\(\int \frac {1}{(c+d x) (2 c^3+d^3 x^3)^{2/3}} \, dx\) [140]

Optimal result

Integrand size = 25, antiderivative size = 187 \[ \int \frac {1}{(c+d x) \left (2 c^3+d^3 x^3\right )^{2/3}} \, dx=-\frac {\arctan \left (\frac {1+\frac {2 d x}{\sqrt [3]{2 c^3+d^3 x^3}}}{\sqrt {3}}\right )}{2 \sqrt {3} c^2 d}+\frac {\sqrt {3} \arctan \left (\frac {1+\frac {2 (2 c+d x)}{\sqrt [3]{2 c^3+d^3 x^3}}}{\sqrt {3}}\right )}{2 c^2 d}-\frac {\log (c+d x)}{2 c^2 d}-\frac {\log \left (d x-\sqrt [3]{2 c^3+d^3 x^3}\right )}{4 c^2 d}+\frac {3 \log \left (d (2 c+d x)-d \sqrt [3]{2 c^3+d^3 x^3}\right )}{4 c^2 d} \] Output:

-1/6*arctan(1/3*(1+2*d*x/(d^3*x^3+2*c^3)^(1/3))*3^(1/2))*3^(1/2)/c^2/d+1/2 
*3^(1/2)*arctan(1/3*(1+2*(d*x+2*c)/(d^3*x^3+2*c^3)^(1/3))*3^(1/2))/c^2/d-1 
/2*ln(d*x+c)/c^2/d-1/4*ln(d*x-(d^3*x^3+2*c^3)^(1/3))/c^2/d+3/4*ln(d*(d*x+2 
*c)-d*(d^3*x^3+2*c^3)^(1/3))/c^2/d
 

Mathematica [F]

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

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

Output:

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

Rubi [A] (verified)

Time = 0.47 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {2579}

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.

\(\displaystyle \int \frac {1}{(c+d x) \left (2 c^3+d^3 x^3\right )^{2/3}} \, dx\)

\(\Big \downarrow \) 2579

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 2579

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

Maple [F]

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

Input:

int(1/(d*x+c)/(d^3*x^3+2*c^3)^(2/3),x)
 

Output:

int(1/(d*x+c)/(d^3*x^3+2*c^3)^(2/3),x)
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

integrate(1/(d*x+c)/(d**3*x**3+2*c**3)**(2/3),x)
 

Output:

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

Maxima [F]

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

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

Output:

integrate(1/((d^3*x^3 + 2*c^3)^(2/3)*(d*x + c)), x)
 

Giac [F]

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

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

Output:

integrate(1/((d^3*x^3 + 2*c^3)^(2/3)*(d*x + c)), x)
 

Mupad [F(-1)]

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

int(1/((2*c^3 + d^3*x^3)^(2/3)*(c + d*x)),x)
 

Output:

int(1/((2*c^3 + d^3*x^3)^(2/3)*(c + d*x)), x)
 

Reduce [F]

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

int(1/(d*x+c)/(d^3*x^3+2*c^3)^(2/3),x)
 

Output:

int(1/((2*c**3 + d**3*x**3)**(2/3)*c + (2*c**3 + d**3*x**3)**(2/3)*d*x),x)