Integrand size = 27, antiderivative size = 526 \[ \int \frac {\sqrt [3]{b^2 x^2+a^3 x^3}}{b+a x} \, dx=\frac {\sqrt [3]{b^2 x^2+a^3 x^3}}{a}+\frac {\left (3 a^2 b-b^2\right ) \arctan \left (\frac {\sqrt {3} a x}{a x+2 \sqrt [3]{b^2 x^2+a^3 x^3}}\right )}{\sqrt {3} a^3}+\frac {\sqrt {-3-3 i \sqrt {3}} \sqrt [3]{a^2-b} b \arctan \left (\frac {\sqrt {3} \sqrt [3]{a} \sqrt [3]{a^2-b} x}{\sqrt [3]{a} \sqrt [3]{a^2-b} x-2 \sqrt [3]{-1} \sqrt [3]{b^2 x^2+a^3 x^3}}\right )}{\sqrt {2} a^{5/3}}+\frac {\left (3 a^2 b-b^2\right ) \log \left (-a x+\sqrt [3]{b^2 x^2+a^3 x^3}\right )}{3 a^3}+\frac {\left (\sqrt [3]{a^2-b} b-i \sqrt {3} \sqrt [3]{a^2-b} b\right ) \log \left (\sqrt [3]{a} \sqrt [3]{a^2-b} x+\sqrt [3]{-1} \sqrt [3]{b^2 x^2+a^3 x^3}\right )}{2 a^{5/3}}+\frac {\left (-3 a^2 b+b^2\right ) \log \left (a^2 x^2+a x \sqrt [3]{b^2 x^2+a^3 x^3}+\left (b^2 x^2+a^3 x^3\right )^{2/3}\right )}{6 a^3}+\frac {i \left (i \sqrt [3]{a^2-b} b+\sqrt {3} \sqrt [3]{a^2-b} b\right ) \log \left (a^{2/3} \left (a^2-b\right )^{2/3} x^2-\sqrt [3]{-1} \sqrt [3]{a} \sqrt [3]{a^2-b} x \sqrt [3]{b^2 x^2+a^3 x^3}+(-1)^{2/3} \left (b^2 x^2+a^3 x^3\right )^{2/3}\right )}{4 a^{5/3}} \]
(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)*(a^2-b)^(1/3) *b*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*((a^2-b)^(1/3)*b-I*3^(1/2)*(a^2-b)^(1/3)* b)*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*I*(I*(a^2-b)^(1/3)*b+3^(1/2)*(a^2-b)^(1/3)*b)*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 = 2.12 (sec) , antiderivative size = 495, normalized size of antiderivative = 0.94 \[ \int \frac {\sqrt [3]{b^2 x^2+a^3 x^3}}{b+a x} \, dx=\frac {x^{4/3} \left (b^2+a^3 x\right )^{2/3} \left (12 a^2 x^{2/3} \sqrt [3]{b^2+a^3 x}+4 \sqrt {3} \left (3 a^2-b\right ) b \arctan \left (\frac {\sqrt {3} a \sqrt [3]{x}}{a \sqrt [3]{x}+2 \sqrt [3]{b^2+a^3 x}}\right )+6 \left (3+i \sqrt {3}\right ) a^{4/3} \sqrt [3]{a^2-b} b \text {arctanh}\left (\frac {i+\frac {\left (-i+\sqrt {3}\right ) \sqrt [3]{b^2+a^3 x}}{\sqrt [3]{a} \sqrt [3]{a^2-b} \sqrt [3]{x}}}{\sqrt {3}}\right )+4 \left (3 a^2-b\right ) b \log \left (-a \sqrt [3]{x}+\sqrt [3]{b^2+a^3 x}\right )+6 \left (1-i \sqrt {3}\right ) a^{4/3} \sqrt [3]{a^2-b} b \log \left (2 \sqrt [3]{a} \sqrt [3]{a^2-b} \sqrt [3]{x}+\left (1+i \sqrt {3}\right ) \sqrt [3]{b^2+a^3 x}\right )+3 i \left (i+\sqrt {3}\right ) a^{4/3} \sqrt [3]{a^2-b} b \log \left (\left (-\sqrt [3]{a} \sqrt [3]{a^2-b} \sqrt [3]{x}+\sqrt [3]{b^2+a^3 x}\right ) \left (2 i \sqrt [3]{a} \sqrt [3]{a^2-b} \sqrt [3]{x}+\left (i+\sqrt {3}\right ) \sqrt [3]{b^2+a^3 x}\right )\right )-2 \left (3 a^2-b\right ) b \log \left (a^2 x^{2/3}+a \sqrt [3]{x} \sqrt [3]{b^2+a^3 x}+\left (b^2+a^3 x\right )^{2/3}\right )\right )}{12 a^3 \left (x^2 \left (b^2+a^3 x\right )\right )^{2/3}} \]
(x^(4/3)*(b^2 + a^3*x)^(2/3)*(12*a^2*x^(2/3)*(b^2 + a^3*x)^(1/3) + 4*Sqrt[ 3]*(3*a^2 - b)*b*ArcTan[(Sqrt[3]*a*x^(1/3))/(a*x^(1/3) + 2*(b^2 + a^3*x)^( 1/3))] + 6*(3 + I*Sqrt[3])*a^(4/3)*(a^2 - b)^(1/3)*b*ArcTanh[(I + ((-I + S qrt[3])*(b^2 + a^3*x)^(1/3))/(a^(1/3)*(a^2 - b)^(1/3)*x^(1/3)))/Sqrt[3]] + 4*(3*a^2 - b)*b*Log[-(a*x^(1/3)) + (b^2 + a^3*x)^(1/3)] + 6*(1 - I*Sqrt[3 ])*a^(4/3)*(a^2 - b)^(1/3)*b*Log[2*a^(1/3)*(a^2 - b)^(1/3)*x^(1/3) + (1 + I*Sqrt[3])*(b^2 + a^3*x)^(1/3)] + (3*I)*(I + Sqrt[3])*a^(4/3)*(a^2 - b)^(1 /3)*b*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))] - 2*(3*a^2 - b)*b*Log[a^2*x^(2/3) + a*x^(1/3)*(b^2 + a^3*x)^(1/3) + (b^2 + a ^3*x)^(2/3)]))/(12*a^3*(x^2*(b^2 + a^3*x))^(2/3))
Time = 0.41 (sec) , antiderivative size = 347, normalized size of antiderivative = 0.66, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2467, 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 |
| \(\displaystyle \frac {\sqrt [3]{a^3 x^3+b^2 x^2} \int \frac {x^{2/3} \sqrt [3]{x a^3+b^2}}{b+a x}dx}{x^{2/3} \sqrt [3]{a^3 x+b^2}}\) |
| \(\Big \downarrow \) 112 |
| \(\displaystyle \frac {\sqrt [3]{a^3 x^3+b^2 x^2} \left (\frac {x^{2/3} \sqrt [3]{a^3 x+b^2}}{a}-\frac {\int \frac {b \left (2 b^2+a \left (3 a^2-b\right ) x\right )}{3 \sqrt [3]{x} (b+a x) \left (x a^3+b^2\right )^{2/3}}dx}{a}\right )}{x^{2/3} \sqrt [3]{a^3 x+b^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt [3]{a^3 x^3+b^2 x^2} \left (\frac {x^{2/3} \sqrt [3]{a^3 x+b^2}}{a}-\frac {b \int \frac {2 b^2+a \left (3 a^2-b\right ) x}{\sqrt [3]{x} (b+a x) \left (x a^3+b^2\right )^{2/3}}dx}{3 a}\right )}{x^{2/3} \sqrt [3]{a^3 x+b^2}}\) |
| \(\Big \downarrow \) 175 |
| \(\displaystyle \frac {\sqrt [3]{a^3 x^3+b^2 x^2} \left (\frac {x^{2/3} \sqrt [3]{a^3 x+b^2}}{a}-\frac {b \left (\left (3 a^2-b\right ) \int \frac {1}{\sqrt [3]{x} \left (x a^3+b^2\right )^{2/3}}dx-3 b \left (a^2-b\right ) \int \frac {1}{\sqrt [3]{x} (b+a x) \left (x a^3+b^2\right )^{2/3}}dx\right )}{3 a}\right )}{x^{2/3} \sqrt [3]{a^3 x+b^2}}\) |
| \(\Big \downarrow \) 71 |
| \(\displaystyle \frac {\sqrt [3]{a^3 x^3+b^2 x^2} \left (\frac {x^{2/3} \sqrt [3]{a^3 x+b^2}}{a}-\frac {b \left (\left (3 a^2-b\right ) \left (-\frac {\sqrt {3} \arctan \left (\frac {2 a \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{a^3 x+b^2}}+\frac {1}{\sqrt {3}}\right )}{a^2}-\frac {\log \left (a^3 x+b^2\right )}{2 a^2}-\frac {3 \log \left (\frac {a \sqrt [3]{x}}{\sqrt [3]{a^3 x+b^2}}-1\right )}{2 a^2}\right )-3 b \left (a^2-b\right ) \int \frac {1}{\sqrt [3]{x} (b+a x) \left (x a^3+b^2\right )^{2/3}}dx\right )}{3 a}\right )}{x^{2/3} \sqrt [3]{a^3 x+b^2}}\) |
| \(\Big \downarrow \) 102 |
| \(\displaystyle \frac {\sqrt [3]{a^3 x^3+b^2 x^2} \left (\frac {x^{2/3} \sqrt [3]{a^3 x+b^2}}{a}-\frac {b \left (\left (3 a^2-b\right ) \left (-\frac {\sqrt {3} \arctan \left (\frac {2 a \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{a^3 x+b^2}}+\frac {1}{\sqrt {3}}\right )}{a^2}-\frac {\log \left (a^3 x+b^2\right )}{2 a^2}-\frac {3 \log \left (\frac {a \sqrt [3]{x}}{\sqrt [3]{a^3 x+b^2}}-1\right )}{2 a^2}\right )-3 b \left (a^2-b\right ) \left (\frac {\log (a x+b)}{2 a^{2/3} b \left (a^2-b\right )^{2/3}}-\frac {\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{a} \sqrt [3]{x} \sqrt [3]{a^2-b}}{\sqrt {3} \sqrt [3]{a^3 x+b^2}}+\frac {1}{\sqrt {3}}\right )}{a^{2/3} b \left (a^2-b\right )^{2/3}}-\frac {3 \log \left (\sqrt [3]{a} \sqrt [3]{x} \sqrt [3]{a^2-b}-\sqrt [3]{a^3 x+b^2}\right )}{2 a^{2/3} b \left (a^2-b\right )^{2/3}}\right )\right )}{3 a}\right )}{x^{2/3} \sqrt [3]{a^3 x+b^2}}\) |
((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*(a^2 - b)*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)*(a ^2 - b)^(2/3)*b)) + Log[b + a*x]/(2*a^(2/3)*(a^2 - b)^(2/3)*b) - (3*Log[a^ (1/3)*(a^2 - b)^(1/3)*x^(1/3) - (b^2 + a^3*x)^(1/3)])/(2*a^(2/3)*(a^2 - b) ^(2/3)*b))))/(3*a)))/(x^(2/3)*(b^2 + a^3*x)^(1/3))
3.31.88.3.1 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 = 0.54 (sec) , antiderivative size = 429, normalized size of antiderivative = 0.82
| method | result | size |
| pseudoelliptic | \(-\frac {b^{2} x^{2} \left (2 \left (x^{2} \left (a^{3} x +b^{2}\right )\right )^{\frac {1}{3}} {\left (a \left (a^{2}-b \right )\right )}^{\frac {2}{3}} a^{2}+\frac {b \left (-\left (3 a^{2}-b \right ) \left (2 \sqrt {3}\, \arctan \left (\frac {\left (a x +2 \left (x^{2} \left (a^{3} x +b^{2}\right )\right )^{\frac {1}{3}}\right ) \sqrt {3}}{3 a x}\right )+\ln \left (\frac {a^{2} x^{2}+\left (x^{2} \left (a^{3} x +b^{2}\right )\right )^{\frac {1}{3}} a x +\left (x^{2} \left (a^{3} x +b^{2}\right )\right )^{\frac {2}{3}}}{x^{2}}\right )-2 \ln \left (\frac {-a x +\left (x^{2} \left (a^{3} x +b^{2}\right )\right )^{\frac {1}{3}}}{x}\right )\right ) {\left (a \left (a^{2}-b \right )\right )}^{\frac {2}{3}}+3 a^{2} \left (a^{2}-b \right ) \left (2 \arctan \left (\frac {\sqrt {3}\, \left ({\left (a \left (a^{2}-b \right )\right )}^{\frac {1}{3}} x +2 \left (x^{2} \left (a^{3} x +b^{2}\right )\right )^{\frac {1}{3}}\right )}{3 {\left (a \left (a^{2}-b \right )\right )}^{\frac {1}{3}} x}\right ) \sqrt {3}+\ln \left (\frac {{\left (a \left (a^{2}-b \right )\right )}^{\frac {2}{3}} x^{2}+{\left (a \left (a^{2}-b \right )\right )}^{\frac {1}{3}} \left (x^{2} \left (a^{3} x +b^{2}\right )\right )^{\frac {1}{3}} x +\left (x^{2} \left (a^{3} x +b^{2}\right )\right )^{\frac {2}{3}}}{x^{2}}\right )-2 \ln \left (\frac {-{\left (a \left (a^{2}-b \right )\right )}^{\frac {1}{3}} x +\left (x^{2} \left (a^{3} x +b^{2}\right )\right )^{\frac {1}{3}}}{x}\right )\right )\right )}{3}\right )}{2 {\left (a \left (a^{2}-b \right )\right )}^{\frac {2}{3}} \left (a x -\left (x^{2} \left (a^{3} x +b^{2}\right )\right )^{\frac {1}{3}}\right ) a^{3} \left (a^{2} x^{2}+\left (x^{2} \left (a^{3} x +b^{2}\right )\right )^{\frac {1}{3}} a x +\left (x^{2} \left (a^{3} x +b^{2}\right )\right )^{\frac {2}{3}}\right )}\) | \(429\) |
-1/2*b^2/(a*(a^2-b))^(2/3)*x^2*(2*(x^2*(a^3*x+b^2))^(1/3)*(a*(a^2-b))^(2/3 )*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))))/(a*x-(x^2*(a^3*x+b^2))^(1/3) )/a^3/(a^2*x^2+(x^2*(a^3*x+b^2))^(1/3)*a*x+(x^2*(a^3*x+b^2))^(2/3))
Time = 0.29 (sec) , antiderivative size = 437, normalized size of antiderivative = 0.83 \[ \int \frac {\sqrt [3]{b^2 x^2+a^3 x^3}}{b+a x} \, dx=\frac {6 \, \sqrt {3} a^{2} b \left (-\frac {a^{2} - b}{a^{2}}\right )^{\frac {1}{3}} \arctan \left (-\frac {\sqrt {3} {\left (a^{2} - b\right )} x + 2 \, \sqrt {3} {\left (a^{3} x^{3} + b^{2} x^{2}\right )}^{\frac {1}{3}} a \left (-\frac {a^{2} - b}{a^{2}}\right )^{\frac {2}{3}}}{3 \, {\left (a^{2} - b\right )} x}\right ) + 6 \, a^{2} b \left (-\frac {a^{2} - b}{a^{2}}\right )^{\frac {1}{3}} \log \left (\frac {a x \left (-\frac {a^{2} - b}{a^{2}}\right )^{\frac {1}{3}} + {\left (a^{3} x^{3} + b^{2} x^{2}\right )}^{\frac {1}{3}}}{x}\right ) - 3 \, a^{2} b \left (-\frac {a^{2} - b}{a^{2}}\right )^{\frac {1}{3}} \log \left (\frac {a^{2} x^{2} \left (-\frac {a^{2} - b}{a^{2}}\right )^{\frac {2}{3}} - {\left (a^{3} x^{3} + b^{2} x^{2}\right )}^{\frac {1}{3}} a x \left (-\frac {a^{2} - b}{a^{2}}\right )^{\frac {1}{3}} + {\left (a^{3} x^{3} + b^{2} x^{2}\right )}^{\frac {2}{3}}}{x^{2}}\right ) - 2 \, \sqrt {3} {\left (3 \, a^{2} b - b^{2}\right )} \arctan \left (\frac {\sqrt {3} a x + 2 \, \sqrt {3} {\left (a^{3} x^{3} + b^{2} x^{2}\right )}^{\frac {1}{3}}}{3 \, a x}\right ) + 6 \, {\left (a^{3} x^{3} + b^{2} x^{2}\right )}^{\frac {1}{3}} a^{2} + 2 \, {\left (3 \, a^{2} b - b^{2}\right )} \log \left (-\frac {a x - {\left (a^{3} x^{3} + b^{2} x^{2}\right )}^{\frac {1}{3}}}{x}\right ) - {\left (3 \, a^{2} b - b^{2}\right )} \log \left (\frac {a^{2} x^{2} + {\left (a^{3} x^{3} + b^{2} x^{2}\right )}^{\frac {1}{3}} a x + {\left (a^{3} x^{3} + b^{2} x^{2}\right )}^{\frac {2}{3}}}{x^{2}}\right )}{6 \, a^{3}} \]
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)*arcta n(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 = 19.27 (sec) , antiderivative size = 339, 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. \[ \int \frac {\sqrt [3]{b^2 x^2+a^3 x^3}}{b+a x} \, dx=\int \frac {{\left (a^3\,x^3+b^2\,x^2\right )}^{1/3}}{b+a\,x} \,d x \]