinimumMonomialExponent[P, x]}, Dist[P^FracPart[p]/(x^(m*FracPart
[p])*Distrib[1/x^m, P]^FracPart[p]), Int[u*x^(m*p)*Distrib[1/x^m, P]^p, x], x]] /; FreeQ[p, x] &&  !IntegerQ[p
] && SumQ[P] && EveryQ[BinomialQ[#1, x] & , P] &&  !PolyQ[P, x, 2]

Rule 6725

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rubi steps

\begin {align*} \int \frac {x}{\left (1+x^2\right ) \sqrt {-x-x^2+x^3}} \, dx &=\frac {\left (\sqrt {x} \sqrt {-1-x+x^2}\right ) \int \frac {\sqrt {x}}{\left (1+x^2\right ) \sqrt {-1-x+x^2}} \, dx}{\sqrt {-x-x^2+x^3}}\\ &=\frac {\left (\sqrt {x} \sqrt {-1-x+x^2}\right ) \int \left (\frac {i \sqrt {x}}{2 (i-x) \sqrt {-1-x+x^2}}+\frac {i \sqrt {x}}{2 (i+x) \sqrt {-1-x+x^2}}\right ) \, dx}{\sqrt {-x-x^2+x^3}}\\ &=\frac {\left (i \sqrt {x} \sqrt {-1-x+x^2}\right ) \int \frac {\sqrt {x}}{(i-x) \sqrt {-1-x+x^2}} \, dx}{2 \sqrt {-x-x^2+x^3}}+\frac {\left (i \sqrt {x} \sqrt {-1-x+x^2}\right ) \int \frac {\sqrt {x}}{(i+x) \sqrt {-1-x+x^2}} \, dx}{2 \sqrt {-x-x^2+x^3}}\\ &=-\frac {\left (\sqrt {x} \sqrt {-1-x+x^2}\right ) \int \frac {1}{(i-x) \sqrt {x} \sqrt {-1-x+x^2}} \, dx}{2 \sqrt {-x-x^2+x^3}}+\frac {\left (\sqrt {x} \sqrt {-1-x+x^2}\right ) \int \frac {1}{\sqrt {x} (i+x) \sqrt {-1-x+x^2}} \, dx}{2 \sqrt {-x-x^2+x^3}}\\ &=-\frac {\left (\sqrt {x} \sqrt {-1-\sqrt {5}+2 x} \sqrt {-1+\sqrt {5}+2 x}\right ) \int \frac {1}{(i-x) \sqrt {x} \sqrt {-1-\sqrt {5}+2 x} \sqrt {-1+\sqrt {5}+2 x}} \, dx}{2 \sqrt {-x-x^2+x^3}}+\frac {\left (\sqrt {x} \sqrt {-1-\sqrt {5}+2 x} \sqrt {-1+\sqrt {5}+2 x}\right ) \int \frac {1}{\sqrt {x} (i+x) \sqrt {-1-\sqrt {5}+2 x} \sqrt {-1+\sqrt {5}+2 x}} \, dx}{2 \sqrt {-x-x^2+x^3}}\\ &=-\frac {\left (\sqrt {x} \sqrt {-1-\sqrt {5}+2 x} \sqrt {-1+\sqrt {5}+2 x}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-i-x^2\right ) \sqrt {-1-\sqrt {5}+2 x^2} \sqrt {-1+\sqrt {5}+2 x^2}} \, dx,x,\sqrt {x}\right )}{\sqrt {-x-x^2+x^3}}+\frac {\left (\sqrt {x} \sqrt {-1-\sqrt {5}+2 x} \sqrt {-1+\sqrt {5}+2 x}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-i+x^2\right ) \sqrt {-1-\sqrt {5}+2 x^2} \sqrt {-1+\sqrt {5}+2 x^2}} \, dx,x,\sqrt {x}\right )}{\sqrt {-x-x^2+x^3}}\\ &=-\frac {\left (\sqrt {x} \sqrt {-1+\sqrt {5}+2 x} \sqrt {1-\frac {2 x}{1+\sqrt {5}}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-i-x^2\right ) \sqrt {-1+\sqrt {5}+2 x^2} \sqrt {1+\frac {2 x^2}{-1-\sqrt {5}}}} \, dx,x,\sqrt {x}\right )}{\sqrt {-x-x^2+x^3}}+\frac {\left (\sqrt {x} \sqrt {-1+\sqrt {5}+2 x} \sqrt {1-\frac {2 x}{1+\sqrt {5}}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-i+x^2\right ) \sqrt {-1+\sqrt {5}+2 x^2} \sqrt {1+\frac {2 x^2}{-1-\sqrt {5}}}} \, dx,x,\sqrt {x}\right )}{\sqrt {-x-x^2+x^3}}\\ &=\frac {i \sqrt {3+\sqrt {5}} \sqrt {x} \sqrt {-1+\sqrt {5}+2 x} \sqrt {1-\frac {2 x}{1+\sqrt {5}}} \Pi \left (-\frac {1}{2} i \left (1+\sqrt {5}\right );\sin ^{-1}\left (\sqrt {\frac {2}{1+\sqrt {5}}} \sqrt {x}\right )|\frac {1}{2} \left (-3-\sqrt {5}\right )\right )}{2 \sqrt {-x-x^2+x^3}}-\frac {i \sqrt {3+\sqrt {5}} \sqrt {x} \sqrt {-1+\sqrt {5}+2 x} \sqrt {1-\frac {2 x}{1+\sqrt {5}}} \Pi \left (\frac {1}{2} i \left (1+\sqrt {5}\right );\sin ^{-1}\left (\sqrt {\frac {2}{1+\sqrt {5}}} \sqrt {x}\right )|\frac {1}{2} \left (-3-\sqrt {5}\right )\right )}{2 \sqrt {-x-x^2+x^3}}\\ \end {align*}

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Mathematica [C]  time = 0.64, size = 152, normalized size = 1.50 \begin {gather*} \frac {\sqrt {\frac {2}{1+\sqrt {5}}} \sqrt {x} \sqrt {-x^2+x+1} \left (\Pi \left (\frac {1}{2} i \left (-1+\sqrt {5}\right );i \sinh ^{-1}\left (\sqrt {\frac {2}{-1+\sqrt {5}}} \sqrt {x}\right )|\frac {1}{2} \left (-3+\sqrt {5}\right )\right )-\Pi \left (-\frac {1}{2} i \left (-1+\sqrt {5}\right );i \sinh ^{-1}\left (\sqrt {\frac {2}{-1+\sqrt {5}}} \sqrt {x}\right )|\frac {1}{2} \left (-3+\sqrt {5}\right )\right )\right )}{\sqrt {x \left (x^2-x-1\right )}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(Sqrt[2/(1 + Sqrt[5])]*Sqrt[x]*Sqrt[1 + x - x^2]*(-EllipticPi[(-1/2*I)*(-1 + Sqrt[5]), I*ArcSinh[Sqrt[2/(-1 +
Sqrt[5])]*Sqrt[x]], (-3 + Sqrt[5])/2] + EllipticPi[(I/2)*(-1 + Sqrt[5]), I*ArcSinh[Sqrt[2/(-1 + Sqrt[5])]*Sqrt
[x]], (-3 + Sqrt[5])/2]))/Sqrt[x*(-1 - x + x^2)]

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IntegrateAlgebraic [A]  time = 0.31, size = 101, normalized size = 1.00 \begin {gather*} -\frac {1}{2} \sqrt {-\frac {1}{5}-\frac {2 i}{5}} \tan ^{-1}\left (\frac {\sqrt {1-2 i} \sqrt {-x-x^2+x^3}}{-1-x+x^2}\right )-\frac {1}{2} \sqrt {-\frac {1}{5}+\frac {2 i}{5}} \tan ^{-1}\left (\frac {\sqrt {1+2 i} \sqrt {-x-x^2+x^3}}{-1-x+x^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*(Sqrt[-1/5 - (2*I)/5]*ArcTan[(Sqrt[1 - 2*I]*Sqrt[-x - x^2 + x^3])/(-1 - x + x^2)]) - (Sqrt[-1/5 + (2*I)/5
]*ArcTan[(Sqrt[1 + 2*I]*Sqrt[-x - x^2 + x^3])/(-1 - x + x^2)])/2

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fricas [B]  time = 1.01, size = 2408, normalized size = 23.84

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/160*5^(1/4)*(sqrt(5)*sqrt(2) + sqrt(2))*sqrt(-sqrt(5) + 5)*log(5*(5*x^4 - 20*x^3 + 2*5^(1/4)*sqrt(x^3 - x^2
- x)*(sqrt(5)*sqrt(2)*(x^2 - x - 1) + 5*sqrt(2)*x)*sqrt(-sqrt(5) + 5) + 30*x^2 + 20*sqrt(5)*(x^3 - x^2 - x) +
20*x + 5)/(x^4 + 2*x^2 + 1)) - 1/160*5^(1/4)*(sqrt(5)*sqrt(2) + sqrt(2))*sqrt(-sqrt(5) + 5)*log(5*(5*x^4 - 20*
x^3 - 2*5^(1/4)*sqrt(x^3 - x^2 - x)*(sqrt(5)*sqrt(2)*(x^2 - x - 1) + 5*sqrt(2)*x)*sqrt(-sqrt(5) + 5) + 30*x^2
+ 20*sqrt(5)*(x^3 - x^2 - x) + 20*x + 5)/(x^4 + 2*x^2 + 1)) + 1/20*5^(1/4)*sqrt(2)*sqrt(-sqrt(5) + 5)*arctan(-
1/100*(50*x^11 + 650*x^10 - 3350*x^9 - 2200*x^8 + 14200*x^7 + 700*x^6 - 14200*x^5 - 2200*x^4 + 3350*x^3 + 650*
x^2 + 5*sqrt(x^3 - x^2 - x)*(5^(3/4)*(sqrt(5)*sqrt(2)*(3*x^9 - 10*x^8 - 20*x^7 + 54*x^6 + 50*x^5 - 54*x^4 - 20
*x^3 + 10*x^2 + 3*x) - sqrt(2)*(x^10 - 7*x^9 - 7*x^8 + 76*x^7 + 24*x^6 - 186*x^5 - 24*x^4 + 76*x^3 + 7*x^2 - 7
*x - 1)) + 2*5^(1/4)*(sqrt(5)*sqrt(2)*(x^9 + 19*x^8 - 52*x^7 + 3*x^6 + 70*x^5 - 3*x^4 - 52*x^3 - 19*x^2 + x) -
5*sqrt(2)*(x^9 - x^8 + 16*x^7 - 33*x^6 - 18*x^5 + 33*x^4 + 16*x^3 + x^2 + x)))*sqrt(-sqrt(5) + 5) - sqrt(5)*(
120*x^10 + 80*x^9 - 840*x^8 - 240*x^7 + 1600*x^6 + 240*x^5 - 840*x^4 - 80*x^3 + 120*x^2 + sqrt(x^3 - x^2 - x)*
(5^(3/4)*(sqrt(5)*sqrt(2)*(x^9 + 2*x^8 + 24*x^7 - 62*x^6 - 50*x^5 + 62*x^4 + 24*x^3 - 2*x^2 + x) - sqrt(2)*(x^
10 - 7*x^9 - 27*x^8 + 96*x^7 + 4*x^6 - 146*x^5 - 4*x^4 + 96*x^3 + 27*x^2 - 7*x - 1)) + 2*5^(1/4)*(sqrt(5)*sqrt
(2)*(x^9 + 9*x^8 - 42*x^7 - 7*x^6 + 90*x^5 + 7*x^4 - 42*x^3 - 9*x^2 + x) - 5*sqrt(2)*(x^9 - 3*x^8 - 14*x^7 + 2
9*x^6 + 18*x^5 - 29*x^4 - 14*x^3 + 3*x^2 + x)))*sqrt(-sqrt(5) + 5) + 2*sqrt(5)*(5*x^11 - 25*x^10 - 105*x^9 + 4
40*x^8 + 50*x^7 - 830*x^6 - 50*x^5 + 440*x^4 + 105*x^3 - 25*x^2 - sqrt(5)*(x^11 + 3*x^10 - 37*x^9 + 96*x^8 - 6
*x^7 - 166*x^6 + 6*x^5 + 96*x^4 + 37*x^3 + 3*x^2 - x) - 5*x) - 40*sqrt(5)*(x^10 + 6*x^9 - 23*x^8 - 2*x^7 + 40*
x^6 + 2*x^5 - 23*x^4 - 6*x^3 + x^2))*sqrt((5*x^4 - 20*x^3 + 2*5^(1/4)*sqrt(x^3 - x^2 - x)*(sqrt(5)*sqrt(2)*(x^
2 - x - 1) + 5*sqrt(2)*x)*sqrt(-sqrt(5) + 5) + 30*x^2 + 20*sqrt(5)*(x^3 - x^2 - x) + 20*x + 5)/(x^4 + 2*x^2 +
1)) + 10*sqrt(5)*(5*x^11 - 15*x^10 - 15*x^9 + 20*x^8 - 20*x^7 + 70*x^6 + 20*x^5 + 20*x^4 + 15*x^3 - 15*x^2 - s
qrt(5)*(x^11 + 13*x^10 - 67*x^9 - 44*x^8 + 284*x^7 + 14*x^6 - 284*x^5 - 44*x^4 + 67*x^3 + 13*x^2 - x) - 5*x) -
50*sqrt(5)*(x^11 - 3*x^10 - 3*x^9 + 4*x^8 - 4*x^7 + 14*x^6 + 4*x^5 + 4*x^4 + 3*x^3 - 3*x^2 - x) - 50*x)/(x^11
- 9*x^10 - 45*x^9 + 180*x^8 + 18*x^7 - 326*x^6 - 18*x^5 + 180*x^4 + 45*x^3 - 9*x^2 - x)) + 1/20*5^(1/4)*sqrt(
2)*sqrt(-sqrt(5) + 5)*arctan(1/100*(50*x^11 + 650*x^10 - 3350*x^9 - 2200*x^8 + 14200*x^7 + 700*x^6 - 14200*x^5
- 2200*x^4 + 3350*x^3 + 650*x^2 - 5*sqrt(x^3 - x^2 - x)*(5^(3/4)*(sqrt(5)*sqrt(2)*(3*x^9 - 10*x^8 - 20*x^7 +
54*x^6 + 50*x^5 - 54*x^4 - 20*x^3 + 10*x^2 + 3*x) - sqrt(2)*(x^10 - 7*x^9 - 7*x^8 + 76*x^7 + 24*x^6 - 186*x^5
- 24*x^4 + 76*x^3 + 7*x^2 - 7*x - 1)) + 2*5^(1/4)*(sqrt(5)*sqrt(2)*(x^9 + 19*x^8 - 52*x^7 + 3*x^6 + 70*x^5 - 3
*x^4 - 52*x^3 - 19*x^2 + x) - 5*sqrt(2)*(x^9 - x^8 + 16*x^7 - 33*x^6 - 18*x^5 + 33*x^4 + 16*x^3 + x^2 + x)))*s
qrt(-sqrt(5) + 5) - sqrt(5)*(120*x^10 + 80*x^9 - 840*x^8 - 240*x^7 + 1600*x^6 + 240*x^5 - 840*x^4 - 80*x^3 + 1
20*x^2 - sqrt(x^3 - x^2 - x)*(5^(3/4)*(sqrt(5)*sqrt(2)*(x^9 + 2*x^8 + 24*x^7 - 62*x^6 - 50*x^5 + 62*x^4 + 24*x
^3 - 2*x^2 + x) - sqrt(2)*(x^10 - 7*x^9 - 27*x^8 + 96*x^7 + 4*x^6 - 146*x^5 - 4*x^4 + 96*x^3 + 27*x^2 - 7*x -
1)) + 2*5^(1/4)*(sqrt(5)*sqrt(2)*(x^9 + 9*x^8 - 42*x^7 - 7*x^6 + 90*x^5 + 7*x^4 - 42*x^3 - 9*x^2 + x) - 5*sqrt
(2)*(x^9 - 3*x^8 - 14*x^7 + 29*x^6 + 18*x^5 - 29*x^4 - 14*x^3 + 3*x^2 + x)))*sqrt(-sqrt(5) + 5) + 2*sqrt(5)*(5
*x^11 - 25*x^10 - 105*x^9 + 440*x^8 + 50*x^7 - 830*x^6 - 50*x^5 + 440*x^4 + 105*x^3 - 25*x^2 - sqrt(5)*(x^11 +
3*x^10 - 37*x^9 + 96*x^8 - 6*x^7 - 166*x^6 + 6*x^5 + 96*x^4 + 37*x^3 + 3*x^2 - x) - 5*x) - 40*sqrt(5)*(x^10 +
6*x^9 - 23*x^8 - 2*x^7 + 40*x^6 + 2*x^5 - 23*x^4 - 6*x^3 + x^2))*sqrt((5*x^4 - 20*x^3 - 2*5^(1/4)*sqrt(x^3 -
x^2 - x)*(sqrt(5)*sqrt(2)*(x^2 - x - 1) + 5*sqrt(2)*x)*sqrt(-sqrt(5) + 5) + 30*x^2 + 20*sqrt(5)*(x^3 - x^2 - x
) + 20*x + 5)/(x^4 + 2*x^2 + 1)) + 10*sqrt(5)*(5*x^11 - 15*x^10 - 15*x^9 + 20*x^8 - 20*x^7 + 70*x^6 + 20*x^5 +
20*x^4 + 15*x^3 - 15*x^2 - sqrt(5)*(x^11 + 13*x^10 - 67*x^9 - 44*x^8 + 284*x^7 + 14*x^6 - 284*x^5 - 44*x^4 +
67*x^3 + 13*x^2 - x) - 5*x) - 50*sqrt(5)*(x^11 - 3*x^10 - 3*x^9 + 4*x^8 - 4*x^7 + 14*x^6 + 4*x^5 + 4*x^4 + 3*x
^3 - 3*x^2 - x) - 50*x)/(x^11 - 9*x^10 - 45*x^9 + 180*x^8 + 18*x^7 - 326*x^6 - 18*x^5 + 180*x^4 + 45*x^3 - 9*x
^2 - x))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{\sqrt {x^{3} - x^{2} - x} {\left (x^{2} + 1\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 1.02, size = 620, normalized size = 6.14

method	result	size
default	\(\frac {\sqrt {\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}-\frac {1}{2 \left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right )}+\frac {\sqrt {5}}{\sqrt {5}-1}}\, \sqrt {-5 x \sqrt {5}+\frac {5 \sqrt {5}}{2}+\frac {25}{2}}\, \sqrt {-\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\, \EllipticPi \left (\sqrt {\frac {x -\frac {1}{2}+\frac {\sqrt {5}}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}, \frac {\frac {1}{2}-\frac {\sqrt {5}}{2}}{\frac {1}{2}-i-\frac {\sqrt {5}}{2}}, \frac {\sqrt {5}\, \sqrt {\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) \sqrt {5}}}{5}\right ) \sqrt {5}}{10 \sqrt {x^{3}-x^{2}-x}\, \left (\frac {1}{2}-i-\frac {\sqrt {5}}{2}\right )}-\frac {\sqrt {\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}-\frac {1}{2 \left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right )}+\frac {\sqrt {5}}{\sqrt {5}-1}}\, \sqrt {-5 x \sqrt {5}+\frac {5 \sqrt {5}}{2}+\frac {25}{2}}\, \sqrt {-\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\, \EllipticPi \left (\sqrt {\frac {x -\frac {1}{2}+\frac {\sqrt {5}}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}, \frac {\frac {1}{2}-\frac {\sqrt {5}}{2}}{\frac {1}{2}-i-\frac {\sqrt {5}}{2}}, \frac {\sqrt {5}\, \sqrt {\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) \sqrt {5}}}{5}\right )}{10 \sqrt {x^{3}-x^{2}-x}\, \left (\frac {1}{2}-i-\frac {\sqrt {5}}{2}\right )}+\frac {\sqrt {\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}-\frac {1}{2 \left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right )}+\frac {\sqrt {5}}{\sqrt {5}-1}}\, \sqrt {-5 x \sqrt {5}+\frac {5 \sqrt {5}}{2}+\frac {25}{2}}\, \sqrt {-\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\, \EllipticPi \left (\sqrt {\frac {x -\frac {1}{2}+\frac {\sqrt {5}}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}, \frac {\frac {1}{2}-\frac {\sqrt {5}}{2}}{\frac {1}{2}+i-\frac {\sqrt {5}}{2}}, \frac {\sqrt {5}\, \sqrt {\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) \sqrt {5}}}{5}\right ) \sqrt {5}}{10 \sqrt {x^{3}-x^{2}-x}\, \left (\frac {1}{2}+i-\frac {\sqrt {5}}{2}\right )}-\frac {\sqrt {\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}-\frac {1}{2 \left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right )}+\frac {\sqrt {5}}{\sqrt {5}-1}}\, \sqrt {-5 x \sqrt {5}+\frac {5 \sqrt {5}}{2}+\frac {25}{2}}\, \sqrt {-\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\, \EllipticPi \left (\sqrt {\frac {x -\frac {1}{2}+\frac {\sqrt {5}}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}, \frac {\frac {1}{2}-\frac {\sqrt {5}}{2}}{\frac {1}{2}+i-\frac {\sqrt {5}}{2}}, \frac {\sqrt {5}\, \sqrt {\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) \sqrt {5}}}{5}\right )}{10 \sqrt {x^{3}-x^{2}-x}\, \left (\frac {1}{2}+i-\frac {\sqrt {5}}{2}\right )}\)	\(620\)
elliptic	\(\frac {\sqrt {\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}-\frac {1}{2 \left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right )}+\frac {\sqrt {5}}{\sqrt {5}-1}}\, \sqrt {-5 x \sqrt {5}+\frac {5 \sqrt {5}}{2}+\frac {25}{2}}\, \sqrt {-\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\, \EllipticPi \left (\sqrt {\frac {x -\frac {1}{2}+\frac {\sqrt {5}}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}, \frac {\frac {1}{2}-\frac {\sqrt {5}}{2}}{\frac {1}{2}-i-\frac {\sqrt {5}}{2}}, \frac {\sqrt {5}\, \sqrt {\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) \sqrt {5}}}{5}\right ) \sqrt {5}}{10 \sqrt {x^{3}-x^{2}-x}\, \left (\frac {1}{2}-i-\frac {\sqrt {5}}{2}\right )}-\frac {\sqrt {\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}-\frac {1}{2 \left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right )}+\frac {\sqrt {5}}{\sqrt {5}-1}}\, \sqrt {-5 x \sqrt {5}+\frac {5 \sqrt {5}}{2}+\frac {25}{2}}\, \sqrt {-\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\, \EllipticPi \left (\sqrt {\frac {x -\frac {1}{2}+\frac {\sqrt {5}}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}, \frac {\frac {1}{2}-\frac {\sqrt {5}}{2}}{\frac {1}{2}-i-\frac {\sqrt {5}}{2}}, \frac {\sqrt {5}\, \sqrt {\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) \sqrt {5}}}{5}\right )}{10 \sqrt {x^{3}-x^{2}-x}\, \left (\frac {1}{2}-i-\frac {\sqrt {5}}{2}\right )}+\frac {\sqrt {\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}-\frac {1}{2 \left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right )}+\frac {\sqrt {5}}{\sqrt {5}-1}}\, \sqrt {-5 x \sqrt {5}+\frac {5 \sqrt {5}}{2}+\frac {25}{2}}\, \sqrt {-\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\, \EllipticPi \left (\sqrt {\frac {x -\frac {1}{2}+\frac {\sqrt {5}}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}, \frac {\frac {1}{2}-\frac {\sqrt {5}}{2}}{\frac {1}{2}+i-\frac {\sqrt {5}}{2}}, \frac {\sqrt {5}\, \sqrt {\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) \sqrt {5}}}{5}\right ) \sqrt {5}}{10 \sqrt {x^{3}-x^{2}-x}\, \left (\frac {1}{2}+i-\frac {\sqrt {5}}{2}\right )}-\frac {\sqrt {\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}-\frac {1}{2 \left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right )}+\frac {\sqrt {5}}{\sqrt {5}-1}}\, \sqrt {-5 x \sqrt {5}+\frac {5 \sqrt {5}}{2}+\frac {25}{2}}\, \sqrt {-\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\, \EllipticPi \left (\sqrt {\frac {x -\frac {1}{2}+\frac {\sqrt {5}}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}, \frac {\frac {1}{2}-\frac {\sqrt {5}}{2}}{\frac {1}{2}+i-\frac {\sqrt {5}}{2}}, \frac {\sqrt {5}\, \sqrt {\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) \sqrt {5}}}{5}\right )}{10 \sqrt {x^{3}-x^{2}-x}\, \left (\frac {1}{2}+i-\frac {\sqrt {5}}{2}\right )}\)	\(620\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*(x/(1/2*5^(1/2)-1/2)-1/2/(1/2*5^(1/2)-1/2)+1/2/(1/2*5^(1/2)-1/2)*5^(1/2))^(1/2)*(-5*x*5^(1/2)+5/2*5^(1/2)
+25/2)^(1/2)*(-x/(1/2*5^(1/2)-1/2))^(1/2)/(x^3-x^2-x)^(1/2)/(1/2-I-1/2*5^(1/2))*EllipticPi(((x-1/2+1/2*5^(1/2)
)/(1/2*5^(1/2)-1/2))^(1/2),(1/2-1/2*5^(1/2))/(1/2-I-1/2*5^(1/2)),1/5*5^(1/2)*((1/2*5^(1/2)-1/2)*5^(1/2))^(1/2)
)*5^(1/2)-1/10*(x/(1/2*5^(1/2)-1/2)-1/2/(1/2*5^(1/2)-1/2)+1/2/(1/2*5^(1/2)-1/2)*5^(1/2))^(1/2)*(-5*x*5^(1/2)+5
/2*5^(1/2)+25/2)^(1/2)*(-x/(1/2*5^(1/2)-1/2))^(1/2)/(x^3-x^2-x)^(1/2)/(1/2-I-1/2*5^(1/2))*EllipticPi(((x-1/2+1
/2*5^(1/2))/(1/2*5^(1/2)-1/2))^(1/2),(1/2-1/2*5^(1/2))/(1/2-I-1/2*5^(1/2)),1/5*5^(1/2)*((1/2*5^(1/2)-1/2)*5^(1
/2))^(1/2))+1/10*(x/(1/2*5^(1/2)-1/2)-1/2/(1/2*5^(1/2)-1/2)+1/2/(1/2*5^(1/2)-1/2)*5^(1/2))^(1/2)*(-5*x*5^(1/2)
+5/2*5^(1/2)+25/2)^(1/2)*(-x/(1/2*5^(1/2)-1/2))^(1/2)/(x^3-x^2-x)^(1/2)/(1/2+I-1/2*5^(1/2))*EllipticPi(((x-1/2
+1/2*5^(1/2))/(1/2*5^(1/2)-1/2))^(1/2),(1/2-1/2*5^(1/2))/(1/2+I-1/2*5^(1/2)),1/5*5^(1/2)*((1/2*5^(1/2)-1/2)*5^
(1/2))^(1/2))*5^(1/2)-1/10*(x/(1/2*5^(1/2)-1/2)-1/2/(1/2*5^(1/2)-1/2)+1/2/(1/2*5^(1/2)-1/2)*5^(1/2))^(1/2)*(-5
*x*5^(1/2)+5/2*5^(1/2)+25/2)^(1/2)*(-x/(1/2*5^(1/2)-1/2))^(1/2)/(x^3-x^2-x)^(1/2)/(1/2+I-1/2*5^(1/2))*Elliptic
Pi(((x-1/2+1/2*5^(1/2))/(1/2*5^(1/2)-1/2))^(1/2),(1/2-1/2*5^(1/2))/(1/2+I-1/2*5^(1/2)),1/5*5^(1/2)*((1/2*5^(1/
2)-1/2)*5^(1/2))^(1/2))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{\sqrt {x^{3} - x^{2} - x} {\left (x^{2} + 1\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 1.04, size = 178, normalized size = 1.76 \begin {gather*} \frac {\sqrt {\frac {x}{\frac {\sqrt {5}}{2}+\frac {1}{2}}}\,\sqrt {\frac {x+\frac {\sqrt {5}}{2}-\frac {1}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\,\left (\sqrt {5}+1\right )\,\sqrt {\frac {\frac {\sqrt {5}}{2}-x+\frac {1}{2}}{\frac {\sqrt {5}}{2}+\frac {1}{2}}}\,\left (\Pi \left (-\frac {\sqrt {5}\,1{}\mathrm {i}}{2}-\frac {1}{2}{}\mathrm {i};\mathrm {asin}\left (\sqrt {\frac {x}{\frac {\sqrt {5}}{2}+\frac {1}{2}}}\right )\middle |-\frac {\frac {\sqrt {5}}{2}+\frac {1}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}\right )-\Pi \left (\frac {\sqrt {5}\,1{}\mathrm {i}}{2}+\frac {1}{2}{}\mathrm {i};\mathrm {asin}\left (\sqrt {\frac {x}{\frac {\sqrt {5}}{2}+\frac {1}{2}}}\right )\middle |-\frac {\frac {\sqrt {5}}{2}+\frac {1}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}\right )\right )\,1{}\mathrm {i}}{2\,\sqrt {x^3-x^2-\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right )\,\left (\frac {\sqrt {5}}{2}+\frac {1}{2}\right )\,x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((x/(5^(1/2)/2 + 1/2))^(1/2)*((x + 5^(1/2)/2 - 1/2)/(5^(1/2)/2 - 1/2))^(1/2)*(5^(1/2) + 1)*((5^(1/2)/2 - x + 1
/2)/(5^(1/2)/2 + 1/2))^(1/2)*(ellipticPi(- (5^(1/2)*1i)/2 - 1i/2, asin((x/(5^(1/2)/2 + 1/2))^(1/2)), -(5^(1/2)
/2 + 1/2)/(5^(1/2)/2 - 1/2)) - ellipticPi((5^(1/2)*1i)/2 + 1i/2, asin((x/(5^(1/2)/2 + 1/2))^(1/2)), -(5^(1/2)/
2 + 1/2)/(5^(1/2)/2 - 1/2)))*1i)/(2*(x^3 - x^2 - x*(5^(1/2)/2 - 1/2)*(5^(1/2)/2 + 1/2))^(1/2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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