\(\int \frac {\sqrt [3]{a+b x^3}}{x (a d-b d x^3)} \, dx\) [785]

Optimal result

Integrand size = 28, antiderivative size = 214 \[ \int \frac {\sqrt [3]{a+b x^3}}{x \left (a d-b d x^3\right )} \, dx=-\frac {\arctan \left (\frac {\sqrt [3]{a}+2 \sqrt [3]{a+b x^3}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{2/3} d}+\frac {\sqrt [3]{2} \arctan \left (\frac {\sqrt [3]{a}+2^{2/3} \sqrt [3]{a+b x^3}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{2/3} d}-\frac {\log (x)}{2 a^{2/3} d}+\frac {\log \left (a-b x^3\right )}{3\ 2^{2/3} a^{2/3} d}+\frac {\log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{2 a^{2/3} d}-\frac {\log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{2^{2/3} a^{2/3} d} \] 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^(2/ 
3)/d+1/3*2^(1/3)*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^(2/3)/d-1/2*ln(x)/a^(2/3)/d+1/6*ln(-b*x^3+a)*2^(1/3)/a^(2/3 
)/d+1/2*ln(a^(1/3)-(b*x^3+a)^(1/3))/a^(2/3)/d-1/2*ln(2^(1/3)*a^(1/3)-(b*x^ 
3+a)^(1/3))*2^(1/3)/a^(2/3)/d
 

Mathematica [A] (verified)

Time = 0.29 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.10 \[ \int \frac {\sqrt [3]{a+b x^3}}{x \left (a d-b d x^3\right )} \, dx=-\frac {2 \sqrt {3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}}{\sqrt {3}}\right )-2 \sqrt [3]{2} \sqrt {3} \arctan \left (\frac {1+\frac {2^{2/3} \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}}{\sqrt {3}}\right )-2 \log \left (-\sqrt [3]{a}+\sqrt [3]{a+b x^3}\right )+2 \sqrt [3]{2} \log \left (-2 \sqrt [3]{a}+2^{2/3} \sqrt [3]{a+b x^3}\right )+\log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^3}+\left (a+b x^3\right )^{2/3}\right )-\sqrt [3]{2} \log \left (2 a^{2/3}+2^{2/3} \sqrt [3]{a} \sqrt [3]{a+b x^3}+\sqrt [3]{2} \left (a+b x^3\right )^{2/3}\right )}{6 a^{2/3} d} \] Input:

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

Output:

-1/6*(2*Sqrt[3]*ArcTan[(1 + (2*(a + b*x^3)^(1/3))/a^(1/3))/Sqrt[3]] - 2*2^ 
(1/3)*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^(1/3)*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^(1/3)*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)])/(a^(2/3)*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, 69, 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.

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

\(\Big \downarrow \) 948

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 94

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

\(\Big \downarrow \) 69

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

\(\Big \downarrow \) 16

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

\(\Big \downarrow \) 1082

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

\(\Big \downarrow \) 217

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

Input:

Int[(a + b*x^3)^(1/3)/(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^(2/3)) 
- Log[x^3]/(2*a^(2/3)) + (3*Log[a^(1/3) - (a + b*x^3)^(1/3)])/(2*a^(2/3)) 
+ 2*b*((Sqrt[3]*ArcTan[(1 + (2^(2/3)*(a + b*x^3)^(1/3))/a^(1/3))/Sqrt[3]]) 
/(2^(2/3)*a^(2/3)*b) + Log[a - b*x^3]/(2*2^(2/3)*a^(2/3)*b) - (3*Log[2^(1/ 
3)*a^(1/3) - (a + b*x^3)^(1/3)])/(2*2^(2/3)*a^(2/3)*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 69

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(2/3)), x_Symbol] :> With[ 
{q = Rt[(b*c - a*d)/b, 3]}, Simp[-Log[RemoveContent[a + b*x, x]]/(2*b*q^2), 
 x] + (-Simp[3/(2*b*q)   Subst[Int[1/(q^2 + q*x + x^2), x], x, (c + d*x)^(1 
/3)], x] - Simp[3/(2*b*q^2)   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 = 1.59 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.85

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

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 634 vs. \(2 (161) = 322\).

Time = 0.10 (sec) , antiderivative size = 634, normalized size of antiderivative = 2.96 \[ \int \frac {\sqrt [3]{a+b x^3}}{x \left (a d-b d x^3\right )} \, dx =\text {Too large to display} \] Input:

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

Output:

-1/6*(1/2)^(1/3)*(sqrt(-3) + 1)*(-(3*a^2*d^3*sqrt(1/(a^4*d^6)) + 1)/(a^2*d 
^3))^(1/3)*log(-(1/2)^(1/3)*(sqrt(-3)*a^3*d^4 + a^3*d^4)*(-(3*a^2*d^3*sqrt 
(1/(a^4*d^6)) + 1)/(a^2*d^3))^(1/3)*sqrt(1/(a^4*d^6)) + 2*(b*x^3 + a)^(1/3 
)) + 1/6*(1/2)^(1/3)*(sqrt(-3) - 1)*(-(3*a^2*d^3*sqrt(1/(a^4*d^6)) + 1)/(a 
^2*d^3))^(1/3)*log((1/2)^(1/3)*(sqrt(-3)*a^3*d^4 - a^3*d^4)*(-(3*a^2*d^3*s 
qrt(1/(a^4*d^6)) + 1)/(a^2*d^3))^(1/3)*sqrt(1/(a^4*d^6)) + 2*(b*x^3 + a)^( 
1/3)) - 1/6*(1/2)^(1/3)*(sqrt(-3) + 1)*((3*a^2*d^3*sqrt(1/(a^4*d^6)) - 1)/ 
(a^2*d^3))^(1/3)*log((1/2)^(1/3)*(sqrt(-3)*a^3*d^4 + a^3*d^4)*((3*a^2*d^3* 
sqrt(1/(a^4*d^6)) - 1)/(a^2*d^3))^(1/3)*sqrt(1/(a^4*d^6)) + 2*(b*x^3 + a)^ 
(1/3)) + 1/6*(1/2)^(1/3)*(sqrt(-3) - 1)*((3*a^2*d^3*sqrt(1/(a^4*d^6)) - 1) 
/(a^2*d^3))^(1/3)*log(-(1/2)^(1/3)*(sqrt(-3)*a^3*d^4 - a^3*d^4)*((3*a^2*d^ 
3*sqrt(1/(a^4*d^6)) - 1)/(a^2*d^3))^(1/3)*sqrt(1/(a^4*d^6)) + 2*(b*x^3 + a 
)^(1/3)) + 1/3*(1/2)^(1/3)*(-(3*a^2*d^3*sqrt(1/(a^4*d^6)) + 1)/(a^2*d^3))^ 
(1/3)*log((1/2)^(1/3)*a^3*d^4*(-(3*a^2*d^3*sqrt(1/(a^4*d^6)) + 1)/(a^2*d^3 
))^(1/3)*sqrt(1/(a^4*d^6)) + (b*x^3 + a)^(1/3)) + 1/3*(1/2)^(1/3)*((3*a^2* 
d^3*sqrt(1/(a^4*d^6)) - 1)/(a^2*d^3))^(1/3)*log(-(1/2)^(1/3)*a^3*d^4*((3*a 
^2*d^3*sqrt(1/(a^4*d^6)) - 1)/(a^2*d^3))^(1/3)*sqrt(1/(a^4*d^6)) + (b*x^3 
+ a)^(1/3))
 

Sympy [F]

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

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

Output:

-Integral((a + b*x**3)**(1/3)/(-a*x + b*x**4), x)/d
 

Maxima [F]

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

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.94 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.01 \[ \int \frac {\sqrt [3]{a+b x^3}}{x \left (a d-b d x^3\right )} \, dx=\frac {\sqrt {3} 2^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} 2^{\frac {2}{3}} {\left (2^{\frac {1}{3}} a^{\frac {1}{3}} + 2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}}\right )}}{6 \, a^{\frac {1}{3}}}\right )}{3 \, a^{\frac {2}{3}} d} - \frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{3 \, a^{\frac {2}{3}} d} + \frac {2^{\frac {1}{3}} \log \left (2^{\frac {2}{3}} a^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (b x^{3} + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + {\left (b x^{3} + a\right )}^{\frac {2}{3}}\right )}{6 \, a^{\frac {2}{3}} d} - \frac {2^{\frac {1}{3}} \log \left ({\left | -2^{\frac {1}{3}} a^{\frac {1}{3}} + {\left (b x^{3} + a\right )}^{\frac {1}{3}} \right |}\right )}{3 \, a^{\frac {2}{3}} d} - \frac {\log \left ({\left (b x^{3} + a\right )}^{\frac {2}{3}} + {\left (b x^{3} + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right )}{6 \, a^{\frac {2}{3}} d} + \frac {\log \left ({\left | {\left (b x^{3} + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}} \right |}\right )}{3 \, a^{\frac {2}{3}} d} \] Input:

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

Output:

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

Mupad [B] (verification not implemented)

Time = 3.77 (sec) , antiderivative size = 345, normalized size of antiderivative = 1.61 \[ \int \frac {\sqrt [3]{a+b x^3}}{x \left (a d-b d x^3\right )} \, dx=\ln \left ({\left (b\,x^3+a\right )}^{1/3}-a\,d\,{\left (\frac {1}{a^2\,d^3}\right )}^{1/3}\right )\,{\left (\frac {1}{27\,a^2\,d^3}\right )}^{1/3}+\ln \left ({\left (b\,x^3+a\right )}^{1/3}+2^{1/3}\,a\,d\,{\left (-\frac {1}{a^2\,d^3}\right )}^{1/3}\right )\,{\left (-\frac {2}{27\,a^2\,d^3}\right )}^{1/3}-\ln \left (2^{1/3}\,a\,d\,{\left (-\frac {1}{a^2\,d^3}\right )}^{1/3}-2\,{\left (b\,x^3+a\right )}^{1/3}+2^{1/3}\,\sqrt {3}\,a\,d\,{\left (-\frac {1}{a^2\,d^3}\right )}^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (-\frac {2}{27\,a^2\,d^3}\right )}^{1/3}+\ln \left (2\,{\left (b\,x^3+a\right )}^{1/3}-2^{1/3}\,a\,d\,{\left (-\frac {1}{a^2\,d^3}\right )}^{1/3}+2^{1/3}\,\sqrt {3}\,a\,d\,{\left (-\frac {1}{a^2\,d^3}\right )}^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (-\frac {2}{27\,a^2\,d^3}\right )}^{1/3}+\ln \left (2\,{\left (b\,x^3+a\right )}^{1/3}+a\,d\,{\left (\frac {1}{a^2\,d^3}\right )}^{1/3}-\sqrt {3}\,a\,d\,{\left (\frac {1}{a^2\,d^3}\right )}^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (\frac {1}{27\,a^2\,d^3}\right )}^{1/3}-\ln \left (2\,{\left (b\,x^3+a\right )}^{1/3}+a\,d\,{\left (\frac {1}{a^2\,d^3}\right )}^{1/3}+\sqrt {3}\,a\,d\,{\left (\frac {1}{a^2\,d^3}\right )}^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (\frac {1}{27\,a^2\,d^3}\right )}^{1/3} \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {\sqrt [3]{a+b x^3}}{x \left (a d-b d x^3\right )} \, dx=\frac {\int \frac {\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{-b \,x^{4}+a x}d x}{d} \] Input:

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

Output:

int((a + b*x**3)**(1/3)/(a*x - b*x**4),x)/d