Integrand size = 24, antiderivative size = 78 \[ \int \frac {x^8}{a^2+2 a b x^2+b^2 x^4} \, dx=\frac {3 a^2 x}{b^4}-\frac {2 a x^3}{3 b^3}+\frac {x^5}{5 b^2}+\frac {a^3 x}{2 b^4 \left (a+b x^2\right )}-\frac {7 a^{5/2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 b^{9/2}} \] Output:
3*a^2*x/b^4-2/3*a*x^3/b^3+1/5*x^5/b^2+1/2*a^3*x/b^4/(b*x^2+a)-7/2*a^(5/2)* arctan(b^(1/2)*x/a^(1/2))/b^(9/2)
Time = 0.03 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.91 \[ \int \frac {x^8}{a^2+2 a b x^2+b^2 x^4} \, dx=\frac {x \left (90 a^2-20 a b x^2+6 b^2 x^4+\frac {15 a^3}{a+b x^2}\right )}{30 b^4}-\frac {7 a^{5/2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 b^{9/2}} \] Input:
Integrate[x^8/(a^2 + 2*a*b*x^2 + b^2*x^4),x]
Output:
(x*(90*a^2 - 20*a*b*x^2 + 6*b^2*x^4 + (15*a^3)/(a + b*x^2)))/(30*b^4) - (7 *a^(5/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*b^(9/2))
Time = 0.38 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.05, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {1380, 27, 252, 254, 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.
\(\displaystyle \int \frac {x^8}{a^2+2 a b x^2+b^2 x^4} \, dx\) |
\(\Big \downarrow \) 1380 |
\(\displaystyle b^2 \int \frac {x^8}{b^2 \left (b x^2+a\right )^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \int \frac {x^8}{\left (a+b x^2\right )^2}dx\) |
\(\Big \downarrow \) 252 |
\(\displaystyle \frac {7 \int \frac {x^6}{b x^2+a}dx}{2 b}-\frac {x^7}{2 b \left (a+b x^2\right )}\) |
\(\Big \downarrow \) 254 |
\(\displaystyle \frac {7 \int \left (\frac {x^4}{b}-\frac {a x^2}{b^2}-\frac {a^3}{b^3 \left (b x^2+a\right )}+\frac {a^2}{b^3}\right )dx}{2 b}-\frac {x^7}{2 b \left (a+b x^2\right )}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {7 \left (-\frac {a^{5/2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{7/2}}+\frac {a^2 x}{b^3}-\frac {a x^3}{3 b^2}+\frac {x^5}{5 b}\right )}{2 b}-\frac {x^7}{2 b \left (a+b x^2\right )}\) |
Input:
Int[x^8/(a^2 + 2*a*b*x^2 + b^2*x^4),x]
Output:
-1/2*x^7/(b*(a + b*x^2)) + (7*((a^2*x)/b^3 - (a*x^3)/(3*b^2) + x^5/(5*b) - (a^(5/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/b^(7/2)))/(2*b)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x )^(m - 1)*((a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] - Simp[c^2*((m - 1)/(2*b* (p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c }, x] && LtQ[p, -1] && GtQ[m, 1] && !ILtQ[(m + 2*p + 3)/2, 0] && IntBinomi alQ[a, b, c, 2, m, p, x]
Int[(x_)^(m_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^2, x], x] /; FreeQ[{a, b}, x] && IGtQ[m, 3]
Int[(u_)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> S imp[1/c^p Int[u*(b/2 + c*x^n)^(2*p), x], x] /; FreeQ[{a, b, c, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]
Time = 0.09 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.83
method | result | size |
default | \(\frac {\frac {1}{5} x^{5} b^{2}-\frac {2}{3} a b \,x^{3}+3 a^{2} x}{b^{4}}-\frac {a^{3} \left (-\frac {x}{2 \left (b \,x^{2}+a \right )}+\frac {7 \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{b^{4}}\) | \(65\) |
risch | \(\frac {x^{5}}{5 b^{2}}-\frac {2 a \,x^{3}}{3 b^{3}}+\frac {3 a^{2} x}{b^{4}}+\frac {a^{3} x}{2 b^{4} \left (b \,x^{2}+a \right )}+\frac {7 \sqrt {-a b}\, a^{2} \ln \left (-\sqrt {-a b}\, x -a \right )}{4 b^{5}}-\frac {7 \sqrt {-a b}\, a^{2} \ln \left (\sqrt {-a b}\, x -a \right )}{4 b^{5}}\) | \(101\) |
Input:
int(x^8/(b^2*x^4+2*a*b*x^2+a^2),x,method=_RETURNVERBOSE)
Output:
1/b^4*(1/5*x^5*b^2-2/3*a*b*x^3+3*a^2*x)-a^3/b^4*(-1/2*x/(b*x^2+a)+7/2/(a*b )^(1/2)*arctan(b/(a*b)^(1/2)*x))
Time = 0.27 (sec) , antiderivative size = 190, normalized size of antiderivative = 2.44 \[ \int \frac {x^8}{a^2+2 a b x^2+b^2 x^4} \, dx=\left [\frac {12 \, b^{3} x^{7} - 28 \, a b^{2} x^{5} + 140 \, a^{2} b x^{3} + 210 \, a^{3} x + 105 \, {\left (a^{2} b x^{2} + a^{3}\right )} \sqrt {-\frac {a}{b}} \log \left (\frac {b x^{2} - 2 \, b x \sqrt {-\frac {a}{b}} - a}{b x^{2} + a}\right )}{60 \, {\left (b^{5} x^{2} + a b^{4}\right )}}, \frac {6 \, b^{3} x^{7} - 14 \, a b^{2} x^{5} + 70 \, a^{2} b x^{3} + 105 \, a^{3} x - 105 \, {\left (a^{2} b x^{2} + a^{3}\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b x \sqrt {\frac {a}{b}}}{a}\right )}{30 \, {\left (b^{5} x^{2} + a b^{4}\right )}}\right ] \] Input:
integrate(x^8/(b^2*x^4+2*a*b*x^2+a^2),x, algorithm="fricas")
Output:
[1/60*(12*b^3*x^7 - 28*a*b^2*x^5 + 140*a^2*b*x^3 + 210*a^3*x + 105*(a^2*b* x^2 + a^3)*sqrt(-a/b)*log((b*x^2 - 2*b*x*sqrt(-a/b) - a)/(b*x^2 + a)))/(b^ 5*x^2 + a*b^4), 1/30*(6*b^3*x^7 - 14*a*b^2*x^5 + 70*a^2*b*x^3 + 105*a^3*x - 105*(a^2*b*x^2 + a^3)*sqrt(a/b)*arctan(b*x*sqrt(a/b)/a))/(b^5*x^2 + a*b^ 4)]
Time = 0.14 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.59 \[ \int \frac {x^8}{a^2+2 a b x^2+b^2 x^4} \, dx=\frac {a^{3} x}{2 a b^{4} + 2 b^{5} x^{2}} + \frac {3 a^{2} x}{b^{4}} - \frac {2 a x^{3}}{3 b^{3}} + \frac {7 \sqrt {- \frac {a^{5}}{b^{9}}} \log {\left (x - \frac {b^{4} \sqrt {- \frac {a^{5}}{b^{9}}}}{a^{2}} \right )}}{4} - \frac {7 \sqrt {- \frac {a^{5}}{b^{9}}} \log {\left (x + \frac {b^{4} \sqrt {- \frac {a^{5}}{b^{9}}}}{a^{2}} \right )}}{4} + \frac {x^{5}}{5 b^{2}} \] Input:
integrate(x**8/(b**2*x**4+2*a*b*x**2+a**2),x)
Output:
a**3*x/(2*a*b**4 + 2*b**5*x**2) + 3*a**2*x/b**4 - 2*a*x**3/(3*b**3) + 7*sq rt(-a**5/b**9)*log(x - b**4*sqrt(-a**5/b**9)/a**2)/4 - 7*sqrt(-a**5/b**9)* log(x + b**4*sqrt(-a**5/b**9)/a**2)/4 + x**5/(5*b**2)
Time = 0.11 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.91 \[ \int \frac {x^8}{a^2+2 a b x^2+b^2 x^4} \, dx=\frac {a^{3} x}{2 \, {\left (b^{5} x^{2} + a b^{4}\right )}} - \frac {7 \, a^{3} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} b^{4}} + \frac {3 \, b^{2} x^{5} - 10 \, a b x^{3} + 45 \, a^{2} x}{15 \, b^{4}} \] Input:
integrate(x^8/(b^2*x^4+2*a*b*x^2+a^2),x, algorithm="maxima")
Output:
1/2*a^3*x/(b^5*x^2 + a*b^4) - 7/2*a^3*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*b^4 ) + 1/15*(3*b^2*x^5 - 10*a*b*x^3 + 45*a^2*x)/b^4
Time = 0.11 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.94 \[ \int \frac {x^8}{a^2+2 a b x^2+b^2 x^4} \, dx=-\frac {7 \, a^{3} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} b^{4}} + \frac {a^{3} x}{2 \, {\left (b x^{2} + a\right )} b^{4}} + \frac {3 \, b^{8} x^{5} - 10 \, a b^{7} x^{3} + 45 \, a^{2} b^{6} x}{15 \, b^{10}} \] Input:
integrate(x^8/(b^2*x^4+2*a*b*x^2+a^2),x, algorithm="giac")
Output:
-7/2*a^3*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*b^4) + 1/2*a^3*x/((b*x^2 + a)*b^ 4) + 1/15*(3*b^8*x^5 - 10*a*b^7*x^3 + 45*a^2*b^6*x)/b^10
Time = 0.04 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.85 \[ \int \frac {x^8}{a^2+2 a b x^2+b^2 x^4} \, dx=\frac {x^5}{5\,b^2}-\frac {2\,a\,x^3}{3\,b^3}+\frac {3\,a^2\,x}{b^4}-\frac {7\,a^{5/2}\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{2\,b^{9/2}}+\frac {a^3\,x}{2\,\left (b^5\,x^2+a\,b^4\right )} \] Input:
int(x^8/(a^2 + b^2*x^4 + 2*a*b*x^2),x)
Output:
x^5/(5*b^2) - (2*a*x^3)/(3*b^3) + (3*a^2*x)/b^4 - (7*a^(5/2)*atan((b^(1/2) *x)/a^(1/2)))/(2*b^(9/2)) + (a^3*x)/(2*(a*b^4 + b^5*x^2))
Time = 0.17 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.23 \[ \int \frac {x^8}{a^2+2 a b x^2+b^2 x^4} \, dx=\frac {-105 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{3}-105 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{2} b \,x^{2}+105 a^{3} b x +70 a^{2} b^{2} x^{3}-14 a \,b^{3} x^{5}+6 b^{4} x^{7}}{30 b^{5} \left (b \,x^{2}+a \right )} \] Input:
int(x^8/(b^2*x^4+2*a*b*x^2+a^2),x)
Output:
( - 105*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**3 - 105*sqrt(b)*s qrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**2*b*x**2 + 105*a**3*b*x + 70*a**2* b**2*x**3 - 14*a*b**3*x**5 + 6*b**4*x**7)/(30*b**5*(a + b*x**2))