\(\int \frac {(a+b x^3)^{2/3}}{x^3 (c+d x^3)} \, dx\) [702]

Optimal result

Integrand size = 24, antiderivative size = 169 \[ \int \frac {\left (a+b x^3\right )^{2/3}}{x^3 \left (c+d x^3\right )} \, dx=-\frac {\left (a+b x^3\right )^{2/3}}{2 c x^2}+\frac {(b c-a d)^{2/3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{b c-a d} x}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{\sqrt {3} c^{5/3}}+\frac {(b c-a d)^{2/3} \log \left (c+d x^3\right )}{6 c^{5/3}}-\frac {(b c-a d)^{2/3} \log \left (\frac {\sqrt [3]{b c-a d} x}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{2 c^{5/3}} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 1.85 (sec) , antiderivative size = 309, normalized size of antiderivative = 1.83 \[ \int \frac {\left (a+b x^3\right )^{2/3}}{x^3 \left (c+d x^3\right )} \, dx=\frac {-\frac {6 c^{2/3} \left (a+b x^3\right )^{2/3}}{x^2}-2 \sqrt {-6+6 i \sqrt {3}} (b c-a d)^{2/3} \arctan \left (\frac {3 \sqrt [3]{b c-a d} x}{\sqrt {3} \sqrt [3]{b c-a d} x-\left (3 i+\sqrt {3}\right ) \sqrt [3]{c} \sqrt [3]{a+b x^3}}\right )+2 \left (1+i \sqrt {3}\right ) (b c-a d)^{2/3} \log \left (2 \sqrt [3]{b c-a d} x+\left (1+i \sqrt {3}\right ) \sqrt [3]{c} \sqrt [3]{a+b x^3}\right )-i \left (-i+\sqrt {3}\right ) (b c-a d)^{2/3} \log \left (2 (b c-a d)^{2/3} x^2+\left (-1-i \sqrt {3}\right ) \sqrt [3]{c} \sqrt [3]{b c-a d} x \sqrt [3]{a+b x^3}+i \left (i+\sqrt {3}\right ) c^{2/3} \left (a+b x^3\right )^{2/3}\right )}{12 c^{5/3}} \] Input:

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

Output:

((-6*c^(2/3)*(a + b*x^3)^(2/3))/x^2 - 2*Sqrt[-6 + (6*I)*Sqrt[3]]*(b*c - a* 
d)^(2/3)*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))] + 2*(1 + I*Sqrt[3])*(b*c - a*d)^( 
2/3)*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])*(b*c - a*d)^(2/3)*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)])/(12*c^(5/3))
 

Rubi [A] (verified)

Time = 0.45 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.08, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {975, 27, 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.

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

\(\Big \downarrow \) 975

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 901

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

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 975

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_) 
)^(q_), x_Symbol] :> Simp[(e*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^q/ 
(a*e*(m + 1))), x] - Simp[1/(a*e^n*(m + 1))   Int[(e*x)^(m + n)*(a + b*x^n) 
^p*(c + d*x^n)^(q - 1)*Simp[c*b*(m + 1) + n*(b*c*(p + 1) + a*d*q) + d*(b*(m 
 + 1) + b*n*(p + q + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && 
 NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[0, q, 1] && LtQ[m, -1] && IntBinomi 
alQ[a, b, c, d, e, m, n, p, q, x]
 

Maple [A] (verified)

Time = 2.14 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.22

method	result	size
pseudoelliptic	\(\frac {-2 \ln \left (\frac {\left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{x}\right ) \left (a d -b c \right ) x^{2}-3 \left (b \,x^{3}+a \right )^{\frac {2}{3}} c \left (\frac {a d -b c}{c}\right )^{\frac {1}{3}}+\left (a d -b c \right ) \left (-2 \arctan \left (\frac {\sqrt {3}\, \left (-\frac {2 \left (b \,x^{3}+a \right )^{\frac {1}{3}}}{\left (\frac {a d -b c}{c}\right )^{\frac {1}{3}}}+x \right )}{3 x}\right ) \sqrt {3}+\ln \left (\frac {\left (\frac {a d -b c}{c}\right )^{\frac {2}{3}} x^{2}-\left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {2}{3}}}{x^{2}}\right )\right ) x^{2}}{6 \left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} c^{2} x^{2}}\)	\(207\)

Input:

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

Output:

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

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\left (a+b x^3\right )^{2/3}}{x^3 \left (c+d x^3\right )} \, dx=\frac {-\left (b \,x^{3}+a \right )^{\frac {2}{3}} b +2 \left (\int \frac {\left (b \,x^{3}+a \right )^{\frac {2}{3}}}{b d \,x^{9}+a d \,x^{6}+b c \,x^{6}+a c \,x^{3}}d x \right ) a^{2} d \,x^{2}-2 \left (\int \frac {\left (b \,x^{3}+a \right )^{\frac {2}{3}}}{b d \,x^{9}+a d \,x^{6}+b c \,x^{6}+a c \,x^{3}}d x \right ) a b c \,x^{2}}{2 a d \,x^{2}} \] Input:

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

Output:

( - (a + b*x**3)**(2/3)*b + 2*int((a + b*x**3)**(2/3)/(a*c*x**3 + a*d*x**6 
 + b*c*x**6 + b*d*x**9),x)*a**2*d*x**2 - 2*int((a + b*x**3)**(2/3)/(a*c*x* 
*3 + a*d*x**6 + b*c*x**6 + b*d*x**9),x)*a*b*c*x**2)/(2*a*d*x**2)