\(\int \frac {\sqrt [4]{-b x^3+a x^4}}{-d-2 c x+x^2} \, dx\) [2818]

Optimal result

Integrand size = 30, antiderivative size = 279 \[ \int \frac {\sqrt [4]{-b x^3+a x^4}}{-d-2 c x+x^2} \, dx=-2 \sqrt [4]{a} \arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b x^3+a x^4}}\right )+2 \sqrt [4]{a} \text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b x^3+a x^4}}\right )+\frac {1}{2} \text {RootSum}\left [b^2-2 a b c-a^2 d+2 b c \text {$\#$1}^4+2 a d \text {$\#$1}^4-d \text {$\#$1}^8\&,\frac {b^2 \log (x)-2 a b c \log (x)-a^2 d \log (x)-b^2 \log \left (\sqrt [4]{-b x^3+a x^4}-x \text {$\#$1}\right )+2 a b c \log \left (\sqrt [4]{-b x^3+a x^4}-x \text {$\#$1}\right )+a^2 d \log \left (\sqrt [4]{-b x^3+a x^4}-x \text {$\#$1}\right )+a d \log (x) \text {$\#$1}^4-a d \log \left (\sqrt [4]{-b x^3+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{b c \text {$\#$1}^3+a d \text {$\#$1}^3-d \text {$\#$1}^7}\&\right ] \]

[Out]

Unintegrable

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(753\) vs. \(2(279)=558\).

Time = 2.96 (sec) , antiderivative size = 753, normalized size of antiderivative = 2.70, 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} \[ \int \frac {\sqrt [4]{-b x^3+a x^4}}{-d-2 c x+x^2} \, dx=\frac {\sqrt [4]{a x^4-b x^3} \left (-\frac {b c-a \left (2 c^2+d\right )}{\sqrt {c^2+d}}-2 a c+b\right ) \arctan \left (\frac {\sqrt [4]{x} \sqrt [4]{b-a \left (c-\sqrt {c^2+d}\right )}}{\sqrt [4]{\sqrt {c^2+d}-c} \sqrt [4]{a x-b}}\right )}{x^{3/4} \sqrt [4]{\sqrt {c^2+d}-c} \sqrt [4]{a x-b} \left (b-a \left (c-\sqrt {c^2+d}\right )\right )^{3/4}}-\frac {\sqrt [4]{a x^4-b x^3} \left (\frac {b c-a \left (2 c^2+d\right )}{\sqrt {c^2+d}}-2 a c+b\right ) \arctan \left (\frac {\sqrt [4]{x} \sqrt [4]{a \left (\sqrt {c^2+d}+c\right )-b}}{\sqrt [4]{\sqrt {c^2+d}+c} \sqrt [4]{a x-b}}\right )}{x^{3/4} \sqrt [4]{\sqrt {c^2+d}+c} \sqrt [4]{a x-b} \left (a \left (\sqrt {c^2+d}+c\right )-b\right )^{3/4}}-\frac {2 \sqrt [4]{a} \sqrt [4]{a x^4-b x^3} \arctan \left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x-b}}\right )}{x^{3/4} \sqrt [4]{a x-b}}-\frac {\sqrt [4]{a x^4-b x^3} \left (-\frac {b c-a \left (2 c^2+d\right )}{\sqrt {c^2+d}}-2 a c+b\right ) \text {arctanh}\left (\frac {\sqrt [4]{x} \sqrt [4]{b-a \left (c-\sqrt {c^2+d}\right )}}{\sqrt [4]{\sqrt {c^2+d}-c} \sqrt [4]{a x-b}}\right )}{x^{3/4} \sqrt [4]{\sqrt {c^2+d}-c} \sqrt [4]{a x-b} \left (b-a \left (c-\sqrt {c^2+d}\right )\right )^{3/4}}+\frac {\sqrt [4]{a x^4-b x^3} \left (\frac {b c-a \left (2 c^2+d\right )}{\sqrt {c^2+d}}-2 a c+b\right ) \text {arctanh}\left (\frac {\sqrt [4]{x} \sqrt [4]{a \left (\sqrt {c^2+d}+c\right )-b}}{\sqrt [4]{\sqrt {c^2+d}+c} \sqrt [4]{a x-b}}\right )}{x^{3/4} \sqrt [4]{\sqrt {c^2+d}+c} \sqrt [4]{a x-b} \left (a \left (\sqrt {c^2+d}+c\right )-b\right )^{3/4}}+\frac {2 \sqrt [4]{a} \sqrt [4]{a x^4-b x^3} \text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x-b}}\right )}{x^{3/4} \sqrt [4]{a x-b}} \]

[In]

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

[Out]

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

Mathematica [A] (verified)

Time = 0.63 (sec) , antiderivative size = 302, normalized size of antiderivative = 1.08 \[ \int \frac {\sqrt [4]{-b x^3+a x^4}}{-d-2 c x+x^2} \, dx=-\frac {x^{9/4} (-b+a x)^{3/4} \left (16 \sqrt [4]{a} \left (\arctan \left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )-\text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )\right )+\text {RootSum}\left [b^2-2 a b c-a^2 d+2 b c \text {$\#$1}^4+2 a d \text {$\#$1}^4-d \text {$\#$1}^8\&,\frac {b^2 \log (x)-2 a b c \log (x)-a^2 d \log (x)-4 b^2 \log \left (\sqrt [4]{-b+a x}-\sqrt [4]{x} \text {$\#$1}\right )+8 a b c \log \left (\sqrt [4]{-b+a x}-\sqrt [4]{x} \text {$\#$1}\right )+4 a^2 d \log \left (\sqrt [4]{-b+a x}-\sqrt [4]{x} \text {$\#$1}\right )+a d \log (x) \text {$\#$1}^4-4 a d \log \left (\sqrt [4]{-b+a x}-\sqrt [4]{x} \text {$\#$1}\right ) \text {$\#$1}^4}{-b c \text {$\#$1}^3-a d \text {$\#$1}^3+d \text {$\#$1}^7}\&\right ]\right )}{8 \left (x^3 (-b+a x)\right )^{3/4}} \]

[In]

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

[Out]

-1/8*(x^(9/4)*(-b + a*x)^(3/4)*(16*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[b^2 - 2*a*b*c - a^2*d + 2*b*c*#1^4 + 2*a*d*#1^4 - d*#1^8 & , (b^2*Log[x] - 2
*a*b*c*Log[x] - a^2*d*Log[x] - 4*b^2*Log[(-b + a*x)^(1/4) - x^(1/4)*#1] + 8*a*b*c*Log[(-b + a*x)^(1/4) - x^(1/
4)*#1] + 4*a^2*d*Log[(-b + a*x)^(1/4) - x^(1/4)*#1] + a*d*Log[x]*#1^4 - 4*a*d*Log[(-b + a*x)^(1/4) - x^(1/4)*#
1]*#1^4)/(-(b*c*#1^3) - a*d*#1^3 + d*#1^7) & ]))/(x^3*(-b + a*x))^(3/4)

Maple [N/A] (verified)

Time = 0.46 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.64

method	result	size
pseudoelliptic	\(a^{\frac {1}{4}} \ln \left (\frac {a^{\frac {1}{4}} x +\left (x^{3} \left (a x -b \right )\right )^{\frac {1}{4}}}{-a^{\frac {1}{4}} x +\left (x^{3} \left (a x -b \right )\right )^{\frac {1}{4}}}\right )+2 a^{\frac {1}{4}} \arctan \left (\frac {\left (x^{3} \left (a x -b \right )\right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x}\right )+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (d \,\textit {\_Z}^{8}+\left (-2 a d -2 b c \right ) \textit {\_Z}^{4}+a^{2} d +2 a b c -b^{2}\right )}{\sum }\frac {\left (\textit {\_R}^{4} a d -a^{2} d -2 a b c +b^{2}\right ) \ln \left (\frac {-\textit {\_R} x +\left (x^{3} \left (a x -b \right )\right )^{\frac {1}{4}}}{x}\right )}{\textit {\_R}^{3} \left (\textit {\_R}^{4} d -a d -b c \right )}\right )}{2}\)	\(179\)

[In]

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

[Out]

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

Fricas [C] (verification not implemented)

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

Time = 2.24 (sec) , antiderivative size = 4415, normalized size of antiderivative = 15.82 \[ \int \frac {\sqrt [4]{-b x^3+a x^4}}{-d-2 c x+x^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/2*sqrt(-sqrt((8*a*c^4 - 4*b*c^3 + a*d^2 + (8*a*c^2 - 3*b*c)*d + (c^4 + 2*c^2*d + d^2)*sqrt((64*a^2*c^6 - 64*
a*b*c^5 + 16*b^2*c^4 + (16*a^2*c^2 - 8*a*b*c + b^2)*d^2 + 8*(8*a^2*c^4 - 6*a*b*c^3 + b^2*c^2)*d)/(c^6 + 3*c^4*
d + 3*c^2*d^2 + d^3)))/(c^4 + 2*c^2*d + d^2)))*log((((c^5 + 2*c^3*d + c*d^2)*x*sqrt((64*a^2*c^6 - 64*a*b*c^5 +
16*b^2*c^4 + (16*a^2*c^2 - 8*a*b*c + b^2)*d^2 + 8*(8*a^2*c^4 - 6*a*b*c^3 + b^2*c^2)*d)/(c^6 + 3*c^4*d + 3*c^2
*d^2 + d^3)) - (8*a*c^5 - 4*b*c^4 + (4*a*c - b)*d^2 + (12*a*c^3 - 5*b*c^2)*d)*x)*sqrt(-sqrt((8*a*c^4 - 4*b*c^3
+ a*d^2 + (8*a*c^2 - 3*b*c)*d + (c^4 + 2*c^2*d + d^2)*sqrt((64*a^2*c^6 - 64*a*b*c^5 + 16*b^2*c^4 + (16*a^2*c^
2 - 8*a*b*c + b^2)*d^2 + 8*(8*a^2*c^4 - 6*a*b*c^3 + b^2*c^2)*d)/(c^6 + 3*c^4*d + 3*c^2*d^2 + d^3)))/(c^4 + 2*c
^2*d + d^2))) + (a*x^4 - b*x^3)^(1/4)*((4*a*c - b)*d^2 + 4*(2*a*c^3 - b*c^2)*d))/x) - 1/2*sqrt(-sqrt((8*a*c^4
- 4*b*c^3 + a*d^2 + (8*a*c^2 - 3*b*c)*d + (c^4 + 2*c^2*d + d^2)*sqrt((64*a^2*c^6 - 64*a*b*c^5 + 16*b^2*c^4 + (
16*a^2*c^2 - 8*a*b*c + b^2)*d^2 + 8*(8*a^2*c^4 - 6*a*b*c^3 + b^2*c^2)*d)/(c^6 + 3*c^4*d + 3*c^2*d^2 + d^3)))/(
c^4 + 2*c^2*d + d^2)))*log(-(((c^5 + 2*c^3*d + c*d^2)*x*sqrt((64*a^2*c^6 - 64*a*b*c^5 + 16*b^2*c^4 + (16*a^2*c
^2 - 8*a*b*c + b^2)*d^2 + 8*(8*a^2*c^4 - 6*a*b*c^3 + b^2*c^2)*d)/(c^6 + 3*c^4*d + 3*c^2*d^2 + d^3)) - (8*a*c^5
- 4*b*c^4 + (4*a*c - b)*d^2 + (12*a*c^3 - 5*b*c^2)*d)*x)*sqrt(-sqrt((8*a*c^4 - 4*b*c^3 + a*d^2 + (8*a*c^2 - 3
*b*c)*d + (c^4 + 2*c^2*d + d^2)*sqrt((64*a^2*c^6 - 64*a*b*c^5 + 16*b^2*c^4 + (16*a^2*c^2 - 8*a*b*c + b^2)*d^2
+ 8*(8*a^2*c^4 - 6*a*b*c^3 + b^2*c^2)*d)/(c^6 + 3*c^4*d + 3*c^2*d^2 + d^3)))/(c^4 + 2*c^2*d + d^2))) - (a*x^4
- b*x^3)^(1/4)*((4*a*c - b)*d^2 + 4*(2*a*c^3 - b*c^2)*d))/x) - 1/2*sqrt(-sqrt((8*a*c^4 - 4*b*c^3 + a*d^2 + (8*
a*c^2 - 3*b*c)*d - (c^4 + 2*c^2*d + d^2)*sqrt((64*a^2*c^6 - 64*a*b*c^5 + 16*b^2*c^4 + (16*a^2*c^2 - 8*a*b*c +
b^2)*d^2 + 8*(8*a^2*c^4 - 6*a*b*c^3 + b^2*c^2)*d)/(c^6 + 3*c^4*d + 3*c^2*d^2 + d^3)))/(c^4 + 2*c^2*d + d^2)))*
log((((c^5 + 2*c^3*d + c*d^2)*x*sqrt((64*a^2*c^6 - 64*a*b*c^5 + 16*b^2*c^4 + (16*a^2*c^2 - 8*a*b*c + b^2)*d^2
+ 8*(8*a^2*c^4 - 6*a*b*c^3 + b^2*c^2)*d)/(c^6 + 3*c^4*d + 3*c^2*d^2 + d^3)) + (8*a*c^5 - 4*b*c^4 + (4*a*c - b)
*d^2 + (12*a*c^3 - 5*b*c^2)*d)*x)*sqrt(-sqrt((8*a*c^4 - 4*b*c^3 + a*d^2 + (8*a*c^2 - 3*b*c)*d - (c^4 + 2*c^2*d
+ d^2)*sqrt((64*a^2*c^6 - 64*a*b*c^5 + 16*b^2*c^4 + (16*a^2*c^2 - 8*a*b*c + b^2)*d^2 + 8*(8*a^2*c^4 - 6*a*b*c
^3 + b^2*c^2)*d)/(c^6 + 3*c^4*d + 3*c^2*d^2 + d^3)))/(c^4 + 2*c^2*d + d^2))) + (a*x^4 - b*x^3)^(1/4)*((4*a*c -
b)*d^2 + 4*(2*a*c^3 - b*c^2)*d))/x) + 1/2*sqrt(-sqrt((8*a*c^4 - 4*b*c^3 + a*d^2 + (8*a*c^2 - 3*b*c)*d - (c^4
+ 2*c^2*d + d^2)*sqrt((64*a^2*c^6 - 64*a*b*c^5 + 16*b^2*c^4 + (16*a^2*c^2 - 8*a*b*c + b^2)*d^2 + 8*(8*a^2*c^4
- 6*a*b*c^3 + b^2*c^2)*d)/(c^6 + 3*c^4*d + 3*c^2*d^2 + d^3)))/(c^4 + 2*c^2*d + d^2)))*log(-(((c^5 + 2*c^3*d +
c*d^2)*x*sqrt((64*a^2*c^6 - 64*a*b*c^5 + 16*b^2*c^4 + (16*a^2*c^2 - 8*a*b*c + b^2)*d^2 + 8*(8*a^2*c^4 - 6*a*b*
c^3 + b^2*c^2)*d)/(c^6 + 3*c^4*d + 3*c^2*d^2 + d^3)) + (8*a*c^5 - 4*b*c^4 + (4*a*c - b)*d^2 + (12*a*c^3 - 5*b*
c^2)*d)*x)*sqrt(-sqrt((8*a*c^4 - 4*b*c^3 + a*d^2 + (8*a*c^2 - 3*b*c)*d - (c^4 + 2*c^2*d + d^2)*sqrt((64*a^2*c^
6 - 64*a*b*c^5 + 16*b^2*c^4 + (16*a^2*c^2 - 8*a*b*c + b^2)*d^2 + 8*(8*a^2*c^4 - 6*a*b*c^3 + b^2*c^2)*d)/(c^6 +
3*c^4*d + 3*c^2*d^2 + d^3)))/(c^4 + 2*c^2*d + d^2))) - (a*x^4 - b*x^3)^(1/4)*((4*a*c - b)*d^2 + 4*(2*a*c^3 -
b*c^2)*d))/x) + 1/2*((8*a*c^4 - 4*b*c^3 + a*d^2 + (8*a*c^2 - 3*b*c)*d + (c^4 + 2*c^2*d + d^2)*sqrt((64*a^2*c^6
- 64*a*b*c^5 + 16*b^2*c^4 + (16*a^2*c^2 - 8*a*b*c + b^2)*d^2 + 8*(8*a^2*c^4 - 6*a*b*c^3 + b^2*c^2)*d)/(c^6 +
3*c^4*d + 3*c^2*d^2 + d^3)))/(c^4 + 2*c^2*d + d^2))^(1/4)*log(((a*x^4 - b*x^3)^(1/4)*((4*a*c - b)*d^2 + 4*(2*a
*c^3 - b*c^2)*d) + ((c^5 + 2*c^3*d + c*d^2)*x*sqrt((64*a^2*c^6 - 64*a*b*c^5 + 16*b^2*c^4 + (16*a^2*c^2 - 8*a*b
*c + b^2)*d^2 + 8*(8*a^2*c^4 - 6*a*b*c^3 + b^2*c^2)*d)/(c^6 + 3*c^4*d + 3*c^2*d^2 + d^3)) - (8*a*c^5 - 4*b*c^4
+ (4*a*c - b)*d^2 + (12*a*c^3 - 5*b*c^2)*d)*x)*((8*a*c^4 - 4*b*c^3 + a*d^2 + (8*a*c^2 - 3*b*c)*d + (c^4 + 2*c
^2*d + d^2)*sqrt((64*a^2*c^6 - 64*a*b*c^5 + 16*b^2*c^4 + (16*a^2*c^2 - 8*a*b*c + b^2)*d^2 + 8*(8*a^2*c^4 - 6*a
*b*c^3 + b^2*c^2)*d)/(c^6 + 3*c^4*d + 3*c^2*d^2 + d^3)))/(c^4 + 2*c^2*d + d^2))^(1/4))/x) - 1/2*((8*a*c^4 - 4*
b*c^3 + a*d^2 + (8*a*c^2 - 3*b*c)*d + (c^4 + 2*c^2*d + d^2)*sqrt((64*a^2*c^6 - 64*a*b*c^5 + 16*b^2*c^4 + (16*a
^2*c^2 - 8*a*b*c + b^2)*d^2 + 8*(8*a^2*c^4 - 6*a*b*c^3 + b^2*c^2)*d)/(c^6 + 3*c^4*d + 3*c^2*d^2 + d^3)))/(c^4
+ 2*c^2*d + d^2))^(1/4)*log(((a*x^4 - b*x^3)^(1/4)*((4*a*c - b)*d^2 + 4*(2*a*c^3 - b*c^2)*d) - ((c^5 + 2*c^3*d
+ c*d^2)*x*sqrt((64*a^2*c^6 - 64*a*b*c^5 + 16*b^2*c^4 + (16*a^2*c^2 - 8*a*b*c + b^2)*d^2 + 8*(8*a^2*c^4 - 6*a
*b*c^3 + b^2*c^2)*d)/(c^6 + 3*c^4*d + 3*c^2*d^2 + d^3)) - (8*a*c^5 - 4*b*c^4 + (4*a*c - b)*d^2 + (12*a*c^3 - 5
*b*c^2)*d)*x)*((8*a*c^4 - 4*b*c^3 + a*d^2 + (8*a*c^2 - 3*b*c)*d + (c^4 + 2*c^2*d + d^2)*sqrt((64*a^2*c^6 - 64*
a*b*c^5 + 16*b^2*c^4 + (16*a^2*c^2 - 8*a*b*c + b^2)*d^2 + 8*(8*a^2*c^4 - 6*a*b*c^3 + b^2*c^2)*d)/(c^6 + 3*c^4*
d + 3*c^2*d^2 + d^3)))/(c^4 + 2*c^2*d + d^2))^(1/4))/x) - 1/2*((8*a*c^4 - 4*b*c^3 + a*d^2 + (8*a*c^2 - 3*b*c)*
d - (c^4 + 2*c^2*d + d^2)*sqrt((64*a^2*c^6 - 64*a*b*c^5 + 16*b^2*c^4 + (16*a^2*c^2 - 8*a*b*c + b^2)*d^2 + 8*(8
*a^2*c^4 - 6*a*b*c^3 + b^2*c^2)*d)/(c^6 + 3*c^4*d + 3*c^2*d^2 + d^3)))/(c^4 + 2*c^2*d + d^2))^(1/4)*log(((a*x^
4 - b*x^3)^(1/4)*((4*a*c - b)*d^2 + 4*(2*a*c^3 - b*c^2)*d) + ((c^5 + 2*c^3*d + c*d^2)*x*sqrt((64*a^2*c^6 - 64*
a*b*c^5 + 16*b^2*c^4 + (16*a^2*c^2 - 8*a*b*c + b^2)*d^2 + 8*(8*a^2*c^4 - 6*a*b*c^3 + b^2*c^2)*d)/(c^6 + 3*c^4*
d + 3*c^2*d^2 + d^3)) + (8*a*c^5 - 4*b*c^4 + (4*a*c - b)*d^2 + (12*a*c^3 - 5*b*c^2)*d)*x)*((8*a*c^4 - 4*b*c^3
+ a*d^2 + (8*a*c^2 - 3*b*c)*d - (c^4 + 2*c^2*d + d^2)*sqrt((64*a^2*c^6 - 64*a*b*c^5 + 16*b^2*c^4 + (16*a^2*c^2
- 8*a*b*c + b^2)*d^2 + 8*(8*a^2*c^4 - 6*a*b*c^3 + b^2*c^2)*d)/(c^6 + 3*c^4*d + 3*c^2*d^2 + d^3)))/(c^4 + 2*c^
2*d + d^2))^(1/4))/x) + 1/2*((8*a*c^4 - 4*b*c^3 + a*d^2 + (8*a*c^2 - 3*b*c)*d - (c^4 + 2*c^2*d + d^2)*sqrt((64
*a^2*c^6 - 64*a*b*c^5 + 16*b^2*c^4 + (16*a^2*c^2 - 8*a*b*c + b^2)*d^2 + 8*(8*a^2*c^4 - 6*a*b*c^3 + b^2*c^2)*d)
/(c^6 + 3*c^4*d + 3*c^2*d^2 + d^3)))/(c^4 + 2*c^2*d + d^2))^(1/4)*log(((a*x^4 - b*x^3)^(1/4)*((4*a*c - b)*d^2
+ 4*(2*a*c^3 - b*c^2)*d) - ((c^5 + 2*c^3*d + c*d^2)*x*sqrt((64*a^2*c^6 - 64*a*b*c^5 + 16*b^2*c^4 + (16*a^2*c^2
- 8*a*b*c + b^2)*d^2 + 8*(8*a^2*c^4 - 6*a*b*c^3 + b^2*c^2)*d)/(c^6 + 3*c^4*d + 3*c^2*d^2 + d^3)) + (8*a*c^5 -
4*b*c^4 + (4*a*c - b)*d^2 + (12*a*c^3 - 5*b*c^2)*d)*x)*((8*a*c^4 - 4*b*c^3 + a*d^2 + (8*a*c^2 - 3*b*c)*d - (c
^4 + 2*c^2*d + d^2)*sqrt((64*a^2*c^6 - 64*a*b*c^5 + 16*b^2*c^4 + (16*a^2*c^2 - 8*a*b*c + b^2)*d^2 + 8*(8*a^2*c
^4 - 6*a*b*c^3 + b^2*c^2)*d)/(c^6 + 3*c^4*d + 3*c^2*d^2 + d^3)))/(c^4 + 2*c^2*d + d^2))^(1/4))/x) + a^(1/4)*lo
g((a^(1/4)*x + (a*x^4 - b*x^3)^(1/4))/x) - a^(1/4)*log(-(a^(1/4)*x - (a*x^4 - b*x^3)^(1/4))/x) + I*a^(1/4)*log
((I*a^(1/4)*x + (a*x^4 - b*x^3)^(1/4))/x) - 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 = 2.11 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.08 \[ \int \frac {\sqrt [4]{-b x^3+a x^4}}{-d-2 c x+x^2} \, dx=\int \frac {\sqrt [4]{x^{3} \left (a x - b\right )}}{- 2 c x - d + x^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.23 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.11 \[ \int \frac {\sqrt [4]{-b x^3+a x^4}}{-d-2 c x+x^2} \, dx=\int { -\frac {{\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}}}{2 \, c x - x^{2} + d} \,d x } \]

[In]

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

[Out]

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

Giac [F(-1)]

Timed out. \[ \int \frac {\sqrt [4]{-b x^3+a x^4}}{-d-2 c x+x^2} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Mupad [N/A]

Not integrable

Time = 6.86 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.11 \[ \int \frac {\sqrt [4]{-b x^3+a x^4}}{-d-2 c x+x^2} \, dx=-\int \frac {{\left (a\,x^4-b\,x^3\right )}^{1/4}}{-x^2+2\,c\,x+d} \,d x \]

[In]

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

[Out]

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