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

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

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Rubi [B]  time = 2.91, 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 = {2056, 905, 63, 331, 298, 203, 206, 6728, 93, 205, 208} \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 ) \tan ^{-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 ) \tan ^{-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 {\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} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x-b}}\right )}{x^{3/4} \sqrt [4]{a x-b}}+\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 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 205

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

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 208

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

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]{-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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 [F]  time = 61.07, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt [4]{-b x^3+a x^4}}{-d-2 c x+x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 1.44, size = 280, normalized size = 1.00 \begin {gather*} -2 \sqrt [4]{a} \tan ^{-1}\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 ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-2*a^(1/4)*ArcTan[(a^(1/4)*x)/(-(b*x^3) + a*x^4)^(1/4)] + 2*a^(1/4)*ArcTanh[(a^(1/4)*x)/(-(b*x^3) + a*x^4)^(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] + b^2*Log[(-(b*x^3) + a*x^4)^(1/4) - x*#1] - 2*a*b*c*Log[(-(b*x^3) + a*x^4)^(1/4) - x*#1] - a^2*d*
Log[(-(b*x^3) + a*x^4)^(1/4) - x*#1] - a*d*Log[x]*#1^4 + a*d*Log[(-(b*x^3) + a*x^4)^(1/4) - x*#1]*#1^4)/(-(b*c
*#1^3) - a*d*#1^3 + d*#1^7) & ]/2

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fricas [B]  time = 68.83, size = 6476, normalized size = 23.21

result too large to display

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 + (112*a^2*c^4 - 72*a*b*c^3 + 11*b^2*c^2)*d^3 + 2*(144*a^2*c^6 - 1
12*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)*((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))/(((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)) - 4*a^(1/4)*arctan((a^(3/4)*x*sqrt((sqrt(a)*x^2 + sqrt(a*x^4 - b*x^3))/x^2) - (a*x^4 - b*x^3)^(1/4)*a^(3/
4))/(a*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)*log((
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)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \mathit {sage}_{2} \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]

sage2

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maple [F]  time = 0.18, 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*} -\int \frac {{\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}}}{2 \, c x - x^{2} + d}\,{d x} \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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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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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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