Integrand size = 24, antiderivative size = 80 \[ \int \frac {1}{x^6 \left (a^2+2 a b x^2+b^2 x^4\right )} \, dx=-\frac {1}{5 a^2 x^5}+\frac {2 b}{3 a^3 x^3}-\frac {3 b^2}{a^4 x}-\frac {b^3 x}{2 a^4 \left (a+b x^2\right )}-\frac {7 b^{5/2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{9/2}} \] Output:
-1/5/a^2/x^5+2/3*b/a^3/x^3-3*b^2/a^4/x-1/2*b^3*x/a^4/(b*x^2+a)-7/2*b^(5/2) *arctan(b^(1/2)*x/a^(1/2))/a^(9/2)
Time = 0.03 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^6 \left (a^2+2 a b x^2+b^2 x^4\right )} \, dx=-\frac {1}{5 a^2 x^5}+\frac {2 b}{3 a^3 x^3}-\frac {3 b^2}{a^4 x}-\frac {b^3 x}{2 a^4 \left (a+b x^2\right )}-\frac {7 b^{5/2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{9/2}} \] Input:
Integrate[1/(x^6*(a^2 + 2*a*b*x^2 + b^2*x^4)),x]
Output:
-1/5*1/(a^2*x^5) + (2*b)/(3*a^3*x^3) - (3*b^2)/(a^4*x) - (b^3*x)/(2*a^4*(a + b*x^2)) - (7*b^(5/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(9/2))
Time = 0.35 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.19, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {1380, 27, 253, 264, 264, 264, 218}
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.
\(\displaystyle \int \frac {1}{x^6 \left (a^2+2 a b x^2+b^2 x^4\right )} \, dx\) |
\(\Big \downarrow \) 1380 |
\(\displaystyle b^2 \int \frac {1}{b^2 x^6 \left (b x^2+a\right )^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \int \frac {1}{x^6 \left (a+b x^2\right )^2}dx\) |
\(\Big \downarrow \) 253 |
\(\displaystyle \frac {7 \int \frac {1}{x^6 \left (b x^2+a\right )}dx}{2 a}+\frac {1}{2 a x^5 \left (a+b x^2\right )}\) |
\(\Big \downarrow \) 264 |
\(\displaystyle \frac {7 \left (-\frac {b \int \frac {1}{x^4 \left (b x^2+a\right )}dx}{a}-\frac {1}{5 a x^5}\right )}{2 a}+\frac {1}{2 a x^5 \left (a+b x^2\right )}\) |
\(\Big \downarrow \) 264 |
\(\displaystyle \frac {7 \left (-\frac {b \left (-\frac {b \int \frac {1}{x^2 \left (b x^2+a\right )}dx}{a}-\frac {1}{3 a x^3}\right )}{a}-\frac {1}{5 a x^5}\right )}{2 a}+\frac {1}{2 a x^5 \left (a+b x^2\right )}\) |
\(\Big \downarrow \) 264 |
\(\displaystyle \frac {7 \left (-\frac {b \left (-\frac {b \left (-\frac {b \int \frac {1}{b x^2+a}dx}{a}-\frac {1}{a x}\right )}{a}-\frac {1}{3 a x^3}\right )}{a}-\frac {1}{5 a x^5}\right )}{2 a}+\frac {1}{2 a x^5 \left (a+b x^2\right )}\) |
\(\Big \downarrow \) 218 |
\(\displaystyle \frac {7 \left (-\frac {b \left (-\frac {b \left (-\frac {\sqrt {b} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{3/2}}-\frac {1}{a x}\right )}{a}-\frac {1}{3 a x^3}\right )}{a}-\frac {1}{5 a x^5}\right )}{2 a}+\frac {1}{2 a x^5 \left (a+b x^2\right )}\) |
Input:
Int[1/(x^6*(a^2 + 2*a*b*x^2 + b^2*x^4)),x]
Output:
1/(2*a*x^5*(a + b*x^2)) + (7*(-1/5*1/(a*x^5) - (b*(-1/3*1/(a*x^3) - (b*(-( 1/(a*x)) - (Sqrt[b]*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/a^(3/2)))/a))/a))/(2*a)
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/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(c*x )^(m + 1))*((a + b*x^2)^(p + 1)/(2*a*c*(p + 1))), x] + Simp[(m + 2*p + 3)/( 2*a*(p + 1)) Int[(c*x)^m*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, m }, x] && LtQ[p, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^( m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + 2*p + 3)/(a*c ^2*(m + 1))) Int[(c*x)^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, p }, x] && LtQ[m, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
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 = 0.12 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.84
method | result | size |
default | \(-\frac {b^{3} \left (\frac {x}{2 b \,x^{2}+2 a}+\frac {7 \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{a^{4}}-\frac {1}{5 a^{2} x^{5}}-\frac {3 b^{2}}{a^{4} x}+\frac {2 b}{3 a^{3} x^{3}}\) | \(67\) |
risch | \(\frac {-\frac {7 b^{3} x^{6}}{2 a^{4}}-\frac {7 b^{2} x^{4}}{3 a^{3}}+\frac {7 b \,x^{2}}{15 a^{2}}-\frac {1}{5 a}}{x^{5} \left (b \,x^{2}+a \right )}+\frac {7 \sqrt {-a b}\, b^{2} \ln \left (-b x +\sqrt {-a b}\right )}{4 a^{5}}-\frac {7 \sqrt {-a b}\, b^{2} \ln \left (-b x -\sqrt {-a b}\right )}{4 a^{5}}\) | \(106\) |
Input:
int(1/x^6/(b^2*x^4+2*a*b*x^2+a^2),x,method=_RETURNVERBOSE)
Output:
-b^3/a^4*(1/2*x/(b*x^2+a)+7/2/(a*b)^(1/2)*arctan(b/(a*b)^(1/2)*x))-1/5/a^2 /x^5-3*b^2/a^4/x+2/3*b/a^3/x^3
Time = 0.09 (sec) , antiderivative size = 198, normalized size of antiderivative = 2.48 \[ \int \frac {1}{x^6 \left (a^2+2 a b x^2+b^2 x^4\right )} \, dx=\left [-\frac {210 \, b^{3} x^{6} + 140 \, a b^{2} x^{4} - 28 \, a^{2} b x^{2} + 12 \, a^{3} - 105 \, {\left (b^{3} x^{7} + a b^{2} x^{5}\right )} \sqrt {-\frac {b}{a}} \log \left (\frac {b x^{2} - 2 \, a x \sqrt {-\frac {b}{a}} - a}{b x^{2} + a}\right )}{60 \, {\left (a^{4} b x^{7} + a^{5} x^{5}\right )}}, -\frac {105 \, b^{3} x^{6} + 70 \, a b^{2} x^{4} - 14 \, a^{2} b x^{2} + 6 \, a^{3} + 105 \, {\left (b^{3} x^{7} + a b^{2} x^{5}\right )} \sqrt {\frac {b}{a}} \arctan \left (x \sqrt {\frac {b}{a}}\right )}{30 \, {\left (a^{4} b x^{7} + a^{5} x^{5}\right )}}\right ] \] Input:
integrate(1/x^6/(b^2*x^4+2*a*b*x^2+a^2),x, algorithm="fricas")
Output:
[-1/60*(210*b^3*x^6 + 140*a*b^2*x^4 - 28*a^2*b*x^2 + 12*a^3 - 105*(b^3*x^7 + a*b^2*x^5)*sqrt(-b/a)*log((b*x^2 - 2*a*x*sqrt(-b/a) - a)/(b*x^2 + a)))/ (a^4*b*x^7 + a^5*x^5), -1/30*(105*b^3*x^6 + 70*a*b^2*x^4 - 14*a^2*b*x^2 + 6*a^3 + 105*(b^3*x^7 + a*b^2*x^5)*sqrt(b/a)*arctan(x*sqrt(b/a)))/(a^4*b*x^ 7 + a^5*x^5)]
Time = 0.19 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.58 \[ \int \frac {1}{x^6 \left (a^2+2 a b x^2+b^2 x^4\right )} \, dx=\frac {7 \sqrt {- \frac {b^{5}}{a^{9}}} \log {\left (- \frac {a^{5} \sqrt {- \frac {b^{5}}{a^{9}}}}{b^{3}} + x \right )}}{4} - \frac {7 \sqrt {- \frac {b^{5}}{a^{9}}} \log {\left (\frac {a^{5} \sqrt {- \frac {b^{5}}{a^{9}}}}{b^{3}} + x \right )}}{4} + \frac {- 6 a^{3} + 14 a^{2} b x^{2} - 70 a b^{2} x^{4} - 105 b^{3} x^{6}}{30 a^{5} x^{5} + 30 a^{4} b x^{7}} \] Input:
integrate(1/x**6/(b**2*x**4+2*a*b*x**2+a**2),x)
Output:
7*sqrt(-b**5/a**9)*log(-a**5*sqrt(-b**5/a**9)/b**3 + x)/4 - 7*sqrt(-b**5/a **9)*log(a**5*sqrt(-b**5/a**9)/b**3 + x)/4 + (-6*a**3 + 14*a**2*b*x**2 - 7 0*a*b**2*x**4 - 105*b**3*x**6)/(30*a**5*x**5 + 30*a**4*b*x**7)
Time = 0.11 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.94 \[ \int \frac {1}{x^6 \left (a^2+2 a b x^2+b^2 x^4\right )} \, dx=-\frac {105 \, b^{3} x^{6} + 70 \, a b^{2} x^{4} - 14 \, a^{2} b x^{2} + 6 \, a^{3}}{30 \, {\left (a^{4} b x^{7} + a^{5} x^{5}\right )}} - \frac {7 \, b^{3} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a^{4}} \] Input:
integrate(1/x^6/(b^2*x^4+2*a*b*x^2+a^2),x, algorithm="maxima")
Output:
-1/30*(105*b^3*x^6 + 70*a*b^2*x^4 - 14*a^2*b*x^2 + 6*a^3)/(a^4*b*x^7 + a^5 *x^5) - 7/2*b^3*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a^4)
Time = 0.17 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.88 \[ \int \frac {1}{x^6 \left (a^2+2 a b x^2+b^2 x^4\right )} \, dx=-\frac {7 \, b^{3} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a^{4}} - \frac {b^{3} x}{2 \, {\left (b x^{2} + a\right )} a^{4}} - \frac {45 \, b^{2} x^{4} - 10 \, a b x^{2} + 3 \, a^{2}}{15 \, a^{4} x^{5}} \] Input:
integrate(1/x^6/(b^2*x^4+2*a*b*x^2+a^2),x, algorithm="giac")
Output:
-7/2*b^3*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a^4) - 1/2*b^3*x/((b*x^2 + a)*a^ 4) - 1/15*(45*b^2*x^4 - 10*a*b*x^2 + 3*a^2)/(a^4*x^5)
Time = 18.23 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.88 \[ \int \frac {1}{x^6 \left (a^2+2 a b x^2+b^2 x^4\right )} \, dx=-\frac {\frac {1}{5\,a}-\frac {7\,b\,x^2}{15\,a^2}+\frac {7\,b^2\,x^4}{3\,a^3}+\frac {7\,b^3\,x^6}{2\,a^4}}{b\,x^7+a\,x^5}-\frac {7\,b^{5/2}\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{2\,a^{9/2}} \] Input:
int(1/(x^6*(a^2 + b^2*x^4 + 2*a*b*x^2)),x)
Output:
- (1/(5*a) - (7*b*x^2)/(15*a^2) + (7*b^2*x^4)/(3*a^3) + (7*b^3*x^6)/(2*a^4 ))/(a*x^5 + b*x^7) - (7*b^(5/2)*atan((b^(1/2)*x)/a^(1/2)))/(2*a^(9/2))
Time = 0.17 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.26 \[ \int \frac {1}{x^6 \left (a^2+2 a b x^2+b^2 x^4\right )} \, dx=\frac {-105 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a \,b^{2} x^{5}-105 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) b^{3} x^{7}-6 a^{4}+14 a^{3} b \,x^{2}-70 a^{2} b^{2} x^{4}-105 a \,b^{3} x^{6}}{30 a^{5} x^{5} \left (b \,x^{2}+a \right )} \] Input:
int(1/x^6/(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*b**2*x**5 - 105*sq rt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*b**3*x**7 - 6*a**4 + 14*a**3*b *x**2 - 70*a**2*b**2*x**4 - 105*a*b**3*x**6)/(30*a**5*x**5*(a + b*x**2))