Integrand size = 30, antiderivative size = 279 \[ \int \frac {\sqrt [4]{-b x^3+a x^4}}{-d-2 c x+x^2} \, dx=-2 \sqrt [4]{a} \arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b x^3+a x^4}}\right )+2 \sqrt [4]{a} \text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b x^3+a x^4}}\right )+\frac {1}{2} \text {RootSum}\left [b^2-2 a b c-a^2 d+2 b c \text {$\#$1}^4+2 a d \text {$\#$1}^4-d \text {$\#$1}^8\&,\frac {b^2 \log (x)-2 a b c \log (x)-a^2 d \log (x)-b^2 \log \left (\sqrt [4]{-b x^3+a x^4}-x \text {$\#$1}\right )+2 a b c \log \left (\sqrt [4]{-b x^3+a x^4}-x \text {$\#$1}\right )+a^2 d \log \left (\sqrt [4]{-b x^3+a x^4}-x \text {$\#$1}\right )+a d \log (x) \text {$\#$1}^4-a d \log \left (\sqrt [4]{-b x^3+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{b c \text {$\#$1}^3+a d \text {$\#$1}^3-d \text {$\#$1}^7}\&\right ] \]
Time = 0.00 (sec) , antiderivative size = 302, normalized size of antiderivative = 1.08 \[ \int \frac {\sqrt [4]{-b x^3+a x^4}}{-d-2 c x+x^2} \, dx=-\frac {x^{9/4} (-b+a x)^{3/4} \left (16 \sqrt [4]{a} \left (\arctan \left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )-\text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )\right )+\text {RootSum}\left [b^2-2 a b c-a^2 d+2 b c \text {$\#$1}^4+2 a d \text {$\#$1}^4-d \text {$\#$1}^8\&,\frac {b^2 \log (x)-2 a b c \log (x)-a^2 d \log (x)-4 b^2 \log \left (\sqrt [4]{-b+a x}-\sqrt [4]{x} \text {$\#$1}\right )+8 a b c \log \left (\sqrt [4]{-b+a x}-\sqrt [4]{x} \text {$\#$1}\right )+4 a^2 d \log \left (\sqrt [4]{-b+a x}-\sqrt [4]{x} \text {$\#$1}\right )+a d \log (x) \text {$\#$1}^4-4 a d \log \left (\sqrt [4]{-b+a x}-\sqrt [4]{x} \text {$\#$1}\right ) \text {$\#$1}^4}{-b c \text {$\#$1}^3-a d \text {$\#$1}^3+d \text {$\#$1}^7}\&\right ]\right )}{8 \left (x^3 (-b+a x)\right )^{3/4}} \]
-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)
Leaf count is larger than twice the leaf count of optimal. \(986\) vs. \(2(279)=558\).
Time = 2.50 (sec) , antiderivative size = 986, normalized size of antiderivative = 3.53, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {2467, 25, 1202, 25, 73, 854, 827, 216, 219, 2035, 7279, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\sqrt [4]{a x^4-b x^3}}{-2 c x-d+x^2} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt [4]{a x^4-b x^3} \int -\frac {x^{3/4} \sqrt [4]{a x-b}}{-x^2+2 c x+d}dx}{x^{3/4} \sqrt [4]{a x-b}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt [4]{a x^4-b x^3} \int \frac {x^{3/4} \sqrt [4]{a x-b}}{-x^2+2 c x+d}dx}{x^{3/4} \sqrt [4]{a x-b}}\) |
| \(\Big \downarrow \) 1202 |
| \(\displaystyle -\frac {\sqrt [4]{a x^4-b x^3} \left (-\int -\frac {a d-(b-2 a c) x}{\sqrt [4]{x} (a x-b)^{3/4} \left (-x^2+2 c x+d\right )}dx-a \int \frac {1}{\sqrt [4]{x} (a x-b)^{3/4}}dx\right )}{x^{3/4} \sqrt [4]{a x-b}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt [4]{a x^4-b x^3} \left (\int \frac {a d-(b-2 a c) x}{\sqrt [4]{x} (a x-b)^{3/4} \left (-x^2+2 c x+d\right )}dx-a \int \frac {1}{\sqrt [4]{x} (a x-b)^{3/4}}dx\right )}{x^{3/4} \sqrt [4]{a x-b}}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {\sqrt [4]{a x^4-b x^3} \left (\int \frac {a d-(b-2 a c) x}{\sqrt [4]{x} (a x-b)^{3/4} \left (-x^2+2 c x+d\right )}dx-4 a \int \frac {\sqrt {x}}{(a x-b)^{3/4}}d\sqrt [4]{x}\right )}{x^{3/4} \sqrt [4]{a x-b}}\) |
| \(\Big \downarrow \) 854 |
| \(\displaystyle -\frac {\sqrt [4]{a x^4-b x^3} \left (\int \frac {a d-(b-2 a c) x}{\sqrt [4]{x} (a x-b)^{3/4} \left (-x^2+2 c x+d\right )}dx-4 a \int \frac {\sqrt {x}}{1-a x}d\frac {\sqrt [4]{x}}{\sqrt [4]{a x-b}}\right )}{x^{3/4} \sqrt [4]{a x-b}}\) |
| \(\Big \downarrow \) 827 |
| \(\displaystyle -\frac {\sqrt [4]{a x^4-b x^3} \left (\int \frac {a d-(b-2 a c) x}{\sqrt [4]{x} (a x-b)^{3/4} \left (-x^2+2 c x+d\right )}dx-4 a \left (\frac {\int \frac {1}{1-\sqrt {a} \sqrt {x}}d\frac {\sqrt [4]{x}}{\sqrt [4]{a x-b}}}{2 \sqrt {a}}-\frac {\int \frac {1}{\sqrt {a} \sqrt {x}+1}d\frac {\sqrt [4]{x}}{\sqrt [4]{a x-b}}}{2 \sqrt {a}}\right )\right )}{x^{3/4} \sqrt [4]{a x-b}}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle -\frac {\sqrt [4]{a x^4-b x^3} \left (\int \frac {a d-(b-2 a c) x}{\sqrt [4]{x} (a x-b)^{3/4} \left (-x^2+2 c x+d\right )}dx-4 a \left (\frac {\int \frac {1}{1-\sqrt {a} \sqrt {x}}d\frac {\sqrt [4]{x}}{\sqrt [4]{a x-b}}}{2 \sqrt {a}}-\frac {\arctan \left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x-b}}\right )}{2 a^{3/4}}\right )\right )}{x^{3/4} \sqrt [4]{a x-b}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {\sqrt [4]{a x^4-b x^3} \left (\int \frac {a d-(b-2 a c) x}{\sqrt [4]{x} (a x-b)^{3/4} \left (-x^2+2 c x+d\right )}dx-4 a \left (\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x-b}}\right )}{2 a^{3/4}}-\frac {\arctan \left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x-b}}\right )}{2 a^{3/4}}\right )\right )}{x^{3/4} \sqrt [4]{a x-b}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {\sqrt [4]{a x^4-b x^3} \left (4 \int \frac {\sqrt {x} (a d-(b-2 a c) x)}{(a x-b)^{3/4} \left (-x^2+2 c x+d\right )}d\sqrt [4]{x}-4 a \left (\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x-b}}\right )}{2 a^{3/4}}-\frac {\arctan \left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x-b}}\right )}{2 a^{3/4}}\right )\right )}{x^{3/4} \sqrt [4]{a x-b}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle -\frac {\sqrt [4]{a x^4-b x^3} \left (4 \int \left (\frac {a d \sqrt {x}}{(a x-b)^{3/4} \left (-x^2+2 c x+d\right )}-\frac {(b-2 a c) x^{3/2}}{(a x-b)^{3/4} \left (-x^2+2 c x+d\right )}\right )d\sqrt [4]{x}-4 a \left (\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x-b}}\right )}{2 a^{3/4}}-\frac {\arctan \left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x-b}}\right )}{2 a^{3/4}}\right )\right )}{x^{3/4} \sqrt [4]{a x-b}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\sqrt [4]{a x^4-b x^3} \left (4 \left (-\frac {(b-2 a c) \left (\sqrt {c^2+d}-c\right )^{3/4} \arctan \left (\frac {\sqrt [4]{b-a \left (c-\sqrt {c^2+d}\right )} \sqrt [4]{x}}{\sqrt [4]{\sqrt {c^2+d}-c} \sqrt [4]{a x-b}}\right )}{4 \sqrt {c^2+d} \left (b-a \left (c-\sqrt {c^2+d}\right )\right )^{3/4}}-\frac {a d \arctan \left (\frac {\sqrt [4]{b-a \left (c-\sqrt {c^2+d}\right )} \sqrt [4]{x}}{\sqrt [4]{\sqrt {c^2+d}-c} \sqrt [4]{a x-b}}\right )}{4 \sqrt {c^2+d} \sqrt [4]{\sqrt {c^2+d}-c} \left (b-a \left (c-\sqrt {c^2+d}\right )\right )^{3/4}}+\frac {(b-2 a c) \left (c+\sqrt {c^2+d}\right )^{3/4} \arctan \left (\frac {\sqrt [4]{a \left (c+\sqrt {c^2+d}\right )-b} \sqrt [4]{x}}{\sqrt [4]{c+\sqrt {c^2+d}} \sqrt [4]{a x-b}}\right )}{4 \sqrt {c^2+d} \left (a \left (c+\sqrt {c^2+d}\right )-b\right )^{3/4}}-\frac {a d \arctan \left (\frac {\sqrt [4]{a \left (c+\sqrt {c^2+d}\right )-b} \sqrt [4]{x}}{\sqrt [4]{c+\sqrt {c^2+d}} \sqrt [4]{a x-b}}\right )}{4 \sqrt {c^2+d} \sqrt [4]{c+\sqrt {c^2+d}} \left (a \left (c+\sqrt {c^2+d}\right )-b\right )^{3/4}}+\frac {(b-2 a c) \left (\sqrt {c^2+d}-c\right )^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{b-a \left (c-\sqrt {c^2+d}\right )} \sqrt [4]{x}}{\sqrt [4]{\sqrt {c^2+d}-c} \sqrt [4]{a x-b}}\right )}{4 \sqrt {c^2+d} \left (b-a \left (c-\sqrt {c^2+d}\right )\right )^{3/4}}+\frac {a d \text {arctanh}\left (\frac {\sqrt [4]{b-a \left (c-\sqrt {c^2+d}\right )} \sqrt [4]{x}}{\sqrt [4]{\sqrt {c^2+d}-c} \sqrt [4]{a x-b}}\right )}{4 \sqrt {c^2+d} \sqrt [4]{\sqrt {c^2+d}-c} \left (b-a \left (c-\sqrt {c^2+d}\right )\right )^{3/4}}-\frac {(b-2 a c) \left (c+\sqrt {c^2+d}\right )^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{a \left (c+\sqrt {c^2+d}\right )-b} \sqrt [4]{x}}{\sqrt [4]{c+\sqrt {c^2+d}} \sqrt [4]{a x-b}}\right )}{4 \sqrt {c^2+d} \left (a \left (c+\sqrt {c^2+d}\right )-b\right )^{3/4}}+\frac {a d \text {arctanh}\left (\frac {\sqrt [4]{a \left (c+\sqrt {c^2+d}\right )-b} \sqrt [4]{x}}{\sqrt [4]{c+\sqrt {c^2+d}} \sqrt [4]{a x-b}}\right )}{4 \sqrt {c^2+d} \sqrt [4]{c+\sqrt {c^2+d}} \left (a \left (c+\sqrt {c^2+d}\right )-b\right )^{3/4}}\right )-4 a \left (\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x-b}}\right )}{2 a^{3/4}}-\frac {\arctan \left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x-b}}\right )}{2 a^{3/4}}\right )\right )}{x^{3/4} \sqrt [4]{a x-b}}\) |
-(((-(b*x^3) + a*x^4)^(1/4)*(-4*a*(-1/2*ArcTan[(a^(1/4)*x^(1/4))/(-b + a*x )^(1/4)]/a^(3/4) + ArcTanh[(a^(1/4)*x^(1/4))/(-b + a*x)^(1/4)]/(2*a^(3/4)) ) + 4*(-1/4*(a*d*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))])/(Sqrt[c^2 + d]*(-c + Sqrt[c^2 + d])^(1/4)*(b - a*(c - Sqrt[c^2 + d]))^(3/4)) - ((b - 2*a*c)*(-c + Sqrt[c^2 + d])^(3/4)*ArcTan[((b - a*(c - Sqrt[c^2 + d]))^(1/4)*x^(1/4))/((-c + Sqr t[c^2 + d])^(1/4)*(-b + a*x)^(1/4))])/(4*Sqrt[c^2 + d]*(b - a*(c - Sqrt[c^ 2 + d]))^(3/4)) - (a*d*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))])/(4*Sqrt[c^2 + d]*(c + Sqrt [c^2 + d])^(1/4)*(-b + a*(c + Sqrt[c^2 + d]))^(3/4)) + ((b - 2*a*c)*(c + S qrt[c^2 + d])^(3/4)*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))])/(4*Sqrt[c^2 + d]*(-b + a*(c + Sqrt[c^2 + d]))^(3/4)) + (a*d*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))])/(4*Sqrt[c^2 + d]* (-c + Sqrt[c^2 + d])^(1/4)*(b - a*(c - Sqrt[c^2 + d]))^(3/4)) + ((b - 2*a* c)*(-c + Sqrt[c^2 + d])^(3/4)*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))])/(4*Sqrt[c^2 + d]*( b - a*(c - Sqrt[c^2 + d]))^(3/4)) + (a*d*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))])/(4*Sqrt [c^2 + d]*(c + Sqrt[c^2 + d])^(1/4)*(-b + a*(c + Sqrt[c^2 + d]))^(3/4))...
3.29.19.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[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] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
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] && (Gt Q[a, 0] || LtQ[b, 0])
Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[s/(2*b) Int[1/(r + s*x^2), x], x] - Simp[s/(2*b) Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ [a/b, 0]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[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]
Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_))/((a_.) + (b_.)*(x _) + (c_.)*(x_)^2), x_Symbol] :> Simp[e*(g/c) Int[(d + e*x)^(m - 1)*(f + g*x)^(n - 1), x], x] + Simp[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] && !IntegerQ[m] && !IntegerQ[n] && GtQ[m, 0] && GtQ[n, 0]
Int[(Fx_)*(x_)^(m_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k Subst [Int[x^(k*(m + 1) - 1)*SubstPower[Fx, x, k], x], x, x^(1/k)], x]] /; Fracti onQ[m] && AlgebraicFunctionQ[Fx, x]
Int[(Fx_.)*(Px_)^(p_), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[Px^F racPart[p]/(x^(r*FracPart[p])*ExpandToSum[Px/x^r, x]^FracPart[p]) Int[x^( p*r)*ExpandToSum[Px/x^r, x]^p*Fx, x], x] /; IGtQ[r, 0]] /; FreeQ[p, x] && P olyQ[Px, x] && !IntegerQ[p] && !MonomialQ[Px, x] && !PolyQ[Fx, x]
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] /; Su mQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]
Time = 0.37 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.64
| method | result | size |
| pseudoelliptic | \(a^{\frac {1}{4}} \ln \left (\frac {a^{\frac {1}{4}} x +\left (x^{3} \left (a x -b \right )\right )^{\frac {1}{4}}}{-a^{\frac {1}{4}} x +\left (x^{3} \left (a x -b \right )\right )^{\frac {1}{4}}}\right )+2 a^{\frac {1}{4}} \arctan \left (\frac {\left (x^{3} \left (a x -b \right )\right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x}\right )+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (d \,\textit {\_Z}^{8}+\left (-2 a d -2 b c \right ) \textit {\_Z}^{4}+a^{2} d +2 a b c -b^{2}\right )}{\sum }\frac {\left (\textit {\_R}^{4} a d -a^{2} d -2 a b c +b^{2}\right ) \ln \left (\frac {-\textit {\_R} x +\left (x^{3} \left (a x -b \right )\right )^{\frac {1}{4}}}{x}\right )}{\textit {\_R}^{3} \left (\textit {\_R}^{4} d -a d -b c \right )}\right )}{2}\) | \(179\) |
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))
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 2.28 (sec) , antiderivative size = 4415, normalized size of antiderivative = 15.82 \[ \int \frac {\sqrt [4]{-b x^3+a x^4}}{-d-2 c x+x^2} \, dx=\text {Too large to display} \]
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^...
Not integrable
Time = 2.12 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.08 \[ \int \frac {\sqrt [4]{-b x^3+a x^4}}{-d-2 c x+x^2} \, dx=\int \frac {\sqrt [4]{x^{3} \left (a x - b\right )}}{- 2 c x - d + x^{2}}\, dx \]
Not integrable
Time = 0.27 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.11 \[ \int \frac {\sqrt [4]{-b x^3+a x^4}}{-d-2 c x+x^2} \, dx=\int { -\frac {{\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}}}{2 \, c x - x^{2} + d} \,d x } \]
Timed out. \[ \int \frac {\sqrt [4]{-b x^3+a x^4}}{-d-2 c x+x^2} \, dx=\text {Timed out} \]
Not integrable
Time = 0.00 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.11 \[ \int \frac {\sqrt [4]{-b x^3+a x^4}}{-d-2 c x+x^2} \, dx=-\int \frac {{\left (a\,x^4-b\,x^3\right )}^{1/4}}{-x^2+2\,c\,x+d} \,d x \]