Integrand size = 30, antiderivative size = 323 \[ \int \frac {x^2 \sqrt [4]{b x^3+a x^4}}{-b+a x^4} \, dx=-\frac {2 \arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b x^3+a x^4}}\right )}{a^{3/4}}+\frac {2 \text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b x^3+a x^4}}\right )}{a^{3/4}}+\frac {1}{4} \text {RootSum}\left [a^4-a b^3-4 a^3 \text {$\#$1}^4+6 a^2 \text {$\#$1}^8-4 a \text {$\#$1}^{12}+\text {$\#$1}^{16}\&,\frac {-a^3 \log (x)+b^3 \log (x)+a^3 \log \left (\sqrt [4]{b x^3+a x^4}-x \text {$\#$1}\right )-b^3 \log \left (\sqrt [4]{b x^3+a x^4}-x \text {$\#$1}\right )+3 a^2 \log (x) \text {$\#$1}^4-3 a^2 \log \left (\sqrt [4]{b x^3+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4-3 a \log (x) \text {$\#$1}^8+3 a \log \left (\sqrt [4]{b x^3+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^8+\log (x) \text {$\#$1}^{12}-\log \left (\sqrt [4]{b x^3+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^{12}}{a^3 \text {$\#$1}^3-3 a^2 \text {$\#$1}^7+3 a \text {$\#$1}^{11}-\text {$\#$1}^{15}}\&\right ] \]
Time = 0.00 (sec) , antiderivative size = 340, normalized size of antiderivative = 1.05 \[ \int \frac {x^2 \sqrt [4]{b x^3+a x^4}}{-b+a x^4} \, dx=-\frac {x^{9/4} (b+a x)^{3/4} \left (32 \left (\arctan \left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )-\text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )\right )+a^{3/4} \text {RootSum}\left [a^4-a b^3-4 a^3 \text {$\#$1}^4+6 a^2 \text {$\#$1}^8-4 a \text {$\#$1}^{12}+\text {$\#$1}^{16}\&,\frac {a^3 \log (x)-b^3 \log (x)-4 a^3 \log \left (\sqrt [4]{b+a x}-\sqrt [4]{x} \text {$\#$1}\right )+4 b^3 \log \left (\sqrt [4]{b+a x}-\sqrt [4]{x} \text {$\#$1}\right )-3 a^2 \log (x) \text {$\#$1}^4+12 a^2 \log \left (\sqrt [4]{b+a x}-\sqrt [4]{x} \text {$\#$1}\right ) \text {$\#$1}^4+3 a \log (x) \text {$\#$1}^8-12 a \log \left (\sqrt [4]{b+a x}-\sqrt [4]{x} \text {$\#$1}\right ) \text {$\#$1}^8-\log (x) \text {$\#$1}^{12}+4 \log \left (\sqrt [4]{b+a x}-\sqrt [4]{x} \text {$\#$1}\right ) \text {$\#$1}^{12}}{a^3 \text {$\#$1}^3-3 a^2 \text {$\#$1}^7+3 a \text {$\#$1}^{11}-\text {$\#$1}^{15}}\&\right ]\right )}{16 a^{3/4} \left (x^3 (b+a x)\right )^{3/4}} \]
-1/16*(x^(9/4)*(b + a*x)^(3/4)*(32*(ArcTan[(a^(1/4)*x^(1/4))/(b + a*x)^(1/ 4)] - ArcTanh[(a^(1/4)*x^(1/4))/(b + a*x)^(1/4)]) + a^(3/4)*RootSum[a^4 - a*b^3 - 4*a^3*#1^4 + 6*a^2*#1^8 - 4*a*#1^12 + #1^16 & , (a^3*Log[x] - b^3* Log[x] - 4*a^3*Log[(b + a*x)^(1/4) - x^(1/4)*#1] + 4*b^3*Log[(b + a*x)^(1/ 4) - x^(1/4)*#1] - 3*a^2*Log[x]*#1^4 + 12*a^2*Log[(b + a*x)^(1/4) - x^(1/4 )*#1]*#1^4 + 3*a*Log[x]*#1^8 - 12*a*Log[(b + a*x)^(1/4) - x^(1/4)*#1]*#1^8 - Log[x]*#1^12 + 4*Log[(b + a*x)^(1/4) - x^(1/4)*#1]*#1^12)/(a^3*#1^3 - 3 *a^2*#1^7 + 3*a*#1^11 - #1^15) & ]))/(a^(3/4)*(x^3*(b + a*x))^(3/4))
Leaf count is larger than twice the leaf count of optimal. \(1348\) vs. \(2(323)=646\).
Time = 4.82 (sec) , antiderivative size = 1348, normalized size of antiderivative = 4.17, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2467, 25, 2035, 7276, 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.
| \(\displaystyle \int \frac {x^2 \sqrt [4]{a x^4+b x^3}}{a x^4-b} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt [4]{a x^4+b x^3} \int -\frac {x^{11/4} \sqrt [4]{b+a x}}{b-a x^4}dx}{x^{3/4} \sqrt [4]{a x+b}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt [4]{a x^4+b x^3} \int \frac {x^{11/4} \sqrt [4]{b+a x}}{b-a x^4}dx}{x^{3/4} \sqrt [4]{a x+b}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {4 \sqrt [4]{a x^4+b x^3} \int \frac {x^{7/2} \sqrt [4]{b+a x}}{b-a x^4}d\sqrt [4]{x}}{x^{3/4} \sqrt [4]{a x+b}}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle -\frac {4 \sqrt [4]{a x^4+b x^3} \int \left (-\frac {\sqrt [4]{b+a x} x^{3/2}}{2 \left (a x^2-\sqrt {a} \sqrt {b}\right )}-\frac {\sqrt [4]{b+a x} x^{3/2}}{2 \left (a x^2+\sqrt {a} \sqrt {b}\right )}\right )d\sqrt [4]{x}}{x^{3/4} \sqrt [4]{a x+b}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {4 \sqrt [4]{a x^4+b x^3} \left (\frac {\arctan \left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{2 a^{3/4}}+\frac {b^{3/4} \arctan \left (\frac {\sqrt [16]{a} \sqrt [4]{a^{3/4}-b^{3/4}} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{8 a^{15/16} \left (a^{3/4}-b^{3/4}\right )^{3/4}}-\frac {\arctan \left (\frac {\sqrt [16]{a} \sqrt [4]{a^{3/4}-b^{3/4}} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{8 a^{3/16} \left (a^{3/4}-b^{3/4}\right )^{3/4}}+\frac {b^{3/4} \arctan \left (\frac {\sqrt [8]{-\sqrt {a}} \sqrt [4]{\frac {a}{\sqrt {-\sqrt {a}}}-b^{3/4}} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{8 \left (-\sqrt {a}\right )^{15/8} \left (\frac {a}{\sqrt {-\sqrt {a}}}-b^{3/4}\right )^{3/4}}-\frac {\arctan \left (\frac {\sqrt [8]{-\sqrt {a}} \sqrt [4]{\frac {a}{\sqrt {-\sqrt {a}}}-b^{3/4}} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{8 \left (-\sqrt {a}\right )^{3/8} \left (\frac {a}{\sqrt {-\sqrt {a}}}-b^{3/4}\right )^{3/4}}-\frac {b^{3/4} \arctan \left (\frac {\sqrt [16]{a} \sqrt [4]{a^{3/4}+b^{3/4}} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{8 a^{15/16} \left (a^{3/4}+b^{3/4}\right )^{3/4}}-\frac {\arctan \left (\frac {\sqrt [16]{a} \sqrt [4]{a^{3/4}+b^{3/4}} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{8 a^{3/16} \left (a^{3/4}+b^{3/4}\right )^{3/4}}-\frac {b^{3/4} \arctan \left (\frac {\sqrt [8]{-\sqrt {a}} \sqrt [4]{\frac {a}{\sqrt {-\sqrt {a}}}+b^{3/4}} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{8 \left (-\sqrt {a}\right )^{15/8} \left (\frac {a}{\sqrt {-\sqrt {a}}}+b^{3/4}\right )^{3/4}}-\frac {\arctan \left (\frac {\sqrt [8]{-\sqrt {a}} \sqrt [4]{\frac {a}{\sqrt {-\sqrt {a}}}+b^{3/4}} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{8 \left (-\sqrt {a}\right )^{3/8} \left (\frac {a}{\sqrt {-\sqrt {a}}}+b^{3/4}\right )^{3/4}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{2 a^{3/4}}-\frac {b^{3/4} \text {arctanh}\left (\frac {\sqrt [16]{a} \sqrt [4]{a^{3/4}-b^{3/4}} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{8 a^{15/16} \left (a^{3/4}-b^{3/4}\right )^{3/4}}+\frac {\text {arctanh}\left (\frac {\sqrt [16]{a} \sqrt [4]{a^{3/4}-b^{3/4}} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{8 a^{3/16} \left (a^{3/4}-b^{3/4}\right )^{3/4}}-\frac {b^{3/4} \text {arctanh}\left (\frac {\sqrt [8]{-\sqrt {a}} \sqrt [4]{\frac {a}{\sqrt {-\sqrt {a}}}-b^{3/4}} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{8 \left (-\sqrt {a}\right )^{15/8} \left (\frac {a}{\sqrt {-\sqrt {a}}}-b^{3/4}\right )^{3/4}}+\frac {\text {arctanh}\left (\frac {\sqrt [8]{-\sqrt {a}} \sqrt [4]{\frac {a}{\sqrt {-\sqrt {a}}}-b^{3/4}} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{8 \left (-\sqrt {a}\right )^{3/8} \left (\frac {a}{\sqrt {-\sqrt {a}}}-b^{3/4}\right )^{3/4}}+\frac {b^{3/4} \text {arctanh}\left (\frac {\sqrt [16]{a} \sqrt [4]{a^{3/4}+b^{3/4}} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{8 a^{15/16} \left (a^{3/4}+b^{3/4}\right )^{3/4}}+\frac {\text {arctanh}\left (\frac {\sqrt [16]{a} \sqrt [4]{a^{3/4}+b^{3/4}} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{8 a^{3/16} \left (a^{3/4}+b^{3/4}\right )^{3/4}}+\frac {b^{3/4} \text {arctanh}\left (\frac {\sqrt [8]{-\sqrt {a}} \sqrt [4]{\frac {a}{\sqrt {-\sqrt {a}}}+b^{3/4}} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{8 \left (-\sqrt {a}\right )^{15/8} \left (\frac {a}{\sqrt {-\sqrt {a}}}+b^{3/4}\right )^{3/4}}+\frac {\text {arctanh}\left (\frac {\sqrt [8]{-\sqrt {a}} \sqrt [4]{\frac {a}{\sqrt {-\sqrt {a}}}+b^{3/4}} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{8 \left (-\sqrt {a}\right )^{3/8} \left (\frac {a}{\sqrt {-\sqrt {a}}}+b^{3/4}\right )^{3/4}}\right )}{x^{3/4} \sqrt [4]{b+a x}}\) |
(-4*(b*x^3 + a*x^4)^(1/4)*(ArcTan[(a^(1/4)*x^(1/4))/(b + a*x)^(1/4)]/(2*a^ (3/4)) - ArcTan[(a^(1/16)*(a^(3/4) - b^(3/4))^(1/4)*x^(1/4))/(b + a*x)^(1/ 4)]/(8*a^(3/16)*(a^(3/4) - b^(3/4))^(3/4)) + (b^(3/4)*ArcTan[(a^(1/16)*(a^ (3/4) - b^(3/4))^(1/4)*x^(1/4))/(b + a*x)^(1/4)])/(8*a^(15/16)*(a^(3/4) - b^(3/4))^(3/4)) - ArcTan[((-Sqrt[a])^(1/8)*(a/Sqrt[-Sqrt[a]] - b^(3/4))^(1 /4)*x^(1/4))/(b + a*x)^(1/4)]/(8*(-Sqrt[a])^(3/8)*(a/Sqrt[-Sqrt[a]] - b^(3 /4))^(3/4)) + (b^(3/4)*ArcTan[((-Sqrt[a])^(1/8)*(a/Sqrt[-Sqrt[a]] - b^(3/4 ))^(1/4)*x^(1/4))/(b + a*x)^(1/4)])/(8*(-Sqrt[a])^(15/8)*(a/Sqrt[-Sqrt[a]] - b^(3/4))^(3/4)) - ArcTan[(a^(1/16)*(a^(3/4) + b^(3/4))^(1/4)*x^(1/4))/( b + a*x)^(1/4)]/(8*a^(3/16)*(a^(3/4) + b^(3/4))^(3/4)) - (b^(3/4)*ArcTan[( a^(1/16)*(a^(3/4) + b^(3/4))^(1/4)*x^(1/4))/(b + a*x)^(1/4)])/(8*a^(15/16) *(a^(3/4) + b^(3/4))^(3/4)) - ArcTan[((-Sqrt[a])^(1/8)*(a/Sqrt[-Sqrt[a]] + b^(3/4))^(1/4)*x^(1/4))/(b + a*x)^(1/4)]/(8*(-Sqrt[a])^(3/8)*(a/Sqrt[-Sqr t[a]] + b^(3/4))^(3/4)) - (b^(3/4)*ArcTan[((-Sqrt[a])^(1/8)*(a/Sqrt[-Sqrt[ a]] + b^(3/4))^(1/4)*x^(1/4))/(b + a*x)^(1/4)])/(8*(-Sqrt[a])^(15/8)*(a/Sq rt[-Sqrt[a]] + b^(3/4))^(3/4)) - ArcTanh[(a^(1/4)*x^(1/4))/(b + a*x)^(1/4) ]/(2*a^(3/4)) + ArcTanh[(a^(1/16)*(a^(3/4) - b^(3/4))^(1/4)*x^(1/4))/(b + a*x)^(1/4)]/(8*a^(3/16)*(a^(3/4) - b^(3/4))^(3/4)) - (b^(3/4)*ArcTanh[(a^( 1/16)*(a^(3/4) - b^(3/4))^(1/4)*x^(1/4))/(b + a*x)^(1/4)])/(8*a^(15/16)*(a ^(3/4) - b^(3/4))^(3/4)) + ArcTanh[((-Sqrt[a])^(1/8)*(a/Sqrt[-Sqrt[a]] ...
3.30.4.3.1 Defintions of rubi rules used
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_)), x_Symbol] :> With[{v = RationalFunctionE xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ [n, 0]
Time = 0.54 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.55
| method | result | size |
| pseudoelliptic | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{16}-4 a \,\textit {\_Z}^{12}+6 a^{2} \textit {\_Z}^{8}-4 a^{3} \textit {\_Z}^{4}+a^{4}-a \,b^{3}\right )}{\sum }\frac {\left (-\textit {\_R}^{12}+3 \textit {\_R}^{8} a -3 \textit {\_R}^{4} a^{2}+a^{3}-b^{3}\right ) \ln \left (\frac {-\textit {\_R} x +\left (x^{3} \left (a x +b \right )\right )^{\frac {1}{4}}}{x}\right )}{\textit {\_R}^{3} \left (-\textit {\_R}^{4}+a \right )^{3}}\right ) a^{\frac {3}{4}}+4 \ln \left (\frac {-a^{\frac {1}{4}} x -\left (x^{3} \left (a x +b \right )\right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x -\left (x^{3} \left (a x +b \right )\right )^{\frac {1}{4}}}\right )+8 \arctan \left (\frac {\left (x^{3} \left (a x +b \right )\right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x}\right )}{4 a^{\frac {3}{4}}}\) | \(178\) |
1/4*(sum((-_R^12+3*_R^8*a-3*_R^4*a^2+a^3-b^3)*ln((-_R*x+(x^3*(a*x+b))^(1/4 ))/x)/_R^3/(-_R^4+a)^3,_R=RootOf(_Z^16-4*_Z^12*a+6*_Z^8*a^2-4*_Z^4*a^3+a^4 -a*b^3))*a^(3/4)+4*ln((-a^(1/4)*x-(x^3*(a*x+b))^(1/4))/(a^(1/4)*x-(x^3*(a* x+b))^(1/4)))+8*arctan(1/a^(1/4)/x*(x^3*(a*x+b))^(1/4)))/a^(3/4)
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.30 (sec) , antiderivative size = 1365, normalized size of antiderivative = 4.23 \[ \int \frac {x^2 \sqrt [4]{b x^3+a x^4}}{-b+a x^4} \, dx=\text {Too large to display} \]
-1/4*sqrt(-sqrt((a^3*sqrt(-sqrt(b^3/a^15)) + 1)/a^3))*log((a*x*sqrt(-sqrt( (a^3*sqrt(-sqrt(b^3/a^15)) + 1)/a^3)) + (a*x^4 + b*x^3)^(1/4))/x) + 1/4*sq rt(-sqrt((a^3*sqrt(-sqrt(b^3/a^15)) + 1)/a^3))*log(-(a*x*sqrt(-sqrt((a^3*s qrt(-sqrt(b^3/a^15)) + 1)/a^3)) - (a*x^4 + b*x^3)^(1/4))/x) - 1/4*sqrt(-sq rt(-(a^3*sqrt(-sqrt(b^3/a^15)) - 1)/a^3))*log((a*x*sqrt(-sqrt(-(a^3*sqrt(- sqrt(b^3/a^15)) - 1)/a^3)) + (a*x^4 + b*x^3)^(1/4))/x) + 1/4*sqrt(-sqrt(-( a^3*sqrt(-sqrt(b^3/a^15)) - 1)/a^3))*log(-(a*x*sqrt(-sqrt(-(a^3*sqrt(-sqrt (b^3/a^15)) - 1)/a^3)) - (a*x^4 + b*x^3)^(1/4))/x) - 1/4*sqrt(-sqrt((a^3*( b^3/a^15)^(1/4) + 1)/a^3))*log((a*x*sqrt(-sqrt((a^3*(b^3/a^15)^(1/4) + 1)/ a^3)) + (a*x^4 + b*x^3)^(1/4))/x) + 1/4*sqrt(-sqrt((a^3*(b^3/a^15)^(1/4) + 1)/a^3))*log(-(a*x*sqrt(-sqrt((a^3*(b^3/a^15)^(1/4) + 1)/a^3)) - (a*x^4 + b*x^3)^(1/4))/x) - 1/4*sqrt(-sqrt(-(a^3*(b^3/a^15)^(1/4) - 1)/a^3))*log(( a*x*sqrt(-sqrt(-(a^3*(b^3/a^15)^(1/4) - 1)/a^3)) + (a*x^4 + b*x^3)^(1/4))/ x) + 1/4*sqrt(-sqrt(-(a^3*(b^3/a^15)^(1/4) - 1)/a^3))*log(-(a*x*sqrt(-sqrt (-(a^3*(b^3/a^15)^(1/4) - 1)/a^3)) - (a*x^4 + b*x^3)^(1/4))/x) + (a^(-3))^ (1/4)*log((a*(a^(-3))^(1/4)*x + (a*x^4 + b*x^3)^(1/4))/x) - (a^(-3))^(1/4) *log(-(a*(a^(-3))^(1/4)*x - (a*x^4 + b*x^3)^(1/4))/x) + I*(a^(-3))^(1/4)*l og((I*a*(a^(-3))^(1/4)*x + (a*x^4 + b*x^3)^(1/4))/x) - I*(a^(-3))^(1/4)*lo g((-I*a*(a^(-3))^(1/4)*x + (a*x^4 + b*x^3)^(1/4))/x) - 1/4*((a^3*sqrt(-sqr t(b^3/a^15)) + 1)/a^3)^(1/4)*log((a*x*((a^3*sqrt(-sqrt(b^3/a^15)) + 1)/...
Not integrable
Time = 1.96 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.07 \[ \int \frac {x^2 \sqrt [4]{b x^3+a x^4}}{-b+a x^4} \, dx=\int \frac {x^{2} \sqrt [4]{x^{3} \left (a x + b\right )}}{a x^{4} - b}\, dx \]
Not integrable
Time = 0.25 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.09 \[ \int \frac {x^2 \sqrt [4]{b x^3+a x^4}}{-b+a x^4} \, dx=\int { \frac {{\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}} x^{2}}{a x^{4} - b} \,d x } \]
Timed out. \[ \int \frac {x^2 \sqrt [4]{b x^3+a x^4}}{-b+a x^4} \, dx=\text {Timed out} \]
Not integrable
Time = 0.00 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.10 \[ \int \frac {x^2 \sqrt [4]{b x^3+a x^4}}{-b+a x^4} \, dx=-\int \frac {x^2\,{\left (a\,x^4+b\,x^3\right )}^{1/4}}{b-a\,x^4} \,d x \]