Integrand size = 20, antiderivative size = 124 \[ \int \frac {A+B x^2}{x^5 \left (a+b x^2\right )^3} \, dx=-\frac {A}{4 a^3 x^4}+\frac {3 A b-a B}{2 a^4 x^2}+\frac {b (A b-a B)}{4 a^3 \left (a+b x^2\right )^2}+\frac {b (3 A b-2 a B)}{2 a^4 \left (a+b x^2\right )}+\frac {3 b (2 A b-a B) \log (x)}{a^5}-\frac {3 b (2 A b-a B) \log \left (a+b x^2\right )}{2 a^5} \]
-1/4*A/a^3/x^4+1/2*(3*A*b-B*a)/a^4/x^2+1/4*b*(A*b-B*a)/a^3/(b*x^2+a)^2+1/2 *b*(3*A*b-2*B*a)/a^4/(b*x^2+a)+3*b*(2*A*b-B*a)*ln(x)/a^5-3/2*b*(2*A*b-B*a) *ln(b*x^2+a)/a^5
Time = 0.05 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.87 \[ \int \frac {A+B x^2}{x^5 \left (a+b x^2\right )^3} \, dx=\frac {-\frac {a^2 A}{x^4}-\frac {2 a (-3 A b+a B)}{x^2}+\frac {a^2 b (A b-a B)}{\left (a+b x^2\right )^2}+\frac {2 a b (3 A b-2 a B)}{a+b x^2}+12 b (2 A b-a B) \log (x)+6 b (-2 A b+a B) \log \left (a+b x^2\right )}{4 a^5} \]
(-((a^2*A)/x^4) - (2*a*(-3*A*b + a*B))/x^2 + (a^2*b*(A*b - a*B))/(a + b*x^ 2)^2 + (2*a*b*(3*A*b - 2*a*B))/(a + b*x^2) + 12*b*(2*A*b - a*B)*Log[x] + 6 *b*(-2*A*b + a*B)*Log[a + b*x^2])/(4*a^5)
Time = 0.30 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.98, 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.
\(\displaystyle \int \frac {A+B x^2}{x^5 \left (a+b x^2\right )^3} \, dx\) |
\(\Big \downarrow \) 354 |
\(\displaystyle \frac {1}{2} \int \frac {B x^2+A}{x^6 \left (b x^2+a\right )^3}dx^2\) |
\(\Big \downarrow \) 86 |
\(\displaystyle \frac {1}{2} \int \left (\frac {3 (a B-2 A b) b^2}{a^5 \left (b x^2+a\right )}+\frac {(2 a B-3 A b) b^2}{a^4 \left (b x^2+a\right )^2}+\frac {(a B-A b) b^2}{a^3 \left (b x^2+a\right )^3}-\frac {3 (a B-2 A b) b}{a^5 x^2}+\frac {a B-3 A b}{a^4 x^4}+\frac {A}{a^3 x^6}\right )dx^2\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} \left (\frac {3 b \log \left (x^2\right ) (2 A b-a B)}{a^5}-\frac {3 b (2 A b-a B) \log \left (a+b x^2\right )}{a^5}+\frac {b (3 A b-2 a B)}{a^4 \left (a+b x^2\right )}+\frac {3 A b-a B}{a^4 x^2}+\frac {b (A b-a B)}{2 a^3 \left (a+b x^2\right )^2}-\frac {A}{2 a^3 x^4}\right )\) |
(-1/2*A/(a^3*x^4) + (3*A*b - a*B)/(a^4*x^2) + (b*(A*b - a*B))/(2*a^3*(a + b*x^2)^2) + (b*(3*A*b - 2*a*B))/(a^4*(a + b*x^2)) + (3*b*(2*A*b - a*B)*Log [x^2])/a^5 - (3*b*(2*A*b - a*B)*Log[a + b*x^2])/a^5)/2
3.1.96.3.1 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 = 2.55 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.99
method | result | size |
default | \(-\frac {A}{4 a^{3} x^{4}}-\frac {-3 A b +B a}{2 a^{4} x^{2}}+\frac {3 b \left (2 A b -B a \right ) \ln \left (x \right )}{a^{5}}-\frac {b^{2} \left (\frac {\left (6 A b -3 B a \right ) \ln \left (b \,x^{2}+a \right )}{b}-\frac {a^{2} \left (A b -B a \right )}{2 b \left (b \,x^{2}+a \right )^{2}}-\frac {a \left (3 A b -2 B a \right )}{b \left (b \,x^{2}+a \right )}\right )}{2 a^{5}}\) | \(123\) |
norman | \(\frac {-\frac {A}{4 a}+\frac {\left (2 A b -B a \right ) x^{2}}{2 a^{2}}-\frac {b \left (6 b^{2} A -3 a b B \right ) x^{6}}{a^{4}}-\frac {b^{2} \left (18 b^{2} A -9 a b B \right ) x^{8}}{4 a^{5}}}{x^{4} \left (b \,x^{2}+a \right )^{2}}+\frac {3 b \left (2 A b -B a \right ) \ln \left (x \right )}{a^{5}}-\frac {3 b \left (2 A b -B a \right ) \ln \left (b \,x^{2}+a \right )}{2 a^{5}}\) | \(123\) |
risch | \(\frac {\frac {3 b^{2} \left (2 A b -B a \right ) x^{6}}{2 a^{4}}+\frac {9 b \left (2 A b -B a \right ) x^{4}}{4 a^{3}}+\frac {\left (2 A b -B a \right ) x^{2}}{2 a^{2}}-\frac {A}{4 a}}{x^{4} \left (b \,x^{2}+a \right )^{2}}+\frac {6 b^{2} \ln \left (x \right ) A}{a^{5}}-\frac {3 b \ln \left (x \right ) B}{a^{4}}-\frac {3 b^{2} \ln \left (b \,x^{2}+a \right ) A}{a^{5}}+\frac {3 b \ln \left (b \,x^{2}+a \right ) B}{2 a^{4}}\) | \(129\) |
parallelrisch | \(\frac {24 A \ln \left (x \right ) x^{8} b^{4}-12 A \ln \left (b \,x^{2}+a \right ) x^{8} b^{4}-12 B \ln \left (x \right ) x^{8} a \,b^{3}+6 B \ln \left (b \,x^{2}+a \right ) x^{8} a \,b^{3}-18 A \,x^{8} b^{4}+9 B \,x^{8} a \,b^{3}+48 A \ln \left (x \right ) x^{6} a \,b^{3}-24 A \ln \left (b \,x^{2}+a \right ) x^{6} a \,b^{3}-24 B \ln \left (x \right ) x^{6} a^{2} b^{2}+12 B \ln \left (b \,x^{2}+a \right ) x^{6} a^{2} b^{2}-24 A \,x^{6} a \,b^{3}+12 B \,x^{6} a^{2} b^{2}+24 A \ln \left (x \right ) x^{4} a^{2} b^{2}-12 A \ln \left (b \,x^{2}+a \right ) x^{4} a^{2} b^{2}-12 B \ln \left (x \right ) x^{4} a^{3} b +6 B \ln \left (b \,x^{2}+a \right ) x^{4} a^{3} b +4 A \,x^{2} a^{3} b -2 B \,x^{2} a^{4}-A \,a^{4}}{4 a^{5} x^{4} \left (b \,x^{2}+a \right )^{2}}\) | \(271\) |
-1/4*A/a^3/x^4-1/2*(-3*A*b+B*a)/a^4/x^2+3*b*(2*A*b-B*a)*ln(x)/a^5-1/2/a^5* b^2*((6*A*b-3*B*a)/b*ln(b*x^2+a)-1/2*a^2*(A*b-B*a)/b/(b*x^2+a)^2-a*(3*A*b- 2*B*a)/b/(b*x^2+a))
Leaf count of result is larger than twice the leaf count of optimal. 229 vs. \(2 (111) = 222\).
Time = 0.31 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.85 \[ \int \frac {A+B x^2}{x^5 \left (a+b x^2\right )^3} \, dx=-\frac {6 \, {\left (B a^{2} b^{2} - 2 \, A a b^{3}\right )} x^{6} + A a^{4} + 9 \, {\left (B a^{3} b - 2 \, A a^{2} b^{2}\right )} x^{4} + 2 \, {\left (B a^{4} - 2 \, A a^{3} b\right )} x^{2} - 6 \, {\left ({\left (B a b^{3} - 2 \, A b^{4}\right )} x^{8} + 2 \, {\left (B a^{2} b^{2} - 2 \, A a b^{3}\right )} x^{6} + {\left (B a^{3} b - 2 \, A a^{2} b^{2}\right )} x^{4}\right )} \log \left (b x^{2} + a\right ) + 12 \, {\left ({\left (B a b^{3} - 2 \, A b^{4}\right )} x^{8} + 2 \, {\left (B a^{2} b^{2} - 2 \, A a b^{3}\right )} x^{6} + {\left (B a^{3} b - 2 \, A a^{2} b^{2}\right )} x^{4}\right )} \log \left (x\right )}{4 \, {\left (a^{5} b^{2} x^{8} + 2 \, a^{6} b x^{6} + a^{7} x^{4}\right )}} \]
-1/4*(6*(B*a^2*b^2 - 2*A*a*b^3)*x^6 + A*a^4 + 9*(B*a^3*b - 2*A*a^2*b^2)*x^ 4 + 2*(B*a^4 - 2*A*a^3*b)*x^2 - 6*((B*a*b^3 - 2*A*b^4)*x^8 + 2*(B*a^2*b^2 - 2*A*a*b^3)*x^6 + (B*a^3*b - 2*A*a^2*b^2)*x^4)*log(b*x^2 + a) + 12*((B*a* b^3 - 2*A*b^4)*x^8 + 2*(B*a^2*b^2 - 2*A*a*b^3)*x^6 + (B*a^3*b - 2*A*a^2*b^ 2)*x^4)*log(x))/(a^5*b^2*x^8 + 2*a^6*b*x^6 + a^7*x^4)
Time = 0.68 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.10 \[ \int \frac {A+B x^2}{x^5 \left (a+b x^2\right )^3} \, dx=\frac {- A a^{3} + x^{6} \cdot \left (12 A b^{3} - 6 B a b^{2}\right ) + x^{4} \cdot \left (18 A a b^{2} - 9 B a^{2} b\right ) + x^{2} \cdot \left (4 A a^{2} b - 2 B a^{3}\right )}{4 a^{6} x^{4} + 8 a^{5} b x^{6} + 4 a^{4} b^{2} x^{8}} - \frac {3 b \left (- 2 A b + B a\right ) \log {\left (x \right )}}{a^{5}} + \frac {3 b \left (- 2 A b + B a\right ) \log {\left (\frac {a}{b} + x^{2} \right )}}{2 a^{5}} \]
(-A*a**3 + x**6*(12*A*b**3 - 6*B*a*b**2) + x**4*(18*A*a*b**2 - 9*B*a**2*b) + x**2*(4*A*a**2*b - 2*B*a**3))/(4*a**6*x**4 + 8*a**5*b*x**6 + 4*a**4*b** 2*x**8) - 3*b*(-2*A*b + B*a)*log(x)/a**5 + 3*b*(-2*A*b + B*a)*log(a/b + x* *2)/(2*a**5)
Time = 0.19 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.10 \[ \int \frac {A+B x^2}{x^5 \left (a+b x^2\right )^3} \, dx=-\frac {6 \, {\left (B a b^{2} - 2 \, A b^{3}\right )} x^{6} + 9 \, {\left (B a^{2} b - 2 \, A a b^{2}\right )} x^{4} + A a^{3} + 2 \, {\left (B a^{3} - 2 \, A a^{2} b\right )} x^{2}}{4 \, {\left (a^{4} b^{2} x^{8} + 2 \, a^{5} b x^{6} + a^{6} x^{4}\right )}} + \frac {3 \, {\left (B a b - 2 \, A b^{2}\right )} \log \left (b x^{2} + a\right )}{2 \, a^{5}} - \frac {3 \, {\left (B a b - 2 \, A b^{2}\right )} \log \left (x^{2}\right )}{2 \, a^{5}} \]
-1/4*(6*(B*a*b^2 - 2*A*b^3)*x^6 + 9*(B*a^2*b - 2*A*a*b^2)*x^4 + A*a^3 + 2* (B*a^3 - 2*A*a^2*b)*x^2)/(a^4*b^2*x^8 + 2*a^5*b*x^6 + a^6*x^4) + 3/2*(B*a* b - 2*A*b^2)*log(b*x^2 + a)/a^5 - 3/2*(B*a*b - 2*A*b^2)*log(x^2)/a^5
Time = 0.29 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.07 \[ \int \frac {A+B x^2}{x^5 \left (a+b x^2\right )^3} \, dx=-\frac {3 \, {\left (B a b - 2 \, A b^{2}\right )} \log \left (x^{2}\right )}{2 \, a^{5}} + \frac {3 \, {\left (B a b^{2} - 2 \, A b^{3}\right )} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, a^{5} b} - \frac {6 \, B a b^{2} x^{6} - 12 \, A b^{3} x^{6} + 9 \, B a^{2} b x^{4} - 18 \, A a b^{2} x^{4} + 2 \, B a^{3} x^{2} - 4 \, A a^{2} b x^{2} + A a^{3}}{4 \, {\left (b x^{4} + a x^{2}\right )}^{2} a^{4}} \]
-3/2*(B*a*b - 2*A*b^2)*log(x^2)/a^5 + 3/2*(B*a*b^2 - 2*A*b^3)*log(abs(b*x^ 2 + a))/(a^5*b) - 1/4*(6*B*a*b^2*x^6 - 12*A*b^3*x^6 + 9*B*a^2*b*x^4 - 18*A *a*b^2*x^4 + 2*B*a^3*x^2 - 4*A*a^2*b*x^2 + A*a^3)/((b*x^4 + a*x^2)^2*a^4)
Time = 4.94 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.06 \[ \int \frac {A+B x^2}{x^5 \left (a+b x^2\right )^3} \, dx=\frac {\frac {x^2\,\left (2\,A\,b-B\,a\right )}{2\,a^2}-\frac {A}{4\,a}+\frac {3\,b^2\,x^6\,\left (2\,A\,b-B\,a\right )}{2\,a^4}+\frac {9\,b\,x^4\,\left (2\,A\,b-B\,a\right )}{4\,a^3}}{a^2\,x^4+2\,a\,b\,x^6+b^2\,x^8}-\frac {\ln \left (b\,x^2+a\right )\,\left (6\,A\,b^2-3\,B\,a\,b\right )}{2\,a^5}+\frac {\ln \left (x\right )\,\left (6\,A\,b^2-3\,B\,a\,b\right )}{a^5} \]