\(\int \frac {a+b x}{d+e x^3} \, dx\) [1]

Optimal result

Integrand size = 15, antiderivative size = 161 \[ \int \frac {a+b x}{d+e x^3} \, dx=-\frac {\left (b \sqrt [3]{d}+a \sqrt [3]{e}\right ) \arctan \left (\frac {\sqrt [3]{d}-2 \sqrt [3]{e} x}{\sqrt {3} \sqrt [3]{d}}\right )}{\sqrt {3} d^{2/3} e^{2/3}}-\frac {\left (b \sqrt [3]{d}-a \sqrt [3]{e}\right ) \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{3 d^{2/3} e^{2/3}}-\frac {\left (a-\frac {b \sqrt [3]{d}}{\sqrt [3]{e}}\right ) \log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )}{6 d^{2/3} \sqrt [3]{e}} \] Output:

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

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.78 \[ \int \frac {a+b x}{d+e x^3} \, dx=\frac {-2 \sqrt {3} \left (b \sqrt [3]{d}+a \sqrt [3]{e}\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{e} x}{\sqrt [3]{d}}}{\sqrt {3}}\right )-\left (b \sqrt [3]{d}-a \sqrt [3]{e}\right ) \left (2 \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )-\log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )\right )}{6 d^{2/3} e^{2/3}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.56 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.97, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.533, Rules used = {2399, 16, 1142, 25, 27, 1082, 217, 1103}

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.

\(\displaystyle \int \frac {a+b x}{d+e x^3} \, dx\)

\(\Big \downarrow \) 2399

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

\(\Big \downarrow \) 16

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

\(\Big \downarrow \) 1142

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1082

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

\(\Big \downarrow \) 217

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

\(\Big \downarrow \) 1103

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

Input:

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

Output:

((a - (b*d^(1/3))/e^(1/3))*Log[d^(1/3) + e^(1/3)*x])/(3*d^(2/3)*e^(1/3)) + 
 (-((Sqrt[3]*(b*d^(1/3) + a*e^(1/3))*ArcTan[(1 - (2*e^(1/3)*x)/d^(1/3))/Sq 
rt[3]])/e^(1/3)) - ((a - (b*d^(1/3))/e^(1/3))*Log[d^(2/3) - d^(1/3)*e^(1/3 
)*x + e^(2/3)*x^2])/2)/(3*d^(2/3)*e^(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 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], 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 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]
 

rule 1103

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, 
e}, x] && EqQ[2*c*d - b*e, 0]
 

rule 1142

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[(2*c*d - b*e)/(2*c)   Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) 
Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
 

rule 2399

Int[((A_) + (B_.)*(x_))/((a_) + (b_.)*(x_)^3), x_Symbol] :> With[{r = Numer 
ator[Rt[a/b, 3]], s = Denominator[Rt[a/b, 3]]}, Simp[(-r)*((B*r - A*s)/(3*a 
*s))   Int[1/(r + s*x), x], x] + Simp[r/(3*a*s)   Int[(r*(B*r + 2*A*s) + s* 
(B*r - A*s)*x)/(r^2 - r*s*x + s^2*x^2), x], x]] /; FreeQ[{a, b, A, B}, x] & 
& NeQ[a*B^3 - b*A^3, 0] && PosQ[a/b]
 

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.09 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.20

method	result	size
risch	\(\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{3}+d \right )}{\sum }\frac {\left (\textit {\_R} b +a \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2}}}{3 e}\)	\(32\)
default	\(a \left (\frac {\ln \left (x +\left (\frac {d}{e}\right )^{\frac {1}{3}}\right )}{3 e \left (\frac {d}{e}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {d}{e}\right )^{\frac {1}{3}} x +\left (\frac {d}{e}\right )^{\frac {2}{3}}\right )}{6 e \left (\frac {d}{e}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {d}{e}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 e \left (\frac {d}{e}\right )^{\frac {2}{3}}}\right )+b \left (-\frac {\ln \left (x +\left (\frac {d}{e}\right )^{\frac {1}{3}}\right )}{3 e \left (\frac {d}{e}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {d}{e}\right )^{\frac {1}{3}} x +\left (\frac {d}{e}\right )^{\frac {2}{3}}\right )}{6 e \left (\frac {d}{e}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {d}{e}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 e \left (\frac {d}{e}\right )^{\frac {1}{3}}}\right )\)	\(186\)

Input:

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

Output:

1/3/e*sum((_R*b+a)/_R^2*ln(x-_R),_R=RootOf(_Z^3*e+d))
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.87 (sec) , antiderivative size = 1961, normalized size of antiderivative = 12.18 \[ \int \frac {a+b x}{d+e x^3} \, dx=\text {Too large to display} \] Input:

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

Output:

-1/6*((1/2)^(1/3)*(I*sqrt(3) + 1)*((b^3*d + a^3*e)/(d^2*e^2) - (b^3*d - a^ 
3*e)/(d^2*e^2))^(1/3) - 2*(1/2)^(2/3)*a*b*(-I*sqrt(3) + 1)/(d*e*((b^3*d + 
a^3*e)/(d^2*e^2) - (b^3*d - a^3*e)/(d^2*e^2))^(1/3)))*log(1/4*((1/2)^(1/3) 
*(I*sqrt(3) + 1)*((b^3*d + a^3*e)/(d^2*e^2) - (b^3*d - a^3*e)/(d^2*e^2))^( 
1/3) - 2*(1/2)^(2/3)*a*b*(-I*sqrt(3) + 1)/(d*e*((b^3*d + a^3*e)/(d^2*e^2) 
- (b^3*d - a^3*e)/(d^2*e^2))^(1/3)))^2*b*d^2*e - 1/2*((1/2)^(1/3)*(I*sqrt( 
3) + 1)*((b^3*d + a^3*e)/(d^2*e^2) - (b^3*d - a^3*e)/(d^2*e^2))^(1/3) - 2* 
(1/2)^(2/3)*a*b*(-I*sqrt(3) + 1)/(d*e*((b^3*d + a^3*e)/(d^2*e^2) - (b^3*d 
- a^3*e)/(d^2*e^2))^(1/3)))*a^2*d*e + 2*a*b^2*d + (b^3*d + a^3*e)*x) + 1/1 
2*((1/2)^(1/3)*(I*sqrt(3) + 1)*((b^3*d + a^3*e)/(d^2*e^2) - (b^3*d - a^3*e 
)/(d^2*e^2))^(1/3) - 2*(1/2)^(2/3)*a*b*(-I*sqrt(3) + 1)/(d*e*((b^3*d + a^3 
*e)/(d^2*e^2) - (b^3*d - a^3*e)/(d^2*e^2))^(1/3)) + 3*sqrt(1/3)*sqrt(-(((1 
/2)^(1/3)*(I*sqrt(3) + 1)*((b^3*d + a^3*e)/(d^2*e^2) - (b^3*d - a^3*e)/(d^ 
2*e^2))^(1/3) - 2*(1/2)^(2/3)*a*b*(-I*sqrt(3) + 1)/(d*e*((b^3*d + a^3*e)/( 
d^2*e^2) - (b^3*d - a^3*e)/(d^2*e^2))^(1/3)))^2*d*e + 16*a*b)/(d*e)))*log( 
-1/4*((1/2)^(1/3)*(I*sqrt(3) + 1)*((b^3*d + a^3*e)/(d^2*e^2) - (b^3*d - a^ 
3*e)/(d^2*e^2))^(1/3) - 2*(1/2)^(2/3)*a*b*(-I*sqrt(3) + 1)/(d*e*((b^3*d + 
a^3*e)/(d^2*e^2) - (b^3*d - a^3*e)/(d^2*e^2))^(1/3)))^2*b*d^2*e + 1/2*((1/ 
2)^(1/3)*(I*sqrt(3) + 1)*((b^3*d + a^3*e)/(d^2*e^2) - (b^3*d - a^3*e)/(d^2 
*e^2))^(1/3) - 2*(1/2)^(2/3)*a*b*(-I*sqrt(3) + 1)/(d*e*((b^3*d + a^3*e)...
 

Sympy [A] (verification not implemented)

Time = 0.39 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.47 \[ \int \frac {a+b x}{d+e x^3} \, dx=\operatorname {RootSum} {\left (27 t^{3} d^{2} e^{2} + 9 t a b d e - a^{3} e + b^{3} d, \left ( t \mapsto t \log {\left (x + \frac {9 t^{2} b d^{2} e + 3 t a^{2} d e + 2 a b^{2} d}{a^{3} e + b^{3} d} \right )} \right )\right )} \] Input:

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

Output:

RootSum(27*_t**3*d**2*e**2 + 9*_t*a*b*d*e - a**3*e + b**3*d, Lambda(_t, _t 
*log(x + (9*_t**2*b*d**2*e + 3*_t*a**2*d*e + 2*a*b**2*d)/(a**3*e + b**3*d) 
)))
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {a+b x}{d+e x^3} \, dx=\text {Exception raised: ValueError} \] Input:

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

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(e>0)', see `assume?` for more de 
tails)Is e
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.88 \[ \int \frac {a+b x}{d+e x^3} \, dx=-\frac {\sqrt {3} {\left (a e - \left (-d e^{2}\right )^{\frac {1}{3}} b\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {d}{e}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {d}{e}\right )^{\frac {1}{3}}}\right )}{3 \, \left (-d e^{2}\right )^{\frac {2}{3}}} - \frac {{\left (a e + \left (-d e^{2}\right )^{\frac {1}{3}} b\right )} \log \left (x^{2} + x \left (-\frac {d}{e}\right )^{\frac {1}{3}} + \left (-\frac {d}{e}\right )^{\frac {2}{3}}\right )}{6 \, \left (-d e^{2}\right )^{\frac {2}{3}}} - \frac {{\left (b \left (-\frac {d}{e}\right )^{\frac {1}{3}} + a\right )} \left (-\frac {d}{e}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {d}{e}\right )^{\frac {1}{3}} \right |}\right )}{3 \, d} \] Input:

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

Output:

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

Mupad [B] (verification not implemented)

Time = 5.95 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.79 \[ \int \frac {a+b x}{d+e x^3} \, dx=\sum _{k=1}^3\ln \left (e\,\left (a\,b+b^2\,x+{\mathrm {root}\left (27\,d^2\,e^2\,z^3+9\,a\,b\,d\,e\,z+b^3\,d-a^3\,e,z,k\right )}^2\,d\,e\,9+\mathrm {root}\left (27\,d^2\,e^2\,z^3+9\,a\,b\,d\,e\,z+b^3\,d-a^3\,e,z,k\right )\,a\,e\,x\,3\right )\right )\,\mathrm {root}\left (27\,d^2\,e^2\,z^3+9\,a\,b\,d\,e\,z+b^3\,d-a^3\,e,z,k\right ) \] Input:

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

Output:

symsum(log(e*(a*b + b^2*x + 9*root(27*d^2*e^2*z^3 + 9*a*b*d*e*z + b^3*d - 
a^3*e, z, k)^2*d*e + 3*root(27*d^2*e^2*z^3 + 9*a*b*d*e*z + b^3*d - a^3*e, 
z, k)*a*e*x))*root(27*d^2*e^2*z^3 + 9*a*b*d*e*z + b^3*d - a^3*e, z, k), k, 
 1, 3)
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.94 \[ \int \frac {a+b x}{d+e x^3} \, dx=\frac {-2 e^{\frac {1}{3}} d^{\frac {2}{3}} \sqrt {3}\, \mathit {atan} \left (\frac {d^{\frac {1}{3}}-2 e^{\frac {1}{3}} x}{d^{\frac {1}{3}} \sqrt {3}}\right ) a -2 \sqrt {3}\, \mathit {atan} \left (\frac {d^{\frac {1}{3}}-2 e^{\frac {1}{3}} x}{d^{\frac {1}{3}} \sqrt {3}}\right ) b d -e^{\frac {1}{3}} d^{\frac {2}{3}} \mathrm {log}\left (d^{\frac {2}{3}}-e^{\frac {1}{3}} d^{\frac {1}{3}} x +e^{\frac {2}{3}} x^{2}\right ) a +2 e^{\frac {1}{3}} d^{\frac {2}{3}} \mathrm {log}\left (d^{\frac {1}{3}}+e^{\frac {1}{3}} x \right ) a +\mathrm {log}\left (d^{\frac {2}{3}}-e^{\frac {1}{3}} d^{\frac {1}{3}} x +e^{\frac {2}{3}} x^{2}\right ) b d -2 \,\mathrm {log}\left (d^{\frac {1}{3}}+e^{\frac {1}{3}} x \right ) b d}{6 e^{\frac {2}{3}} d^{\frac {4}{3}}} \] Input:

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

Output:

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