Integrand size = 13, antiderivative size = 100 \[ \int \frac {1}{x^6 \left (a+b x^2\right )^3} \, dx=-\frac {63}{40 a^3 x^5}+\frac {21 b}{8 a^4 x^3}-\frac {63 b^2}{8 a^5 x}+\frac {1}{4 a x^5 \left (a+b x^2\right )^2}+\frac {9}{8 a^2 x^5 \left (a+b x^2\right )}-\frac {63 b^{5/2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 a^{11/2}} \]
-63/40/a^3/x^5+21/8*b/a^4/x^3-63/8*b^2/a^5/x+1/4/a/x^5/(b*x^2+a)^2+9/8/a^2 /x^5/(b*x^2+a)-63/8*b^(5/2)*arctan(x*b^(1/2)/a^(1/2))/a^(11/2)
Time = 0.04 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.90 \[ \int \frac {1}{x^6 \left (a+b x^2\right )^3} \, dx=-\frac {8 a^4-24 a^3 b x^2+168 a^2 b^2 x^4+525 a b^3 x^6+315 b^4 x^8}{40 a^5 x^5 \left (a+b x^2\right )^2}-\frac {63 b^{5/2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 a^{11/2}} \]
-1/40*(8*a^4 - 24*a^3*b*x^2 + 168*a^2*b^2*x^4 + 525*a*b^3*x^6 + 315*b^4*x^ 8)/(a^5*x^5*(a + b*x^2)^2) - (63*b^(5/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*a ^(11/2))
Time = 0.21 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.22, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {253, 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.
\(\displaystyle \int \frac {1}{x^6 \left (a+b x^2\right )^3} \, dx\) |
\(\Big \downarrow \) 253 |
\(\displaystyle \frac {9 \int \frac {1}{x^6 \left (b x^2+a\right )^2}dx}{4 a}+\frac {1}{4 a x^5 \left (a+b x^2\right )^2}\) |
\(\Big \downarrow \) 253 |
\(\displaystyle \frac {9 \left (\frac {7 \int \frac {1}{x^6 \left (b x^2+a\right )}dx}{2 a}+\frac {1}{2 a x^5 \left (a+b x^2\right )}\right )}{4 a}+\frac {1}{4 a x^5 \left (a+b x^2\right )^2}\) |
\(\Big \downarrow \) 264 |
\(\displaystyle \frac {9 \left (\frac {7 \left (-\frac {b \int \frac {1}{x^4 \left (b x^2+a\right )}dx}{a}-\frac {1}{5 a x^5}\right )}{2 a}+\frac {1}{2 a x^5 \left (a+b x^2\right )}\right )}{4 a}+\frac {1}{4 a x^5 \left (a+b x^2\right )^2}\) |
\(\Big \downarrow \) 264 |
\(\displaystyle \frac {9 \left (\frac {7 \left (-\frac {b \left (-\frac {b \int \frac {1}{x^2 \left (b x^2+a\right )}dx}{a}-\frac {1}{3 a x^3}\right )}{a}-\frac {1}{5 a x^5}\right )}{2 a}+\frac {1}{2 a x^5 \left (a+b x^2\right )}\right )}{4 a}+\frac {1}{4 a x^5 \left (a+b x^2\right )^2}\) |
\(\Big \downarrow \) 264 |
\(\displaystyle \frac {9 \left (\frac {7 \left (-\frac {b \left (-\frac {b \left (-\frac {b \int \frac {1}{b x^2+a}dx}{a}-\frac {1}{a x}\right )}{a}-\frac {1}{3 a x^3}\right )}{a}-\frac {1}{5 a x^5}\right )}{2 a}+\frac {1}{2 a x^5 \left (a+b x^2\right )}\right )}{4 a}+\frac {1}{4 a x^5 \left (a+b x^2\right )^2}\) |
\(\Big \downarrow \) 218 |
\(\displaystyle \frac {9 \left (\frac {7 \left (-\frac {b \left (-\frac {b \left (-\frac {\sqrt {b} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{3/2}}-\frac {1}{a x}\right )}{a}-\frac {1}{3 a x^3}\right )}{a}-\frac {1}{5 a x^5}\right )}{2 a}+\frac {1}{2 a x^5 \left (a+b x^2\right )}\right )}{4 a}+\frac {1}{4 a x^5 \left (a+b x^2\right )^2}\) |
1/(4*a*x^5*(a + b*x^2)^2) + (9*(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)/Sqr t[a]])/a^(3/2)))/a))/a))/(2*a)))/(4*a)
3.2.90.3.1 Defintions of rubi rules used
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]
Time = 1.78 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.75
method | result | size |
default | \(-\frac {1}{5 a^{3} x^{5}}+\frac {b}{a^{4} x^{3}}-\frac {6 b^{2}}{a^{5} x}-\frac {b^{3} \left (\frac {\frac {15}{8} b \,x^{3}+\frac {17}{8} a x}{\left (b \,x^{2}+a \right )^{2}}+\frac {63 \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \sqrt {a b}}\right )}{a^{5}}\) | \(75\) |
risch | \(\frac {-\frac {63 b^{4} x^{8}}{8 a^{5}}-\frac {105 b^{3} x^{6}}{8 a^{4}}-\frac {21 b^{2} x^{4}}{5 a^{3}}+\frac {3 b \,x^{2}}{5 a^{2}}-\frac {1}{5 a}}{x^{5} \left (b \,x^{2}+a \right )^{2}}+\frac {63 \sqrt {-a b}\, b^{2} \ln \left (-b x +\sqrt {-a b}\right )}{16 a^{6}}-\frac {63 \sqrt {-a b}\, b^{2} \ln \left (-b x -\sqrt {-a b}\right )}{16 a^{6}}\) | \(117\) |
-1/5/a^3/x^5+b/a^4/x^3-6*b^2/a^5/x-1/a^5*b^3*((15/8*b*x^3+17/8*a*x)/(b*x^2 +a)^2+63/8/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2)))
Time = 0.26 (sec) , antiderivative size = 264, normalized size of antiderivative = 2.64 \[ \int \frac {1}{x^6 \left (a+b x^2\right )^3} \, dx=\left [-\frac {630 \, b^{4} x^{8} + 1050 \, a b^{3} x^{6} + 336 \, a^{2} b^{2} x^{4} - 48 \, a^{3} b x^{2} + 16 \, a^{4} - 315 \, {\left (b^{4} x^{9} + 2 \, a b^{3} x^{7} + a^{2} b^{2} x^{5}\right )} \sqrt {-\frac {b}{a}} \log \left (\frac {b x^{2} - 2 \, a x \sqrt {-\frac {b}{a}} - a}{b x^{2} + a}\right )}{80 \, {\left (a^{5} b^{2} x^{9} + 2 \, a^{6} b x^{7} + a^{7} x^{5}\right )}}, -\frac {315 \, b^{4} x^{8} + 525 \, a b^{3} x^{6} + 168 \, a^{2} b^{2} x^{4} - 24 \, a^{3} b x^{2} + 8 \, a^{4} + 315 \, {\left (b^{4} x^{9} + 2 \, a b^{3} x^{7} + a^{2} b^{2} x^{5}\right )} \sqrt {\frac {b}{a}} \arctan \left (x \sqrt {\frac {b}{a}}\right )}{40 \, {\left (a^{5} b^{2} x^{9} + 2 \, a^{6} b x^{7} + a^{7} x^{5}\right )}}\right ] \]
[-1/80*(630*b^4*x^8 + 1050*a*b^3*x^6 + 336*a^2*b^2*x^4 - 48*a^3*b*x^2 + 16 *a^4 - 315*(b^4*x^9 + 2*a*b^3*x^7 + a^2*b^2*x^5)*sqrt(-b/a)*log((b*x^2 - 2 *a*x*sqrt(-b/a) - a)/(b*x^2 + a)))/(a^5*b^2*x^9 + 2*a^6*b*x^7 + a^7*x^5), -1/40*(315*b^4*x^8 + 525*a*b^3*x^6 + 168*a^2*b^2*x^4 - 24*a^3*b*x^2 + 8*a^ 4 + 315*(b^4*x^9 + 2*a*b^3*x^7 + a^2*b^2*x^5)*sqrt(b/a)*arctan(x*sqrt(b/a) ))/(a^5*b^2*x^9 + 2*a^6*b*x^7 + a^7*x^5)]
Time = 0.28 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.50 \[ \int \frac {1}{x^6 \left (a+b x^2\right )^3} \, dx=\frac {63 \sqrt {- \frac {b^{5}}{a^{11}}} \log {\left (- \frac {a^{6} \sqrt {- \frac {b^{5}}{a^{11}}}}{b^{3}} + x \right )}}{16} - \frac {63 \sqrt {- \frac {b^{5}}{a^{11}}} \log {\left (\frac {a^{6} \sqrt {- \frac {b^{5}}{a^{11}}}}{b^{3}} + x \right )}}{16} + \frac {- 8 a^{4} + 24 a^{3} b x^{2} - 168 a^{2} b^{2} x^{4} - 525 a b^{3} x^{6} - 315 b^{4} x^{8}}{40 a^{7} x^{5} + 80 a^{6} b x^{7} + 40 a^{5} b^{2} x^{9}} \]
63*sqrt(-b**5/a**11)*log(-a**6*sqrt(-b**5/a**11)/b**3 + x)/16 - 63*sqrt(-b **5/a**11)*log(a**6*sqrt(-b**5/a**11)/b**3 + x)/16 + (-8*a**4 + 24*a**3*b* x**2 - 168*a**2*b**2*x**4 - 525*a*b**3*x**6 - 315*b**4*x**8)/(40*a**7*x**5 + 80*a**6*b*x**7 + 40*a**5*b**2*x**9)
Time = 0.28 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.97 \[ \int \frac {1}{x^6 \left (a+b x^2\right )^3} \, dx=-\frac {315 \, b^{4} x^{8} + 525 \, a b^{3} x^{6} + 168 \, a^{2} b^{2} x^{4} - 24 \, a^{3} b x^{2} + 8 \, a^{4}}{40 \, {\left (a^{5} b^{2} x^{9} + 2 \, a^{6} b x^{7} + a^{7} x^{5}\right )}} - \frac {63 \, b^{3} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} a^{5}} \]
-1/40*(315*b^4*x^8 + 525*a*b^3*x^6 + 168*a^2*b^2*x^4 - 24*a^3*b*x^2 + 8*a^ 4)/(a^5*b^2*x^9 + 2*a^6*b*x^7 + a^7*x^5) - 63/8*b^3*arctan(b*x/sqrt(a*b))/ (sqrt(a*b)*a^5)
Time = 0.30 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.80 \[ \int \frac {1}{x^6 \left (a+b x^2\right )^3} \, dx=-\frac {63 \, b^{3} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} a^{5}} - \frac {15 \, b^{4} x^{3} + 17 \, a b^{3} x}{8 \, {\left (b x^{2} + a\right )}^{2} a^{5}} - \frac {30 \, b^{2} x^{4} - 5 \, a b x^{2} + a^{2}}{5 \, a^{5} x^{5}} \]
-63/8*b^3*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a^5) - 1/8*(15*b^4*x^3 + 17*a*b ^3*x)/((b*x^2 + a)^2*a^5) - 1/5*(30*b^2*x^4 - 5*a*b*x^2 + a^2)/(a^5*x^5)
Time = 4.64 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.92 \[ \int \frac {1}{x^6 \left (a+b x^2\right )^3} \, dx=-\frac {\frac {1}{5\,a}-\frac {3\,b\,x^2}{5\,a^2}+\frac {21\,b^2\,x^4}{5\,a^3}+\frac {105\,b^3\,x^6}{8\,a^4}+\frac {63\,b^4\,x^8}{8\,a^5}}{a^2\,x^5+2\,a\,b\,x^7+b^2\,x^9}-\frac {63\,b^{5/2}\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{8\,a^{11/2}} \]