Integrand size = 20, antiderivative size = 69 \[ \int \frac {x^5}{a+b+2 a x^2+a x^4} \, dx=\frac {x^2}{2 a}+\frac {(a-b) \arctan \left (\frac {\sqrt {a} \left (1+x^2\right )}{\sqrt {b}}\right )}{2 a^{3/2} \sqrt {b}}-\frac {\log \left (a+b+2 a x^2+a x^4\right )}{2 a} \] Output:
1/2*x^2/a+1/2*(a-b)*arctan(a^(1/2)*(x^2+1)/b^(1/2))/a^(3/2)/b^(1/2)-1/2*ln (a*x^4+2*a*x^2+a+b)/a
Time = 0.02 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.90 \[ \int \frac {x^5}{a+b+2 a x^2+a x^4} \, dx=\frac {\frac {(a-b) \arctan \left (\frac {\sqrt {a} \left (1+x^2\right )}{\sqrt {b}}\right )}{\sqrt {b}}+\sqrt {a} \left (x^2-\log \left (b+a \left (1+x^2\right )^2\right )\right )}{2 a^{3/2}} \] Input:
Integrate[x^5/(a + b + 2*a*x^2 + a*x^4),x]
Output:
(((a - b)*ArcTan[(Sqrt[a]*(1 + x^2))/Sqrt[b]])/Sqrt[b] + Sqrt[a]*(x^2 - Lo g[b + a*(1 + x^2)^2]))/(2*a^(3/2))
Time = 0.40 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.94, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, 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 |
\(\displaystyle \frac {1}{2} \left (\frac {(a-b) \arctan \left (\frac {\sqrt {a} \left (x^2+1\right )}{\sqrt {b}}\right )}{a^{3/2} \sqrt {b}}-\frac {\log \left (a x^4+2 a x^2+a+b\right )}{a}+\frac {x^2}{a}\right )\) |
Input:
Int[x^5/(a + b + 2*a*x^2 + a*x^4),x]
Output:
(x^2/a + ((a - b)*ArcTan[(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 = 62, normalized size of antiderivative = 0.90
method | result | size |
default | \(\frac {x^{2}}{2 a}+\frac {-\ln \left (a \,x^{4}+2 a \,x^{2}+a +b \right )+\frac {\left (a -b \right ) \arctan \left (\frac {2 a \,x^{2}+2 a}{2 \sqrt {a b}}\right )}{\sqrt {a b}}}{2 a}\) | \(62\) |
risch | \(\frac {x^{2}}{2 a}-\frac {\ln \left (\left (a^{2} b -b^{2} a +\sqrt {-a b \left (a -b \right )^{2}}\, a \right ) x^{2}+\sqrt {-a b \left (a -b \right )^{2}}\, a +\sqrt {-a b \left (a -b \right )^{2}}\, b \right )}{2 a}+\frac {\ln \left (\left (a^{2} b -b^{2} a +\sqrt {-a b \left (a -b \right )^{2}}\, a \right ) x^{2}+\sqrt {-a b \left (a -b \right )^{2}}\, a +\sqrt {-a b \left (a -b \right )^{2}}\, b \right ) \sqrt {-a b \left (a -b \right )^{2}}}{4 a^{2} b}-\frac {\ln \left (\left (a^{2} b -b^{2} a -\sqrt {-a b \left (a -b \right )^{2}}\, a \right ) x^{2}-\sqrt {-a b \left (a -b \right )^{2}}\, a -\sqrt {-a b \left (a -b \right )^{2}}\, b \right )}{2 a}-\frac {\ln \left (\left (a^{2} b -b^{2} a -\sqrt {-a b \left (a -b \right )^{2}}\, a \right ) x^{2}-\sqrt {-a b \left (a -b \right )^{2}}\, a -\sqrt {-a b \left (a -b \right )^{2}}\, b \right ) \sqrt {-a b \left (a -b \right )^{2}}}{4 a^{2} b}\) | \(320\) |
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)*arctan(1/2*(2*a* x^2+2*a)/(a*b)^(1/2)))
Time = 0.08 (sec) , antiderivative size = 157, normalized size of antiderivative = 2.28 \[ \int \frac {x^5}{a+b+2 a x^2+a x^4} \, dx=\left [\frac {2 \, a b x^{2} - 2 \, a b \log \left (a x^{4} + 2 \, a x^{2} + a + b\right ) + \sqrt {-a b} {\left (a - b\right )} \log \left (\frac {a x^{4} + 2 \, a x^{2} + 2 \, \sqrt {-a b} {\left (x^{2} + 1\right )} + a - b}{a x^{4} + 2 \, a x^{2} + a + b}\right )}{4 \, a^{2} b}, \frac {a b x^{2} - a b \log \left (a x^{4} + 2 \, a x^{2} + a + b\right ) - \sqrt {a b} {\left (a - b\right )} \arctan \left (\frac {\sqrt {a b}}{a x^{2} + a}\right )}{2 \, a^{2} b}\right ] \] 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)* log((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. 144 vs. \(2 (60) = 120\).
Time = 0.62 (sec) , antiderivative size = 144, normalized size of antiderivative = 2.09 \[ \int \frac {x^5}{a+b+2 a x^2+a x^4} \, dx=\left (- \frac {1}{2 a} - \frac {\sqrt {- a^{3} b} \left (a - b\right )}{4 a^{3} b}\right ) \log {\left (x^{2} + \frac {4 a b \left (- \frac {1}{2 a} - \frac {\sqrt {- a^{3} b} \left (a - b\right )}{4 a^{3} b}\right ) + a + b}{a - b} \right )} + \left (- \frac {1}{2 a} + \frac {\sqrt {- a^{3} b} \left (a - b\right )}{4 a^{3} b}\right ) \log {\left (x^{2} + \frac {4 a b \left (- \frac {1}{2 a} + \frac {\sqrt {- a^{3} b} \left (a - b\right )}{4 a^{3} b}\right ) + a + b}{a - b} \right )} + \frac {x^{2}}{2 a} \] 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 = 58, normalized size of antiderivative = 0.84 \[ \int \frac {x^5}{a+b+2 a x^2+a x^4} \, dx=\frac {x^{2}}{2 \, a} + \frac {{\left (a - b\right )} \arctan \left (\frac {a x^{2} + a}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a} - \frac {\log \left (a x^{4} + 2 \, a x^{2} + a + b\right )}{2 \, a} \] Input:
integrate(x^5/(a*x^4+2*a*x^2+a+b),x, algorithm="maxima")
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 = 0.17 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.84 \[ \int \frac {x^5}{a+b+2 a x^2+a x^4} \, dx=\frac {x^{2}}{2 \, a} + \frac {{\left (a - b\right )} \arctan \left (\frac {a x^{2} + a}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a} - \frac {\log \left (a x^{4} + 2 \, a x^{2} + a + b\right )}{2 \, a} \] 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 = 0.19 (sec) , antiderivative size = 302, normalized size of antiderivative = 4.38 \[ \int \frac {x^5}{a+b+2 a x^2+a x^4} \, dx=\frac {x^2}{2\,a}-\frac {\ln \left (a\,x^4+2\,a\,x^2+a+b\right )}{2\,a}-\frac {\mathrm {atan}\left (\frac {a\,b\,\left (x^2\,\left (\frac {\frac {\sqrt {a}\,\left (2\,a-2\,b\right )}{\sqrt {b}}+\frac {\left (a-b\right )\,\left (4\,a\,b-12\,a^2\right )}{4\,a^{3/2}\,\sqrt {b}}}{a+b}+\frac {\sqrt {a}\,\left (6\,a-2\,b-\frac {{\left (a-b\right )}^2}{b}+\frac {2\,a\,b-6\,a^2}{a}\right )}{\sqrt {b}\,\left (a+b\right )}\right )-\frac {\frac {\left (a-b\right )\,\left (16\,a\,b-\frac {8\,a^3+8\,b\,a^2}{a}+16\,a^2\right )}{4\,a^{3/2}\,\sqrt {b}}-\frac {\left (16\,a^3+16\,b\,a^2\right )\,\left (a-b\right )}{8\,a^{5/2}\,\sqrt {b}}}{a+b}+\frac {\sqrt {a}\,\left (4\,a+4\,b-\frac {8\,a\,b-\frac {8\,a^3+8\,b\,a^2}{2\,a}+8\,a^2}{a}-\frac {{\left (a-b\right )}^2\,\left (a^3+b\,a^2\right )}{a^3\,b}\right )}{\sqrt {b}\,\left (a+b\right )}\right )}{a^2-2\,a\,b+b^2}\right )\,\left (a-b\right )}{2\,a^{3/2}\,\sqrt {b}} \] Input:
int(x^5/(a + b + 2*a*x^2 + a*x^4),x)
Output:
x^2/(2*a) - log(a + b + 2*a*x^2 + a*x^4)/(2*a) - (atan((a*b*(x^2*(((a^(1/2 )*(2*a - 2*b))/b^(1/2) + ((a - b)*(4*a*b - 12*a^2))/(4*a^(3/2)*b^(1/2)))/( a + b) + (a^(1/2)*(6*a - 2*b - (a - b)^2/b + (2*a*b - 6*a^2)/a))/(b^(1/2)* (a + b))) - (((a - b)*(16*a*b - (8*a^2*b + 8*a^3)/a + 16*a^2))/(4*a^(3/2)* b^(1/2)) - ((16*a^2*b + 16*a^3)*(a - b))/(8*a^(5/2)*b^(1/2)))/(a + b) + (a ^(1/2)*(4*a + 4*b - (8*a*b - (8*a^2*b + 8*a^3)/(2*a) + 8*a^2)/a - ((a - b) ^2*(a^2*b + a^3))/(a^3*b)))/(b^(1/2)*(a + b))))/(a^2 - 2*a*b + b^2))*(a - b))/(2*a^(3/2)*b^(1/2))
Time = 0.15 (sec) , antiderivative size = 334, normalized size of antiderivative = 4.84 \[ \int \frac {x^5}{a+b+2 a x^2+a x^4} \, dx=\frac {-\sqrt {\sqrt {a}\, \sqrt {a +b}+a}\, \sqrt {\sqrt {a}\, \sqrt {a +b}-a}\, \mathit {atan} \left (\frac {\sqrt {\sqrt {a}\, \sqrt {a +b}-a}\, \sqrt {2}-2 \sqrt {a}\, x}{\sqrt {\sqrt {a}\, \sqrt {a +b}+a}\, \sqrt {2}}\right ) a +\sqrt {\sqrt {a}\, \sqrt {a +b}+a}\, \sqrt {\sqrt {a}\, \sqrt {a +b}-a}\, \mathit {atan} \left (\frac {\sqrt {\sqrt {a}\, \sqrt {a +b}-a}\, \sqrt {2}-2 \sqrt {a}\, x}{\sqrt {\sqrt {a}\, \sqrt {a +b}+a}\, \sqrt {2}}\right ) b -\sqrt {\sqrt {a}\, \sqrt {a +b}+a}\, \sqrt {\sqrt {a}\, \sqrt {a +b}-a}\, \mathit {atan} \left (\frac {\sqrt {\sqrt {a}\, \sqrt {a +b}-a}\, \sqrt {2}+2 \sqrt {a}\, x}{\sqrt {\sqrt {a}\, \sqrt {a +b}+a}\, \sqrt {2}}\right ) a +\sqrt {\sqrt {a}\, \sqrt {a +b}+a}\, \sqrt {\sqrt {a}\, \sqrt {a +b}-a}\, \mathit {atan} \left (\frac {\sqrt {\sqrt {a}\, \sqrt {a +b}-a}\, \sqrt {2}+2 \sqrt {a}\, x}{\sqrt {\sqrt {a}\, \sqrt {a +b}+a}\, \sqrt {2}}\right ) b -\mathrm {log}\left (-\sqrt {\sqrt {a}\, \sqrt {a +b}-a}\, \sqrt {2}\, x +\sqrt {a +b}+\sqrt {a}\, x^{2}\right ) a b -\mathrm {log}\left (\sqrt {\sqrt {a}\, \sqrt {a +b}-a}\, \sqrt {2}\, x +\sqrt {a +b}+\sqrt {a}\, x^{2}\right ) a b +a b \,x^{2}}{2 a^{2} b} \] Input:
int(x^5/(a*x^4+2*a*x^2+a+b),x)
Output:
( - sqrt(sqrt(a)*sqrt(a + b) + a)*sqrt(sqrt(a)*sqrt(a + b) - a)*atan((sqrt (sqrt(a)*sqrt(a + b) - a)*sqrt(2) - 2*sqrt(a)*x)/(sqrt(sqrt(a)*sqrt(a + b) + a)*sqrt(2)))*a + sqrt(sqrt(a)*sqrt(a + b) + a)*sqrt(sqrt(a)*sqrt(a + b) - a)*atan((sqrt(sqrt(a)*sqrt(a + b) - a)*sqrt(2) - 2*sqrt(a)*x)/(sqrt(sqr t(a)*sqrt(a + b) + a)*sqrt(2)))*b - sqrt(sqrt(a)*sqrt(a + b) + a)*sqrt(sqr t(a)*sqrt(a + b) - a)*atan((sqrt(sqrt(a)*sqrt(a + b) - a)*sqrt(2) + 2*sqrt (a)*x)/(sqrt(sqrt(a)*sqrt(a + b) + a)*sqrt(2)))*a + sqrt(sqrt(a)*sqrt(a + b) + a)*sqrt(sqrt(a)*sqrt(a + b) - a)*atan((sqrt(sqrt(a)*sqrt(a + b) - a)* sqrt(2) + 2*sqrt(a)*x)/(sqrt(sqrt(a)*sqrt(a + b) + a)*sqrt(2)))*b - log( - sqrt(sqrt(a)*sqrt(a + b) - a)*sqrt(2)*x + sqrt(a + b) + sqrt(a)*x**2)*a*b - log(sqrt(sqrt(a)*sqrt(a + b) - a)*sqrt(2)*x + sqrt(a + b) + sqrt(a)*x** 2)*a*b + a*b*x**2)/(2*a**2*b)