\(\int \frac {x^2 \sqrt [4]{x^3+x^4}}{-1+x} \, dx\) [1701]

Optimal result

Integrand size = 20, antiderivative size = 114 \[ \int \frac {x^2 \sqrt [4]{x^3+x^4}}{-1+x} \, dx=\frac {1}{96} \left (101+52 x+32 x^2\right ) \sqrt [4]{x^3+x^4}-\frac {155}{64} \arctan \left (\frac {x}{\sqrt [4]{x^3+x^4}}\right )+2 \sqrt [4]{2} \arctan \left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^3+x^4}}\right )+\frac {155}{64} \text {arctanh}\left (\frac {x}{\sqrt [4]{x^3+x^4}}\right )-2 \sqrt [4]{2} \text {arctanh}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^3+x^4}}\right ) \]

[Out]

1/96*(32*x^2+52*x+101)*(x^4+x^3)^(1/4)-155/64*arctan(x/(x^4+x^3)^(1/4))+2*2^(1/4)*arctan(2^(1/4)*x/(x^4+x^3)^(
1/4))+155/64*arctanh(x/(x^4+x^3)^(1/4))-2*2^(1/4)*arctanh(2^(1/4)*x/(x^4+x^3)^(1/4))

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(317\) vs. \(2(114)=228\).

Time = 0.13 (sec) , antiderivative size = 317, normalized size of antiderivative = 2.78, number of steps used = 18, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.650, Rules used = {2067, 103, 161, 96, 95, 304, 209, 212, 963, 79, 52, 65, 338} \[ \int \frac {x^2 \sqrt [4]{x^3+x^4}}{-1+x} \, dx=-\frac {155 \sqrt [4]{x^4+x^3} \arctan \left (\frac {\sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{64 \sqrt [4]{x+1} x^{3/4}}+\frac {2 \sqrt [4]{2} \sqrt [4]{x^4+x^3} \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{\sqrt [4]{x+1} x^{3/4}}+\frac {155 \sqrt [4]{x^4+x^3} \text {arctanh}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{64 \sqrt [4]{x+1} x^{3/4}}-\frac {2 \sqrt [4]{2} \sqrt [4]{x^4+x^3} \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{\sqrt [4]{x+1} x^{3/4}}+\frac {16 \sqrt [4]{x^4+x^3} x}{7 (x+1)^2}+\frac {155}{168} \sqrt [4]{x^4+x^3} x+\frac {8 \sqrt [4]{x^4+x^3}}{3 (x+1)}-\frac {155}{96} \sqrt [4]{x^4+x^3}-\frac {8 \sqrt [4]{x^4+x^3} x^2}{21 (x+1)}+\frac {16 \sqrt [4]{x^4+x^3} x^2}{7 (x+1)^2}+\frac {1}{3} \sqrt [4]{x^4+x^3} x^2 \]

[In]

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

[Out]

(-155*(x^3 + x^4)^(1/4))/96 + (155*x*(x^3 + x^4)^(1/4))/168 + (x^2*(x^3 + x^4)^(1/4))/3 + (16*x*(x^3 + x^4)^(1
/4))/(7*(1 + x)^2) + (16*x^2*(x^3 + x^4)^(1/4))/(7*(1 + x)^2) + (8*(x^3 + x^4)^(1/4))/(3*(1 + x)) - (8*x^2*(x^
3 + x^4)^(1/4))/(21*(1 + x)) - (155*(x^3 + x^4)^(1/4)*ArcTan[x^(1/4)/(1 + x)^(1/4)])/(64*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)) + (155*(x^3
+ x^4)^(1/4)*ArcTanh[x^(1/4)/(1 + x)^(1/4)])/(64*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 52

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 65

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 79

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(-(b*e - a*f
))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1
) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e,
f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || L
tQ[p, n]))))

Rule 95

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 96

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

Rule 103

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(a + b
*x)^m*(c + d*x)^n*((e + f*x)^(p + 1)/(f*(m + n + p + 1))), x] - Dist[1/(f*(m + n + p + 1)), Int[(a + b*x)^(m -
1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[c*m*(b*e - a*f) + a*n*(d*e - c*f) + (d*m*(b*e - a*f) + b*n*(d*e - c*f))
*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && GtQ[m, 0] && GtQ[n, 0] && NeQ[m + n + p + 1, 0] && (Integ
ersQ[2*m, 2*n, 2*p] || (IntegersQ[m, n + p] || IntegersQ[p, m + n]))

Rule 161

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((g_.) + (h_.)*(x_)))/((e_.) + (f_.)*(x_)), x_Symbol]
:> Dist[(f*g - e*h)*((c*f - d*e)^(m + n + 1)/f^(m + n + 2)), Int[(a + b*x)^m/((c + d*x)^(m + 1)*(e + f*x)), x
], x] + Dist[1/f^(m + n + 2), Int[((a + b*x)^m/(c + d*x)^(m + 1))*ExpandToSum[(f^(m + n + 2)*(c + d*x)^(m + n
+ 1)*(g + h*x) - (f*g - e*h)*(c*f - d*e)^(m + n + 1))/(e + f*x), x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h},
x] && IGtQ[m + n + 1, 0] && (LtQ[m, 0] || SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

Rule 209

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

Rule 212

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

Rule 304

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 338

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 963

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> With[{Qx = PolynomialQuotient[(a + b*x + c*x^2)^p, d + e*x, x], R = PolynomialRemainder[(a + b*x + c*x^2)^p,
d + e*x, x]}, Simp[R*(d + e*x)^(m + 1)*((f + g*x)^(n + 1)/((m + 1)*(e*f - d*g))), x] + Dist[1/((m + 1)*(e*f -
d*g)), Int[(d + e*x)^(m + 1)*(f + g*x)^n*ExpandToSum[(m + 1)*(e*f - d*g)*Qx - g*R*(m + n + 2), x], x], x]] /;
FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]
&& IGtQ[p, 0] && LtQ[m, -1]

Rule 2067

Int[((e_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(jn_.))^(p_)*((c_) + (d_.)*(x_)^(n_.))^(q_.), x_Symbol]
:> Dist[e^IntPart[m]*(e*x)^FracPart[m]*((a*x^j + b*x^(j + n))^FracPart[p]/(x^(FracPart[m] + j*FracPart[p])*(a
+ b*x^n)^FracPart[p])), Int[x^(m + j*p)*(a + b*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, e, j, m, n,
p, q}, x] && EqQ[jn, j + n] &&  !IntegerQ[p] && NeQ[b*c - a*d, 0] &&  !(EqQ[n, 1] && EqQ[j, 1])

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt [4]{x^3+x^4} \int \frac {x^{11/4} \sqrt [4]{1+x}}{-1+x} \, dx}{x^{3/4} \sqrt [4]{1+x}} \\ & = \frac {1}{3} x^2 \sqrt [4]{x^3+x^4}-\frac {\sqrt [4]{x^3+x^4} \int \frac {\left (-\frac {11}{4}-\frac {13 x}{4}\right ) x^{7/4}}{(-1+x) (1+x)^{3/4}} \, dx}{3 x^{3/4} \sqrt [4]{1+x}} \\ & = \frac {1}{3} x^2 \sqrt [4]{x^3+x^4}-\frac {\sqrt [4]{x^3+x^4} \int \frac {x^{7/4} \left (-\frac {85}{4}-\frac {25 x}{2}-\frac {13 x^2}{4}\right )}{(1+x)^{11/4}} \, dx}{3 x^{3/4} \sqrt [4]{1+x}}+\frac {\left (8 \sqrt [4]{x^3+x^4}\right ) \int \frac {x^{7/4}}{(-1+x) (1+x)^{11/4}} \, dx}{x^{3/4} \sqrt [4]{1+x}} \\ & = \frac {1}{3} x^2 \sqrt [4]{x^3+x^4}+\frac {16 x \sqrt [4]{x^3+x^4}}{7 (1+x)^2}+\frac {16 x^2 \sqrt [4]{x^3+x^4}}{7 (1+x)^2}-\frac {\left (4 \sqrt [4]{x^3+x^4}\right ) \int \frac {\left (-\frac {67}{16}-\frac {91 x}{16}\right ) x^{7/4}}{(1+x)^{7/4}} \, dx}{21 x^{3/4} \sqrt [4]{1+x}}+\frac {\left (4 \sqrt [4]{x^3+x^4}\right ) \int \frac {x^{3/4}}{(-1+x) (1+x)^{7/4}} \, dx}{x^{3/4} \sqrt [4]{1+x}} \\ & = \frac {1}{3} x^2 \sqrt [4]{x^3+x^4}+\frac {16 x \sqrt [4]{x^3+x^4}}{7 (1+x)^2}+\frac {16 x^2 \sqrt [4]{x^3+x^4}}{7 (1+x)^2}+\frac {8 \sqrt [4]{x^3+x^4}}{3 (1+x)}-\frac {8 x^2 \sqrt [4]{x^3+x^4}}{21 (1+x)}+\frac {\left (155 \sqrt [4]{x^3+x^4}\right ) \int \frac {x^{7/4}}{(1+x)^{3/4}} \, dx}{84 x^{3/4} \sqrt [4]{1+x}}+\frac {\left (2 \sqrt [4]{x^3+x^4}\right ) \int \frac {1}{(-1+x) \sqrt [4]{x} (1+x)^{3/4}} \, dx}{x^{3/4} \sqrt [4]{1+x}} \\ & = \frac {155}{168} x \sqrt [4]{x^3+x^4}+\frac {1}{3} x^2 \sqrt [4]{x^3+x^4}+\frac {16 x \sqrt [4]{x^3+x^4}}{7 (1+x)^2}+\frac {16 x^2 \sqrt [4]{x^3+x^4}}{7 (1+x)^2}+\frac {8 \sqrt [4]{x^3+x^4}}{3 (1+x)}-\frac {8 x^2 \sqrt [4]{x^3+x^4}}{21 (1+x)}-\frac {\left (155 \sqrt [4]{x^3+x^4}\right ) \int \frac {x^{3/4}}{(1+x)^{3/4}} \, dx}{96 x^{3/4} \sqrt [4]{1+x}}+\frac {\left (8 \sqrt [4]{x^3+x^4}\right ) \text {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 {155}{96} \sqrt [4]{x^3+x^4}+\frac {155}{168} x \sqrt [4]{x^3+x^4}+\frac {1}{3} x^2 \sqrt [4]{x^3+x^4}+\frac {16 x \sqrt [4]{x^3+x^4}}{7 (1+x)^2}+\frac {16 x^2 \sqrt [4]{x^3+x^4}}{7 (1+x)^2}+\frac {8 \sqrt [4]{x^3+x^4}}{3 (1+x)}-\frac {8 x^2 \sqrt [4]{x^3+x^4}}{21 (1+x)}+\frac {\left (155 \sqrt [4]{x^3+x^4}\right ) \int \frac {1}{\sqrt [4]{x} (1+x)^{3/4}} \, dx}{128 x^{3/4} \sqrt [4]{1+x}}-\frac {\left (2 \sqrt {2} \sqrt [4]{x^3+x^4}\right ) \text {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 (2 \sqrt {2} \sqrt [4]{x^3+x^4}\right ) \text {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 {155}{96} \sqrt [4]{x^3+x^4}+\frac {155}{168} x \sqrt [4]{x^3+x^4}+\frac {1}{3} x^2 \sqrt [4]{x^3+x^4}+\frac {16 x \sqrt [4]{x^3+x^4}}{7 (1+x)^2}+\frac {16 x^2 \sqrt [4]{x^3+x^4}}{7 (1+x)^2}+\frac {8 \sqrt [4]{x^3+x^4}}{3 (1+x)}-\frac {8 x^2 \sqrt [4]{x^3+x^4}}{21 (1+x)}+\frac {2 \sqrt [4]{2} \sqrt [4]{x^3+x^4} \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}-\frac {2 \sqrt [4]{2} \sqrt [4]{x^3+x^4} \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {\left (155 \sqrt [4]{x^3+x^4}\right ) \text {Subst}\left (\int \frac {x^2}{\left (1+x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{32 x^{3/4} \sqrt [4]{1+x}} \\ & = -\frac {155}{96} \sqrt [4]{x^3+x^4}+\frac {155}{168} x \sqrt [4]{x^3+x^4}+\frac {1}{3} x^2 \sqrt [4]{x^3+x^4}+\frac {16 x \sqrt [4]{x^3+x^4}}{7 (1+x)^2}+\frac {16 x^2 \sqrt [4]{x^3+x^4}}{7 (1+x)^2}+\frac {8 \sqrt [4]{x^3+x^4}}{3 (1+x)}-\frac {8 x^2 \sqrt [4]{x^3+x^4}}{21 (1+x)}+\frac {2 \sqrt [4]{2} \sqrt [4]{x^3+x^4} \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}-\frac {2 \sqrt [4]{2} \sqrt [4]{x^3+x^4} \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {\left (155 \sqrt [4]{x^3+x^4}\right ) \text {Subst}\left (\int \frac {x^2}{1-x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{32 x^{3/4} \sqrt [4]{1+x}} \\ & = -\frac {155}{96} \sqrt [4]{x^3+x^4}+\frac {155}{168} x \sqrt [4]{x^3+x^4}+\frac {1}{3} x^2 \sqrt [4]{x^3+x^4}+\frac {16 x \sqrt [4]{x^3+x^4}}{7 (1+x)^2}+\frac {16 x^2 \sqrt [4]{x^3+x^4}}{7 (1+x)^2}+\frac {8 \sqrt [4]{x^3+x^4}}{3 (1+x)}-\frac {8 x^2 \sqrt [4]{x^3+x^4}}{21 (1+x)}+\frac {2 \sqrt [4]{2} \sqrt [4]{x^3+x^4} \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}-\frac {2 \sqrt [4]{2} \sqrt [4]{x^3+x^4} \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {\left (155 \sqrt [4]{x^3+x^4}\right ) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{64 x^{3/4} \sqrt [4]{1+x}}-\frac {\left (155 \sqrt [4]{x^3+x^4}\right ) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{64 x^{3/4} \sqrt [4]{1+x}} \\ & = -\frac {155}{96} \sqrt [4]{x^3+x^4}+\frac {155}{168} x \sqrt [4]{x^3+x^4}+\frac {1}{3} x^2 \sqrt [4]{x^3+x^4}+\frac {16 x \sqrt [4]{x^3+x^4}}{7 (1+x)^2}+\frac {16 x^2 \sqrt [4]{x^3+x^4}}{7 (1+x)^2}+\frac {8 \sqrt [4]{x^3+x^4}}{3 (1+x)}-\frac {8 x^2 \sqrt [4]{x^3+x^4}}{21 (1+x)}-\frac {155 \sqrt [4]{x^3+x^4} \arctan \left (\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{64 x^{3/4} \sqrt [4]{1+x}}+\frac {2 \sqrt [4]{2} \sqrt [4]{x^3+x^4} \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {155 \sqrt [4]{x^3+x^4} \text {arctanh}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{64 x^{3/4} \sqrt [4]{1+x}}-\frac {2 \sqrt [4]{2} \sqrt [4]{x^3+x^4} \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.48 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.30 \[ \int \frac {x^2 \sqrt [4]{x^3+x^4}}{-1+x} \, dx=\frac {x^{9/4} (1+x)^{3/4} \left (202 x^{3/4} \sqrt [4]{1+x}+104 x^{7/4} \sqrt [4]{1+x}+64 x^{11/4} \sqrt [4]{1+x}-465 \arctan \left (\sqrt [4]{\frac {x}{1+x}}\right )+384 \sqrt [4]{2} \arctan \left (\sqrt [4]{2} \sqrt [4]{\frac {x}{1+x}}\right )+465 \text {arctanh}\left (\sqrt [4]{\frac {x}{1+x}}\right )-384 \sqrt [4]{2} \text {arctanh}\left (\sqrt [4]{2} \sqrt [4]{\frac {x}{1+x}}\right )\right )}{192 \left (x^3 (1+x)\right )^{3/4}} \]

[In]

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

[Out]

(x^(9/4)*(1 + x)^(3/4)*(202*x^(3/4)*(1 + x)^(1/4) + 104*x^(7/4)*(1 + x)^(1/4) + 64*x^(11/4)*(1 + x)^(1/4) - 46
5*ArcTan[(x/(1 + x))^(1/4)] + 384*2^(1/4)*ArcTan[2^(1/4)*(x/(1 + x))^(1/4)] + 465*ArcTanh[(x/(1 + x))^(1/4)] -
384*2^(1/4)*ArcTanh[2^(1/4)*(x/(1 + x))^(1/4)]))/(192*(x^3*(1 + x))^(3/4))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(207\) vs. \(2(90)=180\).

Time = 4.78 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.82

method	result	size
pseudoelliptic	\(-\frac {x^{9} \left (-128 x^{2} \left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}+384 \ln \left (\frac {-2^{\frac {1}{4}} x -\left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}}{2^{\frac {1}{4}} x -\left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}}\right ) 2^{\frac {1}{4}}+768 \arctan \left (\frac {2^{\frac {3}{4}} \left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}}{2 x}\right ) 2^{\frac {1}{4}}-208 \left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}} x -465 \ln \left (\frac {x +\left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}}{x}\right )+465 \ln \left (\frac {\left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}-x}{x}\right )-930 \arctan \left (\frac {\left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}}{x}\right )-404 \left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}\right )}{384 {\left (x +\left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}\right )}^{3} {\left (\left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}-x \right )}^{3} \left (x^{2}+\sqrt {x^{3} \left (1+x \right )}\right )^{3}}\)	\(208\)
trager	\(\left (\frac {1}{3} x^{2}+\frac {13}{24} x +\frac {101}{96}\right ) \left (x^{4}+x^{3}\right )^{\frac {1}{4}}-\frac {155 \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 )}{128}-\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{3} x^{3}+4 \left (x^{4}+x^{3}\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2} x^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{3} x^{2}+4 \sqrt {x^{4}+x^{3}}\, \operatorname {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 )+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2} x^{3}-4 \left (x^{4}+x^{3}\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2} x^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right ) x^{2}-4 \sqrt {x^{4}+x^{3}}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {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 )-\frac {155 \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{3} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right ) \ln \left (\frac {-2 \sqrt {x^{4}+x^{3}}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{3} x +2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{3} x^{3}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right ) \operatorname {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 )}{256}\)	\(444\)
risch	\(\text {Expression too large to display}\)	\(876\)

[In]

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

[Out]

-1/384*x^9*(-128*x^2*(x^3*(1+x))^(1/4)+384*ln((-2^(1/4)*x-(x^3*(1+x))^(1/4))/(2^(1/4)*x-(x^3*(1+x))^(1/4)))*2^
(1/4)+768*arctan(1/2*2^(3/4)/x*(x^3*(1+x))^(1/4))*2^(1/4)-208*(x^3*(1+x))^(1/4)*x-465*ln((x+(x^3*(1+x))^(1/4))
/x)+465*ln(((x^3*(1+x))^(1/4)-x)/x)-930*arctan((x^3*(1+x))^(1/4)/x)-404*(x^3*(1+x))^(1/4))/(x+(x^3*(1+x))^(1/4
))^3/((x^3*(1+x))^(1/4)-x)^3/(x^2+(x^3*(1+x))^(1/2))^3

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.27 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.59 \[ \int \frac {x^2 \sqrt [4]{x^3+x^4}}{-1+x} \, dx=\frac {1}{96} \, {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}} {\left (32 \, x^{2} + 52 \, x + 101\right )} - 2^{\frac {1}{4}} \log \left (\frac {2^{\frac {1}{4}} x + {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + 2^{\frac {1}{4}} \log \left (-\frac {2^{\frac {1}{4}} x - {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - i \cdot 2^{\frac {1}{4}} \log \left (\frac {i \cdot 2^{\frac {1}{4}} x + {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + i \cdot 2^{\frac {1}{4}} \log \left (\frac {-i \cdot 2^{\frac {1}{4}} x + {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {155}{64} \, \arctan \left (\frac {{\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {155}{128} \, \log \left (\frac {x + {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \frac {155}{128} \, \log \left (-\frac {x - {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) \]

[In]

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

[Out]

1/96*(x^4 + x^3)^(1/4)*(32*x^2 + 52*x + 101) - 2^(1/4)*log((2^(1/4)*x + (x^4 + x^3)^(1/4))/x) + 2^(1/4)*log(-(
2^(1/4)*x - (x^4 + x^3)^(1/4))/x) - I*2^(1/4)*log((I*2^(1/4)*x + (x^4 + x^3)^(1/4))/x) + I*2^(1/4)*log((-I*2^(
1/4)*x + (x^4 + x^3)^(1/4))/x) + 155/64*arctan((x^4 + x^3)^(1/4)/x) + 155/128*log((x + (x^4 + x^3)^(1/4))/x) -
155/128*log(-(x - (x^4 + x^3)^(1/4))/x)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.08 \[ \int \frac {x^2 \sqrt [4]{x^3+x^4}}{-1+x} \, dx=\frac {1}{96} \, {\left (101 \, {\left (\frac {1}{x} + 1\right )}^{\frac {9}{4}} - 150 \, {\left (\frac {1}{x} + 1\right )}^{\frac {5}{4}} + 81 \, {\left (\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right )} x^{3} - 2 \cdot 2^{\frac {1}{4}} \arctan \left (\frac {1}{2} \cdot 2^{\frac {3}{4}} {\left (\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) - 2^{\frac {1}{4}} \log \left (2^{\frac {1}{4}} + {\left (\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) + 2^{\frac {1}{4}} \log \left ({\left | -2^{\frac {1}{4}} + {\left (\frac {1}{x} + 1\right )}^{\frac {1}{4}} \right |}\right ) + \frac {155}{64} \, \arctan \left ({\left (\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) + \frac {155}{128} \, \log \left ({\left (\frac {1}{x} + 1\right )}^{\frac {1}{4}} + 1\right ) - \frac {155}{128} \, \log \left ({\left | {\left (\frac {1}{x} + 1\right )}^{\frac {1}{4}} - 1 \right |}\right ) \]

[In]

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

[Out]

1/96*(101*(1/x + 1)^(9/4) - 150*(1/x + 1)^(5/4) + 81*(1/x + 1)^(1/4))*x^3 - 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))) + 155/64*a
rctan((1/x + 1)^(1/4)) + 155/128*log((1/x + 1)^(1/4) + 1) - 155/128*log(abs((1/x + 1)^(1/4) - 1))

Mupad [F(-1)]

Timed out. \[ \int \frac {x^2 \sqrt [4]{x^3+x^4}}{-1+x} \, dx=\int \frac {x^2\,{\left (x^4+x^3\right )}^{1/4}}{x-1} \,d x \]

[In]

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

[Out]

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