3.221 \(\int \frac{x^2 \sqrt{1+x} \sqrt [4]{1-x^2}}{\sqrt{1-x} (\sqrt{1-x}-\sqrt{1+x})} \, dx\)

Optimal. Leaf size=304 \[ \frac{1}{6} \sqrt{x+1} \left (1-x^2\right )^{5/4}+\frac{x \left (1-x^2\right )^{5/4}}{6 \sqrt{1-x}}+\frac{7 \left (1-x^2\right )^{5/4}}{24 \sqrt{1-x}}+\frac{1}{24} (x+1)^{3/4} (1-x)^{5/4}+\frac{5}{16} \sqrt [4]{x+1} (1-x)^{3/4}-\frac{1}{16} (x+1)^{3/4} \sqrt [4]{1-x}+\frac{\log \left (\frac{\sqrt{1-x}}{\sqrt{x+1}}-\frac{\sqrt{2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}+1\right )}{8 \sqrt{2}}-\frac{\log \left (\frac{\sqrt{1-x}}{\sqrt{x+1}}+\frac{\sqrt{2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}+1\right )}{8 \sqrt{2}}-\frac{3 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}\right )}{8 \sqrt{2}}+\frac{3 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}+1\right )}{8 \sqrt{2}} \]

[Out]

(5*(1 - x)^(3/4)*(1 + x)^(1/4))/16 - ((1 - x)^(1/4)*(1 + x)^(3/4))/16 + ((1 - x)^(5/4)*(1 + x)^(3/4))/24 + (7*
(1 - x^2)^(5/4))/(24*Sqrt[1 - x]) + (x*(1 - x^2)^(5/4))/(6*Sqrt[1 - x]) + (Sqrt[1 + x]*(1 - x^2)^(5/4))/6 - (3
*ArcTan[1 - (Sqrt[2]*(1 - x)^(1/4))/(1 + x)^(1/4)])/(8*Sqrt[2]) + (3*ArcTan[1 + (Sqrt[2]*(1 - x)^(1/4))/(1 + x
)^(1/4)])/(8*Sqrt[2]) + Log[1 + Sqrt[1 - x]/Sqrt[1 + x] - (Sqrt[2]*(1 - x)^(1/4))/(1 + x)^(1/4)]/(8*Sqrt[2]) -
Log[1 + Sqrt[1 - x]/Sqrt[1 + x] + (Sqrt[2]*(1 - x)^(1/4))/(1 + x)^(1/4)]/(8*Sqrt[2])

________________________________________________________________________________________

Rubi [A]  time = 0.822812, antiderivative size = 319, normalized size of antiderivative = 1.05, number of steps used = 33, number of rules used = 16, integrand size = 52, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {2103, 795, 675, 50, 63, 240, 211, 1165, 628, 1162, 617, 204, 1633, 793, 331, 297} \[ \frac{\left (1-x^2\right )^{9/4}}{3 (1-x)^{3/2}}+\frac{1}{6} \sqrt{x+1} \left (1-x^2\right )^{5/4}+\frac{1}{6} (1-x)^{7/4} (x+1)^{5/4}+\frac{1}{24} (1-x)^{5/4} (x+1)^{3/4}-\frac{1}{16} \sqrt [4]{1-x} (x+1)^{3/4}+\frac{5}{24} (1-x)^{7/4} \sqrt [4]{x+1}-\frac{5}{48} (1-x)^{3/4} \sqrt [4]{x+1}+\frac{\log \left (\frac{\sqrt{1-x}}{\sqrt{x+1}}-\frac{\sqrt{2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}+1\right )}{8 \sqrt{2}}-\frac{\log \left (\frac{\sqrt{1-x}}{\sqrt{x+1}}+\frac{\sqrt{2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}+1\right )}{8 \sqrt{2}}-\frac{3 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}\right )}{8 \sqrt{2}}+\frac{3 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}+1\right )}{8 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-5*(1 - x)^(3/4)*(1 + x)^(1/4))/48 + (5*(1 - x)^(7/4)*(1 + x)^(1/4))/24 - ((1 - x)^(1/4)*(1 + x)^(3/4))/16 +
((1 - x)^(5/4)*(1 + x)^(3/4))/24 + ((1 - x)^(7/4)*(1 + x)^(5/4))/6 + (Sqrt[1 + x]*(1 - x^2)^(5/4))/6 + (1 - x^
2)^(9/4)/(3*(1 - x)^(3/2)) - (3*ArcTan[1 - (Sqrt[2]*(1 - x)^(1/4))/(1 + x)^(1/4)])/(8*Sqrt[2]) + (3*ArcTan[1 +
(Sqrt[2]*(1 - x)^(1/4))/(1 + x)^(1/4)])/(8*Sqrt[2]) + Log[1 + Sqrt[1 - x]/Sqrt[1 + x] - (Sqrt[2]*(1 - x)^(1/4
))/(1 + x)^(1/4)]/(8*Sqrt[2]) - Log[1 + Sqrt[1 - x]/Sqrt[1 + x] + (Sqrt[2]*(1 - x)^(1/4))/(1 + x)^(1/4)]/(8*Sq
rt[2])

Rule 2103

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

Rule 795

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(g*(d + e*x)^m
*(a + c*x^2)^(p + 1))/(c*(m + 2*p + 2)), x] + Dist[(m*(d*g + e*f) + 2*e*f*(p + 1))/(e*(m + 2*p + 2)), Int[(d +
e*x)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] && NeQ[m + 2*p +
2, 0] && NeQ[m, 2]

Rule 675

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Int[(d + e*x)^(m + p)*(a/d + (c*x)/e)^p,
x] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && GtQ[a, 0] && GtQ[d, 0] &&  !I
GtQ[m, 0]

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 240

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

Rule 211

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Di
st[1/(2*r), Int[(r - s*x^2)/(a + b*x^4), x], x] + Dist[1/(2*r), Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[
{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ, b
]]))

Rule 1165

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(-2*d)/e, 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1162

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

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

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

Rule 1633

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[d*e, Int[(d + e*x)^(m - 1)*
PolynomialQuotient[Pq, a*e + c*d*x, x]*(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e, m, p}, x] && PolyQ[Pq
, x] && EqQ[c*d^2 + a*e^2, 0] && EqQ[PolynomialRemainder[Pq, a*e + c*d*x, x], 0]

Rule 793

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d*g - e*f)*(
d + e*x)^m*(a + c*x^2)^(p + 1))/(2*c*d*(m + p + 1)), x] + Dist[(m*(g*c*d + c*e*f) + 2*e*c*f*(p + 1))/(e*(2*c*d
)*(m + p + 1)), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && EqQ[c*d^2
+ a*e^2, 0] && ((LtQ[m, -1] &&  !IGtQ[m + p + 1, 0]) || (LtQ[m, 0] && LtQ[p, -1]) || EqQ[m + 2*p + 2, 0]) &&
NeQ[m + p + 1, 0]

Rule 331

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^(p + (m + 1)/n), Subst[Int[x^m/(1 - b*x^n)^(
p + (m + 1)/n + 1), x], x, x/(a + b*x^n)^(1/n)], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[
p, -2^(-1)] && IntegersQ[m, p + (m + 1)/n]

Rule 297

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]},
Dist[1/(2*s), Int[(r + s*x^2)/(a + b*x^4), x], x] - Dist[1/(2*s), Int[(r - s*x^2)/(a + b*x^4), x], x]] /; Free
Q[{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ,
b]]))

Rubi steps

\begin{align*} \int \frac{x^2 \sqrt{1+x} \sqrt [4]{1-x^2}}{\sqrt{1-x} \left (\sqrt{1-x}-\sqrt{1+x}\right )} \, dx &=-\left (\frac{1}{2} \int x \sqrt{1+x} \sqrt [4]{1-x^2} \, dx\right )-\frac{1}{2} \int \frac{x (1+x) \sqrt [4]{1-x^2}}{\sqrt{1-x}} \, dx\\ &=\frac{1}{6} \sqrt{1+x} \left (1-x^2\right )^{5/4}-\frac{1}{12} \int \sqrt{1+x} \sqrt [4]{1-x^2} \, dx-\frac{1}{2} \int \frac{x \left (1-x^2\right )^{5/4}}{(1-x)^{3/2}} \, dx\\ &=\frac{1}{6} \sqrt{1+x} \left (1-x^2\right )^{5/4}+\frac{\left (1-x^2\right )^{9/4}}{3 (1-x)^{3/2}}-\frac{1}{12} \int \sqrt [4]{1-x} (1+x)^{3/4} \, dx-\frac{1}{2} \int \frac{\left (1-x^2\right )^{5/4}}{\sqrt{1-x}} \, dx\\ &=\frac{1}{24} (1-x)^{5/4} (1+x)^{3/4}+\frac{1}{6} \sqrt{1+x} \left (1-x^2\right )^{5/4}+\frac{\left (1-x^2\right )^{9/4}}{3 (1-x)^{3/2}}-\frac{1}{16} \int \frac{\sqrt [4]{1-x}}{\sqrt [4]{1+x}} \, dx-\frac{1}{2} \int (1-x)^{3/4} (1+x)^{5/4} \, dx\\ &=-\frac{1}{16} \sqrt [4]{1-x} (1+x)^{3/4}+\frac{1}{24} (1-x)^{5/4} (1+x)^{3/4}+\frac{1}{6} (1-x)^{7/4} (1+x)^{5/4}+\frac{1}{6} \sqrt{1+x} \left (1-x^2\right )^{5/4}+\frac{\left (1-x^2\right )^{9/4}}{3 (1-x)^{3/2}}-\frac{1}{32} \int \frac{1}{(1-x)^{3/4} \sqrt [4]{1+x}} \, dx-\frac{5}{12} \int (1-x)^{3/4} \sqrt [4]{1+x} \, dx\\ &=\frac{5}{24} (1-x)^{7/4} \sqrt [4]{1+x}-\frac{1}{16} \sqrt [4]{1-x} (1+x)^{3/4}+\frac{1}{24} (1-x)^{5/4} (1+x)^{3/4}+\frac{1}{6} (1-x)^{7/4} (1+x)^{5/4}+\frac{1}{6} \sqrt{1+x} \left (1-x^2\right )^{5/4}+\frac{\left (1-x^2\right )^{9/4}}{3 (1-x)^{3/2}}-\frac{5}{48} \int \frac{(1-x)^{3/4}}{(1+x)^{3/4}} \, dx+\frac{1}{8} \operatorname{Subst}\left (\int \frac{1}{\sqrt [4]{2-x^4}} \, dx,x,\sqrt [4]{1-x}\right )\\ &=-\frac{5}{48} (1-x)^{3/4} \sqrt [4]{1+x}+\frac{5}{24} (1-x)^{7/4} \sqrt [4]{1+x}-\frac{1}{16} \sqrt [4]{1-x} (1+x)^{3/4}+\frac{1}{24} (1-x)^{5/4} (1+x)^{3/4}+\frac{1}{6} (1-x)^{7/4} (1+x)^{5/4}+\frac{1}{6} \sqrt{1+x} \left (1-x^2\right )^{5/4}+\frac{\left (1-x^2\right )^{9/4}}{3 (1-x)^{3/2}}+\frac{1}{8} \operatorname{Subst}\left (\int \frac{1}{1+x^4} \, dx,x,\frac{\sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )-\frac{5}{32} \int \frac{1}{\sqrt [4]{1-x} (1+x)^{3/4}} \, dx\\ &=-\frac{5}{48} (1-x)^{3/4} \sqrt [4]{1+x}+\frac{5}{24} (1-x)^{7/4} \sqrt [4]{1+x}-\frac{1}{16} \sqrt [4]{1-x} (1+x)^{3/4}+\frac{1}{24} (1-x)^{5/4} (1+x)^{3/4}+\frac{1}{6} (1-x)^{7/4} (1+x)^{5/4}+\frac{1}{6} \sqrt{1+x} \left (1-x^2\right )^{5/4}+\frac{\left (1-x^2\right )^{9/4}}{3 (1-x)^{3/2}}+\frac{1}{16} \operatorname{Subst}\left (\int \frac{1-x^2}{1+x^4} \, dx,x,\frac{\sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )+\frac{1}{16} \operatorname{Subst}\left (\int \frac{1+x^2}{1+x^4} \, dx,x,\frac{\sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )+\frac{5}{8} \operatorname{Subst}\left (\int \frac{x^2}{\left (2-x^4\right )^{3/4}} \, dx,x,\sqrt [4]{1-x}\right )\\ &=-\frac{5}{48} (1-x)^{3/4} \sqrt [4]{1+x}+\frac{5}{24} (1-x)^{7/4} \sqrt [4]{1+x}-\frac{1}{16} \sqrt [4]{1-x} (1+x)^{3/4}+\frac{1}{24} (1-x)^{5/4} (1+x)^{3/4}+\frac{1}{6} (1-x)^{7/4} (1+x)^{5/4}+\frac{1}{6} \sqrt{1+x} \left (1-x^2\right )^{5/4}+\frac{\left (1-x^2\right )^{9/4}}{3 (1-x)^{3/2}}+\frac{1}{32} \operatorname{Subst}\left (\int \frac{1}{1-\sqrt{2} x+x^2} \, dx,x,\frac{\sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )+\frac{1}{32} \operatorname{Subst}\left (\int \frac{1}{1+\sqrt{2} x+x^2} \, dx,x,\frac{\sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )+\frac{5}{8} \operatorname{Subst}\left (\int \frac{x^2}{1+x^4} \, dx,x,\frac{\sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )-\frac{\operatorname{Subst}\left (\int \frac{\sqrt{2}+2 x}{-1-\sqrt{2} x-x^2} \, dx,x,\frac{\sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )}{32 \sqrt{2}}-\frac{\operatorname{Subst}\left (\int \frac{\sqrt{2}-2 x}{-1+\sqrt{2} x-x^2} \, dx,x,\frac{\sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )}{32 \sqrt{2}}\\ &=-\frac{5}{48} (1-x)^{3/4} \sqrt [4]{1+x}+\frac{5}{24} (1-x)^{7/4} \sqrt [4]{1+x}-\frac{1}{16} \sqrt [4]{1-x} (1+x)^{3/4}+\frac{1}{24} (1-x)^{5/4} (1+x)^{3/4}+\frac{1}{6} (1-x)^{7/4} (1+x)^{5/4}+\frac{1}{6} \sqrt{1+x} \left (1-x^2\right )^{5/4}+\frac{\left (1-x^2\right )^{9/4}}{3 (1-x)^{3/2}}-\frac{\log \left (1+\frac{\sqrt{1-x}}{\sqrt{1+x}}-\frac{\sqrt{2} \sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )}{32 \sqrt{2}}+\frac{\log \left (1+\frac{\sqrt{1-x}}{\sqrt{1+x}}+\frac{\sqrt{2} \sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )}{32 \sqrt{2}}-\frac{5}{16} \operatorname{Subst}\left (\int \frac{1-x^2}{1+x^4} \, dx,x,\frac{\sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )+\frac{5}{16} \operatorname{Subst}\left (\int \frac{1+x^2}{1+x^4} \, dx,x,\frac{\sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )+\frac{\operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )}{16 \sqrt{2}}-\frac{\operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )}{16 \sqrt{2}}\\ &=-\frac{5}{48} (1-x)^{3/4} \sqrt [4]{1+x}+\frac{5}{24} (1-x)^{7/4} \sqrt [4]{1+x}-\frac{1}{16} \sqrt [4]{1-x} (1+x)^{3/4}+\frac{1}{24} (1-x)^{5/4} (1+x)^{3/4}+\frac{1}{6} (1-x)^{7/4} (1+x)^{5/4}+\frac{1}{6} \sqrt{1+x} \left (1-x^2\right )^{5/4}+\frac{\left (1-x^2\right )^{9/4}}{3 (1-x)^{3/2}}-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )}{16 \sqrt{2}}+\frac{\tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )}{16 \sqrt{2}}-\frac{\log \left (1+\frac{\sqrt{1-x}}{\sqrt{1+x}}-\frac{\sqrt{2} \sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )}{32 \sqrt{2}}+\frac{\log \left (1+\frac{\sqrt{1-x}}{\sqrt{1+x}}+\frac{\sqrt{2} \sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )}{32 \sqrt{2}}+\frac{5}{32} \operatorname{Subst}\left (\int \frac{1}{1-\sqrt{2} x+x^2} \, dx,x,\frac{\sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )+\frac{5}{32} \operatorname{Subst}\left (\int \frac{1}{1+\sqrt{2} x+x^2} \, dx,x,\frac{\sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )+\frac{5 \operatorname{Subst}\left (\int \frac{\sqrt{2}+2 x}{-1-\sqrt{2} x-x^2} \, dx,x,\frac{\sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )}{32 \sqrt{2}}+\frac{5 \operatorname{Subst}\left (\int \frac{\sqrt{2}-2 x}{-1+\sqrt{2} x-x^2} \, dx,x,\frac{\sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )}{32 \sqrt{2}}\\ &=-\frac{5}{48} (1-x)^{3/4} \sqrt [4]{1+x}+\frac{5}{24} (1-x)^{7/4} \sqrt [4]{1+x}-\frac{1}{16} \sqrt [4]{1-x} (1+x)^{3/4}+\frac{1}{24} (1-x)^{5/4} (1+x)^{3/4}+\frac{1}{6} (1-x)^{7/4} (1+x)^{5/4}+\frac{1}{6} \sqrt{1+x} \left (1-x^2\right )^{5/4}+\frac{\left (1-x^2\right )^{9/4}}{3 (1-x)^{3/2}}-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )}{16 \sqrt{2}}+\frac{\tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )}{16 \sqrt{2}}+\frac{\log \left (1+\frac{\sqrt{1-x}}{\sqrt{1+x}}-\frac{\sqrt{2} \sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )}{8 \sqrt{2}}-\frac{\log \left (1+\frac{\sqrt{1-x}}{\sqrt{1+x}}+\frac{\sqrt{2} \sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )}{8 \sqrt{2}}+\frac{5 \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )}{16 \sqrt{2}}-\frac{5 \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )}{16 \sqrt{2}}\\ &=-\frac{5}{48} (1-x)^{3/4} \sqrt [4]{1+x}+\frac{5}{24} (1-x)^{7/4} \sqrt [4]{1+x}-\frac{1}{16} \sqrt [4]{1-x} (1+x)^{3/4}+\frac{1}{24} (1-x)^{5/4} (1+x)^{3/4}+\frac{1}{6} (1-x)^{7/4} (1+x)^{5/4}+\frac{1}{6} \sqrt{1+x} \left (1-x^2\right )^{5/4}+\frac{\left (1-x^2\right )^{9/4}}{3 (1-x)^{3/2}}-\frac{3 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )}{8 \sqrt{2}}+\frac{3 \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )}{8 \sqrt{2}}+\frac{\log \left (1+\frac{\sqrt{1-x}}{\sqrt{1+x}}-\frac{\sqrt{2} \sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )}{8 \sqrt{2}}-\frac{\log \left (1+\frac{\sqrt{1-x}}{\sqrt{1+x}}+\frac{\sqrt{2} \sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )}{8 \sqrt{2}}\\ \end{align*}

Mathematica [C]  time = 0.484503, size = 153, normalized size = 0.5 \[ \frac{\sqrt [4]{1-x^2} \, _2F_1\left (\frac{1}{4},\frac{1}{4};\frac{5}{4};\frac{1-x}{2}\right )}{8 \sqrt [4]{2} \sqrt [4]{x+1}}+\frac{5 \left (1-x^2\right )^{3/4} \, _2F_1\left (\frac{3}{4},\frac{3}{4};\frac{7}{4};\frac{1-x}{2}\right )}{24\ 2^{3/4} (x+1)^{3/4}}-\frac{1}{48} \sqrt{x+1} \sqrt [4]{1-x^2} \left (8 x^2-\frac{\sqrt{1-x^2} \left (8 x^2+22 x+29\right )}{x+1}+2 x-7\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[1 + x]*(1 - x^2)^(1/4)*(-7 + 2*x + 8*x^2 - (Sqrt[1 - x^2]*(29 + 22*x + 8*x^2))/(1 + x)))/48 + ((1 - x^2
)^(1/4)*Hypergeometric2F1[1/4, 1/4, 5/4, (1 - x)/2])/(8*2^(1/4)*(1 + x)^(1/4)) + (5*(1 - x^2)^(3/4)*Hypergeome
tric2F1[3/4, 3/4, 7/4, (1 - x)/2])/(24*2^(3/4)*(1 + x)^(3/4))

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Maple [F]  time = 0.049, size = 0, normalized size = 0. \begin{align*} \int{{x}^{2}\sqrt [4]{-{x}^{2}+1}\sqrt{1+x}{\frac{1}{\sqrt{1-x}}} \left ( \sqrt{1-x}-\sqrt{1+x} \right ) ^{-1}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(-x^2+1)^(1/4)*(1+x)^(1/2)/(1-x)^(1/2)/((1-x)^(1/2)-(1+x)^(1/2)),x)

[Out]

int(x^2*(-x^2+1)^(1/4)*(1+x)^(1/2)/(1-x)^(1/2)/((1-x)^(1/2)-(1+x)^(1/2)),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{{\left (-x^{2} + 1\right )}^{\frac{1}{4}} \sqrt{x + 1} x^{2}}{\sqrt{-x + 1}{\left (\sqrt{x + 1} - \sqrt{-x + 1}\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(-x^2+1)^(1/4)*(1+x)^(1/2)/(1-x)^(1/2)/((1-x)^(1/2)-(1+x)^(1/2)),x, algorithm="maxima")

[Out]

-integrate((-x^2 + 1)^(1/4)*sqrt(x + 1)*x^2/(sqrt(-x + 1)*(sqrt(x + 1) - sqrt(-x + 1))), x)

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Fricas [B]  time = 2.19136, size = 1611, normalized size = 5.3 \begin{align*} -\frac{1}{48} \,{\left (8 \, x^{2} + 2 \, x - 7\right )}{\left (-x^{2} + 1\right )}^{\frac{1}{4}} \sqrt{x + 1} + \frac{1}{48} \,{\left (8 \, x^{2} + 22 \, x + 29\right )}{\left (-x^{2} + 1\right )}^{\frac{1}{4}} \sqrt{-x + 1} - \frac{1}{16} \, \sqrt{2} \arctan \left (\frac{\sqrt{2}{\left (x + 1\right )} \sqrt{\frac{\sqrt{2}{\left (-x^{2} + 1\right )}^{\frac{1}{4}} \sqrt{x + 1} + x + \sqrt{-x^{2} + 1} + 1}{x + 1}} - \sqrt{2}{\left (-x^{2} + 1\right )}^{\frac{1}{4}} \sqrt{x + 1} - x - 1}{x + 1}\right ) - \frac{1}{16} \, \sqrt{2} \arctan \left (\frac{\sqrt{2}{\left (x + 1\right )} \sqrt{-\frac{\sqrt{2}{\left (-x^{2} + 1\right )}^{\frac{1}{4}} \sqrt{x + 1} - x - \sqrt{-x^{2} + 1} - 1}{x + 1}} - \sqrt{2}{\left (-x^{2} + 1\right )}^{\frac{1}{4}} \sqrt{x + 1} + x + 1}{x + 1}\right ) - \frac{5}{16} \, \sqrt{2} \arctan \left (\frac{\sqrt{2}{\left (x - 1\right )} \sqrt{\frac{\sqrt{2}{\left (-x^{2} + 1\right )}^{\frac{1}{4}} \sqrt{-x + 1} + x - \sqrt{-x^{2} + 1} - 1}{x - 1}} - \sqrt{2}{\left (-x^{2} + 1\right )}^{\frac{1}{4}} \sqrt{-x + 1} - x + 1}{x - 1}\right ) - \frac{5}{16} \, \sqrt{2} \arctan \left (\frac{\sqrt{2}{\left (x - 1\right )} \sqrt{-\frac{\sqrt{2}{\left (-x^{2} + 1\right )}^{\frac{1}{4}} \sqrt{-x + 1} - x + \sqrt{-x^{2} + 1} + 1}{x - 1}} - \sqrt{2}{\left (-x^{2} + 1\right )}^{\frac{1}{4}} \sqrt{-x + 1} + x - 1}{x - 1}\right ) + \frac{1}{64} \, \sqrt{2} \log \left (\frac{4 \,{\left (\sqrt{2}{\left (-x^{2} + 1\right )}^{\frac{1}{4}} \sqrt{x + 1} + x + \sqrt{-x^{2} + 1} + 1\right )}}{x + 1}\right ) - \frac{1}{64} \, \sqrt{2} \log \left (-\frac{4 \,{\left (\sqrt{2}{\left (-x^{2} + 1\right )}^{\frac{1}{4}} \sqrt{x + 1} - x - \sqrt{-x^{2} + 1} - 1\right )}}{x + 1}\right ) + \frac{5}{64} \, \sqrt{2} \log \left (\frac{4 \,{\left (\sqrt{2}{\left (-x^{2} + 1\right )}^{\frac{1}{4}} \sqrt{-x + 1} + x - \sqrt{-x^{2} + 1} - 1\right )}}{x - 1}\right ) - \frac{5}{64} \, \sqrt{2} \log \left (-\frac{4 \,{\left (\sqrt{2}{\left (-x^{2} + 1\right )}^{\frac{1}{4}} \sqrt{-x + 1} - x + \sqrt{-x^{2} + 1} + 1\right )}}{x - 1}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(-x^2+1)^(1/4)*(1+x)^(1/2)/(1-x)^(1/2)/((1-x)^(1/2)-(1+x)^(1/2)),x, algorithm="fricas")

[Out]

-1/48*(8*x^2 + 2*x - 7)*(-x^2 + 1)^(1/4)*sqrt(x + 1) + 1/48*(8*x^2 + 22*x + 29)*(-x^2 + 1)^(1/4)*sqrt(-x + 1)
- 1/16*sqrt(2)*arctan((sqrt(2)*(x + 1)*sqrt((sqrt(2)*(-x^2 + 1)^(1/4)*sqrt(x + 1) + x + sqrt(-x^2 + 1) + 1)/(x
+ 1)) - sqrt(2)*(-x^2 + 1)^(1/4)*sqrt(x + 1) - x - 1)/(x + 1)) - 1/16*sqrt(2)*arctan((sqrt(2)*(x + 1)*sqrt(-(
sqrt(2)*(-x^2 + 1)^(1/4)*sqrt(x + 1) - x - sqrt(-x^2 + 1) - 1)/(x + 1)) - sqrt(2)*(-x^2 + 1)^(1/4)*sqrt(x + 1)
+ x + 1)/(x + 1)) - 5/16*sqrt(2)*arctan((sqrt(2)*(x - 1)*sqrt((sqrt(2)*(-x^2 + 1)^(1/4)*sqrt(-x + 1) + x - sq
rt(-x^2 + 1) - 1)/(x - 1)) - sqrt(2)*(-x^2 + 1)^(1/4)*sqrt(-x + 1) - x + 1)/(x - 1)) - 5/16*sqrt(2)*arctan((sq
rt(2)*(x - 1)*sqrt(-(sqrt(2)*(-x^2 + 1)^(1/4)*sqrt(-x + 1) - x + sqrt(-x^2 + 1) + 1)/(x - 1)) - sqrt(2)*(-x^2
+ 1)^(1/4)*sqrt(-x + 1) + x - 1)/(x - 1)) + 1/64*sqrt(2)*log(4*(sqrt(2)*(-x^2 + 1)^(1/4)*sqrt(x + 1) + x + sqr
t(-x^2 + 1) + 1)/(x + 1)) - 1/64*sqrt(2)*log(-4*(sqrt(2)*(-x^2 + 1)^(1/4)*sqrt(x + 1) - x - sqrt(-x^2 + 1) - 1
)/(x + 1)) + 5/64*sqrt(2)*log(4*(sqrt(2)*(-x^2 + 1)^(1/4)*sqrt(-x + 1) + x - sqrt(-x^2 + 1) - 1)/(x - 1)) - 5/
64*sqrt(2)*log(-4*(sqrt(2)*(-x^2 + 1)^(1/4)*sqrt(-x + 1) - x + sqrt(-x^2 + 1) + 1)/(x - 1))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(-x**2+1)**(1/4)*(1+x)**(1/2)/(1-x)**(1/2)/((1-x)**(1/2)-(1+x)**(1/2)),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{{\left (-x^{2} + 1\right )}^{\frac{1}{4}} \sqrt{x + 1} x^{2}}{\sqrt{-x + 1}{\left (\sqrt{x + 1} - \sqrt{-x + 1}\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(-x^2+1)^(1/4)*(1+x)^(1/2)/(1-x)^(1/2)/((1-x)^(1/2)-(1+x)^(1/2)),x, algorithm="giac")

[Out]

integrate(-(-x^2 + 1)^(1/4)*sqrt(x + 1)*x^2/(sqrt(-x + 1)*(sqrt(x + 1) - sqrt(-x + 1))), x)