Integrand size = 13, antiderivative size = 88 \[ \int \frac {x^6}{\left (a+\frac {b}{x^2}\right )^2} \, dx=-\frac {4 b^3 x}{a^5}+\frac {b^2 x^3}{a^4}-\frac {2 b x^5}{5 a^3}+\frac {x^7}{7 a^2}-\frac {b^4 x}{2 a^5 \left (b+a x^2\right )}+\frac {9 b^{7/2} \arctan \left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{2 a^{11/2}} \] Output:
-4*b^3*x/a^5+b^2*x^3/a^4-2/5*b*x^5/a^3+1/7*x^7/a^2-1/2*b^4*x/a^5/(a*x^2+b) +9/2*b^(7/2)*arctan(a^(1/2)*x/b^(1/2))/a^(11/2)
Time = 0.04 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.93 \[ \int \frac {x^6}{\left (a+\frac {b}{x^2}\right )^2} \, dx=\frac {x \left (-280 b^3+70 a b^2 x^2-28 a^2 b x^4+10 a^3 x^6-\frac {35 b^4}{b+a x^2}\right )}{70 a^5}+\frac {9 b^{7/2} \arctan \left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{2 a^{11/2}} \] Input:
Integrate[x^6/(a + b/x^2)^2,x]
Output:
(x*(-280*b^3 + 70*a*b^2*x^2 - 28*a^2*b*x^4 + 10*a^3*x^6 - (35*b^4)/(b + a* x^2)))/(70*a^5) + (9*b^(7/2)*ArcTan[(Sqrt[a]*x)/Sqrt[b]])/(2*a^(11/2))
Time = 0.35 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.08, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {795, 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.
\(\displaystyle \int \frac {x^6}{\left (a+\frac {b}{x^2}\right )^2} \, dx\) |
\(\Big \downarrow \) 795 |
\(\displaystyle \int \frac {x^{10}}{\left (a x^2+b\right )^2}dx\) |
\(\Big \downarrow \) 252 |
\(\displaystyle \frac {9 \int \frac {x^8}{a x^2+b}dx}{2 a}-\frac {x^9}{2 a \left (a x^2+b\right )}\) |
\(\Big \downarrow \) 254 |
\(\displaystyle \frac {9 \int \left (\frac {x^6}{a}-\frac {b x^4}{a^2}+\frac {b^2 x^2}{a^3}-\frac {b^3}{a^4}+\frac {b^4}{a^4 \left (a x^2+b\right )}\right )dx}{2 a}-\frac {x^9}{2 a \left (a x^2+b\right )}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {9 \left (\frac {b^{7/2} \arctan \left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{a^{9/2}}-\frac {b^3 x}{a^4}+\frac {b^2 x^3}{3 a^3}-\frac {b x^5}{5 a^2}+\frac {x^7}{7 a}\right )}{2 a}-\frac {x^9}{2 a \left (a x^2+b\right )}\) |
Input:
Int[x^6/(a + b/x^2)^2,x]
Output:
-1/2*x^9/(a*(b + a*x^2)) + (9*(-((b^3*x)/a^4) + (b^2*x^3)/(3*a^3) - (b*x^5 )/(5*a^2) + x^7/(7*a) + (b^(7/2)*ArcTan[(Sqrt[a]*x)/Sqrt[b]])/a^(9/2)))/(2 *a)
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[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(m + n*p)* (b + a/x^n)^p, x] /; FreeQ[{a, b, m, n}, x] && IntegerQ[p] && NegQ[n]
Time = 0.07 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.84
method | result | size |
default | \(\frac {\frac {1}{7} a^{3} x^{7}-\frac {2}{5} a^{2} b \,x^{5}+a \,b^{2} x^{3}-4 b^{3} x}{a^{5}}+\frac {b^{4} \left (-\frac {x}{2 \left (a \,x^{2}+b \right )}+\frac {9 \arctan \left (\frac {a x}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{a^{5}}\) | \(74\) |
risch | \(\frac {x^{7}}{7 a^{2}}-\frac {2 b \,x^{5}}{5 a^{3}}+\frac {b^{2} x^{3}}{a^{4}}-\frac {4 b^{3} x}{a^{5}}-\frac {b^{4} x}{2 a^{5} \left (a \,x^{2}+b \right )}+\frac {9 \sqrt {-a b}\, b^{3} \ln \left (-\sqrt {-a b}\, x +b \right )}{4 a^{6}}-\frac {9 \sqrt {-a b}\, b^{3} \ln \left (\sqrt {-a b}\, x +b \right )}{4 a^{6}}\) | \(107\) |
Input:
int(x^6/(a+b/x^2)^2,x,method=_RETURNVERBOSE)
Output:
1/a^5*(1/7*a^3*x^7-2/5*a^2*b*x^5+a*b^2*x^3-4*b^3*x)+b^4/a^5*(-1/2*x/(a*x^2 +b)+9/2/(a*b)^(1/2)*arctan(a*x/(a*b)^(1/2)))
Time = 0.08 (sec) , antiderivative size = 212, normalized size of antiderivative = 2.41 \[ \int \frac {x^6}{\left (a+\frac {b}{x^2}\right )^2} \, dx=\left [\frac {20 \, a^{4} x^{9} - 36 \, a^{3} b x^{7} + 84 \, a^{2} b^{2} x^{5} - 420 \, a b^{3} x^{3} - 630 \, b^{4} x + 315 \, {\left (a b^{3} x^{2} + b^{4}\right )} \sqrt {-\frac {b}{a}} \log \left (\frac {a x^{2} + 2 \, a x \sqrt {-\frac {b}{a}} - b}{a x^{2} + b}\right )}{140 \, {\left (a^{6} x^{2} + a^{5} b\right )}}, \frac {10 \, a^{4} x^{9} - 18 \, a^{3} b x^{7} + 42 \, a^{2} b^{2} x^{5} - 210 \, a b^{3} x^{3} - 315 \, b^{4} x + 315 \, {\left (a b^{3} x^{2} + b^{4}\right )} \sqrt {\frac {b}{a}} \arctan \left (\frac {a x \sqrt {\frac {b}{a}}}{b}\right )}{70 \, {\left (a^{6} x^{2} + a^{5} b\right )}}\right ] \] Input:
integrate(x^6/(a+b/x^2)^2,x, algorithm="fricas")
Output:
[1/140*(20*a^4*x^9 - 36*a^3*b*x^7 + 84*a^2*b^2*x^5 - 420*a*b^3*x^3 - 630*b ^4*x + 315*(a*b^3*x^2 + b^4)*sqrt(-b/a)*log((a*x^2 + 2*a*x*sqrt(-b/a) - b) /(a*x^2 + b)))/(a^6*x^2 + a^5*b), 1/70*(10*a^4*x^9 - 18*a^3*b*x^7 + 42*a^2 *b^2*x^5 - 210*a*b^3*x^3 - 315*b^4*x + 315*(a*b^3*x^2 + b^4)*sqrt(b/a)*arc tan(a*x*sqrt(b/a)/b))/(a^6*x^2 + a^5*b)]
Time = 0.19 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.52 \[ \int \frac {x^6}{\left (a+\frac {b}{x^2}\right )^2} \, dx=- \frac {b^{4} x}{2 a^{6} x^{2} + 2 a^{5} b} - \frac {9 \sqrt {- \frac {b^{7}}{a^{11}}} \log {\left (- \frac {a^{5} \sqrt {- \frac {b^{7}}{a^{11}}}}{b^{3}} + x \right )}}{4} + \frac {9 \sqrt {- \frac {b^{7}}{a^{11}}} \log {\left (\frac {a^{5} \sqrt {- \frac {b^{7}}{a^{11}}}}{b^{3}} + x \right )}}{4} + \frac {x^{7}}{7 a^{2}} - \frac {2 b x^{5}}{5 a^{3}} + \frac {b^{2} x^{3}}{a^{4}} - \frac {4 b^{3} x}{a^{5}} \] Input:
integrate(x**6/(a+b/x**2)**2,x)
Output:
-b**4*x/(2*a**6*x**2 + 2*a**5*b) - 9*sqrt(-b**7/a**11)*log(-a**5*sqrt(-b** 7/a**11)/b**3 + x)/4 + 9*sqrt(-b**7/a**11)*log(a**5*sqrt(-b**7/a**11)/b**3 + x)/4 + x**7/(7*a**2) - 2*b*x**5/(5*a**3) + b**2*x**3/a**4 - 4*b**3*x/a* *5
Time = 0.11 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.93 \[ \int \frac {x^6}{\left (a+\frac {b}{x^2}\right )^2} \, dx=-\frac {b^{4} x}{2 \, {\left (a^{6} x^{2} + a^{5} b\right )}} + \frac {9 \, b^{4} \arctan \left (\frac {a x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a^{5}} + \frac {5 \, a^{3} x^{7} - 14 \, a^{2} b x^{5} + 35 \, a b^{2} x^{3} - 140 \, b^{3} x}{35 \, a^{5}} \] Input:
integrate(x^6/(a+b/x^2)^2,x, algorithm="maxima")
Output:
-1/2*b^4*x/(a^6*x^2 + a^5*b) + 9/2*b^4*arctan(a*x/sqrt(a*b))/(sqrt(a*b)*a^ 5) + 1/35*(5*a^3*x^7 - 14*a^2*b*x^5 + 35*a*b^2*x^3 - 140*b^3*x)/a^5
Time = 0.12 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.95 \[ \int \frac {x^6}{\left (a+\frac {b}{x^2}\right )^2} \, dx=\frac {9 \, b^{4} \arctan \left (\frac {a x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a^{5}} - \frac {b^{4} x}{2 \, {\left (a x^{2} + b\right )} a^{5}} + \frac {5 \, a^{12} x^{7} - 14 \, a^{11} b x^{5} + 35 \, a^{10} b^{2} x^{3} - 140 \, a^{9} b^{3} x}{35 \, a^{14}} \] Input:
integrate(x^6/(a+b/x^2)^2,x, algorithm="giac")
Output:
9/2*b^4*arctan(a*x/sqrt(a*b))/(sqrt(a*b)*a^5) - 1/2*b^4*x/((a*x^2 + b)*a^5 ) + 1/35*(5*a^12*x^7 - 14*a^11*b*x^5 + 35*a^10*b^2*x^3 - 140*a^9*b^3*x)/a^ 14
Time = 0.28 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.88 \[ \int \frac {x^6}{\left (a+\frac {b}{x^2}\right )^2} \, dx=\frac {x^7}{7\,a^2}-\frac {2\,b\,x^5}{5\,a^3}-\frac {4\,b^3\,x}{a^5}+\frac {9\,b^{7/2}\,\mathrm {atan}\left (\frac {\sqrt {a}\,x}{\sqrt {b}}\right )}{2\,a^{11/2}}+\frac {b^2\,x^3}{a^4}-\frac {b^4\,x}{2\,\left (a^6\,x^2+b\,a^5\right )} \] Input:
int(x^6/(a + b/x^2)^2,x)
Output:
x^7/(7*a^2) - (2*b*x^5)/(5*a^3) - (4*b^3*x)/a^5 + (9*b^(7/2)*atan((a^(1/2) *x)/b^(1/2)))/(2*a^(11/2)) + (b^2*x^3)/a^4 - (b^4*x)/(2*(a^5*b + a^6*x^2))
Time = 0.25 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.22 \[ \int \frac {x^6}{\left (a+\frac {b}{x^2}\right )^2} \, dx=\frac {315 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {a x}{\sqrt {b}\, \sqrt {a}}\right ) a \,b^{3} x^{2}+315 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {a x}{\sqrt {b}\, \sqrt {a}}\right ) b^{4}+10 a^{5} x^{9}-18 a^{4} b \,x^{7}+42 a^{3} b^{2} x^{5}-210 a^{2} b^{3} x^{3}-315 a \,b^{4} x}{70 a^{6} \left (a \,x^{2}+b \right )} \] Input:
int(x^6/(a+b/x^2)^2,x)
Output:
(315*sqrt(b)*sqrt(a)*atan((a*x)/(sqrt(b)*sqrt(a)))*a*b**3*x**2 + 315*sqrt( b)*sqrt(a)*atan((a*x)/(sqrt(b)*sqrt(a)))*b**4 + 10*a**5*x**9 - 18*a**4*b*x **7 + 42*a**3*b**2*x**5 - 210*a**2*b**3*x**3 - 315*a*b**4*x)/(70*a**6*(a*x **2 + b))