\(\int \frac {\sqrt [4]{b x^3+a x^4}}{-2 b+a x+2 x^2} \, dx\) [2507]

Optimal result

Integrand size = 30, antiderivative size = 209 \[ \int \frac {\sqrt [4]{b x^3+a x^4}}{-2 b+a x+2 x^2} \, dx=-\sqrt [4]{a} \arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b x^3+a x^4}}\right )+\sqrt [4]{a} \text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b x^3+a x^4}}\right )+\frac {1}{2} \text {RootSum}\left [3 a^2-2 b-5 a \text {$\#$1}^4+2 \text {$\#$1}^8\&,\frac {-3 a^2 \log (x)+2 b \log (x)+3 a^2 \log \left (\sqrt [4]{b x^3+a x^4}-x \text {$\#$1}\right )-2 b \log \left (\sqrt [4]{b x^3+a x^4}-x \text {$\#$1}\right )+2 a \log (x) \text {$\#$1}^4-2 a \log \left (\sqrt [4]{b x^3+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{5 a \text {$\#$1}^3-4 \text {$\#$1}^7}\&\right ] \]

[Out]

Unintegrable

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(750\) vs. \(2(209)=418\).

Time = 2.26 (sec) , antiderivative size = 750, normalized size of antiderivative = 3.59, number of steps used = 17, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.367, Rules used = {2081, 919, 65, 338, 304, 209, 212, 6860, 95, 211, 214} \[ \int \frac {\sqrt [4]{b x^3+a x^4}}{-2 b+a x+2 x^2} \, dx=\frac {\left (-\frac {a \left (a^2+6 b\right )}{\sqrt {a^2+16 b}}+a^2-2 b\right ) \sqrt [4]{a x^4+b x^3} \arctan \left (\frac {\sqrt [4]{x} \sqrt [4]{-a \sqrt {a^2+16 b}+a^2-4 b}}{\sqrt [4]{a-\sqrt {a^2+16 b}} \sqrt [4]{a x+b}}\right )}{x^{3/4} \sqrt [4]{a-\sqrt {a^2+16 b}} \left (-a \sqrt {a^2+16 b}+a^2-4 b\right )^{3/4} \sqrt [4]{a x+b}}+\frac {\left (\frac {a \left (a^2+6 b\right )}{\sqrt {a^2+16 b}}+a^2-2 b\right ) \sqrt [4]{a x^4+b x^3} \arctan \left (\frac {\sqrt [4]{x} \sqrt [4]{a \sqrt {a^2+16 b}+a^2-4 b}}{\sqrt [4]{\sqrt {a^2+16 b}+a} \sqrt [4]{a x+b}}\right )}{x^{3/4} \sqrt [4]{\sqrt {a^2+16 b}+a} \left (a \sqrt {a^2+16 b}+a^2-4 b\right )^{3/4} \sqrt [4]{a x+b}}-\frac {\left (-\frac {a \left (a^2+6 b\right )}{\sqrt {a^2+16 b}}+a^2-2 b\right ) \sqrt [4]{a x^4+b x^3} \text {arctanh}\left (\frac {\sqrt [4]{x} \sqrt [4]{-a \sqrt {a^2+16 b}+a^2-4 b}}{\sqrt [4]{a-\sqrt {a^2+16 b}} \sqrt [4]{a x+b}}\right )}{x^{3/4} \sqrt [4]{a-\sqrt {a^2+16 b}} \left (-a \sqrt {a^2+16 b}+a^2-4 b\right )^{3/4} \sqrt [4]{a x+b}}-\frac {\left (\frac {a \left (a^2+6 b\right )}{\sqrt {a^2+16 b}}+a^2-2 b\right ) \sqrt [4]{a x^4+b x^3} \text {arctanh}\left (\frac {\sqrt [4]{x} \sqrt [4]{a \sqrt {a^2+16 b}+a^2-4 b}}{\sqrt [4]{\sqrt {a^2+16 b}+a} \sqrt [4]{a x+b}}\right )}{x^{3/4} \sqrt [4]{\sqrt {a^2+16 b}+a} \left (a \sqrt {a^2+16 b}+a^2-4 b\right )^{3/4} \sqrt [4]{a x+b}}-\frac {\sqrt [4]{a} \sqrt [4]{a x^4+b x^3} \arctan \left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x+b}}\right )}{x^{3/4} \sqrt [4]{a x+b}}+\frac {\sqrt [4]{a} \sqrt [4]{a x^4+b x^3} \text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x+b}}\right )}{x^{3/4} \sqrt [4]{a x+b}} \]

[In]

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

[Out]

-((a^(1/4)*(b*x^3 + a*x^4)^(1/4)*ArcTan[(a^(1/4)*x^(1/4))/(b + a*x)^(1/4)])/(x^(3/4)*(b + a*x)^(1/4))) + ((a^2
- 2*b - (a*(a^2 + 6*b))/Sqrt[a^2 + 16*b])*(b*x^3 + a*x^4)^(1/4)*ArcTan[((a^2 - 4*b - a*Sqrt[a^2 + 16*b])^(1/4
)*x^(1/4))/((a - Sqrt[a^2 + 16*b])^(1/4)*(b + a*x)^(1/4))])/((a - Sqrt[a^2 + 16*b])^(1/4)*(a^2 - 4*b - a*Sqrt[
a^2 + 16*b])^(3/4)*x^(3/4)*(b + a*x)^(1/4)) + ((a^2 - 2*b + (a*(a^2 + 6*b))/Sqrt[a^2 + 16*b])*(b*x^3 + a*x^4)^
(1/4)*ArcTan[((a^2 - 4*b + a*Sqrt[a^2 + 16*b])^(1/4)*x^(1/4))/((a + Sqrt[a^2 + 16*b])^(1/4)*(b + a*x)^(1/4))])
/((a + Sqrt[a^2 + 16*b])^(1/4)*(a^2 - 4*b + a*Sqrt[a^2 + 16*b])^(3/4)*x^(3/4)*(b + a*x)^(1/4)) + (a^(1/4)*(b*x
^3 + a*x^4)^(1/4)*ArcTanh[(a^(1/4)*x^(1/4))/(b + a*x)^(1/4)])/(x^(3/4)*(b + a*x)^(1/4)) - ((a^2 - 2*b - (a*(a^
2 + 6*b))/Sqrt[a^2 + 16*b])*(b*x^3 + a*x^4)^(1/4)*ArcTanh[((a^2 - 4*b - a*Sqrt[a^2 + 16*b])^(1/4)*x^(1/4))/((a
- Sqrt[a^2 + 16*b])^(1/4)*(b + a*x)^(1/4))])/((a - Sqrt[a^2 + 16*b])^(1/4)*(a^2 - 4*b - a*Sqrt[a^2 + 16*b])^(
3/4)*x^(3/4)*(b + a*x)^(1/4)) - ((a^2 - 2*b + (a*(a^2 + 6*b))/Sqrt[a^2 + 16*b])*(b*x^3 + a*x^4)^(1/4)*ArcTanh[
((a^2 - 4*b + a*Sqrt[a^2 + 16*b])^(1/4)*x^(1/4))/((a + Sqrt[a^2 + 16*b])^(1/4)*(b + a*x)^(1/4))])/((a + Sqrt[a
^2 + 16*b])^(1/4)*(a^2 - 4*b + a*Sqrt[a^2 + 16*b])^(3/4)*x^(3/4)*(b + a*x)^(1/4))

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 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 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 211

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

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 214

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

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 919

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 2081

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 6860

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*} \text {integral}& = \frac {\sqrt [4]{b x^3+a x^4} \int \frac {x^{3/4} \sqrt [4]{b+a x}}{-2 b+a x+2 x^2} \, dx}{x^{3/4} \sqrt [4]{b+a x}} \\ & = \frac {\sqrt [4]{b x^3+a x^4} \int \frac {2 a b-\left (a^2-2 b\right ) x}{\sqrt [4]{x} (b+a x)^{3/4} \left (-2 b+a x+2 x^2\right )} \, dx}{2 x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (a \sqrt [4]{b x^3+a x^4}\right ) \int \frac {1}{\sqrt [4]{x} (b+a x)^{3/4}} \, dx}{2 x^{3/4} \sqrt [4]{b+a x}} \\ & = \frac {\sqrt [4]{b x^3+a x^4} \int \left (\frac {-a^2+2 b+\frac {a \left (a^2+6 b\right )}{\sqrt {a^2+16 b}}}{\sqrt [4]{x} \left (a-\sqrt {a^2+16 b}+4 x\right ) (b+a x)^{3/4}}+\frac {-a^2+2 b-\frac {a \left (a^2+6 b\right )}{\sqrt {a^2+16 b}}}{\sqrt [4]{x} \left (a+\sqrt {a^2+16 b}+4 x\right ) (b+a x)^{3/4}}\right ) \, dx}{2 x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (2 a \sqrt [4]{b x^3+a x^4}\right ) \text {Subst}\left (\int \frac {x^2}{\left (b+a x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{x^{3/4} \sqrt [4]{b+a x}} \\ & = \frac {\left (2 a \sqrt [4]{b x^3+a x^4}\right ) \text {Subst}\left (\int \frac {x^2}{1-a x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (\left (-a^2+2 b-\frac {a \left (a^2+6 b\right )}{\sqrt {a^2+16 b}}\right ) \sqrt [4]{b x^3+a x^4}\right ) \int \frac {1}{\sqrt [4]{x} \left (a+\sqrt {a^2+16 b}+4 x\right ) (b+a x)^{3/4}} \, dx}{2 x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (\left (-a^2+2 b+\frac {a \left (a^2+6 b\right )}{\sqrt {a^2+16 b}}\right ) \sqrt [4]{b x^3+a x^4}\right ) \int \frac {1}{\sqrt [4]{x} \left (a-\sqrt {a^2+16 b}+4 x\right ) (b+a x)^{3/4}} \, dx}{2 x^{3/4} \sqrt [4]{b+a x}} \\ & = \frac {\left (\sqrt {a} \sqrt [4]{b x^3+a x^4}\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {a} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (\sqrt {a} \sqrt [4]{b x^3+a x^4}\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {a} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (2 \left (-a^2+2 b-\frac {a \left (a^2+6 b\right )}{\sqrt {a^2+16 b}}\right ) \sqrt [4]{b x^3+a x^4}\right ) \text {Subst}\left (\int \frac {x^2}{a+\sqrt {a^2+16 b}-\left (-4 b+a \left (a+\sqrt {a^2+16 b}\right )\right ) x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (2 \left (-a^2+2 b+\frac {a \left (a^2+6 b\right )}{\sqrt {a^2+16 b}}\right ) \sqrt [4]{b x^3+a x^4}\right ) \text {Subst}\left (\int \frac {x^2}{a-\sqrt {a^2+16 b}-\left (-4 b+a \left (a-\sqrt {a^2+16 b}\right )\right ) x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{x^{3/4} \sqrt [4]{b+a x}} \\ & = -\frac {\sqrt [4]{a} \sqrt [4]{b x^3+a x^4} \arctan \left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{x^{3/4} \sqrt [4]{b+a x}}+\frac {\sqrt [4]{a} \sqrt [4]{b x^3+a x^4} \text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (\left (-a^2+2 b+\frac {a \left (a^2+6 b\right )}{\sqrt {a^2+16 b}}\right ) \sqrt [4]{b x^3+a x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a-\sqrt {a^2+16 b}}-\sqrt {a^2-4 b-a \sqrt {a^2+16 b}} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{\sqrt {a^2-4 b-a \sqrt {a^2+16 b}} x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (\left (-a^2+2 b+\frac {a \left (a^2+6 b\right )}{\sqrt {a^2+16 b}}\right ) \sqrt [4]{b x^3+a x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a-\sqrt {a^2+16 b}}+\sqrt {a^2-4 b-a \sqrt {a^2+16 b}} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{\sqrt {a^2-4 b-a \sqrt {a^2+16 b}} x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (\left (-a^2+2 b-\frac {a \left (a^2+6 b\right )}{\sqrt {a^2+16 b}}\right ) \sqrt [4]{b x^3+a x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+\sqrt {a^2+16 b}}-\sqrt {a^2-4 b+a \sqrt {a^2+16 b}} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{\sqrt {a^2-4 b+a \sqrt {a^2+16 b}} x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (\left (-a^2+2 b-\frac {a \left (a^2+6 b\right )}{\sqrt {a^2+16 b}}\right ) \sqrt [4]{b x^3+a x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+\sqrt {a^2+16 b}}+\sqrt {a^2-4 b+a \sqrt {a^2+16 b}} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{\sqrt {a^2-4 b+a \sqrt {a^2+16 b}} x^{3/4} \sqrt [4]{b+a x}} \\ & = -\frac {\sqrt [4]{a} \sqrt [4]{b x^3+a x^4} \arctan \left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (a^2-2 b-\frac {a \left (a^2+6 b\right )}{\sqrt {a^2+16 b}}\right ) \sqrt [4]{b x^3+a x^4} \arctan \left (\frac {\sqrt [4]{a^2-4 b-a \sqrt {a^2+16 b}} \sqrt [4]{x}}{\sqrt [4]{a-\sqrt {a^2+16 b}} \sqrt [4]{b+a x}}\right )}{\sqrt [4]{a-\sqrt {a^2+16 b}} \left (a^2-4 b-a \sqrt {a^2+16 b}\right )^{3/4} x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (a^2-2 b+\frac {a \left (a^2+6 b\right )}{\sqrt {a^2+16 b}}\right ) \sqrt [4]{b x^3+a x^4} \arctan \left (\frac {\sqrt [4]{a^2-4 b+a \sqrt {a^2+16 b}} \sqrt [4]{x}}{\sqrt [4]{a+\sqrt {a^2+16 b}} \sqrt [4]{b+a x}}\right )}{\sqrt [4]{a+\sqrt {a^2+16 b}} \left (a^2-4 b+a \sqrt {a^2+16 b}\right )^{3/4} x^{3/4} \sqrt [4]{b+a x}}+\frac {\sqrt [4]{a} \sqrt [4]{b x^3+a x^4} \text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (a^2-2 b-\frac {a \left (a^2+6 b\right )}{\sqrt {a^2+16 b}}\right ) \sqrt [4]{b x^3+a x^4} \text {arctanh}\left (\frac {\sqrt [4]{a^2-4 b-a \sqrt {a^2+16 b}} \sqrt [4]{x}}{\sqrt [4]{a-\sqrt {a^2+16 b}} \sqrt [4]{b+a x}}\right )}{\sqrt [4]{a-\sqrt {a^2+16 b}} \left (a^2-4 b-a \sqrt {a^2+16 b}\right )^{3/4} x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (a^2-2 b+\frac {a \left (a^2+6 b\right )}{\sqrt {a^2+16 b}}\right ) \sqrt [4]{b x^3+a x^4} \text {arctanh}\left (\frac {\sqrt [4]{a^2-4 b+a \sqrt {a^2+16 b}} \sqrt [4]{x}}{\sqrt [4]{a+\sqrt {a^2+16 b}} \sqrt [4]{b+a x}}\right )}{\sqrt [4]{a+\sqrt {a^2+16 b}} \left (a^2-4 b+a \sqrt {a^2+16 b}\right )^{3/4} x^{3/4} \sqrt [4]{b+a x}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.49 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.07 \[ \int \frac {\sqrt [4]{b x^3+a x^4}}{-2 b+a x+2 x^2} \, dx=-\frac {x^{9/4} (b+a x)^{3/4} \left (8 \sqrt [4]{a} \left (\arctan \left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )-\text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )\right )+\text {RootSum}\left [3 a^2-2 b-5 a \text {$\#$1}^4+2 \text {$\#$1}^8\&,\frac {3 a^2 \log (x)-2 b \log (x)-12 a^2 \log \left (\sqrt [4]{b+a x}-\sqrt [4]{x} \text {$\#$1}\right )+8 b \log \left (\sqrt [4]{b+a x}-\sqrt [4]{x} \text {$\#$1}\right )-2 a \log (x) \text {$\#$1}^4+8 a \log \left (\sqrt [4]{b+a x}-\sqrt [4]{x} \text {$\#$1}\right ) \text {$\#$1}^4}{5 a \text {$\#$1}^3-4 \text {$\#$1}^7}\&\right ]\right )}{8 \left (x^3 (b+a x)\right )^{3/4}} \]

[In]

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

[Out]

-1/8*(x^(9/4)*(b + a*x)^(3/4)*(8*a^(1/4)*(ArcTan[(a^(1/4)*x^(1/4))/(b + a*x)^(1/4)] - ArcTanh[(a^(1/4)*x^(1/4)
)/(b + a*x)^(1/4)]) + RootSum[3*a^2 - 2*b - 5*a*#1^4 + 2*#1^8 & , (3*a^2*Log[x] - 2*b*Log[x] - 12*a^2*Log[(b +
a*x)^(1/4) - x^(1/4)*#1] + 8*b*Log[(b + a*x)^(1/4) - x^(1/4)*#1] - 2*a*Log[x]*#1^4 + 8*a*Log[(b + a*x)^(1/4)
- x^(1/4)*#1]*#1^4)/(5*a*#1^3 - 4*#1^7) & ]))/(x^3*(b + a*x))^(3/4)

Maple [N/A] (verified)

Time = 0.49 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.70

method	result	size
pseudoelliptic	\(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (2 \textit {\_Z}^{8}-5 a \,\textit {\_Z}^{4}+3 a^{2}-2 b \right )}{\sum }\frac {\left (2 \textit {\_R}^{4} a -3 a^{2}+2 b \right ) \ln \left (\frac {-\textit {\_R} x +\left (x^{3} \left (a x +b \right )\right )^{\frac {1}{4}}}{x}\right )}{\textit {\_R}^{3} \left (4 \textit {\_R}^{4}-5 a \right )}\right )}{2}+\frac {a^{\frac {1}{4}} \ln \left (\frac {x \,a^{\frac {1}{4}}+\left (x^{3} \left (a x +b \right )\right )^{\frac {1}{4}}}{-x \,a^{\frac {1}{4}}+\left (x^{3} \left (a x +b \right )\right )^{\frac {1}{4}}}\right )}{2}+a^{\frac {1}{4}} \arctan \left (\frac {\left (x^{3} \left (a x +b \right )\right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x}\right )\)	\(146\)

[In]

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

[Out]

1/2*sum((2*_R^4*a-3*a^2+2*b)*ln((-_R*x+(x^3*(a*x+b))^(1/4))/x)/_R^3/(4*_R^4-5*a),_R=RootOf(2*_Z^8-5*_Z^4*a+3*a
^2-2*b))+1/2*a^(1/4)*ln((x*a^(1/4)+(x^3*(a*x+b))^(1/4))/(-x*a^(1/4)+(x^3*(a*x+b))^(1/4)))+a^(1/4)*arctan(1/a^(
1/4)/x*(x^3*(a*x+b))^(1/4))

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 3 vs. order 1.

Time = 0.49 (sec) , antiderivative size = 3186, normalized size of antiderivative = 15.24 \[ \int \frac {\sqrt [4]{b x^3+a x^4}}{-2 b+a x+2 x^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/2*sqrt(sqrt(1/2)*sqrt((a^5 + 14*a^3*b + 8*a*b^2 + (a^4 + 32*a^2*b + 256*b^2)*sqrt((a^8 + 12*a^6*b + 20*a^4*b
^2 - 96*a^2*b^3 + 64*b^4)/(a^6 + 48*a^4*b + 768*a^2*b^2 + 4096*b^3)))/(a^4 + 32*a^2*b + 256*b^2)))*log(-(((a^5
+ 32*a^3*b + 256*a*b^2)*x*sqrt((a^8 + 12*a^6*b + 20*a^4*b^2 - 96*a^2*b^3 + 64*b^4)/(a^6 + 48*a^4*b + 768*a^2*
b^2 + 4096*b^3)) - (a^6 + 22*a^4*b + 88*a^2*b^2 - 128*b^3)*x)*sqrt(sqrt(1/2)*sqrt((a^5 + 14*a^3*b + 8*a*b^2 +
(a^4 + 32*a^2*b + 256*b^2)*sqrt((a^8 + 12*a^6*b + 20*a^4*b^2 - 96*a^2*b^3 + 64*b^4)/(a^6 + 48*a^4*b + 768*a^2*
b^2 + 4096*b^3)))/(a^4 + 32*a^2*b + 256*b^2))) + 8*(a^4*b + 6*a^2*b^2 - 8*b^3)*(a*x^4 + b*x^3)^(1/4))/x) - 1/2
*sqrt(sqrt(1/2)*sqrt((a^5 + 14*a^3*b + 8*a*b^2 + (a^4 + 32*a^2*b + 256*b^2)*sqrt((a^8 + 12*a^6*b + 20*a^4*b^2
- 96*a^2*b^3 + 64*b^4)/(a^6 + 48*a^4*b + 768*a^2*b^2 + 4096*b^3)))/(a^4 + 32*a^2*b + 256*b^2)))*log((((a^5 + 3
2*a^3*b + 256*a*b^2)*x*sqrt((a^8 + 12*a^6*b + 20*a^4*b^2 - 96*a^2*b^3 + 64*b^4)/(a^6 + 48*a^4*b + 768*a^2*b^2
+ 4096*b^3)) - (a^6 + 22*a^4*b + 88*a^2*b^2 - 128*b^3)*x)*sqrt(sqrt(1/2)*sqrt((a^5 + 14*a^3*b + 8*a*b^2 + (a^4
+ 32*a^2*b + 256*b^2)*sqrt((a^8 + 12*a^6*b + 20*a^4*b^2 - 96*a^2*b^3 + 64*b^4)/(a^6 + 48*a^4*b + 768*a^2*b^2
+ 4096*b^3)))/(a^4 + 32*a^2*b + 256*b^2))) - 8*(a^4*b + 6*a^2*b^2 - 8*b^3)*(a*x^4 + b*x^3)^(1/4))/x) + 1/2*sqr
t(-sqrt(1/2)*sqrt((a^5 + 14*a^3*b + 8*a*b^2 + (a^4 + 32*a^2*b + 256*b^2)*sqrt((a^8 + 12*a^6*b + 20*a^4*b^2 - 9
6*a^2*b^3 + 64*b^4)/(a^6 + 48*a^4*b + 768*a^2*b^2 + 4096*b^3)))/(a^4 + 32*a^2*b + 256*b^2)))*log(-(((a^5 + 32*
a^3*b + 256*a*b^2)*x*sqrt((a^8 + 12*a^6*b + 20*a^4*b^2 - 96*a^2*b^3 + 64*b^4)/(a^6 + 48*a^4*b + 768*a^2*b^2 +
4096*b^3)) - (a^6 + 22*a^4*b + 88*a^2*b^2 - 128*b^3)*x)*sqrt(-sqrt(1/2)*sqrt((a^5 + 14*a^3*b + 8*a*b^2 + (a^4
+ 32*a^2*b + 256*b^2)*sqrt((a^8 + 12*a^6*b + 20*a^4*b^2 - 96*a^2*b^3 + 64*b^4)/(a^6 + 48*a^4*b + 768*a^2*b^2 +
4096*b^3)))/(a^4 + 32*a^2*b + 256*b^2))) + 8*(a^4*b + 6*a^2*b^2 - 8*b^3)*(a*x^4 + b*x^3)^(1/4))/x) - 1/2*sqrt
(-sqrt(1/2)*sqrt((a^5 + 14*a^3*b + 8*a*b^2 + (a^4 + 32*a^2*b + 256*b^2)*sqrt((a^8 + 12*a^6*b + 20*a^4*b^2 - 96
*a^2*b^3 + 64*b^4)/(a^6 + 48*a^4*b + 768*a^2*b^2 + 4096*b^3)))/(a^4 + 32*a^2*b + 256*b^2)))*log((((a^5 + 32*a^
3*b + 256*a*b^2)*x*sqrt((a^8 + 12*a^6*b + 20*a^4*b^2 - 96*a^2*b^3 + 64*b^4)/(a^6 + 48*a^4*b + 768*a^2*b^2 + 40
96*b^3)) - (a^6 + 22*a^4*b + 88*a^2*b^2 - 128*b^3)*x)*sqrt(-sqrt(1/2)*sqrt((a^5 + 14*a^3*b + 8*a*b^2 + (a^4 +
32*a^2*b + 256*b^2)*sqrt((a^8 + 12*a^6*b + 20*a^4*b^2 - 96*a^2*b^3 + 64*b^4)/(a^6 + 48*a^4*b + 768*a^2*b^2 + 4
096*b^3)))/(a^4 + 32*a^2*b + 256*b^2))) - 8*(a^4*b + 6*a^2*b^2 - 8*b^3)*(a*x^4 + b*x^3)^(1/4))/x) - 1/2*sqrt(s
qrt(1/2)*sqrt((a^5 + 14*a^3*b + 8*a*b^2 - (a^4 + 32*a^2*b + 256*b^2)*sqrt((a^8 + 12*a^6*b + 20*a^4*b^2 - 96*a^
2*b^3 + 64*b^4)/(a^6 + 48*a^4*b + 768*a^2*b^2 + 4096*b^3)))/(a^4 + 32*a^2*b + 256*b^2)))*log(-(((a^5 + 32*a^3*
b + 256*a*b^2)*x*sqrt((a^8 + 12*a^6*b + 20*a^4*b^2 - 96*a^2*b^3 + 64*b^4)/(a^6 + 48*a^4*b + 768*a^2*b^2 + 4096
*b^3)) + (a^6 + 22*a^4*b + 88*a^2*b^2 - 128*b^3)*x)*sqrt(sqrt(1/2)*sqrt((a^5 + 14*a^3*b + 8*a*b^2 - (a^4 + 32*
a^2*b + 256*b^2)*sqrt((a^8 + 12*a^6*b + 20*a^4*b^2 - 96*a^2*b^3 + 64*b^4)/(a^6 + 48*a^4*b + 768*a^2*b^2 + 4096
*b^3)))/(a^4 + 32*a^2*b + 256*b^2))) + 8*(a^4*b + 6*a^2*b^2 - 8*b^3)*(a*x^4 + b*x^3)^(1/4))/x) + 1/2*sqrt(sqrt
(1/2)*sqrt((a^5 + 14*a^3*b + 8*a*b^2 - (a^4 + 32*a^2*b + 256*b^2)*sqrt((a^8 + 12*a^6*b + 20*a^4*b^2 - 96*a^2*b
^3 + 64*b^4)/(a^6 + 48*a^4*b + 768*a^2*b^2 + 4096*b^3)))/(a^4 + 32*a^2*b + 256*b^2)))*log((((a^5 + 32*a^3*b +
256*a*b^2)*x*sqrt((a^8 + 12*a^6*b + 20*a^4*b^2 - 96*a^2*b^3 + 64*b^4)/(a^6 + 48*a^4*b + 768*a^2*b^2 + 4096*b^3
)) + (a^6 + 22*a^4*b + 88*a^2*b^2 - 128*b^3)*x)*sqrt(sqrt(1/2)*sqrt((a^5 + 14*a^3*b + 8*a*b^2 - (a^4 + 32*a^2*
b + 256*b^2)*sqrt((a^8 + 12*a^6*b + 20*a^4*b^2 - 96*a^2*b^3 + 64*b^4)/(a^6 + 48*a^4*b + 768*a^2*b^2 + 4096*b^3
)))/(a^4 + 32*a^2*b + 256*b^2))) - 8*(a^4*b + 6*a^2*b^2 - 8*b^3)*(a*x^4 + b*x^3)^(1/4))/x) - 1/2*sqrt(-sqrt(1/
2)*sqrt((a^5 + 14*a^3*b + 8*a*b^2 - (a^4 + 32*a^2*b + 256*b^2)*sqrt((a^8 + 12*a^6*b + 20*a^4*b^2 - 96*a^2*b^3
+ 64*b^4)/(a^6 + 48*a^4*b + 768*a^2*b^2 + 4096*b^3)))/(a^4 + 32*a^2*b + 256*b^2)))*log(-(((a^5 + 32*a^3*b + 25
6*a*b^2)*x*sqrt((a^8 + 12*a^6*b + 20*a^4*b^2 - 96*a^2*b^3 + 64*b^4)/(a^6 + 48*a^4*b + 768*a^2*b^2 + 4096*b^3))
+ (a^6 + 22*a^4*b + 88*a^2*b^2 - 128*b^3)*x)*sqrt(-sqrt(1/2)*sqrt((a^5 + 14*a^3*b + 8*a*b^2 - (a^4 + 32*a^2*b
+ 256*b^2)*sqrt((a^8 + 12*a^6*b + 20*a^4*b^2 - 96*a^2*b^3 + 64*b^4)/(a^6 + 48*a^4*b + 768*a^2*b^2 + 4096*b^3)
))/(a^4 + 32*a^2*b + 256*b^2))) + 8*(a^4*b + 6*a^2*b^2 - 8*b^3)*(a*x^4 + b*x^3)^(1/4))/x) + 1/2*sqrt(-sqrt(1/2
)*sqrt((a^5 + 14*a^3*b + 8*a*b^2 - (a^4 + 32*a^2*b + 256*b^2)*sqrt((a^8 + 12*a^6*b + 20*a^4*b^2 - 96*a^2*b^3 +
64*b^4)/(a^6 + 48*a^4*b + 768*a^2*b^2 + 4096*b^3)))/(a^4 + 32*a^2*b + 256*b^2)))*log((((a^5 + 32*a^3*b + 256*
a*b^2)*x*sqrt((a^8 + 12*a^6*b + 20*a^4*b^2 - 96*a^2*b^3 + 64*b^4)/(a^6 + 48*a^4*b + 768*a^2*b^2 + 4096*b^3)) +
(a^6 + 22*a^4*b + 88*a^2*b^2 - 128*b^3)*x)*sqrt(-sqrt(1/2)*sqrt((a^5 + 14*a^3*b + 8*a*b^2 - (a^4 + 32*a^2*b +
256*b^2)*sqrt((a^8 + 12*a^6*b + 20*a^4*b^2 - 96*a^2*b^3 + 64*b^4)/(a^6 + 48*a^4*b + 768*a^2*b^2 + 4096*b^3)))
/(a^4 + 32*a^2*b + 256*b^2))) - 8*(a^4*b + 6*a^2*b^2 - 8*b^3)*(a*x^4 + b*x^3)^(1/4))/x) + 1/2*a^(1/4)*log((a^(
1/4)*x + (a*x^4 + b*x^3)^(1/4))/x) - 1/2*a^(1/4)*log(-(a^(1/4)*x - (a*x^4 + b*x^3)^(1/4))/x) + 1/2*I*a^(1/4)*l
og((I*a^(1/4)*x + (a*x^4 + b*x^3)^(1/4))/x) - 1/2*I*a^(1/4)*log((-I*a^(1/4)*x + (a*x^4 + b*x^3)^(1/4))/x)

Sympy [N/A]

Not integrable

Time = 1.61 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.11 \[ \int \frac {\sqrt [4]{b x^3+a x^4}}{-2 b+a x+2 x^2} \, dx=\int \frac {\sqrt [4]{x^{3} \left (a x + b\right )}}{a x - 2 b + 2 x^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.21 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.14 \[ \int \frac {\sqrt [4]{b x^3+a x^4}}{-2 b+a x+2 x^2} \, dx=\int { \frac {{\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}}}{a x + 2 \, x^{2} - 2 \, b} \,d x } \]

[In]

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

[Out]

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

Giac [C] (verification not implemented)

Result contains higher order function than in optimal. Order 3 vs. order 1.

Time = 1.57 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.83 \[ \int \frac {\sqrt [4]{b x^3+a x^4}}{-2 b+a x+2 x^2} \, dx=\frac {1}{2} \, \sqrt {2} \left (-a\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} + 2 \, {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right ) + \frac {1}{2} \, \sqrt {2} \left (-a\right )^{\frac {1}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} - 2 \, {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right ) + \frac {1}{4} \, \sqrt {2} \left (-a\right )^{\frac {1}{4}} \log \left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}} + \sqrt {-a} + \sqrt {a + \frac {b}{x}}\right ) - \frac {1}{4} \, \sqrt {2} \left (-a\right )^{\frac {1}{4}} \log \left (-\sqrt {2} \left (-a\right )^{\frac {1}{4}} {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}} + \sqrt {-a} + \sqrt {a + \frac {b}{x}}\right ) \]

[In]

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

[Out]

1/2*sqrt(2)*(-a)^(1/4)*arctan(1/2*sqrt(2)*(sqrt(2)*(-a)^(1/4) + 2*(a + b/x)^(1/4))/(-a)^(1/4)) + 1/2*sqrt(2)*(
-a)^(1/4)*arctan(-1/2*sqrt(2)*(sqrt(2)*(-a)^(1/4) - 2*(a + b/x)^(1/4))/(-a)^(1/4)) + 1/4*sqrt(2)*(-a)^(1/4)*lo
g(sqrt(2)*(-a)^(1/4)*(a + b/x)^(1/4) + sqrt(-a) + sqrt(a + b/x)) - 1/4*sqrt(2)*(-a)^(1/4)*log(-sqrt(2)*(-a)^(1
/4)*(a + b/x)^(1/4) + sqrt(-a) + sqrt(a + b/x))

Mupad [N/A]

Not integrable

Time = 6.77 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.14 \[ \int \frac {\sqrt [4]{b x^3+a x^4}}{-2 b+a x+2 x^2} \, dx=\int \frac {{\left (a\,x^4+b\,x^3\right )}^{1/4}}{2\,x^2+a\,x-2\,b} \,d x \]

[In]

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

[Out]

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