Integrand size = 18, antiderivative size = 134 \[ \int \frac {a+c x^2}{d-e x^3} \, dx=\frac {a \arctan \left (\frac {\sqrt [3]{d}+2 \sqrt [3]{e} x}{\sqrt {3} \sqrt [3]{d}}\right )}{\sqrt {3} d^{2/3} \sqrt [3]{e}}-\frac {a \log \left (\sqrt [3]{d}-\sqrt [3]{e} x\right )}{3 d^{2/3} \sqrt [3]{e}}+\frac {a \log \left (d^{2/3}+\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )}{6 d^{2/3} \sqrt [3]{e}}-\frac {c \log \left (d-e x^3\right )}{3 e} \] Output:
1/3*a*arctan(1/3*(d^(1/3)+2*e^(1/3)*x)*3^(1/2)/d^(1/3))*3^(1/2)/d^(2/3)/e^ (1/3)-1/3*a*ln(d^(1/3)-e^(1/3)*x)/d^(2/3)/e^(1/3)+1/6*a*ln(d^(2/3)+d^(1/3) *e^(1/3)*x+e^(2/3)*x^2)/d^(2/3)/e^(1/3)-1/3*c*ln(-e*x^3+d)/e
Time = 0.02 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.92 \[ \int \frac {a+c x^2}{d-e x^3} \, dx=\frac {2 \sqrt {3} a e^{2/3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{e} x}{\sqrt [3]{d}}}{\sqrt {3}}\right )-2 a e^{2/3} \log \left (\sqrt [3]{d}-\sqrt [3]{e} x\right )+a e^{2/3} \log \left (d^{2/3}+\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )-2 c d^{2/3} \log \left (d-e x^3\right )}{6 d^{2/3} e} \] Input:
Integrate[(a + c*x^2)/(d - e*x^3),x]
Output:
(2*Sqrt[3]*a*e^(2/3)*ArcTan[(1 + (2*e^(1/3)*x)/d^(1/3))/Sqrt[3]] - 2*a*e^( 2/3)*Log[d^(1/3) - e^(1/3)*x] + a*e^(2/3)*Log[d^(2/3) + d^(1/3)*e^(1/3)*x + e^(2/3)*x^2] - 2*c*d^(2/3)*Log[d - e*x^3])/(6*d^(2/3)*e)
Time = 0.54 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.97, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {2410, 27, 750, 16, 792, 1142, 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.
\(\displaystyle \int \frac {a+c x^2}{d-e x^3} \, dx\) |
\(\Big \downarrow \) 2410 |
\(\displaystyle \int \frac {a}{d-e x^3}dx+c \int \frac {x^2}{d-e x^3}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle a \int \frac {1}{d-e x^3}dx+c \int \frac {x^2}{d-e x^3}dx\) |
\(\Big \downarrow \) 750 |
\(\displaystyle a \left (\frac {\int \frac {\sqrt [3]{e} x+2 \sqrt [3]{d}}{e^{2/3} x^2+\sqrt [3]{d} \sqrt [3]{e} x+d^{2/3}}dx}{3 d^{2/3}}+\frac {\int \frac {1}{\sqrt [3]{d}-\sqrt [3]{e} x}dx}{3 d^{2/3}}\right )+c \int \frac {x^2}{d-e x^3}dx\) |
\(\Big \downarrow \) 16 |
\(\displaystyle a \left (\frac {\int \frac {\sqrt [3]{e} x+2 \sqrt [3]{d}}{e^{2/3} x^2+\sqrt [3]{d} \sqrt [3]{e} x+d^{2/3}}dx}{3 d^{2/3}}-\frac {\log \left (\sqrt [3]{d}-\sqrt [3]{e} x\right )}{3 d^{2/3} \sqrt [3]{e}}\right )+c \int \frac {x^2}{d-e x^3}dx\) |
\(\Big \downarrow \) 792 |
\(\displaystyle a \left (\frac {\int \frac {\sqrt [3]{e} x+2 \sqrt [3]{d}}{e^{2/3} x^2+\sqrt [3]{d} \sqrt [3]{e} x+d^{2/3}}dx}{3 d^{2/3}}-\frac {\log \left (\sqrt [3]{d}-\sqrt [3]{e} x\right )}{3 d^{2/3} \sqrt [3]{e}}\right )-\frac {c \log \left (d-e x^3\right )}{3 e}\) |
\(\Big \downarrow \) 1142 |
\(\displaystyle a \left (\frac {\frac {3}{2} \sqrt [3]{d} \int \frac {1}{e^{2/3} x^2+\sqrt [3]{d} \sqrt [3]{e} x+d^{2/3}}dx+\frac {\int \frac {\sqrt [3]{e} \left (2 \sqrt [3]{e} x+\sqrt [3]{d}\right )}{e^{2/3} x^2+\sqrt [3]{d} \sqrt [3]{e} x+d^{2/3}}dx}{2 \sqrt [3]{e}}}{3 d^{2/3}}-\frac {\log \left (\sqrt [3]{d}-\sqrt [3]{e} x\right )}{3 d^{2/3} \sqrt [3]{e}}\right )-\frac {c \log \left (d-e x^3\right )}{3 e}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle a \left (\frac {\frac {3}{2} \sqrt [3]{d} \int \frac {1}{e^{2/3} x^2+\sqrt [3]{d} \sqrt [3]{e} x+d^{2/3}}dx+\frac {1}{2} \int \frac {2 \sqrt [3]{e} x+\sqrt [3]{d}}{e^{2/3} x^2+\sqrt [3]{d} \sqrt [3]{e} x+d^{2/3}}dx}{3 d^{2/3}}-\frac {\log \left (\sqrt [3]{d}-\sqrt [3]{e} x\right )}{3 d^{2/3} \sqrt [3]{e}}\right )-\frac {c \log \left (d-e x^3\right )}{3 e}\) |
\(\Big \downarrow \) 1082 |
\(\displaystyle a \left (\frac {\frac {1}{2} \int \frac {2 \sqrt [3]{e} x+\sqrt [3]{d}}{e^{2/3} x^2+\sqrt [3]{d} \sqrt [3]{e} x+d^{2/3}}dx-\frac {3 \int \frac {1}{-\left (\frac {2 \sqrt [3]{e} x}{\sqrt [3]{d}}+1\right )^2-3}d\left (\frac {2 \sqrt [3]{e} x}{\sqrt [3]{d}}+1\right )}{\sqrt [3]{e}}}{3 d^{2/3}}-\frac {\log \left (\sqrt [3]{d}-\sqrt [3]{e} x\right )}{3 d^{2/3} \sqrt [3]{e}}\right )-\frac {c \log \left (d-e x^3\right )}{3 e}\) |
\(\Big \downarrow \) 217 |
\(\displaystyle a \left (\frac {\frac {1}{2} \int \frac {2 \sqrt [3]{e} x+\sqrt [3]{d}}{e^{2/3} x^2+\sqrt [3]{d} \sqrt [3]{e} x+d^{2/3}}dx+\frac {\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [3]{e} x}{\sqrt [3]{d}}+1}{\sqrt {3}}\right )}{\sqrt [3]{e}}}{3 d^{2/3}}-\frac {\log \left (\sqrt [3]{d}-\sqrt [3]{e} x\right )}{3 d^{2/3} \sqrt [3]{e}}\right )-\frac {c \log \left (d-e x^3\right )}{3 e}\) |
\(\Big \downarrow \) 1103 |
\(\displaystyle a \left (\frac {\frac {\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [3]{e} x}{\sqrt [3]{d}}+1}{\sqrt {3}}\right )}{\sqrt [3]{e}}+\frac {\log \left (d^{2/3}+\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )}{2 \sqrt [3]{e}}}{3 d^{2/3}}-\frac {\log \left (\sqrt [3]{d}-\sqrt [3]{e} x\right )}{3 d^{2/3} \sqrt [3]{e}}\right )-\frac {c \log \left (d-e x^3\right )}{3 e}\) |
Input:
Int[(a + c*x^2)/(d - e*x^3),x]
Output:
a*(-1/3*Log[d^(1/3) - e^(1/3)*x]/(d^(2/3)*e^(1/3)) + ((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^(2/3))) - (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[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Simp[1/(3*Rt[a, 3]^2) Int[1/ (Rt[a, 3] + Rt[b, 3]*x), x], x] + Simp[1/(3*Rt[a, 3]^2) Int[(2*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[(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[((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[(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 | \(-\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{3} e -d \right )}{\sum }\frac {\left (\textit {\_R}^{2} c +a \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2}}}{3 e}\) | \(36\) |
default | \(a \left (-\frac {\ln \left (x -\left (\frac {d}{e}\right )^{\frac {1}{3}}\right )}{3 e \left (\frac {d}{e}\right )^{\frac {2}{3}}}+\frac {\ln \left (x^{2}+\left (\frac {d}{e}\right )^{\frac {1}{3}} x +\left (\frac {d}{e}\right )^{\frac {2}{3}}\right )}{6 e \left (\frac {d}{e}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {d}{e}\right )^{\frac {1}{3}}}+1\right )}{3}\right )}{3 e \left (\frac {d}{e}\right )^{\frac {2}{3}}}\right )-\frac {c \ln \left (-x^{3} e +d \right )}{3 e}\) | \(110\) |
Input:
int((c*x^2+a)/(-e*x^3+d),x,method=_RETURNVERBOSE)
Output:
-1/3/e*sum((_R^2*c+a)/_R^2*ln(x-_R),_R=RootOf(_Z^3*e-d))
Result contains complex when optimal does not.
Time = 0.87 (sec) , antiderivative size = 1040, normalized size of antiderivative = 7.76 \[ \int \frac {a+c x^2}{d-e x^3} \, dx=\text {Too large to display} \] Input:
integrate((c*x^2+a)/(-e*x^3+d),x, algorithm="fricas")
Output:
-1/12*(2*((1/2)^(1/3)*(I*sqrt(3) + 1)*(c^3/e^3 + a^3/(d^2*e) - (c^3*d^2 + a^3*e^2)/(d^2*e^3))^(1/3) + 2*c/e)*e*log(-1/2*((1/2)^(1/3)*(I*sqrt(3) + 1) *(c^3/e^3 + a^3/(d^2*e) - (c^3*d^2 + a^3*e^2)/(d^2*e^3))^(1/3) + 2*c/e)*d* e + a*e*x + c*d) - (((1/2)^(1/3)*(I*sqrt(3) + 1)*(c^3/e^3 + a^3/(d^2*e) - (c^3*d^2 + a^3*e^2)/(d^2*e^3))^(1/3) + 2*c/e)*e + 3*sqrt(1/3)*e*sqrt(-(((1 /2)^(1/3)*(I*sqrt(3) + 1)*(c^3/e^3 + a^3/(d^2*e) - (c^3*d^2 + a^3*e^2)/(d^ 2*e^3))^(1/3) + 2*c/e)^2*e^2 - 4*((1/2)^(1/3)*(I*sqrt(3) + 1)*(c^3/e^3 + a ^3/(d^2*e) - (c^3*d^2 + a^3*e^2)/(d^2*e^3))^(1/3) + 2*c/e)*c*e + 4*c^2)/e^ 2) - 6*c)*log(1/2*((1/2)^(1/3)*(I*sqrt(3) + 1)*(c^3/e^3 + a^3/(d^2*e) - (c ^3*d^2 + a^3*e^2)/(d^2*e^3))^(1/3) + 2*c/e)*d*e + 2*a*e*x + 3/2*sqrt(1/3)* d*e*sqrt(-(((1/2)^(1/3)*(I*sqrt(3) + 1)*(c^3/e^3 + a^3/(d^2*e) - (c^3*d^2 + a^3*e^2)/(d^2*e^3))^(1/3) + 2*c/e)^2*e^2 - 4*((1/2)^(1/3)*(I*sqrt(3) + 1 )*(c^3/e^3 + a^3/(d^2*e) - (c^3*d^2 + a^3*e^2)/(d^2*e^3))^(1/3) + 2*c/e)*c *e + 4*c^2)/e^2) - c*d) - (((1/2)^(1/3)*(I*sqrt(3) + 1)*(c^3/e^3 + a^3/(d^ 2*e) - (c^3*d^2 + a^3*e^2)/(d^2*e^3))^(1/3) + 2*c/e)*e - 3*sqrt(1/3)*e*sqr t(-(((1/2)^(1/3)*(I*sqrt(3) + 1)*(c^3/e^3 + a^3/(d^2*e) - (c^3*d^2 + a^3*e ^2)/(d^2*e^3))^(1/3) + 2*c/e)^2*e^2 - 4*((1/2)^(1/3)*(I*sqrt(3) + 1)*(c^3/ e^3 + a^3/(d^2*e) - (c^3*d^2 + a^3*e^2)/(d^2*e^3))^(1/3) + 2*c/e)*c*e + 4* c^2)/e^2) - 6*c)*log(1/2*((1/2)^(1/3)*(I*sqrt(3) + 1)*(c^3/e^3 + a^3/(d^2* e) - (c^3*d^2 + a^3*e^2)/(d^2*e^3))^(1/3) + 2*c/e)*d*e + 2*a*e*x - 3/2*...
Time = 0.29 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.52 \[ \int \frac {a+c x^2}{d-e x^3} \, dx=- \operatorname {RootSum} {\left (27 t^{3} d^{2} e^{3} - 27 t^{2} c d^{2} e^{2} + 9 t c^{2} d^{2} e - a^{3} e^{2} - c^{3} d^{2}, \left ( t \mapsto t \log {\left (x + \frac {- 3 t d e + c d}{a e} \right )} \right )\right )} \] Input:
integrate((c*x**2+a)/(-e*x**3+d),x)
Output:
-RootSum(27*_t**3*d**2*e**3 - 27*_t**2*c*d**2*e**2 + 9*_t*c**2*d**2*e - a* *3*e**2 - c**3*d**2, Lambda(_t, _t*log(x + (-3*_t*d*e + c*d)/(a*e))))
Exception generated. \[ \int \frac {a+c x^2}{d-e x^3} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((c*x^2+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
Time = 0.13 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.93 \[ \int \frac {a+c x^2}{d-e x^3} \, dx=-\frac {a \left (\frac {d}{e}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (\frac {d}{e}\right )^{\frac {1}{3}} \right |}\right )}{3 \, d} - \frac {c \log \left ({\left | e x^{3} - d \right |}\right )}{3 \, e} + \frac {\sqrt {3} \left (d e^{2}\right )^{\frac {1}{3}} a \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (\frac {d}{e}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {d}{e}\right )^{\frac {1}{3}}}\right )}{3 \, d e} + \frac {\left (d e^{2}\right )^{\frac {1}{3}} a \log \left (x^{2} + x \left (\frac {d}{e}\right )^{\frac {1}{3}} + \left (\frac {d}{e}\right )^{\frac {2}{3}}\right )}{6 \, d e} \] Input:
integrate((c*x^2+a)/(-e*x^3+d),x, algorithm="giac")
Output:
-1/3*a*(d/e)^(1/3)*log(abs(x - (d/e)^(1/3)))/d - 1/3*c*log(abs(e*x^3 - d)) /e + 1/3*sqrt(3)*(d*e^2)^(1/3)*a*arctan(1/3*sqrt(3)*(2*x + (d/e)^(1/3))/(d /e)^(1/3))/(d*e) + 1/6*(d*e^2)^(1/3)*a*log(x^2 + x*(d/e)^(1/3) + (d/e)^(2/ 3))/(d*e)
Time = 0.27 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.33 \[ \int \frac {a+c x^2}{d-e x^3} \, dx=\sum _{k=1}^3\ln \left (-\left (c+\mathrm {root}\left (27\,d^2\,e^3\,z^3+27\,c\,d^2\,e^2\,z^2+9\,c^2\,d^2\,e\,z+c^3\,d^2+a^3\,e^2,z,k\right )\,e\,3\right )\,\left (c\,d+\mathrm {root}\left (27\,d^2\,e^3\,z^3+27\,c\,d^2\,e^2\,z^2+9\,c^2\,d^2\,e\,z+c^3\,d^2+a^3\,e^2,z,k\right )\,d\,e\,3+a\,e\,x\right )\right )\,\mathrm {root}\left (27\,d^2\,e^3\,z^3+27\,c\,d^2\,e^2\,z^2+9\,c^2\,d^2\,e\,z+c^3\,d^2+a^3\,e^2,z,k\right ) \] Input:
int((a + c*x^2)/(d - e*x^3),x)
Output:
symsum(log(-(c + 3*root(27*d^2*e^3*z^3 + 27*c*d^2*e^2*z^2 + 9*c^2*d^2*e*z + c^3*d^2 + a^3*e^2, z, k)*e)*(c*d + 3*root(27*d^2*e^3*z^3 + 27*c*d^2*e^2* z^2 + 9*c^2*d^2*e*z + c^3*d^2 + a^3*e^2, z, k)*d*e + a*e*x))*root(27*d^2*e ^3*z^3 + 27*c*d^2*e^2*z^2 + 9*c^2*d^2*e*z + c^3*d^2 + a^3*e^2, z, k), k, 1 , 3)
Time = 0.15 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.94 \[ \int \frac {a+c x^2}{d-e x^3} \, dx=\frac {2 d^{\frac {1}{3}} \sqrt {3}\, \mathit {atan} \left (\frac {d^{\frac {1}{3}}+2 e^{\frac {1}{3}} x}{d^{\frac {1}{3}} \sqrt {3}}\right ) a e +d^{\frac {1}{3}} \mathrm {log}\left (d^{\frac {2}{3}}+e^{\frac {1}{3}} d^{\frac {1}{3}} x +e^{\frac {2}{3}} x^{2}\right ) a e -2 d^{\frac {1}{3}} \mathrm {log}\left (d^{\frac {1}{3}}-e^{\frac {1}{3}} x \right ) a e -2 e^{\frac {1}{3}} \mathrm {log}\left (d^{\frac {2}{3}}+e^{\frac {1}{3}} d^{\frac {1}{3}} x +e^{\frac {2}{3}} x^{2}\right ) c d -2 e^{\frac {1}{3}} \mathrm {log}\left (d^{\frac {1}{3}}-e^{\frac {1}{3}} x \right ) c d}{6 e^{\frac {4}{3}} d} \] Input:
int((c*x^2+a)/(-e*x^3+d),x)
Output:
(2*d**(1/3)*sqrt(3)*atan((d**(1/3) + 2*e**(1/3)*x)/(d**(1/3)*sqrt(3)))*a*e + d**(1/3)*log(d**(2/3) + e**(1/3)*d**(1/3)*x + e**(2/3)*x**2)*a*e - 2*d* *(1/3)*log(d**(1/3) - e**(1/3)*x)*a*e - 2*e**(1/3)*log(d**(2/3) + e**(1/3) *d**(1/3)*x + e**(2/3)*x**2)*c*d - 2*e**(1/3)*log(d**(1/3) - e**(1/3)*x)*c *d)/(6*e**(1/3)*d*e)