\(\int \frac {1}{\sqrt [3]{a+b x^3} (c+d x^3)^2} \, dx\) [102]

Optimal result

Integrand size = 21, antiderivative size = 217 \[ \int \frac {1}{\sqrt [3]{a+b x^3} \left (c+d x^3\right )^2} \, dx=-\frac {d x \left (a+b x^3\right )^{2/3}}{3 c (b c-a d) \left (c+d x^3\right )}+\frac {(3 b c-2 a d) \arctan \left (\frac {1+\frac {2 \sqrt [3]{b c-a d} x}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3} c^{5/3} (b c-a d)^{4/3}}+\frac {(3 b c-2 a d) \log \left (c+d x^3\right )}{18 c^{5/3} (b c-a d)^{4/3}}-\frac {(3 b c-2 a d) \log \left (\frac {\sqrt [3]{b c-a d} x}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{6 c^{5/3} (b c-a d)^{4/3}} \] Output:

-1/3*d*x*(b*x^3+a)^(2/3)/c/(-a*d+b*c)/(d*x^3+c)+1/9*(-2*a*d+3*b*c)*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^(5 
/3)/(-a*d+b*c)^(4/3)+1/18*(-2*a*d+3*b*c)*ln(d*x^3+c)/c^(5/3)/(-a*d+b*c)^(4 
/3)-1/6*(-2*a*d+3*b*c)*ln((-a*d+b*c)^(1/3)*x/c^(1/3)-(b*x^3+a)^(1/3))/c^(5 
/3)/(-a*d+b*c)^(4/3)
                                                                                    
                                                                                    
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 2.17 (sec) , antiderivative size = 336, normalized size of antiderivative = 1.55 \[ \int \frac {1}{\sqrt [3]{a+b x^3} \left (c+d x^3\right )^2} \, dx=\frac {-12 c^{2/3} d \sqrt [3]{b c-a d} x \left (a+b x^3\right )^{2/3}+2 \left (3-i \sqrt {3}\right ) (3 b c-2 a d) \left (c+d x^3\right ) \text {arctanh}\left (\frac {i+\frac {\left (-i+\sqrt {3}\right ) \sqrt [3]{c} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d} x}}{\sqrt {3}}\right )+2 \left (1+i \sqrt {3}\right ) (3 b c-2 a d) \left (c+d x^3\right ) \log \left (2 \sqrt [3]{b c-a d} x+\left (1+i \sqrt {3}\right ) \sqrt [3]{c} \sqrt [3]{a+b x^3}\right )-i \left (-i+\sqrt {3}\right ) (3 b c-2 a d) \left (c+d x^3\right ) \log \left (2 (b c-a d)^{2/3} x^2+\left (-1-i \sqrt {3}\right ) \sqrt [3]{c} \sqrt [3]{b c-a d} x \sqrt [3]{a+b x^3}+i \left (i+\sqrt {3}\right ) c^{2/3} \left (a+b x^3\right )^{2/3}\right )}{36 c^{5/3} (b c-a d)^{4/3} \left (c+d x^3\right )} \] Input:

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

Output:

(-12*c^(2/3)*d*(b*c - a*d)^(1/3)*x*(a + b*x^3)^(2/3) + 2*(3 - I*Sqrt[3])*( 
3*b*c - 2*a*d)*(c + d*x^3)*ArcTanh[(I + ((-I + Sqrt[3])*c^(1/3)*(a + b*x^3 
)^(1/3))/((b*c - a*d)^(1/3)*x))/Sqrt[3]] + 2*(1 + I*Sqrt[3])*(3*b*c - 2*a* 
d)*(c + d*x^3)*Log[2*(b*c - a*d)^(1/3)*x + (1 + I*Sqrt[3])*c^(1/3)*(a + b* 
x^3)^(1/3)] - I*(-I + Sqrt[3])*(3*b*c - 2*a*d)*(c + d*x^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)])/(36*c^(5/3)*(b*c - a*d)^ 
(4/3)*(c + d*x^3))
 

Rubi [A] (verified)

Time = 0.47 (sec) , antiderivative size = 214, normalized size of antiderivative = 0.99, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {907, 901}

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.

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

\(\Big \downarrow \) 907

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

\(\Big \downarrow \) 901

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 901

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

rule 907

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] 
:> Simp[(-b)*x*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*n*(p + 1)*(b*c - 
 a*d))), x] + Simp[(b*c + n*(p + 1)*(b*c - a*d))/(a*n*(p + 1)*(b*c - a*d)) 
  Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n, q} 
, x] && NeQ[b*c - a*d, 0] && EqQ[n*(p + q + 2) + 1, 0] && (LtQ[p, -1] ||  ! 
LtQ[q, -1]) && NeQ[p, -1]
 

Maple [A] (verified)

Time = 4.41 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.13

method	result	size
pseudoelliptic	\(\frac {-\frac {\left (d \,x^{3}+c \right ) \left (a d -\frac {3 b c}{2}\right ) \ln \left (\frac {\left (\frac {a d -b c}{c}\right )^{\frac {2}{3}} x^{2}-\left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {2}{3}}}{x^{2}}\right )}{9}+\frac {2 \left (d \,x^{3}+c \right ) \left (a d -\frac {3 b c}{2}\right ) \ln \left (\frac {\left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{x}\right )}{9}+\frac {d \left (b \,x^{3}+a \right )^{\frac {2}{3}} x c \left (\frac {a d -b c}{c}\right )^{\frac {1}{3}}}{3}+\frac {2 \left (d \,x^{3}+c \right ) \arctan \left (\frac {\sqrt {3}\, \left (-\frac {2 \left (b \,x^{3}+a \right )^{\frac {1}{3}}}{\left (\frac {a d -b c}{c}\right )^{\frac {1}{3}}}+x \right )}{3 x}\right ) \sqrt {3}\, \left (a d -\frac {3 b c}{2}\right )}{9}}{\left (a d -b c \right ) c^{2} \left (d \,x^{3}+c \right ) \left (\frac {a d -b c}{c}\right )^{\frac {1}{3}}}\)	\(246\)

Input:

int(1/(b*x^3+a)^(1/3)/(d*x^3+c)^2,x,method=_RETURNVERBOSE)
 

Output:

2/9/((a*d-b*c)/c)^(1/3)*(-1/2*(d*x^3+c)*(a*d-3/2*b*c)*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)+(d*x^3 
+c)*(a*d-3/2*b*c)*ln((((a*d-b*c)/c)^(1/3)*x+(b*x^3+a)^(1/3))/x)+3/2*d*(b*x 
^3+a)^(2/3)*x*c*((a*d-b*c)/c)^(1/3)+(d*x^3+c)*arctan(1/3*3^(1/2)*(-2/((a*d 
-b*c)/c)^(1/3)*(b*x^3+a)^(1/3)+x)/x)*3^(1/2)*(a*d-3/2*b*c))/(a*d-b*c)/c^2/ 
(d*x^3+c)
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {1}{\sqrt [3]{a+b x^3} \left (c+d x^3\right )^2} \, dx=\int \frac {1}{\left (b \,x^{3}+a \right )^{\frac {1}{3}} c^{2}+2 \left (b \,x^{3}+a \right )^{\frac {1}{3}} c d \,x^{3}+\left (b \,x^{3}+a \right )^{\frac {1}{3}} d^{2} x^{6}}d x \] Input:

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

Output:

int(1/((a + b*x**3)**(1/3)*c**2 + 2*(a + b*x**3)**(1/3)*c*d*x**3 + (a + b* 
x**3)**(1/3)*d**2*x**6),x)