Integrand size = 26, antiderivative size = 256 \[ \int \frac {\sqrt [3]{a+b x}}{\sqrt [3]{c+d x} (e+f x)^2} \, dx=\frac {\sqrt [3]{a+b x} (c+d x)^{2/3}}{(d e-c f) (e+f x)}+\frac {(b c-a d) \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{b e-a f} \sqrt [3]{c+d x}}{\sqrt {3} \sqrt [3]{d e-c f} \sqrt [3]{a+b x}}\right )}{\sqrt {3} (b e-a f)^{2/3} (d e-c f)^{4/3}}-\frac {(b c-a d) \log (e+f x)}{6 (b e-a f)^{2/3} (d e-c f)^{4/3}}+\frac {(b c-a d) \log \left (-\sqrt [3]{a+b x}+\frac {\sqrt [3]{b e-a f} \sqrt [3]{c+d x}}{\sqrt [3]{d e-c f}}\right )}{2 (b e-a f)^{2/3} (d e-c f)^{4/3}} \]
(b*x+a)^(1/3)*(d*x+c)^(2/3)/(-c*f+d*e)/(f*x+e)-1/6*(-a*d+b*c)*ln(f*x+e)/(- a*f+b*e)^(2/3)/(-c*f+d*e)^(4/3)+1/2*(-a*d+b*c)*ln(-(b*x+a)^(1/3)+(-a*f+b*e )^(1/3)*(d*x+c)^(1/3)/(-c*f+d*e)^(1/3))/(-a*f+b*e)^(2/3)/(-c*f+d*e)^(4/3)+ 1/3*(-a*d+b*c)*arctan(1/3*3^(1/2)+2/3*(-a*f+b*e)^(1/3)*(d*x+c)^(1/3)/(-c*f +d*e)^(1/3)/(b*x+a)^(1/3)*3^(1/2))/(-a*f+b*e)^(2/3)/(-c*f+d*e)^(4/3)*3^(1/ 2)
Time = 9.45 (sec) , antiderivative size = 361, normalized size of antiderivative = 1.41 \[ \int \frac {\sqrt [3]{a+b x}}{\sqrt [3]{c+d x} (e+f x)^2} \, dx=\frac {1}{6} \left (\frac {6 \sqrt [3]{a+b x} (c+d x)^{2/3}}{(d e-c f) (e+f x)}-\frac {2 \sqrt {3} (b c-a d) \arctan \left (\frac {\sqrt {3} \sqrt [3]{-d e+c f} \sqrt [3]{a+b x}}{\sqrt [3]{-d e+c f} \sqrt [3]{a+b x}-2 \sqrt [3]{b e-a f} \sqrt [3]{c+d x}}\right )}{(b e-a f)^{2/3} (-d e+c f)^{4/3}}+\frac {2 (b c-a d) \log \left (\sqrt [3]{-d e+c f} \sqrt [3]{a+b x}+\sqrt [3]{b e-a f} \sqrt [3]{c+d x}\right )}{(b e-a f)^{2/3} (-d e+c f)^{4/3}}-\frac {(b c-a d) \log \left ((-d e+c f)^{2/3} (a+b x)^{2/3}-\sqrt [3]{b e-a f} \sqrt [3]{-d e+c f} \sqrt [3]{a+b x} \sqrt [3]{c+d x}+(b e-a f)^{2/3} (c+d x)^{2/3}\right )}{(b e-a f)^{2/3} (-d e+c f)^{4/3}}\right ) \]
((6*(a + b*x)^(1/3)*(c + d*x)^(2/3))/((d*e - c*f)*(e + f*x)) - (2*Sqrt[3]* (b*c - a*d)*ArcTan[(Sqrt[3]*(-(d*e) + c*f)^(1/3)*(a + b*x)^(1/3))/((-(d*e) + c*f)^(1/3)*(a + b*x)^(1/3) - 2*(b*e - a*f)^(1/3)*(c + d*x)^(1/3))])/((b *e - a*f)^(2/3)*(-(d*e) + c*f)^(4/3)) + (2*(b*c - a*d)*Log[(-(d*e) + c*f)^ (1/3)*(a + b*x)^(1/3) + (b*e - a*f)^(1/3)*(c + d*x)^(1/3)])/((b*e - a*f)^( 2/3)*(-(d*e) + c*f)^(4/3)) - ((b*c - a*d)*Log[(-(d*e) + c*f)^(2/3)*(a + b* x)^(2/3) - (b*e - a*f)^(1/3)*(-(d*e) + c*f)^(1/3)*(a + b*x)^(1/3)*(c + d*x )^(1/3) + (b*e - a*f)^(2/3)*(c + d*x)^(2/3)])/((b*e - a*f)^(2/3)*(-(d*e) + c*f)^(4/3)))/6
Time = 0.27 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {105, 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.
\(\displaystyle \int \frac {\sqrt [3]{a+b x}}{\sqrt [3]{c+d x} (e+f x)^2} \, dx\) |
\(\Big \downarrow \) 105 |
\(\displaystyle \frac {\sqrt [3]{a+b x} (c+d x)^{2/3}}{(e+f x) (d e-c f)}-\frac {(b c-a d) \int \frac {1}{(a+b x)^{2/3} \sqrt [3]{c+d x} (e+f x)}dx}{3 (d e-c f)}\) |
\(\Big \downarrow \) 102 |
\(\displaystyle \frac {\sqrt [3]{a+b x} (c+d x)^{2/3}}{(e+f x) (d e-c f)}-\frac {(b c-a d) \left (-\frac {\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt {3} \sqrt [3]{a+b x} \sqrt [3]{d e-c f}}+\frac {1}{\sqrt {3}}\right )}{(b e-a f)^{2/3} \sqrt [3]{d e-c f}}+\frac {\log (e+f x)}{2 (b e-a f)^{2/3} \sqrt [3]{d e-c f}}-\frac {3 \log \left (\frac {\sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt [3]{d e-c f}}-\sqrt [3]{a+b x}\right )}{2 (b e-a f)^{2/3} \sqrt [3]{d e-c f}}\right )}{3 (d e-c f)}\) |
((a + b*x)^(1/3)*(c + d*x)^(2/3))/((d*e - c*f)*(e + f*x)) - ((b*c - a*d)*( -((Sqrt[3]*ArcTan[1/Sqrt[3] + (2*(b*e - a*f)^(1/3)*(c + d*x)^(1/3))/(Sqrt[ 3]*(d*e - c*f)^(1/3)*(a + b*x)^(1/3))])/((b*e - a*f)^(2/3)*(d*e - c*f)^(1/ 3))) + Log[e + f*x]/(2*(b*e - a*f)^(2/3)*(d*e - c*f)^(1/3)) - (3*Log[-(a + b*x)^(1/3) + ((b*e - a*f)^(1/3)*(c + d*x)^(1/3))/(d*e - c*f)^(1/3)])/(2*( b*e - a*f)^(2/3)*(d*e - c*f)^(1/3))))/(3*(d*e - c*f))
3.31.12.3.1 Defintions of rubi rules used
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 + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((m + 1)*(b*e - a*f))), x] - Simp[n*((d*e - c*f)/((m + 1)*(b*e - a*f))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[m + n + p + 2, 0] && GtQ[n, 0] && (SumSimplerQ[m, 1] || !SumSimplerQ[p, 1]) && NeQ[m, -1]
\[\int \frac {\left (b x +a \right )^{\frac {1}{3}}}{\left (d x +c \right )^{\frac {1}{3}} \left (f x +e \right )^{2}}d x\]
Leaf count of result is larger than twice the leaf count of optimal. 1171 vs. \(2 (217) = 434\).
Time = 0.34 (sec) , antiderivative size = 2504, normalized size of antiderivative = 9.78 \[ \int \frac {\sqrt [3]{a+b x}}{\sqrt [3]{c+d x} (e+f x)^2} \, dx=\text {Too large to display} \]
[1/6*(3*sqrt(1/3)*((b^2*c*d - a*b*d^2)*e^3 - (b^2*c^2 - a^2*d^2)*e^2*f + ( a*b*c^2 - a^2*c*d)*e*f^2 + ((b^2*c*d - a*b*d^2)*e^2*f - (b^2*c^2 - a^2*d^2 )*e*f^2 + (a*b*c^2 - a^2*c*d)*f^3)*x)*sqrt(-(b^2*d*e^3 - a^2*c*f^3 - (b^2* c + 2*a*b*d)*e^2*f + (2*a*b*c + a^2*d)*e*f^2)^(1/3)/(d*e - c*f))*log(-(3*a ^2*c*f^2 + (b^2*c + 2*a*b*d)*e^2 - 2*(2*a*b*c + a^2*d)*e*f - 3*(b^2*d*e^3 - a^2*c*f^3 - (b^2*c + 2*a*b*d)*e^2*f + (2*a*b*c + a^2*d)*e*f^2)^(1/3)*(b* e - a*f)*(b*x + a)^(1/3)*(d*x + c)^(2/3) + (3*b^2*d*e^2 - 2*(b^2*c + 2*a*b *d)*e*f + (2*a*b*c + a^2*d)*f^2)*x - 3*sqrt(1/3)*(2*(b*d*e^2 + a*c*f^2 - ( b*c + a*d)*e*f)*(b*x + a)^(2/3)*(d*x + c)^(1/3) - (b^2*d*e^3 - a^2*c*f^3 - (b^2*c + 2*a*b*d)*e^2*f + (2*a*b*c + a^2*d)*e*f^2)^(2/3)*(b*x + a)^(1/3)* (d*x + c)^(2/3) - (b^2*d*e^3 - a^2*c*f^3 - (b^2*c + 2*a*b*d)*e^2*f + (2*a* b*c + a^2*d)*e*f^2)^(1/3)*(b*c*e - a*c*f + (b*d*e - a*d*f)*x))*sqrt(-(b^2* d*e^3 - a^2*c*f^3 - (b^2*c + 2*a*b*d)*e^2*f + (2*a*b*c + a^2*d)*e*f^2)^(1/ 3)/(d*e - c*f)))/(f*x + e)) - (b^2*d*e^3 - a^2*c*f^3 - (b^2*c + 2*a*b*d)*e ^2*f + (2*a*b*c + a^2*d)*e*f^2)^(2/3)*((b*c - a*d)*f*x + (b*c - a*d)*e)*lo g(((b*d*e^2 + a*c*f^2 - (b*c + a*d)*e*f)*(b*x + a)^(2/3)*(d*x + c)^(1/3) + (b^2*d*e^3 - a^2*c*f^3 - (b^2*c + 2*a*b*d)*e^2*f + (2*a*b*c + a^2*d)*e*f^ 2)^(2/3)*(b*x + a)^(1/3)*(d*x + c)^(2/3) + (b^2*d*e^3 - a^2*c*f^3 - (b^2*c + 2*a*b*d)*e^2*f + (2*a*b*c + a^2*d)*e*f^2)^(1/3)*(b*c*e - a*c*f + (b*d*e - a*d*f)*x))/(d*x + c)) + 2*(b^2*d*e^3 - a^2*c*f^3 - (b^2*c + 2*a*b*d)...
\[ \int \frac {\sqrt [3]{a+b x}}{\sqrt [3]{c+d x} (e+f x)^2} \, dx=\int \frac {\sqrt [3]{a + b x}}{\sqrt [3]{c + d x} \left (e + f x\right )^{2}}\, dx \]
\[ \int \frac {\sqrt [3]{a+b x}}{\sqrt [3]{c+d x} (e+f x)^2} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {1}{3}}}{{\left (d x + c\right )}^{\frac {1}{3}} {\left (f x + e\right )}^{2}} \,d x } \]
\[ \int \frac {\sqrt [3]{a+b x}}{\sqrt [3]{c+d x} (e+f x)^2} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {1}{3}}}{{\left (d x + c\right )}^{\frac {1}{3}} {\left (f x + e\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {\sqrt [3]{a+b x}}{\sqrt [3]{c+d x} (e+f x)^2} \, dx=\int \frac {{\left (a+b\,x\right )}^{1/3}}{{\left (e+f\,x\right )}^2\,{\left (c+d\,x\right )}^{1/3}} \,d x \]