Integrand size = 22, antiderivative size = 69 \[ \int \frac {x^5}{a-b+2 a x^2+a x^4} \, dx=\frac {x^2}{2 a}-\frac {(a+b) \text {arctanh}\left (\frac {\sqrt {a} \left (1+x^2\right )}{\sqrt {b}}\right )}{2 a^{3/2} \sqrt {b}}-\frac {\log \left (a-b+2 a x^2+a x^4\right )}{2 a} \] Output:
1/2*x^2/a-1/2*(a+b)*arctanh(a^(1/2)*(x^2+1)/b^(1/2))/a^(3/2)/b^(1/2)-1/2*l n(a*x^4+2*a*x^2+a-b)/a
Time = 0.03 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.90 \[ \int \frac {x^5}{a-b+2 a x^2+a x^4} \, dx=-\frac {(a+b) \text {arctanh}\left (\frac {\sqrt {a} \left (1+x^2\right )}{\sqrt {b}}\right )}{2 a^{3/2} \sqrt {b}}+\frac {x^2-\log \left (-b+a \left (1+x^2\right )^2\right )}{2 a} \] Input:
Integrate[x^5/(a - b + 2*a*x^2 + a*x^4),x]
Output:
-1/2*((a + b)*ArcTanh[(Sqrt[a]*(1 + x^2))/Sqrt[b]])/(a^(3/2)*Sqrt[b]) + (x ^2 - Log[-b + a*(1 + x^2)^2])/(2*a)
Time = 0.41 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.96, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {1434, 1143, 2009}
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.
\(\displaystyle \int \frac {x^5}{a x^4+2 a x^2+a-b} \, dx\) |
\(\Big \downarrow \) 1434 |
\(\displaystyle \frac {1}{2} \int \frac {x^4}{a x^4+2 a x^2+a-b}dx^2\) |
\(\Big \downarrow \) 1143 |
\(\displaystyle \frac {1}{2} \int \left (\frac {1}{a}-\frac {2 a x^2+a-b}{a \left (a x^4+2 a x^2+a-b\right )}\right )dx^2\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} \left (-\frac {(a+b) \text {arctanh}\left (\frac {\sqrt {a} \left (x^2+1\right )}{\sqrt {b}}\right )}{a^{3/2} \sqrt {b}}-\frac {\log \left (a x^4+2 a x^2+a-b\right )}{a}+\frac {x^2}{a}\right )\) |
Input:
Int[x^5/(a - b + 2*a*x^2 + a*x^4),x]
Output:
(x^2/a - ((a + b)*ArcTanh[(Sqrt[a]*(1 + x^2))/Sqrt[b]])/(a^(3/2)*Sqrt[b]) - Log[a - b + 2*a*x^2 + a*x^4]/a)/2
Int[((d_.) + (e_.)*(x_))^(m_)/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[m, 1]
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 = 63, normalized size of antiderivative = 0.91
method | result | size |
default | \(\frac {x^{2}}{2 a}+\frac {-\ln \left (a \,x^{4}+2 a \,x^{2}+a -b \right )-\frac {\left (a +b \right ) \operatorname {arctanh}\left (\frac {2 a \,x^{2}+2 a}{2 \sqrt {a b}}\right )}{\sqrt {a b}}}{2 a}\) | \(63\) |
risch | \(\frac {x^{2}}{2 a}-\frac {\ln \left (\left (a^{2} b +b^{2} a +\sqrt {a b \left (a +b \right )^{2}}\, a \right ) x^{2}+\sqrt {a b \left (a +b \right )^{2}}\, a -\sqrt {a b \left (a +b \right )^{2}}\, b \right )}{2 a}+\frac {\ln \left (\left (a^{2} b +b^{2} a +\sqrt {a b \left (a +b \right )^{2}}\, a \right ) x^{2}+\sqrt {a b \left (a +b \right )^{2}}\, a -\sqrt {a b \left (a +b \right )^{2}}\, b \right ) \sqrt {a b \left (a +b \right )^{2}}}{4 a^{2} b}-\frac {\ln \left (\left (a^{2} b +b^{2} a -\sqrt {a b \left (a +b \right )^{2}}\, a \right ) x^{2}-\sqrt {a b \left (a +b \right )^{2}}\, a +\sqrt {a b \left (a +b \right )^{2}}\, b \right )}{2 a}-\frac {\ln \left (\left (a^{2} b +b^{2} a -\sqrt {a b \left (a +b \right )^{2}}\, a \right ) x^{2}-\sqrt {a b \left (a +b \right )^{2}}\, a +\sqrt {a b \left (a +b \right )^{2}}\, b \right ) \sqrt {a b \left (a +b \right )^{2}}}{4 a^{2} b}\) | \(274\) |
Input:
int(x^5/(a*x^4+2*a*x^2+a-b),x,method=_RETURNVERBOSE)
Output:
1/2*x^2/a+1/2/a*(-ln(a*x^4+2*a*x^2+a-b)-(a+b)/(a*b)^(1/2)*arctanh(1/2*(2*a *x^2+2*a)/(a*b)^(1/2)))
Time = 0.10 (sec) , antiderivative size = 156, normalized size of antiderivative = 2.26 \[ \int \frac {x^5}{a-b+2 a x^2+a x^4} \, dx=\left [\frac {2 \, a b x^{2} - 2 \, a b \log \left (a x^{4} + 2 \, a x^{2} + a - b\right ) + \sqrt {a b} {\left (a + b\right )} \log \left (\frac {a x^{4} + 2 \, a x^{2} - 2 \, \sqrt {a b} {\left (x^{2} + 1\right )} + a + b}{a x^{4} + 2 \, a x^{2} + a - b}\right )}{4 \, a^{2} b}, \frac {a b x^{2} - a b \log \left (a x^{4} + 2 \, a x^{2} + a - b\right ) + \sqrt {-a b} {\left (a + b\right )} \arctan \left (\frac {\sqrt {-a b}}{a x^{2} + a}\right )}{2 \, a^{2} b}\right ] \] Input:
integrate(x^5/(a*x^4+2*a*x^2+a-b),x, algorithm="fricas")
Output:
[1/4*(2*a*b*x^2 - 2*a*b*log(a*x^4 + 2*a*x^2 + a - b) + sqrt(a*b)*(a + b)*l og((a*x^4 + 2*a*x^2 - 2*sqrt(a*b)*(x^2 + 1) + a + b)/(a*x^4 + 2*a*x^2 + a - b)))/(a^2*b), 1/2*(a*b*x^2 - a*b*log(a*x^4 + 2*a*x^2 + a - b) + sqrt(-a* b)*(a + b)*arctan(sqrt(-a*b)/(a*x^2 + a)))/(a^2*b)]
Leaf count of result is larger than twice the leaf count of optimal. 138 vs. \(2 (60) = 120\).
Time = 0.63 (sec) , antiderivative size = 138, normalized size of antiderivative = 2.00 \[ \int \frac {x^5}{a-b+2 a x^2+a x^4} \, dx=\left (- \frac {1}{2 a} - \frac {\sqrt {a^{3} b} \left (a + b\right )}{4 a^{3} b}\right ) \log {\left (x^{2} + \frac {- 4 a b \left (- \frac {1}{2 a} - \frac {\sqrt {a^{3} b} \left (a + b\right )}{4 a^{3} b}\right ) + a - b}{a + b} \right )} + \left (- \frac {1}{2 a} + \frac {\sqrt {a^{3} b} \left (a + b\right )}{4 a^{3} b}\right ) \log {\left (x^{2} + \frac {- 4 a b \left (- \frac {1}{2 a} + \frac {\sqrt {a^{3} b} \left (a + b\right )}{4 a^{3} b}\right ) + a - b}{a + b} \right )} + \frac {x^{2}}{2 a} \] Input:
integrate(x**5/(a*x**4+2*a*x**2+a-b),x)
Output:
(-1/(2*a) - sqrt(a**3*b)*(a + b)/(4*a**3*b))*log(x**2 + (-4*a*b*(-1/(2*a) - sqrt(a**3*b)*(a + b)/(4*a**3*b)) + a - b)/(a + b)) + (-1/(2*a) + sqrt(a* *3*b)*(a + b)/(4*a**3*b))*log(x**2 + (-4*a*b*(-1/(2*a) + sqrt(a**3*b)*(a + b)/(4*a**3*b)) + a - b)/(a + b)) + x**2/(2*a)
Time = 0.11 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.07 \[ \int \frac {x^5}{a-b+2 a x^2+a x^4} \, dx=\frac {x^{2}}{2 \, a} + \frac {{\left (a + b\right )} \log \left (\frac {a x^{2} + a - \sqrt {a b}}{a x^{2} + a + \sqrt {a b}}\right )}{4 \, \sqrt {a b} a} - \frac {\log \left (a x^{4} + 2 \, a x^{2} + a - b\right )}{2 \, a} \] Input:
integrate(x^5/(a*x^4+2*a*x^2+a-b),x, algorithm="maxima")
Output:
1/2*x^2/a + 1/4*(a + b)*log((a*x^2 + a - sqrt(a*b))/(a*x^2 + a + sqrt(a*b) ))/(sqrt(a*b)*a) - 1/2*log(a*x^4 + 2*a*x^2 + a - b)/a
Time = 0.14 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.87 \[ \int \frac {x^5}{a-b+2 a x^2+a x^4} \, dx=\frac {x^{2}}{2 \, a} + \frac {{\left (a + b\right )} \arctan \left (\frac {a x^{2} + a}{\sqrt {-a b}}\right )}{2 \, \sqrt {-a b} a} - \frac {\log \left (a x^{4} + 2 \, a x^{2} + a - b\right )}{2 \, a} \] Input:
integrate(x^5/(a*x^4+2*a*x^2+a-b),x, algorithm="giac")
Output:
1/2*x^2/a + 1/2*(a + b)*arctan((a*x^2 + a)/sqrt(-a*b))/(sqrt(-a*b)*a) - 1/ 2*log(a*x^4 + 2*a*x^2 + a - b)/a
Time = 0.42 (sec) , antiderivative size = 166, normalized size of antiderivative = 2.41 \[ \int \frac {x^5}{a-b+2 a x^2+a x^4} \, dx=\frac {x^2}{2\,a}-\ln \left (a\,\sqrt {a^3\,b}-b\,\sqrt {a^3\,b}-a^2\,b\,x^2+a\,x^2\,\sqrt {a^3\,b}\right )\,\left (\frac {\frac {a^2}{2}+\frac {\sqrt {a^3\,b}}{4}}{a^3}+\frac {\sqrt {a^3\,b}}{4\,a^2\,b}\right )-\ln \left (a\,\sqrt {a^3\,b}-b\,\sqrt {a^3\,b}+a^2\,b\,x^2+a\,x^2\,\sqrt {a^3\,b}\right )\,\left (\frac {\frac {a^2}{2}-\frac {\sqrt {a^3\,b}}{4}}{a^3}-\frac {\sqrt {a^3\,b}}{4\,a^2\,b}\right ) \] Input:
int(x^5/(a - b + 2*a*x^2 + a*x^4),x)
Output:
x^2/(2*a) - log(a*(a^3*b)^(1/2) - b*(a^3*b)^(1/2) - a^2*b*x^2 + a*x^2*(a^3 *b)^(1/2))*((a^2/2 + (a^3*b)^(1/2)/4)/a^3 + (a^3*b)^(1/2)/(4*a^2*b)) - log (a*(a^3*b)^(1/2) - b*(a^3*b)^(1/2) + a^2*b*x^2 + a*x^2*(a^3*b)^(1/2))*((a^ 2/2 - (a^3*b)^(1/2)/4)/a^3 - (a^3*b)^(1/2)/(4*a^2*b))
Time = 0.15 (sec) , antiderivative size = 207, normalized size of antiderivative = 3.00 \[ \int \frac {x^5}{a-b+2 a x^2+a x^4} \, dx=\frac {\sqrt {b}\, \sqrt {a}\, \mathrm {log}\left (-\sqrt {\sqrt {b}\, \sqrt {a}-a}+\sqrt {a}\, x \right ) a +\sqrt {b}\, \sqrt {a}\, \mathrm {log}\left (-\sqrt {\sqrt {b}\, \sqrt {a}-a}+\sqrt {a}\, x \right ) b +\sqrt {b}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {\sqrt {b}\, \sqrt {a}-a}+\sqrt {a}\, x \right ) a +\sqrt {b}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {\sqrt {b}\, \sqrt {a}-a}+\sqrt {a}\, x \right ) b -\sqrt {b}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {b}\, \sqrt {a}+a \,x^{2}+a \right ) a -\sqrt {b}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {b}\, \sqrt {a}+a \,x^{2}+a \right ) b -2 \,\mathrm {log}\left (-\sqrt {\sqrt {b}\, \sqrt {a}-a}+\sqrt {a}\, x \right ) a b -2 \,\mathrm {log}\left (\sqrt {\sqrt {b}\, \sqrt {a}-a}+\sqrt {a}\, x \right ) a b -2 \,\mathrm {log}\left (\sqrt {b}\, \sqrt {a}+a \,x^{2}+a \right ) a b +2 a b \,x^{2}}{4 a^{2} b} \] Input:
int(x^5/(a*x^4+2*a*x^2+a-b),x)
Output:
(sqrt(b)*sqrt(a)*log( - sqrt(sqrt(b)*sqrt(a) - a) + sqrt(a)*x)*a + sqrt(b) *sqrt(a)*log( - sqrt(sqrt(b)*sqrt(a) - a) + sqrt(a)*x)*b + sqrt(b)*sqrt(a) *log(sqrt(sqrt(b)*sqrt(a) - a) + sqrt(a)*x)*a + sqrt(b)*sqrt(a)*log(sqrt(s qrt(b)*sqrt(a) - a) + sqrt(a)*x)*b - sqrt(b)*sqrt(a)*log(sqrt(b)*sqrt(a) + a*x**2 + a)*a - sqrt(b)*sqrt(a)*log(sqrt(b)*sqrt(a) + a*x**2 + a)*b - 2*l og( - sqrt(sqrt(b)*sqrt(a) - a) + sqrt(a)*x)*a*b - 2*log(sqrt(sqrt(b)*sqrt (a) - a) + sqrt(a)*x)*a*b - 2*log(sqrt(b)*sqrt(a) + a*x**2 + a)*a*b + 2*a* b*x**2)/(4*a**2*b)