\(\int \frac {e+f x}{(c+d x) \sqrt [3]{-c^3+d^3 x^3}} \, dx\) [219]

Optimal result

Integrand size = 30, antiderivative size = 234 \[ \int \frac {e+f x}{(c+d x) \sqrt [3]{-c^3+d^3 x^3}} \, dx=\frac {f \arctan \left (\frac {1+\frac {2 d x}{\sqrt [3]{-c^3+d^3 x^3}}}{\sqrt {3}}\right )}{\sqrt {3} d^2}+\frac {\sqrt {3} (d e-c f) \arctan \left (\frac {1-\frac {\sqrt [3]{2} (c-d x)}{\sqrt [3]{-c^3+d^3 x^3}}}{\sqrt {3}}\right )}{2 \sqrt [3]{2} c d^2}+\frac {(d e-c f) \log \left ((c-d x) (c+d x)^2\right )}{4 \sqrt [3]{2} c d^2}-\frac {f \log \left (-d x+\sqrt [3]{-c^3+d^3 x^3}\right )}{2 d^2}-\frac {3 (d e-c f) \log \left (d (c-d x)+2^{2/3} d \sqrt [3]{-c^3+d^3 x^3}\right )}{4 \sqrt [3]{2} c d^2} \] Output:

1/3*f*arctan(1/3*(1+2*d*x/(d^3*x^3-c^3)^(1/3))*3^(1/2))*3^(1/2)/d^2+1/4*3^ 
(1/2)*(-c*f+d*e)*arctan(1/3*(1-2^(1/3)*(-d*x+c)/(d^3*x^3-c^3)^(1/3))*3^(1/ 
2))*2^(2/3)/c/d^2+1/8*(-c*f+d*e)*ln((-d*x+c)*(d*x+c)^2)*2^(2/3)/c/d^2-1/2* 
f*ln(-d*x+(d^3*x^3-c^3)^(1/3))/d^2-3/8*(-c*f+d*e)*ln(d*(-d*x+c)+2^(2/3)*d* 
(d^3*x^3-c^3)^(1/3))*2^(2/3)/c/d^2
                                                                                    
                                                                                    
 

Mathematica [F]

\[ \int \frac {e+f x}{(c+d x) \sqrt [3]{-c^3+d^3 x^3}} \, dx=\int \frac {e+f x}{(c+d x) \sqrt [3]{-c^3+d^3 x^3}} \, dx \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.77 (sec) , antiderivative size = 227, normalized size of antiderivative = 0.97, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2577, 769, 2574}

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.

\(\displaystyle \int \frac {e+f x}{(c+d x) \sqrt [3]{d^3 x^3-c^3}} \, dx\)

\(\Big \downarrow \) 2577

\(\displaystyle \frac {(d e-c f) \int \frac {1}{(c+d x) \sqrt [3]{d^3 x^3-c^3}}dx}{d}+\frac {f \int \frac {1}{\sqrt [3]{d^3 x^3-c^3}}dx}{d}\)

\(\Big \downarrow \) 769

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

\(\Big \downarrow \) 2574

\(\displaystyle \frac {(d e-c f) \left (\frac {\sqrt {3} \arctan \left (\frac {1-\frac {\sqrt [3]{2} (c-d x)}{\sqrt [3]{d^3 x^3-c^3}}}{\sqrt {3}}\right )}{2 \sqrt [3]{2} c d}-\frac {3 \log \left (2^{2/3} d \sqrt [3]{d^3 x^3-c^3}+d (c-d x)\right )}{4 \sqrt [3]{2} c d}+\frac {\log \left ((c-d x) (c+d x)^2\right )}{4 \sqrt [3]{2} c d}\right )}{d}+\frac {f \left (\frac {\arctan \left (\frac {\frac {2 d x}{\sqrt [3]{d^3 x^3-c^3}}+1}{\sqrt {3}}\right )}{\sqrt {3} d}-\frac {\log \left (\sqrt [3]{d^3 x^3-c^3}-d x\right )}{2 d}\right )}{d}\)

Input:

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

Output:

(f*(ArcTan[(1 + (2*d*x)/(-c^3 + d^3*x^3)^(1/3))/Sqrt[3]]/(Sqrt[3]*d) - Log 
[-(d*x) + (-c^3 + d^3*x^3)^(1/3)]/(2*d)))/d + ((d*e - c*f)*((Sqrt[3]*ArcTa 
n[(1 - (2^(1/3)*(c - d*x))/(-c^3 + d^3*x^3)^(1/3))/Sqrt[3]])/(2*2^(1/3)*c* 
d) + Log[(c - d*x)*(c + d*x)^2]/(4*2^(1/3)*c*d) - (3*Log[d*(c - d*x) + 2^( 
2/3)*d*(-c^3 + d^3*x^3)^(1/3)])/(4*2^(1/3)*c*d)))/d
 

Defintions of rubi rules used

rule 769

Int[((a_) + (b_.)*(x_)^3)^(-1/3), x_Symbol] :> Simp[ArcTan[(1 + 2*Rt[b, 3]* 
(x/(a + b*x^3)^(1/3)))/Sqrt[3]]/(Sqrt[3]*Rt[b, 3]), x] - Simp[Log[(a + b*x^ 
3)^(1/3) - Rt[b, 3]*x]/(2*Rt[b, 3]), x] /; FreeQ[{a, b}, x]
 

rule 2574

Int[1/(((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^3)^(1/3)), x_Symbol] :> Simp[ 
Sqrt[3]*(ArcTan[(1 - 2^(1/3)*Rt[b, 3]*((c - d*x)/(d*(a + b*x^3)^(1/3))))/Sq 
rt[3]]/(2^(4/3)*Rt[b, 3]*c)), x] + (Simp[Log[(c + d*x)^2*(c - d*x)]/(2^(7/3 
)*Rt[b, 3]*c), x] - Simp[(3*Log[Rt[b, 3]*(c - d*x) + 2^(2/3)*d*(a + b*x^3)^ 
(1/3)])/(2^(7/3)*Rt[b, 3]*c), x]) /; FreeQ[{a, b, c, d}, x] && EqQ[b*c^3 + 
a*d^3, 0]
 

rule 2577

Int[((e_.) + (f_.)*(x_))/(((c_.) + (d_.)*(x_))*((a_) + (b_.)*(x_)^3)^(1/3)) 
, x_Symbol] :> Simp[f/d   Int[1/(a + b*x^3)^(1/3), x], x] + Simp[(d*e - c*f 
)/d   Int[1/((c + d*x)*(a + b*x^3)^(1/3)), x], x] /; FreeQ[{a, b, c, d, e, 
f}, x]
 

Maple [F]

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

Input:

int((f*x+e)/(d*x+c)/(d^3*x^3-c^3)^(1/3),x)
 

Output:

int((f*x+e)/(d*x+c)/(d^3*x^3-c^3)^(1/3),x)
 

Fricas [F(-1)]

Timed out. \[ \int \frac {e+f x}{(c+d x) \sqrt [3]{-c^3+d^3 x^3}} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Sympy [F]

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

integrate((f*x+e)/(d*x+c)/(d**3*x**3-c**3)**(1/3),x)
 

Output:

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

Maxima [F]

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

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

Output:

integrate((f*x + e)/((d^3*x^3 - c^3)^(1/3)*(d*x + c)), x)
 

Giac [F]

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

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

Output:

integrate((f*x + e)/((d^3*x^3 - c^3)^(1/3)*(d*x + c)), x)
 

Mupad [F(-1)]

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

int((e + f*x)/((d^3*x^3 - c^3)^(1/3)*(c + d*x)),x)
 

Output:

int((e + f*x)/((d^3*x^3 - c^3)^(1/3)*(c + d*x)), x)
 

Reduce [F]

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

int((f*x+e)/(d*x+c)/(d^3*x^3-c^3)^(1/3),x)
 

Output:

int(x/(( - c**3 + d**3*x**3)**(1/3)*c + ( - c**3 + d**3*x**3)**(1/3)*d*x), 
x)*f + int(1/(( - c**3 + d**3*x**3)**(1/3)*c + ( - c**3 + d**3*x**3)**(1/3 
)*d*x),x)*e