Integrand size = 34, antiderivative size = 433 \[ \int \frac {-b+a x}{(b+a x) \sqrt [3]{b^2 x^2+a^3 x^3}} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} a x}{a x+2 \sqrt [3]{b^2 x^2+a^3 x^3}}\right )}{a}+\frac {\sqrt {6 \left (-1+i \sqrt {3}\right )} \arctan \left (\frac {\sqrt {3} \sqrt [3]{a} \sqrt [3]{a^2-b} x}{\sqrt [3]{a} \sqrt [3]{a^2-b} x-2 \sqrt [3]{-1} \sqrt [3]{b^2 x^2+a^3 x^3}}\right )}{\sqrt [3]{a} \sqrt [3]{a^2-b}}-\frac {\log \left (-a x+\sqrt [3]{b^2 x^2+a^3 x^3}\right )}{a}-\frac {i \left (-i+\sqrt {3}\right ) \log \left (\sqrt [3]{a} \sqrt [3]{a^2-b} x+\sqrt [3]{-1} \sqrt [3]{b^2 x^2+a^3 x^3}\right )}{\sqrt [3]{a} \sqrt [3]{a^2-b}}+\frac {\log \left (a^2 x^2+a x \sqrt [3]{b^2 x^2+a^3 x^3}+\left (b^2 x^2+a^3 x^3\right )^{2/3}\right )}{2 a}+\frac {\left (1+i \sqrt {3}\right ) \log \left (a^{2/3} \left (a^2-b\right )^{2/3} x^2-\sqrt [3]{-1} \sqrt [3]{a} \sqrt [3]{a^2-b} x \sqrt [3]{b^2 x^2+a^3 x^3}+(-1)^{2/3} \left (b^2 x^2+a^3 x^3\right )^{2/3}\right )}{2 \sqrt [3]{a} \sqrt [3]{a^2-b}} \]
3^(1/2)*arctan(3^(1/2)*a*x/(a*x+2*(a^3*x^3+b^2*x^2)^(1/3)))/a+(-6+6*I*3^(1 /2))^(1/2)*arctan(3^(1/2)*a^(1/3)*(a^2-b)^(1/3)*x/(a^(1/3)*(a^2-b)^(1/3)*x -2*(-1)^(1/3)*(a^3*x^3+b^2*x^2)^(1/3)))/a^(1/3)/(a^2-b)^(1/3)-ln(-a*x+(a^3 *x^3+b^2*x^2)^(1/3))/a-I*(-I+3^(1/2))*ln(a^(1/3)*(a^2-b)^(1/3)*x+(-1)^(1/3 )*(a^3*x^3+b^2*x^2)^(1/3))/a^(1/3)/(a^2-b)^(1/3)+1/2*ln(a^2*x^2+a*x*(a^3*x ^3+b^2*x^2)^(1/3)+(a^3*x^3+b^2*x^2)^(2/3))/a+1/2*(1+I*3^(1/2))*ln(a^(2/3)* (a^2-b)^(2/3)*x^2-(-1)^(1/3)*a^(1/3)*(a^2-b)^(1/3)*x*(a^3*x^3+b^2*x^2)^(1/ 3)+(-1)^(2/3)*(a^3*x^3+b^2*x^2)^(2/3))/a^(1/3)/(a^2-b)^(1/3)
Time = 2.22 (sec) , antiderivative size = 622, normalized size of antiderivative = 1.44 \[ \int \frac {-b+a x}{(b+a x) \sqrt [3]{b^2 x^2+a^3 x^3}} \, dx=\frac {x^{2/3} \sqrt [3]{b^2+a^3 x} \left (2 \sqrt {3} \sqrt [3]{a^2-b} \arctan \left (\frac {\sqrt {3} a \sqrt [3]{x}}{a \sqrt [3]{x}+2 \sqrt [3]{b^2+a^3 x}}\right )+2 \sqrt {-6+6 i \sqrt {3}} a^{2/3} \arctan \left (\frac {3 \sqrt [3]{a} \sqrt [3]{a^2-b} \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{a} \sqrt [3]{a^2-b} \sqrt [3]{x}-\left (3 i+\sqrt {3}\right ) \sqrt [3]{b^2+a^3 x}}\right )-2 \sqrt [3]{a^2-b} \log \left (a \left (a \sqrt [3]{x}-\sqrt [3]{b^2+a^3 x}\right )\right )-2 a^{2/3} \log \left (2 \sqrt [3]{a} \sqrt [3]{a^2-b} \sqrt [3]{x}+\left (1+i \sqrt {3}\right ) \sqrt [3]{b^2+a^3 x}\right )-2 i \sqrt {3} a^{2/3} \log \left (2 \sqrt [3]{a} \sqrt [3]{a^2-b} \sqrt [3]{x}+\left (1+i \sqrt {3}\right ) \sqrt [3]{b^2+a^3 x}\right )+a^{2/3} \log \left (\left (-\sqrt [3]{a} \sqrt [3]{a^2-b} \sqrt [3]{x}+\sqrt [3]{b^2+a^3 x}\right ) \left (2 i \sqrt [3]{a} \sqrt [3]{a^2-b} \sqrt [3]{x}+\left (i+\sqrt {3}\right ) \sqrt [3]{b^2+a^3 x}\right )\right )+i \sqrt {3} a^{2/3} \log \left (\left (-\sqrt [3]{a} \sqrt [3]{a^2-b} \sqrt [3]{x}+\sqrt [3]{b^2+a^3 x}\right ) \left (2 i \sqrt [3]{a} \sqrt [3]{a^2-b} \sqrt [3]{x}+\left (i+\sqrt {3}\right ) \sqrt [3]{b^2+a^3 x}\right )\right )+\sqrt [3]{a^2-b} \log \left (a^2 x^{2/3}+a \sqrt [3]{x} \sqrt [3]{b^2+a^3 x}+\left (b^2+a^3 x\right )^{2/3}\right )\right )}{2 a \sqrt [3]{a^2-b} \sqrt [3]{x^2 \left (b^2+a^3 x\right )}} \]
(x^(2/3)*(b^2 + a^3*x)^(1/3)*(2*Sqrt[3]*(a^2 - b)^(1/3)*ArcTan[(Sqrt[3]*a* x^(1/3))/(a*x^(1/3) + 2*(b^2 + a^3*x)^(1/3))] + 2*Sqrt[-6 + (6*I)*Sqrt[3]] *a^(2/3)*ArcTan[(3*a^(1/3)*(a^2 - b)^(1/3)*x^(1/3))/(Sqrt[3]*a^(1/3)*(a^2 - b)^(1/3)*x^(1/3) - (3*I + Sqrt[3])*(b^2 + a^3*x)^(1/3))] - 2*(a^2 - b)^( 1/3)*Log[a*(a*x^(1/3) - (b^2 + a^3*x)^(1/3))] - 2*a^(2/3)*Log[2*a^(1/3)*(a ^2 - b)^(1/3)*x^(1/3) + (1 + I*Sqrt[3])*(b^2 + a^3*x)^(1/3)] - (2*I)*Sqrt[ 3]*a^(2/3)*Log[2*a^(1/3)*(a^2 - b)^(1/3)*x^(1/3) + (1 + I*Sqrt[3])*(b^2 + a^3*x)^(1/3)] + a^(2/3)*Log[(-(a^(1/3)*(a^2 - b)^(1/3)*x^(1/3)) + (b^2 + a ^3*x)^(1/3))*((2*I)*a^(1/3)*(a^2 - b)^(1/3)*x^(1/3) + (I + Sqrt[3])*(b^2 + a^3*x)^(1/3))] + I*Sqrt[3]*a^(2/3)*Log[(-(a^(1/3)*(a^2 - b)^(1/3)*x^(1/3) ) + (b^2 + a^3*x)^(1/3))*((2*I)*a^(1/3)*(a^2 - b)^(1/3)*x^(1/3) + (I + Sqr t[3])*(b^2 + a^3*x)^(1/3))] + (a^2 - b)^(1/3)*Log[a^2*x^(2/3) + a*x^(1/3)* (b^2 + a^3*x)^(1/3) + (b^2 + a^3*x)^(2/3)]))/(2*a*(a^2 - b)^(1/3)*(x^2*(b^ 2 + a^3*x))^(1/3))
Time = 0.41 (sec) , antiderivative size = 294, normalized size of antiderivative = 0.68, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.147, Rules used = {2467, 25, 175, 71, 102}
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.
| \(\displaystyle \int \frac {a x-b}{(a x+b) \sqrt [3]{a^3 x^3+b^2 x^2}} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {x^{2/3} \sqrt [3]{a^3 x+b^2} \int -\frac {b-a x}{x^{2/3} (b+a x) \sqrt [3]{x a^3+b^2}}dx}{\sqrt [3]{a^3 x^3+b^2 x^2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {x^{2/3} \sqrt [3]{a^3 x+b^2} \int \frac {b-a x}{x^{2/3} (b+a x) \sqrt [3]{x a^3+b^2}}dx}{\sqrt [3]{a^3 x^3+b^2 x^2}}\) |
| \(\Big \downarrow \) 175 |
| \(\displaystyle -\frac {x^{2/3} \sqrt [3]{a^3 x+b^2} \left (2 b \int \frac {1}{x^{2/3} (b+a x) \sqrt [3]{x a^3+b^2}}dx-\int \frac {1}{x^{2/3} \sqrt [3]{x a^3+b^2}}dx\right )}{\sqrt [3]{a^3 x^3+b^2 x^2}}\) |
| \(\Big \downarrow \) 71 |
| \(\displaystyle -\frac {x^{2/3} \sqrt [3]{a^3 x+b^2} \left (2 b \int \frac {1}{x^{2/3} (b+a x) \sqrt [3]{x a^3+b^2}}dx+\frac {\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{a^3 x+b^2}}{\sqrt {3} a \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{a}+\frac {3 \log \left (\frac {\sqrt [3]{a^3 x+b^2}}{a \sqrt [3]{x}}-1\right )}{2 a}+\frac {\log (x)}{2 a}\right )}{\sqrt [3]{a^3 x^3+b^2 x^2}}\) |
| \(\Big \downarrow \) 102 |
| \(\displaystyle -\frac {x^{2/3} \sqrt [3]{a^3 x+b^2} \left (\frac {\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{a^3 x+b^2}}{\sqrt {3} a \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{a}+\frac {3 \log \left (\frac {\sqrt [3]{a^3 x+b^2}}{a \sqrt [3]{x}}-1\right )}{2 a}+2 b \left (\frac {\log (a x+b)}{2 \sqrt [3]{a} b \sqrt [3]{a^2-b}}-\frac {\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{a^3 x+b^2}}{\sqrt {3} \sqrt [3]{a} \sqrt [3]{x} \sqrt [3]{a^2-b}}+\frac {1}{\sqrt {3}}\right )}{\sqrt [3]{a} b \sqrt [3]{a^2-b}}-\frac {3 \log \left (\frac {\sqrt [3]{a^3 x+b^2}}{\sqrt [3]{a} \sqrt [3]{a^2-b}}-\sqrt [3]{x}\right )}{2 \sqrt [3]{a} b \sqrt [3]{a^2-b}}\right )+\frac {\log (x)}{2 a}\right )}{\sqrt [3]{a^3 x^3+b^2 x^2}}\) |
-((x^(2/3)*(b^2 + a^3*x)^(1/3)*((Sqrt[3]*ArcTan[1/Sqrt[3] + (2*(b^2 + a^3* x)^(1/3))/(Sqrt[3]*a*x^(1/3))])/a + Log[x]/(2*a) + 2*b*(-((Sqrt[3]*ArcTan[ 1/Sqrt[3] + (2*(b^2 + a^3*x)^(1/3))/(Sqrt[3]*a^(1/3)*(a^2 - b)^(1/3)*x^(1/ 3))])/(a^(1/3)*(a^2 - b)^(1/3)*b)) + Log[b + a*x]/(2*a^(1/3)*(a^2 - b)^(1/ 3)*b) - (3*Log[-x^(1/3) + (b^2 + a^3*x)^(1/3)/(a^(1/3)*(a^2 - b)^(1/3))])/ (2*a^(1/3)*(a^2 - b)^(1/3)*b)) + (3*Log[-1 + (b^2 + a^3*x)^(1/3)/(a*x^(1/3 ))])/(2*a)))/(b^2*x^2 + a^3*x^3)^(1/3))
3.31.29.3.1 Defintions of rubi rules used
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[1/(((a_.) + (b_.)*(x_))^(1/3)*((c_.) + (d_.)*(x_))^(2/3)*((e_.) + (f_.) *(x_))), x_] :> With[{q = Rt[(d*e - c*f)/(b*e - a*f), 3]}, Simp[(-Sqrt[3])* q*(ArcTan[1/Sqrt[3] + 2*q*((a + b*x)^(1/3)/(Sqrt[3]*(c + d*x)^(1/3)))]/(d*e - c*f)), x] + (Simp[q*(Log[e + f*x]/(2*(d*e - c*f))), x] - Simp[3*q*(Log[q *(a + b*x)^(1/3) - (c + d*x)^(1/3)]/(2*(d*e - c*f))), x])] /; FreeQ[{a, b, c, d, e, f}, x]
Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_ )))/((a_.) + (b_.)*(x_)), x_] :> Simp[h/b Int[(c + d*x)^n*(e + f*x)^p, x] , x] + Simp[(b*g - a*h)/b Int[(c + d*x)^n*((e + f*x)^p/(a + b*x)), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, 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 = 0.68 (sec) , antiderivative size = 325, normalized size of antiderivative = 0.75
| method | result | size |
| pseudoelliptic | \(\frac {-2 \sqrt {3}\, \arctan \left (\frac {\left (a x +2 \left (x^{2} \left (a^{3} x +b^{2}\right )\right )^{\frac {1}{3}}\right ) \sqrt {3}}{3 a x}\right ) {\left (a \left (a^{2}-b \right )\right )}^{\frac {1}{3}}+4 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left ({\left (a \left (a^{2}-b \right )\right )}^{\frac {1}{3}} x +2 \left (x^{2} \left (a^{3} x +b^{2}\right )\right )^{\frac {1}{3}}\right )}{3 {\left (a \left (a^{2}-b \right )\right )}^{\frac {1}{3}} x}\right ) a -2 \ln \left (\frac {-a x +\left (x^{2} \left (a^{3} x +b^{2}\right )\right )^{\frac {1}{3}}}{x}\right ) {\left (a \left (a^{2}-b \right )\right )}^{\frac {1}{3}}+\ln \left (\frac {a^{2} x^{2}+\left (x^{2} \left (a^{3} x +b^{2}\right )\right )^{\frac {1}{3}} a x +\left (x^{2} \left (a^{3} x +b^{2}\right )\right )^{\frac {2}{3}}}{x^{2}}\right ) {\left (a \left (a^{2}-b \right )\right )}^{\frac {1}{3}}-2 \ln \left (\frac {{\left (a \left (a^{2}-b \right )\right )}^{\frac {2}{3}} x^{2}+{\left (a \left (a^{2}-b \right )\right )}^{\frac {1}{3}} \left (x^{2} \left (a^{3} x +b^{2}\right )\right )^{\frac {1}{3}} x +\left (x^{2} \left (a^{3} x +b^{2}\right )\right )^{\frac {2}{3}}}{x^{2}}\right ) a +4 \ln \left (\frac {-{\left (a \left (a^{2}-b \right )\right )}^{\frac {1}{3}} x +\left (x^{2} \left (a^{3} x +b^{2}\right )\right )^{\frac {1}{3}}}{x}\right ) a}{2 a {\left (a \left (a^{2}-b \right )\right )}^{\frac {1}{3}}}\) | \(325\) |
1/2*(-2*3^(1/2)*arctan(1/3*(a*x+2*(x^2*(a^3*x+b^2))^(1/3))*3^(1/2)/a/x)*(a *(a^2-b))^(1/3)+4*3^(1/2)*arctan(1/3*3^(1/2)*((a*(a^2-b))^(1/3)*x+2*(x^2*( a^3*x+b^2))^(1/3))/(a*(a^2-b))^(1/3)/x)*a-2*ln((-a*x+(x^2*(a^3*x+b^2))^(1/ 3))/x)*(a*(a^2-b))^(1/3)+ln((a^2*x^2+(x^2*(a^3*x+b^2))^(1/3)*a*x+(x^2*(a^3 *x+b^2))^(2/3))/x^2)*(a*(a^2-b))^(1/3)-2*ln(((a*(a^2-b))^(2/3)*x^2+(a*(a^2 -b))^(1/3)*(x^2*(a^3*x+b^2))^(1/3)*x+(x^2*(a^3*x+b^2))^(2/3))/x^2)*a+4*ln( (-(a*(a^2-b))^(1/3)*x+(x^2*(a^3*x+b^2))^(1/3))/x)*a)/a/(a*(a^2-b))^(1/3)
Time = 0.25 (sec) , antiderivative size = 823, normalized size of antiderivative = 1.90 \[ \int \frac {-b+a x}{(b+a x) \sqrt [3]{b^2 x^2+a^3 x^3}} \, dx=\left [\frac {2 \, \sqrt {3} {\left (a^{3} - a b\right )} \sqrt {-\frac {1}{{\left (a^{3} - a b\right )}^{\frac {2}{3}}}} \log \left (\frac {2 \, b^{2} x + {\left (3 \, a^{3} - a b\right )} x^{2} - 3 \, {\left (a^{3} x^{3} + b^{2} x^{2}\right )}^{\frac {1}{3}} {\left (a^{3} - a b\right )}^{\frac {2}{3}} x - \sqrt {3} {\left ({\left (a^{3} - a b\right )}^{\frac {4}{3}} x^{2} + {\left (a^{3} x^{3} + b^{2} x^{2}\right )}^{\frac {1}{3}} {\left (a^{3} - a b\right )} x - 2 \, {\left (a^{3} x^{3} + b^{2} x^{2}\right )}^{\frac {2}{3}} {\left (a^{3} - a b\right )}^{\frac {2}{3}}\right )} \sqrt {-\frac {1}{{\left (a^{3} - a b\right )}^{\frac {2}{3}}}}}{a x^{2} + b x}\right ) - 2 \, \sqrt {3} {\left (a^{2} - b\right )} \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 ) - 2 \, {\left (a^{2} - b\right )} \log \left (-\frac {a x - {\left (a^{3} x^{3} + b^{2} x^{2}\right )}^{\frac {1}{3}}}{x}\right ) + {\left (a^{2} - b\right )} \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 ) + 4 \, {\left (a^{3} - a b\right )}^{\frac {2}{3}} \log \left (-\frac {{\left (a^{3} - a b\right )}^{\frac {1}{3}} x - {\left (a^{3} x^{3} + b^{2} x^{2}\right )}^{\frac {1}{3}}}{x}\right ) - 2 \, {\left (a^{3} - a b\right )}^{\frac {2}{3}} \log \left (\frac {{\left (a^{3} - a b\right )}^{\frac {2}{3}} x^{2} + {\left (a^{3} x^{3} + b^{2} x^{2}\right )}^{\frac {1}{3}} {\left (a^{3} - a b\right )}^{\frac {1}{3}} x + {\left (a^{3} x^{3} + b^{2} x^{2}\right )}^{\frac {2}{3}}}{x^{2}}\right )}{2 \, {\left (a^{3} - a b\right )}}, -\frac {2 \, \sqrt {3} {\left (a^{2} - b\right )} \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 ) - 4 \, \sqrt {3} {\left (a^{3} - a b\right )}^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left ({\left (a^{3} - a b\right )}^{\frac {1}{3}} x + 2 \, {\left (a^{3} x^{3} + b^{2} x^{2}\right )}^{\frac {1}{3}}\right )}}{3 \, {\left (a^{3} - a b\right )}^{\frac {1}{3}} x}\right ) + 2 \, {\left (a^{2} - b\right )} \log \left (-\frac {a x - {\left (a^{3} x^{3} + b^{2} x^{2}\right )}^{\frac {1}{3}}}{x}\right ) - {\left (a^{2} - b\right )} \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 ) - 4 \, {\left (a^{3} - a b\right )}^{\frac {2}{3}} \log \left (-\frac {{\left (a^{3} - a b\right )}^{\frac {1}{3}} x - {\left (a^{3} x^{3} + b^{2} x^{2}\right )}^{\frac {1}{3}}}{x}\right ) + 2 \, {\left (a^{3} - a b\right )}^{\frac {2}{3}} \log \left (\frac {{\left (a^{3} - a b\right )}^{\frac {2}{3}} x^{2} + {\left (a^{3} x^{3} + b^{2} x^{2}\right )}^{\frac {1}{3}} {\left (a^{3} - a b\right )}^{\frac {1}{3}} x + {\left (a^{3} x^{3} + b^{2} x^{2}\right )}^{\frac {2}{3}}}{x^{2}}\right )}{2 \, {\left (a^{3} - a b\right )}}\right ] \]
[1/2*(2*sqrt(3)*(a^3 - a*b)*sqrt(-1/(a^3 - a*b)^(2/3))*log((2*b^2*x + (3*a ^3 - a*b)*x^2 - 3*(a^3*x^3 + b^2*x^2)^(1/3)*(a^3 - a*b)^(2/3)*x - sqrt(3)* ((a^3 - a*b)^(4/3)*x^2 + (a^3*x^3 + b^2*x^2)^(1/3)*(a^3 - a*b)*x - 2*(a^3* x^3 + b^2*x^2)^(2/3)*(a^3 - a*b)^(2/3))*sqrt(-1/(a^3 - a*b)^(2/3)))/(a*x^2 + b*x)) - 2*sqrt(3)*(a^2 - b)*arctan(1/3*(sqrt(3)*a*x + 2*sqrt(3)*(a^3*x^ 3 + b^2*x^2)^(1/3))/(a*x)) - 2*(a^2 - b)*log(-(a*x - (a^3*x^3 + b^2*x^2)^( 1/3))/x) + (a^2 - b)*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) + 4*(a^3 - a*b)^(2/3)*log(-((a^3 - a*b)^(1/3)*x - (a^3*x^3 + b^2*x^2)^(1/3))/x) - 2*(a^3 - a*b)^(2/3)*log(((a^3 - a*b)^(2/ 3)*x^2 + (a^3*x^3 + b^2*x^2)^(1/3)*(a^3 - a*b)^(1/3)*x + (a^3*x^3 + b^2*x^ 2)^(2/3))/x^2))/(a^3 - a*b), -1/2*(2*sqrt(3)*(a^2 - b)*arctan(1/3*(sqrt(3) *a*x + 2*sqrt(3)*(a^3*x^3 + b^2*x^2)^(1/3))/(a*x)) - 4*sqrt(3)*(a^3 - a*b) ^(2/3)*arctan(1/3*sqrt(3)*((a^3 - a*b)^(1/3)*x + 2*(a^3*x^3 + b^2*x^2)^(1/ 3))/((a^3 - a*b)^(1/3)*x)) + 2*(a^2 - b)*log(-(a*x - (a^3*x^3 + b^2*x^2)^( 1/3))/x) - (a^2 - b)*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) - 4*(a^3 - a*b)^(2/3)*log(-((a^3 - a*b)^(1/3)*x - (a^3*x^3 + b^2*x^2)^(1/3))/x) + 2*(a^3 - a*b)^(2/3)*log(((a^3 - a*b)^(2/ 3)*x^2 + (a^3*x^3 + b^2*x^2)^(1/3)*(a^3 - a*b)^(1/3)*x + (a^3*x^3 + b^2*x^ 2)^(2/3))/x^2))/(a^3 - a*b)]
\[ \int \frac {-b+a x}{(b+a x) \sqrt [3]{b^2 x^2+a^3 x^3}} \, dx=\int \frac {a x - b}{\sqrt [3]{x^{2} \left (a^{3} x + b^{2}\right )} \left (a x + b\right )}\, dx \]
\[ \int \frac {-b+a x}{(b+a x) \sqrt [3]{b^2 x^2+a^3 x^3}} \, dx=\int { \frac {a x - b}{{\left (a^{3} x^{3} + b^{2} x^{2}\right )}^{\frac {1}{3}} {\left (a x + b\right )}} \,d x } \]
Time = 20.20 (sec) , antiderivative size = 255, normalized size of antiderivative = 0.59 \[ \int \frac {-b+a x}{(b+a x) \sqrt [3]{b^2 x^2+a^3 x^3}} \, dx=\frac {2 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left ({\left (a^{3} - a b\right )}^{\frac {1}{3}} + 2 \, {\left (a^{3} + \frac {b^{2}}{x}\right )}^{\frac {1}{3}}\right )}}{3 \, {\left (a^{3} - a b\right )}^{\frac {1}{3}}}\right )}{{\left (a^{3} - a b\right )}^{\frac {1}{3}}} - \frac {\log \left ({\left (a^{3} - a b\right )}^{\frac {2}{3}} + {\left (a^{3} - a b\right )}^{\frac {1}{3}} {\left (a^{3} + \frac {b^{2}}{x}\right )}^{\frac {1}{3}} + {\left (a^{3} + \frac {b^{2}}{x}\right )}^{\frac {2}{3}}\right )}{{\left (a^{3} - a b\right )}^{\frac {1}{3}}} + \frac {2 \, \log \left ({\left | -{\left (a^{3} - a b\right )}^{\frac {1}{3}} + {\left (a^{3} + \frac {b^{2}}{x}\right )}^{\frac {1}{3}} \right |}\right )}{{\left (a^{3} - a b\right )}^{\frac {1}{3}}} - \frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (a + 2 \, {\left (a^{3} + \frac {b^{2}}{x}\right )}^{\frac {1}{3}}\right )}}{3 \, a}\right )}{a} + \frac {\log \left (a^{2} + {\left (a^{3} + \frac {b^{2}}{x}\right )}^{\frac {1}{3}} a + {\left (a^{3} + \frac {b^{2}}{x}\right )}^{\frac {2}{3}}\right )}{2 \, a} - \frac {\log \left ({\left | -a + {\left (a^{3} + \frac {b^{2}}{x}\right )}^{\frac {1}{3}} \right |}\right )}{a} \]
2*sqrt(3)*arctan(1/3*sqrt(3)*((a^3 - a*b)^(1/3) + 2*(a^3 + b^2/x)^(1/3))/( a^3 - a*b)^(1/3))/(a^3 - a*b)^(1/3) - log((a^3 - a*b)^(2/3) + (a^3 - a*b)^ (1/3)*(a^3 + b^2/x)^(1/3) + (a^3 + b^2/x)^(2/3))/(a^3 - a*b)^(1/3) + 2*log (abs(-(a^3 - a*b)^(1/3) + (a^3 + b^2/x)^(1/3)))/(a^3 - a*b)^(1/3) - sqrt(3 )*arctan(1/3*sqrt(3)*(a + 2*(a^3 + b^2/x)^(1/3))/a)/a + 1/2*log(a^2 + (a^3 + b^2/x)^(1/3)*a + (a^3 + b^2/x)^(2/3))/a - log(abs(-a + (a^3 + b^2/x)^(1 /3)))/a
Timed out. \[ \int \frac {-b+a x}{(b+a x) \sqrt [3]{b^2 x^2+a^3 x^3}} \, dx=\int -\frac {b-a\,x}{{\left (a^3\,x^3+b^2\,x^2\right )}^{1/3}\,\left (b+a\,x\right )} \,d x \]