\(\int \frac {1}{(a+b x) \sqrt [3]{-c+d x}} \, dx\) [289]

Optimal result

Integrand size = 19, antiderivative size = 138 \[ \int \frac {1}{(a+b x) \sqrt [3]{-c+d x}} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} \sqrt [3]{-c+d x}}{\sqrt [3]{b c+a d}}}{\sqrt {3}}\right )}{b^{2/3} \sqrt [3]{b c+a d}}+\frac {\log (a+b x)}{2 b^{2/3} \sqrt [3]{b c+a d}}-\frac {3 \log \left (\sqrt [3]{b c+a d}+\sqrt [3]{b} \sqrt [3]{-c+d x}\right )}{2 b^{2/3} \sqrt [3]{b c+a d}} \] Output:

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

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.14 \[ \int \frac {1}{(a+b x) \sqrt [3]{-c+d x}} \, dx=\frac {-2 \sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} \sqrt [3]{-c+d x}}{\sqrt [3]{b c+a d}}}{\sqrt {3}}\right )-2 \log \left (\sqrt [3]{b c+a d}+\sqrt [3]{b} \sqrt [3]{-c+d x}\right )+\log \left ((b c+a d)^{2/3}-\sqrt [3]{b} \sqrt [3]{b c+a d} \sqrt [3]{-c+d x}+b^{2/3} (-c+d x)^{2/3}\right )}{2 b^{2/3} \sqrt [3]{b c+a d}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {68, 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.

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

\(\Big \downarrow \) 68

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

\(\Big \downarrow \) 16

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

\(\Big \downarrow \) 1082

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

\(\Big \downarrow \) 217

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

Input:

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

Output:

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

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 68

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] && NegQ[(b*c - a*d)/b]
 

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 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 = 0.60 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.03

method	result	size
pseudoelliptic	\(-\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (-2 \left (x d -c \right )^{\frac {1}{3}}+\left (\frac {a d +b c}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a d +b c}{b}\right )^{\frac {1}{3}}}\right )+\ln \left (\left (x d -c \right )^{\frac {1}{3}}+\left (\frac {a d +b c}{b}\right )^{\frac {1}{3}}\right )-\frac {\ln \left (\left (x d -c \right )^{\frac {2}{3}}-\left (\frac {a d +b c}{b}\right )^{\frac {1}{3}} \left (x d -c \right )^{\frac {1}{3}}+\left (\frac {a d +b c}{b}\right )^{\frac {2}{3}}\right )}{2}}{\left (\frac {a d +b c}{b}\right )^{\frac {1}{3}} b}\)	\(142\)
derivativedivides	\(-\frac {\ln \left (\left (x d -c \right )^{\frac {1}{3}}+\left (\frac {a d +b c}{b}\right )^{\frac {1}{3}}\right )}{b \left (\frac {a d +b c}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (\left (x d -c \right )^{\frac {2}{3}}-\left (\frac {a d +b c}{b}\right )^{\frac {1}{3}} \left (x d -c \right )^{\frac {1}{3}}+\left (\frac {a d +b c}{b}\right )^{\frac {2}{3}}\right )}{2 b \left (\frac {a d +b c}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (x d -c \right )^{\frac {1}{3}}}{\left (\frac {a d +b c}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{b \left (\frac {a d +b c}{b}\right )^{\frac {1}{3}}}\)	\(162\)
default	\(-\frac {\ln \left (\left (x d -c \right )^{\frac {1}{3}}+\left (\frac {a d +b c}{b}\right )^{\frac {1}{3}}\right )}{b \left (\frac {a d +b c}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (\left (x d -c \right )^{\frac {2}{3}}-\left (\frac {a d +b c}{b}\right )^{\frac {1}{3}} \left (x d -c \right )^{\frac {1}{3}}+\left (\frac {a d +b c}{b}\right )^{\frac {2}{3}}\right )}{2 b \left (\frac {a d +b c}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (x d -c \right )^{\frac {1}{3}}}{\left (\frac {a d +b c}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{b \left (\frac {a d +b c}{b}\right )^{\frac {1}{3}}}\)	\(162\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 252 vs. \(2 (107) = 214\).

Time = 0.13 (sec) , antiderivative size = 595, normalized size of antiderivative = 4.31 \[ \int \frac {1}{(a+b x) \sqrt [3]{-c+d x}} \, dx=\left [\frac {\sqrt {3} {\left (b^{2} c + a b d\right )} \sqrt {\frac {{\left (-b^{3} c - a b^{2} d\right )}^{\frac {1}{3}}}{b c + a d}} \log \left (\frac {2 \, b^{2} d x - 3 \, b^{2} c - a b d + \sqrt {3} {\left ({\left (-b^{3} c - a b^{2} d\right )}^{\frac {1}{3}} {\left (b c + a d\right )} + {\left (b^{2} c + a b d\right )} {\left (d x - c\right )}^{\frac {1}{3}} + 2 \, {\left (-b^{3} c - a b^{2} d\right )}^{\frac {2}{3}} {\left (d x - c\right )}^{\frac {2}{3}}\right )} \sqrt {\frac {{\left (-b^{3} c - a b^{2} d\right )}^{\frac {1}{3}}}{b c + a d}} - 3 \, {\left (-b^{3} c - a b^{2} d\right )}^{\frac {2}{3}} {\left (d x - c\right )}^{\frac {1}{3}}}{b x + a}\right ) + {\left (-b^{3} c - a b^{2} d\right )}^{\frac {2}{3}} \log \left ({\left (d x - c\right )}^{\frac {2}{3}} b^{2} + {\left (-b^{3} c - a b^{2} d\right )}^{\frac {1}{3}} {\left (d x - c\right )}^{\frac {1}{3}} b + {\left (-b^{3} c - a b^{2} d\right )}^{\frac {2}{3}}\right ) - 2 \, {\left (-b^{3} c - a b^{2} d\right )}^{\frac {2}{3}} \log \left ({\left (d x - c\right )}^{\frac {1}{3}} b - {\left (-b^{3} c - a b^{2} d\right )}^{\frac {1}{3}}\right )}{2 \, {\left (b^{3} c + a b^{2} d\right )}}, \frac {2 \, \sqrt {3} {\left (b^{2} c + a b d\right )} \sqrt {-\frac {{\left (-b^{3} c - a b^{2} d\right )}^{\frac {1}{3}}}{b c + a d}} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (d x - c\right )}^{\frac {1}{3}} b + {\left (-b^{3} c - a b^{2} d\right )}^{\frac {1}{3}}\right )} \sqrt {-\frac {{\left (-b^{3} c - a b^{2} d\right )}^{\frac {1}{3}}}{b c + a d}}}{3 \, b}\right ) + {\left (-b^{3} c - a b^{2} d\right )}^{\frac {2}{3}} \log \left ({\left (d x - c\right )}^{\frac {2}{3}} b^{2} + {\left (-b^{3} c - a b^{2} d\right )}^{\frac {1}{3}} {\left (d x - c\right )}^{\frac {1}{3}} b + {\left (-b^{3} c - a b^{2} d\right )}^{\frac {2}{3}}\right ) - 2 \, {\left (-b^{3} c - a b^{2} d\right )}^{\frac {2}{3}} \log \left ({\left (d x - c\right )}^{\frac {1}{3}} b - {\left (-b^{3} c - a b^{2} d\right )}^{\frac {1}{3}}\right )}{2 \, {\left (b^{3} c + a b^{2} d\right )}}\right ] \] Input:

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

Output:

[1/2*(sqrt(3)*(b^2*c + a*b*d)*sqrt((-b^3*c - a*b^2*d)^(1/3)/(b*c + a*d))*l 
og((2*b^2*d*x - 3*b^2*c - a*b*d + sqrt(3)*((-b^3*c - a*b^2*d)^(1/3)*(b*c + 
 a*d) + (b^2*c + a*b*d)*(d*x - c)^(1/3) + 2*(-b^3*c - a*b^2*d)^(2/3)*(d*x 
- c)^(2/3))*sqrt((-b^3*c - a*b^2*d)^(1/3)/(b*c + a*d)) - 3*(-b^3*c - a*b^2 
*d)^(2/3)*(d*x - c)^(1/3))/(b*x + a)) + (-b^3*c - a*b^2*d)^(2/3)*log((d*x 
- c)^(2/3)*b^2 + (-b^3*c - a*b^2*d)^(1/3)*(d*x - c)^(1/3)*b + (-b^3*c - a* 
b^2*d)^(2/3)) - 2*(-b^3*c - a*b^2*d)^(2/3)*log((d*x - c)^(1/3)*b - (-b^3*c 
 - a*b^2*d)^(1/3)))/(b^3*c + a*b^2*d), 1/2*(2*sqrt(3)*(b^2*c + a*b*d)*sqrt 
(-(-b^3*c - a*b^2*d)^(1/3)/(b*c + a*d))*arctan(1/3*sqrt(3)*(2*(d*x - c)^(1 
/3)*b + (-b^3*c - a*b^2*d)^(1/3))*sqrt(-(-b^3*c - a*b^2*d)^(1/3)/(b*c + a* 
d))/b) + (-b^3*c - a*b^2*d)^(2/3)*log((d*x - c)^(2/3)*b^2 + (-b^3*c - a*b^ 
2*d)^(1/3)*(d*x - c)^(1/3)*b + (-b^3*c - a*b^2*d)^(2/3)) - 2*(-b^3*c - a*b 
^2*d)^(2/3)*log((d*x - c)^(1/3)*b - (-b^3*c - a*b^2*d)^(1/3)))/(b^3*c + a* 
b^2*d)]
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.33 \[ \int \frac {1}{(a+b x) \sqrt [3]{-c+d x}} \, dx=\frac {\frac {2 \, \sqrt {3} d \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (d x - c\right )}^{\frac {1}{3}} - \left (\frac {b c + a d}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {b c + a d}{b}\right )^{\frac {1}{3}}}\right )}{b \left (\frac {b c + a d}{b}\right )^{\frac {1}{3}}} + \frac {d \log \left ({\left (d x - c\right )}^{\frac {2}{3}} - {\left (d x - c\right )}^{\frac {1}{3}} \left (\frac {b c + a d}{b}\right )^{\frac {1}{3}} + \left (\frac {b c + a d}{b}\right )^{\frac {2}{3}}\right )}{b \left (\frac {b c + a d}{b}\right )^{\frac {1}{3}}} - \frac {2 \, d \log \left ({\left (d x - c\right )}^{\frac {1}{3}} + \left (\frac {b c + a d}{b}\right )^{\frac {1}{3}}\right )}{b \left (\frac {b c + a d}{b}\right )^{\frac {1}{3}}}}{2 \, d} \] Input:

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 219 vs. \(2 (107) = 214\).

Time = 0.15 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.59 \[ \int \frac {1}{(a+b x) \sqrt [3]{-c+d x}} \, dx=-\frac {3 \, {\left (-b^{3} c - a b^{2} d\right )}^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (d x - c\right )}^{\frac {1}{3}} + \left (-\frac {b c + a d}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {b c + a d}{b}\right )^{\frac {1}{3}}}\right )}{\sqrt {3} b^{3} c + \sqrt {3} a b^{2} d} + \frac {{\left (-b^{3} c - a b^{2} d\right )}^{\frac {2}{3}} \log \left ({\left (d x - c\right )}^{\frac {2}{3}} + {\left (d x - c\right )}^{\frac {1}{3}} \left (-\frac {b c + a d}{b}\right )^{\frac {1}{3}} + \left (-\frac {b c + a d}{b}\right )^{\frac {2}{3}}\right )}{2 \, {\left (b^{3} c + a b^{2} d\right )}} - \frac {\left (-\frac {b c + a d}{b}\right )^{\frac {2}{3}} \log \left ({\left | {\left (d x - c\right )}^{\frac {1}{3}} - \left (-\frac {b c + a d}{b}\right )^{\frac {1}{3}} \right |}\right )}{b c + a d} \] Input:

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

Output:

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

Mupad [B] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.56 \[ \int \frac {1}{(a+b x) \sqrt [3]{-c+d x}} \, dx=\frac {\ln \left (9\,b\,{\left (d\,x-c\right )}^{1/3}+\frac {9\,c\,b^3+9\,a\,d\,b^2}{b^{4/3}\,{\left (-a\,d-b\,c\right )}^{2/3}}\right )}{b^{2/3}\,{\left (-a\,d-b\,c\right )}^{1/3}}+\frac {\ln \left (9\,b\,{\left (d\,x-c\right )}^{1/3}+\frac {{\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}^2\,\left (9\,c\,b^3+9\,a\,d\,b^2\right )}{4\,b^{4/3}\,{\left (-a\,d-b\,c\right )}^{2/3}}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{2\,b^{2/3}\,{\left (-a\,d-b\,c\right )}^{1/3}}-\frac {\ln \left (9\,b\,{\left (d\,x-c\right )}^{1/3}+\frac {{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^2\,\left (9\,c\,b^3+9\,a\,d\,b^2\right )}{4\,b^{4/3}\,{\left (-a\,d-b\,c\right )}^{2/3}}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{2\,b^{2/3}\,{\left (-a\,d-b\,c\right )}^{1/3}} \] Input:

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

Output:

log(9*b*(d*x - c)^(1/3) + (9*b^3*c + 9*a*b^2*d)/(b^(4/3)*(- a*d - b*c)^(2/ 
3)))/(b^(2/3)*(- a*d - b*c)^(1/3)) + (log(9*b*(d*x - c)^(1/3) + ((3^(1/2)* 
1i - 1)^2*(9*b^3*c + 9*a*b^2*d))/(4*b^(4/3)*(- a*d - b*c)^(2/3)))*(3^(1/2) 
*1i - 1))/(2*b^(2/3)*(- a*d - b*c)^(1/3)) - (log(9*b*(d*x - c)^(1/3) + ((3 
^(1/2)*1i + 1)^2*(9*b^3*c + 9*a*b^2*d))/(4*b^(4/3)*(- a*d - b*c)^(2/3)))*( 
3^(1/2)*1i + 1))/(2*b^(2/3)*(- a*d - b*c)^(1/3))
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.70 \[ \int \frac {1}{(a+b x) \sqrt [3]{-c+d x}} \, dx=\frac {2 \sqrt {3}\, \mathit {atan} \left (\frac {2 b^{\frac {1}{3}} \left (d x -c \right )^{\frac {1}{6}}-b^{\frac {1}{6}} \left (a d +b c \right )^{\frac {1}{6}} \sqrt {3}}{b^{\frac {1}{6}} \left (a d +b c \right )^{\frac {1}{6}}}\right )-2 \sqrt {3}\, \mathit {atan} \left (\frac {2 b^{\frac {1}{3}} \left (d x -c \right )^{\frac {1}{6}}+b^{\frac {1}{6}} \left (a d +b c \right )^{\frac {1}{6}} \sqrt {3}}{b^{\frac {1}{6}} \left (a d +b c \right )^{\frac {1}{6}}}\right )+\mathrm {log}\left (-b^{\frac {1}{6}} \left (a d +b c \right )^{\frac {1}{6}} \left (d x -c \right )^{\frac {1}{6}} \sqrt {3}+b^{\frac {1}{3}} \left (d x -c \right )^{\frac {1}{3}}+\left (a d +b c \right )^{\frac {1}{3}}\right )+\mathrm {log}\left (b^{\frac {1}{6}} \left (a d +b c \right )^{\frac {1}{6}} \left (d x -c \right )^{\frac {1}{6}} \sqrt {3}+b^{\frac {1}{3}} \left (d x -c \right )^{\frac {1}{3}}+\left (a d +b c \right )^{\frac {1}{3}}\right )-2 \,\mathrm {log}\left (b^{\frac {1}{3}} \left (d x -c \right )^{\frac {1}{3}}+\left (a d +b c \right )^{\frac {1}{3}}\right )}{2 b^{\frac {2}{3}} \left (a d +b c \right )^{\frac {1}{3}}} \] Input:

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

Output:

(2*sqrt(3)*atan((2*b**(1/3)*( - c + d*x)**(1/6) - b**(1/6)*(a*d + b*c)**(1 
/6)*sqrt(3))/(b**(1/6)*(a*d + b*c)**(1/6))) - 2*sqrt(3)*atan((2*b**(1/3)*( 
 - c + d*x)**(1/6) + b**(1/6)*(a*d + b*c)**(1/6)*sqrt(3))/(b**(1/6)*(a*d + 
 b*c)**(1/6))) + log( - b**(1/6)*(a*d + b*c)**(1/6)*( - c + d*x)**(1/6)*sq 
rt(3) + b**(1/3)*( - c + d*x)**(1/3) + (a*d + b*c)**(1/3)) + log(b**(1/6)* 
(a*d + b*c)**(1/6)*( - c + d*x)**(1/6)*sqrt(3) + b**(1/3)*( - c + d*x)**(1 
/3) + (a*d + b*c)**(1/3)) - 2*log(b**(1/3)*( - c + d*x)**(1/3) + (a*d + b* 
c)**(1/3)))/(2*b**(2/3)*(a*d + b*c)**(1/3))