Integrand size = 20, antiderivative size = 134 \[ \int \frac {A+B x^2}{x^8 \left (a+b x^2\right )^2} \, dx=-\frac {A}{7 a^2 x^7}+\frac {2 A b-a B}{5 a^3 x^5}-\frac {b (3 A b-2 a B)}{3 a^4 x^3}+\frac {b^2 (4 A b-3 a B)}{a^5 x}+\frac {b^3 (A b-a B) x}{2 a^5 \left (a+b x^2\right )}+\frac {b^{5/2} (9 A b-7 a B) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{11/2}} \] Output:
-1/7*A/a^2/x^7+1/5*(2*A*b-B*a)/a^3/x^5-1/3*b*(3*A*b-2*B*a)/a^4/x^3+b^2*(4* A*b-3*B*a)/a^5/x+1/2*b^3*(A*b-B*a)*x/a^5/(b*x^2+a)+1/2*b^(5/2)*(9*A*b-7*B* a)*arctan(b^(1/2)*x/a^(1/2))/a^(11/2)
Time = 0.07 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.01 \[ \int \frac {A+B x^2}{x^8 \left (a+b x^2\right )^2} \, dx=-\frac {A}{7 a^2 x^7}+\frac {2 A b-a B}{5 a^3 x^5}+\frac {b (-3 A b+2 a B)}{3 a^4 x^3}-\frac {b^2 (-4 A b+3 a B)}{a^5 x}-\frac {b^3 (-A b+a B) x}{2 a^5 \left (a+b x^2\right )}-\frac {b^{5/2} (-9 A b+7 a B) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{11/2}} \] Input:
Integrate[(A + B*x^2)/(x^8*(a + b*x^2)^2),x]
Output:
-1/7*A/(a^2*x^7) + (2*A*b - a*B)/(5*a^3*x^5) + (b*(-3*A*b + 2*a*B))/(3*a^4 *x^3) - (b^2*(-4*A*b + 3*a*B))/(a^5*x) - (b^3*(-(A*b) + a*B)*x)/(2*a^5*(a + b*x^2)) - (b^(5/2)*(-9*A*b + 7*a*B)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(1 1/2))
Time = 0.53 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.10, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {361, 25, 2333, 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.
\(\displaystyle \int \frac {A+B x^2}{x^8 \left (a+b x^2\right )^2} \, dx\) |
\(\Big \downarrow \) 361 |
\(\displaystyle \frac {b^3 x (A b-a B)}{2 a^5 \left (a+b x^2\right )}-\frac {1}{2} b^3 \int -\frac {\frac {(A b-a B) x^8}{a^5}-\frac {2 (A b-a B) x^6}{a^4 b}+\frac {2 (A b-a B) x^4}{a^3 b^2}-\frac {2 (A b-a B) x^2}{a^2 b^3}+\frac {2 A}{a b^3}}{x^8 \left (b x^2+a\right )}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {1}{2} b^3 \int \frac {\frac {(A b-a B) x^8}{a^5}-\frac {2 (A b-a B) x^6}{a^4 b}+\frac {2 (A b-a B) x^4}{a^3 b^2}-\frac {2 (A b-a B) x^2}{a^2 b^3}+\frac {2 A}{a b^3}}{x^8 \left (b x^2+a\right )}dx+\frac {b^3 x (A b-a B)}{2 a^5 \left (a+b x^2\right )}\) |
\(\Big \downarrow \) 2333 |
\(\displaystyle \frac {1}{2} b^3 \int \left (\frac {2 A}{a^2 b^3 x^8}+\frac {9 A b-7 a B}{a^5 \left (b x^2+a\right )}+\frac {2 (3 a B-4 A b)}{a^5 b x^2}-\frac {2 (2 a B-3 A b)}{a^4 b^2 x^4}+\frac {2 (a B-2 A b)}{a^3 b^3 x^6}\right )dx+\frac {b^3 x (A b-a B)}{2 a^5 \left (a+b x^2\right )}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {b^3 x (A b-a B)}{2 a^5 \left (a+b x^2\right )}+\frac {1}{2} b^3 \left (\frac {(9 A b-7 a B) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{11/2} \sqrt {b}}+\frac {2 (4 A b-3 a B)}{a^5 b x}-\frac {2 (3 A b-2 a B)}{3 a^4 b^2 x^3}+\frac {2 (2 A b-a B)}{5 a^3 b^3 x^5}-\frac {2 A}{7 a^2 b^3 x^7}\right )\) |
Input:
Int[(A + B*x^2)/(x^8*(a + b*x^2)^2),x]
Output:
(b^3*(A*b - a*B)*x)/(2*a^5*(a + b*x^2)) + (b^3*((-2*A)/(7*a^2*b^3*x^7) + ( 2*(2*A*b - a*B))/(5*a^3*b^3*x^5) - (2*(3*A*b - 2*a*B))/(3*a^4*b^2*x^3) + ( 2*(4*A*b - 3*a*B))/(a^5*b*x) + ((9*A*b - 7*a*B)*ArcTan[(Sqrt[b]*x)/Sqrt[a] ])/(a^(11/2)*Sqrt[b])))/2
Int[(x_)^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] : > Simp[(-a)^(m/2 - 1)*(b*c - a*d)*x*((a + b*x^2)^(p + 1)/(2*b^(m/2 + 1)*(p + 1))), x] + Simp[1/(2*b^(m/2 + 1)*(p + 1)) Int[x^m*(a + b*x^2)^(p + 1)*E xpandToSum[2*b*(p + 1)*Together[(b^(m/2)*(c + d*x^2) - (-a)^(m/2 - 1)*(b*c - a*d)*x^(-m + 2))/(a + b*x^2)] - ((-a)^(m/2 - 1)*(b*c - a*d))/x^m, x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && ILtQ[m/ 2, 0] && (IntegerQ[p] || EqQ[m + 2*p + 1, 0])
Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ ExpandIntegrand[(c*x)^m*Pq*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]
Time = 0.42 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.87
method | result | size |
default | \(\frac {b^{3} \left (\frac {\left (\frac {A b}{2}-\frac {B a}{2}\right ) x}{b \,x^{2}+a}+\frac {\left (9 A b -7 B a \right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{a^{5}}-\frac {A}{7 a^{2} x^{7}}-\frac {-2 A b +B a}{5 a^{3} x^{5}}-\frac {b \left (3 A b -2 B a \right )}{3 a^{4} x^{3}}+\frac {b^{2} \left (4 A b -3 B a \right )}{a^{5} x}\) | \(117\) |
risch | \(\frac {\frac {b^{3} \left (9 A b -7 B a \right ) x^{8}}{2 a^{5}}+\frac {b^{2} \left (9 A b -7 B a \right ) x^{6}}{3 a^{4}}-\frac {b \left (9 A b -7 B a \right ) x^{4}}{15 a^{3}}+\frac {\left (9 A b -7 B a \right ) x^{2}}{35 a^{2}}-\frac {A}{7 a}}{x^{7} \left (b \,x^{2}+a \right )}+\frac {9 \sqrt {-a b}\, b^{3} \ln \left (-b x -\sqrt {-a b}\right ) A}{4 a^{6}}-\frac {7 \sqrt {-a b}\, b^{2} \ln \left (-b x -\sqrt {-a b}\right ) B}{4 a^{5}}-\frac {9 \sqrt {-a b}\, b^{3} \ln \left (-b x +\sqrt {-a b}\right ) A}{4 a^{6}}+\frac {7 \sqrt {-a b}\, b^{2} \ln \left (-b x +\sqrt {-a b}\right ) B}{4 a^{5}}\) | \(209\) |
Input:
int((B*x^2+A)/x^8/(b*x^2+a)^2,x,method=_RETURNVERBOSE)
Output:
1/a^5*b^3*((1/2*A*b-1/2*B*a)*x/(b*x^2+a)+1/2*(9*A*b-7*B*a)/(a*b)^(1/2)*arc tan(b*x/(a*b)^(1/2)))-1/7*A/a^2/x^7-1/5*(-2*A*b+B*a)/a^3/x^5-1/3*b*(3*A*b- 2*B*a)/a^4/x^3+b^2*(4*A*b-3*B*a)/a^5/x
Time = 0.09 (sec) , antiderivative size = 360, normalized size of antiderivative = 2.69 \[ \int \frac {A+B x^2}{x^8 \left (a+b x^2\right )^2} \, dx=\left [-\frac {210 \, {\left (7 \, B a b^{3} - 9 \, A b^{4}\right )} x^{8} + 140 \, {\left (7 \, B a^{2} b^{2} - 9 \, A a b^{3}\right )} x^{6} + 60 \, A a^{4} - 28 \, {\left (7 \, B a^{3} b - 9 \, A a^{2} b^{2}\right )} x^{4} + 12 \, {\left (7 \, B a^{4} - 9 \, A a^{3} b\right )} x^{2} + 105 \, {\left ({\left (7 \, B a b^{3} - 9 \, A b^{4}\right )} x^{9} + {\left (7 \, B a^{2} b^{2} - 9 \, A a b^{3}\right )} x^{7}\right )} \sqrt {-\frac {b}{a}} \log \left (\frac {b x^{2} + 2 \, a x \sqrt {-\frac {b}{a}} - a}{b x^{2} + a}\right )}{420 \, {\left (a^{5} b x^{9} + a^{6} x^{7}\right )}}, -\frac {105 \, {\left (7 \, B a b^{3} - 9 \, A b^{4}\right )} x^{8} + 70 \, {\left (7 \, B a^{2} b^{2} - 9 \, A a b^{3}\right )} x^{6} + 30 \, A a^{4} - 14 \, {\left (7 \, B a^{3} b - 9 \, A a^{2} b^{2}\right )} x^{4} + 6 \, {\left (7 \, B a^{4} - 9 \, A a^{3} b\right )} x^{2} + 105 \, {\left ({\left (7 \, B a b^{3} - 9 \, A b^{4}\right )} x^{9} + {\left (7 \, B a^{2} b^{2} - 9 \, A a b^{3}\right )} x^{7}\right )} \sqrt {\frac {b}{a}} \arctan \left (x \sqrt {\frac {b}{a}}\right )}{210 \, {\left (a^{5} b x^{9} + a^{6} x^{7}\right )}}\right ] \] Input:
integrate((B*x^2+A)/x^8/(b*x^2+a)^2,x, algorithm="fricas")
Output:
[-1/420*(210*(7*B*a*b^3 - 9*A*b^4)*x^8 + 140*(7*B*a^2*b^2 - 9*A*a*b^3)*x^6 + 60*A*a^4 - 28*(7*B*a^3*b - 9*A*a^2*b^2)*x^4 + 12*(7*B*a^4 - 9*A*a^3*b)* x^2 + 105*((7*B*a*b^3 - 9*A*b^4)*x^9 + (7*B*a^2*b^2 - 9*A*a*b^3)*x^7)*sqrt (-b/a)*log((b*x^2 + 2*a*x*sqrt(-b/a) - a)/(b*x^2 + a)))/(a^5*b*x^9 + a^6*x ^7), -1/210*(105*(7*B*a*b^3 - 9*A*b^4)*x^8 + 70*(7*B*a^2*b^2 - 9*A*a*b^3)* x^6 + 30*A*a^4 - 14*(7*B*a^3*b - 9*A*a^2*b^2)*x^4 + 6*(7*B*a^4 - 9*A*a^3*b )*x^2 + 105*((7*B*a*b^3 - 9*A*b^4)*x^9 + (7*B*a^2*b^2 - 9*A*a*b^3)*x^7)*sq rt(b/a)*arctan(x*sqrt(b/a)))/(a^5*b*x^9 + a^6*x^7)]
Time = 0.38 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.80 \[ \int \frac {A+B x^2}{x^8 \left (a+b x^2\right )^2} \, dx=\frac {\sqrt {- \frac {b^{5}}{a^{11}}} \left (- 9 A b + 7 B a\right ) \log {\left (- \frac {a^{6} \sqrt {- \frac {b^{5}}{a^{11}}} \left (- 9 A b + 7 B a\right )}{- 9 A b^{4} + 7 B a b^{3}} + x \right )}}{4} - \frac {\sqrt {- \frac {b^{5}}{a^{11}}} \left (- 9 A b + 7 B a\right ) \log {\left (\frac {a^{6} \sqrt {- \frac {b^{5}}{a^{11}}} \left (- 9 A b + 7 B a\right )}{- 9 A b^{4} + 7 B a b^{3}} + x \right )}}{4} + \frac {- 30 A a^{4} + x^{8} \cdot \left (945 A b^{4} - 735 B a b^{3}\right ) + x^{6} \cdot \left (630 A a b^{3} - 490 B a^{2} b^{2}\right ) + x^{4} \left (- 126 A a^{2} b^{2} + 98 B a^{3} b\right ) + x^{2} \cdot \left (54 A a^{3} b - 42 B a^{4}\right )}{210 a^{6} x^{7} + 210 a^{5} b x^{9}} \] Input:
integrate((B*x**2+A)/x**8/(b*x**2+a)**2,x)
Output:
sqrt(-b**5/a**11)*(-9*A*b + 7*B*a)*log(-a**6*sqrt(-b**5/a**11)*(-9*A*b + 7 *B*a)/(-9*A*b**4 + 7*B*a*b**3) + x)/4 - sqrt(-b**5/a**11)*(-9*A*b + 7*B*a) *log(a**6*sqrt(-b**5/a**11)*(-9*A*b + 7*B*a)/(-9*A*b**4 + 7*B*a*b**3) + x) /4 + (-30*A*a**4 + x**8*(945*A*b**4 - 735*B*a*b**3) + x**6*(630*A*a*b**3 - 490*B*a**2*b**2) + x**4*(-126*A*a**2*b**2 + 98*B*a**3*b) + x**2*(54*A*a** 3*b - 42*B*a**4))/(210*a**6*x**7 + 210*a**5*b*x**9)
Time = 0.14 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.07 \[ \int \frac {A+B x^2}{x^8 \left (a+b x^2\right )^2} \, dx=-\frac {105 \, {\left (7 \, B a b^{3} - 9 \, A b^{4}\right )} x^{8} + 70 \, {\left (7 \, B a^{2} b^{2} - 9 \, A a b^{3}\right )} x^{6} + 30 \, A a^{4} - 14 \, {\left (7 \, B a^{3} b - 9 \, A a^{2} b^{2}\right )} x^{4} + 6 \, {\left (7 \, B a^{4} - 9 \, A a^{3} b\right )} x^{2}}{210 \, {\left (a^{5} b x^{9} + a^{6} x^{7}\right )}} - \frac {{\left (7 \, B a b^{3} - 9 \, A b^{4}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a^{5}} \] Input:
integrate((B*x^2+A)/x^8/(b*x^2+a)^2,x, algorithm="maxima")
Output:
-1/210*(105*(7*B*a*b^3 - 9*A*b^4)*x^8 + 70*(7*B*a^2*b^2 - 9*A*a*b^3)*x^6 + 30*A*a^4 - 14*(7*B*a^3*b - 9*A*a^2*b^2)*x^4 + 6*(7*B*a^4 - 9*A*a^3*b)*x^2 )/(a^5*b*x^9 + a^6*x^7) - 1/2*(7*B*a*b^3 - 9*A*b^4)*arctan(b*x/sqrt(a*b))/ (sqrt(a*b)*a^5)
Time = 0.12 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.01 \[ \int \frac {A+B x^2}{x^8 \left (a+b x^2\right )^2} \, dx=-\frac {{\left (7 \, B a b^{3} - 9 \, A b^{4}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a^{5}} - \frac {B a b^{3} x - A b^{4} x}{2 \, {\left (b x^{2} + a\right )} a^{5}} - \frac {315 \, B a b^{2} x^{6} - 420 \, A b^{3} x^{6} - 70 \, B a^{2} b x^{4} + 105 \, A a b^{2} x^{4} + 21 \, B a^{3} x^{2} - 42 \, A a^{2} b x^{2} + 15 \, A a^{3}}{105 \, a^{5} x^{7}} \] Input:
integrate((B*x^2+A)/x^8/(b*x^2+a)^2,x, algorithm="giac")
Output:
-1/2*(7*B*a*b^3 - 9*A*b^4)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a^5) - 1/2*(B* a*b^3*x - A*b^4*x)/((b*x^2 + a)*a^5) - 1/105*(315*B*a*b^2*x^6 - 420*A*b^3* x^6 - 70*B*a^2*b*x^4 + 105*A*a*b^2*x^4 + 21*B*a^3*x^2 - 42*A*a^2*b*x^2 + 1 5*A*a^3)/(a^5*x^7)
Time = 0.39 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.92 \[ \int \frac {A+B x^2}{x^8 \left (a+b x^2\right )^2} \, dx=\frac {\frac {x^2\,\left (9\,A\,b-7\,B\,a\right )}{35\,a^2}-\frac {A}{7\,a}+\frac {b^2\,x^6\,\left (9\,A\,b-7\,B\,a\right )}{3\,a^4}+\frac {b^3\,x^8\,\left (9\,A\,b-7\,B\,a\right )}{2\,a^5}-\frac {b\,x^4\,\left (9\,A\,b-7\,B\,a\right )}{15\,a^3}}{b\,x^9+a\,x^7}+\frac {b^{5/2}\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )\,\left (9\,A\,b-7\,B\,a\right )}{2\,a^{11/2}} \] Input:
int((A + B*x^2)/(x^8*(a + b*x^2)^2),x)
Output:
((x^2*(9*A*b - 7*B*a))/(35*a^2) - A/(7*a) + (b^2*x^6*(9*A*b - 7*B*a))/(3*a ^4) + (b^3*x^8*(9*A*b - 7*B*a))/(2*a^5) - (b*x^4*(9*A*b - 7*B*a))/(15*a^3) )/(a*x^7 + b*x^9) + (b^(5/2)*atan((b^(1/2)*x)/a^(1/2))*(9*A*b - 7*B*a))/(2 *a^(11/2))
Time = 0.20 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.50 \[ \int \frac {A+B x^2}{x^8 \left (a+b x^2\right )^2} \, dx=\frac {105 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) b^{3} x^{7}-15 a^{4}+21 a^{3} b \,x^{2}-35 a^{2} b^{2} x^{4}+105 a \,b^{3} x^{6}}{105 a^{5} x^{7}} \] Input:
int((B*x^2+A)/x^8/(b*x^2+a)^2,x)
Output:
(105*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*b**3*x**7 - 15*a**4 + 2 1*a**3*b*x**2 - 35*a**2*b**2*x**4 + 105*a*b**3*x**6)/(105*a**5*x**7)