\(\int \frac {1+k^{3/2} x^3}{\sqrt {(1-x^2) (1-k^2 x^2)} (-1+k^{3/2} x^3)} \, dx\) [1824]

Optimal result

Integrand size = 47, antiderivative size = 124 \[ \int \frac {1+k^{3/2} x^3}{\sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (-1+k^{3/2} x^3\right )} \, dx=-\frac {2 \arctan \left (\frac {(-1+k) x}{1-2 \sqrt {k} x+k x^2+\sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}}\right )}{3 (-1+k)}-\frac {4 \arctan \left (\frac {\sqrt {1+k+k^2} x}{1+\sqrt {k} x+k x^2+\sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}}\right )}{3 \sqrt {1+k+k^2}} \]

[Out]

-2*arctan((-1+k)*x/(1-2*k^(1/2)*x+k*x^2+(1+(-k^2-1)*x^2+k^2*x^4)^(1/2)))/(-3+3*k)-4/3*arctan((k^2+k+1)^(1/2)*x
/(1+k^(1/2)*x+k*x^2+(1+(-k^2-1)*x^2+k^2*x^4)^(1/2)))/(k^2+k+1)^(1/2)

Rubi [C] (verified)

Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.

Time = 1.62 (sec) , antiderivative size = 709, normalized size of antiderivative = 5.72, number of steps used = 28, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.277, Rules used = {1976, 6857, 1117, 1738, 1224, 1712, 209, 1261, 738, 210, 1230, 1720, 212} \[ \int \frac {1+k^{3/2} x^3}{\sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (-1+k^{3/2} x^3\right )} \, dx=-\frac {\arctan \left (\frac {(1-k) x}{\sqrt {k^2 x^4-\left (k^2+1\right ) x^2+1}}\right )}{3 (1-k)}-\frac {2 \arctan \left (\frac {\sqrt {k^2+k+1} x}{\sqrt {k^2 x^4-\left (k^2+1\right ) x^2+1}}\right )}{3 \sqrt {k^2+k+1}}-\frac {\arctan \left (\frac {(1-k) \left (k x^2+1\right )}{2 \sqrt {k} \sqrt {k^2 x^4-\left (k^2+1\right ) x^2+1}}\right )}{3 (1-k)}-\frac {\left (1-\sqrt [3]{-1}\right ) \left (k x^2+1\right ) \sqrt {\frac {k^2 x^4-\left (k^2+1\right ) x^2+1}{\left (k x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt {k} x\right ),\frac {(k+1)^2}{4 k}\right )}{3 \sqrt {k} \sqrt {k^2 x^4-\left (k^2+1\right ) x^2+1}}-\frac {\left (k x^2+1\right ) \sqrt {\frac {k^2 x^4-\left (k^2+1\right ) x^2+1}{\left (k x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt {k} x\right ),\frac {(k+1)^2}{4 k}\right )}{3 \left (1-\sqrt [3]{-1}\right ) \sqrt {k} \sqrt {k^2 x^4-\left (k^2+1\right ) x^2+1}}+\frac {\left (k x^2+1\right ) \sqrt {\frac {k^2 x^4-\left (k^2+1\right ) x^2+1}{\left (k x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt {k} x\right ),\frac {(k+1)^2}{4 k}\right )}{3 \sqrt {k} \sqrt {k^2 x^4-\left (k^2+1\right ) x^2+1}}+\frac {(-1)^{2/3} \sqrt {2} \text {arctanh}\left (\frac {-\left (\sqrt [3]{-1} \left (k^2+1\right )+2 k\right ) k x^2+k^2+2 \sqrt [3]{-1} k+1}{\sqrt {2} \sqrt {k} \sqrt {\left (1+i \sqrt {3}\right ) \left (k^2+k+1\right )} \sqrt {k^2 x^4-\left (k^2+1\right ) x^2+1}}\right )}{3 \sqrt {\left (1+i \sqrt {3}\right ) \left (k^2+k+1\right )}}-\frac {\sqrt [3]{-1} \sqrt {2} \text {arctanh}\left (\frac {-\left (2 k-(-1)^{2/3} \left (k^2+1\right )\right ) k x^2+k^2-2 (-1)^{2/3} k+1}{\sqrt {2} \sqrt {k} \sqrt {\left (1-i \sqrt {3}\right ) \left (k^2+k+1\right )} \sqrt {k^2 x^4-\left (k^2+1\right ) x^2+1}}\right )}{3 \sqrt {\left (1-i \sqrt {3}\right ) \left (k^2+k+1\right )}} \]

[In]

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

[Out]

-1/3*ArcTan[((1 - k)*x)/Sqrt[1 - (1 + k^2)*x^2 + k^2*x^4]]/(1 - k) - (2*ArcTan[(Sqrt[1 + k + k^2]*x)/Sqrt[1 -
(1 + k^2)*x^2 + k^2*x^4]])/(3*Sqrt[1 + k + k^2]) - ArcTan[((1 - k)*(1 + k*x^2))/(2*Sqrt[k]*Sqrt[1 - (1 + k^2)*
x^2 + k^2*x^4])]/(3*(1 - k)) + ((-1)^(2/3)*Sqrt[2]*ArcTanh[(1 + 2*(-1)^(1/3)*k + k^2 - k*(2*k + (-1)^(1/3)*(1
+ k^2))*x^2)/(Sqrt[2]*Sqrt[k]*Sqrt[(1 + I*Sqrt[3])*(1 + k + k^2)]*Sqrt[1 - (1 + k^2)*x^2 + k^2*x^4])])/(3*Sqrt
[(1 + I*Sqrt[3])*(1 + k + k^2)]) - ((-1)^(1/3)*Sqrt[2]*ArcTanh[(1 - 2*(-1)^(2/3)*k + k^2 - k*(2*k - (-1)^(2/3)
*(1 + k^2))*x^2)/(Sqrt[2]*Sqrt[k]*Sqrt[(1 - I*Sqrt[3])*(1 + k + k^2)]*Sqrt[1 - (1 + k^2)*x^2 + k^2*x^4])])/(3*
Sqrt[(1 - I*Sqrt[3])*(1 + k + k^2)]) + ((1 + k*x^2)*Sqrt[(1 - (1 + k^2)*x^2 + k^2*x^4)/(1 + k*x^2)^2]*Elliptic
F[2*ArcTan[Sqrt[k]*x], (1 + k)^2/(4*k)])/(3*Sqrt[k]*Sqrt[1 - (1 + k^2)*x^2 + k^2*x^4]) - ((1 + k*x^2)*Sqrt[(1
- (1 + k^2)*x^2 + k^2*x^4)/(1 + k*x^2)^2]*EllipticF[2*ArcTan[Sqrt[k]*x], (1 + k)^2/(4*k)])/(3*(1 - (-1)^(1/3))
*Sqrt[k]*Sqrt[1 - (1 + k^2)*x^2 + k^2*x^4]) - ((1 - (-1)^(1/3))*(1 + k*x^2)*Sqrt[(1 - (1 + k^2)*x^2 + k^2*x^4)
/(1 + k*x^2)^2]*EllipticF[2*ArcTan[Sqrt[k]*x], (1 + k)^2/(4*k)])/(3*Sqrt[k]*Sqrt[1 - (1 + k^2)*x^2 + k^2*x^4])

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 738

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

Rule 1117

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

Rule 1224

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

Rule 1230

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

Rule 1261

Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[
Int[(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x]

Rule 1712

Int[((A_) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4]), x_Symbol] :> Dist[
A, Subst[Int[1/(d - (b*d - 2*a*e)*x^2), x], x, x/Sqrt[a + b*x^2 + c*x^4]], x] /; FreeQ[{a, b, c, d, e, A, B},
x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[c*d^2 - a*e^2, 0] && EqQ[B*d + A*e, 0]

Rule 1720

Int[((A_) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4]), x_Symbol] :> With[
{q = Rt[B/A, 2]}, Simp[(-(B*d - A*e))*(ArcTan[Rt[-b + c*(d/e) + a*(e/d), 2]*(x/Sqrt[a + b*x^2 + c*x^4])]/(2*d*
e*Rt[-b + c*(d/e) + a*(e/d), 2])), x] + Simp[(B*d + A*e)*(A + B*x^2)*(Sqrt[A^2*((a + b*x^2 + c*x^4)/(a*(A + B*
x^2)^2))]/(4*d*e*A*q*Sqrt[a + b*x^2 + c*x^4]))*EllipticPi[Cancel[-(B*d - A*e)^2/(4*d*e*A*B)], 2*ArcTan[q*x], 1
/2 - b*(A/(4*a*B))], x]] /; FreeQ[{a, b, c, d, e, A, B}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^
2, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a] && EqQ[c*A^2 - a*B^2, 0]

Rule 1738

Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4]), x_Symbol] :> Dist[d, Int[1/((d^2 - e^2*x
^2)*Sqrt[a + b*x^2 + c*x^4]), x], x] - Dist[e, Int[x/((d^2 - e^2*x^2)*Sqrt[a + b*x^2 + c*x^4]), x], x] /; Free
Q[{a, b, c, d, e}, x]

Rule 1976

Int[(u_.)*((e_.)*((a_.) + (b_.)*(x_)^(n_.))*((c_) + (d_.)*(x_)^(n_.)))^(p_), x_Symbol] :> Int[u*(a*c*e + (b*c
+ a*d)*e*x^n + b*d*e*x^(2*n))^p, x] /; FreeQ[{a, b, c, d, e, n, p}, x]

Rule 6857

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*} \text {integral}& = \int \frac {1+k^{3/2} x^3}{\left (-1+k^{3/2} x^3\right ) \sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}} \, dx \\ & = \int \left (\frac {1}{\sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}}+\frac {2}{\left (-1+k^{3/2} x^3\right ) \sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}}\right ) \, dx \\ & = 2 \int \frac {1}{\left (-1+k^{3/2} x^3\right ) \sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}} \, dx+\int \frac {1}{\sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}} \, dx \\ & = \frac {\left (1+k x^2\right ) \sqrt {\frac {1-\left (1+k^2\right ) x^2+k^2 x^4}{\left (1+k x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt {k} x\right ),\frac {(1+k)^2}{4 k}\right )}{2 \sqrt {k} \sqrt {1-\left (1+k^2\right ) x^2+k^2 x^4}}+2 \int \left (-\frac {1}{3 \left (1-\sqrt {k} x\right ) \sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}}-\frac {1}{3 \left (1+\sqrt [3]{-1} \sqrt {k} x\right ) \sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}}-\frac {1}{3 \left (1-(-1)^{2/3} \sqrt {k} x\right ) \sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}}\right ) \, dx \\ & = \frac {\left (1+k x^2\right ) \sqrt {\frac {1-\left (1+k^2\right ) x^2+k^2 x^4}{\left (1+k x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt {k} x\right ),\frac {(1+k)^2}{4 k}\right )}{2 \sqrt {k} \sqrt {1-\left (1+k^2\right ) x^2+k^2 x^4}}-\frac {2}{3} \int \frac {1}{\left (1-\sqrt {k} x\right ) \sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}} \, dx-\frac {2}{3} \int \frac {1}{\left (1+\sqrt [3]{-1} \sqrt {k} x\right ) \sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}} \, dx-\frac {2}{3} \int \frac {1}{\left (1-(-1)^{2/3} \sqrt {k} x\right ) \sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}} \, dx \\ & = \frac {\left (1+k x^2\right ) \sqrt {\frac {1-\left (1+k^2\right ) x^2+k^2 x^4}{\left (1+k x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt {k} x\right ),\frac {(1+k)^2}{4 k}\right )}{2 \sqrt {k} \sqrt {1-\left (1+k^2\right ) x^2+k^2 x^4}}-\frac {2}{3} \int \frac {1}{\left (1-k x^2\right ) \sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}} \, dx-\frac {2}{3} \int \frac {1}{\left (1+\sqrt [3]{-1} k x^2\right ) \sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}} \, dx-\frac {2}{3} \int \frac {1}{\left (1-(-1)^{2/3} k x^2\right ) \sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}} \, dx-\frac {1}{3} \left (2 \sqrt {k}\right ) \int \frac {x}{\left (1-k x^2\right ) \sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}} \, dx+\frac {1}{3} \left (2 \sqrt [3]{-1} \sqrt {k}\right ) \int \frac {x}{\left (1-(-1)^{2/3} k x^2\right ) \sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}} \, dx-\frac {1}{3} \left (2 (-1)^{2/3} \sqrt {k}\right ) \int \frac {x}{\left (1+\sqrt [3]{-1} k x^2\right ) \sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}} \, dx \\ & = \frac {\left (1+k x^2\right ) \sqrt {\frac {1-\left (1+k^2\right ) x^2+k^2 x^4}{\left (1+k x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt {k} x\right ),\frac {(1+k)^2}{4 k}\right )}{2 \sqrt {k} \sqrt {1-\left (1+k^2\right ) x^2+k^2 x^4}}-\frac {1}{3} \int \frac {1}{\sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}} \, dx-\frac {1}{3} \int \frac {1+k x^2}{\left (1-k x^2\right ) \sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}} \, dx-\frac {2 \int \frac {1}{\sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}} \, dx}{3 \left (1-\sqrt [3]{-1}\right )}-\frac {1}{3} \left (2 \left (1-\sqrt [3]{-1}\right )\right ) \int \frac {1}{\sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}} \, dx-\frac {1}{3} \left (2 \left (1+(-1)^{2/3}\right )\right ) \int \frac {1+k x^2}{\left (1-(-1)^{2/3} k x^2\right ) \sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}} \, dx-\frac {1}{3} \sqrt {k} \text {Subst}\left (\int \frac {1}{(1-k x) \sqrt {1+\left (-1-k^2\right ) x+k^2 x^2}} \, dx,x,x^2\right )+\frac {1}{3} \left (\sqrt [3]{-1} \sqrt {k}\right ) \text {Subst}\left (\int \frac {1}{\left (1-(-1)^{2/3} k x\right ) \sqrt {1+\left (-1-k^2\right ) x+k^2 x^2}} \, dx,x,x^2\right )-\frac {1}{3} \left ((-1)^{2/3} \sqrt {k}\right ) \text {Subst}\left (\int \frac {1}{\left (1+\sqrt [3]{-1} k x\right ) \sqrt {1+\left (-1-k^2\right ) x+k^2 x^2}} \, dx,x,x^2\right )+\frac {\left (2 \sqrt [3]{-1} k \left (k+\sqrt [3]{-1} k\right )\right ) \int \frac {1+k x^2}{\left (1+\sqrt [3]{-1} k x^2\right ) \sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}} \, dx}{3 \left (k^2-(-1)^{2/3} k^2\right )} \\ & = -\frac {2 \arctan \left (\frac {\sqrt {1+k+k^2} x}{\sqrt {1-\left (1+k^2\right ) x^2+k^2 x^4}}\right )}{3 \sqrt {1+k+k^2}}+\frac {\left (1+k x^2\right ) \sqrt {\frac {1-\left (1+k^2\right ) x^2+k^2 x^4}{\left (1+k x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt {k} x\right ),\frac {(1+k)^2}{4 k}\right )}{3 \sqrt {k} \sqrt {1-\left (1+k^2\right ) x^2+k^2 x^4}}-\frac {\left (1+k x^2\right ) \sqrt {\frac {1-\left (1+k^2\right ) x^2+k^2 x^4}{\left (1+k x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt {k} x\right ),\frac {(1+k)^2}{4 k}\right )}{3 \left (1-\sqrt [3]{-1}\right ) \sqrt {k} \sqrt {1-\left (1+k^2\right ) x^2+k^2 x^4}}-\frac {\left (1-\sqrt [3]{-1}\right ) \left (1+k x^2\right ) \sqrt {\frac {1-\left (1+k^2\right ) x^2+k^2 x^4}{\left (1+k x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt {k} x\right ),\frac {(1+k)^2}{4 k}\right )}{3 \sqrt {k} \sqrt {1-\left (1+k^2\right ) x^2+k^2 x^4}}-\frac {1}{3} \text {Subst}\left (\int \frac {1}{1-\left (-1+2 k-k^2\right ) x^2} \, dx,x,\frac {x}{\sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}}\right )+\frac {1}{3} \left (2 \sqrt {k}\right ) \text {Subst}\left (\int \frac {1}{8 k^2+4 k \left (-1-k^2\right )-x^2} \, dx,x,\frac {1-2 k+k^2+(1-k)^2 k x^2}{\sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}}\right )-\frac {1}{3} \left (2 \sqrt [3]{-1} \sqrt {k}\right ) \text {Subst}\left (\int \frac {1}{4 k^2-4 \sqrt [3]{-1} k^2+4 (-1)^{2/3} k \left (-1-k^2\right )-x^2} \, dx,x,\frac {1-2 (-1)^{2/3} k+k^2-\left (2 k^2+(-1)^{2/3} k \left (-1-k^2\right )\right ) x^2}{\sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}}\right )+\frac {1}{3} \left (2 (-1)^{2/3} \sqrt {k}\right ) \text {Subst}\left (\int \frac {1}{4 k^2+4 (-1)^{2/3} k^2-4 \sqrt [3]{-1} k \left (-1-k^2\right )-x^2} \, dx,x,\frac {1+2 \sqrt [3]{-1} k+k^2-\left (2 k^2-\sqrt [3]{-1} k \left (-1-k^2\right )\right ) x^2}{\sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}}\right ) \\ & = -\frac {\arctan \left (\frac {(1-k) x}{\sqrt {1-\left (1+k^2\right ) x^2+k^2 x^4}}\right )}{3 (1-k)}-\frac {2 \arctan \left (\frac {\sqrt {1+k+k^2} x}{\sqrt {1-\left (1+k^2\right ) x^2+k^2 x^4}}\right )}{3 \sqrt {1+k+k^2}}-\frac {\arctan \left (\frac {(1-k) \left (1+k x^2\right )}{2 \sqrt {k} \sqrt {1-\left (1+k^2\right ) x^2+k^2 x^4}}\right )}{3 (1-k)}+\frac {(-1)^{2/3} \sqrt {2} \text {arctanh}\left (\frac {1+2 \sqrt [3]{-1} k+k^2-k \left (2 k+\sqrt [3]{-1} \left (1+k^2\right )\right ) x^2}{\sqrt {2} \sqrt {k} \sqrt {\left (1+i \sqrt {3}\right ) \left (1+k+k^2\right )} \sqrt {1-\left (1+k^2\right ) x^2+k^2 x^4}}\right )}{3 \sqrt {\left (1+i \sqrt {3}\right ) \left (1+k+k^2\right )}}-\frac {\sqrt [3]{-1} \sqrt {2} \text {arctanh}\left (\frac {1-2 (-1)^{2/3} k+k^2-k \left (2 k-(-1)^{2/3} \left (1+k^2\right )\right ) x^2}{\sqrt {2} \sqrt {k} \sqrt {\left (1-i \sqrt {3}\right ) \left (1+k+k^2\right )} \sqrt {1-\left (1+k^2\right ) x^2+k^2 x^4}}\right )}{3 \sqrt {\left (1-i \sqrt {3}\right ) \left (1+k+k^2\right )}}+\frac {\left (1+k x^2\right ) \sqrt {\frac {1-\left (1+k^2\right ) x^2+k^2 x^4}{\left (1+k x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt {k} x\right ),\frac {(1+k)^2}{4 k}\right )}{3 \sqrt {k} \sqrt {1-\left (1+k^2\right ) x^2+k^2 x^4}}-\frac {\left (1+k x^2\right ) \sqrt {\frac {1-\left (1+k^2\right ) x^2+k^2 x^4}{\left (1+k x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt {k} x\right ),\frac {(1+k)^2}{4 k}\right )}{3 \left (1-\sqrt [3]{-1}\right ) \sqrt {k} \sqrt {1-\left (1+k^2\right ) x^2+k^2 x^4}}-\frac {\left (1-\sqrt [3]{-1}\right ) \left (1+k x^2\right ) \sqrt {\frac {1-\left (1+k^2\right ) x^2+k^2 x^4}{\left (1+k x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt {k} x\right ),\frac {(1+k)^2}{4 k}\right )}{3 \sqrt {k} \sqrt {1-\left (1+k^2\right ) x^2+k^2 x^4}} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.

Time = 23.96 (sec) , antiderivative size = 558, normalized size of antiderivative = 4.50 \[ \int \frac {1+k^{3/2} x^3}{\sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (-1+k^{3/2} x^3\right )} \, dx=\frac {(1+i) \sqrt {-1+x^2} \sqrt {-1+k^2 x^2} \left (-\frac {(1-i) \arctan \left (\frac {\sqrt {-1+k^2 x^2}}{\sqrt {k} \sqrt {-1+x^2}}\right )}{-1+k}+\frac {\left (-1-i \sqrt {3}+k-i \sqrt {3} k\right ) \arctan \left (\frac {(1+i) \sqrt {1+k+k^2} \sqrt {-1+k^2 x^2}}{\sqrt {k} \sqrt {-\sqrt {2+2 i \sqrt {3}}-4 i k+\left (-i+\sqrt {3}\right ) k^2} \sqrt {-1+x^2}}\right )}{\sqrt {1+k+k^2} \sqrt {-\sqrt {2+2 i \sqrt {3}}-4 i k+\left (-i+\sqrt {3}\right ) k^2}}+\frac {\left (1-i \sqrt {3}+\left (-1-i \sqrt {3}\right ) k\right ) \text {arctanh}\left (\frac {(1+i) \sqrt {1+k+k^2} \sqrt {-1+k^2 x^2}}{\sqrt {k} \sqrt {-\sqrt {2-2 i \sqrt {3}}+4 i k+\left (i+\sqrt {3}\right ) k^2} \sqrt {-1+x^2}}\right )}{\sqrt {1+k+k^2} \sqrt {-\sqrt {2-2 i \sqrt {3}}+4 i k+\left (i+\sqrt {3}\right ) k^2}}\right )+3 \sqrt {1-x^2} \sqrt {1-k^2 x^2} \operatorname {EllipticF}\left (\arcsin (x),k^2\right )-2 \sqrt {1-x^2} \sqrt {1-k^2 x^2} \operatorname {EllipticPi}\left (k,\arcsin (x),k^2\right )-2 \sqrt {1-x^2} \sqrt {1-k^2 x^2} \operatorname {EllipticPi}\left (-\sqrt [3]{-1} k,\arcsin (x),k^2\right )-2 \sqrt {1-x^2} \sqrt {1-k^2 x^2} \operatorname {EllipticPi}\left (\frac {1}{2} i \left (i+\sqrt {3}\right ) k,\arcsin (x),k^2\right )}{3 \sqrt {\left (-1+x^2\right ) \left (-1+k^2 x^2\right )}} \]

[In]

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

[Out]

((1 + I)*Sqrt[-1 + x^2]*Sqrt[-1 + k^2*x^2]*(((-1 + I)*ArcTan[Sqrt[-1 + k^2*x^2]/(Sqrt[k]*Sqrt[-1 + x^2])])/(-1
+ k) + ((-1 - I*Sqrt[3] + k - I*Sqrt[3]*k)*ArcTan[((1 + I)*Sqrt[1 + k + k^2]*Sqrt[-1 + k^2*x^2])/(Sqrt[k]*Sqr
t[-Sqrt[2 + (2*I)*Sqrt[3]] - (4*I)*k + (-I + Sqrt[3])*k^2]*Sqrt[-1 + x^2])])/(Sqrt[1 + k + k^2]*Sqrt[-Sqrt[2 +
(2*I)*Sqrt[3]] - (4*I)*k + (-I + Sqrt[3])*k^2]) + ((1 - I*Sqrt[3] + (-1 - I*Sqrt[3])*k)*ArcTanh[((1 + I)*Sqrt
[1 + k + k^2]*Sqrt[-1 + k^2*x^2])/(Sqrt[k]*Sqrt[-Sqrt[2 - (2*I)*Sqrt[3]] + (4*I)*k + (I + Sqrt[3])*k^2]*Sqrt[-
1 + x^2])])/(Sqrt[1 + k + k^2]*Sqrt[-Sqrt[2 - (2*I)*Sqrt[3]] + (4*I)*k + (I + Sqrt[3])*k^2])) + 3*Sqrt[1 - x^2
]*Sqrt[1 - k^2*x^2]*EllipticF[ArcSin[x], k^2] - 2*Sqrt[1 - x^2]*Sqrt[1 - k^2*x^2]*EllipticPi[k, ArcSin[x], k^2
] - 2*Sqrt[1 - x^2]*Sqrt[1 - k^2*x^2]*EllipticPi[-((-1)^(1/3)*k), ArcSin[x], k^2] - 2*Sqrt[1 - x^2]*Sqrt[1 - k
^2*x^2]*EllipticPi[(I/2)*(I + Sqrt[3])*k, ArcSin[x], k^2])/(3*Sqrt[(-1 + x^2)*(-1 + k^2*x^2)])

Maple [A] (verified)

Time = 1.17 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.76

method	result	size
pseudoelliptic	\(-\frac {2 \left (\sqrt {-\left (-1+k \right )^{2}}\, \ln \left (\frac {\sqrt {-k^{2}-k -1}\, \sqrt {\left (x^{2}-1\right ) \left (k^{2} x^{2}-1\right )}-k^{\frac {3}{2}} x^{2}-\sqrt {k}+\left (-k^{2}-2 k -1\right ) x}{1+\sqrt {k}\, x +k \,x^{2}}\right )+\frac {\sqrt {-k^{2}-k -1}\, \ln \left (\frac {\sqrt {-\left (-1+k \right )^{2}}\, \sqrt {\left (x^{2}-1\right ) \left (k^{2} x^{2}-1\right )}+2 k^{\frac {3}{2}} x^{2}+2 \sqrt {k}+\left (-k^{2}-2 k -1\right ) x}{1-2 \sqrt {k}\, x +k \,x^{2}}\right )}{2}+\ln \left (2\right ) \left (\sqrt {-\left (-1+k \right )^{2}}+\frac {\sqrt {-k^{2}-k -1}}{2}\right )\right )}{3 \sqrt {-k^{2}-k -1}\, \sqrt {-\left (-1+k \right )^{2}}}\)	\(218\)
elliptic	\(\text {Expression too large to display}\)	\(1045\)
default	\(\text {Expression too large to display}\)	\(1398\)

[In]

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

[Out]

-2/3*((-(-1+k)^2)^(1/2)*ln(((-k^2-k-1)^(1/2)*((x^2-1)*(k^2*x^2-1))^(1/2)-k^(3/2)*x^2-k^(1/2)+(-k^2-2*k-1)*x)/(
1+k^(1/2)*x+k*x^2))+1/2*(-k^2-k-1)^(1/2)*ln(((-(-1+k)^2)^(1/2)*((x^2-1)*(k^2*x^2-1))^(1/2)+2*k^(3/2)*x^2+2*k^(
1/2)+(-k^2-2*k-1)*x)/(1-2*k^(1/2)*x+k*x^2))+ln(2)*((-(-1+k)^2)^(1/2)+1/2*(-k^2-k-1)^(1/2)))/(-k^2-k-1)^(1/2)/(
-(-1+k)^2)^(1/2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 213 vs. \(2 (106) = 212\).

Time = 0.55 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.72 \[ \int \frac {1+k^{3/2} x^3}{\sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (-1+k^{3/2} x^3\right )} \, dx=\frac {2 \, \sqrt {k^{2} + k + 1} {\left (k - 1\right )} \arctan \left (-\frac {\sqrt {k^{2} x^{4} - {\left (k^{2} + 1\right )} x^{2} + 1} \sqrt {k^{2} + k + 1} {\left ({\left (k^{2} + 2 \, k + 1\right )} x - {\left (k x^{2} + 1\right )} \sqrt {k}\right )}}{k^{3} x^{4} - {\left (k^{4} + 4 \, k^{3} + 4 \, k^{2} + 4 \, k + 1\right )} x^{2} + k}\right ) - {\left (k^{2} + k + 1\right )} \arctan \left (\frac {\sqrt {k^{2} x^{4} - {\left (k^{2} + 1\right )} x^{2} + 1} {\left ({\left (k^{3} + k^{2} - k - 1\right )} x + 2 \, {\left ({\left (k^{2} - k\right )} x^{2} + k - 1\right )} \sqrt {k}\right )}}{4 \, k^{3} x^{4} - {\left (k^{4} + 4 \, k^{3} - 2 \, k^{2} + 4 \, k + 1\right )} x^{2} + 4 \, k}\right )}{3 \, {\left (k^{3} - 1\right )}} \]

[In]

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

[Out]

1/3*(2*sqrt(k^2 + k + 1)*(k - 1)*arctan(-sqrt(k^2*x^4 - (k^2 + 1)*x^2 + 1)*sqrt(k^2 + k + 1)*((k^2 + 2*k + 1)*
x - (k*x^2 + 1)*sqrt(k))/(k^3*x^4 - (k^4 + 4*k^3 + 4*k^2 + 4*k + 1)*x^2 + k)) - (k^2 + k + 1)*arctan(sqrt(k^2*
x^4 - (k^2 + 1)*x^2 + 1)*((k^3 + k^2 - k - 1)*x + 2*((k^2 - k)*x^2 + k - 1)*sqrt(k))/(4*k^3*x^4 - (k^4 + 4*k^3
- 2*k^2 + 4*k + 1)*x^2 + 4*k)))/(k^3 - 1)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F(-2)]

Exception generated. \[ \int \frac {1+k^{3/2} x^3}{\sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (-1+k^{3/2} x^3\right )} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [F(-1)]

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

[In]

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

[Out]

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