3.15.18 \(\int \frac {(1+x^2) \sqrt [4]{x^3+x^4}}{x^2 (-1+x^2)} \, dx\)

Optimal. Leaf size=101 \[ \frac {4 \sqrt [4]{x^4+x^3}}{x}-2 \tan ^{-1}\left (\frac {x}{\sqrt [4]{x^4+x^3}}\right )+2 \sqrt [4]{2} \tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^4+x^3}}\right )+2 \tanh ^{-1}\left (\frac {x}{\sqrt [4]{x^4+x^3}}\right )-2 \sqrt [4]{2} \tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^4+x^3}}\right ) \]

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Rubi [A]  time = 1.25, antiderivative size = 193, normalized size of antiderivative = 1.91, number of steps used = 44, number of rules used = 20, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.741, Rules used = {2056, 1586, 6733, 6725, 264, 331, 298, 203, 206, 1240, 410, 237, 335, 275, 231, 407, 409, 1213, 537, 494} \begin {gather*} \frac {4 \sqrt [4]{x^4+x^3}}{x}-\frac {2 \sqrt [4]{x^4+x^3} \tan ^{-1}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{x^{3/4} \sqrt [4]{x+1}}+\frac {2 \sqrt [4]{2} \sqrt [4]{x^4+x^3} \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{x^{3/4} \sqrt [4]{x+1}}+\frac {2 \sqrt [4]{x^4+x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{x^{3/4} \sqrt [4]{x+1}}-\frac {2 \sqrt [4]{2} \sqrt [4]{x^4+x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{x^{3/4} \sqrt [4]{x+1}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((1 + x^2)*(x^3 + x^4)^(1/4))/(x^2*(-1 + x^2)),x]

[Out]

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

Rule 203

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

Rule 206

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

Rule 231

Int[((a_) + (b_.)*(x_)^2)^(-3/4), x_Symbol] :> Simp[(2*EllipticF[(1*ArcTan[Rt[b/a, 2]*x])/2, 2])/(a^(3/4)*Rt[b
/a, 2]), x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b/a]

Rule 237

Int[((a_) + (b_.)*(x_)^4)^(-3/4), x_Symbol] :> Dist[(x^3*(1 + a/(b*x^4))^(3/4))/(a + b*x^4)^(3/4), Int[1/(x^3*
(1 + a/(b*x^4))^(3/4)), x], x] /; FreeQ[{a, b}, x]

Rule 264

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a
*c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rule 275

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

Rule 298

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-(a/b), 2]], s = Denominator[Rt[-(a/b),
2]]}, Dist[s/(2*b), Int[1/(r + s*x^2), x], x] - Dist[s/(2*b), Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] &
&  !GtQ[a/b, 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 335

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

Rule 407

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

Rule 409

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

Rule 410

Int[1/(((a_) + (b_.)*(x_)^4)^(3/4)*((c_) + (d_.)*(x_)^4)), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x^4)^
(3/4), x], x] - Dist[d/(b*c - a*d), Int[(a + b*x^4)^(1/4)/(c + d*x^4), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ
[b*c - a*d, 0]

Rule 494

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> With[{k = Denominato
r[p]}, Dist[(k*a^(p + (m + 1)/n))/n, Subst[Int[(x^((k*(m + 1))/n - 1)*(c - (b*c - a*d)*x^k)^q)/(1 - b*x^k)^(p
+ q + (m + 1)/n + 1), x], x, x^(n/k)/(a + b*x^n)^(1/k)], x]] /; FreeQ[{a, b, c, d}, x] && IGtQ[n, 0] && Ration
alQ[m, p] && IntegersQ[p + (m + 1)/n, q] && LtQ[-1, p, 0]

Rule 537

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

Rule 1213

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

Rule 1240

Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (c_.)*(x_)^4)^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + c*x^4)^p, (d/
(d^2 - e^2*x^4) - (e*x^2)/(d^2 - e^2*x^4))^(-q), x], x] /; FreeQ[{a, c, d, e, p}, x] && NeQ[c*d^2 + a*e^2, 0]
&&  !IntegerQ[p] && ILtQ[q, 0]

Rule 1586

Int[(u_.)*(Px_)^(p_.)*(Qx_)^(q_.), x_Symbol] :> Int[u*PolynomialQuotient[Px, Qx, x]^p*Qx^(p + q), x] /; FreeQ[
q, x] && PolyQ[Px, x] && PolyQ[Qx, x] && EqQ[PolynomialRemainder[Px, Qx, x], 0] && IntegerQ[p] && LtQ[p*q, 0]

Rule 2056

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

Rule 6725

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

Rule 6733

Int[(u_)*(x_)^(m_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k, Subst[Int[x^(k*(m + 1) - 1)*(u /. x -> x^k
), x], x, x^(1/k)], x]] /; FractionQ[m]

Rubi steps

\begin {align*} \int \frac {\left (1+x^2\right ) \sqrt [4]{x^3+x^4}}{x^2 \left (-1+x^2\right )} \, dx &=\frac {\sqrt [4]{x^3+x^4} \int \frac {\sqrt [4]{1+x} \left (1+x^2\right )}{x^{5/4} \left (-1+x^2\right )} \, dx}{x^{3/4} \sqrt [4]{1+x}}\\ &=\frac {\sqrt [4]{x^3+x^4} \int \frac {1+x^2}{(-1+x) x^{5/4} (1+x)^{3/4}} \, dx}{x^{3/4} \sqrt [4]{1+x}}\\ &=\frac {\left (4 \sqrt [4]{x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1+x^8}{x^2 \left (-1+x^4\right ) \left (1+x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{x^{3/4} \sqrt [4]{1+x}}\\ &=\frac {\left (4 \sqrt [4]{x^3+x^4}\right ) \operatorname {Subst}\left (\int \left (-\frac {1}{x^2 \left (1+x^4\right )^{3/4}}+\frac {x^2}{\left (1+x^4\right )^{3/4}}+\frac {1}{\left (-1+x^2\right ) \left (1+x^4\right )^{3/4}}+\frac {1}{\left (1+x^2\right ) \left (1+x^4\right )^{3/4}}\right ) \, dx,x,\sqrt [4]{x}\right )}{x^{3/4} \sqrt [4]{1+x}}\\ &=-\frac {\left (4 \sqrt [4]{x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{x^2 \left (1+x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {\left (4 \sqrt [4]{x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (1+x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {\left (4 \sqrt [4]{x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-1+x^2\right ) \left (1+x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {\left (4 \sqrt [4]{x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1+x^2\right ) \left (1+x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{x^{3/4} \sqrt [4]{1+x}}\\ &=\frac {4 \sqrt [4]{x^3+x^4}}{x}+\frac {\left (4 \sqrt [4]{x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{1-x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {\left (4 \sqrt [4]{x^3+x^4}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{\left (1-x^4\right ) \left (1+x^4\right )^{3/4}}+\frac {x^2}{\left (-1+x^4\right ) \left (1+x^4\right )^{3/4}}\right ) \, dx,x,\sqrt [4]{x}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {\left (4 \sqrt [4]{x^3+x^4}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{\left (-1+x^4\right ) \left (1+x^4\right )^{3/4}}+\frac {x^2}{\left (-1+x^4\right ) \left (1+x^4\right )^{3/4}}\right ) \, dx,x,\sqrt [4]{x}\right )}{x^{3/4} \sqrt [4]{1+x}}\\ &=\frac {4 \sqrt [4]{x^3+x^4}}{x}+\frac {\left (2 \sqrt [4]{x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}-\frac {\left (2 \sqrt [4]{x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {\left (4 \sqrt [4]{x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1-x^4\right ) \left (1+x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {\left (4 \sqrt [4]{x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-1+x^4\right ) \left (1+x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{x^{3/4} \sqrt [4]{1+x}}+2 \frac {\left (4 \sqrt [4]{x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (-1+x^4\right ) \left (1+x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{x^{3/4} \sqrt [4]{1+x}}\\ &=\frac {4 \sqrt [4]{x^3+x^4}}{x}-\frac {2 \sqrt [4]{x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {2 \sqrt [4]{x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {\left (2 \sqrt [4]{x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [4]{1+x^4}}{1-x^4} \, dx,x,\sqrt [4]{x}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {\left (2 \sqrt [4]{x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [4]{1+x^4}}{-1+x^4} \, dx,x,\sqrt [4]{x}\right )}{x^{3/4} \sqrt [4]{1+x}}+2 \frac {\left (4 \sqrt [4]{x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{-1+2 x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}\\ &=\frac {4 \sqrt [4]{x^3+x^4}}{x}-\frac {2 \sqrt [4]{x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {2 \sqrt [4]{x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}+2 \left (-\frac {\left (\sqrt {2} \sqrt [4]{x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {\left (\sqrt {2} \sqrt [4]{x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {2} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}\right )+\frac {\left (2 \sqrt {\frac {1}{1+x}} \sqrt [4]{1+x} \sqrt [4]{x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1-2 x^4\right ) \sqrt {1-x^4}} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4}}+\frac {\left (2 \sqrt {\frac {1}{1+x}} \sqrt [4]{1+x} \sqrt [4]{x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x^4} \left (-1+2 x^4\right )} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4}}\\ &=\frac {4 \sqrt [4]{x^3+x^4}}{x}-\frac {2 \sqrt [4]{x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {2 \sqrt [4]{x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}+2 \left (\frac {\sqrt [4]{2} \sqrt [4]{x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}-\frac {\sqrt [4]{2} \sqrt [4]{x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}\right )\\ \end {align*}

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Mathematica [C]  time = 0.06, size = 77, normalized size = 0.76 \begin {gather*} -\frac {4 \sqrt [4]{x^3 (x+1)} \left (3 (x+1) \, _2F_1\left (-\frac {1}{4},-\frac {1}{4};\frac {3}{4};-x\right )+2 \sqrt [4]{x+1} \left (x \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};\frac {2 x}{x+1}\right )-3 (x+1)\right )\right )}{3 x (x+1)^{5/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((1 + x^2)*(x^3 + x^4)^(1/4))/(x^2*(-1 + x^2)),x]

[Out]

(-4*(x^3*(1 + x))^(1/4)*(3*(1 + x)*Hypergeometric2F1[-1/4, -1/4, 3/4, -x] + 2*(1 + x)^(1/4)*(-3*(1 + x) + x*Hy
pergeometric2F1[3/4, 1, 7/4, (2*x)/(1 + x)])))/(3*x*(1 + x)^(5/4))

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IntegrateAlgebraic [A]  time = 0.38, size = 101, normalized size = 1.00 \begin {gather*} \frac {4 \sqrt [4]{x^3+x^4}}{x}-2 \tan ^{-1}\left (\frac {x}{\sqrt [4]{x^3+x^4}}\right )+2 \sqrt [4]{2} \tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^3+x^4}}\right )+2 \tanh ^{-1}\left (\frac {x}{\sqrt [4]{x^3+x^4}}\right )-2 \sqrt [4]{2} \tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^3+x^4}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((1 + x^2)*(x^3 + x^4)^(1/4))/(x^2*(-1 + x^2)),x]

[Out]

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

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fricas [B]  time = 0.46, size = 182, normalized size = 1.80 \begin {gather*} \frac {4 \cdot 2^{\frac {1}{4}} x \arctan \left (\frac {2^{\frac {3}{4}} x \sqrt {\frac {\sqrt {2} x^{2} + \sqrt {x^{4} + x^{3}}}{x^{2}}} - 2^{\frac {3}{4}} {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{2 \, x}\right ) - 2^{\frac {1}{4}} x \log \left (\frac {2^{\frac {1}{4}} x + {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + 2^{\frac {1}{4}} x \log \left (-\frac {2^{\frac {1}{4}} x - {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + 2 \, x \arctan \left (\frac {{\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + x \log \left (\frac {x + {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - x \log \left (-\frac {x - {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + 4 \, {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+1)*(x^4+x^3)^(1/4)/x^2/(x^2-1),x, algorithm="fricas")

[Out]

(4*2^(1/4)*x*arctan(1/2*(2^(3/4)*x*sqrt((sqrt(2)*x^2 + sqrt(x^4 + x^3))/x^2) - 2^(3/4)*(x^4 + x^3)^(1/4))/x) -
2^(1/4)*x*log((2^(1/4)*x + (x^4 + x^3)^(1/4))/x) + 2^(1/4)*x*log(-(2^(1/4)*x - (x^4 + x^3)^(1/4))/x) + 2*x*ar
ctan((x^4 + x^3)^(1/4)/x) + x*log((x + (x^4 + x^3)^(1/4))/x) - x*log(-(x - (x^4 + x^3)^(1/4))/x) + 4*(x^4 + x^
3)^(1/4))/x

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giac [A]  time = 0.26, size = 97, normalized size = 0.96 \begin {gather*} -2 \cdot 2^{\frac {1}{4}} \arctan \left (\frac {1}{2} \cdot 2^{\frac {3}{4}} {\left (\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) - 2^{\frac {1}{4}} \log \left (2^{\frac {1}{4}} + {\left (\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) + 2^{\frac {1}{4}} \log \left ({\left | -2^{\frac {1}{4}} + {\left (\frac {1}{x} + 1\right )}^{\frac {1}{4}} \right |}\right ) + 4 \, {\left (\frac {1}{x} + 1\right )}^{\frac {1}{4}} + 2 \, \arctan \left ({\left (\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) + \log \left ({\left (\frac {1}{x} + 1\right )}^{\frac {1}{4}} + 1\right ) - \log \left ({\left | {\left (\frac {1}{x} + 1\right )}^{\frac {1}{4}} - 1 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+1)*(x^4+x^3)^(1/4)/x^2/(x^2-1),x, algorithm="giac")

[Out]

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

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maple [C]  time = 3.46, size = 439, normalized size = 4.35

method	result	size
trager	\(\frac {4 \left (x^{4}+x^{3}\right )^{\frac {1}{4}}}{x}-\ln \left (\frac {2 \left (x^{4}+x^{3}\right )^{\frac {3}{4}}-2 \sqrt {x^{4}+x^{3}}\, x +2 x^{2} \left (x^{4}+x^{3}\right )^{\frac {1}{4}}-2 x^{3}-x^{2}}{x^{2}}\right )+\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) \ln \left (\frac {3 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} x^{3}+\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} x^{2}-4 \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} \left (x^{4}+x^{3}\right )^{\frac {1}{4}} x^{2}-4 \sqrt {x^{4}+x^{3}}\, \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) x +4 \left (x^{4}+x^{3}\right )^{\frac {3}{4}}}{\left (-1+x \right ) x^{2}}\right )-\RootOf \left (\textit {\_Z}^{4}-2\right ) \ln \left (\frac {3 \RootOf \left (\textit {\_Z}^{4}-2\right )^{3} x^{3}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{3} x^{2}+4 \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} \left (x^{4}+x^{3}\right )^{\frac {1}{4}} x^{2}+4 \sqrt {x^{4}+x^{3}}\, \RootOf \left (\textit {\_Z}^{4}-2\right ) x +4 \left (x^{4}+x^{3}\right )^{\frac {3}{4}}}{\left (-1+x \right ) x^{2}}\right )+\frac {\RootOf \left (\textit {\_Z}^{4}-2\right )^{3} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) \ln \left (\frac {2 \sqrt {x^{4}+x^{3}}\, \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}-2\right )^{3} x -2 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}-2\right )^{3} x^{3}-\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}-2\right )^{3} x^{2}+4 \left (x^{4}+x^{3}\right )^{\frac {3}{4}}-4 x^{2} \left (x^{4}+x^{3}\right )^{\frac {1}{4}}}{x^{2}}\right )}{2}\)	\(439\)
risch	\(\frac {4 \left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}}{x}+\frac {\left (-\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {2 \sqrt {x^{4}+3 x^{3}+3 x^{2}+x}\, \RootOf \left (\textit {\_Z}^{2}+1\right ) x -2 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{3}+2 \left (x^{4}+3 x^{3}+3 x^{2}+x \right )^{\frac {3}{4}}+2 \sqrt {x^{4}+3 x^{3}+3 x^{2}+x}\, \RootOf \left (\textit {\_Z}^{2}+1\right )-2 \left (x^{4}+3 x^{3}+3 x^{2}+x \right )^{\frac {1}{4}} x^{2}-5 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{2}-4 \left (x^{4}+3 x^{3}+3 x^{2}+x \right )^{\frac {1}{4}} x -4 \RootOf \left (\textit {\_Z}^{2}+1\right ) x -2 \left (x^{4}+3 x^{3}+3 x^{2}+x \right )^{\frac {1}{4}}-\RootOf \left (\textit {\_Z}^{2}+1\right )}{\left (1+x \right )^{2}}\right )+\ln \left (\frac {2 \left (x^{4}+3 x^{3}+3 x^{2}+x \right )^{\frac {3}{4}}+2 \sqrt {x^{4}+3 x^{3}+3 x^{2}+x}\, x +2 \left (x^{4}+3 x^{3}+3 x^{2}+x \right )^{\frac {1}{4}} x^{2}+2 x^{3}+2 \sqrt {x^{4}+3 x^{3}+3 x^{2}+x}+4 \left (x^{4}+3 x^{3}+3 x^{2}+x \right )^{\frac {1}{4}} x +5 x^{2}+2 \left (x^{4}+3 x^{3}+3 x^{2}+x \right )^{\frac {1}{4}}+4 x +1}{\left (1+x \right )^{2}}\right )-\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) \ln \left (-\frac {2 \sqrt {x^{4}+3 x^{3}+3 x^{2}+x}\, \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) x +2 \sqrt {x^{4}+3 x^{3}+3 x^{2}+x}\, \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right )+2 \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} \left (x^{4}+3 x^{3}+3 x^{2}+x \right )^{\frac {1}{4}} x^{2}+4 \left (x^{4}+3 x^{3}+3 x^{2}+x \right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} x -3 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) x^{3}+2 \left (x^{4}+3 x^{3}+3 x^{2}+x \right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-2\right )^{2}-4 \left (x^{4}+3 x^{3}+3 x^{2}+x \right )^{\frac {3}{4}}-7 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) x^{2}-5 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) x -\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right )}{\left (-1+x \right ) \left (1+x \right )^{2}}\right )-\RootOf \left (\textit {\_Z}^{4}-2\right ) \ln \left (\frac {2 \sqrt {x^{4}+3 x^{3}+3 x^{2}+x}\, \RootOf \left (\textit {\_Z}^{4}-2\right )^{3} x +2 \sqrt {x^{4}+3 x^{3}+3 x^{2}+x}\, \RootOf \left (\textit {\_Z}^{4}-2\right )^{3}+2 \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} \left (x^{4}+3 x^{3}+3 x^{2}+x \right )^{\frac {1}{4}} x^{2}+4 \left (x^{4}+3 x^{3}+3 x^{2}+x \right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} x +3 \RootOf \left (\textit {\_Z}^{4}-2\right ) x^{3}+2 \left (x^{4}+3 x^{3}+3 x^{2}+x \right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-2\right )^{2}+7 \RootOf \left (\textit {\_Z}^{4}-2\right ) x^{2}+4 \left (x^{4}+3 x^{3}+3 x^{2}+x \right )^{\frac {3}{4}}+5 \RootOf \left (\textit {\_Z}^{4}-2\right ) x +\RootOf \left (\textit {\_Z}^{4}-2\right )}{\left (-1+x \right ) \left (1+x \right )^{2}}\right )\right ) \left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}} \left (\left (1+x \right )^{3} x \right )^{\frac {1}{4}}}{x \left (1+x \right )}\)	\(869\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2+1)*(x^4+x^3)^(1/4)/x^2/(x^2-1),x,method=_RETURNVERBOSE)

[Out]

4*(x^4+x^3)^(1/4)/x-ln((2*(x^4+x^3)^(3/4)-2*(x^4+x^3)^(1/2)*x+2*x^2*(x^4+x^3)^(1/4)-2*x^3-x^2)/x^2)+RootOf(_Z^
2+RootOf(_Z^4-2)^2)*ln((3*RootOf(_Z^2+RootOf(_Z^4-2)^2)*RootOf(_Z^4-2)^2*x^3+RootOf(_Z^2+RootOf(_Z^4-2)^2)*Roo
tOf(_Z^4-2)^2*x^2-4*RootOf(_Z^4-2)^2*(x^4+x^3)^(1/4)*x^2-4*(x^4+x^3)^(1/2)*RootOf(_Z^2+RootOf(_Z^4-2)^2)*x+4*(
x^4+x^3)^(3/4))/(-1+x)/x^2)-RootOf(_Z^4-2)*ln((3*RootOf(_Z^4-2)^3*x^3+RootOf(_Z^4-2)^3*x^2+4*RootOf(_Z^4-2)^2*
(x^4+x^3)^(1/4)*x^2+4*(x^4+x^3)^(1/2)*RootOf(_Z^4-2)*x+4*(x^4+x^3)^(3/4))/(-1+x)/x^2)+1/2*RootOf(_Z^4-2)^3*Roo
tOf(_Z^2+RootOf(_Z^4-2)^2)*ln((2*(x^4+x^3)^(1/2)*RootOf(_Z^2+RootOf(_Z^4-2)^2)*RootOf(_Z^4-2)^3*x-2*RootOf(_Z^
2+RootOf(_Z^4-2)^2)*RootOf(_Z^4-2)^3*x^3-RootOf(_Z^2+RootOf(_Z^4-2)^2)*RootOf(_Z^4-2)^3*x^2+4*(x^4+x^3)^(3/4)-
4*x^2*(x^4+x^3)^(1/4))/x^2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{4} + x^{3}\right )}^{\frac {1}{4}} {\left (x^{2} + 1\right )}}{{\left (x^{2} - 1\right )} x^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+1)*(x^4+x^3)^(1/4)/x^2/(x^2-1),x, algorithm="maxima")

[Out]

integrate((x^4 + x^3)^(1/4)*(x^2 + 1)/((x^2 - 1)*x^2), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (x^4+x^3\right )}^{1/4}\,\left (x^2+1\right )}{x^2\,\left (x^2-1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^3 + x^4)^(1/4)*(x^2 + 1))/(x^2*(x^2 - 1)),x)

[Out]

int(((x^3 + x^4)^(1/4)*(x^2 + 1))/(x^2*(x^2 - 1)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt [4]{x^{3} \left (x + 1\right )} \left (x^{2} + 1\right )}{x^{2} \left (x - 1\right ) \left (x + 1\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**2+1)*(x**4+x**3)**(1/4)/x**2/(x**2-1),x)

[Out]

Integral((x**3*(x + 1))**(1/4)*(x**2 + 1)/(x**2*(x - 1)*(x + 1)), x)

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