Integrand size = 19, antiderivative size = 138 \[ \int \frac {1}{(a+b x) (-c+d x)^{2/3}} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} \sqrt [3]{-c+d x}}{\sqrt [3]{b c+a d}}}{\sqrt {3}}\right )}{\sqrt [3]{b} (b c+a d)^{2/3}}-\frac {\log (a+b x)}{2 \sqrt [3]{b} (b c+a d)^{2/3}}+\frac {3 \log \left (\sqrt [3]{b c+a d}+\sqrt [3]{b} \sqrt [3]{-c+d x}\right )}{2 \sqrt [3]{b} (b c+a d)^{2/3}} \] 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 ^(1/3)/(a*d+b*c)^(2/3)-1/2*ln(b*x+a)/b^(1/3)/(a*d+b*c)^(2/3)+3/2*ln((a*d+b *c)^(1/3)+b^(1/3)*(d*x-c)^(1/3))/b^(1/3)/(a*d+b*c)^(2/3)
Time = 0.13 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.14 \[ \int \frac {1}{(a+b x) (-c+d x)^{2/3}} \, dx=-\frac {2 \sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} \sqrt [3]{-c+d x}}{\sqrt [3]{b c+a d}}}{\sqrt {3}}\right )-2 \log \left (\sqrt [3]{b c+a d}+\sqrt [3]{b} \sqrt [3]{-c+d x}\right )+\log \left ((b c+a d)^{2/3}-\sqrt [3]{b} \sqrt [3]{b c+a d} \sqrt [3]{-c+d x}+b^{2/3} (-c+d x)^{2/3}\right )}{2 \sqrt [3]{b} (b c+a d)^{2/3}} \] Input:
Integrate[1/((a + b*x)*(-c + d*x)^(2/3)),x]
Output:
-1/2*(2*Sqrt[3]*ArcTan[(1 - (2*b^(1/3)*(-c + d*x)^(1/3))/(b*c + a*d)^(1/3) )/Sqrt[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)])/(b^(1/3)*(b*c + a*d)^(2/3))
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 = {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.
| \(\displaystyle \int \frac {1}{(a+b x) (d x-c)^{2/3}} \, dx\) |
| \(\Big \downarrow \) 70 |
| \(\displaystyle \frac {3 \int \frac {1}{\frac {(b c+a d)^{2/3}}{b^{2/3}}-\frac {\sqrt [3]{d x-c} \sqrt [3]{b c+a d}}{\sqrt [3]{b}}+(d x-c)^{2/3}}d\sqrt [3]{d x-c}}{2 b^{2/3} \sqrt [3]{a d+b c}}+\frac {3 \int \frac {1}{\frac {\sqrt [3]{b c+a d}}{\sqrt [3]{b}}+\sqrt [3]{d x-c}}d\sqrt [3]{d x-c}}{2 \sqrt [3]{b} (a d+b c)^{2/3}}-\frac {\log (a+b x)}{2 \sqrt [3]{b} (a d+b c)^{2/3}}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {3 \int \frac {1}{\frac {(b c+a d)^{2/3}}{b^{2/3}}-\frac {\sqrt [3]{d x-c} \sqrt [3]{b c+a d}}{\sqrt [3]{b}}+(d x-c)^{2/3}}d\sqrt [3]{d x-c}}{2 b^{2/3} \sqrt [3]{a d+b c}}-\frac {\log (a+b x)}{2 \sqrt [3]{b} (a d+b c)^{2/3}}+\frac {3 \log \left (\sqrt [3]{a d+b c}+\sqrt [3]{b} \sqrt [3]{d x-c}\right )}{2 \sqrt [3]{b} (a d+b c)^{2/3}}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {3 \int \frac {1}{-(d x-c)^{2/3}-3}d\left (1-\frac {2 \sqrt [3]{b} \sqrt [3]{d x-c}}{\sqrt [3]{b c+a d}}\right )}{\sqrt [3]{b} (a d+b c)^{2/3}}-\frac {\log (a+b x)}{2 \sqrt [3]{b} (a d+b c)^{2/3}}+\frac {3 \log \left (\sqrt [3]{a d+b c}+\sqrt [3]{b} \sqrt [3]{d x-c}\right )}{2 \sqrt [3]{b} (a d+b c)^{2/3}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} \sqrt [3]{d x-c}}{\sqrt [3]{a d+b c}}}{\sqrt {3}}\right )}{\sqrt [3]{b} (a d+b c)^{2/3}}-\frac {\log (a+b x)}{2 \sqrt [3]{b} (a d+b c)^{2/3}}+\frac {3 \log \left (\sqrt [3]{a d+b c}+\sqrt [3]{b} \sqrt [3]{d x-c}\right )}{2 \sqrt [3]{b} (a d+b c)^{2/3}}\) |
Input:
Int[1/((a + b*x)*(-c + d*x)^(2/3)),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^(1/3)*(b*c + a*d)^(2/3))) - Log[a + b*x]/(2*b^(1/3)*(b*c + a*d)^ (2/3)) + (3*Log[(b*c + a*d)^(1/3) + b^(1/3)*(-c + d*x)^(1/3)])/(2*b^(1/3)* (b*c + a*d)^(2/3))
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
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]
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])
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]
Time = 0.42 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.05
| method | result | size |
| pseudoelliptic | \(\frac {-2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (-2 \left (x d -c \right )^{\frac {1}{3}}+\left (\frac {a d +b c}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a d +b c}{b}\right )^{\frac {1}{3}}}\right )+2 \ln \left (\left (x d -c \right )^{\frac {1}{3}}+\left (\frac {a d +b c}{b}\right )^{\frac {1}{3}}\right )-\ln \left (\left (x d -c \right )^{\frac {2}{3}}-\left (\frac {a d +b c}{b}\right )^{\frac {1}{3}} \left (x d -c \right )^{\frac {1}{3}}+\left (\frac {a d +b c}{b}\right )^{\frac {2}{3}}\right )}{2 b \left (\frac {a d +b c}{b}\right )^{\frac {2}{3}}}\) | \(145\) |
| derivativedivides | \(\frac {\ln \left (\left (x d -c \right )^{\frac {1}{3}}+\left (\frac {a d +b c}{b}\right )^{\frac {1}{3}}\right )}{b \left (\frac {a d +b c}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (\left (x d -c \right )^{\frac {2}{3}}-\left (\frac {a d +b c}{b}\right )^{\frac {1}{3}} \left (x d -c \right )^{\frac {1}{3}}+\left (\frac {a d +b c}{b}\right )^{\frac {2}{3}}\right )}{2 b \left (\frac {a d +b c}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (x d -c \right )^{\frac {1}{3}}}{\left (\frac {a d +b c}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{b \left (\frac {a d +b c}{b}\right )^{\frac {2}{3}}}\) | \(161\) |
| default | \(\frac {\ln \left (\left (x d -c \right )^{\frac {1}{3}}+\left (\frac {a d +b c}{b}\right )^{\frac {1}{3}}\right )}{b \left (\frac {a d +b c}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (\left (x d -c \right )^{\frac {2}{3}}-\left (\frac {a d +b c}{b}\right )^{\frac {1}{3}} \left (x d -c \right )^{\frac {1}{3}}+\left (\frac {a d +b c}{b}\right )^{\frac {2}{3}}\right )}{2 b \left (\frac {a d +b c}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (x d -c \right )^{\frac {1}{3}}}{\left (\frac {a d +b c}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{b \left (\frac {a d +b c}{b}\right )^{\frac {2}{3}}}\) | \(161\) |
Input:
int(1/(b*x+a)/(d*x-c)^(2/3),x,method=_RETURNVERBOSE)
Output:
1/2*(-2*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))+2*ln((d*x-c)^(1/3)+((a*d+b*c)/b)^(1/3))-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)))/b/((a*d+b*c)/b )^(2/3)
Leaf count of result is larger than twice the leaf count of optimal. 406 vs. \(2 (107) = 214\).
Time = 0.14 (sec) , antiderivative size = 903, normalized size of antiderivative = 6.54 \[ \int \frac {1}{(a+b x) (-c+d x)^{2/3}} \, dx =\text {Too large to display} \] Input:
integrate(1/(b*x+a)/(d*x-c)^(2/3),x, algorithm="fricas")
Output:
[1/2*(sqrt(3)*(b^2*c + a*b*d)*sqrt(-(b^3*c^2 + 2*a*b^2*c*d + a^2*b*d^2)^(1 /3)/b)*log(-(3*b^2*c^2 + 4*a*b*c*d + a^2*d^2 - 2*(b^2*c*d + a*b*d^2)*x - s qrt(3)*(2*(b^2*c + a*b*d)*(d*x - c)^(2/3) - (b^3*c^2 + 2*a*b^2*c*d + a^2*b *d^2)^(1/3)*(b*c + a*d) + (b^3*c^2 + 2*a*b^2*c*d + a^2*b*d^2)^(2/3)*(d*x - c)^(1/3))*sqrt(-(b^3*c^2 + 2*a*b^2*c*d + a^2*b*d^2)^(1/3)/b) + 3*(b^3*c^2 + 2*a*b^2*c*d + a^2*b*d^2)^(1/3)*(b*c + a*d)*(d*x - c)^(1/3))/(b*x + a)) - (b^3*c^2 + 2*a*b^2*c*d + a^2*b*d^2)^(2/3)*log((b^2*c + a*b*d)*(d*x - c)^ (2/3) + (b^3*c^2 + 2*a*b^2*c*d + a^2*b*d^2)^(1/3)*(b*c + a*d) - (b^3*c^2 + 2*a*b^2*c*d + a^2*b*d^2)^(2/3)*(d*x - c)^(1/3)) + 2*(b^3*c^2 + 2*a*b^2*c* d + a^2*b*d^2)^(2/3)*log((b^2*c + a*b*d)*(d*x - c)^(1/3) + (b^3*c^2 + 2*a* b^2*c*d + a^2*b*d^2)^(2/3)))/(b^3*c^2 + 2*a*b^2*c*d + a^2*b*d^2), 1/2*(2*s qrt(3)*(b^2*c + a*b*d)*sqrt((b^3*c^2 + 2*a*b^2*c*d + a^2*b*d^2)^(1/3)/b)*a rctan(-1/3*sqrt(3)*((b^3*c^2 + 2*a*b^2*c*d + a^2*b*d^2)^(1/3)*(b*c + a*d) - 2*(b^3*c^2 + 2*a*b^2*c*d + a^2*b*d^2)^(2/3)*(d*x - c)^(1/3))*sqrt((b^3*c ^2 + 2*a*b^2*c*d + a^2*b*d^2)^(1/3)/b)/(b^2*c^2 + 2*a*b*c*d + a^2*d^2)) - (b^3*c^2 + 2*a*b^2*c*d + a^2*b*d^2)^(2/3)*log((b^2*c + a*b*d)*(d*x - c)^(2 /3) + (b^3*c^2 + 2*a*b^2*c*d + a^2*b*d^2)^(1/3)*(b*c + a*d) - (b^3*c^2 + 2 *a*b^2*c*d + a^2*b*d^2)^(2/3)*(d*x - c)^(1/3)) + 2*(b^3*c^2 + 2*a*b^2*c*d + a^2*b*d^2)^(2/3)*log((b^2*c + a*b*d)*(d*x - c)^(1/3) + (b^3*c^2 + 2*a*b^ 2*c*d + a^2*b*d^2)^(2/3)))/(b^3*c^2 + 2*a*b^2*c*d + a^2*b*d^2)]
\[ \int \frac {1}{(a+b x) (-c+d x)^{2/3}} \, dx=\int \frac {1}{\left (a + b x\right ) \left (- c + d x\right )^{\frac {2}{3}}}\, dx \] Input:
integrate(1/(b*x+a)/(d*x-c)**(2/3),x)
Output:
Integral(1/((a + b*x)*(-c + d*x)**(2/3)), x)
Time = 0.17 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.33 \[ \int \frac {1}{(a+b x) (-c+d x)^{2/3}} \, dx=\frac {\frac {2 \, \sqrt {3} d \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (d x - c\right )}^{\frac {1}{3}} - \left (\frac {b c + a d}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {b c + a d}{b}\right )^{\frac {1}{3}}}\right )}{b \left (\frac {b c + a d}{b}\right )^{\frac {2}{3}}} - \frac {d \log \left ({\left (d x - c\right )}^{\frac {2}{3}} - {\left (d x - c\right )}^{\frac {1}{3}} \left (\frac {b c + a d}{b}\right )^{\frac {1}{3}} + \left (\frac {b c + a d}{b}\right )^{\frac {2}{3}}\right )}{b \left (\frac {b c + a d}{b}\right )^{\frac {2}{3}}} + \frac {2 \, d \log \left ({\left (d x - c\right )}^{\frac {1}{3}} + \left (\frac {b c + a d}{b}\right )^{\frac {1}{3}}\right )}{b \left (\frac {b c + a d}{b}\right )^{\frac {2}{3}}}}{2 \, d} \] Input:
integrate(1/(b*x+a)/(d*x-c)^(2/3),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)^(2/3)) - 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)^(2/3)) + 2*d*log((d*x - c)^(1/3) + ((b*c + a*d)/b)^(1/3))/( b*((b*c + a*d)/b)^(2/3)))/d
Leaf count of result is larger than twice the leaf count of optimal. 215 vs. \(2 (107) = 214\).
Time = 0.14 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.56 \[ \int \frac {1}{(a+b x) (-c+d x)^{2/3}} \, dx=\frac {3 \, {\left (-b^{3} c - a b^{2} d\right )}^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (d x - c\right )}^{\frac {1}{3}} + \left (-\frac {b c + a d}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {b c + a d}{b}\right )^{\frac {1}{3}}}\right )}{\sqrt {3} b^{2} c + \sqrt {3} a b d} + \frac {{\left (-b^{3} c - a b^{2} d\right )}^{\frac {1}{3}} \log \left ({\left (d x - c\right )}^{\frac {2}{3}} + {\left (d x - c\right )}^{\frac {1}{3}} \left (-\frac {b c + a d}{b}\right )^{\frac {1}{3}} + \left (-\frac {b c + a d}{b}\right )^{\frac {2}{3}}\right )}{2 \, {\left (b^{2} c + a b d\right )}} - \frac {\left (-\frac {b c + a d}{b}\right )^{\frac {1}{3}} \log \left ({\left | {\left (d x - c\right )}^{\frac {1}{3}} - \left (-\frac {b c + a d}{b}\right )^{\frac {1}{3}} \right |}\right )}{b c + a d} \] Input:
integrate(1/(b*x+a)/(d*x-c)^(2/3),x, algorithm="giac")
Output:
3*(-b^3*c - a*b^2*d)^(1/3)*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^2*c + sqrt(3)*a*b*d) + 1/2*(-b^3*c - a*b^2*d)^(1/3)*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^2*c + a*b*d) - (-(b*c + a*d) /b)^(1/3)*log(abs((d*x - c)^(1/3) - (-(b*c + a*d)/b)^(1/3)))/(b*c + a*d)
Time = 0.08 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.49 \[ \int \frac {1}{(a+b x) (-c+d x)^{2/3}} \, dx=\frac {\ln \left (9\,b^2\,{\left (d\,x-c\right )}^{1/3}+\frac {9\,c\,b^3+9\,a\,d\,b^2}{b^{1/3}\,{\left (a\,d+b\,c\right )}^{2/3}}\right )}{b^{1/3}\,{\left (a\,d+b\,c\right )}^{2/3}}+\frac {\ln \left (9\,b^2\,{\left (d\,x-c\right )}^{1/3}+\frac {\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\,\left (9\,c\,b^3+9\,a\,d\,b^2\right )}{2\,b^{1/3}\,{\left (a\,d+b\,c\right )}^{2/3}}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{2\,b^{1/3}\,{\left (a\,d+b\,c\right )}^{2/3}}-\frac {\ln \left (9\,b^2\,{\left (d\,x-c\right )}^{1/3}-\frac {\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\,\left (9\,c\,b^3+9\,a\,d\,b^2\right )}{2\,b^{1/3}\,{\left (a\,d+b\,c\right )}^{2/3}}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{2\,b^{1/3}\,{\left (a\,d+b\,c\right )}^{2/3}} \] Input:
int(1/((a + b*x)*(d*x - c)^(2/3)),x)
Output:
log(9*b^2*(d*x - c)^(1/3) + (9*b^3*c + 9*a*b^2*d)/(b^(1/3)*(a*d + b*c)^(2/ 3)))/(b^(1/3)*(a*d + b*c)^(2/3)) + (log(9*b^2*(d*x - c)^(1/3) + ((3^(1/2)* 1i - 1)*(9*b^3*c + 9*a*b^2*d))/(2*b^(1/3)*(a*d + b*c)^(2/3)))*(3^(1/2)*1i - 1))/(2*b^(1/3)*(a*d + b*c)^(2/3)) - (log(9*b^2*(d*x - c)^(1/3) - ((3^(1/ 2)*1i + 1)*(9*b^3*c + 9*a*b^2*d))/(2*b^(1/3)*(a*d + b*c)^(2/3)))*(3^(1/2)* 1i + 1))/(2*b^(1/3)*(a*d + b*c)^(2/3))
Time = 0.18 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.73 \[ \int \frac {1}{(a+b x) (-c+d x)^{2/3}} \, dx=\frac {2 \sqrt {3}\, \mathit {atan} \left (\frac {2 b^{\frac {1}{3}} \left (d x -c \right )^{\frac {1}{6}}-b^{\frac {1}{6}} \left (a d +b c \right )^{\frac {1}{6}} \sqrt {3}}{b^{\frac {1}{6}} \left (a d +b c \right )^{\frac {1}{6}}}\right )-2 \sqrt {3}\, \mathit {atan} \left (\frac {2 b^{\frac {1}{3}} \left (d x -c \right )^{\frac {1}{6}}+b^{\frac {1}{6}} \left (a d +b c \right )^{\frac {1}{6}} \sqrt {3}}{b^{\frac {1}{6}} \left (a d +b c \right )^{\frac {1}{6}}}\right )-\mathrm {log}\left (-b^{\frac {1}{6}} \left (a d +b c \right )^{\frac {1}{6}} \left (d x -c \right )^{\frac {1}{6}} \sqrt {3}+b^{\frac {1}{3}} \left (d x -c \right )^{\frac {1}{3}}+\left (a d +b c \right )^{\frac {1}{3}}\right )-\mathrm {log}\left (b^{\frac {1}{6}} \left (a d +b c \right )^{\frac {1}{6}} \left (d x -c \right )^{\frac {1}{6}} \sqrt {3}+b^{\frac {1}{3}} \left (d x -c \right )^{\frac {1}{3}}+\left (a d +b c \right )^{\frac {1}{3}}\right )+2 \,\mathrm {log}\left (b^{\frac {1}{3}} \left (d x -c \right )^{\frac {1}{3}}+\left (a d +b c \right )^{\frac {1}{3}}\right )}{2 b^{\frac {1}{3}} \left (a d +b c \right )^{\frac {2}{3}}} \] Input:
int(1/(b*x+a)/(d*x-c)^(2/3),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**(1/3)*(a*d + b*c)**(2/3))