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

Optimal. Leaf size=125 \[ -\frac {4 \tan ^{-1}\left (\frac {\sqrt {k^2+k+1} x}{\sqrt {k^2 x^4+\left (-k^2-1\right ) x^2+1}+k x^2-\sqrt {k} x+1}\right )}{3 \sqrt {k^2+k+1}}-\frac {2 \tan ^{-1}\left (\frac {(k-1) x}{\sqrt {k^2 x^4+\left (-k^2-1\right ) x^2+1}+k x^2+2 \sqrt {k} x+1}\right )}{3 (k-1)} \]

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Rubi [C]  time = 2.94, antiderivative size = 610, normalized size of antiderivative = 4.88, number of steps used = 21, number of rules used = 9, integrand size = 47, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {6719, 6725, 419, 2113, 537, 571, 93, 205, 208} \begin {gather*} -\frac {2 \sqrt {1-x^2} \sqrt {1-k^2 x^2} \tan ^{-1}\left (\frac {\sqrt {k} \sqrt {1-x^2}}{\sqrt {1-k^2 x^2}}\right )}{3 (1-k) \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}-\frac {2 (-1)^{2/3} \sqrt {1-x^2} \sqrt {1-k^2 x^2} \tanh ^{-1}\left (\frac {\sqrt {k} \sqrt {k+\sqrt [3]{-1}} \sqrt {1-x^2}}{\sqrt {\sqrt [3]{-1} k+1} \sqrt {1-k^2 x^2}}\right )}{3 \sqrt {k+\sqrt [3]{-1}} \sqrt {\sqrt [3]{-1} k+1} \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}+\frac {2 \sqrt [3]{-1} \sqrt {1-x^2} \sqrt {1-k^2 x^2} \tanh ^{-1}\left (\frac {\sqrt {k} \sqrt {k-(-1)^{2/3}} \sqrt {1-x^2}}{\sqrt {1-(-1)^{2/3} k} \sqrt {1-k^2 x^2}}\right )}{3 \sqrt {k-(-1)^{2/3}} \sqrt {1-(-1)^{2/3} k} \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}+\frac {\sqrt {1-x^2} \sqrt {1-k^2 x^2} F\left (\sin ^{-1}(x)|k^2\right )}{\sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}-\frac {2 \sqrt {1-x^2} \sqrt {1-k^2 x^2} \Pi \left (k;\sin ^{-1}(x)|k^2\right )}{3 \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}-\frac {2 \sqrt {1-x^2} \sqrt {1-k^2 x^2} \Pi \left (-\sqrt [3]{-1} k;\sin ^{-1}(x)|k^2\right )}{3 \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}-\frac {2 \sqrt {1-x^2} \sqrt {1-k^2 x^2} \Pi \left ((-1)^{2/3} k;\sin ^{-1}(x)|k^2\right )}{3 \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}} \end {gather*}

Antiderivative was successfully verified.

[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]

(-2*Sqrt[1 - x^2]*Sqrt[1 - k^2*x^2]*ArcTan[(Sqrt[k]*Sqrt[1 - x^2])/Sqrt[1 - k^2*x^2]])/(3*(1 - k)*Sqrt[(1 - x^
2)*(1 - k^2*x^2)]) - (2*(-1)^(2/3)*Sqrt[1 - x^2]*Sqrt[1 - k^2*x^2]*ArcTanh[(Sqrt[k]*Sqrt[(-1)^(1/3) + k]*Sqrt[
1 - x^2])/(Sqrt[1 + (-1)^(1/3)*k]*Sqrt[1 - k^2*x^2])])/(3*Sqrt[(-1)^(1/3) + k]*Sqrt[1 + (-1)^(1/3)*k]*Sqrt[(1
- x^2)*(1 - k^2*x^2)]) + (2*(-1)^(1/3)*Sqrt[1 - x^2]*Sqrt[1 - k^2*x^2]*ArcTanh[(Sqrt[k]*Sqrt[-(-1)^(2/3) + k]*
Sqrt[1 - x^2])/(Sqrt[1 - (-1)^(2/3)*k]*Sqrt[1 - k^2*x^2])])/(3*Sqrt[-(-1)^(2/3) + k]*Sqrt[1 - (-1)^(2/3)*k]*Sq
rt[(1 - x^2)*(1 - k^2*x^2)]) + (Sqrt[1 - x^2]*Sqrt[1 - k^2*x^2]*EllipticF[ArcSin[x], k^2])/Sqrt[(1 - x^2)*(1 -
k^2*x^2)] - (2*Sqrt[1 - x^2]*Sqrt[1 - k^2*x^2]*EllipticPi[k, ArcSin[x], k^2])/(3*Sqrt[(1 - x^2)*(1 - k^2*x^2)
]) - (2*Sqrt[1 - x^2]*Sqrt[1 - k^2*x^2]*EllipticPi[-((-1)^(1/3)*k), ArcSin[x], k^2])/(3*Sqrt[(1 - x^2)*(1 - k^
2*x^2)]) - (2*Sqrt[1 - x^2]*Sqrt[1 - k^2*x^2]*EllipticPi[(-1)^(2/3)*k, ArcSin[x], k^2])/(3*Sqrt[(1 - x^2)*(1 -
k^2*x^2)])

Rule 93

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 205

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

Rule 208

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

Rule 419

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

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 571

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_))^(r_.), x
_Symbol] :> Dist[1/n, Subst[Int[(a + b*x)^p*(c + d*x)^q*(e + f*x)^r, x], x, x^n], x] /; FreeQ[{a, b, c, d, e,
f, m, n, p, q, r}, x] && EqQ[m - n + 1, 0]

Rule 2113

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

Rule 6719

Int[(u_.)*((a_.)*(v_)^(m_.)*(w_)^(n_.))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a*v^m*w^n)^FracPart[p])/(v^(m*F
racPart[p])*w^(n*FracPart[p])), Int[u*v^(m*p)*w^(n*p), x], x] /; FreeQ[{a, m, n, p}, x] &&  !IntegerQ[p] &&  !
FreeQ[v, x] &&  !FreeQ[w, x]

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 {-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 {\left (\sqrt {1-x^2} \sqrt {1-k^2 x^2}\right ) \int \frac {-1+k^{3/2} x^3}{\sqrt {1-x^2} \sqrt {1-k^2 x^2} \left (1+k^{3/2} x^3\right )} \, dx}{\sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}\\ &=\frac {\left (\sqrt {1-x^2} \sqrt {1-k^2 x^2}\right ) \int \left (\frac {1}{\sqrt {1-x^2} \sqrt {1-k^2 x^2}}-\frac {2}{\sqrt {1-x^2} \sqrt {1-k^2 x^2} \left (1+k^{3/2} x^3\right )}\right ) \, dx}{\sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}\\ &=\frac {\left (\sqrt {1-x^2} \sqrt {1-k^2 x^2}\right ) \int \frac {1}{\sqrt {1-x^2} \sqrt {1-k^2 x^2}} \, dx}{\sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}-\frac {\left (2 \sqrt {1-x^2} \sqrt {1-k^2 x^2}\right ) \int \frac {1}{\sqrt {1-x^2} \sqrt {1-k^2 x^2} \left (1+k^{3/2} x^3\right )} \, dx}{\sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}\\ &=\frac {\sqrt {1-x^2} \sqrt {1-k^2 x^2} F\left (\sin ^{-1}(x)|k^2\right )}{\sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}-\frac {\left (2 \sqrt {1-x^2} \sqrt {1-k^2 x^2}\right ) \int \left (-\frac {1}{3 \left (-1-\sqrt {k} x\right ) \sqrt {1-x^2} \sqrt {1-k^2 x^2}}-\frac {1}{3 \left (-1+\sqrt [3]{-1} \sqrt {k} x\right ) \sqrt {1-x^2} \sqrt {1-k^2 x^2}}-\frac {1}{3 \left (-1-(-1)^{2/3} \sqrt {k} x\right ) \sqrt {1-x^2} \sqrt {1-k^2 x^2}}\right ) \, dx}{\sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}\\ &=\frac {\sqrt {1-x^2} \sqrt {1-k^2 x^2} F\left (\sin ^{-1}(x)|k^2\right )}{\sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}+\frac {\left (2 \sqrt {1-x^2} \sqrt {1-k^2 x^2}\right ) \int \frac {1}{\left (-1-\sqrt {k} x\right ) \sqrt {1-x^2} \sqrt {1-k^2 x^2}} \, dx}{3 \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}+\frac {\left (2 \sqrt {1-x^2} \sqrt {1-k^2 x^2}\right ) \int \frac {1}{\left (-1+\sqrt [3]{-1} \sqrt {k} x\right ) \sqrt {1-x^2} \sqrt {1-k^2 x^2}} \, dx}{3 \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}+\frac {\left (2 \sqrt {1-x^2} \sqrt {1-k^2 x^2}\right ) \int \frac {1}{\left (-1-(-1)^{2/3} \sqrt {k} x\right ) \sqrt {1-x^2} \sqrt {1-k^2 x^2}} \, dx}{3 \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}\\ &=\frac {\sqrt {1-x^2} \sqrt {1-k^2 x^2} F\left (\sin ^{-1}(x)|k^2\right )}{\sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}-\frac {\left (2 \sqrt {1-x^2} \sqrt {1-k^2 x^2}\right ) \int \frac {1}{\sqrt {1-x^2} \left (1-k x^2\right ) \sqrt {1-k^2 x^2}} \, dx}{3 \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}-\frac {\left (2 \sqrt {1-x^2} \sqrt {1-k^2 x^2}\right ) \int \frac {1}{\sqrt {1-x^2} \left (1+\sqrt [3]{-1} k x^2\right ) \sqrt {1-k^2 x^2}} \, dx}{3 \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}-\frac {\left (2 \sqrt {1-x^2} \sqrt {1-k^2 x^2}\right ) \int \frac {1}{\sqrt {1-x^2} \left (1-(-1)^{2/3} k x^2\right ) \sqrt {1-k^2 x^2}} \, dx}{3 \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}+\frac {\left (2 \sqrt {k} \sqrt {1-x^2} \sqrt {1-k^2 x^2}\right ) \int \frac {x}{\sqrt {1-x^2} \left (1-k x^2\right ) \sqrt {1-k^2 x^2}} \, dx}{3 \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}-\frac {\left (2 \sqrt [3]{-1} \sqrt {k} \sqrt {1-x^2} \sqrt {1-k^2 x^2}\right ) \int \frac {x}{\sqrt {1-x^2} \left (1-(-1)^{2/3} k x^2\right ) \sqrt {1-k^2 x^2}} \, dx}{3 \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}+\frac {\left (2 (-1)^{2/3} \sqrt {k} \sqrt {1-x^2} \sqrt {1-k^2 x^2}\right ) \int \frac {x}{\sqrt {1-x^2} \left (1+\sqrt [3]{-1} k x^2\right ) \sqrt {1-k^2 x^2}} \, dx}{3 \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}\\ &=\frac {\sqrt {1-x^2} \sqrt {1-k^2 x^2} F\left (\sin ^{-1}(x)|k^2\right )}{\sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}-\frac {2 \sqrt {1-x^2} \sqrt {1-k^2 x^2} \Pi \left (k;\sin ^{-1}(x)|k^2\right )}{3 \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}-\frac {2 \sqrt {1-x^2} \sqrt {1-k^2 x^2} \Pi \left (-\sqrt [3]{-1} k;\sin ^{-1}(x)|k^2\right )}{3 \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}-\frac {2 \sqrt {1-x^2} \sqrt {1-k^2 x^2} \Pi \left ((-1)^{2/3} k;\sin ^{-1}(x)|k^2\right )}{3 \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}+\frac {\left (\sqrt {k} \sqrt {1-x^2} \sqrt {1-k^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x} (1-k x) \sqrt {1-k^2 x}} \, dx,x,x^2\right )}{3 \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}-\frac {\left (\sqrt [3]{-1} \sqrt {k} \sqrt {1-x^2} \sqrt {1-k^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x} \left (1-(-1)^{2/3} k x\right ) \sqrt {1-k^2 x}} \, dx,x,x^2\right )}{3 \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}+\frac {\left ((-1)^{2/3} \sqrt {k} \sqrt {1-x^2} \sqrt {1-k^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x} \left (1+\sqrt [3]{-1} k x\right ) \sqrt {1-k^2 x}} \, dx,x,x^2\right )}{3 \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}\\ &=\frac {\sqrt {1-x^2} \sqrt {1-k^2 x^2} F\left (\sin ^{-1}(x)|k^2\right )}{\sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}-\frac {2 \sqrt {1-x^2} \sqrt {1-k^2 x^2} \Pi \left (k;\sin ^{-1}(x)|k^2\right )}{3 \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}-\frac {2 \sqrt {1-x^2} \sqrt {1-k^2 x^2} \Pi \left (-\sqrt [3]{-1} k;\sin ^{-1}(x)|k^2\right )}{3 \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}-\frac {2 \sqrt {1-x^2} \sqrt {1-k^2 x^2} \Pi \left ((-1)^{2/3} k;\sin ^{-1}(x)|k^2\right )}{3 \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}+\frac {\left (2 \sqrt {k} \sqrt {1-x^2} \sqrt {1-k^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{-1+k-\left (k-k^2\right ) x^2} \, dx,x,\frac {\sqrt {1-x^2}}{\sqrt {1-k^2 x^2}}\right )}{3 \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}-\frac {\left (2 \sqrt [3]{-1} \sqrt {k} \sqrt {1-x^2} \sqrt {1-k^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{-1+(-1)^{2/3} k-\left ((-1)^{2/3} k-k^2\right ) x^2} \, dx,x,\frac {\sqrt {1-x^2}}{\sqrt {1-k^2 x^2}}\right )}{3 \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}+\frac {\left (2 (-1)^{2/3} \sqrt {k} \sqrt {1-x^2} \sqrt {1-k^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{-1-\sqrt [3]{-1} k-\left (-\sqrt [3]{-1} k-k^2\right ) x^2} \, dx,x,\frac {\sqrt {1-x^2}}{\sqrt {1-k^2 x^2}}\right )}{3 \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}\\ &=-\frac {2 \sqrt {1-x^2} \sqrt {1-k^2 x^2} \tan ^{-1}\left (\frac {\sqrt {k} \sqrt {1-x^2}}{\sqrt {1-k^2 x^2}}\right )}{3 (1-k) \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}-\frac {2 (-1)^{2/3} \sqrt {1-x^2} \sqrt {1-k^2 x^2} \tanh ^{-1}\left (\frac {\sqrt {k} \sqrt {\sqrt [3]{-1}+k} \sqrt {1-x^2}}{\sqrt {1+\sqrt [3]{-1} k} \sqrt {1-k^2 x^2}}\right )}{3 \sqrt {\sqrt [3]{-1}+k} \sqrt {1+\sqrt [3]{-1} k} \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}+\frac {2 \sqrt [3]{-1} \sqrt {1-x^2} \sqrt {1-k^2 x^2} \tanh ^{-1}\left (\frac {\sqrt {k} \sqrt {-(-1)^{2/3}+k} \sqrt {1-x^2}}{\sqrt {1-(-1)^{2/3} k} \sqrt {1-k^2 x^2}}\right )}{3 \sqrt {-(-1)^{2/3}+k} \sqrt {1-(-1)^{2/3} k} \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}+\frac {\sqrt {1-x^2} \sqrt {1-k^2 x^2} F\left (\sin ^{-1}(x)|k^2\right )}{\sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}-\frac {2 \sqrt {1-x^2} \sqrt {1-k^2 x^2} \Pi \left (k;\sin ^{-1}(x)|k^2\right )}{3 \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}-\frac {2 \sqrt {1-x^2} \sqrt {1-k^2 x^2} \Pi \left (-\sqrt [3]{-1} k;\sin ^{-1}(x)|k^2\right )}{3 \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}-\frac {2 \sqrt {1-x^2} \sqrt {1-k^2 x^2} \Pi \left ((-1)^{2/3} k;\sin ^{-1}(x)|k^2\right )}{3 \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C]  time = 4.92, size = 444, normalized size = 3.55 \begin {gather*} \frac {-2 \sqrt {k} \sqrt {x^2-1} \sqrt {k^2 x^2-1} \left (-\frac {\tanh ^{-1}\left (\frac {\sqrt {(k-1) k} \sqrt {x^2-1}}{\sqrt {1-k} \sqrt {k^2 x^2-1}}\right )}{\sqrt {1-k} \sqrt {(k-1) k}}-\frac {(-1)^{2/3} \tanh ^{-1}\left (\frac {\sqrt {k \left (k+\sqrt [3]{-1}\right )} \sqrt {x^2-1}}{\sqrt {\sqrt [3]{-1} k+1} \sqrt {k^2 x^2-1}}\right )}{\sqrt {k \left (k+\sqrt [3]{-1}\right )} \sqrt {\sqrt [3]{-1} k+1}}+\frac {\sqrt [3]{-1} \tanh ^{-1}\left (\frac {\sqrt {k \left (k-(-1)^{2/3}\right )} \sqrt {x^2-1}}{\sqrt {1-(-1)^{2/3} k} \sqrt {k^2 x^2-1}}\right )}{\sqrt {k \left (k-(-1)^{2/3}\right )} \sqrt {1-(-1)^{2/3} k}}\right )+3 \sqrt {1-x^2} \sqrt {1-k^2 x^2} F\left (\sin ^{-1}(x)|k^2\right )-2 \sqrt {1-x^2} \sqrt {1-k^2 x^2} \Pi \left (k;\sin ^{-1}(x)|k^2\right )-2 \sqrt {1-x^2} \sqrt {1-k^2 x^2} \Pi \left (-\sqrt [3]{-1} k;\sin ^{-1}(x)|k^2\right )-2 \sqrt {1-x^2} \sqrt {1-k^2 x^2} \Pi \left (\frac {1}{2} i \left (i+\sqrt {3}\right ) k;\sin ^{-1}(x)|k^2\right )}{3 \sqrt {\left (x^2-1\right ) \left (k^2 x^2-1\right )}} \end {gather*}

Antiderivative was successfully verified.

[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]

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

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IntegrateAlgebraic [A]  time = 4.26, size = 125, normalized size = 1.00 \begin {gather*} -\frac {4 \tan ^{-1}\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}}-\frac {2 \tan ^{-1}\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)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 1.45, size = 212, normalized size = 1.70 \begin {gather*} -\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 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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,x):;OUTPUT:sym2
poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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maple [B]  time = 0.41, size = 1540, normalized size = 12.32

method	result	size
elliptic	\(\frac {\left (\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {\left (-x^{2}+1\right ) \left (-k^{2} x^{2}+1\right )}}{x \left (-1+k \right )}\right )}{-3+3 k}+\frac {4 \arctan \left (\frac {\sqrt {\left (-x^{2}+1\right ) \left (-k^{2} x^{2}+1\right )}\, \sqrt {2}}{x \sqrt {2 k^{2}+2 k +2}}\right )}{3 \sqrt {2 k^{2}+2 k +2}}\right ) \sqrt {2}}{2}-\frac {k \sqrt {2}\, \ln \left (\frac {\frac {\sqrt {-3 k^{2}}\, k^{2}+k^{3}+\sqrt {-3 k^{2}}\, k +k^{2}+\sqrt {-3 k^{2}}+k}{k}-k \left (k^{2}+\sqrt {-3 k^{2}}+k +1\right ) \left (x^{2}+\frac {\sqrt {-3 k^{2}}+k}{2 k^{2}}\right )+\frac {\sqrt {2}\, \sqrt {\frac {\sqrt {-3 k^{2}}\, k^{2}+k^{3}+\sqrt {-3 k^{2}}\, k +k^{2}+\sqrt {-3 k^{2}}+k}{k}}\, \sqrt {4 k^{3} \left (x^{2}+\frac {\sqrt {-3 k^{2}}+k}{2 k^{2}}\right )^{2}-4 k \left (k^{2}+\sqrt {-3 k^{2}}+k +1\right ) \left (x^{2}+\frac {\sqrt {-3 k^{2}}+k}{2 k^{2}}\right )+\frac {2 \sqrt {-3 k^{2}}\, k^{2}+2 k^{3}+2 k^{2}+2 \sqrt {-3 k^{2}}\, k +2 \sqrt {-3 k^{2}}+2 k}{k}}}{2}}{x^{2}+\frac {\sqrt {-3 k^{2}}+k}{2 k^{2}}}\right )}{\left (\sqrt {-3 k^{2}}+3 k \right ) \sqrt {\frac {\sqrt {-3 k^{2}}\, k^{2}+k^{3}+\sqrt {-3 k^{2}}\, k +k^{2}+\sqrt {-3 k^{2}}+k}{k}}}-\frac {k^{2} \sqrt {2}\, \ln \left (\frac {\frac {\sqrt {-3 k^{2}}\, k^{2}+k^{3}+\sqrt {-3 k^{2}}\, k +k^{2}+\sqrt {-3 k^{2}}+k}{k}-k \left (k^{2}+\sqrt {-3 k^{2}}+k +1\right ) \left (x^{2}+\frac {\sqrt {-3 k^{2}}+k}{2 k^{2}}\right )+\frac {\sqrt {2}\, \sqrt {\frac {\sqrt {-3 k^{2}}\, k^{2}+k^{3}+\sqrt {-3 k^{2}}\, k +k^{2}+\sqrt {-3 k^{2}}+k}{k}}\, \sqrt {4 k^{3} \left (x^{2}+\frac {\sqrt {-3 k^{2}}+k}{2 k^{2}}\right )^{2}-4 k \left (k^{2}+\sqrt {-3 k^{2}}+k +1\right ) \left (x^{2}+\frac {\sqrt {-3 k^{2}}+k}{2 k^{2}}\right )+\frac {2 \sqrt {-3 k^{2}}\, k^{2}+2 k^{3}+2 k^{2}+2 \sqrt {-3 k^{2}}\, k +2 \sqrt {-3 k^{2}}+2 k}{k}}}{2}}{x^{2}+\frac {\sqrt {-3 k^{2}}+k}{2 k^{2}}}\right )}{\left (\sqrt {-3 k^{2}}+3 k \right ) \sqrt {-3 k^{2}}\, \sqrt {\frac {\sqrt {-3 k^{2}}\, k^{2}+k^{3}+\sqrt {-3 k^{2}}\, k +k^{2}+\sqrt {-3 k^{2}}+k}{k}}}+\frac {k \sqrt {2}\, \ln \left (\frac {\frac {-\sqrt {-3 k^{2}}\, k^{2}+k^{3}-\sqrt {-3 k^{2}}\, k +k^{2}-\sqrt {-3 k^{2}}+k}{k}-k \left (k^{2}+1+k -\sqrt {-3 k^{2}}\right ) \left (x^{2}-\frac {-k +\sqrt {-3 k^{2}}}{2 k^{2}}\right )+\frac {\sqrt {2}\, \sqrt {\frac {-\sqrt {-3 k^{2}}\, k^{2}+k^{3}-\sqrt {-3 k^{2}}\, k +k^{2}-\sqrt {-3 k^{2}}+k}{k}}\, \sqrt {4 k^{3} \left (x^{2}-\frac {-k +\sqrt {-3 k^{2}}}{2 k^{2}}\right )^{2}-4 k \left (k^{2}+1+k -\sqrt {-3 k^{2}}\right ) \left (x^{2}-\frac {-k +\sqrt {-3 k^{2}}}{2 k^{2}}\right )+\frac {-2 \sqrt {-3 k^{2}}\, k^{2}+2 k^{3}+2 k^{2}-2 \sqrt {-3 k^{2}}\, k -2 \sqrt {-3 k^{2}}+2 k}{k}}}{2}}{x^{2}-\frac {-k +\sqrt {-3 k^{2}}}{2 k^{2}}}\right )}{\left (\sqrt {-3 k^{2}}-3 k \right ) \sqrt {\frac {-\sqrt {-3 k^{2}}\, k^{2}+k^{3}-\sqrt {-3 k^{2}}\, k +k^{2}-\sqrt {-3 k^{2}}+k}{k}}}-\frac {k^{2} \sqrt {2}\, \ln \left (\frac {\frac {-\sqrt {-3 k^{2}}\, k^{2}+k^{3}-\sqrt {-3 k^{2}}\, k +k^{2}-\sqrt {-3 k^{2}}+k}{k}-k \left (k^{2}+1+k -\sqrt {-3 k^{2}}\right ) \left (x^{2}-\frac {-k +\sqrt {-3 k^{2}}}{2 k^{2}}\right )+\frac {\sqrt {2}\, \sqrt {\frac {-\sqrt {-3 k^{2}}\, k^{2}+k^{3}-\sqrt {-3 k^{2}}\, k +k^{2}-\sqrt {-3 k^{2}}+k}{k}}\, \sqrt {4 k^{3} \left (x^{2}-\frac {-k +\sqrt {-3 k^{2}}}{2 k^{2}}\right )^{2}-4 k \left (k^{2}+1+k -\sqrt {-3 k^{2}}\right ) \left (x^{2}-\frac {-k +\sqrt {-3 k^{2}}}{2 k^{2}}\right )+\frac {-2 \sqrt {-3 k^{2}}\, k^{2}+2 k^{3}+2 k^{2}-2 \sqrt {-3 k^{2}}\, k -2 \sqrt {-3 k^{2}}+2 k}{k}}}{2}}{x^{2}-\frac {-k +\sqrt {-3 k^{2}}}{2 k^{2}}}\right )}{\left (\sqrt {-3 k^{2}}-3 k \right ) \sqrt {-3 k^{2}}\, \sqrt {\frac {-\sqrt {-3 k^{2}}\, k^{2}+k^{3}-\sqrt {-3 k^{2}}\, k +k^{2}-\sqrt {-3 k^{2}}+k}{k}}}-\frac {4 k^{2} \ln \left (\frac {-2 k^{2}+4 k -2+\left (-k^{3}+2 k^{2}-k \right ) \left (x^{2}-\frac {1}{k}\right )+2 \sqrt {-\left (-1+k \right )^{2}}\, \sqrt {k^{3} \left (x^{2}-\frac {1}{k}\right )^{2}+\left (-k^{3}+2 k^{2}-k \right ) \left (x^{2}-\frac {1}{k}\right )-k^{2}+2 k -1}}{x^{2}-\frac {1}{k}}\right )}{\left (\sqrt {-3 k^{2}}-3 k \right ) \left (\sqrt {-3 k^{2}}+3 k \right ) \sqrt {-\left (-1+k \right )^{2}}}\)	\(1540\)
default	\(\frac {\sqrt {-x^{2}+1}\, \sqrt {-k^{2} x^{2}+1}\, \EllipticF \left (x , k\right )}{\sqrt {k^{2} x^{4}-k^{2} x^{2}-x^{2}+1}}+\frac {-\frac {\sqrt {2}\, \ln \left (\frac {\frac {\sqrt {-3 k^{2}}\, k^{2}+k^{3}+\sqrt {-3 k^{2}}\, k +k^{2}+\sqrt {-3 k^{2}}+k}{k^{2}}+\left (-k^{2}-1-\sqrt {-3 k^{2}}-k \right ) \left (x^{2}+\frac {\sqrt {-3 k^{2}}+k}{2 k^{2}}\right )+\frac {\sqrt {2}\, \sqrt {\frac {\sqrt {-3 k^{2}}\, k^{2}+k^{3}+\sqrt {-3 k^{2}}\, k +k^{2}+\sqrt {-3 k^{2}}+k}{k^{2}}}\, \sqrt {4 k^{2} \left (x^{2}+\frac {\sqrt {-3 k^{2}}+k}{2 k^{2}}\right )^{2}+4 \left (-k^{2}-1-\sqrt {-3 k^{2}}-k \right ) \left (x^{2}+\frac {\sqrt {-3 k^{2}}+k}{2 k^{2}}\right )+\frac {2 \sqrt {-3 k^{2}}\, k^{2}+2 k^{3}+2 k^{2}+2 \sqrt {-3 k^{2}}\, k +2 \sqrt {-3 k^{2}}+2 k}{k^{2}}}}{2}}{x^{2}+\frac {\sqrt {-3 k^{2}}+k}{2 k^{2}}}\right )}{6 \sqrt {\frac {\sqrt {-3 k^{2}}\, k^{2}+k^{3}+\sqrt {-3 k^{2}}\, k +k^{2}+\sqrt {-3 k^{2}}+k}{k^{2}}}}-\frac {k \sqrt {2}\, \ln \left (\frac {\frac {\sqrt {-3 k^{2}}\, k^{2}+k^{3}+\sqrt {-3 k^{2}}\, k +k^{2}+\sqrt {-3 k^{2}}+k}{k^{2}}+\left (-k^{2}-1-\sqrt {-3 k^{2}}-k \right ) \left (x^{2}+\frac {\sqrt {-3 k^{2}}+k}{2 k^{2}}\right )+\frac {\sqrt {2}\, \sqrt {\frac {\sqrt {-3 k^{2}}\, k^{2}+k^{3}+\sqrt {-3 k^{2}}\, k +k^{2}+\sqrt {-3 k^{2}}+k}{k^{2}}}\, \sqrt {4 k^{2} \left (x^{2}+\frac {\sqrt {-3 k^{2}}+k}{2 k^{2}}\right )^{2}+4 \left (-k^{2}-1-\sqrt {-3 k^{2}}-k \right ) \left (x^{2}+\frac {\sqrt {-3 k^{2}}+k}{2 k^{2}}\right )+\frac {2 \sqrt {-3 k^{2}}\, k^{2}+2 k^{3}+2 k^{2}+2 \sqrt {-3 k^{2}}\, k +2 \sqrt {-3 k^{2}}+2 k}{k^{2}}}}{2}}{x^{2}+\frac {\sqrt {-3 k^{2}}+k}{2 k^{2}}}\right )}{2 \sqrt {-3 k^{2}}\, \sqrt {\frac {\sqrt {-3 k^{2}}\, k^{2}+k^{3}+\sqrt {-3 k^{2}}\, k +k^{2}+\sqrt {-3 k^{2}}+k}{k^{2}}}}-\frac {\sqrt {2}\, \ln \left (\frac {\frac {-\sqrt {-3 k^{2}}\, k^{2}+k^{3}-\sqrt {-3 k^{2}}\, k +k^{2}-\sqrt {-3 k^{2}}+k}{k^{2}}+\left (-k^{2}-1-k +\sqrt {-3 k^{2}}\right ) \left (x^{2}-\frac {-k +\sqrt {-3 k^{2}}}{2 k^{2}}\right )+\frac {\sqrt {2}\, \sqrt {\frac {-\sqrt {-3 k^{2}}\, k^{2}+k^{3}-\sqrt {-3 k^{2}}\, k +k^{2}-\sqrt {-3 k^{2}}+k}{k^{2}}}\, \sqrt {4 k^{2} \left (x^{2}-\frac {-k +\sqrt {-3 k^{2}}}{2 k^{2}}\right )^{2}+4 \left (-k^{2}-1-k +\sqrt {-3 k^{2}}\right ) \left (x^{2}-\frac {-k +\sqrt {-3 k^{2}}}{2 k^{2}}\right )+\frac {-2 \sqrt {-3 k^{2}}\, k^{2}+2 k^{3}+2 k^{2}-2 \sqrt {-3 k^{2}}\, k -2 \sqrt {-3 k^{2}}+2 k}{k^{2}}}}{2}}{x^{2}-\frac {-k +\sqrt {-3 k^{2}}}{2 k^{2}}}\right )}{6 \sqrt {\frac {-\sqrt {-3 k^{2}}\, k^{2}+k^{3}-\sqrt {-3 k^{2}}\, k +k^{2}-\sqrt {-3 k^{2}}+k}{k^{2}}}}+\frac {k \sqrt {2}\, \ln \left (\frac {\frac {-\sqrt {-3 k^{2}}\, k^{2}+k^{3}-\sqrt {-3 k^{2}}\, k +k^{2}-\sqrt {-3 k^{2}}+k}{k^{2}}+\left (-k^{2}-1-k +\sqrt {-3 k^{2}}\right ) \left (x^{2}-\frac {-k +\sqrt {-3 k^{2}}}{2 k^{2}}\right )+\frac {\sqrt {2}\, \sqrt {\frac {-\sqrt {-3 k^{2}}\, k^{2}+k^{3}-\sqrt {-3 k^{2}}\, k +k^{2}-\sqrt {-3 k^{2}}+k}{k^{2}}}\, \sqrt {4 k^{2} \left (x^{2}-\frac {-k +\sqrt {-3 k^{2}}}{2 k^{2}}\right )^{2}+4 \left (-k^{2}-1-k +\sqrt {-3 k^{2}}\right ) \left (x^{2}-\frac {-k +\sqrt {-3 k^{2}}}{2 k^{2}}\right )+\frac {-2 \sqrt {-3 k^{2}}\, k^{2}+2 k^{3}+2 k^{2}-2 \sqrt {-3 k^{2}}\, k -2 \sqrt {-3 k^{2}}+2 k}{k^{2}}}}{2}}{x^{2}-\frac {-k +\sqrt {-3 k^{2}}}{2 k^{2}}}\right )}{2 \sqrt {-3 k^{2}}\, \sqrt {\frac {-\sqrt {-3 k^{2}}\, k^{2}+k^{3}-\sqrt {-3 k^{2}}\, k +k^{2}-\sqrt {-3 k^{2}}+k}{k^{2}}}}-\frac {\left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (k^{2} \textit {\_Z}^{4}+k \,\textit {\_Z}^{2}+1\right )}{\sum }\frac {\left (k \,\underline {\hspace {1.25 ex}}\alpha ^{2}+2\right ) \left (-\frac {\arctanh \left (\frac {k \left (2 k^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}-k^{2}-1\right ) \left (\underline {\hspace {1.25 ex}}\alpha ^{2} k^{4}+k^{4} x^{2}+2 k^{3} x^{2}-2 k^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}+6 k^{2} x^{2}+2 k \,x^{2}+\underline {\hspace {1.25 ex}}\alpha ^{2}-4 k^{2}+x^{2}-4 k -4\right )}{2 \left (k^{4}+2 k^{3}+6 k^{2}+2 k +1\right ) \sqrt {-k \,\underline {\hspace {1.25 ex}}\alpha ^{2} \left (k^{2}+k +1\right )}\, \sqrt {k^{3} x^{4}-k^{3} x^{2}-k \,x^{2}+k}}\right )}{\sqrt {-k \,\underline {\hspace {1.25 ex}}\alpha ^{2} \left (k^{2}+k +1\right )}}+\frac {2 k \underline {\hspace {1.25 ex}}\alpha \left (k \,\underline {\hspace {1.25 ex}}\alpha ^{2}+1\right ) \sqrt {-k^{2} x^{2}+1}\, \sqrt {-x^{2}+1}\, \EllipticPi \left (x \sqrt {k^{2}}, -\frac {k \,\underline {\hspace {1.25 ex}}\alpha ^{2}+1}{k}, \frac {1}{\sqrt {k^{2}}}\right )}{\sqrt {k^{2}}\, \sqrt {\left (k^{2} x^{4}-k^{2} x^{2}-x^{2}+1\right ) k}}\right )}{\underline {\hspace {1.25 ex}}\alpha \left (2 k \,\underline {\hspace {1.25 ex}}\alpha ^{2}+1\right )}\right )}{6}}{\sqrt {k}}-\frac {2 \sqrt {-x^{2}+1}\, \sqrt {-k^{2} x^{2}+1}\, \EllipticPi \left (x , k , k\right )}{3 \sqrt {k^{2} x^{4}-k^{2} x^{2}-x^{2}+1}}+\frac {\ln \left (\frac {-2 k^{2}+4 k -2+\left (-k^{3}+2 k^{2}-k \right ) \left (x^{2}-\frac {1}{k}\right )+2 \sqrt {-\left (-1+k \right )^{2}}\, \sqrt {k^{3} \left (x^{2}-\frac {1}{k}\right )^{2}+\left (-k^{3}+2 k^{2}-k \right ) \left (x^{2}-\frac {1}{k}\right )-k^{2}+2 k -1}}{x^{2}-\frac {1}{k}}\right )}{3 \sqrt {-\left (-1+k \right )^{2}}}\)	\(1764\)

Verification of antiderivative is not currently implemented for this CAS.

[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]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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