Integrand size = 30, antiderivative size = 299 \[ \int \frac {\sqrt [3]{-b^2 x^2+a^3 x^3}}{b+a x^2} \, dx=-\sqrt {3} \arctan \left (\frac {\sqrt {3} a x}{a x+2 \sqrt [3]{-b^2 x^2+a^3 x^3}}\right )-\log \left (-a x+\sqrt [3]{-b^2 x^2+a^3 x^3}\right )+\frac {1}{2} \log \left (a^2 x^2+a x \sqrt [3]{-b^2 x^2+a^3 x^3}+\left (-b^2 x^2+a^3 x^3\right )^{2/3}\right )+\frac {1}{2} \text {RootSum}\left [a^6+a b^3-2 a^3 \text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-a^5 \log (x)-b^3 \log (x)+a^5 \log \left (\sqrt [3]{-b^2 x^2+a^3 x^3}-x \text {$\#$1}\right )+b^3 \log \left (\sqrt [3]{-b^2 x^2+a^3 x^3}-x \text {$\#$1}\right )+a^2 \log (x) \text {$\#$1}^3-a^2 \log \left (\sqrt [3]{-b^2 x^2+a^3 x^3}-x \text {$\#$1}\right ) \text {$\#$1}^3}{a^3 \text {$\#$1}^2-\text {$\#$1}^5}\&\right ] \]
Time = 0.41 (sec) , antiderivative size = 332, normalized size of antiderivative = 1.11 \[ \int \frac {\sqrt [3]{-b^2 x^2+a^3 x^3}}{b+a x^2} \, dx=-\frac {x^{4/3} \left (-b^2+a^3 x\right )^{2/3} \left (6 \sqrt {3} \arctan \left (\frac {\sqrt {3} a \sqrt [3]{x}}{a \sqrt [3]{x}+2 \sqrt [3]{-b^2+a^3 x}}\right )+6 \log \left (-a \sqrt [3]{x}+\sqrt [3]{-b^2+a^3 x}\right )-3 \log \left (a^2 x^{2/3}+a \sqrt [3]{x} \sqrt [3]{-b^2+a^3 x}+\left (-b^2+a^3 x\right )^{2/3}\right )+\text {RootSum}\left [a^6+a b^3-2 a^3 \text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-a^5 \log (x)-b^3 \log (x)+3 a^5 \log \left (\sqrt [3]{-b^2+a^3 x}-\sqrt [3]{x} \text {$\#$1}\right )+3 b^3 \log \left (\sqrt [3]{-b^2+a^3 x}-\sqrt [3]{x} \text {$\#$1}\right )+a^2 \log (x) \text {$\#$1}^3-3 a^2 \log \left (\sqrt [3]{-b^2+a^3 x}-\sqrt [3]{x} \text {$\#$1}\right ) \text {$\#$1}^3}{-a^3 \text {$\#$1}^2+\text {$\#$1}^5}\&\right ]\right )}{6 \left (x^2 \left (-b^2+a^3 x\right )\right )^{2/3}} \]
-1/6*(x^(4/3)*(-b^2 + a^3*x)^(2/3)*(6*Sqrt[3]*ArcTan[(Sqrt[3]*a*x^(1/3))/( a*x^(1/3) + 2*(-b^2 + a^3*x)^(1/3))] + 6*Log[-(a*x^(1/3)) + (-b^2 + a^3*x) ^(1/3)] - 3*Log[a^2*x^(2/3) + a*x^(1/3)*(-b^2 + a^3*x)^(1/3) + (-b^2 + a^3 *x)^(2/3)] + RootSum[a^6 + a*b^3 - 2*a^3*#1^3 + #1^6 & , (-(a^5*Log[x]) - b^3*Log[x] + 3*a^5*Log[(-b^2 + a^3*x)^(1/3) - x^(1/3)*#1] + 3*b^3*Log[(-b^ 2 + a^3*x)^(1/3) - x^(1/3)*#1] + a^2*Log[x]*#1^3 - 3*a^2*Log[(-b^2 + a^3*x )^(1/3) - x^(1/3)*#1]*#1^3)/(-(a^3*#1^2) + #1^5) & ]))/(x^2*(-b^2 + a^3*x) )^(2/3)
Leaf count is larger than twice the leaf count of optimal. \(1051\) vs. \(2(299)=598\).
Time = 1.77 (sec) , antiderivative size = 1051, normalized size of antiderivative = 3.52, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.233, Rules used = {2467, 609, 27, 71, 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.
| \(\displaystyle \int \frac {\sqrt [3]{a^3 x^3-b^2 x^2}}{a x^2+b} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt [3]{a^3 x^3-b^2 x^2} \int \frac {x^{2/3} \sqrt [3]{a^3 x-b^2}}{a x^2+b}dx}{x^{2/3} \sqrt [3]{a^3 x-b^2}}\) |
| \(\Big \downarrow \) 609 |
| \(\displaystyle \frac {\sqrt [3]{a^3 x^3-b^2 x^2} \left (a^2 \int \frac {1}{\sqrt [3]{x} \left (a^3 x-b^2\right )^{2/3}}dx-\frac {\int \frac {a b \left (a^2+b x\right )}{\sqrt [3]{x} \left (a^3 x-b^2\right )^{2/3} \left (a x^2+b\right )}dx}{a}\right )}{x^{2/3} \sqrt [3]{a^3 x-b^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt [3]{a^3 x^3-b^2 x^2} \left (a^2 \int \frac {1}{\sqrt [3]{x} \left (a^3 x-b^2\right )^{2/3}}dx-b \int \frac {a^2+b x}{\sqrt [3]{x} \left (a^3 x-b^2\right )^{2/3} \left (a x^2+b\right )}dx\right )}{x^{2/3} \sqrt [3]{a^3 x-b^2}}\) |
| \(\Big \downarrow \) 71 |
| \(\displaystyle \frac {\sqrt [3]{a^3 x^3-b^2 x^2} \left (a^2 \left (-\frac {\sqrt {3} \arctan \left (\frac {2 a \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{a^3 x-b^2}}+\frac {1}{\sqrt {3}}\right )}{a^2}-\frac {\log \left (a^3 x-b^2\right )}{2 a^2}-\frac {3 \log \left (\frac {a \sqrt [3]{x}}{\sqrt [3]{a^3 x-b^2}}-1\right )}{2 a^2}\right )-b \int \frac {a^2+b x}{\sqrt [3]{x} \left (a^3 x-b^2\right )^{2/3} \left (a x^2+b\right )}dx\right )}{x^{2/3} \sqrt [3]{a^3 x-b^2}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle \frac {\sqrt [3]{a^3 x^3-b^2 x^2} \left (a^2 \left (-\frac {\sqrt {3} \arctan \left (\frac {2 a \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{a^3 x-b^2}}+\frac {1}{\sqrt {3}}\right )}{a^2}-\frac {\log \left (a^3 x-b^2\right )}{2 a^2}-\frac {3 \log \left (\frac {a \sqrt [3]{x}}{\sqrt [3]{a^3 x-b^2}}-1\right )}{2 a^2}\right )-3 b \int \frac {\sqrt [3]{x} \left (a^2+b x\right )}{\left (a^3 x-b^2\right )^{2/3} \left (a x^2+b\right )}d\sqrt [3]{x}\right )}{x^{2/3} \sqrt [3]{a^3 x-b^2}}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \frac {\sqrt [3]{a^3 x^3-b^2 x^2} \left (a^2 \left (-\frac {\sqrt {3} \arctan \left (\frac {2 a \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{a^3 x-b^2}}+\frac {1}{\sqrt {3}}\right )}{a^2}-\frac {\log \left (a^3 x-b^2\right )}{2 a^2}-\frac {3 \log \left (\frac {a \sqrt [3]{x}}{\sqrt [3]{a^3 x-b^2}}-1\right )}{2 a^2}\right )-3 b \int \left (\frac {\sqrt [3]{x} a^2}{\left (a^3 x-b^2\right )^{2/3} \left (a x^2+b\right )}+\frac {b x^{4/3}}{\left (a^3 x-b^2\right )^{2/3} \left (a x^2+b\right )}\right )d\sqrt [3]{x}\right )}{x^{2/3} \sqrt [3]{a^3 x-b^2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt [3]{a^3 x^3-b^2 x^2} \left (a^2 \left (-\frac {\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{x} a}{\sqrt {3} \sqrt [3]{a^3 x-b^2}}+\frac {1}{\sqrt {3}}\right )}{a^2}-\frac {\log \left (a^3 x-b^2\right )}{2 a^2}-\frac {3 \log \left (\frac {a \sqrt [3]{x}}{\sqrt [3]{a^3 x-b^2}}-1\right )}{2 a^2}\right )-3 b \left (-\frac {\arctan \left (\frac {\frac {2 \sqrt [3]{a^3-\sqrt {-a} b^{3/2}} \sqrt [3]{x}}{\sqrt [3]{a^3 x-b^2}}+1}{\sqrt {3}}\right ) a^2}{2 \sqrt {3} b \left (a^3-\sqrt {-a} b^{3/2}\right )^{2/3}}-\frac {\arctan \left (\frac {\frac {2 \sqrt [3]{a^3+\sqrt {-a} b^{3/2}} \sqrt [3]{x}}{\sqrt [3]{a^3 x-b^2}}+1}{\sqrt {3}}\right ) a^2}{2 \sqrt {3} b \left (a^3+\sqrt {-a} b^{3/2}\right )^{2/3}}+\frac {\log \left (\sqrt {-a} \sqrt {b}-a x\right ) a^2}{12 b \left (a^3+\sqrt {-a} b^{3/2}\right )^{2/3}}+\frac {\log \left (a x+\sqrt {-a} \sqrt {b}\right ) a^2}{12 b \left (a^3-\sqrt {-a} b^{3/2}\right )^{2/3}}-\frac {\log \left (\sqrt [3]{a^3-\sqrt {-a} b^{3/2}} \sqrt [3]{x}-\sqrt [3]{a^3 x-b^2}\right ) a^2}{4 b \left (a^3-\sqrt {-a} b^{3/2}\right )^{2/3}}-\frac {\log \left (\sqrt [3]{a^3+\sqrt {-a} b^{3/2}} \sqrt [3]{x}-\sqrt [3]{a^3 x-b^2}\right ) a^2}{4 b \left (a^3+\sqrt {-a} b^{3/2}\right )^{2/3}}-\frac {\sqrt {b} \arctan \left (\frac {\frac {2 \sqrt [3]{a^3-\sqrt {-a} b^{3/2}} \sqrt [3]{x}}{\sqrt [3]{a^3 x-b^2}}+1}{\sqrt {3}}\right )}{2 \sqrt {3} \sqrt {-a} \left (a^3-\sqrt {-a} b^{3/2}\right )^{2/3}}+\frac {\sqrt {b} \arctan \left (\frac {\frac {2 \sqrt [3]{a^3+\sqrt {-a} b^{3/2}} \sqrt [3]{x}}{\sqrt [3]{a^3 x-b^2}}+1}{\sqrt {3}}\right )}{2 \sqrt {3} \sqrt {-a} \left (a^3+\sqrt {-a} b^{3/2}\right )^{2/3}}-\frac {\sqrt {b} \log \left (a x-\sqrt {-a} \sqrt {b}\right )}{12 \sqrt {-a} \left (a^3+\sqrt {-a} b^{3/2}\right )^{2/3}}+\frac {\sqrt {b} \log \left (a x+\sqrt {-a} \sqrt {b}\right )}{12 \sqrt {-a} \left (a^3-\sqrt {-a} b^{3/2}\right )^{2/3}}-\frac {\sqrt {b} \log \left (\sqrt [3]{a^3-\sqrt {-a} b^{3/2}} \sqrt [3]{x}-\sqrt [3]{a^3 x-b^2}\right )}{4 \sqrt {-a} \left (a^3-\sqrt {-a} b^{3/2}\right )^{2/3}}+\frac {\sqrt {b} \log \left (\sqrt [3]{a^3+\sqrt {-a} b^{3/2}} \sqrt [3]{x}-\sqrt [3]{a^3 x-b^2}\right )}{4 \sqrt {-a} \left (a^3+\sqrt {-a} b^{3/2}\right )^{2/3}}\right )\right )}{x^{2/3} \sqrt [3]{a^3 x-b^2}}\) |
((-(b^2*x^2) + a^3*x^3)^(1/3)*(a^2*(-((Sqrt[3]*ArcTan[1/Sqrt[3] + (2*a*x^( 1/3))/(Sqrt[3]*(-b^2 + a^3*x)^(1/3))])/a^2) - Log[-b^2 + a^3*x]/(2*a^2) - (3*Log[-1 + (a*x^(1/3))/(-b^2 + a^3*x)^(1/3)])/(2*a^2)) - 3*b*(-1/2*(a^2*A rcTan[(1 + (2*(a^3 - Sqrt[-a]*b^(3/2))^(1/3)*x^(1/3))/(-b^2 + a^3*x)^(1/3) )/Sqrt[3]])/(Sqrt[3]*b*(a^3 - Sqrt[-a]*b^(3/2))^(2/3)) - (Sqrt[b]*ArcTan[( 1 + (2*(a^3 - Sqrt[-a]*b^(3/2))^(1/3)*x^(1/3))/(-b^2 + a^3*x)^(1/3))/Sqrt[ 3]])/(2*Sqrt[3]*Sqrt[-a]*(a^3 - Sqrt[-a]*b^(3/2))^(2/3)) - (a^2*ArcTan[(1 + (2*(a^3 + Sqrt[-a]*b^(3/2))^(1/3)*x^(1/3))/(-b^2 + a^3*x)^(1/3))/Sqrt[3] ])/(2*Sqrt[3]*b*(a^3 + Sqrt[-a]*b^(3/2))^(2/3)) + (Sqrt[b]*ArcTan[(1 + (2* (a^3 + Sqrt[-a]*b^(3/2))^(1/3)*x^(1/3))/(-b^2 + a^3*x)^(1/3))/Sqrt[3]])/(2 *Sqrt[3]*Sqrt[-a]*(a^3 + Sqrt[-a]*b^(3/2))^(2/3)) + (a^2*Log[Sqrt[-a]*Sqrt [b] - a*x])/(12*b*(a^3 + Sqrt[-a]*b^(3/2))^(2/3)) - (Sqrt[b]*Log[-(Sqrt[-a ]*Sqrt[b]) + a*x])/(12*Sqrt[-a]*(a^3 + Sqrt[-a]*b^(3/2))^(2/3)) + (a^2*Log [Sqrt[-a]*Sqrt[b] + a*x])/(12*b*(a^3 - Sqrt[-a]*b^(3/2))^(2/3)) + (Sqrt[b] *Log[Sqrt[-a]*Sqrt[b] + a*x])/(12*Sqrt[-a]*(a^3 - Sqrt[-a]*b^(3/2))^(2/3)) - (a^2*Log[(a^3 - Sqrt[-a]*b^(3/2))^(1/3)*x^(1/3) - (-b^2 + a^3*x)^(1/3)] )/(4*b*(a^3 - Sqrt[-a]*b^(3/2))^(2/3)) - (Sqrt[b]*Log[(a^3 - Sqrt[-a]*b^(3 /2))^(1/3)*x^(1/3) - (-b^2 + a^3*x)^(1/3)])/(4*Sqrt[-a]*(a^3 - Sqrt[-a]*b^ (3/2))^(2/3)) - (a^2*Log[(a^3 + Sqrt[-a]*b^(3/2))^(1/3)*x^(1/3) - (-b^2 + a^3*x)^(1/3)])/(4*b*(a^3 + Sqrt[-a]*b^(3/2))^(2/3)) + (Sqrt[b]*Log[(a^3...
3.29.57.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/(((a_.) + (b_.)*(x_))^(1/3)*((c_.) + (d_.)*(x_))^(2/3)), x_Symbol] :> With[{q = Rt[d/b, 3]}, Simp[(-Sqrt[3])*(q/d)*ArcTan[2*q*((a + b*x)^(1/3)/( Sqrt[3]*(c + d*x)^(1/3))) + 1/Sqrt[3]], x] + (-Simp[3*(q/(2*d))*Log[q*((a + b*x)^(1/3)/(c + d*x)^(1/3)) - 1], x] - Simp[(q/(2*d))*Log[c + d*x], x])] / ; FreeQ[{a, b, c, d}, x] && PosQ[d/b]
Int[(((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_))/((a_) + (b_.)*(x_)^2), x_S ymbol] :> Simp[d*(e/b) Int[(e*x)^(m - 1)*(c + d*x)^(n - 1), x], x] - Simp [e/b Int[(e*x)^(m - 1)*(c + d*x)^(n - 1)*((a*d - b*c*x)/(a + b*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && LtQ[0, n, 1] && LtQ[0, m, 1] && !Integ erQ[m] && !IntegerQ[n]
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 = 0.68 (sec) , antiderivative size = 206, normalized size of antiderivative = 0.69
| method | result | size |
| pseudoelliptic | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}-2 a^{3} \textit {\_Z}^{3}+a^{6}+a \,b^{3}\right )}{\sum }\frac {\left (-\textit {\_R}^{3} a^{2}+a^{5}+b^{3}\right ) \ln \left (\frac {-\textit {\_R} x +\left (x^{2} \left (a^{3} x -b^{2}\right )\right )^{\frac {1}{3}}}{x}\right )}{-\textit {\_R}^{5}+\textit {\_R}^{2} a^{3}}\right )}{2}-\ln \left (\frac {-a x +\left (x^{2} \left (a^{3} x -b^{2}\right )\right )^{\frac {1}{3}}}{x}\right )+\frac {\ln \left (\frac {a^{2} x^{2}+a \left (x^{2} \left (a^{3} x -b^{2}\right )\right )^{\frac {1}{3}} x +\left (x^{2} \left (a^{3} x -b^{2}\right )\right )^{\frac {2}{3}}}{x^{2}}\right )}{2}+\sqrt {3}\, \arctan \left (\frac {\left (a x +2 \left (x^{2} \left (a^{3} x -b^{2}\right )\right )^{\frac {1}{3}}\right ) \sqrt {3}}{3 a x}\right )\) | \(206\) |
1/2*sum((-_R^3*a^2+a^5+b^3)*ln((-_R*x+(x^2*(a^3*x-b^2))^(1/3))/x)/(-_R^5+_ R^2*a^3),_R=RootOf(_Z^6-2*_Z^3*a^3+a^6+a*b^3))-ln((-a*x+(x^2*(a^3*x-b^2))^ (1/3))/x)+1/2*ln((a^2*x^2+a*(x^2*(a^3*x-b^2))^(1/3)*x+(x^2*(a^3*x-b^2))^(2 /3))/x^2)+3^(1/2)*arctan(1/3*(a*x+2*(x^2*(a^3*x-b^2))^(1/3))*3^(1/2)/a/x)
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.28 (sec) , antiderivative size = 554, normalized size of antiderivative = 1.85 \[ \int \frac {\sqrt [3]{-b^2 x^2+a^3 x^3}}{b+a x^2} \, dx=-\frac {1}{4} \, {\left (\sqrt {-3} + 1\right )} {\left (\sqrt {-\frac {b^{3}}{a^{5}}} + 1\right )}^{\frac {1}{3}} \log \left (\frac {{\left (\sqrt {-3} a x + a x\right )} {\left (\sqrt {-\frac {b^{3}}{a^{5}}} + 1\right )}^{\frac {1}{3}} + 2 \, {\left (a^{3} x^{3} - b^{2} x^{2}\right )}^{\frac {1}{3}}}{x}\right ) + \frac {1}{4} \, {\left (\sqrt {-3} - 1\right )} {\left (\sqrt {-\frac {b^{3}}{a^{5}}} + 1\right )}^{\frac {1}{3}} \log \left (-\frac {{\left (\sqrt {-3} a x - a x\right )} {\left (\sqrt {-\frac {b^{3}}{a^{5}}} + 1\right )}^{\frac {1}{3}} - 2 \, {\left (a^{3} x^{3} - b^{2} x^{2}\right )}^{\frac {1}{3}}}{x}\right ) - \frac {1}{4} \, {\left (\sqrt {-3} + 1\right )} {\left (-\sqrt {-\frac {b^{3}}{a^{5}}} + 1\right )}^{\frac {1}{3}} \log \left (\frac {{\left (\sqrt {-3} a x + a x\right )} {\left (-\sqrt {-\frac {b^{3}}{a^{5}}} + 1\right )}^{\frac {1}{3}} + 2 \, {\left (a^{3} x^{3} - b^{2} x^{2}\right )}^{\frac {1}{3}}}{x}\right ) + \frac {1}{4} \, {\left (\sqrt {-3} - 1\right )} {\left (-\sqrt {-\frac {b^{3}}{a^{5}}} + 1\right )}^{\frac {1}{3}} \log \left (-\frac {{\left (\sqrt {-3} a x - a x\right )} {\left (-\sqrt {-\frac {b^{3}}{a^{5}}} + 1\right )}^{\frac {1}{3}} - 2 \, {\left (a^{3} x^{3} - b^{2} x^{2}\right )}^{\frac {1}{3}}}{x}\right ) + \sqrt {3} \arctan \left (\frac {\sqrt {3} a x + 2 \, \sqrt {3} {\left (a^{3} x^{3} - b^{2} x^{2}\right )}^{\frac {1}{3}}}{3 \, a x}\right ) + \frac {1}{2} \, {\left (\sqrt {-\frac {b^{3}}{a^{5}}} + 1\right )}^{\frac {1}{3}} \log \left (-\frac {a x {\left (\sqrt {-\frac {b^{3}}{a^{5}}} + 1\right )}^{\frac {1}{3}} - {\left (a^{3} x^{3} - b^{2} x^{2}\right )}^{\frac {1}{3}}}{x}\right ) + \frac {1}{2} \, {\left (-\sqrt {-\frac {b^{3}}{a^{5}}} + 1\right )}^{\frac {1}{3}} \log \left (-\frac {a x {\left (-\sqrt {-\frac {b^{3}}{a^{5}}} + 1\right )}^{\frac {1}{3}} - {\left (a^{3} x^{3} - b^{2} x^{2}\right )}^{\frac {1}{3}}}{x}\right ) - \log \left (-\frac {a x - {\left (a^{3} x^{3} - b^{2} x^{2}\right )}^{\frac {1}{3}}}{x}\right ) + \frac {1}{2} \, \log \left (\frac {a^{2} x^{2} + {\left (a^{3} x^{3} - b^{2} x^{2}\right )}^{\frac {1}{3}} a x + {\left (a^{3} x^{3} - b^{2} x^{2}\right )}^{\frac {2}{3}}}{x^{2}}\right ) \]
-1/4*(sqrt(-3) + 1)*(sqrt(-b^3/a^5) + 1)^(1/3)*log(((sqrt(-3)*a*x + a*x)*( sqrt(-b^3/a^5) + 1)^(1/3) + 2*(a^3*x^3 - b^2*x^2)^(1/3))/x) + 1/4*(sqrt(-3 ) - 1)*(sqrt(-b^3/a^5) + 1)^(1/3)*log(-((sqrt(-3)*a*x - a*x)*(sqrt(-b^3/a^ 5) + 1)^(1/3) - 2*(a^3*x^3 - b^2*x^2)^(1/3))/x) - 1/4*(sqrt(-3) + 1)*(-sqr t(-b^3/a^5) + 1)^(1/3)*log(((sqrt(-3)*a*x + a*x)*(-sqrt(-b^3/a^5) + 1)^(1/ 3) + 2*(a^3*x^3 - b^2*x^2)^(1/3))/x) + 1/4*(sqrt(-3) - 1)*(-sqrt(-b^3/a^5) + 1)^(1/3)*log(-((sqrt(-3)*a*x - a*x)*(-sqrt(-b^3/a^5) + 1)^(1/3) - 2*(a^ 3*x^3 - b^2*x^2)^(1/3))/x) + sqrt(3)*arctan(1/3*(sqrt(3)*a*x + 2*sqrt(3)*( a^3*x^3 - b^2*x^2)^(1/3))/(a*x)) + 1/2*(sqrt(-b^3/a^5) + 1)^(1/3)*log(-(a* x*(sqrt(-b^3/a^5) + 1)^(1/3) - (a^3*x^3 - b^2*x^2)^(1/3))/x) + 1/2*(-sqrt( -b^3/a^5) + 1)^(1/3)*log(-(a*x*(-sqrt(-b^3/a^5) + 1)^(1/3) - (a^3*x^3 - b^ 2*x^2)^(1/3))/x) - log(-(a*x - (a^3*x^3 - b^2*x^2)^(1/3))/x) + 1/2*log((a^ 2*x^2 + (a^3*x^3 - b^2*x^2)^(1/3)*a*x + (a^3*x^3 - b^2*x^2)^(2/3))/x^2)
Not integrable
Time = 0.59 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.07 \[ \int \frac {\sqrt [3]{-b^2 x^2+a^3 x^3}}{b+a x^2} \, dx=\int \frac {\sqrt [3]{x^{2} \left (a^{3} x - b^{2}\right )}}{a x^{2} + b}\, dx \]
Not integrable
Time = 0.25 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.10 \[ \int \frac {\sqrt [3]{-b^2 x^2+a^3 x^3}}{b+a x^2} \, dx=\int { \frac {{\left (a^{3} x^{3} - b^{2} x^{2}\right )}^{\frac {1}{3}}}{a x^{2} + b} \,d x } \]
Not integrable
Time = 0.48 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.10 \[ \int \frac {\sqrt [3]{-b^2 x^2+a^3 x^3}}{b+a x^2} \, dx=\int { \frac {{\left (a^{3} x^{3} - b^{2} x^{2}\right )}^{\frac {1}{3}}}{a x^{2} + b} \,d x } \]
Not integrable
Time = 6.81 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.10 \[ \int \frac {\sqrt [3]{-b^2 x^2+a^3 x^3}}{b+a x^2} \, dx=\int \frac {{\left (a^3\,x^3-b^2\,x^2\right )}^{1/3}}{a\,x^2+b} \,d x \]