\(\int \frac {\sqrt [3]{a+b x}}{\sqrt [3]{c+d x} (e+f x)^2} \, dx\) [1637]

Optimal result

Integrand size = 26, antiderivative size = 256 \[ \int \frac {\sqrt [3]{a+b x}}{\sqrt [3]{c+d x} (e+f x)^2} \, dx=\frac {\sqrt [3]{a+b x} (c+d x)^{2/3}}{(d e-c f) (e+f x)}+\frac {(b c-a d) \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{b e-a f} \sqrt [3]{c+d x}}{\sqrt {3} \sqrt [3]{d e-c f} \sqrt [3]{a+b x}}\right )}{\sqrt {3} (b e-a f)^{2/3} (d e-c f)^{4/3}}-\frac {(b c-a d) \log (e+f x)}{6 (b e-a f)^{2/3} (d e-c f)^{4/3}}+\frac {(b c-a d) \log \left (-\sqrt [3]{a+b x}+\frac {\sqrt [3]{b e-a f} \sqrt [3]{c+d x}}{\sqrt [3]{d e-c f}}\right )}{2 (b e-a f)^{2/3} (d e-c f)^{4/3}} \] Output:

(b*x+a)^(1/3)*(d*x+c)^(2/3)/(-c*f+d*e)/(f*x+e)+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)*3^(1/2)/(-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)-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)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 8.40 (sec) , antiderivative size = 361, normalized size of antiderivative = 1.41 \[ \int \frac {\sqrt [3]{a+b x}}{\sqrt [3]{c+d x} (e+f x)^2} \, dx=\frac {1}{6} \left (\frac {6 \sqrt [3]{a+b x} (c+d x)^{2/3}}{(d e-c f) (e+f x)}-\frac {2 \sqrt {3} (b c-a d) \arctan \left (\frac {\sqrt {3} \sqrt [3]{-d e+c f} \sqrt [3]{a+b x}}{\sqrt [3]{-d e+c f} \sqrt [3]{a+b x}-2 \sqrt [3]{b e-a f} \sqrt [3]{c+d x}}\right )}{(b e-a f)^{2/3} (-d e+c f)^{4/3}}+\frac {2 (b c-a d) \log \left (\sqrt [3]{-d e+c f} \sqrt [3]{a+b x}+\sqrt [3]{b e-a f} \sqrt [3]{c+d x}\right )}{(b e-a f)^{2/3} (-d e+c f)^{4/3}}-\frac {(b c-a d) \log \left ((-d e+c f)^{2/3} (a+b x)^{2/3}-\sqrt [3]{b e-a f} \sqrt [3]{-d e+c f} \sqrt [3]{a+b x} \sqrt [3]{c+d x}+(b e-a f)^{2/3} (c+d x)^{2/3}\right )}{(b e-a f)^{2/3} (-d e+c f)^{4/3}}\right ) \] Input:

Integrate[(a + b*x)^(1/3)/((c + d*x)^(1/3)*(e + f*x)^2),x]
 

Output:

((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
 

Rubi [A] (verified)

Time = 0.26 (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

\(\displaystyle \frac {\sqrt [3]{a+b x} (c+d x)^{2/3}}{(e+f x) (d e-c f)}-\frac {(b c-a d) \left (-\frac {\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt {3} \sqrt [3]{a+b x} \sqrt [3]{d e-c f}}+\frac {1}{\sqrt {3}}\right )}{(b e-a f)^{2/3} \sqrt [3]{d e-c f}}+\frac {\log (e+f x)}{2 (b e-a f)^{2/3} \sqrt [3]{d e-c f}}-\frac {3 \log \left (\frac {\sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt [3]{d e-c f}}-\sqrt [3]{a+b x}\right )}{2 (b e-a f)^{2/3} \sqrt [3]{d e-c f}}\right )}{3 (d e-c f)}\)

Input:

Int[(a + b*x)^(1/3)/((c + d*x)^(1/3)*(e + f*x)^2),x]
 

Output:

((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))
 

Defintions of rubi rules used

rule 102

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]
 

rule 105

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]
 

Maple [F]

\[\int \frac {\left (b x +a \right )^{\frac {1}{3}}}{\left (x d +c \right )^{\frac {1}{3}} \left (f x +e \right )^{2}}d x\]

Input:

int((b*x+a)^(1/3)/(d*x+c)^(1/3)/(f*x+e)^2,x)
 

Output:

int((b*x+a)^(1/3)/(d*x+c)^(1/3)/(f*x+e)^2,x)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1171 vs. \(2 (217) = 434\).

Time = 0.20 (sec) , antiderivative size = 2504, normalized size of antiderivative = 9.78 \[ \int \frac {\sqrt [3]{a+b x}}{\sqrt [3]{c+d x} (e+f x)^2} \, dx=\text {Too large to display} \] Input:

integrate((b*x+a)^(1/3)/(d*x+c)^(1/3)/(f*x+e)^2,x, algorithm="fricas")
 

Output:

[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)...
 

Sympy [F]

\[ \int \frac {\sqrt [3]{a+b x}}{\sqrt [3]{c+d x} (e+f x)^2} \, dx=\int \frac {\sqrt [3]{a + b x}}{\sqrt [3]{c + d x} \left (e + f x\right )^{2}}\, dx \] Input:

integrate((b*x+a)**(1/3)/(d*x+c)**(1/3)/(f*x+e)**2,x)
 

Output:

Integral((a + b*x)**(1/3)/((c + d*x)**(1/3)*(e + f*x)**2), x)
 

Maxima [F]

\[ \int \frac {\sqrt [3]{a+b x}}{\sqrt [3]{c+d x} (e+f x)^2} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {1}{3}}}{{\left (d x + c\right )}^{\frac {1}{3}} {\left (f x + e\right )}^{2}} \,d x } \] Input:

integrate((b*x+a)^(1/3)/(d*x+c)^(1/3)/(f*x+e)^2,x, algorithm="maxima")
 

Output:

integrate((b*x + a)^(1/3)/((d*x + c)^(1/3)*(f*x + e)^2), x)
 

Giac [F]

\[ \int \frac {\sqrt [3]{a+b x}}{\sqrt [3]{c+d x} (e+f x)^2} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {1}{3}}}{{\left (d x + c\right )}^{\frac {1}{3}} {\left (f x + e\right )}^{2}} \,d x } \] Input:

integrate((b*x+a)^(1/3)/(d*x+c)^(1/3)/(f*x+e)^2,x, algorithm="giac")
 

Output:

integrate((b*x + a)^(1/3)/((d*x + c)^(1/3)*(f*x + e)^2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt [3]{a+b x}}{\sqrt [3]{c+d x} (e+f x)^2} \, dx=\int \frac {{\left (a+b\,x\right )}^{1/3}}{{\left (e+f\,x\right )}^2\,{\left (c+d\,x\right )}^{1/3}} \,d x \] Input:

int((a + b*x)^(1/3)/((e + f*x)^2*(c + d*x)^(1/3)),x)
 

Output:

int((a + b*x)^(1/3)/((e + f*x)^2*(c + d*x)^(1/3)), x)
 

Reduce [F]

\[ \int \frac {\sqrt [3]{a+b x}}{\sqrt [3]{c+d x} (e+f x)^2} \, dx=\int \frac {\left (b x +a \right )^{\frac {1}{3}}}{\left (d x +c \right )^{\frac {1}{3}} \left (f x +e \right )^{2}}d x \] Input:

int((b*x+a)^(1/3)/(d*x+c)^(1/3)/(f*x+e)^2,x)
 

Output:

int((b*x+a)^(1/3)/(d*x+c)^(1/3)/(f*x+e)^2,x)