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

Optimal. Leaf size=114 \[ -\frac {155}{64} \tan ^{-1}\left (\frac {x}{\sqrt [4]{x^4+x^3}}\right )+2 \sqrt [4]{2} \tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^4+x^3}}\right )+\frac {155}{64} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{x^4+x^3}}\right )-2 \sqrt [4]{2} \tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^4+x^3}}\right )+\frac {1}{96} \sqrt [4]{x^4+x^3} \left (32 x^2+52 x+101\right ) \]

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Rubi [B]  time = 0.21, antiderivative size = 230, normalized size of antiderivative = 2.02, number of steps used = 27, number of rules used = 11, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.550, Rules used = {2042, 101, 157, 50, 63, 331, 298, 203, 206, 105, 93} \begin {gather*} \frac {13}{24} \sqrt [4]{x^4+x^3} x+\frac {101}{96} \sqrt [4]{x^4+x^3}-\frac {155 \sqrt [4]{x^4+x^3} \tan ^{-1}\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} \tan ^{-1}\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} \tanh ^{-1}\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} \tanh ^{-1}\left (\frac {\sqrt [4]{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))/(-1 + x),x]

[Out]

(101*(x^3 + x^4)^(1/4))/96 + (13*x*(x^3 + x^4)^(1/4))/24 + (x^2*(x^3 + x^4)^(1/4))/3 - (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 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 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 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}}{-1+x} \, dx &=\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 {\left (13 \sqrt [4]{x^3+x^4}\right ) \int \frac {x^{7/4}}{(1+x)^{3/4}} \, dx}{12 x^{3/4} \sqrt [4]{1+x}}+\frac {\left (2 \sqrt [4]{x^3+x^4}\right ) \int \frac {x^{7/4}}{(-1+x) (1+x)^{3/4}} \, dx}{x^{3/4} \sqrt [4]{1+x}}\\ &=\frac {13}{24} x \sqrt [4]{x^3+x^4}+\frac {1}{3} x^2 \sqrt [4]{x^3+x^4}-\frac {\left (91 \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 (2 \sqrt [4]{x^3+x^4}\right ) \int \frac {x^{3/4}}{(1+x)^{3/4}} \, dx}{x^{3/4} \sqrt [4]{1+x}}+\frac {\left (2 \sqrt [4]{x^3+x^4}\right ) \int \frac {x^{3/4}}{(-1+x) (1+x)^{3/4}} \, dx}{x^{3/4} \sqrt [4]{1+x}}\\ &=\frac {101}{96} \sqrt [4]{x^3+x^4}+\frac {13}{24} x \sqrt [4]{x^3+x^4}+\frac {1}{3} x^2 \sqrt [4]{x^3+x^4}+\frac {\left (91 \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 (3 \sqrt [4]{x^3+x^4}\right ) \int \frac {1}{\sqrt [4]{x} (1+x)^{3/4}} \, dx}{2 x^{3/4} \sqrt [4]{1+x}}+\frac {\left (2 \sqrt [4]{x^3+x^4}\right ) \int \frac {1}{\sqrt [4]{x} (1+x)^{3/4}} \, dx}{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 {101}{96} \sqrt [4]{x^3+x^4}+\frac {13}{24} x \sqrt [4]{x^3+x^4}+\frac {1}{3} x^2 \sqrt [4]{x^3+x^4}+\frac {\left (91 \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 )}{32 x^{3/4} \sqrt [4]{1+x}}-\frac {\left (6 \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 (8 \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 (8 \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 {101}{96} \sqrt [4]{x^3+x^4}+\frac {13}{24} x \sqrt [4]{x^3+x^4}+\frac {1}{3} x^2 \sqrt [4]{x^3+x^4}+\frac {\left (91 \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 )}{32 x^{3/4} \sqrt [4]{1+x}}-\frac {\left (6 \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 (8 \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 (2 \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 (2 \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 {101}{96} \sqrt [4]{x^3+x^4}+\frac {13}{24} x \sqrt [4]{x^3+x^4}+\frac {1}{3} x^2 \sqrt [4]{x^3+x^4}+\frac {2 \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 {2 \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}}+\frac {\left (91 \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 )}{64 x^{3/4} \sqrt [4]{1+x}}-\frac {\left (91 \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 )}{64 x^{3/4} \sqrt [4]{1+x}}-\frac {\left (3 \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 (3 \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}{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}{1+x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}\\ &=\frac {101}{96} \sqrt [4]{x^3+x^4}+\frac {13}{24} x \sqrt [4]{x^3+x^4}+\frac {1}{3} x^2 \sqrt [4]{x^3+x^4}-\frac {155 \sqrt [4]{x^3+x^4} \tan ^{-1}\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} \tan ^{-1}\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} \tanh ^{-1}\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} \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}\\ \end {align*}

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Mathematica [C]  time = 0.07, size = 124, normalized size = 1.09 \begin {gather*} \frac {4 \sqrt [4]{x^3 (x+1)} \left (21 (x+1) x^2 \, _2F_1\left (-\frac {1}{4},\frac {11}{4};\frac {15}{4};-x\right )+33 (x+1) x \, _2F_1\left (-\frac {1}{4},\frac {7}{4};\frac {11}{4};-x\right )+77 (x+1) \, _2F_1\left (-\frac {1}{4},\frac {3}{4};\frac {7}{4};-x\right )+77 (x+1) \, _2F_1\left (\frac {3}{4},\frac {3}{4};\frac {7}{4};-x\right )-154 \sqrt [4]{x+1} \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};\frac {2 x}{x+1}\right )\right )}{231 (x+1)^{5/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(4*(x^3*(1 + x))^(1/4)*(77*(1 + x)*Hypergeometric2F1[-1/4, 3/4, 7/4, -x] + 33*x*(1 + x)*Hypergeometric2F1[-1/4
, 7/4, 11/4, -x] + 21*x^2*(1 + x)*Hypergeometric2F1[-1/4, 11/4, 15/4, -x] + 77*(1 + x)*Hypergeometric2F1[3/4,
3/4, 7/4, -x] - 154*(1 + x)^(1/4)*Hypergeometric2F1[3/4, 1, 7/4, (2*x)/(1 + x)]))/(231*(1 + x)^(5/4))

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IntegrateAlgebraic [A]  time = 0.57, size = 114, normalized size = 1.00 \begin {gather*} \frac {1}{96} \left (101+52 x+32 x^2\right ) \sqrt [4]{x^3+x^4}-\frac {155}{64} \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 )+\frac {155}{64} \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[(x^2*(x^3 + x^4)^(1/4))/(-1 + x),x]

[Out]

((101 + 52*x + 32*x^2)*(x^3 + x^4)^(1/4))/96 - (155*ArcTan[x/(x^3 + x^4)^(1/4)])/64 + 2*2^(1/4)*ArcTan[(2^(1/4
)*x)/(x^3 + x^4)^(1/4)] + (155*ArcTanh[x/(x^3 + x^4)^(1/4)])/64 - 2*2^(1/4)*ArcTanh[(2^(1/4)*x)/(x^3 + x^4)^(1
/4)]

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fricas [B]  time = 0.48, size = 183, normalized size = 1.61 \begin {gather*} \frac {1}{96} \, {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}} {\left (32 \, x^{2} + 52 \, x + 101\right )} + 4 \cdot 2^{\frac {1}{4}} \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}} \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 ) + \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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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) + 4*2^(1/4)*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)*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) + 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)

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giac [A]  time = 0.53, size = 123, normalized size = 1.08 \begin {gather*} \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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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maple [C]  time = 3.05, size = 385, normalized size = 3.38

method	result	size
trager	\(\left (\frac {1}{3} x^{2}+\frac {13}{24} x +\frac {101}{96}\right ) \left (x^{4}+x^{3}\right )^{\frac {1}{4}}+\frac {155 \RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {2 \sqrt {x^{4}+x^{3}}\, \RootOf \left (\textit {\_Z}^{2}+1\right ) x -2 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{3}+2 \left (x^{4}+x^{3}\right )^{\frac {3}{4}}-2 x^{2} \left (x^{4}+x^{3}\right )^{\frac {1}{4}}-\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{2}}{x^{2}}\right )}{128}-\RootOf \left (\textit {\_Z}^{4}-2\right ) \ln \left (\frac {3 \RootOf \left (\textit {\_Z}^{4}-2\right )^{3} x^{3}+4 \left (x^{4}+x^{3}\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} x^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{3} x^{2}+4 \sqrt {x^{4}+x^{3}}\, \RootOf \left (\textit {\_Z}^{4}-2\right ) x +4 \left (x^{4}+x^{3}\right )^{\frac {3}{4}}}{\left (-1+x \right ) x^{2}}\right )+\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) \ln \left (-\frac {-3 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} x^{3}-\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} x^{2}+4 \left (x^{4}+x^{3}\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} x^{2}+4 \sqrt {x^{4}+x^{3}}\, \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) x -4 \left (x^{4}+x^{3}\right )^{\frac {3}{4}}}{\left (-1+x \right ) x^{2}}\right )+\frac {155 \ln \left (\frac {2 \left (x^{4}+x^{3}\right )^{\frac {3}{4}}+2 \sqrt {x^{4}+x^{3}}\, x +2 x^{2} \left (x^{4}+x^{3}\right )^{\frac {1}{4}}+2 x^{3}+x^{2}}{x^{2}}\right )}{128}\)	\(385\)
risch	\(\frac {\left (32 x^{2}+52 x +101\right ) \left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}}{96}+\frac {\left (\frac {155 \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}+\frac {155 \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 \sqrt {x^{4}+3 x^{3}+3 x^{2}+x}\, \RootOf \left (\textit {\_Z}^{2}+1\right )-5 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{2}-2 \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}} x^{2}-4 \RootOf \left (\textit {\_Z}^{2}+1\right ) x +4 \left (x^{4}+3 x^{3}+3 x^{2}+x \right )^{\frac {1}{4}} x -\RootOf \left (\textit {\_Z}^{2}+1\right )+2 \left (x^{4}+3 x^{3}+3 x^{2}+x \right )^{\frac {1}{4}}}{\left (1+x \right )^{2}}\right )}{128}-\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 )+\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}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} x +2 \sqrt {x^{4}+3 x^{3}+3 x^{2}+x}\, \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}-2\right )^{2}-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}-3 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) x^{3}-4 \left (x^{4}+3 x^{3}+3 x^{2}+x \right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} x -7 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) x^{2}-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}}-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 )\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 )}\)	\(877\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(1/3*x^2+13/24*x+101/96)*(x^4+x^3)^(1/4)+155/128*RootOf(_Z^2+1)*ln((2*(x^4+x^3)^(1/2)*RootOf(_Z^2+1)*x-2*RootO
f(_Z^2+1)*x^3+2*(x^4+x^3)^(3/4)-2*x^2*(x^4+x^3)^(1/4)-RootOf(_Z^2+1)*x^2)/x^2)-RootOf(_Z^4-2)*ln((3*RootOf(_Z^
4-2)^3*x^3+4*(x^4+x^3)^(1/4)*RootOf(_Z^4-2)^2*x^2+RootOf(_Z^4-2)^3*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)+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)*RootOf(_Z^4-2)^2*x^2+4*(x^4+x^3)^(1/4)*RootOf(_Z^4-2)^2*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)+155/128*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)

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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 - 1}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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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-1} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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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 - 1}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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