Integrand size = 20, antiderivative size = 149 \[ \int \frac {A+B x^2}{x^7 \left (a+b x^2\right )^3} \, dx=-\frac {A}{6 a^3 x^6}+\frac {3 A b-a B}{4 a^4 x^4}-\frac {3 b (2 A b-a B)}{2 a^5 x^2}-\frac {b^2 (A b-a B)}{4 a^4 \left (a+b x^2\right )^2}-\frac {b^2 (4 A b-3 a B)}{2 a^5 \left (a+b x^2\right )}-\frac {2 b^2 (5 A b-3 a B) \log (x)}{a^6}+\frac {b^2 (5 A b-3 a B) \log \left (a+b x^2\right )}{a^6} \]
-1/6*A/a^3/x^6+1/4*(3*A*b-B*a)/a^4/x^4-3/2*b*(2*A*b-B*a)/a^5/x^2-1/4*b^2*( A*b-B*a)/a^4/(b*x^2+a)^2-1/2*b^2*(4*A*b-3*B*a)/a^5/(b*x^2+a)-2*b^2*(5*A*b- 3*B*a)*ln(x)/a^6+b^2*(5*A*b-3*B*a)*ln(b*x^2+a)/a^6
Time = 0.08 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.91 \[ \int \frac {A+B x^2}{x^7 \left (a+b x^2\right )^3} \, dx=\frac {-\frac {2 a^3 A}{x^6}-\frac {3 a^2 (-3 A b+a B)}{x^4}+\frac {18 a b (-2 A b+a B)}{x^2}+\frac {3 a^2 b^2 (-A b+a B)}{\left (a+b x^2\right )^2}+\frac {6 a b^2 (-4 A b+3 a B)}{a+b x^2}+24 b^2 (-5 A b+3 a B) \log (x)+12 b^2 (5 A b-3 a B) \log \left (a+b x^2\right )}{12 a^6} \]
((-2*a^3*A)/x^6 - (3*a^2*(-3*A*b + a*B))/x^4 + (18*a*b*(-2*A*b + a*B))/x^2 + (3*a^2*b^2*(-(A*b) + a*B))/(a + b*x^2)^2 + (6*a*b^2*(-4*A*b + 3*a*B))/( a + b*x^2) + 24*b^2*(-5*A*b + 3*a*B)*Log[x] + 12*b^2*(5*A*b - 3*a*B)*Log[a + b*x^2])/(12*a^6)
Time = 0.34 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.02, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {354, 86, 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.
| \(\displaystyle \int \frac {A+B x^2}{x^7 \left (a+b x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 354 |
| \(\displaystyle \frac {1}{2} \int \frac {B x^2+A}{x^8 \left (b x^2+a\right )^3}dx^2\) |
| \(\Big \downarrow \) 86 |
| \(\displaystyle \frac {1}{2} \int \left (-\frac {2 (3 a B-5 A b) b^3}{a^6 \left (b x^2+a\right )}-\frac {(3 a B-4 A b) b^3}{a^5 \left (b x^2+a\right )^2}-\frac {(a B-A b) b^3}{a^4 \left (b x^2+a\right )^3}+\frac {2 (3 a B-5 A b) b^2}{a^6 x^2}-\frac {3 (a B-2 A b) b}{a^5 x^4}+\frac {a B-3 A b}{a^4 x^6}+\frac {A}{a^3 x^8}\right )dx^2\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{2} \left (-\frac {2 b^2 \log \left (x^2\right ) (5 A b-3 a B)}{a^6}+\frac {2 b^2 (5 A b-3 a B) \log \left (a+b x^2\right )}{a^6}-\frac {b^2 (4 A b-3 a B)}{a^5 \left (a+b x^2\right )}-\frac {3 b (2 A b-a B)}{a^5 x^2}-\frac {b^2 (A b-a B)}{2 a^4 \left (a+b x^2\right )^2}+\frac {3 A b-a B}{2 a^4 x^4}-\frac {A}{3 a^3 x^6}\right )\) |
(-1/3*A/(a^3*x^6) + (3*A*b - a*B)/(2*a^4*x^4) - (3*b*(2*A*b - a*B))/(a^5*x ^2) - (b^2*(A*b - a*B))/(2*a^4*(a + b*x^2)^2) - (b^2*(4*A*b - 3*a*B))/(a^5 *(a + b*x^2)) - (2*b^2*(5*A*b - 3*a*B)*Log[x^2])/a^6 + (2*b^2*(5*A*b - 3*a *B)*Log[a + b*x^2])/a^6)/2
3.1.97.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_ .), x_] :> Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && ((ILtQ[n, 0] && ILtQ[p, 0]) || EqQ[p, 1 ] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_S ymbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2)*(a + b*x)^p*(c + d*x)^q, x], x , x^2], x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IntegerQ [(m - 1)/2]
Time = 2.50 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.96
| method | result | size |
| default | \(-\frac {A}{6 a^{3} x^{6}}-\frac {-3 A b +B a}{4 a^{4} x^{4}}-\frac {3 b \left (2 A b -B a \right )}{2 a^{5} x^{2}}-\frac {2 b^{2} \left (5 A b -3 B a \right ) \ln \left (x \right )}{a^{6}}+\frac {b^{3} \left (\frac {\left (10 A b -6 B a \right ) \ln \left (b \,x^{2}+a \right )}{b}-\frac {a^{2} \left (A b -B a \right )}{2 b \left (b \,x^{2}+a \right )^{2}}-\frac {a \left (4 A b -3 B a \right )}{b \left (b \,x^{2}+a \right )}\right )}{2 a^{6}}\) | \(143\) |
| norman | \(\frac {-\frac {A}{6 a}+\frac {\left (5 A b -3 B a \right ) x^{2}}{12 a^{2}}-\frac {b \left (5 A b -3 B a \right ) x^{4}}{3 a^{3}}+\frac {2 b \left (5 b^{3} A -3 a \,b^{2} B \right ) x^{8}}{a^{5}}+\frac {b^{2} \left (15 b^{3} A -9 a \,b^{2} B \right ) x^{10}}{2 a^{6}}}{x^{6} \left (b \,x^{2}+a \right )^{2}}+\frac {b^{2} \left (5 A b -3 B a \right ) \ln \left (b \,x^{2}+a \right )}{a^{6}}-\frac {2 b^{2} \left (5 A b -3 B a \right ) \ln \left (x \right )}{a^{6}}\) | \(148\) |
| risch | \(\frac {-\frac {b^{3} \left (5 A b -3 B a \right ) x^{8}}{a^{5}}-\frac {3 b^{2} \left (5 A b -3 B a \right ) x^{6}}{2 a^{4}}-\frac {b \left (5 A b -3 B a \right ) x^{4}}{3 a^{3}}+\frac {\left (5 A b -3 B a \right ) x^{2}}{12 a^{2}}-\frac {A}{6 a}}{x^{6} \left (b \,x^{2}+a \right )^{2}}-\frac {10 b^{3} \ln \left (x \right ) A}{a^{6}}+\frac {6 b^{2} \ln \left (x \right ) B}{a^{5}}+\frac {5 b^{3} \ln \left (-b \,x^{2}-a \right ) A}{a^{6}}-\frac {3 b^{2} \ln \left (-b \,x^{2}-a \right ) B}{a^{5}}\) | \(159\) |
| parallelrisch | \(-\frac {120 A \ln \left (x \right ) x^{10} b^{5}-60 A \ln \left (b \,x^{2}+a \right ) x^{10} b^{5}-72 B \ln \left (x \right ) x^{10} a \,b^{4}+36 B \ln \left (b \,x^{2}+a \right ) x^{10} a \,b^{4}-90 A \,b^{5} x^{10}+54 B a \,b^{4} x^{10}+240 A \ln \left (x \right ) x^{8} a \,b^{4}-120 A \ln \left (b \,x^{2}+a \right ) x^{8} a \,b^{4}-144 B \ln \left (x \right ) x^{8} a^{2} b^{3}+72 B \ln \left (b \,x^{2}+a \right ) x^{8} a^{2} b^{3}-120 a A \,b^{4} x^{8}+72 B \,a^{2} b^{3} x^{8}+120 A \ln \left (x \right ) x^{6} a^{2} b^{3}-60 A \ln \left (b \,x^{2}+a \right ) x^{6} a^{2} b^{3}-72 B \ln \left (x \right ) x^{6} a^{3} b^{2}+36 B \ln \left (b \,x^{2}+a \right ) x^{6} a^{3} b^{2}+20 a^{3} A \,b^{2} x^{4}-12 B \,a^{4} b \,x^{4}-5 a^{4} A b \,x^{2}+3 a^{5} B \,x^{2}+2 a^{5} A}{12 a^{6} x^{6} \left (b \,x^{2}+a \right )^{2}}\) | \(297\) |
-1/6*A/a^3/x^6-1/4*(-3*A*b+B*a)/a^4/x^4-3/2*b*(2*A*b-B*a)/a^5/x^2-2*b^2*(5 *A*b-3*B*a)*ln(x)/a^6+1/2/a^6*b^3*((10*A*b-6*B*a)/b*ln(b*x^2+a)-1/2*a^2*(A *b-B*a)/b/(b*x^2+a)^2-a*(4*A*b-3*B*a)/b/(b*x^2+a))
Time = 0.28 (sec) , antiderivative size = 267, normalized size of antiderivative = 1.79 \[ \int \frac {A+B x^2}{x^7 \left (a+b x^2\right )^3} \, dx=\frac {12 \, {\left (3 \, B a^{2} b^{3} - 5 \, A a b^{4}\right )} x^{8} + 18 \, {\left (3 \, B a^{3} b^{2} - 5 \, A a^{2} b^{3}\right )} x^{6} - 2 \, A a^{5} + 4 \, {\left (3 \, B a^{4} b - 5 \, A a^{3} b^{2}\right )} x^{4} - {\left (3 \, B a^{5} - 5 \, A a^{4} b\right )} x^{2} - 12 \, {\left ({\left (3 \, B a b^{4} - 5 \, A b^{5}\right )} x^{10} + 2 \, {\left (3 \, B a^{2} b^{3} - 5 \, A a b^{4}\right )} x^{8} + {\left (3 \, B a^{3} b^{2} - 5 \, A a^{2} b^{3}\right )} x^{6}\right )} \log \left (b x^{2} + a\right ) + 24 \, {\left ({\left (3 \, B a b^{4} - 5 \, A b^{5}\right )} x^{10} + 2 \, {\left (3 \, B a^{2} b^{3} - 5 \, A a b^{4}\right )} x^{8} + {\left (3 \, B a^{3} b^{2} - 5 \, A a^{2} b^{3}\right )} x^{6}\right )} \log \left (x\right )}{12 \, {\left (a^{6} b^{2} x^{10} + 2 \, a^{7} b x^{8} + a^{8} x^{6}\right )}} \]
1/12*(12*(3*B*a^2*b^3 - 5*A*a*b^4)*x^8 + 18*(3*B*a^3*b^2 - 5*A*a^2*b^3)*x^ 6 - 2*A*a^5 + 4*(3*B*a^4*b - 5*A*a^3*b^2)*x^4 - (3*B*a^5 - 5*A*a^4*b)*x^2 - 12*((3*B*a*b^4 - 5*A*b^5)*x^10 + 2*(3*B*a^2*b^3 - 5*A*a*b^4)*x^8 + (3*B* a^3*b^2 - 5*A*a^2*b^3)*x^6)*log(b*x^2 + a) + 24*((3*B*a*b^4 - 5*A*b^5)*x^1 0 + 2*(3*B*a^2*b^3 - 5*A*a*b^4)*x^8 + (3*B*a^3*b^2 - 5*A*a^2*b^3)*x^6)*log (x))/(a^6*b^2*x^10 + 2*a^7*b*x^8 + a^8*x^6)
Time = 0.75 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.11 \[ \int \frac {A+B x^2}{x^7 \left (a+b x^2\right )^3} \, dx=\frac {- 2 A a^{4} + x^{8} \left (- 60 A b^{4} + 36 B a b^{3}\right ) + x^{6} \left (- 90 A a b^{3} + 54 B a^{2} b^{2}\right ) + x^{4} \left (- 20 A a^{2} b^{2} + 12 B a^{3} b\right ) + x^{2} \cdot \left (5 A a^{3} b - 3 B a^{4}\right )}{12 a^{7} x^{6} + 24 a^{6} b x^{8} + 12 a^{5} b^{2} x^{10}} + \frac {2 b^{2} \left (- 5 A b + 3 B a\right ) \log {\left (x \right )}}{a^{6}} - \frac {b^{2} \left (- 5 A b + 3 B a\right ) \log {\left (\frac {a}{b} + x^{2} \right )}}{a^{6}} \]
(-2*A*a**4 + x**8*(-60*A*b**4 + 36*B*a*b**3) + x**6*(-90*A*a*b**3 + 54*B*a **2*b**2) + x**4*(-20*A*a**2*b**2 + 12*B*a**3*b) + x**2*(5*A*a**3*b - 3*B* a**4))/(12*a**7*x**6 + 24*a**6*b*x**8 + 12*a**5*b**2*x**10) + 2*b**2*(-5*A *b + 3*B*a)*log(x)/a**6 - b**2*(-5*A*b + 3*B*a)*log(a/b + x**2)/a**6
Time = 0.20 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.14 \[ \int \frac {A+B x^2}{x^7 \left (a+b x^2\right )^3} \, dx=\frac {12 \, {\left (3 \, B a b^{3} - 5 \, A b^{4}\right )} x^{8} + 18 \, {\left (3 \, B a^{2} b^{2} - 5 \, A a b^{3}\right )} x^{6} - 2 \, A a^{4} + 4 \, {\left (3 \, B a^{3} b - 5 \, A a^{2} b^{2}\right )} x^{4} - {\left (3 \, B a^{4} - 5 \, A a^{3} b\right )} x^{2}}{12 \, {\left (a^{5} b^{2} x^{10} + 2 \, a^{6} b x^{8} + a^{7} x^{6}\right )}} - \frac {{\left (3 \, B a b^{2} - 5 \, A b^{3}\right )} \log \left (b x^{2} + a\right )}{a^{6}} + \frac {{\left (3 \, B a b^{2} - 5 \, A b^{3}\right )} \log \left (x^{2}\right )}{a^{6}} \]
1/12*(12*(3*B*a*b^3 - 5*A*b^4)*x^8 + 18*(3*B*a^2*b^2 - 5*A*a*b^3)*x^6 - 2* A*a^4 + 4*(3*B*a^3*b - 5*A*a^2*b^2)*x^4 - (3*B*a^4 - 5*A*a^3*b)*x^2)/(a^5* b^2*x^10 + 2*a^6*b*x^8 + a^7*x^6) - (3*B*a*b^2 - 5*A*b^3)*log(b*x^2 + a)/a ^6 + (3*B*a*b^2 - 5*A*b^3)*log(x^2)/a^6
Time = 0.30 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.35 \[ \int \frac {A+B x^2}{x^7 \left (a+b x^2\right )^3} \, dx=\frac {{\left (3 \, B a b^{2} - 5 \, A b^{3}\right )} \log \left (x^{2}\right )}{a^{6}} - \frac {{\left (3 \, B a b^{3} - 5 \, A b^{4}\right )} \log \left ({\left | b x^{2} + a \right |}\right )}{a^{6} b} + \frac {18 \, B a b^{4} x^{4} - 30 \, A b^{5} x^{4} + 42 \, B a^{2} b^{3} x^{2} - 68 \, A a b^{4} x^{2} + 25 \, B a^{3} b^{2} - 39 \, A a^{2} b^{3}}{4 \, {\left (b x^{2} + a\right )}^{2} a^{6}} - \frac {66 \, B a b^{2} x^{6} - 110 \, A b^{3} x^{6} - 18 \, B a^{2} b x^{4} + 36 \, A a b^{2} x^{4} + 3 \, B a^{3} x^{2} - 9 \, A a^{2} b x^{2} + 2 \, A a^{3}}{12 \, a^{6} x^{6}} \]
(3*B*a*b^2 - 5*A*b^3)*log(x^2)/a^6 - (3*B*a*b^3 - 5*A*b^4)*log(abs(b*x^2 + a))/(a^6*b) + 1/4*(18*B*a*b^4*x^4 - 30*A*b^5*x^4 + 42*B*a^2*b^3*x^2 - 68* A*a*b^4*x^2 + 25*B*a^3*b^2 - 39*A*a^2*b^3)/((b*x^2 + a)^2*a^6) - 1/12*(66* B*a*b^2*x^6 - 110*A*b^3*x^6 - 18*B*a^2*b*x^4 + 36*A*a*b^2*x^4 + 3*B*a^3*x^ 2 - 9*A*a^2*b*x^2 + 2*A*a^3)/(a^6*x^6)
Time = 4.99 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.04 \[ \int \frac {A+B x^2}{x^7 \left (a+b x^2\right )^3} \, dx=\frac {\ln \left (b\,x^2+a\right )\,\left (5\,A\,b^3-3\,B\,a\,b^2\right )}{a^6}-\frac {\frac {A}{6\,a}-\frac {x^2\,\left (5\,A\,b-3\,B\,a\right )}{12\,a^2}+\frac {3\,b^2\,x^6\,\left (5\,A\,b-3\,B\,a\right )}{2\,a^4}+\frac {b^3\,x^8\,\left (5\,A\,b-3\,B\,a\right )}{a^5}+\frac {b\,x^4\,\left (5\,A\,b-3\,B\,a\right )}{3\,a^3}}{a^2\,x^6+2\,a\,b\,x^8+b^2\,x^{10}}-\frac {\ln \left (x\right )\,\left (10\,A\,b^3-6\,B\,a\,b^2\right )}{a^6} \]