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

Optimal. Leaf size=123 \[ -\frac {1}{2} \text {RootSum}\left [2 \text {$\#$1}^8-5 \text {$\#$1}^4+1\& ,\frac {-2 \text {$\#$1}^4 \log \left (\sqrt [4]{x^4+x^3}-\text {$\#$1} x\right )+2 \text {$\#$1}^4 \log (x)+\log \left (\sqrt [4]{x^4+x^3}-\text {$\#$1} x\right )-\log (x)}{4 \text {$\#$1}^7-5 \text {$\#$1}^3}\& \right ]-\tan ^{-1}\left (\frac {x}{\sqrt [4]{x^4+x^3}}\right )+\tanh ^{-1}\left (\frac {x}{\sqrt [4]{x^4+x^3}}\right ) \]

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Rubi [B]  time = 0.76, antiderivative size = 378, normalized size of antiderivative = 3.07, number of steps used = 17, number of rules used = 9, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {2056, 905, 63, 331, 298, 203, 206, 6728, 93} \begin {gather*} -\frac {\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 {\sqrt [4]{\frac {1}{2} \left (23-\sqrt {17}\right )} \sqrt [4]{x^4+x^3} \tan ^{-1}\left (\frac {\sqrt [4]{\frac {2}{5+\sqrt {17}}} \sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{\sqrt {17} x^{3/4} \sqrt [4]{x+1}}+\frac {\sqrt [4]{\frac {1}{2} \left (23+\sqrt {17}\right )} \sqrt [4]{x^4+x^3} \tan ^{-1}\left (\frac {\sqrt [4]{5+\sqrt {17}} \sqrt [4]{x}}{\sqrt {2} \sqrt [4]{x+1}}\right )}{\sqrt {17} x^{3/4} \sqrt [4]{x+1}}+\frac {\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 {\sqrt [4]{\frac {1}{2} \left (23-\sqrt {17}\right )} \sqrt [4]{x^4+x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {2}{5+\sqrt {17}}} \sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{\sqrt {17} x^{3/4} \sqrt [4]{x+1}}-\frac {\sqrt [4]{\frac {1}{2} \left (23+\sqrt {17}\right )} \sqrt [4]{x^4+x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{5+\sqrt {17}} \sqrt [4]{x}}{\sqrt {2} \sqrt [4]{x+1}}\right )}{\sqrt {17} x^{3/4} \sqrt [4]{x+1}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(((x^3 + x^4)^(1/4)*ArcTan[x^(1/4)/(1 + x)^(1/4)])/(x^(3/4)*(1 + x)^(1/4))) + (((23 - Sqrt[17])/2)^(1/4)*(x^3
+ x^4)^(1/4)*ArcTan[((2/(5 + Sqrt[17]))^(1/4)*x^(1/4))/(1 + x)^(1/4)])/(Sqrt[17]*x^(3/4)*(1 + x)^(1/4)) + (((
23 + Sqrt[17])/2)^(1/4)*(x^3 + x^4)^(1/4)*ArcTan[((5 + Sqrt[17])^(1/4)*x^(1/4))/(Sqrt[2]*(1 + x)^(1/4))])/(Sqr
t[17]*x^(3/4)*(1 + x)^(1/4)) + ((x^3 + x^4)^(1/4)*ArcTanh[x^(1/4)/(1 + x)^(1/4)])/(x^(3/4)*(1 + x)^(1/4)) - ((
(23 - Sqrt[17])/2)^(1/4)*(x^3 + x^4)^(1/4)*ArcTanh[((2/(5 + Sqrt[17]))^(1/4)*x^(1/4))/(1 + x)^(1/4)])/(Sqrt[17
]*x^(3/4)*(1 + x)^(1/4)) - (((23 + Sqrt[17])/2)^(1/4)*(x^3 + x^4)^(1/4)*ArcTanh[((5 + Sqrt[17])^(1/4)*x^(1/4))
/(Sqrt[2]*(1 + x)^(1/4))])/(Sqrt[17]*x^(3/4)*(1 + x)^(1/4))

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 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 905

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Di
st[(e*g)/c, Int[(d + e*x)^(m - 1)*(f + g*x)^(n - 1), x], x] + Dist[1/c, Int[(Simp[c*d*f - a*e*g + (c*e*f + c*d
*g - b*e*g)*x, x]*(d + e*x)^(m - 1)*(f + g*x)^(n - 1))/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e, f, g
}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[m, 0] &
& GtQ[n, 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 6728

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +
b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

Rubi steps

\begin {align*} \int \frac {\sqrt [4]{x^3+x^4}}{-2+x+2 x^2} \, dx &=\frac {\sqrt [4]{x^3+x^4} \int \frac {x^{3/4} \sqrt [4]{1+x}}{-2+x+2 x^2} \, dx}{x^{3/4} \sqrt [4]{1+x}}\\ &=\frac {\sqrt [4]{x^3+x^4} \int \frac {1}{\sqrt [4]{x} (1+x)^{3/4}} \, dx}{2 x^{3/4} \sqrt [4]{1+x}}+\frac {\sqrt [4]{x^3+x^4} \int \frac {2+x}{\sqrt [4]{x} (1+x)^{3/4} \left (-2+x+2 x^2\right )} \, dx}{2 x^{3/4} \sqrt [4]{1+x}}\\ &=\frac {\sqrt [4]{x^3+x^4} \int \left (\frac {1+\frac {7}{\sqrt {17}}}{\sqrt [4]{x} (1+x)^{3/4} \left (1-\sqrt {17}+4 x\right )}+\frac {1-\frac {7}{\sqrt {17}}}{\sqrt [4]{x} (1+x)^{3/4} \left (1+\sqrt {17}+4 x\right )}\right ) \, dx}{2 x^{3/4} \sqrt [4]{1+x}}+\frac {\left (2 \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 (2 \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 (\left (17-7 \sqrt {17}\right ) \sqrt [4]{x^3+x^4}\right ) \int \frac {1}{\sqrt [4]{x} (1+x)^{3/4} \left (1+\sqrt {17}+4 x\right )} \, dx}{34 x^{3/4} \sqrt [4]{1+x}}+\frac {\left (\left (17+7 \sqrt {17}\right ) \sqrt [4]{x^3+x^4}\right ) \int \frac {1}{\sqrt [4]{x} (1+x)^{3/4} \left (1-\sqrt {17}+4 x\right )} \, dx}{34 x^{3/4} \sqrt [4]{1+x}}\\ &=\frac {\sqrt [4]{x^3+x^4} \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 {\sqrt [4]{x^3+x^4} \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 \left (17-7 \sqrt {17}\right ) \sqrt [4]{x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{1+\sqrt {17}-\left (-3+\sqrt {17}\right ) x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{17 x^{3/4} \sqrt [4]{1+x}}+\frac {\left (2 \left (17+7 \sqrt {17}\right ) \sqrt [4]{x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{1-\sqrt {17}-\left (-3-\sqrt {17}\right ) x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{17 x^{3/4} \sqrt [4]{1+x}}\\ &=-\frac {\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 {\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 (\sqrt {2} \left (17-7 \sqrt {17}\right ) \sqrt [4]{x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {5+\sqrt {17}}-\sqrt {2} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{17 \left (3-\sqrt {17}\right ) x^{3/4} \sqrt [4]{1+x}}+\frac {\left (\sqrt {2} \left (17-7 \sqrt {17}\right ) \sqrt [4]{x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {5+\sqrt {17}}+\sqrt {2} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{17 \left (3-\sqrt {17}\right ) x^{3/4} \sqrt [4]{1+x}}-\frac {\left (\sqrt {2} \left (17+7 \sqrt {17}\right ) \sqrt [4]{x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {5-\sqrt {17}}-\sqrt {2} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{17 \left (3+\sqrt {17}\right ) x^{3/4} \sqrt [4]{1+x}}+\frac {\left (\sqrt {2} \left (17+7 \sqrt {17}\right ) \sqrt [4]{x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {5-\sqrt {17}}+\sqrt {2} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{17 \left (3+\sqrt {17}\right ) x^{3/4} \sqrt [4]{1+x}}\\ &=-\frac {\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 {\sqrt [4]{\frac {1}{2} \left (23-\sqrt {17}\right )} \sqrt [4]{x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{\frac {2}{5+\sqrt {17}}} \sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{\sqrt {17} x^{3/4} \sqrt [4]{1+x}}+\frac {\sqrt [4]{\frac {1}{2} \left (23+\sqrt {17}\right )} \sqrt [4]{x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{5+\sqrt {17}} \sqrt [4]{x}}{\sqrt {2} \sqrt [4]{1+x}}\right )}{\sqrt {17} x^{3/4} \sqrt [4]{1+x}}+\frac {\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 {\sqrt [4]{\frac {1}{2} \left (23-\sqrt {17}\right )} \sqrt [4]{x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {2}{5+\sqrt {17}}} \sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{\sqrt {17} x^{3/4} \sqrt [4]{1+x}}-\frac {\sqrt [4]{\frac {1}{2} \left (23+\sqrt {17}\right )} \sqrt [4]{x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{5+\sqrt {17}} \sqrt [4]{x}}{\sqrt {2} \sqrt [4]{1+x}}\right )}{\sqrt {17} x^{3/4} \sqrt [4]{1+x}}\\ \end {align*}

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Mathematica [A]  time = 0.17, size = 124, normalized size = 1.01 \begin {gather*} \frac {x^3 \left (34 (x+1)^{3/4} \, _2F_1\left (\frac {3}{4},\frac {3}{4};\frac {7}{4};-x\right )+\left (3 \sqrt {17}-17\right ) \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};\frac {\left (-3+\sqrt {17}\right ) x}{\left (1+\sqrt {17}\right ) (x+1)}\right )-\left (17+3 \sqrt {17}\right ) \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};\frac {\left (3+\sqrt {17}\right ) x}{\left (-1+\sqrt {17}\right ) (x+1)}\right )\right )}{51 \left (x^3 (x+1)\right )^{3/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x^3*(34*(1 + x)^(3/4)*Hypergeometric2F1[3/4, 3/4, 7/4, -x] + (-17 + 3*Sqrt[17])*Hypergeometric2F1[3/4, 1, 7/4
, ((-3 + Sqrt[17])*x)/((1 + Sqrt[17])*(1 + x))] - (17 + 3*Sqrt[17])*Hypergeometric2F1[3/4, 1, 7/4, ((3 + Sqrt[
17])*x)/((-1 + Sqrt[17])*(1 + x))]))/(51*(x^3*(1 + x))^(3/4))

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IntegrateAlgebraic [A]  time = 0.30, size = 123, normalized size = 1.00 \begin {gather*} -\tan ^{-1}\left (\frac {x}{\sqrt [4]{x^3+x^4}}\right )+\tanh ^{-1}\left (\frac {x}{\sqrt [4]{x^3+x^4}}\right )-\frac {1}{2} \text {RootSum}\left [1-5 \text {$\#$1}^4+2 \text {$\#$1}^8\&,\frac {-\log (x)+\log \left (\sqrt [4]{x^3+x^4}-x \text {$\#$1}\right )+2 \log (x) \text {$\#$1}^4-2 \log \left (\sqrt [4]{x^3+x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-5 \text {$\#$1}^3+4 \text {$\#$1}^7}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-ArcTan[x/(x^3 + x^4)^(1/4)] + ArcTanh[x/(x^3 + x^4)^(1/4)] - RootSum[1 - 5*#1^4 + 2*#1^8 & , (-Log[x] + Log[(
x^3 + x^4)^(1/4) - x*#1] + 2*Log[x]*#1^4 - 2*Log[(x^3 + x^4)^(1/4) - x*#1]*#1^4)/(-5*#1^3 + 4*#1^7) & ]/2

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fricas [B]  time = 0.54, size = 637, normalized size = 5.18 \begin {gather*} -\frac {1}{17} \, \sqrt {34} \sqrt {\sqrt {2} \sqrt {\sqrt {17} + 23}} \arctan \left (\frac {\sqrt {34} {\left (5 \, \sqrt {17} \sqrt {2} x - 51 \, \sqrt {2} x\right )} \sqrt {\sqrt {2} \sqrt {\sqrt {17} + 23}} \sqrt {\sqrt {17} + 23} \sqrt {\frac {{\left (\sqrt {17} \sqrt {2} x^{2} + 9 \, \sqrt {2} x^{2}\right )} \sqrt {\sqrt {17} + 23} + 64 \, \sqrt {x^{4} + x^{3}}}{x^{2}}} - 8 \, \sqrt {34} {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}} {\left (5 \, \sqrt {17} \sqrt {2} - 51 \, \sqrt {2}\right )} \sqrt {\sqrt {2} \sqrt {\sqrt {17} + 23}} \sqrt {\sqrt {17} + 23}}{34816 \, x}\right ) + \frac {1}{17} \, \sqrt {34} \sqrt {\sqrt {2} \sqrt {-\sqrt {17} + 23}} \arctan \left (\frac {\sqrt {34} {\left (5 \, \sqrt {17} \sqrt {2} x + 51 \, \sqrt {2} x\right )} \sqrt {\sqrt {2} \sqrt {-\sqrt {17} + 23}} \sqrt {-\sqrt {17} + 23} \sqrt {-\frac {{\left (\sqrt {17} \sqrt {2} x^{2} - 9 \, \sqrt {2} x^{2}\right )} \sqrt {-\sqrt {17} + 23} - 64 \, \sqrt {x^{4} + x^{3}}}{x^{2}}} - 8 \, \sqrt {34} {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}} {\left (5 \, \sqrt {17} \sqrt {2} + 51 \, \sqrt {2}\right )} \sqrt {\sqrt {2} \sqrt {-\sqrt {17} + 23}} \sqrt {-\sqrt {17} + 23}}{34816 \, x}\right ) - \frac {1}{68} \, \sqrt {34} \sqrt {\sqrt {2} \sqrt {\sqrt {17} + 23}} \log \left (\frac {\sqrt {34} {\left (\sqrt {17} x + 17 \, x\right )} \sqrt {\sqrt {2} \sqrt {\sqrt {17} + 23}} + 272 \, {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{68} \, \sqrt {34} \sqrt {\sqrt {2} \sqrt {\sqrt {17} + 23}} \log \left (-\frac {\sqrt {34} {\left (\sqrt {17} x + 17 \, x\right )} \sqrt {\sqrt {2} \sqrt {\sqrt {17} + 23}} - 272 \, {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{68} \, \sqrt {34} \sqrt {\sqrt {2} \sqrt {-\sqrt {17} + 23}} \log \left (\frac {\sqrt {34} {\left (\sqrt {17} x - 17 \, x\right )} \sqrt {\sqrt {2} \sqrt {-\sqrt {17} + 23}} + 272 \, {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{68} \, \sqrt {34} \sqrt {\sqrt {2} \sqrt {-\sqrt {17} + 23}} \log \left (-\frac {\sqrt {34} {\left (\sqrt {17} x - 17 \, x\right )} \sqrt {\sqrt {2} \sqrt {-\sqrt {17} + 23}} - 272 \, {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \arctan \left (\frac {{\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{2} \, \log \left (\frac {x + {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{2} \, \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^4+x^3)^(1/4)/(2*x^2+x-2),x, algorithm="fricas")

[Out]

-1/17*sqrt(34)*sqrt(sqrt(2)*sqrt(sqrt(17) + 23))*arctan(1/34816*(sqrt(34)*(5*sqrt(17)*sqrt(2)*x - 51*sqrt(2)*x
)*sqrt(sqrt(2)*sqrt(sqrt(17) + 23))*sqrt(sqrt(17) + 23)*sqrt(((sqrt(17)*sqrt(2)*x^2 + 9*sqrt(2)*x^2)*sqrt(sqrt
(17) + 23) + 64*sqrt(x^4 + x^3))/x^2) - 8*sqrt(34)*(x^4 + x^3)^(1/4)*(5*sqrt(17)*sqrt(2) - 51*sqrt(2))*sqrt(sq
rt(2)*sqrt(sqrt(17) + 23))*sqrt(sqrt(17) + 23))/x) + 1/17*sqrt(34)*sqrt(sqrt(2)*sqrt(-sqrt(17) + 23))*arctan(1
/34816*(sqrt(34)*(5*sqrt(17)*sqrt(2)*x + 51*sqrt(2)*x)*sqrt(sqrt(2)*sqrt(-sqrt(17) + 23))*sqrt(-sqrt(17) + 23)
*sqrt(-((sqrt(17)*sqrt(2)*x^2 - 9*sqrt(2)*x^2)*sqrt(-sqrt(17) + 23) - 64*sqrt(x^4 + x^3))/x^2) - 8*sqrt(34)*(x
^4 + x^3)^(1/4)*(5*sqrt(17)*sqrt(2) + 51*sqrt(2))*sqrt(sqrt(2)*sqrt(-sqrt(17) + 23))*sqrt(-sqrt(17) + 23))/x)
- 1/68*sqrt(34)*sqrt(sqrt(2)*sqrt(sqrt(17) + 23))*log((sqrt(34)*(sqrt(17)*x + 17*x)*sqrt(sqrt(2)*sqrt(sqrt(17)
+ 23)) + 272*(x^4 + x^3)^(1/4))/x) + 1/68*sqrt(34)*sqrt(sqrt(2)*sqrt(sqrt(17) + 23))*log(-(sqrt(34)*(sqrt(17)
*x + 17*x)*sqrt(sqrt(2)*sqrt(sqrt(17) + 23)) - 272*(x^4 + x^3)^(1/4))/x) + 1/68*sqrt(34)*sqrt(sqrt(2)*sqrt(-sq
rt(17) + 23))*log((sqrt(34)*(sqrt(17)*x - 17*x)*sqrt(sqrt(2)*sqrt(-sqrt(17) + 23)) + 272*(x^4 + x^3)^(1/4))/x)
- 1/68*sqrt(34)*sqrt(sqrt(2)*sqrt(-sqrt(17) + 23))*log(-(sqrt(34)*(sqrt(17)*x - 17*x)*sqrt(sqrt(2)*sqrt(-sqrt
(17) + 23)) - 272*(x^4 + x^3)^(1/4))/x) + arctan((x^4 + x^3)^(1/4)/x) + 1/2*log((x + (x^4 + x^3)^(1/4))/x) - 1
/2*log(-(x - (x^4 + x^3)^(1/4))/x)

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Unab
le to convert to real 1/4 Error: Bad Argument ValueUnable to convert to real 1/4 Error: Bad Argument Value

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maple [B]  time = 14.54, size = 3014, normalized size = 24.50

method	result	size
trager	\(\text {Expression too large to display}\)	\(3014\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*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)+1/2*RootOf(_Z^2+1)*ln((2*(
x^4+x^3)^(1/2)*RootOf(_Z^2+1)*x-2*RootOf(_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(RootOf(1336336*_Z^8-6647*_Z^4+8)^2+_Z^2)*ln((-195917548288*RootOf(RootOf(1336336*_Z^8-6647*_Z^4
+8)^2+_Z^2)*RootOf(1336336*_Z^8-6647*_Z^4+8)^10*x^3+195917548288*RootOf(RootOf(1336336*_Z^8-6647*_Z^4+8)^2+_Z^
2)*RootOf(1336336*_Z^8-6647*_Z^4+8)^10*x^2+357351968*x^3*RootOf(RootOf(1336336*_Z^8-6647*_Z^4+8)^2+_Z^2)*RootO
f(1336336*_Z^8-6647*_Z^4+8)^6-162561344*(x^4+x^3)^(1/4)*RootOf(1336336*_Z^8-6647*_Z^4+8)^6*x^2-1162376496*Root
Of(RootOf(1336336*_Z^8-6647*_Z^4+8)^2+_Z^2)*RootOf(1336336*_Z^8-6647*_Z^4+8)^6*x^2+30777344*(x^4+x^3)^(1/2)*Ro
otOf(RootOf(1336336*_Z^8-6647*_Z^4+8)^2+_Z^2)*RootOf(1336336*_Z^8-6647*_Z^4+8)^4*x+5826240*(x^4+x^3)^(3/4)*Roo
tOf(1336336*_Z^8-6647*_Z^4+8)^4+86445*x^3*RootOf(RootOf(1336336*_Z^8-6647*_Z^4+8)^2+_Z^2)*RootOf(1336336*_Z^8-
6647*_Z^4+8)^2+331772*(x^4+x^3)^(1/4)*RootOf(1336336*_Z^8-6647*_Z^4+8)^2*x^2+1556010*RootOf(RootOf(1336336*_Z^
8-6647*_Z^4+8)^2+_Z^2)*RootOf(1336336*_Z^8-6647*_Z^4+8)^2*x^2-62808*(x^4+x^3)^(1/2)*RootOf(RootOf(1336336*_Z^8
-6647*_Z^4+8)^2+_Z^2)*x-11888*(x^4+x^3)^(3/4))/(4624*x*RootOf(1336336*_Z^8-6647*_Z^4+8)^4-4624*RootOf(1336336*
_Z^8-6647*_Z^4+8)^4-9*x+10)/x^2)+RootOf(1336336*_Z^8-6647*_Z^4+8)*ln((-195917548288*RootOf(1336336*_Z^8-6647*_
Z^4+8)^11*x^3+195917548288*RootOf(1336336*_Z^8-6647*_Z^4+8)^11*x^2+357351968*RootOf(1336336*_Z^8-6647*_Z^4+8)^
7*x^3+162561344*(x^4+x^3)^(1/4)*RootOf(1336336*_Z^8-6647*_Z^4+8)^6*x^2-1162376496*RootOf(1336336*_Z^8-6647*_Z^
4+8)^7*x^2-30777344*(x^4+x^3)^(1/2)*RootOf(1336336*_Z^8-6647*_Z^4+8)^5*x+5826240*(x^4+x^3)^(3/4)*RootOf(133633
6*_Z^8-6647*_Z^4+8)^4+86445*RootOf(1336336*_Z^8-6647*_Z^4+8)^3*x^3-331772*(x^4+x^3)^(1/4)*RootOf(1336336*_Z^8-
6647*_Z^4+8)^2*x^2+1556010*RootOf(1336336*_Z^8-6647*_Z^4+8)^3*x^2+62808*(x^4+x^3)^(1/2)*RootOf(1336336*_Z^8-66
47*_Z^4+8)*x-11888*(x^4+x^3)^(3/4))/(4624*x*RootOf(1336336*_Z^8-6647*_Z^4+8)^4-4624*RootOf(1336336*_Z^8-6647*_
Z^4+8)^4-9*x+10)/x^2)+289*ln(-(9771288832*RootOf(RootOf(1336336*_Z^8-6647*_Z^4+8)^2+_Z^2)*RootOf(_Z^4+11161211
9056*RootOf(1336336*_Z^8-6647*_Z^4+8)^4-555164087)^3*RootOf(1336336*_Z^8-6647*_Z^4+8)^11*x^3-9771288832*RootOf
(RootOf(1336336*_Z^8-6647*_Z^4+8)^2+_Z^2)*RootOf(_Z^4+111612119056*RootOf(1336336*_Z^8-6647*_Z^4+8)^4-55516408
7)^3*RootOf(1336336*_Z^8-6647*_Z^4+8)^11*x^2-79380208*x^3*RootOf(RootOf(1336336*_Z^8-6647*_Z^4+8)^2+_Z^2)*Root
Of(_Z^4+111612119056*RootOf(1336336*_Z^8-6647*_Z^4+8)^4-555164087)^3*RootOf(1336336*_Z^8-6647*_Z^4+8)^7+392300
16*x^2*RootOf(RootOf(1336336*_Z^8-6647*_Z^4+8)^2+_Z^2)*RootOf(_Z^4+111612119056*RootOf(1336336*_Z^8-6647*_Z^4+
8)^4-555164087)^3*RootOf(1336336*_Z^8-6647*_Z^4+8)^7+512738759840*(x^4+x^3)^(1/2)*RootOf(1336336*_Z^8-6647*_Z^
4+8)^7*RootOf(RootOf(1336336*_Z^8-6647*_Z^4+8)^2+_Z^2)*RootOf(_Z^4+111612119056*RootOf(1336336*_Z^8-6647*_Z^4+
8)^4-555164087)*x+148776*x^3*RootOf(1336336*_Z^8-6647*_Z^4+8)^3*RootOf(_Z^4+111612119056*RootOf(1336336*_Z^8-6
647*_Z^4+8)^4-555164087)^3*RootOf(RootOf(1336336*_Z^8-6647*_Z^4+8)^2+_Z^2)-9562432*(x^4+x^3)^(1/4)*RootOf(1336
336*_Z^8-6647*_Z^4+8)^4*RootOf(_Z^4+111612119056*RootOf(1336336*_Z^8-6647*_Z^4+8)^4-555164087)^2*x^2-30995*x^2
*RootOf(1336336*_Z^8-6647*_Z^4+8)^3*RootOf(_Z^4+111612119056*RootOf(1336336*_Z^8-6647*_Z^4+8)^4-555164087)^3*R
ootOf(RootOf(1336336*_Z^8-6647*_Z^4+8)^2+_Z^2)-1503957734*(x^4+x^3)^(1/2)*RootOf(1336336*_Z^8-6647*_Z^4+8)^3*R
ootOf(RootOf(1336336*_Z^8-6647*_Z^4+8)^2+_Z^2)*RootOf(_Z^4+111612119056*RootOf(1336336*_Z^8-6647*_Z^4+8)^4-555
164087)*x+114497268480*(x^4+x^3)^(3/4)*RootOf(1336336*_Z^8-6647*_Z^4+8)^4+28048*(x^4+x^3)^(1/4)*RootOf(_Z^4+11
1612119056*RootOf(1336336*_Z^8-6647*_Z^4+8)^4-555164087)^2*x^2-335891984*(x^4+x^3)^(3/4))/(4624*x*RootOf(13363
36*_Z^8-6647*_Z^4+8)^4-4624*RootOf(1336336*_Z^8-6647*_Z^4+8)^4-14*x+13)/x^2)*RootOf(1336336*_Z^8-6647*_Z^4+8)^
7*RootOf(RootOf(1336336*_Z^8-6647*_Z^4+8)^2+_Z^2)*RootOf(_Z^4+111612119056*RootOf(1336336*_Z^8-6647*_Z^4+8)^4-
555164087)-23/16*ln(-(9771288832*RootOf(RootOf(1336336*_Z^8-6647*_Z^4+8)^2+_Z^2)*RootOf(_Z^4+111612119056*Root
Of(1336336*_Z^8-6647*_Z^4+8)^4-555164087)^3*RootOf(1336336*_Z^8-6647*_Z^4+8)^11*x^3-9771288832*RootOf(RootOf(1
336336*_Z^8-6647*_Z^4+8)^2+_Z^2)*RootOf(_Z^4+111612119056*RootOf(1336336*_Z^8-6647*_Z^4+8)^4-555164087)^3*Root
Of(1336336*_Z^8-6647*_Z^4+8)^11*x^2-79380208*x^3*RootOf(RootOf(1336336*_Z^8-6647*_Z^4+8)^2+_Z^2)*RootOf(_Z^4+1
11612119056*RootOf(1336336*_Z^8-6647*_Z^4+8)^4-555164087)^3*RootOf(1336336*_Z^8-6647*_Z^4+8)^7+39230016*x^2*Ro
otOf(RootOf(1336336*_Z^8-6647*_Z^4+8)^2+_Z^2)*RootOf(_Z^4+111612119056*RootOf(1336336*_Z^8-6647*_Z^4+8)^4-5551
64087)^3*RootOf(1336336*_Z^8-6647*_Z^4+8)^7+512738759840*(x^4+x^3)^(1/2)*RootOf(1336336*_Z^8-6647*_Z^4+8)^7*Ro
otOf(RootOf(1336336*_Z^8-6647*_Z^4+8)^2+_Z^2)*RootOf(_Z^4+111612119056*RootOf(1336336*_Z^8-6647*_Z^4+8)^4-5551
64087)*x+148776*x^3*RootOf(1336336*_Z^8-6647*_Z^4+8)^3*RootOf(_Z^4+111612119056*RootOf(1336336*_Z^8-6647*_Z^4+
8)^4-555164087)^3*RootOf(RootOf(1336336*_Z^8-6647*_Z^4+8)^2+_Z^2)-9562432*(x^4+x^3)^(1/4)*RootOf(1336336*_Z^8-
6647*_Z^4+8)^4*RootOf(_Z^4+111612119056*RootOf(1336336*_Z^8-6647*_Z^4+8)^4-555164087)^2*x^2-30995*x^2*RootOf(1
336336*_Z^8-6647*_Z^4+8)^3*RootOf(_Z^4+111612119056*RootOf(1336336*_Z^8-6647*_Z^4+8)^4-555164087)^3*RootOf(Roo
tOf(1336336*_Z^8-6647*_Z^4+8)^2+_Z^2)-1503957734*(x^4+x^3)^(1/2)*RootOf(1336336*_Z^8-6647*_Z^4+8)^3*RootOf(Roo
tOf(1336336*_Z^8-6647*_Z^4+8)^2+_Z^2)*RootOf(_Z^4+111612119056*RootOf(1336336*_Z^8-6647*_Z^4+8)^4-555164087)*x
[phone]*(x^4+x^3)^(3/4)*RootOf(1336336*_Z^8-6647*_Z^4+8)^4+28048*(x^4+x^3)^(1/4)*RootOf(_Z^4+11161211905
6*RootOf(1336336*_Z^8-6647*_Z^4+8)^4-555164087)^2*x^2-335891984*(x^4+x^3)^(3/4))/(4624*x*RootOf(1336336*_Z^8-6
647*_Z^4+8)^4-4624*RootOf(1336336*_Z^8-6647*_Z^4+8)^4-14*x+13)/x^2)*RootOf(1336336*_Z^8-6647*_Z^4+8)^3*RootOf(
RootOf(1336336*_Z^8-6647*_Z^4+8)^2+_Z^2)*RootOf(_Z^4+111612119056*RootOf(1336336*_Z^8-6647*_Z^4+8)^4-555164087
)-1/578*RootOf(_Z^4+111612119056*RootOf(1336336*_Z^8-6647*_Z^4+8)^4-555164087)*ln(-(-720285104*RootOf(_Z^4+111
612119056*RootOf(1336336*_Z^8-6647*_Z^4+8)^4-555164087)^3*RootOf(1336336*_Z^8-6647*_Z^4+8)^8*x^3+720285104*Roo
tOf(_Z^4+111612119056*RootOf(1336336*_Z^8-6647*_Z^4+8)^4-555164087)^3*RootOf(1336336*_Z^8-6647*_Z^4+8)^8*x^2+5
851672*x^3*RootOf(_Z^4+111612119056*RootOf(1336336*_Z^8-6647*_Z^4+8)^4-555164087)^3*RootOf(1336336*_Z^8-6647*_
Z^4+8)^4+345442856*(x^4+x^3)^(1/4)*RootOf(1336336*_Z^8-6647*_Z^4+8)^4*RootOf(_Z^4+111612119056*RootOf(1336336*
_Z^8-6647*_Z^4+8)^4-555164087)^2*x^2-2892023*x^2*RootOf(_Z^4+111612119056*RootOf(1336336*_Z^8-6647*_Z^4+8)^4-5
55164087)^3*RootOf(1336336*_Z^8-6647*_Z^4+8)^4+37802272768*(x^4+x^3)^(1/2)*RootOf(_Z^4+111612119056*RootOf(133
6336*_Z^8-6647*_Z^4+8)^4-555164087)*RootOf(1336336*_Z^8-6647*_Z^4+8)^4*x+4136213823840*(x^4+x^3)^(3/4)*RootOf(
1336336*_Z^8-6647*_Z^4+8)^4-10968*x^3*RootOf(_Z^4+111612119056*RootOf(1336336*_Z^8-6647*_Z^4+8)^4-555164087)^3
-1013234*(x^4+x^3)^(1/4)*RootOf(_Z^4+111612119056*RootOf(1336336*_Z^8-6647*_Z^4+8)^4-555164087)^2*x^2+2285*x^2
*RootOf(_Z^4+111612119056*RootOf(1336336*_Z^8-6647*_Z^4+8)^4-555164087)^3-110886410*(x^4+x^3)^(1/2)*RootOf(_Z^
4+111612119056*RootOf(1336336*_Z^8-6647*_Z^4+8)^4-555164087)*x-12134097922*(x^4+x^3)^(3/4))/(4624*x*RootOf(133
6336*_Z^8-6647*_Z^4+8)^4-4624*RootOf(1336336*_Z^8-6647*_Z^4+8)^4-14*x+13)/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}}}{2 \, x^{2} + x - 2}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((x^4 + x^3)^(1/4)/(2*x^2 + 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}}{2\,x^2+x-2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((x^3 + x^4)^(1/4)/(x + 2*x^2 - 2), 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 )}}{2 x^{2} + x - 2}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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