Integrand size = 19, antiderivative size = 133 \[ \int \frac {b x+c x^2}{d+e x^3} \, dx=-\frac {b \arctan \left (\frac {\sqrt [3]{d}-2 \sqrt [3]{e} x}{\sqrt {3} \sqrt [3]{d}}\right )}{\sqrt {3} \sqrt [3]{d} e^{2/3}}+\frac {\left (c-\frac {b \sqrt [3]{e}}{\sqrt [3]{d}}\right ) \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{3 e}+\frac {\left (2 c+\frac {b \sqrt [3]{e}}{\sqrt [3]{d}}\right ) \log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )}{6 e} \] Output:
-1/3*b*arctan(1/3*(d^(1/3)-2*e^(1/3)*x)*3^(1/2)/d^(1/3))*3^(1/2)/d^(1/3)/e ^(2/3)+1/3*(c-b*e^(1/3)/d^(1/3))*ln(d^(1/3)+e^(1/3)*x)/e+1/6*(2*c+b*e^(1/3 )/d^(1/3))*ln(d^(2/3)-d^(1/3)*e^(1/3)*x+e^(2/3)*x^2)/e
Time = 0.02 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.92 \[ \int \frac {b x+c x^2}{d+e x^3} \, dx=\frac {-2 \sqrt {3} b \sqrt [3]{e} \arctan \left (\frac {1-\frac {2 \sqrt [3]{e} x}{\sqrt [3]{d}}}{\sqrt {3}}\right )-2 b \sqrt [3]{e} \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )+b \sqrt [3]{e} \log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )+2 c \sqrt [3]{d} \log \left (d+e x^3\right )}{6 \sqrt [3]{d} e} \] Input:
Integrate[(b*x + c*x^2)/(d + e*x^3),x]
Output:
(-2*Sqrt[3]*b*e^(1/3)*ArcTan[(1 - (2*e^(1/3)*x)/d^(1/3))/Sqrt[3]] - 2*b*e^ (1/3)*Log[d^(1/3) + e^(1/3)*x] + b*e^(1/3)*Log[d^(2/3) - d^(1/3)*e^(1/3)*x + e^(2/3)*x^2] + 2*c*d^(1/3)*Log[d + e*x^3])/(6*d^(1/3)*e)
Time = 0.63 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.02, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.632, Rules used = {2027, 2410, 27, 792, 821, 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.
\(\displaystyle \int \frac {b x+c x^2}{d+e x^3} \, dx\) |
\(\Big \downarrow \) 2027 |
\(\displaystyle \int \frac {x (b+c x)}{d+e x^3}dx\) |
\(\Big \downarrow \) 2410 |
\(\displaystyle \int \frac {b x}{e x^3+d}dx+c \int \frac {x^2}{e x^3+d}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle b \int \frac {x}{e x^3+d}dx+c \int \frac {x^2}{e x^3+d}dx\) |
\(\Big \downarrow \) 792 |
\(\displaystyle b \int \frac {x}{e x^3+d}dx+\frac {c \log \left (d+e x^3\right )}{3 e}\) |
\(\Big \downarrow \) 821 |
\(\displaystyle b \left (\frac {\int \frac {\sqrt [3]{e} x+\sqrt [3]{d}}{e^{2/3} x^2-\sqrt [3]{d} \sqrt [3]{e} x+d^{2/3}}dx}{3 \sqrt [3]{d} \sqrt [3]{e}}-\frac {\int \frac {1}{\sqrt [3]{e} x+\sqrt [3]{d}}dx}{3 \sqrt [3]{d} \sqrt [3]{e}}\right )+\frac {c \log \left (d+e x^3\right )}{3 e}\) |
\(\Big \downarrow \) 16 |
\(\displaystyle b \left (\frac {\int \frac {\sqrt [3]{e} x+\sqrt [3]{d}}{e^{2/3} x^2-\sqrt [3]{d} \sqrt [3]{e} x+d^{2/3}}dx}{3 \sqrt [3]{d} \sqrt [3]{e}}-\frac {\log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{3 \sqrt [3]{d} e^{2/3}}\right )+\frac {c \log \left (d+e x^3\right )}{3 e}\) |
\(\Big \downarrow \) 1142 |
\(\displaystyle b \left (\frac {\frac {3}{2} \sqrt [3]{d} \int \frac {1}{e^{2/3} x^2-\sqrt [3]{d} \sqrt [3]{e} x+d^{2/3}}dx+\frac {\int -\frac {\sqrt [3]{e} \left (\sqrt [3]{d}-2 \sqrt [3]{e} x\right )}{e^{2/3} x^2-\sqrt [3]{d} \sqrt [3]{e} x+d^{2/3}}dx}{2 \sqrt [3]{e}}}{3 \sqrt [3]{d} \sqrt [3]{e}}-\frac {\log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{3 \sqrt [3]{d} e^{2/3}}\right )+\frac {c \log \left (d+e x^3\right )}{3 e}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle b \left (\frac {\frac {3}{2} \sqrt [3]{d} \int \frac {1}{e^{2/3} x^2-\sqrt [3]{d} \sqrt [3]{e} x+d^{2/3}}dx-\frac {\int \frac {\sqrt [3]{e} \left (\sqrt [3]{d}-2 \sqrt [3]{e} x\right )}{e^{2/3} x^2-\sqrt [3]{d} \sqrt [3]{e} x+d^{2/3}}dx}{2 \sqrt [3]{e}}}{3 \sqrt [3]{d} \sqrt [3]{e}}-\frac {\log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{3 \sqrt [3]{d} e^{2/3}}\right )+\frac {c \log \left (d+e x^3\right )}{3 e}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle b \left (\frac {\frac {3}{2} \sqrt [3]{d} \int \frac {1}{e^{2/3} x^2-\sqrt [3]{d} \sqrt [3]{e} x+d^{2/3}}dx-\frac {1}{2} \int \frac {\sqrt [3]{d}-2 \sqrt [3]{e} x}{e^{2/3} x^2-\sqrt [3]{d} \sqrt [3]{e} x+d^{2/3}}dx}{3 \sqrt [3]{d} \sqrt [3]{e}}-\frac {\log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{3 \sqrt [3]{d} e^{2/3}}\right )+\frac {c \log \left (d+e x^3\right )}{3 e}\) |
\(\Big \downarrow \) 1082 |
\(\displaystyle b \left (\frac {\frac {3 \int \frac {1}{-\left (1-\frac {2 \sqrt [3]{e} x}{\sqrt [3]{d}}\right )^2-3}d\left (1-\frac {2 \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{\sqrt [3]{e}}-\frac {1}{2} \int \frac {\sqrt [3]{d}-2 \sqrt [3]{e} x}{e^{2/3} x^2-\sqrt [3]{d} \sqrt [3]{e} x+d^{2/3}}dx}{3 \sqrt [3]{d} \sqrt [3]{e}}-\frac {\log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{3 \sqrt [3]{d} e^{2/3}}\right )+\frac {c \log \left (d+e x^3\right )}{3 e}\) |
\(\Big \downarrow \) 217 |
\(\displaystyle b \left (\frac {-\frac {1}{2} \int \frac {\sqrt [3]{d}-2 \sqrt [3]{e} x}{e^{2/3} x^2-\sqrt [3]{d} \sqrt [3]{e} x+d^{2/3}}dx-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{e} x}{\sqrt [3]{d}}}{\sqrt {3}}\right )}{\sqrt [3]{e}}}{3 \sqrt [3]{d} \sqrt [3]{e}}-\frac {\log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{3 \sqrt [3]{d} e^{2/3}}\right )+\frac {c \log \left (d+e x^3\right )}{3 e}\) |
\(\Big \downarrow \) 1103 |
\(\displaystyle b \left (\frac {\frac {\log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )}{2 \sqrt [3]{e}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{e} x}{\sqrt [3]{d}}}{\sqrt {3}}\right )}{\sqrt [3]{e}}}{3 \sqrt [3]{d} \sqrt [3]{e}}-\frac {\log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{3 \sqrt [3]{d} e^{2/3}}\right )+\frac {c \log \left (d+e x^3\right )}{3 e}\) |
Input:
Int[(b*x + c*x^2)/(d + e*x^3),x]
Output:
b*(-1/3*Log[d^(1/3) + e^(1/3)*x]/(d^(1/3)*e^(2/3)) + (-((Sqrt[3]*ArcTan[(1 - (2*e^(1/3)*x)/d^(1/3))/Sqrt[3]])/e^(1/3)) + Log[d^(2/3) - d^(1/3)*e^(1/ 3)*x + e^(2/3)*x^2]/(2*e^(1/3)))/(3*d^(1/3)*e^(1/3))) + (c*Log[d + e*x^3]) /(3*e)
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveConten t[a + b*x^n, x]]/(b*n), x] /; FreeQ[{a, b, m, n}, x] && EqQ[m, n - 1]
Int[(x_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> Simp[-(3*Rt[a, 3]*Rt[b, 3])^(- 1) Int[1/(Rt[a, 3] + Rt[b, 3]*x), x], x] + Simp[1/(3*Rt[a, 3]*Rt[b, 3]) Int[(Rt[a, 3] + Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3]^2 *x^2), x], x] /; FreeQ[{a, b}, x]
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]
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]
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]
Int[(Fx_.)*((a_.)*(x_)^(r_.) + (b_.)*(x_)^(s_.))^(p_.), x_Symbol] :> Int[x^ (p*r)*(a + b*x^(s - r))^p*Fx, x] /; FreeQ[{a, b, r, s}, x] && IntegerQ[p] & & PosQ[s - r] && !(EqQ[p, 1] && EqQ[u, 1])
Int[(P2_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> With[{A = Coeff[P2, x, 0], B = Coeff[P2, x, 1], C = Coeff[P2, x, 2]}, Int[(A + B*x)/(a + b*x^3), x] + Si mp[C Int[x^2/(a + b*x^3), x], x] /; EqQ[a*B^3 - b*A^3, 0] || !RationalQ[ a/b]] /; FreeQ[{a, b}, x] && PolyQ[P2, x, 2]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.10 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.27
method | result | size |
risch | \(\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{3} e +d \right )}{\sum }\frac {\left (\textit {\_R}^{2} c +\textit {\_R} b \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2}}}{3 e}\) | \(36\) |
default | \(b \left (-\frac {\ln \left (x +\left (\frac {d}{e}\right )^{\frac {1}{3}}\right )}{3 e \left (\frac {d}{e}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {d}{e}\right )^{\frac {1}{3}} x +\left (\frac {d}{e}\right )^{\frac {2}{3}}\right )}{6 e \left (\frac {d}{e}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {d}{e}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 e \left (\frac {d}{e}\right )^{\frac {1}{3}}}\right )+\frac {c \ln \left (x^{3} e +d \right )}{3 e}\) | \(108\) |
Input:
int((c*x^2+b*x)/(e*x^3+d),x,method=_RETURNVERBOSE)
Output:
1/3/e*sum((_R^2*c+_R*b)/_R^2*ln(x-_R),_R=RootOf(_Z^3*e+d))
Result contains complex when optimal does not.
Time = 0.87 (sec) , antiderivative size = 1344, normalized size of antiderivative = 10.11 \[ \int \frac {b x+c x^2}{d+e x^3} \, dx=\text {Too large to display} \] Input:
integrate((c*x^2+b*x)/(e*x^3+d),x, algorithm="fricas")
Output:
-1/12*(2*(3*(I*sqrt(3) + 1)*(-1/54*c^3/e^3 + 1/54*b^3/(d*e^2) + 1/54*(c^3* d - b^3*e)/(d*e^3))^(1/3) - 2*c/e)*e*log(1/4*(3*(I*sqrt(3) + 1)*(-1/54*c^3 /e^3 + 1/54*b^3/(d*e^2) + 1/54*(c^3*d - b^3*e)/(d*e^3))^(1/3) - 2*c/e)^2*d *e^2 + (3*(I*sqrt(3) + 1)*(-1/54*c^3/e^3 + 1/54*b^3/(d*e^2) + 1/54*(c^3*d - b^3*e)/(d*e^3))^(1/3) - 2*c/e)*c*d*e + b^2*e*x + c^2*d) - ((3*(I*sqrt(3) + 1)*(-1/54*c^3/e^3 + 1/54*b^3/(d*e^2) + 1/54*(c^3*d - b^3*e)/(d*e^3))^(1 /3) - 2*c/e)*e + 3*sqrt(1/3)*e*sqrt(-((3*(I*sqrt(3) + 1)*(-1/54*c^3/e^3 + 1/54*b^3/(d*e^2) + 1/54*(c^3*d - b^3*e)/(d*e^3))^(1/3) - 2*c/e)^2*e^2 + 4* (3*(I*sqrt(3) + 1)*(-1/54*c^3/e^3 + 1/54*b^3/(d*e^2) + 1/54*(c^3*d - b^3*e )/(d*e^3))^(1/3) - 2*c/e)*c*e + 4*c^2)/e^2) + 6*c)*log(-1/4*(3*(I*sqrt(3) + 1)*(-1/54*c^3/e^3 + 1/54*b^3/(d*e^2) + 1/54*(c^3*d - b^3*e)/(d*e^3))^(1/ 3) - 2*c/e)^2*d*e^2 - (3*(I*sqrt(3) + 1)*(-1/54*c^3/e^3 + 1/54*b^3/(d*e^2) + 1/54*(c^3*d - b^3*e)/(d*e^3))^(1/3) - 2*c/e)*c*d*e + 2*b^2*e*x - c^2*d + 3/4*sqrt(1/3)*((3*(I*sqrt(3) + 1)*(-1/54*c^3/e^3 + 1/54*b^3/(d*e^2) + 1/ 54*(c^3*d - b^3*e)/(d*e^3))^(1/3) - 2*c/e)*d*e^2 + 2*c*d*e)*sqrt(-((3*(I*s qrt(3) + 1)*(-1/54*c^3/e^3 + 1/54*b^3/(d*e^2) + 1/54*(c^3*d - b^3*e)/(d*e^ 3))^(1/3) - 2*c/e)^2*e^2 + 4*(3*(I*sqrt(3) + 1)*(-1/54*c^3/e^3 + 1/54*b^3/ (d*e^2) + 1/54*(c^3*d - b^3*e)/(d*e^3))^(1/3) - 2*c/e)*c*e + 4*c^2)/e^2)) - ((3*(I*sqrt(3) + 1)*(-1/54*c^3/e^3 + 1/54*b^3/(d*e^2) + 1/54*(c^3*d - b^ 3*e)/(d*e^3))^(1/3) - 2*c/e)*e - 3*sqrt(1/3)*e*sqrt(-((3*(I*sqrt(3) + 1...
Time = 0.22 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.56 \[ \int \frac {b x+c x^2}{d+e x^3} \, dx=\operatorname {RootSum} {\left (27 t^{3} d e^{3} - 27 t^{2} c d e^{2} + 9 t c^{2} d e + b^{3} e - c^{3} d, \left ( t \mapsto t \log {\left (x + \frac {9 t^{2} d e^{2} - 6 t c d e + c^{2} d}{b^{2} e} \right )} \right )\right )} \] Input:
integrate((c*x**2+b*x)/(e*x**3+d),x)
Output:
RootSum(27*_t**3*d*e**3 - 27*_t**2*c*d*e**2 + 9*_t*c**2*d*e + b**3*e - c** 3*d, Lambda(_t, _t*log(x + (9*_t**2*d*e**2 - 6*_t*c*d*e + c**2*d)/(b**2*e) )))
Exception generated. \[ \int \frac {b x+c x^2}{d+e x^3} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((c*x^2+b*x)/(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
Time = 0.13 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.89 \[ \int \frac {b x+c x^2}{d+e x^3} \, dx=\frac {\sqrt {3} b \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {d}{e}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {d}{e}\right )^{\frac {1}{3}}}\right )}{3 \, \left (-d e^{2}\right )^{\frac {1}{3}}} - \frac {b \log \left (x^{2} + x \left (-\frac {d}{e}\right )^{\frac {1}{3}} + \left (-\frac {d}{e}\right )^{\frac {2}{3}}\right )}{6 \, \left (-d e^{2}\right )^{\frac {1}{3}}} - \frac {b \left (-\frac {d}{e}\right )^{\frac {2}{3}} \log \left ({\left | x - \left (-\frac {d}{e}\right )^{\frac {1}{3}} \right |}\right )}{3 \, d} + \frac {c \log \left ({\left | e x^{3} + d \right |}\right )}{3 \, e} \] Input:
integrate((c*x^2+b*x)/(e*x^3+d),x, algorithm="giac")
Output:
1/3*sqrt(3)*b*arctan(1/3*sqrt(3)*(2*x + (-d/e)^(1/3))/(-d/e)^(1/3))/(-d*e^ 2)^(1/3) - 1/6*b*log(x^2 + x*(-d/e)^(1/3) + (-d/e)^(2/3))/(-d*e^2)^(1/3) - 1/3*b*(-d/e)^(2/3)*log(abs(x - (-d/e)^(1/3)))/d + 1/3*c*log(abs(e*x^3 + d ))/e
Time = 5.95 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.19 \[ \int \frac {b x+c x^2}{d+e x^3} \, dx=\sum _{k=1}^3\ln \left (-\mathrm {root}\left (27\,d\,e^3\,z^3-27\,c\,d\,e^2\,z^2+9\,c^2\,d\,e\,z+b^3\,e-c^3\,d,z,k\right )\,\left (6\,c\,d\,e-\mathrm {root}\left (27\,d\,e^3\,z^3-27\,c\,d\,e^2\,z^2+9\,c^2\,d\,e\,z+b^3\,e-c^3\,d,z,k\right )\,d\,e^2\,9\right )+c^2\,d+b^2\,e\,x\right )\,\mathrm {root}\left (27\,d\,e^3\,z^3-27\,c\,d\,e^2\,z^2+9\,c^2\,d\,e\,z+b^3\,e-c^3\,d,z,k\right ) \] Input:
int((b*x + c*x^2)/(d + e*x^3),x)
Output:
symsum(log(c^2*d - root(27*d*e^3*z^3 - 27*c*d*e^2*z^2 + 9*c^2*d*e*z + b^3* e - c^3*d, z, k)*(6*c*d*e - 9*root(27*d*e^3*z^3 - 27*c*d*e^2*z^2 + 9*c^2*d *e*z + b^3*e - c^3*d, z, k)*d*e^2) + b^2*e*x)*root(27*d*e^3*z^3 - 27*c*d*e ^2*z^2 + 9*c^2*d*e*z + b^3*e - c^3*d, z, k), k, 1, 3)
Time = 0.15 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.91 \[ \int \frac {b x+c x^2}{d+e x^3} \, dx=\frac {-2 \sqrt {3}\, \mathit {atan} \left (\frac {d^{\frac {1}{3}}-2 e^{\frac {1}{3}} x}{d^{\frac {1}{3}} \sqrt {3}}\right ) b e +2 e^{\frac {2}{3}} d^{\frac {1}{3}} \mathrm {log}\left (d^{\frac {2}{3}}-e^{\frac {1}{3}} d^{\frac {1}{3}} x +e^{\frac {2}{3}} x^{2}\right ) c +2 e^{\frac {2}{3}} d^{\frac {1}{3}} \mathrm {log}\left (d^{\frac {1}{3}}+e^{\frac {1}{3}} x \right ) c +\mathrm {log}\left (d^{\frac {2}{3}}-e^{\frac {1}{3}} d^{\frac {1}{3}} x +e^{\frac {2}{3}} x^{2}\right ) b e -2 \,\mathrm {log}\left (d^{\frac {1}{3}}+e^{\frac {1}{3}} x \right ) b e}{6 e^{\frac {5}{3}} d^{\frac {1}{3}}} \] Input:
int((c*x^2+b*x)/(e*x^3+d),x)
Output:
( - 2*sqrt(3)*atan((d**(1/3) - 2*e**(1/3)*x)/(d**(1/3)*sqrt(3)))*b*e + 2*e **(2/3)*d**(1/3)*log(d**(2/3) - e**(1/3)*d**(1/3)*x + e**(2/3)*x**2)*c + 2 *e**(2/3)*d**(1/3)*log(d**(1/3) + e**(1/3)*x)*c + log(d**(2/3) - e**(1/3)* d**(1/3)*x + e**(2/3)*x**2)*b*e - 2*log(d**(1/3) + e**(1/3)*x)*b*e)/(6*e** (2/3)*d**(1/3)*e)