\(\int \frac {x}{(a+b x^3)^{2/3} (c+d x^3)} \, dx\) [755]

Optimal result

Integrand size = 22, antiderivative size = 149 \[ \int \frac {x}{\left (a+b x^3\right )^{2/3} \left (c+d x^3\right )} \, dx=-\frac {\arctan \left (\frac {1+\frac {2 \sqrt [3]{b c-a d} x}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{c} (b c-a d)^{2/3}}+\frac {\log \left (c+d x^3\right )}{6 \sqrt [3]{c} (b c-a d)^{2/3}}-\frac {\log \left (\frac {\sqrt [3]{b c-a d} x}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{2 \sqrt [3]{c} (b c-a d)^{2/3}} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 1.26 (sec) , antiderivative size = 255, normalized size of antiderivative = 1.71 \[ \int \frac {x}{\left (a+b x^3\right )^{2/3} \left (c+d x^3\right )} \, dx=\frac {2 \sqrt {-6-6 i \sqrt {3}} \arctan \left (\frac {3 \sqrt [3]{b c-a d} x}{\sqrt {3} \sqrt [3]{b c-a d} x-\left (3 i+\sqrt {3}\right ) \sqrt [3]{c} \sqrt [3]{a+b x^3}}\right )+\left (1-i \sqrt {3}\right ) \left (2 \log \left (2 \sqrt [3]{b c-a d} x+\left (1+i \sqrt {3}\right ) \sqrt [3]{c} \sqrt [3]{a+b x^3}\right )-\log \left (2 (b c-a d)^{2/3} x^2+\left (-1-i \sqrt {3}\right ) \sqrt [3]{c} \sqrt [3]{b c-a d} x \sqrt [3]{a+b x^3}+i \left (i+\sqrt {3}\right ) c^{2/3} \left (a+b x^3\right )^{2/3}\right )\right )}{12 \sqrt [3]{c} (b c-a d)^{2/3}} \] Input:

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

Output:

(2*Sqrt[-6 - (6*I)*Sqrt[3]]*ArcTan[(3*(b*c - a*d)^(1/3)*x)/(Sqrt[3]*(b*c - 
 a*d)^(1/3)*x - (3*I + Sqrt[3])*c^(1/3)*(a + b*x^3)^(1/3))] + (1 - I*Sqrt[ 
3])*(2*Log[2*(b*c - a*d)^(1/3)*x + (1 + I*Sqrt[3])*c^(1/3)*(a + b*x^3)^(1/ 
3)] - Log[2*(b*c - a*d)^(2/3)*x^2 + (-1 - I*Sqrt[3])*c^(1/3)*(b*c - a*d)^( 
1/3)*x*(a + b*x^3)^(1/3) + I*(I + Sqrt[3])*c^(2/3)*(a + b*x^3)^(2/3)]))/(1 
2*c^(1/3)*(b*c - a*d)^(2/3))
 

Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {992}

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.

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

\(\Big \downarrow \) 992

\(\displaystyle -\frac {\arctan \left (\frac {\frac {2 x \sqrt [3]{b c-a d}}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{c} (b c-a d)^{2/3}}+\frac {\log \left (c+d x^3\right )}{6 \sqrt [3]{c} (b c-a d)^{2/3}}-\frac {\log \left (\frac {x \sqrt [3]{b c-a d}}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{2 \sqrt [3]{c} (b c-a d)^{2/3}}\)

Input:

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

Output:

-(ArcTan[(1 + (2*(b*c - a*d)^(1/3)*x)/(c^(1/3)*(a + b*x^3)^(1/3)))/Sqrt[3] 
]/(Sqrt[3]*c^(1/3)*(b*c - a*d)^(2/3))) + Log[c + d*x^3]/(6*c^(1/3)*(b*c - 
a*d)^(2/3)) - Log[((b*c - a*d)^(1/3)*x)/c^(1/3) - (a + b*x^3)^(1/3)]/(2*c^ 
(1/3)*(b*c - a*d)^(2/3))
 

Defintions of rubi rules used

rule 992

Int[(x_)/(((a_) + (b_.)*(x_)^3)^(2/3)*((c_) + (d_.)*(x_)^3)), x_Symbol] :> 
With[{q = Rt[(b*c - a*d)/c, 3]}, Simp[-ArcTan[(1 + (2*q*x)/(a + b*x^3)^(1/3 
))/Sqrt[3]]/(Sqrt[3]*c*q^2), x] + (-Simp[Log[q*x - (a + b*x^3)^(1/3)]/(2*c* 
q^2), x] + Simp[Log[c + d*x^3]/(6*c*q^2), x])] /; FreeQ[{a, b, c, d}, x] && 
 NeQ[b*c - a*d, 0]
 

Maple [A] (verified)

Time = 1.14 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.13

method	result	size
pseudoelliptic	\(\frac {2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} x -2 \left (b \,x^{3}+a \right )^{\frac {1}{3}}\right )}{3 \left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} x}\right )+\ln \left (\frac {\left (\frac {a d -b c}{c}\right )^{\frac {2}{3}} x^{2}-\left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {2}{3}}}{x^{2}}\right )-2 \ln \left (\frac {\left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{x}\right )}{6 \left (\frac {a d -b c}{c}\right )^{\frac {2}{3}} c}\)	\(169\)

Input:

int(x/(b*x^3+a)^(2/3)/(d*x^3+c),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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