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

________________________________________________________________________________________

Rubi [A]  time = 0.82, 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 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 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 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 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 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]]))

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 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 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 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 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 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 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 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 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 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]

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 = 0.51, size = 153, normalized size = 0.50 \[ \frac {\sqrt [4]{1-x^2} \, _2F_1\left (\frac {1}{4},\frac {1}{4};\frac {5}{4};\frac {1-x}{2}\right )}{8 \sqrt [4]{2} \sqrt [4]{x+1}}+\frac {5 \left (1-x^2\right )^{3/4} \, _2F_1\left (\frac {3}{4},\frac {3}{4};\frac {7}{4};\frac {1-x}{2}\right )}{24\ 2^{3/4} (x+1)^{3/4}}-\frac {1}{48} \sqrt {x+1} \sqrt [4]{1-x^2} \left (8 x^2-\frac {\sqrt {1-x^2} \left (8 x^2+22 x+29\right )}{x+1}+2 x-7\right ) \]

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]

-1/48*(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))) + ((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)*Hypergeo
metric2F1[3/4, 3/4, 7/4, (1 - x)/2])/(24*2^(3/4)*(1 + x)^(3/4))

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 42.80, size = 294, normalized size = 0.97 \[ \frac {1}{48} \sqrt {x+1} \sqrt [4]{1-x^2} \left (-8 x^2-2 x+7\right )+\frac {1}{48} \sqrt {1-x} \sqrt [4]{1-x^2} \left (8 x^2+22 x+29\right )-\frac {\tan ^{-1}\left (\frac {-\frac {\sqrt {1-x^2}}{\sqrt {2}}+\frac {x}{\sqrt {2}}+\frac {1}{\sqrt {2}}}{\sqrt {x+1} \sqrt [4]{1-x^2}}\right )}{16 \sqrt {2}}-\frac {5 \tan ^{-1}\left (\frac {\frac {\sqrt {1-x^2}}{\sqrt {2}}+\frac {x}{\sqrt {2}}-\frac {1}{\sqrt {2}}}{\sqrt {1-x} \sqrt [4]{1-x^2}}\right )}{16 \sqrt {2}}+\frac {5 \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {1-x} \sqrt [4]{1-x^2}}{-\sqrt {1-x^2}+x-1}\right )}{16 \sqrt {2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {x+1} \sqrt [4]{1-x^2}}{\sqrt {1-x^2}+x+1}\right )}{16 \sqrt {2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[1 + x]*(7 - 2*x - 8*x^2)*(1 - x^2)^(1/4))/48 + (Sqrt[1 - x]*(1 - x^2)^(1/4)*(29 + 22*x + 8*x^2))/48 - Ar
cTan[(1/Sqrt[2] + x/Sqrt[2] - Sqrt[1 - x^2]/Sqrt[2])/(Sqrt[1 + x]*(1 - x^2)^(1/4))]/(16*Sqrt[2]) - (5*ArcTan[(
-(1/Sqrt[2]) + x/Sqrt[2] + Sqrt[1 - x^2]/Sqrt[2])/(Sqrt[1 - x]*(1 - x^2)^(1/4))])/(16*Sqrt[2]) + (5*ArcTanh[(S
qrt[2]*Sqrt[1 - x]*(1 - x^2)^(1/4))/(-1 + x - Sqrt[1 - x^2])])/(16*Sqrt[2]) + ArcTanh[(Sqrt[2]*Sqrt[1 + x]*(1
- x^2)^(1/4))/(1 + x + Sqrt[1 - x^2])]/(16*Sqrt[2])

________________________________________________________________________________________

fricas [B]  time = 1.29, size = 577, normalized size = 1.90 \[ -\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 ) \]

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))

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int -\frac {{\left (-x^{2} + 1\right )}^{\frac {1}{4}} \sqrt {x + 1} x^{2}}{\sqrt {-x + 1} {\left (\sqrt {x + 1} - \sqrt {-x + 1}\right )}}\,{d x} \]

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)

________________________________________________________________________________________

maple [F]  time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {x^{2} \left (-x^{2}+1\right )^{\frac {1}{4}} \sqrt {1+x}}{\sqrt {1-x}\, \left (\sqrt {1-x}-\sqrt {1+x}\right )}\, dx\]

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)

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\int \frac {{\left (-x^{2} + 1\right )}^{\frac {1}{4}} \sqrt {x + 1} x^{2}}{\sqrt {-x + 1} {\left (\sqrt {x + 1} - \sqrt {-x + 1}\right )}}\,{d x} \]

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)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ -\int \frac {x^2\,{\left (1-x^2\right )}^{1/4}\,\sqrt {x+1}}{\left (\sqrt {x+1}-\sqrt {1-x}\right )\,\sqrt {1-x}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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

________________________________________________________________________________________