Integrand size = 43, antiderivative size = 278 \[ \int \frac {b^2+a^2 x^2}{\left (-b^2+a^2 x^2\right )^3 \sqrt [3]{-b x^2+a x^3}} \, dx=-\frac {\left (-b x^2+a x^3\right )^{2/3} \left (-1190 b^4+91 a b^3 x+1503 a^2 b^2 x^2-67 a^3 b x^3-625 a^4 x^4\right )}{672 b^4 x (b-a x)^3 (b+a x)^2}-\frac {59 \arctan \left (\frac {\sqrt {3} \sqrt [3]{a} x}{\sqrt [3]{a} x+2^{2/3} \sqrt [3]{-b x^2+a x^3}}\right )}{48 \sqrt [3]{2} \sqrt {3} \sqrt [3]{a} b^4}+\frac {59 \log \left (-2 \sqrt [3]{a} x+2^{2/3} \sqrt [3]{-b x^2+a x^3}\right )}{144 \sqrt [3]{2} \sqrt [3]{a} b^4}-\frac {59 \log \left (2 a^{2/3} x^2+2^{2/3} \sqrt [3]{a} x \sqrt [3]{-b x^2+a x^3}+\sqrt [3]{2} \left (-b x^2+a x^3\right )^{2/3}\right )}{288 \sqrt [3]{2} \sqrt [3]{a} b^4} \]
-1/672*(a*x^3-b*x^2)^(2/3)*(-625*a^4*x^4-67*a^3*b*x^3+1503*a^2*b^2*x^2+91* a*b^3*x-1190*b^4)/b^4/x/(-a*x+b)^3/(a*x+b)^2-59/288*arctan(3^(1/2)*a^(1/3) *x/(a^(1/3)*x+2^(2/3)*(a*x^3-b*x^2)^(1/3)))*2^(2/3)*3^(1/2)/a^(1/3)/b^4+59 /288*ln(-2*a^(1/3)*x+2^(2/3)*(a*x^3-b*x^2)^(1/3))*2^(2/3)/a^(1/3)/b^4-59/5 76*ln(2*a^(2/3)*x^2+2^(2/3)*a^(1/3)*x*(a*x^3-b*x^2)^(1/3)+2^(1/3)*(a*x^3-b *x^2)^(2/3))*2^(2/3)/a^(1/3)/b^4
Time = 1.36 (sec) , antiderivative size = 302, normalized size of antiderivative = 1.09 \[ \int \frac {b^2+a^2 x^2}{\left (-b^2+a^2 x^2\right )^3 \sqrt [3]{-b x^2+a x^3}} \, dx=\frac {x^{2/3} \left (-\frac {6 \sqrt [3]{x} \left (1190 b^4-91 a b^3 x-1503 a^2 b^2 x^2+67 a^3 b x^3+625 a^4 x^4\right )}{(b-a x)^2 (b+a x)^2}-\frac {826\ 2^{2/3} \sqrt {3} \sqrt [3]{-b+a x} \arctan \left (\frac {\sqrt {3} \sqrt [3]{a} \sqrt [3]{x}}{\sqrt [3]{a} \sqrt [3]{x}+2^{2/3} \sqrt [3]{-b+a x}}\right )}{\sqrt [3]{a}}+\frac {826\ 2^{2/3} \sqrt [3]{-b+a x} \log \left (-2 \sqrt [3]{a} \sqrt [3]{x}+2^{2/3} \sqrt [3]{-b+a x}\right )}{\sqrt [3]{a}}-\frac {413\ 2^{2/3} \sqrt [3]{-b+a x} \log \left (2 a^{2/3} x^{2/3}+2^{2/3} \sqrt [3]{a} \sqrt [3]{x} \sqrt [3]{-b+a x}+\sqrt [3]{2} (-b+a x)^{2/3}\right )}{\sqrt [3]{a}}\right )}{4032 b^4 \sqrt [3]{x^2 (-b+a x)}} \]
(x^(2/3)*((-6*x^(1/3)*(1190*b^4 - 91*a*b^3*x - 1503*a^2*b^2*x^2 + 67*a^3*b *x^3 + 625*a^4*x^4))/((b - a*x)^2*(b + a*x)^2) - (826*2^(2/3)*Sqrt[3]*(-b + a*x)^(1/3)*ArcTan[(Sqrt[3]*a^(1/3)*x^(1/3))/(a^(1/3)*x^(1/3) + 2^(2/3)*( -b + a*x)^(1/3))])/a^(1/3) + (826*2^(2/3)*(-b + a*x)^(1/3)*Log[-2*a^(1/3)* x^(1/3) + 2^(2/3)*(-b + a*x)^(1/3)])/a^(1/3) - (413*2^(2/3)*(-b + a*x)^(1/ 3)*Log[2*a^(2/3)*x^(2/3) + 2^(2/3)*a^(1/3)*x^(1/3)*(-b + a*x)^(1/3) + 2^(1 /3)*(-b + a*x)^(2/3)])/a^(1/3)))/(4032*b^4*(x^2*(-b + a*x))^(1/3))
Time = 1.39 (sec) , antiderivative size = 381, normalized size of antiderivative = 1.37, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.163, Rules used = {2467, 25, 2003, 2035, 25, 7293, 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.
| \(\displaystyle \int \frac {a^2 x^2+b^2}{\left (a^2 x^2-b^2\right )^3 \sqrt [3]{a x^3-b x^2}} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {x^{2/3} \sqrt [3]{a x-b} \int -\frac {b^2+a^2 x^2}{x^{2/3} \sqrt [3]{a x-b} \left (b^2-a^2 x^2\right )^3}dx}{\sqrt [3]{a x^3-b x^2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {x^{2/3} \sqrt [3]{a x-b} \int \frac {b^2+a^2 x^2}{x^{2/3} \sqrt [3]{a x-b} \left (b^2-a^2 x^2\right )^3}dx}{\sqrt [3]{a x^3-b x^2}}\) |
| \(\Big \downarrow \) 2003 |
| \(\displaystyle -\frac {x^{2/3} \sqrt [3]{a x-b} \int \frac {b^2+a^2 x^2}{x^{2/3} (-b-a x)^3 (a x-b)^{10/3}}dx}{\sqrt [3]{a x^3-b x^2}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {3 x^{2/3} \sqrt [3]{a x-b} \int -\frac {b^2+a^2 x^2}{(a x-b)^{10/3} (b+a x)^3}d\sqrt [3]{x}}{\sqrt [3]{a x^3-b x^2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {3 x^{2/3} \sqrt [3]{a x-b} \int \frac {b^2+a^2 x^2}{(a x-b)^{10/3} (b+a x)^3}d\sqrt [3]{x}}{\sqrt [3]{a x^3-b x^2}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {3 x^{2/3} \sqrt [3]{a x-b} \int \left (\frac {2 b^2}{(a x-b)^{10/3} (b+a x)^3}-\frac {2 b}{(a x-b)^{10/3} (b+a x)^2}+\frac {1}{(a x-b)^{10/3} (b+a x)}\right )d\sqrt [3]{x}}{\sqrt [3]{a x^3-b x^2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {3 x^{2/3} \sqrt [3]{a x-b} \left (\frac {59 \arctan \left (\frac {\frac {2 \sqrt [3]{2} \sqrt [3]{a} \sqrt [3]{x}}{\sqrt [3]{a x-b}}+1}{\sqrt {3}}\right )}{144 \sqrt [3]{2} \sqrt {3} \sqrt [3]{a} b^4}-\frac {19 \sqrt [3]{x}}{48 b^4 \sqrt [3]{a x-b}}+\frac {1423 \sqrt [3]{x} (a x-b)^{2/3}}{2016 b^4 (a x+b)}+\frac {59 \log (a x+b)}{864 \sqrt [3]{2} \sqrt [3]{a} b^4}-\frac {59 \log \left (\sqrt [3]{2} \sqrt [3]{a} \sqrt [3]{x}-\sqrt [3]{a x-b}\right )}{288 \sqrt [3]{2} \sqrt [3]{a} b^4}+\frac {985 \sqrt [3]{x}}{672 b^3 \sqrt [3]{a x-b} (a x+b)}-\frac {\sqrt [3]{x}}{48 b^3 (a x-b)^{4/3}}-\frac {73 \sqrt [3]{x}}{336 b^2 (a x-b)^{4/3} (a x+b)}+\frac {\sqrt [3]{x}}{6 b^2 (a x-b)^{7/3}}-\frac {29 \sqrt [3]{x}}{84 b (a x-b)^{7/3} (a x+b)}+\frac {\sqrt [3]{x}}{6 (a x-b)^{7/3} (a x+b)^2}\right )}{\sqrt [3]{a x^3-b x^2}}\) |
(-3*x^(2/3)*(-b + a*x)^(1/3)*(x^(1/3)/(6*b^2*(-b + a*x)^(7/3)) - x^(1/3)/( 48*b^3*(-b + a*x)^(4/3)) - (19*x^(1/3))/(48*b^4*(-b + a*x)^(1/3)) + x^(1/3 )/(6*(-b + a*x)^(7/3)*(b + a*x)^2) - (29*x^(1/3))/(84*b*(-b + a*x)^(7/3)*( b + a*x)) - (73*x^(1/3))/(336*b^2*(-b + a*x)^(4/3)*(b + a*x)) + (985*x^(1/ 3))/(672*b^3*(-b + a*x)^(1/3)*(b + a*x)) + (1423*x^(1/3)*(-b + a*x)^(2/3)) /(2016*b^4*(b + a*x)) + (59*ArcTan[(1 + (2*2^(1/3)*a^(1/3)*x^(1/3))/(-b + a*x)^(1/3))/Sqrt[3]])/(144*2^(1/3)*Sqrt[3]*a^(1/3)*b^4) + (59*Log[b + a*x] )/(864*2^(1/3)*a^(1/3)*b^4) - (59*Log[2^(1/3)*a^(1/3)*x^(1/3) - (-b + a*x) ^(1/3)])/(288*2^(1/3)*a^(1/3)*b^4)))/(-(b*x^2) + a*x^3)^(1/3)
3.29.16.3.1 Defintions of rubi rules used
Int[(u_)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] : > Int[u*(c + d*x)^(n + p)*(a/c + (b/d)*x)^p, x] /; FreeQ[{a, b, c, d, n, p} , x] && EqQ[b*c^2 + a*d^2, 0] && (IntegerQ[p] || (GtQ[a, 0] && GtQ[c, 0] && !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]
Time = 0.76 (sec) , antiderivative size = 240, normalized size of antiderivative = 0.86
| method | result | size |
| pseudoelliptic | \(-\frac {59 \left (2^{\frac {2}{3}} \left (a^{2} x^{2}-b^{2}\right )^{2} \left (-2 \arctan \left (\frac {\sqrt {3}\, \left (2^{\frac {2}{3}} \left (x^{2} \left (a x -b \right )\right )^{\frac {1}{3}}+a^{\frac {1}{3}} x \right )}{3 a^{\frac {1}{3}} x}\right ) \sqrt {3}+\ln \left (\frac {2^{\frac {2}{3}} a^{\frac {2}{3}} x^{2}+2^{\frac {1}{3}} a^{\frac {1}{3}} \left (x^{2} \left (a x -b \right )\right )^{\frac {1}{3}} x +\left (x^{2} \left (a x -b \right )\right )^{\frac {2}{3}}}{x^{2}}\right )-2 \ln \left (\frac {-2^{\frac {1}{3}} a^{\frac {1}{3}} x +\left (x^{2} \left (a x -b \right )\right )^{\frac {1}{3}}}{x}\right )\right ) \left (x^{2} \left (a x -b \right )\right )^{\frac {1}{3}}-\frac {78 x \left (-\frac {67 a^{\frac {10}{3}} b \,x^{3}}{91}-\frac {625 a^{\frac {13}{3}} x^{4}}{91}+b^{2} \left (a^{\frac {4}{3}} b x +\frac {1503 a^{\frac {7}{3}} x^{2}}{91}-\frac {170 a^{\frac {1}{3}} b^{2}}{13}\right )\right )}{59}\right )}{576 a^{\frac {1}{3}} \left (x^{2} \left (a x -b \right )\right )^{\frac {1}{3}} \left (a x +b \right )^{2} \left (a x -b \right )^{2} b^{4}}\) | \(240\) |
-59/576/a^(1/3)*(2^(2/3)*(a^2*x^2-b^2)^2*(-2*arctan(1/3*3^(1/2)*(2^(2/3)*( x^2*(a*x-b))^(1/3)+a^(1/3)*x)/a^(1/3)/x)*3^(1/2)+ln((2^(2/3)*a^(2/3)*x^2+2 ^(1/3)*a^(1/3)*(x^2*(a*x-b))^(1/3)*x+(x^2*(a*x-b))^(2/3))/x^2)-2*ln((-2^(1 /3)*a^(1/3)*x+(x^2*(a*x-b))^(1/3))/x))*(x^2*(a*x-b))^(1/3)-78/59*x*(-67/91 *a^(10/3)*b*x^3-625/91*a^(13/3)*x^4+b^2*(a^(4/3)*b*x+1503/91*a^(7/3)*x^2-1 70/13*a^(1/3)*b^2)))/(x^2*(a*x-b))^(1/3)/(a*x+b)^2/(a*x-b)^2/b^4
Time = 0.28 (sec) , antiderivative size = 959, normalized size of antiderivative = 3.45 \[ \int \frac {b^2+a^2 x^2}{\left (-b^2+a^2 x^2\right )^3 \sqrt [3]{-b x^2+a x^3}} \, dx=\left [\frac {826 \cdot 2^{\frac {2}{3}} {\left (a^{5} x^{6} - a^{4} b x^{5} - 2 \, a^{3} b^{2} x^{4} + 2 \, a^{2} b^{3} x^{3} + a b^{4} x^{2} - b^{5} x\right )} a^{\frac {2}{3}} \log \left (-\frac {2^{\frac {1}{3}} a^{\frac {1}{3}} x - {\left (a x^{3} - b x^{2}\right )}^{\frac {1}{3}}}{x}\right ) - 413 \cdot 2^{\frac {2}{3}} {\left (a^{5} x^{6} - a^{4} b x^{5} - 2 \, a^{3} b^{2} x^{4} + 2 \, a^{2} b^{3} x^{3} + a b^{4} x^{2} - b^{5} x\right )} a^{\frac {2}{3}} \log \left (\frac {2^{\frac {2}{3}} a^{\frac {2}{3}} x^{2} + 2^{\frac {1}{3}} {\left (a x^{3} - b x^{2}\right )}^{\frac {1}{3}} a^{\frac {1}{3}} x + {\left (a x^{3} - b x^{2}\right )}^{\frac {2}{3}}}{x^{2}}\right ) + 2478 \, \sqrt {\frac {1}{6}} {\left (a^{6} x^{6} - a^{5} b x^{5} - 2 \, a^{4} b^{2} x^{4} + 2 \, a^{3} b^{3} x^{3} + a^{2} b^{4} x^{2} - a b^{5} x\right )} \sqrt {-\frac {2^{\frac {1}{3}}}{a^{\frac {2}{3}}}} \log \left (-\frac {4 \, a x^{2} - 3 \cdot 2^{\frac {2}{3}} {\left (a x^{3} - b x^{2}\right )}^{\frac {1}{3}} a^{\frac {2}{3}} x - 2 \, b x - 6 \, \sqrt {\frac {1}{6}} {\left (2^{\frac {1}{3}} a^{\frac {4}{3}} x^{2} + {\left (a x^{3} - b x^{2}\right )}^{\frac {1}{3}} a x - 2^{\frac {2}{3}} {\left (a x^{3} - b x^{2}\right )}^{\frac {2}{3}} a^{\frac {2}{3}}\right )} \sqrt {-\frac {2^{\frac {1}{3}}}{a^{\frac {2}{3}}}}}{a x^{2} + b x}\right ) - 6 \, {\left (625 \, a^{5} x^{4} + 67 \, a^{4} b x^{3} - 1503 \, a^{3} b^{2} x^{2} - 91 \, a^{2} b^{3} x + 1190 \, a b^{4}\right )} {\left (a x^{3} - b x^{2}\right )}^{\frac {2}{3}}}{4032 \, {\left (a^{6} b^{4} x^{6} - a^{5} b^{5} x^{5} - 2 \, a^{4} b^{6} x^{4} + 2 \, a^{3} b^{7} x^{3} + a^{2} b^{8} x^{2} - a b^{9} x\right )}}, \frac {826 \cdot 2^{\frac {2}{3}} {\left (a^{5} x^{6} - a^{4} b x^{5} - 2 \, a^{3} b^{2} x^{4} + 2 \, a^{2} b^{3} x^{3} + a b^{4} x^{2} - b^{5} x\right )} a^{\frac {2}{3}} \log \left (-\frac {2^{\frac {1}{3}} a^{\frac {1}{3}} x - {\left (a x^{3} - b x^{2}\right )}^{\frac {1}{3}}}{x}\right ) - 413 \cdot 2^{\frac {2}{3}} {\left (a^{5} x^{6} - a^{4} b x^{5} - 2 \, a^{3} b^{2} x^{4} + 2 \, a^{2} b^{3} x^{3} + a b^{4} x^{2} - b^{5} x\right )} a^{\frac {2}{3}} \log \left (\frac {2^{\frac {2}{3}} a^{\frac {2}{3}} x^{2} + 2^{\frac {1}{3}} {\left (a x^{3} - b x^{2}\right )}^{\frac {1}{3}} a^{\frac {1}{3}} x + {\left (a x^{3} - b x^{2}\right )}^{\frac {2}{3}}}{x^{2}}\right ) + 4956 \, \sqrt {\frac {1}{6}} {\left (a^{6} x^{6} - a^{5} b x^{5} - 2 \, a^{4} b^{2} x^{4} + 2 \, a^{3} b^{3} x^{3} + a^{2} b^{4} x^{2} - a b^{5} x\right )} \sqrt {\frac {2^{\frac {1}{3}}}{a^{\frac {2}{3}}}} \arctan \left (\frac {\sqrt {\frac {1}{6}} {\left (2^{\frac {1}{3}} a^{\frac {1}{3}} x + 2 \, {\left (a x^{3} - b x^{2}\right )}^{\frac {1}{3}}\right )} \sqrt {\frac {2^{\frac {1}{3}}}{a^{\frac {2}{3}}}}}{x}\right ) - 6 \, {\left (625 \, a^{5} x^{4} + 67 \, a^{4} b x^{3} - 1503 \, a^{3} b^{2} x^{2} - 91 \, a^{2} b^{3} x + 1190 \, a b^{4}\right )} {\left (a x^{3} - b x^{2}\right )}^{\frac {2}{3}}}{4032 \, {\left (a^{6} b^{4} x^{6} - a^{5} b^{5} x^{5} - 2 \, a^{4} b^{6} x^{4} + 2 \, a^{3} b^{7} x^{3} + a^{2} b^{8} x^{2} - a b^{9} x\right )}}\right ] \]
[1/4032*(826*2^(2/3)*(a^5*x^6 - a^4*b*x^5 - 2*a^3*b^2*x^4 + 2*a^2*b^3*x^3 + a*b^4*x^2 - b^5*x)*a^(2/3)*log(-(2^(1/3)*a^(1/3)*x - (a*x^3 - b*x^2)^(1/ 3))/x) - 413*2^(2/3)*(a^5*x^6 - a^4*b*x^5 - 2*a^3*b^2*x^4 + 2*a^2*b^3*x^3 + a*b^4*x^2 - b^5*x)*a^(2/3)*log((2^(2/3)*a^(2/3)*x^2 + 2^(1/3)*(a*x^3 - b *x^2)^(1/3)*a^(1/3)*x + (a*x^3 - b*x^2)^(2/3))/x^2) + 2478*sqrt(1/6)*(a^6* x^6 - a^5*b*x^5 - 2*a^4*b^2*x^4 + 2*a^3*b^3*x^3 + a^2*b^4*x^2 - a*b^5*x)*s qrt(-2^(1/3)/a^(2/3))*log(-(4*a*x^2 - 3*2^(2/3)*(a*x^3 - b*x^2)^(1/3)*a^(2 /3)*x - 2*b*x - 6*sqrt(1/6)*(2^(1/3)*a^(4/3)*x^2 + (a*x^3 - b*x^2)^(1/3)*a *x - 2^(2/3)*(a*x^3 - b*x^2)^(2/3)*a^(2/3))*sqrt(-2^(1/3)/a^(2/3)))/(a*x^2 + b*x)) - 6*(625*a^5*x^4 + 67*a^4*b*x^3 - 1503*a^3*b^2*x^2 - 91*a^2*b^3*x + 1190*a*b^4)*(a*x^3 - b*x^2)^(2/3))/(a^6*b^4*x^6 - a^5*b^5*x^5 - 2*a^4*b ^6*x^4 + 2*a^3*b^7*x^3 + a^2*b^8*x^2 - a*b^9*x), 1/4032*(826*2^(2/3)*(a^5* x^6 - a^4*b*x^5 - 2*a^3*b^2*x^4 + 2*a^2*b^3*x^3 + a*b^4*x^2 - b^5*x)*a^(2/ 3)*log(-(2^(1/3)*a^(1/3)*x - (a*x^3 - b*x^2)^(1/3))/x) - 413*2^(2/3)*(a^5* x^6 - a^4*b*x^5 - 2*a^3*b^2*x^4 + 2*a^2*b^3*x^3 + a*b^4*x^2 - b^5*x)*a^(2/ 3)*log((2^(2/3)*a^(2/3)*x^2 + 2^(1/3)*(a*x^3 - b*x^2)^(1/3)*a^(1/3)*x + (a *x^3 - b*x^2)^(2/3))/x^2) + 4956*sqrt(1/6)*(a^6*x^6 - a^5*b*x^5 - 2*a^4*b^ 2*x^4 + 2*a^3*b^3*x^3 + a^2*b^4*x^2 - a*b^5*x)*sqrt(2^(1/3)/a^(2/3))*arcta n(sqrt(1/6)*(2^(1/3)*a^(1/3)*x + 2*(a*x^3 - b*x^2)^(1/3))*sqrt(2^(1/3)/a^( 2/3))/x) - 6*(625*a^5*x^4 + 67*a^4*b*x^3 - 1503*a^3*b^2*x^2 - 91*a^2*b^...
\[ \int \frac {b^2+a^2 x^2}{\left (-b^2+a^2 x^2\right )^3 \sqrt [3]{-b x^2+a x^3}} \, dx=\int \frac {a^{2} x^{2} + b^{2}}{\sqrt [3]{x^{2} \left (a x - b\right )} \left (a x - b\right )^{3} \left (a x + b\right )^{3}}\, dx \]
\[ \int \frac {b^2+a^2 x^2}{\left (-b^2+a^2 x^2\right )^3 \sqrt [3]{-b x^2+a x^3}} \, dx=\int { \frac {a^{2} x^{2} + b^{2}}{{\left (a^{2} x^{2} - b^{2}\right )}^{3} {\left (a x^{3} - b x^{2}\right )}^{\frac {1}{3}}} \,d x } \]
Time = 10.68 (sec) , antiderivative size = 210, normalized size of antiderivative = 0.76 \[ \int \frac {b^2+a^2 x^2}{\left (-b^2+a^2 x^2\right )^3 \sqrt [3]{-b x^2+a x^3}} \, dx=\frac {59 \, \sqrt {3} 2^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} 2^{\frac {2}{3}} {\left (2^{\frac {1}{3}} a^{\frac {1}{3}} + 2 \, {\left (a - \frac {b}{x}\right )}^{\frac {1}{3}}\right )}}{6 \, a^{\frac {1}{3}}}\right )}{288 \, a^{\frac {1}{3}} b^{4}} - \frac {59 \cdot 2^{\frac {2}{3}} \log \left (2^{\frac {2}{3}} a^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (a - \frac {b}{x}\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + {\left (a - \frac {b}{x}\right )}^{\frac {2}{3}}\right )}{576 \, a^{\frac {1}{3}} b^{4}} + \frac {59 \cdot 2^{\frac {2}{3}} \log \left ({\left | -2^{\frac {1}{3}} a^{\frac {1}{3}} + {\left (a - \frac {b}{x}\right )}^{\frac {1}{3}} \right |}\right )}{288 \, a^{\frac {1}{3}} b^{4}} - \frac {13 \, {\left (a - \frac {b}{x}\right )}^{\frac {5}{3}} - 23 \, {\left (a - \frac {b}{x}\right )}^{\frac {2}{3}} a}{48 \, {\left (a + \frac {b}{x}\right )}^{2} b^{4}} - \frac {3 \, {\left (112 \, {\left (a - \frac {b}{x}\right )}^{2} - 35 \, {\left (a - \frac {b}{x}\right )} a + 8 \, a^{2}\right )}}{224 \, {\left (a - \frac {b}{x}\right )}^{\frac {7}{3}} b^{4}} \]
59/288*sqrt(3)*2^(2/3)*arctan(1/6*sqrt(3)*2^(2/3)*(2^(1/3)*a^(1/3) + 2*(a - b/x)^(1/3))/a^(1/3))/(a^(1/3)*b^4) - 59/576*2^(2/3)*log(2^(2/3)*a^(2/3) + 2^(1/3)*(a - b/x)^(1/3)*a^(1/3) + (a - b/x)^(2/3))/(a^(1/3)*b^4) + 59/28 8*2^(2/3)*log(abs(-2^(1/3)*a^(1/3) + (a - b/x)^(1/3)))/(a^(1/3)*b^4) - 1/4 8*(13*(a - b/x)^(5/3) - 23*(a - b/x)^(2/3)*a)/((a + b/x)^2*b^4) - 3/224*(1 12*(a - b/x)^2 - 35*(a - b/x)*a + 8*a^2)/((a - b/x)^(7/3)*b^4)
Timed out. \[ \int \frac {b^2+a^2 x^2}{\left (-b^2+a^2 x^2\right )^3 \sqrt [3]{-b x^2+a x^3}} \, dx=\int -\frac {a^2\,x^2+b^2}{{\left (b^2-a^2\,x^2\right )}^3\,{\left (a\,x^3-b\,x^2\right )}^{1/3}} \,d x \]