Integrand size = 20, antiderivative size = 101 \[ \int \frac {A+B x^2}{x^3 \left (a+b x^2\right )^3} \, dx=-\frac {A}{2 a^3 x^2}-\frac {A b-a B}{4 a^2 \left (a+b x^2\right )^2}-\frac {2 A b-a B}{2 a^3 \left (a+b x^2\right )}-\frac {(3 A b-a B) \log (x)}{a^4}+\frac {(3 A b-a B) \log \left (a+b x^2\right )}{2 a^4} \]
-1/2*A/a^3/x^2+1/4*(-A*b+B*a)/a^2/(b*x^2+a)^2+1/2*(-2*A*b+B*a)/a^3/(b*x^2+ a)-(3*A*b-B*a)*ln(x)/a^4+1/2*(3*A*b-B*a)*ln(b*x^2+a)/a^4
Time = 0.04 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.86 \[ \int \frac {A+B x^2}{x^3 \left (a+b x^2\right )^3} \, dx=\frac {-\frac {2 a A}{x^2}+\frac {a^2 (-A b+a B)}{\left (a+b x^2\right )^2}+\frac {2 a (-2 A b+a B)}{a+b x^2}+4 (-3 A b+a B) \log (x)+2 (3 A b-a B) \log \left (a+b x^2\right )}{4 a^4} \]
((-2*a*A)/x^2 + (a^2*(-(A*b) + a*B))/(a + b*x^2)^2 + (2*a*(-2*A*b + a*B))/ (a + b*x^2) + 4*(-3*A*b + a*B)*Log[x] + 2*(3*A*b - a*B)*Log[a + b*x^2])/(4 *a^4)
Time = 0.26 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.99, 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.
\(\displaystyle \int \frac {A+B x^2}{x^3 \left (a+b x^2\right )^3} \, dx\) |
\(\Big \downarrow \) 354 |
\(\displaystyle \frac {1}{2} \int \frac {B x^2+A}{x^4 \left (b x^2+a\right )^3}dx^2\) |
\(\Big \downarrow \) 86 |
\(\displaystyle \frac {1}{2} \int \left (\frac {A}{a^3 x^4}-\frac {b (a B-3 A b)}{a^4 \left (b x^2+a\right )}+\frac {a B-3 A b}{a^4 x^2}-\frac {b (a B-2 A b)}{a^3 \left (b x^2+a\right )^2}-\frac {b (a B-A b)}{a^2 \left (b x^2+a\right )^3}\right )dx^2\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} \left (-\frac {\log \left (x^2\right ) (3 A b-a B)}{a^4}+\frac {(3 A b-a B) \log \left (a+b x^2\right )}{a^4}-\frac {2 A b-a B}{a^3 \left (a+b x^2\right )}-\frac {A}{a^3 x^2}-\frac {A b-a B}{2 a^2 \left (a+b x^2\right )^2}\right )\) |
(-(A/(a^3*x^2)) - (A*b - a*B)/(2*a^2*(a + b*x^2)^2) - (2*A*b - a*B)/(a^3*( a + b*x^2)) - ((3*A*b - a*B)*Log[x^2])/a^4 + ((3*A*b - a*B)*Log[a + b*x^2] )/a^4)/2
3.1.95.3.1 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 = 2.53 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.96
method | result | size |
norman | \(\frac {\frac {b \left (3 A b -B a \right ) x^{4}}{a^{3}}-\frac {A}{2 a}+\frac {b^{2} \left (9 A b -3 B a \right ) x^{6}}{4 a^{4}}}{x^{2} \left (b \,x^{2}+a \right )^{2}}-\frac {\left (3 A b -B a \right ) \ln \left (x \right )}{a^{4}}+\frac {\left (3 A b -B a \right ) \ln \left (b \,x^{2}+a \right )}{2 a^{4}}\) | \(97\) |
default | \(-\frac {A}{2 a^{3} x^{2}}+\frac {\left (-3 A b +B a \right ) \ln \left (x \right )}{a^{4}}+\frac {b \left (\frac {\left (3 A b -B a \right ) \ln \left (b \,x^{2}+a \right )}{b}-\frac {a^{2} \left (A b -B a \right )}{2 b \left (b \,x^{2}+a \right )^{2}}-\frac {a \left (2 A b -B a \right )}{b \left (b \,x^{2}+a \right )}\right )}{2 a^{4}}\) | \(102\) |
risch | \(\frac {-\frac {b \left (3 A b -B a \right ) x^{4}}{2 a^{3}}-\frac {3 \left (3 A b -B a \right ) x^{2}}{4 a^{2}}-\frac {A}{2 a}}{x^{2} \left (b \,x^{2}+a \right )^{2}}-\frac {3 \ln \left (x \right ) A b}{a^{4}}+\frac {\ln \left (x \right ) B}{a^{3}}+\frac {3 \ln \left (-b \,x^{2}-a \right ) A b}{2 a^{4}}-\frac {\ln \left (-b \,x^{2}-a \right ) B}{2 a^{3}}\) | \(108\) |
parallelrisch | \(-\frac {12 A \ln \left (x \right ) x^{6} b^{3}-6 A \ln \left (b \,x^{2}+a \right ) x^{6} b^{3}-4 B \ln \left (x \right ) x^{6} a \,b^{2}+2 B \ln \left (b \,x^{2}+a \right ) x^{6} a \,b^{2}-9 A \,x^{6} b^{3}+3 B \,x^{6} a \,b^{2}+24 A \ln \left (x \right ) x^{4} a \,b^{2}-12 A \ln \left (b \,x^{2}+a \right ) x^{4} a \,b^{2}-8 B \ln \left (x \right ) x^{4} a^{2} b +4 B \ln \left (b \,x^{2}+a \right ) x^{4} a^{2} b -12 A a \,b^{2} x^{4}+4 B \,a^{2} b \,x^{4}+12 A \ln \left (x \right ) x^{2} a^{2} b -6 A \ln \left (b \,x^{2}+a \right ) x^{2} a^{2} b -4 B \ln \left (x \right ) x^{2} a^{3}+2 B \ln \left (b \,x^{2}+a \right ) x^{2} a^{3}+2 a^{3} A}{4 a^{4} x^{2} \left (b \,x^{2}+a \right )^{2}}\) | \(240\) |
(b*(3*A*b-B*a)/a^3*x^4-1/2*A/a+1/4*b^2*(9*A*b-3*B*a)/a^4*x^6)/x^2/(b*x^2+a )^2-(3*A*b-B*a)*ln(x)/a^4+1/2*(3*A*b-B*a)*ln(b*x^2+a)/a^4
Leaf count of result is larger than twice the leaf count of optimal. 197 vs. \(2 (89) = 178\).
Time = 0.29 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.95 \[ \int \frac {A+B x^2}{x^3 \left (a+b x^2\right )^3} \, dx=\frac {2 \, {\left (B a^{2} b - 3 \, A a b^{2}\right )} x^{4} - 2 \, A a^{3} + 3 \, {\left (B a^{3} - 3 \, A a^{2} b\right )} x^{2} - 2 \, {\left ({\left (B a b^{2} - 3 \, A b^{3}\right )} x^{6} + 2 \, {\left (B a^{2} b - 3 \, A a b^{2}\right )} x^{4} + {\left (B a^{3} - 3 \, A a^{2} b\right )} x^{2}\right )} \log \left (b x^{2} + a\right ) + 4 \, {\left ({\left (B a b^{2} - 3 \, A b^{3}\right )} x^{6} + 2 \, {\left (B a^{2} b - 3 \, A a b^{2}\right )} x^{4} + {\left (B a^{3} - 3 \, A a^{2} b\right )} x^{2}\right )} \log \left (x\right )}{4 \, {\left (a^{4} b^{2} x^{6} + 2 \, a^{5} b x^{4} + a^{6} x^{2}\right )}} \]
1/4*(2*(B*a^2*b - 3*A*a*b^2)*x^4 - 2*A*a^3 + 3*(B*a^3 - 3*A*a^2*b)*x^2 - 2 *((B*a*b^2 - 3*A*b^3)*x^6 + 2*(B*a^2*b - 3*A*a*b^2)*x^4 + (B*a^3 - 3*A*a^2 *b)*x^2)*log(b*x^2 + a) + 4*((B*a*b^2 - 3*A*b^3)*x^6 + 2*(B*a^2*b - 3*A*a* b^2)*x^4 + (B*a^3 - 3*A*a^2*b)*x^2)*log(x))/(a^4*b^2*x^6 + 2*a^5*b*x^4 + a ^6*x^2)
Time = 0.65 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.06 \[ \int \frac {A+B x^2}{x^3 \left (a+b x^2\right )^3} \, dx=\frac {- 2 A a^{2} + x^{4} \left (- 6 A b^{2} + 2 B a b\right ) + x^{2} \left (- 9 A a b + 3 B a^{2}\right )}{4 a^{5} x^{2} + 8 a^{4} b x^{4} + 4 a^{3} b^{2} x^{6}} + \frac {\left (- 3 A b + B a\right ) \log {\left (x \right )}}{a^{4}} - \frac {\left (- 3 A b + B a\right ) \log {\left (\frac {a}{b} + x^{2} \right )}}{2 a^{4}} \]
(-2*A*a**2 + x**4*(-6*A*b**2 + 2*B*a*b) + x**2*(-9*A*a*b + 3*B*a**2))/(4*a **5*x**2 + 8*a**4*b*x**4 + 4*a**3*b**2*x**6) + (-3*A*b + B*a)*log(x)/a**4 - (-3*A*b + B*a)*log(a/b + x**2)/(2*a**4)
Time = 0.20 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.08 \[ \int \frac {A+B x^2}{x^3 \left (a+b x^2\right )^3} \, dx=\frac {2 \, {\left (B a b - 3 \, A b^{2}\right )} x^{4} - 2 \, A a^{2} + 3 \, {\left (B a^{2} - 3 \, A a b\right )} x^{2}}{4 \, {\left (a^{3} b^{2} x^{6} + 2 \, a^{4} b x^{4} + a^{5} x^{2}\right )}} - \frac {{\left (B a - 3 \, A b\right )} \log \left (b x^{2} + a\right )}{2 \, a^{4}} + \frac {{\left (B a - 3 \, A b\right )} \log \left (x^{2}\right )}{2 \, a^{4}} \]
1/4*(2*(B*a*b - 3*A*b^2)*x^4 - 2*A*a^2 + 3*(B*a^2 - 3*A*a*b)*x^2)/(a^3*b^2 *x^6 + 2*a^4*b*x^4 + a^5*x^2) - 1/2*(B*a - 3*A*b)*log(b*x^2 + a)/a^4 + 1/2 *(B*a - 3*A*b)*log(x^2)/a^4
Time = 0.28 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.37 \[ \int \frac {A+B x^2}{x^3 \left (a+b x^2\right )^3} \, dx=\frac {{\left (B a - 3 \, A b\right )} \log \left (x^{2}\right )}{2 \, a^{4}} - \frac {{\left (B a b - 3 \, A b^{2}\right )} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, a^{4} b} + \frac {3 \, B a b^{2} x^{4} - 9 \, A b^{3} x^{4} + 8 \, B a^{2} b x^{2} - 22 \, A a b^{2} x^{2} + 6 \, B a^{3} - 14 \, A a^{2} b}{4 \, {\left (b x^{2} + a\right )}^{2} a^{4}} - \frac {B a x^{2} - 3 \, A b x^{2} + A a}{2 \, a^{4} x^{2}} \]
1/2*(B*a - 3*A*b)*log(x^2)/a^4 - 1/2*(B*a*b - 3*A*b^2)*log(abs(b*x^2 + a)) /(a^4*b) + 1/4*(3*B*a*b^2*x^4 - 9*A*b^3*x^4 + 8*B*a^2*b*x^2 - 22*A*a*b^2*x ^2 + 6*B*a^3 - 14*A*a^2*b)/((b*x^2 + a)^2*a^4) - 1/2*(B*a*x^2 - 3*A*b*x^2 + A*a)/(a^4*x^2)
Time = 0.11 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.06 \[ \int \frac {A+B x^2}{x^3 \left (a+b x^2\right )^3} \, dx=\frac {\ln \left (b\,x^2+a\right )\,\left (3\,A\,b-B\,a\right )}{2\,a^4}-\frac {\frac {A}{2\,a}+\frac {3\,x^2\,\left (3\,A\,b-B\,a\right )}{4\,a^2}+\frac {b\,x^4\,\left (3\,A\,b-B\,a\right )}{2\,a^3}}{a^2\,x^2+2\,a\,b\,x^4+b^2\,x^6}-\frac {\ln \left (x\right )\,\left (3\,A\,b-B\,a\right )}{a^4} \]