\(\int \frac {\sqrt [3]{a+b x^3}}{x (c+d x^3)} \, dx\) [677]

Optimal result

Integrand size = 24, antiderivative size = 246 \[ \int \frac {\sqrt [3]{a+b x^3}}{x \left (c+d x^3\right )} \, dx=-\frac {\sqrt [3]{a} \arctan \left (\frac {\sqrt [3]{a}+2 \sqrt [3]{a+b x^3}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} c}-\frac {\sqrt [3]{b c-a d} \arctan \left (\frac {1-\frac {2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}}{\sqrt {3}}\right )}{\sqrt {3} c \sqrt [3]{d}}-\frac {\sqrt [3]{a} \log (x)}{2 c}-\frac {\sqrt [3]{b c-a d} \log \left (c+d x^3\right )}{6 c \sqrt [3]{d}}+\frac {\sqrt [3]{a} \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{2 c}+\frac {\sqrt [3]{b c-a d} \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 c \sqrt [3]{d}} \] Output:

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

Mathematica [A] (verified)

Time = 0.38 (sec) , antiderivative size = 312, normalized size of antiderivative = 1.27 \[ \int \frac {\sqrt [3]{a+b x^3}}{x \left (c+d x^3\right )} \, dx=-\frac {2 \sqrt {3} \sqrt [3]{a} \sqrt [3]{d} \arctan \left (\frac {1+\frac {2 \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}}{\sqrt {3}}\right )+2 \sqrt {3} \sqrt [3]{b c-a d} \arctan \left (\frac {1-\frac {2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}}{\sqrt {3}}\right )-2 \sqrt [3]{a} \sqrt [3]{d} \log \left (-\sqrt [3]{a}+\sqrt [3]{a+b x^3}\right )-2 \sqrt [3]{b c-a d} \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )+\sqrt [3]{a} \sqrt [3]{d} \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^3}+\left (a+b x^3\right )^{2/3}\right )+\sqrt [3]{b c-a d} \log \left ((b c-a d)^{2/3}-\sqrt [3]{d} \sqrt [3]{b c-a d} \sqrt [3]{a+b x^3}+d^{2/3} \left (a+b x^3\right )^{2/3}\right )}{6 c \sqrt [3]{d}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.64 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.01, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {948, 94, 69, 16, 70, 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.

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

\(\Big \downarrow \) 948

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

\(\Big \downarrow \) 94

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

\(\Big \downarrow \) 69

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

\(\Big \downarrow \) 16

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

\(\Big \downarrow \) 70

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

\(\Big \downarrow \) 16

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

\(\Big \downarrow \) 1082

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

\(\Big \downarrow \) 217

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

Input:

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

Output:

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

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 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 70

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] && NegQ[(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 = 1.88 (sec) , antiderivative size = 244, normalized size of antiderivative = 0.99

method	result	size
pseudoelliptic	\(\frac {\left (\frac {a d -b c}{d}\right )^{\frac {2}{3}} d \left (-2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (b \,x^{3}+a \right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )+2 \ln \left (\left (b \,x^{3}+a \right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right )-\ln \left (\left (b \,x^{3}+a \right )^{\frac {2}{3}}+a^{\frac {1}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right )\right ) a^{\frac {1}{3}}+\left (a d -b c \right ) \left (2 \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (b \,x^{3}+a \right )^{\frac {1}{3}}}{\left (\frac {a d -b c}{d}\right )^{\frac {1}{3}}}+1\right )}{3}\right ) \sqrt {3}+\ln \left (\left (b \,x^{3}+a \right )^{\frac {2}{3}}+\left (\frac {a d -b c}{d}\right )^{\frac {1}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}}+\left (\frac {a d -b c}{d}\right )^{\frac {2}{3}}\right )-2 \ln \left (\left (b \,x^{3}+a \right )^{\frac {1}{3}}-\left (\frac {a d -b c}{d}\right )^{\frac {1}{3}}\right )\right )}{6 \left (\frac {a d -b c}{d}\right )^{\frac {2}{3}} d c}\)	\(244\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 276, normalized size of antiderivative = 1.12 \[ \int \frac {\sqrt [3]{a+b x^3}}{x \left (c+d x^3\right )} \, dx=-\frac {2 \, \sqrt {3} \left (\frac {b c - a d}{d}\right )^{\frac {1}{3}} \arctan \left (-\frac {2 \, \sqrt {3} {\left (b x^{3} + a\right )}^{\frac {1}{3}} d \left (\frac {b c - a d}{d}\right )^{\frac {2}{3}} - \sqrt {3} {\left (b c - a d\right )}}{3 \, {\left (b c - a d\right )}}\right ) + 2 \, \sqrt {3} a^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} {\left (b x^{3} + a\right )}^{\frac {1}{3}} a^{\frac {2}{3}} + \sqrt {3} a}{3 \, a}\right ) + a^{\frac {1}{3}} \log \left ({\left (b x^{3} + a\right )}^{\frac {2}{3}} + {\left (b x^{3} + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right ) + \left (\frac {b c - a d}{d}\right )^{\frac {1}{3}} \log \left ({\left (b x^{3} + a\right )}^{\frac {2}{3}} - {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (\frac {b c - a d}{d}\right )^{\frac {1}{3}} + \left (\frac {b c - a d}{d}\right )^{\frac {2}{3}}\right ) - 2 \, a^{\frac {1}{3}} \log \left ({\left (b x^{3} + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right ) - 2 \, \left (\frac {b c - a d}{d}\right )^{\frac {1}{3}} \log \left ({\left (b x^{3} + a\right )}^{\frac {1}{3}} + \left (\frac {b c - a d}{d}\right )^{\frac {1}{3}}\right )}{6 \, c} \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.40 (sec) , antiderivative size = 311, normalized size of antiderivative = 1.26 \[ \int \frac {\sqrt [3]{a+b x^3}}{x \left (c+d x^3\right )} \, dx=-\frac {{\left (b c - a d\right )} \left (-\frac {b c - a d}{d}\right )^{\frac {1}{3}} \log \left ({\left | {\left (b x^{3} + a\right )}^{\frac {1}{3}} - \left (-\frac {b c - a d}{d}\right )^{\frac {1}{3}} \right |}\right )}{3 \, {\left (b c^{2} - a c d\right )}} - \frac {\sqrt {3} a^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{3 \, c} - \frac {a^{\frac {1}{3}} \log \left ({\left (b x^{3} + a\right )}^{\frac {2}{3}} + {\left (b x^{3} + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right )}{6 \, c} + \frac {a^{\frac {1}{3}} \log \left ({\left | {\left (b x^{3} + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}} \right |}\right )}{3 \, c} + \frac {\sqrt {3} {\left (-b c d^{2} + a d^{3}\right )}^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} + \left (-\frac {b c - a d}{d}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {b c - a d}{d}\right )^{\frac {1}{3}}}\right )}{3 \, c d} + \frac {{\left (-b c d^{2} + a d^{3}\right )}^{\frac {1}{3}} \log \left ({\left (b x^{3} + a\right )}^{\frac {2}{3}} + {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-\frac {b c - a d}{d}\right )^{\frac {1}{3}} + \left (-\frac {b c - a d}{d}\right )^{\frac {2}{3}}\right )}{6 \, c d} \] Input:

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

Output:

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

Mupad [B] (verification not implemented)

Time = 3.65 (sec) , antiderivative size = 1607, normalized size of antiderivative = 6.53 \[ \int \frac {\sqrt [3]{a+b x^3}}{x \left (c+d x^3\right )} \, dx=\text {Too large to display} \] Input:

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

Output:

log((a + b*x^3)^(1/3)*(6*a^4*b^4*d^5 - 3*a*b^7*c^3*d^2 - 12*a^3*b^5*c*d^4 
+ 9*a^2*b^6*c^2*d^3) - (a/(27*c^3))^(1/3)*(((243*a*b^6*c^6*d^3 - 729*a^2*b 
^5*c^5*d^4 + 486*a^3*b^4*c^4*d^5)*(a/(27*c^3))^(1/3) - (a + b*x^3)^(1/3)*( 
81*a*b^6*c^5*d^3 - 81*a^2*b^5*c^4*d^4))*(a/(27*c^3))^(2/3) - 9*a*b^7*c^4*d 
^2 + 27*a^2*b^6*c^3*d^3 - 18*a^3*b^5*c^2*d^4))*(a/(27*c^3))^(1/3) + log((a 
 + b*x^3)^(1/3)*(6*a^4*b^4*d^5 - 3*a*b^7*c^3*d^2 - 12*a^3*b^5*c*d^4 + 9*a^ 
2*b^6*c^2*d^3) - (((243*a*b^6*c^6*d^3 - 729*a^2*b^5*c^5*d^4 + 486*a^3*b^4* 
c^4*d^5)*(-(a*d - b*c)/(27*c^3*d))^(1/3) - (a + b*x^3)^(1/3)*(81*a*b^6*c^5 
*d^3 - 81*a^2*b^5*c^4*d^4))*(-(a*d - b*c)/(27*c^3*d))^(2/3) - 9*a*b^7*c^4* 
d^2 + 27*a^2*b^6*c^3*d^3 - 18*a^3*b^5*c^2*d^4)*(-(a*d - b*c)/(27*c^3*d))^( 
1/3))*(-(a*d - b*c)/(27*c^3*d))^(1/3) + log((a + b*x^3)^(1/3)*(6*a^4*b^4*d 
^5 - 3*a*b^7*c^3*d^2 - 12*a^3*b^5*c*d^4 + 9*a^2*b^6*c^2*d^3) + ((3^(1/2)*1 
i)/2 - 1/2)*(-(a*d - b*c)/(27*c^3*d))^(1/3)*(((3^(1/2)*1i)/2 - 1/2)^2*((a 
+ b*x^3)^(1/3)*(81*a*b^6*c^5*d^3 - 81*a^2*b^5*c^4*d^4) - ((3^(1/2)*1i)/2 - 
 1/2)*(243*a*b^6*c^6*d^3 - 729*a^2*b^5*c^5*d^4 + 486*a^3*b^4*c^4*d^5)*(-(a 
*d - b*c)/(27*c^3*d))^(1/3))*(-(a*d - b*c)/(27*c^3*d))^(2/3) + 9*a*b^7*c^4 
*d^2 - 27*a^2*b^6*c^3*d^3 + 18*a^3*b^5*c^2*d^4))*((3^(1/2)*1i)/2 - 1/2)*(- 
(a*d - b*c)/(27*c^3*d))^(1/3) - log((a + b*x^3)^(1/3)*(6*a^4*b^4*d^5 - 3*a 
*b^7*c^3*d^2 - 12*a^3*b^5*c*d^4 + 9*a^2*b^6*c^2*d^3) - ((3^(1/2)*1i)/2 + 1 
/2)*(-(a*d - b*c)/(27*c^3*d))^(1/3)*(((3^(1/2)*1i)/2 + 1/2)^2*((a + b*x...
 

Reduce [F]

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

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

Output:

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