Integrand size = 29, antiderivative size = 506 \[ \int \frac {\sqrt [3]{b^2 x^2+a^3 x^3}}{-b+a x} \, dx=\frac {\sqrt [3]{b^2 x^2+a^3 x^3}}{a}-\frac {\left (3 a^2 b+b^2\right ) \arctan \left (\frac {\sqrt {3} a x}{a x+2 \sqrt [3]{b^2 x^2+a^3 x^3}}\right )}{\sqrt {3} a^3}-\frac {\sqrt {-3-3 i \sqrt {3}} b \sqrt [3]{a^2+b} \arctan \left (\frac {\sqrt {3} \sqrt [3]{a} \sqrt [3]{a^2+b} x}{\sqrt [3]{a} \sqrt [3]{a^2+b} x-2 \sqrt [3]{-1} \sqrt [3]{b^2 x^2+a^3 x^3}}\right )}{\sqrt {2} a^{5/3}}+\frac {\left (-3 a^2 b-b^2\right ) \log \left (-a x+\sqrt [3]{b^2 x^2+a^3 x^3}\right )}{3 a^3}+\frac {i \left (i b \sqrt [3]{a^2+b}+\sqrt {3} b \sqrt [3]{a^2+b}\right ) \log \left (\sqrt [3]{a} \sqrt [3]{a^2+b} x+\sqrt [3]{-1} \sqrt [3]{b^2 x^2+a^3 x^3}\right )}{2 a^{5/3}}+\frac {\left (3 a^2 b+b^2\right ) \log \left (a^2 x^2+a x \sqrt [3]{b^2 x^2+a^3 x^3}+\left (b^2 x^2+a^3 x^3\right )^{2/3}\right )}{6 a^3}+\frac {\left (b \sqrt [3]{a^2+b}-i \sqrt {3} b \sqrt [3]{a^2+b}\right ) \log \left (a^{2/3} \left (a^2+b\right )^{2/3} x^2-\sqrt [3]{-1} \sqrt [3]{a} \sqrt [3]{a^2+b} x \sqrt [3]{b^2 x^2+a^3 x^3}+(-1)^{2/3} \left (b^2 x^2+a^3 x^3\right )^{2/3}\right )}{4 a^{5/3}} \]
(a^3*x^3+b^2*x^2)^(1/3)/a-1/3*(3*a^2*b+b^2)*arctan(3^(1/2)*a*x/(a*x+2*(a^3 *x^3+b^2*x^2)^(1/3)))*3^(1/2)/a^3-1/2*(-3-3*I*3^(1/2))^(1/2)*b*(a^2+b)^(1/ 3)*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)))*2^(1/2)/a^(5/3)+1/3*(-3*a^2*b-b^2)*ln(-a*x +(a^3*x^3+b^2*x^2)^(1/3))/a^3+1/2*I*(I*b*(a^2+b)^(1/3)+3^(1/2)*b*(a^2+b)^( 1/3))*ln(a^(1/3)*(a^2+b)^(1/3)*x+(-1)^(1/3)*(a^3*x^3+b^2*x^2)^(1/3))/a^(5/ 3)+1/6*(3*a^2*b+b^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^3+1/4*(b*(a^2+b)^(1/3)-I*3^(1/2)*b*(a^2+b)^(1/3))*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^(5/3)
Time = 1.95 (sec) , antiderivative size = 694, normalized size of antiderivative = 1.37 \[ \int \frac {\sqrt [3]{b^2 x^2+a^3 x^3}}{-b+a x} \, dx=\frac {\sqrt [3]{x^2 \left (b^2+a^3 x\right )}}{a}-\frac {b \left (3 a^2+b\right ) \sqrt [3]{x^2 \left (b^2+a^3 x\right )} \arctan \left (\frac {\sqrt {3} a \sqrt [3]{x}}{a \sqrt [3]{x}+2 \sqrt [3]{b^2+a^3 x}}\right )}{\sqrt {3} a^3 x^{2/3} \sqrt [3]{b^2+a^3 x}}-\frac {i \left (-3 i+\sqrt {3}\right ) b \sqrt [3]{a^2+b} \sqrt [3]{x^2 \left (b^2+a^3 x\right )} \text {arctanh}\left (\frac {i \sqrt [3]{a} \sqrt [3]{a^2+b} \sqrt [3]{x}+\left (-i+\sqrt {3}\right ) \sqrt [3]{b^2+a^3 x}}{\sqrt {3} \sqrt [3]{a} \sqrt [3]{a^2+b} \sqrt [3]{x}}\right )}{2 a^{5/3} x^{2/3} \sqrt [3]{b^2+a^3 x}}-\frac {b \left (3 a^2+b\right ) \sqrt [3]{x^2 \left (b^2+a^3 x\right )} \log \left (-a \sqrt [3]{x}+\sqrt [3]{b^2+a^3 x}\right )}{3 a^3 x^{2/3} \sqrt [3]{b^2+a^3 x}}+\frac {i \left (i+\sqrt {3}\right ) b \sqrt [3]{a^2+b} \sqrt [3]{x^2 \left (b^2+a^3 x\right )} \log \left (2 \sqrt [3]{a} \sqrt [3]{a^2+b} \sqrt [3]{x}+\left (1+i \sqrt {3}\right ) \sqrt [3]{b^2+a^3 x}\right )}{2 a^{5/3} x^{2/3} \sqrt [3]{b^2+a^3 x}}+\frac {b \left (3 a^2+b\right ) \sqrt [3]{x^2 \left (b^2+a^3 x\right )} \log \left (a^2 x^{2/3}+a \sqrt [3]{x} \sqrt [3]{b^2+a^3 x}+\left (b^2+a^3 x\right )^{2/3}\right )}{6 a^3 x^{2/3} \sqrt [3]{b^2+a^3 x}}+\frac {\left (1-i \sqrt {3}\right ) b \sqrt [3]{a^2+b} \sqrt [3]{x^2 \left (b^2+a^3 x\right )} \log \left (-2 i a^{2/3} \left (a^2+b\right )^{2/3} x^{2/3}+\sqrt [3]{a} \sqrt [3]{a^2+b} \left (i \sqrt [3]{x}-\sqrt {3} \sqrt [3]{x}\right ) \sqrt [3]{b^2+a^3 x}+\left (i+\sqrt {3}\right ) \left (b^2+a^3 x\right )^{2/3}\right )}{4 a^{5/3} x^{2/3} \sqrt [3]{b^2+a^3 x}} \]
(x^2*(b^2 + a^3*x))^(1/3)/a - (b*(3*a^2 + b)*(x^2*(b^2 + a^3*x))^(1/3)*Arc Tan[(Sqrt[3]*a*x^(1/3))/(a*x^(1/3) + 2*(b^2 + a^3*x)^(1/3))])/(Sqrt[3]*a^3 *x^(2/3)*(b^2 + a^3*x)^(1/3)) - ((I/2)*(-3*I + Sqrt[3])*b*(a^2 + b)^(1/3)* (x^2*(b^2 + a^3*x))^(1/3)*ArcTanh[(I*a^(1/3)*(a^2 + b)^(1/3)*x^(1/3) + (-I + Sqrt[3])*(b^2 + a^3*x)^(1/3))/(Sqrt[3]*a^(1/3)*(a^2 + b)^(1/3)*x^(1/3)) ])/(a^(5/3)*x^(2/3)*(b^2 + a^3*x)^(1/3)) - (b*(3*a^2 + b)*(x^2*(b^2 + a^3* x))^(1/3)*Log[-(a*x^(1/3)) + (b^2 + a^3*x)^(1/3)])/(3*a^3*x^(2/3)*(b^2 + a ^3*x)^(1/3)) + ((I/2)*(I + Sqrt[3])*b*(a^2 + b)^(1/3)*(x^2*(b^2 + a^3*x))^ (1/3)*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^(5/3)*x^(2/3)*(b^2 + a^3*x)^(1/3)) + (b*(3*a^2 + b)*(x^2*(b^2 + a^3*x))^(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)])/(6*a^3*x^(2/3)*(b^2 + a^3*x)^(1/3)) + ((1 - I*Sqrt[3])*b*(a ^2 + b)^(1/3)*(x^2*(b^2 + a^3*x))^(1/3)*Log[(-2*I)*a^(2/3)*(a^2 + b)^(2/3) *x^(2/3) + a^(1/3)*(a^2 + b)^(1/3)*(I*x^(1/3) - Sqrt[3]*x^(1/3))*(b^2 + a^ 3*x)^(1/3) + (I + Sqrt[3])*(b^2 + a^3*x)^(2/3)])/(4*a^(5/3)*x^(2/3)*(b^2 + a^3*x)^(1/3))
Time = 0.42 (sec) , antiderivative size = 337, normalized size of antiderivative = 0.67, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {2467, 25, 112, 27, 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.
| \(\displaystyle \int \frac {\sqrt [3]{a^3 x^3+b^2 x^2}}{a x-b} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt [3]{a^3 x^3+b^2 x^2} \int -\frac {x^{2/3} \sqrt [3]{x a^3+b^2}}{b-a x}dx}{x^{2/3} \sqrt [3]{a^3 x+b^2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt [3]{a^3 x^3+b^2 x^2} \int \frac {x^{2/3} \sqrt [3]{x a^3+b^2}}{b-a x}dx}{x^{2/3} \sqrt [3]{a^3 x+b^2}}\) |
| \(\Big \downarrow \) 112 |
| \(\displaystyle -\frac {\sqrt [3]{a^3 x^3+b^2 x^2} \left (\frac {\int \frac {b \left (2 b^2+a \left (3 a^2+b\right ) x\right )}{3 \sqrt [3]{x} (b-a x) \left (x a^3+b^2\right )^{2/3}}dx}{a}-\frac {x^{2/3} \sqrt [3]{a^3 x+b^2}}{a}\right )}{x^{2/3} \sqrt [3]{a^3 x+b^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\sqrt [3]{a^3 x^3+b^2 x^2} \left (\frac {b \int \frac {2 b^2+a \left (3 a^2+b\right ) x}{\sqrt [3]{x} (b-a x) \left (x a^3+b^2\right )^{2/3}}dx}{3 a}-\frac {x^{2/3} \sqrt [3]{a^3 x+b^2}}{a}\right )}{x^{2/3} \sqrt [3]{a^3 x+b^2}}\) |
| \(\Big \downarrow \) 175 |
| \(\displaystyle -\frac {\sqrt [3]{a^3 x^3+b^2 x^2} \left (\frac {b \left (3 b \left (a^2+b\right ) \int \frac {1}{\sqrt [3]{x} (b-a x) \left (x a^3+b^2\right )^{2/3}}dx-\left (3 a^2+b\right ) \int \frac {1}{\sqrt [3]{x} \left (x a^3+b^2\right )^{2/3}}dx\right )}{3 a}-\frac {x^{2/3} \sqrt [3]{a^3 x+b^2}}{a}\right )}{x^{2/3} \sqrt [3]{a^3 x+b^2}}\) |
| \(\Big \downarrow \) 71 |
| \(\displaystyle -\frac {\sqrt [3]{a^3 x^3+b^2 x^2} \left (\frac {b \left (3 b \left (a^2+b\right ) \int \frac {1}{\sqrt [3]{x} (b-a x) \left (x a^3+b^2\right )^{2/3}}dx-\left (3 a^2+b\right ) \left (-\frac {\sqrt {3} \arctan \left (\frac {2 a \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{a^3 x+b^2}}+\frac {1}{\sqrt {3}}\right )}{a^2}-\frac {\log \left (a^3 x+b^2\right )}{2 a^2}-\frac {3 \log \left (\frac {a \sqrt [3]{x}}{\sqrt [3]{a^3 x+b^2}}-1\right )}{2 a^2}\right )\right )}{3 a}-\frac {x^{2/3} \sqrt [3]{a^3 x+b^2}}{a}\right )}{x^{2/3} \sqrt [3]{a^3 x+b^2}}\) |
| \(\Big \downarrow \) 102 |
| \(\displaystyle -\frac {\sqrt [3]{a^3 x^3+b^2 x^2} \left (\frac {b \left (3 b \left (a^2+b\right ) \left (\frac {\log (b-a x)}{2 a^{2/3} b \left (a^2+b\right )^{2/3}}-\frac {\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{a} \sqrt [3]{x} \sqrt [3]{a^2+b}}{\sqrt {3} \sqrt [3]{a^3 x+b^2}}+\frac {1}{\sqrt {3}}\right )}{a^{2/3} b \left (a^2+b\right )^{2/3}}-\frac {3 \log \left (\sqrt [3]{a} \sqrt [3]{x} \sqrt [3]{a^2+b}-\sqrt [3]{a^3 x+b^2}\right )}{2 a^{2/3} b \left (a^2+b\right )^{2/3}}\right )-\left (3 a^2+b\right ) \left (-\frac {\sqrt {3} \arctan \left (\frac {2 a \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{a^3 x+b^2}}+\frac {1}{\sqrt {3}}\right )}{a^2}-\frac {\log \left (a^3 x+b^2\right )}{2 a^2}-\frac {3 \log \left (\frac {a \sqrt [3]{x}}{\sqrt [3]{a^3 x+b^2}}-1\right )}{2 a^2}\right )\right )}{3 a}-\frac {x^{2/3} \sqrt [3]{a^3 x+b^2}}{a}\right )}{x^{2/3} \sqrt [3]{a^3 x+b^2}}\) |
-(((b^2*x^2 + a^3*x^3)^(1/3)*(-((x^(2/3)*(b^2 + a^3*x)^(1/3))/a) + (b*(-(( 3*a^2 + b)*(-((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*(a^2 + b)*(-((Sqrt[3]*ArcTan[1/Sqrt[3] + (2*a^(1/3)*(a^2 + b)^(1/3)*x^(1/3))/(Sqrt[3]*(b^2 + a^3*x)^(1/3))])/(a^ (2/3)*b*(a^2 + b)^(2/3))) + Log[b - a*x]/(2*a^(2/3)*b*(a^2 + b)^(2/3)) - ( 3*Log[a^(1/3)*(a^2 + b)^(1/3)*x^(1/3) - (b^2 + a^3*x)^(1/3)])/(2*a^(2/3)*b *(a^2 + b)^(2/3)))))/(3*a)))/(x^(2/3)*(b^2 + a^3*x)^(1/3)))
3.31.78.3.1 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[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[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[(a + b*x)^m*(c + d*x)^n*((e + f*x)^(p + 1)/(f*(m + n + p + 1))), x] - Simp[1/(f*(m + n + p + 1)) Int[(a + b*x)^(m - 1)*(c + d*x) ^(n - 1)*(e + f*x)^p*Simp[c*m*(b*e - a*f) + a*n*(d*e - c*f) + (d*m*(b*e - a *f) + b*n*(d*e - c*f))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && GtQ[m, 0] && GtQ[n, 0] && NeQ[m + n + p + 1, 0] && (IntegersQ[2*m, 2*n, 2*p ] || (IntegersQ[m, n + p] || IntegersQ[p, m + n]))
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 = 0.77 (sec) , antiderivative size = 409, normalized size of antiderivative = 0.81
| method | result | size |
| pseudoelliptic | \(\frac {\left (-2 {\left (a \left (a^{2}+b \right )\right )}^{\frac {2}{3}} \left (x^{2} \left (a^{3} x +b^{2}\right )\right )^{\frac {1}{3}} a^{2}+\frac {b \left (-\left (3 a^{2}+b \right ) \left (2 \sqrt {3}\, \arctan \left (\frac {\left (a x +2 \left (x^{2} \left (a^{3} x +b^{2}\right )\right )^{\frac {1}{3}}\right ) \sqrt {3}}{3 a x}\right )+\ln \left (\frac {a^{2} x^{2}+\left (x^{2} \left (a^{3} x +b^{2}\right )\right )^{\frac {1}{3}} a x +\left (x^{2} \left (a^{3} x +b^{2}\right )\right )^{\frac {2}{3}}}{x^{2}}\right )-2 \ln \left (\frac {-a x +\left (x^{2} \left (a^{3} x +b^{2}\right )\right )^{\frac {1}{3}}}{x}\right )\right ) {\left (a \left (a^{2}+b \right )\right )}^{\frac {2}{3}}+3 a^{2} \left (a^{2}+b \right ) \left (2 \arctan \left (\frac {\sqrt {3}\, \left ({\left (a \left (a^{2}+b \right )\right )}^{\frac {1}{3}} x +2 \left (x^{2} \left (a^{3} x +b^{2}\right )\right )^{\frac {1}{3}}\right )}{3 {\left (a \left (a^{2}+b \right )\right )}^{\frac {1}{3}} x}\right ) \sqrt {3}+\ln \left (\frac {{\left (a \left (a^{2}+b \right )\right )}^{\frac {2}{3}} x^{2}+{\left (a \left (a^{2}+b \right )\right )}^{\frac {1}{3}} \left (x^{2} \left (a^{3} x +b^{2}\right )\right )^{\frac {1}{3}} x +\left (x^{2} \left (a^{3} x +b^{2}\right )\right )^{\frac {2}{3}}}{x^{2}}\right )-2 \ln \left (\frac {-{\left (a \left (a^{2}+b \right )\right )}^{\frac {1}{3}} x +\left (x^{2} \left (a^{3} x +b^{2}\right )\right )^{\frac {1}{3}}}{x}\right )\right )\right )}{3}\right ) b^{2} x^{2}}{2 {\left (a \left (a^{2}+b \right )\right )}^{\frac {2}{3}} \left (a^{2} x^{2}+\left (x^{2} \left (a^{3} x +b^{2}\right )\right )^{\frac {1}{3}} a x +\left (x^{2} \left (a^{3} x +b^{2}\right )\right )^{\frac {2}{3}}\right ) \left (a x -\left (x^{2} \left (a^{3} x +b^{2}\right )\right )^{\frac {1}{3}}\right ) a^{3}}\) | \(409\) |
1/2*(-2*(a*(a^2+b))^(2/3)*(x^2*(a^3*x+b^2))^(1/3)*a^2+1/3*b*(-(3*a^2+b)*(2 *3^(1/2)*arctan(1/3*(a*x+2*(x^2*(a^3*x+b^2))^(1/3))*3^(1/2)/a/x)+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)-2*ln((-a*x+(x ^2*(a^3*x+b^2))^(1/3))/x))*(a*(a^2+b))^(2/3)+3*a^2*(a^2+b)*(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 )*3^(1/2)+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)-2*ln((-(a*(a^2+b))^(1/3)*x+(x^2*(a^3*x +b^2))^(1/3))/x))))*b^2/(a*(a^2+b))^(2/3)*x^2/(a^2*x^2+(x^2*(a^3*x+b^2))^( 1/3)*a*x+(x^2*(a^3*x+b^2))^(2/3))/(a*x-(x^2*(a^3*x+b^2))^(1/3))/a^3
Time = 0.26 (sec) , antiderivative size = 408, normalized size of antiderivative = 0.81 \[ \int \frac {\sqrt [3]{b^2 x^2+a^3 x^3}}{-b+a x} \, dx=-\frac {6 \, \sqrt {3} a^{2} b \left (\frac {a^{2} + b}{a^{2}}\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (a^{2} + b\right )} x + 2 \, \sqrt {3} {\left (a^{3} x^{3} + b^{2} x^{2}\right )}^{\frac {1}{3}} a \left (\frac {a^{2} + b}{a^{2}}\right )^{\frac {2}{3}}}{3 \, {\left (a^{2} + b\right )} x}\right ) - 6 \, a^{2} b \left (\frac {a^{2} + b}{a^{2}}\right )^{\frac {1}{3}} \log \left (-\frac {a x \left (\frac {a^{2} + b}{a^{2}}\right )^{\frac {1}{3}} - {\left (a^{3} x^{3} + b^{2} x^{2}\right )}^{\frac {1}{3}}}{x}\right ) + 3 \, a^{2} b \left (\frac {a^{2} + b}{a^{2}}\right )^{\frac {1}{3}} \log \left (\frac {a^{2} x^{2} \left (\frac {a^{2} + b}{a^{2}}\right )^{\frac {2}{3}} + {\left (a^{3} x^{3} + b^{2} x^{2}\right )}^{\frac {1}{3}} a x \left (\frac {a^{2} + b}{a^{2}}\right )^{\frac {1}{3}} + {\left (a^{3} x^{3} + b^{2} x^{2}\right )}^{\frac {2}{3}}}{x^{2}}\right ) - 2 \, \sqrt {3} {\left (3 \, a^{2} b + b^{2}\right )} \arctan \left (\frac {\sqrt {3} a x + 2 \, \sqrt {3} {\left (a^{3} x^{3} + b^{2} x^{2}\right )}^{\frac {1}{3}}}{3 \, a x}\right ) - 6 \, {\left (a^{3} x^{3} + b^{2} x^{2}\right )}^{\frac {1}{3}} a^{2} + 2 \, {\left (3 \, a^{2} b + b^{2}\right )} \log \left (-\frac {a x - {\left (a^{3} x^{3} + b^{2} x^{2}\right )}^{\frac {1}{3}}}{x}\right ) - {\left (3 \, a^{2} b + b^{2}\right )} \log \left (\frac {a^{2} x^{2} + {\left (a^{3} x^{3} + b^{2} x^{2}\right )}^{\frac {1}{3}} a x + {\left (a^{3} x^{3} + b^{2} x^{2}\right )}^{\frac {2}{3}}}{x^{2}}\right )}{6 \, a^{3}} \]
-1/6*(6*sqrt(3)*a^2*b*((a^2 + b)/a^2)^(1/3)*arctan(1/3*(sqrt(3)*(a^2 + b)* x + 2*sqrt(3)*(a^3*x^3 + b^2*x^2)^(1/3)*a*((a^2 + b)/a^2)^(2/3))/((a^2 + b )*x)) - 6*a^2*b*((a^2 + b)/a^2)^(1/3)*log(-(a*x*((a^2 + b)/a^2)^(1/3) - (a ^3*x^3 + b^2*x^2)^(1/3))/x) + 3*a^2*b*((a^2 + b)/a^2)^(1/3)*log((a^2*x^2*( (a^2 + b)/a^2)^(2/3) + (a^3*x^3 + b^2*x^2)^(1/3)*a*x*((a^2 + b)/a^2)^(1/3) + (a^3*x^3 + b^2*x^2)^(2/3))/x^2) - 2*sqrt(3)*(3*a^2*b + b^2)*arctan(1/3* (sqrt(3)*a*x + 2*sqrt(3)*(a^3*x^3 + b^2*x^2)^(1/3))/(a*x)) - 6*(a^3*x^3 + b^2*x^2)^(1/3)*a^2 + 2*(3*a^2*b + b^2)*log(-(a*x - (a^3*x^3 + b^2*x^2)^(1/ 3))/x) - (3*a^2*b + b^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))/a^3
\[ \int \frac {\sqrt [3]{b^2 x^2+a^3 x^3}}{-b+a x} \, dx=\int \frac {\sqrt [3]{x^{2} \left (a^{3} x + b^{2}\right )}}{a x - b}\, dx \]
\[ \int \frac {\sqrt [3]{b^2 x^2+a^3 x^3}}{-b+a x} \, dx=\int { \frac {{\left (a^{3} x^{3} + b^{2} x^{2}\right )}^{\frac {1}{3}}}{a x - b} \,d x } \]
Time = 23.25 (sec) , antiderivative size = 322, normalized size of antiderivative = 0.64 \[ \int \frac {\sqrt [3]{b^2 x^2+a^3 x^3}}{-b+a x} \, dx=\frac {{\left (a^{3} + a b\right )}^{\frac {1}{3}} {\left (a^{2} b + b^{2}\right )} \log \left ({\left | -{\left (a^{3} + a b\right )}^{\frac {1}{3}} + {\left (a^{3} + \frac {b^{2}}{x}\right )}^{\frac {1}{3}} \right |}\right )}{a^{4} + a^{2} b} - \frac {\sqrt {3} {\left (a^{3} + a b\right )}^{\frac {1}{3}} b \arctan \left (\frac {\sqrt {3} {\left ({\left (a^{3} + a b\right )}^{\frac {1}{3}} + 2 \, {\left (a^{3} + \frac {b^{2}}{x}\right )}^{\frac {1}{3}}\right )}}{3 \, {\left (a^{3} + a b\right )}^{\frac {1}{3}}}\right )}{a^{2}} + \frac {{\left (a^{3} + \frac {b^{2}}{x}\right )}^{\frac {1}{3}} x}{a} - \frac {{\left (a^{3} + a b\right )}^{\frac {1}{3}} b \log \left ({\left (a^{3} + a b\right )}^{\frac {2}{3}} + {\left (a^{3} + a b\right )}^{\frac {1}{3}} {\left (a^{3} + \frac {b^{2}}{x}\right )}^{\frac {1}{3}} + {\left (a^{3} + \frac {b^{2}}{x}\right )}^{\frac {2}{3}}\right )}{2 \, a^{2}} + \frac {\sqrt {3} {\left (3 \, a^{2} b + b^{2}\right )} \arctan \left (\frac {\sqrt {3} {\left (a + 2 \, {\left (a^{3} + \frac {b^{2}}{x}\right )}^{\frac {1}{3}}\right )}}{3 \, a}\right )}{3 \, a^{3}} + \frac {{\left (3 \, a^{2} b + b^{2}\right )} \log \left (a^{2} + {\left (a^{3} + \frac {b^{2}}{x}\right )}^{\frac {1}{3}} a + {\left (a^{3} + \frac {b^{2}}{x}\right )}^{\frac {2}{3}}\right )}{6 \, a^{3}} - \frac {{\left (3 \, a^{2} b + b^{2}\right )} \log \left ({\left | -a + {\left (a^{3} + \frac {b^{2}}{x}\right )}^{\frac {1}{3}} \right |}\right )}{3 \, a^{3}} \]
(a^3 + a*b)^(1/3)*(a^2*b + b^2)*log(abs(-(a^3 + a*b)^(1/3) + (a^3 + b^2/x) ^(1/3)))/(a^4 + a^2*b) - sqrt(3)*(a^3 + a*b)^(1/3)*b*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^2 + (a^3 + b^2/x)^(1/3)*x/a - 1/2*(a^3 + a*b)^(1/3)*b*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^2 + 1/3*sqrt(3)*(3 *a^2*b + b^2)*arctan(1/3*sqrt(3)*(a + 2*(a^3 + b^2/x)^(1/3))/a)/a^3 + 1/6* (3*a^2*b + b^2)*log(a^2 + (a^3 + b^2/x)^(1/3)*a + (a^3 + b^2/x)^(2/3))/a^3 - 1/3*(3*a^2*b + b^2)*log(abs(-a + (a^3 + b^2/x)^(1/3)))/a^3
Timed out. \[ \int \frac {\sqrt [3]{b^2 x^2+a^3 x^3}}{-b+a x} \, dx=-\int \frac {{\left (a^3\,x^3+b^2\,x^2\right )}^{1/3}}{b-a\,x} \,d x \]