Integrand size = 18, antiderivative size = 67 \[ \int \frac {x^3}{a+f x^2+c x^4} \, dx=-\frac {f \arctan \left (\frac {f+2 c x^2}{\sqrt {4 a c-f^2}}\right )}{2 c \sqrt {4 a c-f^2}}+\frac {\log \left (a+f x^2+c x^4\right )}{4 c} \] Output:
-1/2*f*arctan((2*c*x^2+f)/(4*a*c-f^2)^(1/2))/c/(4*a*c-f^2)^(1/2)+1/4*ln(c* x^4+f*x^2+a)/c
Time = 0.02 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.93 \[ \int \frac {x^3}{a+f x^2+c x^4} \, dx=\frac {-\frac {2 f \arctan \left (\frac {f+2 c x^2}{\sqrt {4 a c-f^2}}\right )}{\sqrt {4 a c-f^2}}+\log \left (a+f x^2+c x^4\right )}{4 c} \] Input:
Integrate[x^3/(a + f*x^2 + c*x^4),x]
Output:
((-2*f*ArcTan[(f + 2*c*x^2)/Sqrt[4*a*c - f^2]])/Sqrt[4*a*c - f^2] + Log[a + f*x^2 + c*x^4])/(4*c)
Time = 0.38 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.03, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {1434, 1142, 1083, 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.
\(\displaystyle \int \frac {x^3}{a+c x^4+f x^2} \, dx\) |
\(\Big \downarrow \) 1434 |
\(\displaystyle \frac {1}{2} \int \frac {x^2}{c x^4+f x^2+a}dx^2\) |
\(\Big \downarrow \) 1142 |
\(\displaystyle \frac {1}{2} \left (\frac {\int \frac {2 c x^2+f}{c x^4+f x^2+a}dx^2}{2 c}-\frac {f \int \frac {1}{c x^4+f x^2+a}dx^2}{2 c}\right )\) |
\(\Big \downarrow \) 1083 |
\(\displaystyle \frac {1}{2} \left (\frac {f \int \frac {1}{-x^4+f^2-4 a c}d\left (2 c x^2+f\right )}{c}+\frac {\int \frac {2 c x^2+f}{c x^4+f x^2+a}dx^2}{2 c}\right )\) |
\(\Big \downarrow \) 217 |
\(\displaystyle \frac {1}{2} \left (\frac {\int \frac {2 c x^2+f}{c x^4+f x^2+a}dx^2}{2 c}-\frac {f \arctan \left (\frac {2 c x^2+f}{\sqrt {4 a c-f^2}}\right )}{c \sqrt {4 a c-f^2}}\right )\) |
\(\Big \downarrow \) 1103 |
\(\displaystyle \frac {1}{2} \left (\frac {\log \left (a+c x^4+f x^2\right )}{2 c}-\frac {f \arctan \left (\frac {2 c x^2+f}{\sqrt {4 a c-f^2}}\right )}{c \sqrt {4 a c-f^2}}\right )\) |
Input:
Int[x^3/(a + f*x^2 + c*x^4),x]
Output:
(-((f*ArcTan[(f + 2*c*x^2)/Sqrt[4*a*c - f^2]])/(c*Sqrt[4*a*c - f^2])) + Lo g[a + f*x^2 + c*x^4]/(2*c))/2
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] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{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[(x_)^(m_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Simp [1/2 Subst[Int[x^((m - 1)/2)*(a + b*x + c*x^2)^p, x], x, x^2], x] /; Free Q[{a, b, c, p}, x] && IntegerQ[(m - 1)/2]
Time = 0.10 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.90
method | result | size |
default | \(-\frac {f \arctan \left (\frac {2 c \,x^{2}+f}{\sqrt {4 a c -f^{2}}}\right )}{2 c \sqrt {4 a c -f^{2}}}+\frac {\ln \left (c \,x^{4}+f \,x^{2}+a \right )}{4 c}\) | \(60\) |
risch | \(\frac {\ln \left (\left (-4 a c f +f^{3}+\sqrt {-f^{2} \left (4 a c -f^{2}\right )}\, f \right ) x^{2}+2 \sqrt {-f^{2} \left (4 a c -f^{2}\right )}\, a \right ) a}{4 a c -f^{2}}-\frac {\ln \left (\left (-4 a c f +f^{3}+\sqrt {-f^{2} \left (4 a c -f^{2}\right )}\, f \right ) x^{2}+2 \sqrt {-f^{2} \left (4 a c -f^{2}\right )}\, a \right ) f^{2}}{4 c \left (4 a c -f^{2}\right )}+\frac {\ln \left (\left (-4 a c f +f^{3}+\sqrt {-f^{2} \left (4 a c -f^{2}\right )}\, f \right ) x^{2}+2 \sqrt {-f^{2} \left (4 a c -f^{2}\right )}\, a \right ) \sqrt {-f^{2} \left (4 a c -f^{2}\right )}}{4 c \left (4 a c -f^{2}\right )}+\frac {\ln \left (\left (-4 a c f +f^{3}-\sqrt {-f^{2} \left (4 a c -f^{2}\right )}\, f \right ) x^{2}-2 \sqrt {-f^{2} \left (4 a c -f^{2}\right )}\, a \right ) a}{4 a c -f^{2}}-\frac {\ln \left (\left (-4 a c f +f^{3}-\sqrt {-f^{2} \left (4 a c -f^{2}\right )}\, f \right ) x^{2}-2 \sqrt {-f^{2} \left (4 a c -f^{2}\right )}\, a \right ) f^{2}}{4 c \left (4 a c -f^{2}\right )}-\frac {\ln \left (\left (-4 a c f +f^{3}-\sqrt {-f^{2} \left (4 a c -f^{2}\right )}\, f \right ) x^{2}-2 \sqrt {-f^{2} \left (4 a c -f^{2}\right )}\, a \right ) \sqrt {-f^{2} \left (4 a c -f^{2}\right )}}{4 c \left (4 a c -f^{2}\right )}\) | \(465\) |
Input:
int(x^3/(c*x^4+f*x^2+a),x,method=_RETURNVERBOSE)
Output:
-1/2*f*arctan((2*c*x^2+f)/(4*a*c-f^2)^(1/2))/c/(4*a*c-f^2)^(1/2)+1/4*ln(c* x^4+f*x^2+a)/c
Time = 0.09 (sec) , antiderivative size = 194, normalized size of antiderivative = 2.90 \[ \int \frac {x^3}{a+f x^2+c x^4} \, dx=\left [-\frac {\sqrt {-4 \, a c + f^{2}} f \log \left (\frac {2 \, c^{2} x^{4} + 2 \, c f x^{2} - 2 \, a c + f^{2} + {\left (2 \, c x^{2} + f\right )} \sqrt {-4 \, a c + f^{2}}}{c x^{4} + f x^{2} + a}\right ) - {\left (4 \, a c - f^{2}\right )} \log \left (c x^{4} + f x^{2} + a\right )}{4 \, {\left (4 \, a c^{2} - c f^{2}\right )}}, \frac {2 \, \sqrt {4 \, a c - f^{2}} f \arctan \left (-\frac {2 \, c x^{2} + f}{\sqrt {4 \, a c - f^{2}}}\right ) + {\left (4 \, a c - f^{2}\right )} \log \left (c x^{4} + f x^{2} + a\right )}{4 \, {\left (4 \, a c^{2} - c f^{2}\right )}}\right ] \] Input:
integrate(x^3/(c*x^4+f*x^2+a),x, algorithm="fricas")
Output:
[-1/4*(sqrt(-4*a*c + f^2)*f*log((2*c^2*x^4 + 2*c*f*x^2 - 2*a*c + f^2 + (2* c*x^2 + f)*sqrt(-4*a*c + f^2))/(c*x^4 + f*x^2 + a)) - (4*a*c - f^2)*log(c* x^4 + f*x^2 + a))/(4*a*c^2 - c*f^2), 1/4*(2*sqrt(4*a*c - f^2)*f*arctan(-(2 *c*x^2 + f)/sqrt(4*a*c - f^2)) + (4*a*c - f^2)*log(c*x^4 + f*x^2 + a))/(4* a*c^2 - c*f^2)]
Leaf count of result is larger than twice the leaf count of optimal. 223 vs. \(2 (54) = 108\).
Time = 0.47 (sec) , antiderivative size = 223, normalized size of antiderivative = 3.33 \[ \int \frac {x^3}{a+f x^2+c x^4} \, dx=\left (- \frac {f \sqrt {- 4 a c + f^{2}}}{4 c \left (4 a c - f^{2}\right )} + \frac {1}{4 c}\right ) \log {\left (x^{2} + \frac {- 8 a c \left (- \frac {f \sqrt {- 4 a c + f^{2}}}{4 c \left (4 a c - f^{2}\right )} + \frac {1}{4 c}\right ) + 2 a + 2 f^{2} \left (- \frac {f \sqrt {- 4 a c + f^{2}}}{4 c \left (4 a c - f^{2}\right )} + \frac {1}{4 c}\right )}{f} \right )} + \left (\frac {f \sqrt {- 4 a c + f^{2}}}{4 c \left (4 a c - f^{2}\right )} + \frac {1}{4 c}\right ) \log {\left (x^{2} + \frac {- 8 a c \left (\frac {f \sqrt {- 4 a c + f^{2}}}{4 c \left (4 a c - f^{2}\right )} + \frac {1}{4 c}\right ) + 2 a + 2 f^{2} \left (\frac {f \sqrt {- 4 a c + f^{2}}}{4 c \left (4 a c - f^{2}\right )} + \frac {1}{4 c}\right )}{f} \right )} \] Input:
integrate(x**3/(c*x**4+f*x**2+a),x)
Output:
(-f*sqrt(-4*a*c + f**2)/(4*c*(4*a*c - f**2)) + 1/(4*c))*log(x**2 + (-8*a*c *(-f*sqrt(-4*a*c + f**2)/(4*c*(4*a*c - f**2)) + 1/(4*c)) + 2*a + 2*f**2*(- f*sqrt(-4*a*c + f**2)/(4*c*(4*a*c - f**2)) + 1/(4*c)))/f) + (f*sqrt(-4*a*c + f**2)/(4*c*(4*a*c - f**2)) + 1/(4*c))*log(x**2 + (-8*a*c*(f*sqrt(-4*a*c + f**2)/(4*c*(4*a*c - f**2)) + 1/(4*c)) + 2*a + 2*f**2*(f*sqrt(-4*a*c + f **2)/(4*c*(4*a*c - f**2)) + 1/(4*c)))/f)
Exception generated. \[ \int \frac {x^3}{a+f x^2+c x^4} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(x^3/(c*x^4+f*x^2+a),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(f^2-4*a*c>0)', see `assume?` for more deta
Time = 0.38 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.88 \[ \int \frac {x^3}{a+f x^2+c x^4} \, dx=-\frac {f \arctan \left (\frac {2 \, c x^{2} + f}{\sqrt {4 \, a c - f^{2}}}\right )}{2 \, \sqrt {4 \, a c - f^{2}} c} + \frac {\log \left (c x^{4} + f x^{2} + a\right )}{4 \, c} \] Input:
integrate(x^3/(c*x^4+f*x^2+a),x, algorithm="giac")
Output:
-1/2*f*arctan((2*c*x^2 + f)/sqrt(4*a*c - f^2))/(sqrt(4*a*c - f^2)*c) + 1/4 *log(c*x^4 + f*x^2 + a)/c
Time = 17.63 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.76 \[ \int \frac {x^3}{a+f x^2+c x^4} \, dx=\frac {4\,a\,c\,\ln \left (c\,x^4+f\,x^2+a\right )}{16\,a\,c^2-4\,c\,f^2}-\frac {f^2\,\ln \left (c\,x^4+f\,x^2+a\right )}{16\,a\,c^2-4\,c\,f^2}-\frac {f\,\mathrm {atan}\left (\frac {f}{\sqrt {4\,a\,c-f^2}}+\frac {2\,c\,x^2}{\sqrt {4\,a\,c-f^2}}\right )}{2\,c\,\sqrt {4\,a\,c-f^2}} \] Input:
int(x^3/(a + c*x^4 + f*x^2),x)
Output:
(4*a*c*log(a + c*x^4 + f*x^2))/(16*a*c^2 - 4*c*f^2) - (f^2*log(a + c*x^4 + f*x^2))/(16*a*c^2 - 4*c*f^2) - (f*atan(f/(4*a*c - f^2)^(1/2) + (2*c*x^2)/ (4*a*c - f^2)^(1/2)))/(2*c*(4*a*c - f^2)^(1/2))
Time = 0.19 (sec) , antiderivative size = 236, normalized size of antiderivative = 3.52 \[ \int \frac {x^3}{a+f x^2+c x^4} \, dx=\frac {2 \sqrt {2 \sqrt {c}\, \sqrt {a}+f}\, \sqrt {2 \sqrt {c}\, \sqrt {a}-f}\, \mathit {atan} \left (\frac {\sqrt {2 \sqrt {c}\, \sqrt {a}-f}-2 \sqrt {c}\, x}{\sqrt {2 \sqrt {c}\, \sqrt {a}+f}}\right ) f +2 \sqrt {2 \sqrt {c}\, \sqrt {a}+f}\, \sqrt {2 \sqrt {c}\, \sqrt {a}-f}\, \mathit {atan} \left (\frac {\sqrt {2 \sqrt {c}\, \sqrt {a}-f}+2 \sqrt {c}\, x}{\sqrt {2 \sqrt {c}\, \sqrt {a}+f}}\right ) f +4 \,\mathrm {log}\left (-\sqrt {2 \sqrt {c}\, \sqrt {a}-f}\, x +\sqrt {a}+\sqrt {c}\, x^{2}\right ) a c -\mathrm {log}\left (-\sqrt {2 \sqrt {c}\, \sqrt {a}-f}\, x +\sqrt {a}+\sqrt {c}\, x^{2}\right ) f^{2}+4 \,\mathrm {log}\left (\sqrt {2 \sqrt {c}\, \sqrt {a}-f}\, x +\sqrt {a}+\sqrt {c}\, x^{2}\right ) a c -\mathrm {log}\left (\sqrt {2 \sqrt {c}\, \sqrt {a}-f}\, x +\sqrt {a}+\sqrt {c}\, x^{2}\right ) f^{2}}{4 c \left (4 a c -f^{2}\right )} \] Input:
int(x^3/(c*x^4+f*x^2+a),x)
Output:
(2*sqrt(2*sqrt(c)*sqrt(a) + f)*sqrt(2*sqrt(c)*sqrt(a) - f)*atan((sqrt(2*sq rt(c)*sqrt(a) - f) - 2*sqrt(c)*x)/sqrt(2*sqrt(c)*sqrt(a) + f))*f + 2*sqrt( 2*sqrt(c)*sqrt(a) + f)*sqrt(2*sqrt(c)*sqrt(a) - f)*atan((sqrt(2*sqrt(c)*sq rt(a) - f) + 2*sqrt(c)*x)/sqrt(2*sqrt(c)*sqrt(a) + f))*f + 4*log( - sqrt(2 *sqrt(c)*sqrt(a) - f)*x + sqrt(a) + sqrt(c)*x**2)*a*c - log( - sqrt(2*sqrt (c)*sqrt(a) - f)*x + sqrt(a) + sqrt(c)*x**2)*f**2 + 4*log(sqrt(2*sqrt(c)*s qrt(a) - f)*x + sqrt(a) + sqrt(c)*x**2)*a*c - log(sqrt(2*sqrt(c)*sqrt(a) - f)*x + sqrt(a) + sqrt(c)*x**2)*f**2)/(4*c*(4*a*c - f**2))