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

Optimal. Leaf size=138 \[ \frac {1135}{64} \tan ^{-1}\left (\frac {x}{\sqrt [4]{x^4-x^3}}\right )-8\ 2^{3/4} \sqrt [4]{3} \tan ^{-1}\left (\frac {\sqrt [4]{\frac {3}{2}} x}{\sqrt [4]{x^4-x^3}}\right )-\frac {1135}{64} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{x^4-x^3}}\right )+8\ 2^{3/4} \sqrt [4]{3} \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {3}{2}} x}{\sqrt [4]{x^4-x^3}}\right )+\frac {1}{96} \sqrt [4]{x^4-x^3} \left (32 x^2-100 x+401\right ) \]

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Rubi [A]  time = 0.23, antiderivative size = 258, normalized size of antiderivative = 1.87, number of steps used = 27, number of rules used = 12, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.546, Rules used = {2042, 101, 157, 50, 63, 240, 212, 206, 203, 105, 93, 298} \begin {gather*} -\frac {25}{24} \sqrt [4]{x^4-x^3} x+\frac {401}{96} \sqrt [4]{x^4-x^3}-\frac {1135 \sqrt [4]{x^4-x^3} \tan ^{-1}\left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}\right )}{64 \sqrt [4]{x-1} x^{3/4}}-\frac {8\ 2^{3/4} \sqrt [4]{3} \sqrt [4]{x^4-x^3} \tan ^{-1}\left (\frac {\sqrt [4]{\frac {3}{2}} \sqrt [4]{x}}{\sqrt [4]{x-1}}\right )}{\sqrt [4]{x-1} x^{3/4}}-\frac {1135 \sqrt [4]{x^4-x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}\right )}{64 \sqrt [4]{x-1} x^{3/4}}+\frac {8\ 2^{3/4} \sqrt [4]{3} \sqrt [4]{x^4-x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {3}{2}} \sqrt [4]{x}}{\sqrt [4]{x-1}}\right )}{\sqrt [4]{x-1} x^{3/4}}+\frac {1}{3} \sqrt [4]{x^4-x^3} x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(401*(-x^3 + x^4)^(1/4))/96 - (25*x*(-x^3 + x^4)^(1/4))/24 + (x^2*(-x^3 + x^4)^(1/4))/3 - (1135*(-x^3 + x^4)^(
1/4)*ArcTan[(-1 + x)^(1/4)/x^(1/4)])/(64*(-1 + x)^(1/4)*x^(3/4)) - (8*2^(3/4)*3^(1/4)*(-x^3 + x^4)^(1/4)*ArcTa
n[((3/2)^(1/4)*x^(1/4))/(-1 + x)^(1/4)])/((-1 + x)^(1/4)*x^(3/4)) - (1135*(-x^3 + x^4)^(1/4)*ArcTanh[(-1 + x)^
(1/4)/x^(1/4)])/(64*(-1 + x)^(1/4)*x^(3/4)) + (8*2^(3/4)*3^(1/4)*(-x^3 + x^4)^(1/4)*ArcTanh[((3/2)^(1/4)*x^(1/
4))/(-1 + x)^(1/4)])/((-1 + x)^(1/4)*x^(3/4))

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 93

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

Rule 101

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 105

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

Rule 157

Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/((a_.) + (b_.)*(x_)), x_Symbol]
:> Dist[h/b, Int[(c + d*x)^n*(e + f*x)^p, x], x] + Dist[(b*g - a*h)/b, Int[((c + d*x)^n*(e + f*x)^p)/(a + b*x
), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x]

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 212

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

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 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 2042

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*} \int \frac {x^2 \sqrt [4]{-x^3+x^4}}{2+x} \, dx &=\frac {\sqrt [4]{-x^3+x^4} \int \frac {\sqrt [4]{-1+x} x^{11/4}}{2+x} \, dx}{\sqrt [4]{-1+x} x^{3/4}}\\ &=\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 {11}{2}+\frac {25 x}{4}\right )}{(-1+x)^{3/4} (2+x)} \, dx}{3 \sqrt [4]{-1+x} x^{3/4}}\\ &=\frac {1}{3} x^2 \sqrt [4]{-x^3+x^4}-\frac {\left (25 \sqrt [4]{-x^3+x^4}\right ) \int \frac {x^{7/4}}{(-1+x)^{3/4}} \, dx}{12 \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (6 \sqrt [4]{-x^3+x^4}\right ) \int \frac {x^{7/4}}{(-1+x)^{3/4} (2+x)} \, dx}{\sqrt [4]{-1+x} x^{3/4}}\\ &=-\frac {25}{24} x \sqrt [4]{-x^3+x^4}+\frac {1}{3} x^2 \sqrt [4]{-x^3+x^4}-\frac {\left (175 \sqrt [4]{-x^3+x^4}\right ) \int \frac {x^{3/4}}{(-1+x)^{3/4}} \, dx}{96 \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (6 \sqrt [4]{-x^3+x^4}\right ) \int \frac {x^{3/4}}{(-1+x)^{3/4}} \, dx}{\sqrt [4]{-1+x} x^{3/4}}-\frac {\left (12 \sqrt [4]{-x^3+x^4}\right ) \int \frac {x^{3/4}}{(-1+x)^{3/4} (2+x)} \, dx}{\sqrt [4]{-1+x} x^{3/4}}\\ &=\frac {401}{96} \sqrt [4]{-x^3+x^4}-\frac {25}{24} x \sqrt [4]{-x^3+x^4}+\frac {1}{3} x^2 \sqrt [4]{-x^3+x^4}-\frac {\left (175 \sqrt [4]{-x^3+x^4}\right ) \int \frac {1}{(-1+x)^{3/4} \sqrt [4]{x}} \, dx}{128 \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (9 \sqrt [4]{-x^3+x^4}\right ) \int \frac {1}{(-1+x)^{3/4} \sqrt [4]{x}} \, dx}{2 \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (12 \sqrt [4]{-x^3+x^4}\right ) \int \frac {1}{(-1+x)^{3/4} \sqrt [4]{x}} \, dx}{\sqrt [4]{-1+x} x^{3/4}}+\frac {\left (24 \sqrt [4]{-x^3+x^4}\right ) \int \frac {1}{(-1+x)^{3/4} \sqrt [4]{x} (2+x)} \, dx}{\sqrt [4]{-1+x} x^{3/4}}\\ &=\frac {401}{96} \sqrt [4]{-x^3+x^4}-\frac {25}{24} x \sqrt [4]{-x^3+x^4}+\frac {1}{3} x^2 \sqrt [4]{-x^3+x^4}-\frac {\left (175 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{1+x^4}} \, dx,x,\sqrt [4]{-1+x}\right )}{32 \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (18 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{1+x^4}} \, dx,x,\sqrt [4]{-1+x}\right )}{\sqrt [4]{-1+x} x^{3/4}}-\frac {\left (48 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{1+x^4}} \, dx,x,\sqrt [4]{-1+x}\right )}{\sqrt [4]{-1+x} x^{3/4}}+\frac {\left (96 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{2-3 x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}\\ &=\frac {401}{96} \sqrt [4]{-x^3+x^4}-\frac {25}{24} x \sqrt [4]{-x^3+x^4}+\frac {1}{3} x^2 \sqrt [4]{-x^3+x^4}-\frac {\left (175 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^4} \, dx,x,\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{32 \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (18 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^4} \, dx,x,\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}-\frac {\left (48 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^4} \, dx,x,\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}+\frac {\left (16 \sqrt {3} \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {2}-\sqrt {3} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}-\frac {\left (16 \sqrt {3} \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {2}+\sqrt {3} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}\\ &=\frac {401}{96} \sqrt [4]{-x^3+x^4}-\frac {25}{24} x \sqrt [4]{-x^3+x^4}+\frac {1}{3} x^2 \sqrt [4]{-x^3+x^4}-\frac {8\ 2^{3/4} \sqrt [4]{3} \sqrt [4]{-x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{\frac {3}{2}} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}+\frac {8\ 2^{3/4} \sqrt [4]{3} \sqrt [4]{-x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {3}{2}} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}-\frac {\left (175 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{64 \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (175 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{64 \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (9 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}+\frac {\left (9 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}-\frac {\left (24 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}-\frac {\left (24 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}\\ &=\frac {401}{96} \sqrt [4]{-x^3+x^4}-\frac {25}{24} x \sqrt [4]{-x^3+x^4}+\frac {1}{3} x^2 \sqrt [4]{-x^3+x^4}-\frac {1135 \sqrt [4]{-x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{64 \sqrt [4]{-1+x} x^{3/4}}-\frac {8\ 2^{3/4} \sqrt [4]{3} \sqrt [4]{-x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{\frac {3}{2}} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}-\frac {1135 \sqrt [4]{-x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{64 \sqrt [4]{-1+x} x^{3/4}}+\frac {8\ 2^{3/4} \sqrt [4]{3} \sqrt [4]{-x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {3}{2}} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}\\ \end {align*}

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Mathematica [C]  time = 0.04, size = 124, normalized size = 0.90 \begin {gather*} \frac {4 \sqrt [4]{(x-1) x^3} \left (\sqrt [4]{x} \, _2F_1\left (-\frac {11}{4},\frac {1}{4};\frac {5}{4};1-x\right )-3 \sqrt [4]{x} \, _2F_1\left (-\frac {7}{4},\frac {1}{4};\frac {5}{4};1-x\right )+6 \sqrt [4]{x} \, _2F_1\left (-\frac {3}{4},\frac {1}{4};\frac {5}{4};1-x\right )-12 \sqrt [4]{x} \, _2F_1\left (\frac {1}{4},\frac {1}{4};\frac {5}{4};1-x\right )+8 \, _2F_1\left (\frac {1}{4},1;\frac {5}{4};\frac {2 (x-1)}{3 x}\right )\right )}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(4*((-1 + x)*x^3)^(1/4)*(x^(1/4)*Hypergeometric2F1[-11/4, 1/4, 5/4, 1 - x] - 3*x^(1/4)*Hypergeometric2F1[-7/4,
1/4, 5/4, 1 - x] + 6*x^(1/4)*Hypergeometric2F1[-3/4, 1/4, 5/4, 1 - x] - 12*x^(1/4)*Hypergeometric2F1[1/4, 1/4
, 5/4, 1 - x] + 8*Hypergeometric2F1[1/4, 1, 5/4, (2*(-1 + x))/(3*x)]))/x

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IntegrateAlgebraic [A]  time = 0.62, size = 138, normalized size = 1.00 \begin {gather*} \frac {1}{96} \left (401-100 x+32 x^2\right ) \sqrt [4]{-x^3+x^4}+\frac {1135}{64} \tan ^{-1}\left (\frac {x}{\sqrt [4]{-x^3+x^4}}\right )-8\ 2^{3/4} \sqrt [4]{3} \tan ^{-1}\left (\frac {\sqrt [4]{\frac {3}{2}} x}{\sqrt [4]{-x^3+x^4}}\right )-\frac {1135}{64} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{-x^3+x^4}}\right )+8\ 2^{3/4} \sqrt [4]{3} \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {3}{2}} x}{\sqrt [4]{-x^3+x^4}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((401 - 100*x + 32*x^2)*(-x^3 + x^4)^(1/4))/96 + (1135*ArcTan[x/(-x^3 + x^4)^(1/4)])/64 - 8*2^(3/4)*3^(1/4)*Ar
cTan[((3/2)^(1/4)*x)/(-x^3 + x^4)^(1/4)] - (1135*ArcTanh[x/(-x^3 + x^4)^(1/4)])/64 + 8*2^(3/4)*3^(1/4)*ArcTanh
[((3/2)^(1/4)*x)/(-x^3 + x^4)^(1/4)]

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fricas [A]  time = 0.63, size = 207, normalized size = 1.50 \begin {gather*} \frac {1}{96} \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}} {\left (32 \, x^{2} - 100 \, x + 401\right )} - 16 \cdot 24^{\frac {1}{4}} \arctan \left (\frac {24^{\frac {3}{4}} \sqrt {2} x \sqrt {\frac {\sqrt {6} x^{2} + 2 \, \sqrt {x^{4} - x^{3}}}{x^{2}}} - 2 \cdot 24^{\frac {3}{4}} {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{24 \, x}\right ) + 4 \cdot 24^{\frac {1}{4}} \log \left (\frac {24^{\frac {1}{4}} x + 2 \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - 4 \cdot 24^{\frac {1}{4}} \log \left (-\frac {24^{\frac {1}{4}} x - 2 \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1135}{64} \, \arctan \left (\frac {{\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1135}{128} \, \log \left (\frac {x + {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1135}{128} \, \log \left (-\frac {x - {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/96*(x^4 - x^3)^(1/4)*(32*x^2 - 100*x + 401) - 16*24^(1/4)*arctan(1/24*(24^(3/4)*sqrt(2)*x*sqrt((sqrt(6)*x^2
+ 2*sqrt(x^4 - x^3))/x^2) - 2*24^(3/4)*(x^4 - x^3)^(1/4))/x) + 4*24^(1/4)*log((24^(1/4)*x + 2*(x^4 - x^3)^(1/4
))/x) - 4*24^(1/4)*log(-(24^(1/4)*x - 2*(x^4 - x^3)^(1/4))/x) - 1135/64*arctan((x^4 - x^3)^(1/4)/x) - 1135/128
*log((x + (x^4 - x^3)^(1/4))/x) + 1135/128*log(-(x - (x^4 - x^3)^(1/4))/x)

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giac [A]  time = 0.63, size = 149, normalized size = 1.08 \begin {gather*} -\frac {1}{96} \, {\left (401 \, {\left (\frac {1}{x} - 1\right )}^{2} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} - 702 \, {\left (-\frac {1}{x} + 1\right )}^{\frac {5}{4}} + 333 \, {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right )} x^{3} - 8 \cdot 24^{\frac {1}{4}} \arctan \left (\frac {2}{3} \, \left (\frac {3}{2}\right )^{\frac {3}{4}} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) - 4 \cdot 24^{\frac {1}{4}} \log \left (\left (\frac {3}{2}\right )^{\frac {1}{4}} + {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) + 4 \cdot 24^{\frac {1}{4}} \log \left ({\left | -\left (\frac {3}{2}\right )^{\frac {1}{4}} + {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} \right |}\right ) + \frac {1135}{64} \, \arctan \left ({\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) + \frac {1135}{128} \, \log \left ({\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} + 1\right ) - \frac {1135}{128} \, \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*(x^4-x^3)^(1/4)/(2+x),x, algorithm="giac")

[Out]

-1/96*(401*(1/x - 1)^2*(-1/x + 1)^(1/4) - 702*(-1/x + 1)^(5/4) + 333*(-1/x + 1)^(1/4))*x^3 - 8*24^(1/4)*arctan
(2/3*(3/2)^(3/4)*(-1/x + 1)^(1/4)) - 4*24^(1/4)*log((3/2)^(1/4) + (-1/x + 1)^(1/4)) + 4*24^(1/4)*log(abs(-(3/2
)^(1/4) + (-1/x + 1)^(1/4))) + 1135/64*arctan((-1/x + 1)^(1/4)) + 1135/128*log((-1/x + 1)^(1/4) + 1) - 1135/12
8*log(abs((-1/x + 1)^(1/4) - 1))

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maple [C]  time = 3.72, size = 474, normalized size = 3.43

method	result	size
trager	\(\left (\frac {1}{3} x^{2}-\frac {25}{24} x +\frac {401}{96}\right ) \left (x^{4}-x^{3}\right )^{\frac {1}{4}}-\frac {1135 \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}-4 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-24\right )^{2}\right ) \ln \left (-\frac {-5 \RootOf \left (\textit {\_Z}^{4}-24\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-24\right )^{2}\right ) x^{3}+12 \RootOf \left (\textit {\_Z}^{4}-24\right )^{2} \left (x^{4}-x^{3}\right )^{\frac {1}{4}} x^{2}+2 \RootOf \left (\textit {\_Z}^{4}-24\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-24\right )^{2}\right ) x^{2}+24 \sqrt {x^{4}-x^{3}}\, \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-24\right )^{2}\right ) x -48 \left (x^{4}-x^{3}\right )^{\frac {3}{4}}}{x^{2} \left (2+x \right )}\right )+4 \RootOf \left (\textit {\_Z}^{4}-24\right ) \ln \left (\frac {5 \RootOf \left (\textit {\_Z}^{4}-24\right )^{3} x^{3}-2 \RootOf \left (\textit {\_Z}^{4}-24\right )^{3} x^{2}+12 \RootOf \left (\textit {\_Z}^{4}-24\right )^{2} \left (x^{4}-x^{3}\right )^{\frac {1}{4}} x^{2}+24 \sqrt {x^{4}-x^{3}}\, \RootOf \left (\textit {\_Z}^{4}-24\right ) x +48 \left (x^{4}-x^{3}\right )^{\frac {3}{4}}}{x^{2} \left (2+x \right )}\right )-\frac {1135 \RootOf \left (\textit {\_Z}^{4}-24\right )^{3} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-24\right )^{2}\right ) \ln \left (\frac {2 \sqrt {x^{4}-x^{3}}\, \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-24\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}-24\right )^{3} x -2 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-24\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}-24\right )^{3} x^{3}+\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-24\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}-24\right )^{3} x^{2}+48 \left (x^{4}-x^{3}\right )^{\frac {3}{4}}-48 x^{2} \left (x^{4}-x^{3}\right )^{\frac {1}{4}}}{x^{2}}\right )}{3072}\)	\(474\)
risch	\(\frac {\left (32 x^{2}-100 x +401\right ) \left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}}{96}+\frac {\left (\frac {1135 \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 )}{128}+\frac {1135 \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 )}{128}+4 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-24\right )^{2}\right ) \ln \left (\frac {-\sqrt {x^{4}-3 x^{3}+3 x^{2}-x}\, \RootOf \left (\textit {\_Z}^{4}-24\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-24\right )^{2}\right ) x +\sqrt {x^{4}-3 x^{3}+3 x^{2}-x}\, \RootOf \left (\textit {\_Z}^{4}-24\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-24\right )^{2}\right )-2 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-24\right )^{2} x^{2}+4 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-24\right )^{2} x +5 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-24\right )^{2}\right ) x^{3}+12 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {3}{4}}-2 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-24\right )^{2}-12 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-24\right )^{2}\right ) x^{2}+9 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-24\right )^{2}\right ) x -2 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-24\right )^{2}\right )}{\left (-1+x \right )^{2} \left (2+x \right )}\right )-4 \RootOf \left (\textit {\_Z}^{4}-24\right ) \ln \left (\frac {-\sqrt {x^{4}-3 x^{3}+3 x^{2}-x}\, \RootOf \left (\textit {\_Z}^{4}-24\right )^{3} x +\sqrt {x^{4}-3 x^{3}+3 x^{2}-x}\, \RootOf \left (\textit {\_Z}^{4}-24\right )^{3}+2 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-24\right )^{2} x^{2}-4 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-24\right )^{2} x -5 \RootOf \left (\textit {\_Z}^{4}-24\right ) x^{3}+12 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {3}{4}}+2 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-24\right )^{2}+12 \RootOf \left (\textit {\_Z}^{4}-24\right ) x^{2}-9 \RootOf \left (\textit {\_Z}^{4}-24\right ) x +2 \RootOf \left (\textit {\_Z}^{4}-24\right )}{\left (-1+x \right )^{2} \left (2+x \right )}\right )\right ) \left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}} \left (x \left (-1+x \right )^{3}\right )^{\frac {1}{4}}}{x \left (-1+x \right )}\)	\(923\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(1/3*x^2-25/24*x+401/96)*(x^4-x^3)^(1/4)-1135/128*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)-4*RootOf(_Z^2+RootOf(_Z^4-24)^2)*ln(-(-5*RootOf(_Z^4-24)^2*RootOf(_Z^2+RootOf(_Z^4-24)^2)*
x^3+12*RootOf(_Z^4-24)^2*(x^4-x^3)^(1/4)*x^2+2*RootOf(_Z^4-24)^2*RootOf(_Z^2+RootOf(_Z^4-24)^2)*x^2+24*(x^4-x^
3)^(1/2)*RootOf(_Z^2+RootOf(_Z^4-24)^2)*x-48*(x^4-x^3)^(3/4))/x^2/(2+x))+4*RootOf(_Z^4-24)*ln((5*RootOf(_Z^4-2
4)^3*x^3-2*RootOf(_Z^4-24)^3*x^2+12*RootOf(_Z^4-24)^2*(x^4-x^3)^(1/4)*x^2+24*(x^4-x^3)^(1/2)*RootOf(_Z^4-24)*x
+48*(x^4-x^3)^(3/4))/x^2/(2+x))-1135/3072*RootOf(_Z^4-24)^3*RootOf(_Z^2+RootOf(_Z^4-24)^2)*ln((2*(x^4-x^3)^(1/
2)*RootOf(_Z^2+RootOf(_Z^4-24)^2)*RootOf(_Z^4-24)^3*x-2*RootOf(_Z^2+RootOf(_Z^4-24)^2)*RootOf(_Z^4-24)^3*x^3+R
ootOf(_Z^2+RootOf(_Z^4-24)^2)*RootOf(_Z^4-24)^3*x^2+48*(x^4-x^3)^(3/4)-48*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}} x^{2}}{x + 2}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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