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

Optimal. Leaf size=279 \[ -2 \sqrt [4]{a} \text {ArcTan}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b x^3+a x^4}}\right )+2 \sqrt [4]{a} \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b x^3+a x^4}}\right )+\frac {1}{2} \text {RootSum}\left [b^2-2 a b c-a^2 d+2 b c \text {$\#$1}^4+2 a d \text {$\#$1}^4-d \text {$\#$1}^8\& ,\frac {b^2 \log (x)-2 a b c \log (x)-a^2 d \log (x)-b^2 \log \left (\sqrt [4]{-b x^3+a x^4}-x \text {$\#$1}\right )+2 a b c \log \left (\sqrt [4]{-b x^3+a x^4}-x \text {$\#$1}\right )+a^2 d \log \left (\sqrt [4]{-b x^3+a x^4}-x \text {$\#$1}\right )+a d \log (x) \text {$\#$1}^4-a d \log \left (\sqrt [4]{-b x^3+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{b c \text {$\#$1}^3+a d \text {$\#$1}^3-d \text {$\#$1}^7}\& \right ] \]

[Out]

Unintegrable

________________________________________________________________________________________

Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(753\) vs. \(2(279)=558\).
time = 2.13, antiderivative size = 753, normalized size of antiderivative = 2.70, number of steps used = 17, number of rules used = 11, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.367, Rules used = {2081, 919, 65, 338, 304, 209, 212, 6860, 95, 211, 214} \begin {gather*} \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 {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 ) \text {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} \text {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 ) \tanh ^{-1}\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 ) \tanh ^{-1}\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} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x-b}}\right )}{x^{3/4} \sqrt [4]{a x-b}} \end {gather*}

Antiderivative was successfully verified.

[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*} \int \frac {\sqrt [4]{-b x^3+a x^4}}{-d-2 c x+x^2} \, dx &=\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} \tan ^{-1}\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} \tanh ^{-1}\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} \tan ^{-1}\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} \tan ^{-1}\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} \tan ^{-1}\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} \tanh ^{-1}\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} \tanh ^{-1}\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} \tanh ^{-1}\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*}

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Mathematica [A]
time = 0.53, size = 302, normalized size = 1.08 \begin {gather*} -\frac {x^{9/4} (-b+a x)^{3/4} \left (16 \sqrt [4]{a} \left (\text {ArcTan}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )-\tanh ^{-1}\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}} \end {gather*}

Antiderivative was successfully verified.

[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)

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Maple [F]
time = 0.13, size = 0, normalized size = 0.00 \[\int \frac {\left (a \,x^{4}-b \,x^{3}\right )^{\frac {1}{4}}}{-2 c x +x^{2}-d}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [C] Result contains higher order function than in optimal. Order 3 vs. order 1.
time = 37.07, size = 6476, normalized size = 23.21 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-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 + 1
6*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)*arctan((((4*a*c^9 - 2*b*c^8 + (3*a*c - b)*d^4 + (13*a*c^3 - 5*b*c^2)*
d^3 + 3*(7*a*c^5 - 3*b*c^4)*d^2 + (15*a*c^7 - 7*b*c^6)*d)*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)) - (32*
a^2*c^9 - 32*a*b*c^8 + 8*b^2*c^7 + (4*a^2*c - a*b)*d^4 + 2*(16*a^2*c^3 - 9*a*b*c^2 + b^2*c)*d^3 + (84*a^2*c^5
- 65*a*b*c^4 + 12*b^2*c^3)*d^2 + 2*(44*a^2*c^7 - 40*a*b*c^6 + 9*b^2*c^5)*d)*x)*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))*sqrt((((16*a^2*c^2 - 8*a*b*c + b^2)*d^4 + 8*(8*a^2*c^4 - 6*a*b*c^3 + b^2*c^2)*d^3 + 16*(4*a^2*c^6 - 4*a*b
*c^5 + b^2*c^4)*d^2)*sqrt(a*x^4 - b*x^3) - (2*(8*a*c^10 - 4*b*c^9 + (4*a*c^2 - b*c)*d^4 + (20*a*c^4 - 7*b*c^3)
*d^3 + 3*(12*a*c^6 - 5*b*c^5)*d^2 + (28*a*c^8 - 13*b*c^7)*d)*x^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)) -
(128*a^2*c^10 - 128*a*b*c^9 + 32*b^2*c^8 + (16*a^2*c^2 - 8*a*b*c + b^2)*d^4 + (112*a^2*c^4 - 72*a*b*c^3 + 11*
b^2*c^2)*d^3 + 2*(144*a^2*c^6 - 112*a*b*c^5 + 21*b^2*c^4)*d^2 + 32*(10*a^2*c^8 - 9*a*b*c^7 + 2*b^2*c^6)*d)*x^2
)*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)))/x^2) + ((16*a^3*c^2 - 8*a^2*b*c + a*b^2)*d^6 + 2*(80*a^3*c^4 - 64*a^2*b
*c^3 + 15*a*b^2*c^2 - b^3*c)*d^5 + (592*a^3*c^6 - 616*a^2*b*c^5 + 201*a*b^2*c^4 - 20*b^3*c^3)*d^4 + 2*(512*a^3
*c^8 - 632*a^2*b*c^7 + 254*a*b^2*c^6 - 33*b^3*c^5)*d^3 + 16*(52*a^3*c^10 - 72*a^2*b*c^9 + 33*a*b^2*c^8 - 5*b^3
*c^7)*d^2 + 32*(8*a^3*c^12 - 12*a^2*b*c^11 + 6*a*b^2*c^10 - b^3*c^9)*d - ((12*a^2*c^2 - 7*a*b*c + b^2)*d^6 + (
76*a^2*c^4 - 53*a*b*c^3 + 9*b^2*c^2)*d^5 + (188*a^2*c^6 - 149*a*b*c^5 + 29*b^2*c^4)*d^4 + (228*a^2*c^8 - 199*a
*b*c^7 + 43*b^2*c^6)*d^3 + 2*(68*a^2*c^10 - 64*a*b*c^9 + 15*b^2*c^8)*d^2 + 8*(4*a^2*c^12 - 4*a*b*c^11 + b^2*c^
10)*d)*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)))*(a*x^4 - b*x^3)^(1/4)*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)))
*((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)/(((16*a^4*c^2 - 8*a^3*b*c + a^2*b^2)*d^6 + (64*a^4*c^4 - 16*a^3*b*c^3 -
24*a^2*b^2*c^2 + 10*a*b^3*c - b^4)*d^5 + 8*(8*a^4*c^6 + 8*a^3*b*c^5 - 18*a^2*b^2*c^4 + 8*a*b^3*c^3 - b^4*c^2)
*d^4 + 16*(8*a^3*b*c^7 - 12*a^2*b^2*c^6 + 6*a*b^3*c^5 - b^4*c^4)*d^3)*x)) + 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)*arctan((((4*a*c^9 - 2*b*c^8 + (3*a*c - b)*d^4 + (13*a*c^3 - 5*b*c^2)*d^3 + 3*(7*a*c^5 - 3*b*c^4)*d^2 + (
15*a*c^7 - 7*b*c^6)*d)*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)) + (32*a^2*c^9 - 32*a*b*c^8 + 8*b^2*c^7 +
(4*a^2*c - a*b)*d^4 + 2*(16*a^2*c^3 - 9*a*b*c^2 + b^2*c)*d^3 + (84*a^2*c^5 - 65*a*b*c^4 + 12*b^2*c^3)*d^2 + 2*
(44*a^2*c^7 - 40*a*b*c^6 + 9*b^2*c^5)*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))^(3/4)*sqrt((((16*a^2*c^2 - 8*a*b*c
+ b^2)*d^4 + 8*(8*a^2*c^4 - 6*a*b*c^3 + b^2*c^2)*d^3 + 16*(4*a^2*c^6 - 4*a*b*c^5 + b^2*c^4)*d^2)*sqrt(a*x^4 -
b*x^3) + (2*(8*a*c^10 - 4*b*c^9 + (4*a*c^2 - b*c)*d^4 + (20*a*c^4 - 7*b*c^3)*d^3 + 3*(12*a*c^6 - 5*b*c^5)*d^2
+ (28*a*c^8 - 13*b*c^7)*d)*x^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)) + (128*a^2*c^10 - 128*a*b*c^9 + 32
*b^2*c^8 + (16*a^2*c^2 - 8*a*b*c + b^2)*d^4 + (...

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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