\(\int \frac {(a+b x^3)^{2/3}}{x (a d-b d x^3)} \, dx\) [804]

Optimal result

Integrand size = 28, antiderivative size = 214 \[ \int \frac {\left (a+b x^3\right )^{2/3}}{x \left (a d-b d x^3\right )} \, dx=\frac {\arctan \left (\frac {\sqrt [3]{a}+2 \sqrt [3]{a+b x^3}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} \sqrt [3]{a} d}-\frac {2^{2/3} \arctan \left (\frac {\sqrt [3]{a}+2^{2/3} \sqrt [3]{a+b x^3}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} \sqrt [3]{a} d}-\frac {\log (x)}{2 \sqrt [3]{a} d}+\frac {\log \left (a-b x^3\right )}{3 \sqrt [3]{2} \sqrt [3]{a} d}+\frac {\log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{2 \sqrt [3]{a} d}-\frac {\log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{\sqrt [3]{2} \sqrt [3]{a} d} \] Output:

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

Mathematica [A] (verified)

Time = 0.33 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.10 \[ \int \frac {\left (a+b x^3\right )^{2/3}}{x \left (a d-b d x^3\right )} \, dx=\frac {2 \sqrt {3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}}{\sqrt {3}}\right )-2\ 2^{2/3} \sqrt {3} \arctan \left (\frac {1+\frac {2^{2/3} \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}}{\sqrt {3}}\right )+2 \log \left (-\sqrt [3]{a}+\sqrt [3]{a+b x^3}\right )-2\ 2^{2/3} \log \left (-2 \sqrt [3]{a}+2^{2/3} \sqrt [3]{a+b x^3}\right )-\log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^3}+\left (a+b x^3\right )^{2/3}\right )+2^{2/3} \log \left (2 a^{2/3}+2^{2/3} \sqrt [3]{a} \sqrt [3]{a+b x^3}+\sqrt [3]{2} \left (a+b x^3\right )^{2/3}\right )}{6 \sqrt [3]{a} d} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.55 (sec) , antiderivative size = 212, normalized size of antiderivative = 0.99, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {948, 27, 94, 67, 16, 1082, 217}

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.

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

\(\Big \downarrow \) 948

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 94

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

\(\Big \downarrow \) 67

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

\(\Big \downarrow \) 16

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

\(\Big \downarrow \) 1082

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

\(\Big \downarrow \) 217

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 16

Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + 
b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
 

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 67

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

rule 94

Int[((e_.) + (f_.)*(x_))^(p_)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), 
x_] :> Simp[(b*e - a*f)/(b*c - a*d)   Int[(e + f*x)^(p - 1)/(a + b*x), x], 
x] - Simp[(d*e - c*f)/(b*c - a*d)   Int[(e + f*x)^(p - 1)/(c + d*x), x], x] 
 /; FreeQ[{a, b, c, d, e, f}, x] && LtQ[0, p, 1]
 

rule 217

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( 
-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & 
& (LtQ[a, 0] || LtQ[b, 0])
 

rule 948

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

rule 1082

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S 
implify[a*(c/b^2)]}, Simp[-2/b   Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b 
)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /; Fre 
eQ[{a, b, c}, x]
 

Maple [A] (verified)

Time = 1.21 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.85

method	result	size
pseudoelliptic	\(\frac {-2 \sqrt {3}\, 2^{\frac {2}{3}} \arctan \left (\frac {\left (a^{\frac {1}{3}}+2^{\frac {2}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}}\right ) \sqrt {3}}{3 a^{\frac {1}{3}}}\right )-2 \,2^{\frac {2}{3}} \ln \left (\left (b \,x^{3}+a \right )^{\frac {1}{3}}-2^{\frac {1}{3}} a^{\frac {1}{3}}\right )+2^{\frac {2}{3}} \ln \left (\left (b \,x^{3}+a \right )^{\frac {2}{3}}+2^{\frac {1}{3}} a^{\frac {1}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}}+2^{\frac {2}{3}} a^{\frac {2}{3}}\right )+2 \arctan \left (\frac {\left (a^{\frac {1}{3}}+2 \left (b \,x^{3}+a \right )^{\frac {1}{3}}\right ) \sqrt {3}}{3 a^{\frac {1}{3}}}\right ) \sqrt {3}+2 \ln \left (\left (b \,x^{3}+a \right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right )-\ln \left (\left (b \,x^{3}+a \right )^{\frac {2}{3}}+a^{\frac {1}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right )}{6 d \,a^{\frac {1}{3}}}\)	\(182\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 530, normalized size of antiderivative = 2.48 \[ \int \frac {\left (a+b x^3\right )^{2/3}}{x \left (a d-b d x^3\right )} \, dx =\text {Too large to display} \] Input:

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

Output:

[-1/6*(2*4^(1/3)*sqrt(3)*a*(-1/a)^(1/3)*arctan(1/3*4^(1/3)*sqrt(3)*(b*x^3 
+ a)^(1/3)*(-1/a)^(1/3) - 1/3*sqrt(3)) - 3*sqrt(1/3)*a*sqrt(-1/a^(2/3))*lo 
g((2*b*x^3 + 3*sqrt(1/3)*(2*(b*x^3 + a)^(2/3)*a^(2/3) - (b*x^3 + a)^(1/3)* 
a - a^(4/3))*sqrt(-1/a^(2/3)) - 3*(b*x^3 + a)^(1/3)*a^(2/3) + 3*a)/x^3) + 
4^(1/3)*a*(-1/a)^(1/3)*log(4^(2/3)*(b*x^3 + a)^(1/3)*a*(-1/a)^(2/3) - 2*4^ 
(1/3)*a*(-1/a)^(1/3) + 2*(b*x^3 + a)^(2/3)) - 2*4^(1/3)*a*(-1/a)^(1/3)*log 
(-4^(2/3)*a*(-1/a)^(2/3) + 2*(b*x^3 + a)^(1/3)) + a^(2/3)*log((b*x^3 + a)^ 
(2/3) + (b*x^3 + a)^(1/3)*a^(1/3) + a^(2/3)) - 2*a^(2/3)*log((b*x^3 + a)^( 
1/3) - a^(1/3)))/(a*d), -1/6*(2*4^(1/3)*sqrt(3)*a*(-1/a)^(1/3)*arctan(1/3* 
4^(1/3)*sqrt(3)*(b*x^3 + a)^(1/3)*(-1/a)^(1/3) - 1/3*sqrt(3)) + 4^(1/3)*a* 
(-1/a)^(1/3)*log(4^(2/3)*(b*x^3 + a)^(1/3)*a*(-1/a)^(2/3) - 2*4^(1/3)*a*(- 
1/a)^(1/3) + 2*(b*x^3 + a)^(2/3)) - 2*4^(1/3)*a*(-1/a)^(1/3)*log(-4^(2/3)* 
a*(-1/a)^(2/3) + 2*(b*x^3 + a)^(1/3)) - 6*sqrt(1/3)*a^(2/3)*arctan(sqrt(1/ 
3)*(2*(b*x^3 + a)^(1/3) + a^(1/3))/a^(1/3)) + a^(2/3)*log((b*x^3 + a)^(2/3 
) + (b*x^3 + a)^(1/3)*a^(1/3) + a^(2/3)) - 2*a^(2/3)*log((b*x^3 + a)^(1/3) 
 - a^(1/3)))/(a*d)]
 

Sympy [F]

\[ \int \frac {\left (a+b x^3\right )^{2/3}}{x \left (a d-b d x^3\right )} \, dx=- \frac {\int \frac {\left (a + b x^{3}\right )^{\frac {2}{3}}}{- a x + b x^{4}}\, dx}{d} \] Input:

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

Output:

-Integral((a + b*x**3)**(2/3)/(-a*x + b*x**4), x)/d
 

Maxima [F]

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

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.88 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.01 \[ \int \frac {\left (a+b x^3\right )^{2/3}}{x \left (a d-b d x^3\right )} \, dx=-\frac {\sqrt {3} 2^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} 2^{\frac {2}{3}} {\left (2^{\frac {1}{3}} a^{\frac {1}{3}} + 2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}}\right )}}{6 \, a^{\frac {1}{3}}}\right )}{3 \, a^{\frac {1}{3}} d} + \frac {2^{\frac {2}{3}} \log \left (2^{\frac {2}{3}} a^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (b x^{3} + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + {\left (b x^{3} + a\right )}^{\frac {2}{3}}\right )}{6 \, a^{\frac {1}{3}} d} - \frac {2^{\frac {2}{3}} \log \left ({\left | -2^{\frac {1}{3}} a^{\frac {1}{3}} + {\left (b x^{3} + a\right )}^{\frac {1}{3}} \right |}\right )}{3 \, a^{\frac {1}{3}} d} + \frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{3 \, a^{\frac {1}{3}} d} - \frac {\log \left ({\left (b x^{3} + a\right )}^{\frac {2}{3}} + {\left (b x^{3} + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right )}{6 \, a^{\frac {1}{3}} d} + \frac {\log \left ({\left | {\left (b x^{3} + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}} \right |}\right )}{3 \, a^{\frac {1}{3}} d} \] Input:

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

Output:

-1/3*sqrt(3)*2^(2/3)*arctan(1/6*sqrt(3)*2^(2/3)*(2^(1/3)*a^(1/3) + 2*(b*x^ 
3 + a)^(1/3))/a^(1/3))/(a^(1/3)*d) + 1/6*2^(2/3)*log(2^(2/3)*a^(2/3) + 2^( 
1/3)*(b*x^3 + a)^(1/3)*a^(1/3) + (b*x^3 + a)^(2/3))/(a^(1/3)*d) - 1/3*2^(2 
/3)*log(abs(-2^(1/3)*a^(1/3) + (b*x^3 + a)^(1/3)))/(a^(1/3)*d) + 1/3*sqrt( 
3)*arctan(1/3*sqrt(3)*(2*(b*x^3 + a)^(1/3) + a^(1/3))/a^(1/3))/(a^(1/3)*d) 
 - 1/6*log((b*x^3 + a)^(2/3) + (b*x^3 + a)^(1/3)*a^(1/3) + a^(2/3))/(a^(1/ 
3)*d) + 1/3*log(abs((b*x^3 + a)^(1/3) - a^(1/3)))/(a^(1/3)*d)
 

Mupad [B] (verification not implemented)

Time = 4.29 (sec) , antiderivative size = 369, normalized size of antiderivative = 1.72 \[ \int \frac {\left (a+b x^3\right )^{2/3}}{x \left (a d-b d x^3\right )} \, dx=\ln \left (2\,{\left (b\,x^3+a\right )}^{1/3}-2\,2^{1/3}\,a\,d^2\,{\left (-\frac {1}{a\,d^3}\right )}^{2/3}\right )\,{\left (-\frac {4}{27\,a\,d^3}\right )}^{1/3}+\ln \left ({\left (b\,x^3+a\right )}^{1/3}-a\,d^2\,{\left (\frac {1}{a\,d^3}\right )}^{2/3}\right )\,{\left (\frac {1}{27\,a\,d^3}\right )}^{1/3}-\ln \left (4\,{\left (b\,x^3+a\right )}^{1/3}+2\,2^{1/3}\,a\,d^2\,{\left (-\frac {1}{a\,d^3}\right )}^{2/3}-2^{1/3}\,\sqrt {3}\,a\,d^2\,{\left (-\frac {1}{a\,d^3}\right )}^{2/3}\,2{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (-\frac {4}{27\,a\,d^3}\right )}^{1/3}+\ln \left (4\,{\left (b\,x^3+a\right )}^{1/3}+2\,2^{1/3}\,a\,d^2\,{\left (-\frac {1}{a\,d^3}\right )}^{2/3}+2^{1/3}\,\sqrt {3}\,a\,d^2\,{\left (-\frac {1}{a\,d^3}\right )}^{2/3}\,2{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (-\frac {4}{27\,a\,d^3}\right )}^{1/3}-\ln \left (2\,{\left (b\,x^3+a\right )}^{1/3}+a\,d^2\,{\left (\frac {1}{a\,d^3}\right )}^{2/3}-\sqrt {3}\,a\,d^2\,{\left (\frac {1}{a\,d^3}\right )}^{2/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (\frac {1}{27\,a\,d^3}\right )}^{1/3}+\ln \left (2\,{\left (b\,x^3+a\right )}^{1/3}+a\,d^2\,{\left (\frac {1}{a\,d^3}\right )}^{2/3}+\sqrt {3}\,a\,d^2\,{\left (\frac {1}{a\,d^3}\right )}^{2/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (\frac {1}{27\,a\,d^3}\right )}^{1/3} \] Input:

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

Output:

log(2*(a + b*x^3)^(1/3) - 2*2^(1/3)*a*d^2*(-1/(a*d^3))^(2/3))*(-4/(27*a*d^ 
3))^(1/3) + log((a + b*x^3)^(1/3) - a*d^2*(1/(a*d^3))^(2/3))*(1/(27*a*d^3) 
)^(1/3) - log(4*(a + b*x^3)^(1/3) + 2*2^(1/3)*a*d^2*(-1/(a*d^3))^(2/3) - 2 
^(1/3)*3^(1/2)*a*d^2*(-1/(a*d^3))^(2/3)*2i)*((3^(1/2)*1i)/2 + 1/2)*(-4/(27 
*a*d^3))^(1/3) + log(4*(a + b*x^3)^(1/3) + 2*2^(1/3)*a*d^2*(-1/(a*d^3))^(2 
/3) + 2^(1/3)*3^(1/2)*a*d^2*(-1/(a*d^3))^(2/3)*2i)*((3^(1/2)*1i)/2 - 1/2)* 
(-4/(27*a*d^3))^(1/3) - log(2*(a + b*x^3)^(1/3) + a*d^2*(1/(a*d^3))^(2/3) 
- 3^(1/2)*a*d^2*(1/(a*d^3))^(2/3)*1i)*((3^(1/2)*1i)/2 + 1/2)*(1/(27*a*d^3) 
)^(1/3) + log(2*(a + b*x^3)^(1/3) + a*d^2*(1/(a*d^3))^(2/3) + 3^(1/2)*a*d^ 
2*(1/(a*d^3))^(2/3)*1i)*((3^(1/2)*1i)/2 - 1/2)*(1/(27*a*d^3))^(1/3)
 

Reduce [F]

\[ \int \frac {\left (a+b x^3\right )^{2/3}}{x \left (a d-b d x^3\right )} \, dx=\frac {\int \frac {\left (b \,x^{3}+a \right )^{\frac {2}{3}}}{-b \,x^{4}+a x}d x}{d} \] Input:

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

Output:

int((a + b*x**3)**(2/3)/(a*x - b*x**4),x)/d