Integrand size = 22, antiderivative size = 81 \[ \int \frac {\left (a+b x^2\right )^2}{x^3 \left (c+d x^2\right )^2} \, dx=-\frac {a^2}{2 c^2 x^2}-\frac {(b c-a d)^2}{2 c^2 d \left (c+d x^2\right )}+\frac {2 a (b c-a d) \log (x)}{c^3}-\frac {a (b c-a d) \log \left (c+d x^2\right )}{c^3} \] Output:
-1/2*a^2/c^2/x^2-1/2*(-a*d+b*c)^2/c^2/d/(d*x^2+c)+2*a*(-a*d+b*c)*ln(x)/c^3 -a*(-a*d+b*c)*ln(d*x^2+c)/c^3
Time = 0.06 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.89 \[ \int \frac {\left (a+b x^2\right )^2}{x^3 \left (c+d x^2\right )^2} \, dx=-\frac {\frac {a^2 c}{x^2}+\frac {c (b c-a d)^2}{d \left (c+d x^2\right )}+4 a (-b c+a d) \log (x)-2 a (-b c+a d) \log \left (c+d x^2\right )}{2 c^3} \] Input:
Integrate[(a + b*x^2)^2/(x^3*(c + d*x^2)^2),x]
Output:
-1/2*((a^2*c)/x^2 + (c*(b*c - a*d)^2)/(d*(c + d*x^2)) + 4*a*(-(b*c) + a*d) *Log[x] - 2*a*(-(b*c) + a*d)*Log[c + d*x^2])/c^3
Time = 0.24 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.02, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {354, 99, 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.
\(\displaystyle \int \frac {\left (a+b x^2\right )^2}{x^3 \left (c+d x^2\right )^2} \, dx\) |
\(\Big \downarrow \) 354 |
\(\displaystyle \frac {1}{2} \int \frac {\left (b x^2+a\right )^2}{x^4 \left (d x^2+c\right )^2}dx^2\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \frac {1}{2} \int \left (\frac {a^2}{c^2 x^4}+\frac {2 d (a d-b c) a}{c^3 \left (d x^2+c\right )}-\frac {2 (a d-b c) a}{c^3 x^2}+\frac {(b c-a d)^2}{c^2 \left (d x^2+c\right )^2}\right )dx^2\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} \left (-\frac {a^2}{c^2 x^2}+\frac {2 a \log \left (x^2\right ) (b c-a d)}{c^3}-\frac {2 a (b c-a d) \log \left (c+d x^2\right )}{c^3}-\frac {(b c-a d)^2}{c^2 d \left (c+d x^2\right )}\right )\) |
Input:
Int[(a + b*x^2)^2/(x^3*(c + d*x^2)^2),x]
Output:
(-(a^2/(c^2*x^2)) - (b*c - a*d)^2/(c^2*d*(c + d*x^2)) + (2*a*(b*c - a*d)*L og[x^2])/c^3 - (2*a*(b*c - a*d)*Log[c + d*x^2])/c^3)/2
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] && (IntegerQ[p] | | (GtQ[m, 0] && GeQ[n, -1]))
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 = 0.44 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.95
method | result | size |
default | \(-\frac {a^{2}}{2 c^{2} x^{2}}-\frac {2 \left (a d -b c \right ) a \ln \left (x \right )}{c^{3}}+\frac {\left (a d -b c \right ) \left (-\frac {\left (a d -b c \right ) c}{d \left (x^{2} d +c \right )}+2 a \ln \left (x^{2} d +c \right )\right )}{2 c^{3}}\) | \(77\) |
norman | \(\frac {-\frac {a^{2}}{2 c}+\frac {\left (2 a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) x^{4}}{2 c^{3}}}{x^{2} \left (x^{2} d +c \right )}+\frac {\left (a d -b c \right ) a \ln \left (x^{2} d +c \right )}{c^{3}}-\frac {2 \left (a d -b c \right ) a \ln \left (x \right )}{c^{3}}\) | \(91\) |
risch | \(\frac {-\frac {\left (2 a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) x^{2}}{2 c^{2} d}-\frac {a^{2}}{2 c}}{x^{2} \left (x^{2} d +c \right )}-\frac {2 a^{2} \ln \left (x \right ) d}{c^{3}}+\frac {2 a \ln \left (x \right ) b}{c^{2}}+\frac {a^{2} \ln \left (-x^{2} d -c \right ) d}{c^{3}}-\frac {a \ln \left (-x^{2} d -c \right ) b}{c^{2}}\) | \(114\) |
parallelrisch | \(-\frac {4 \ln \left (x \right ) x^{4} a^{2} d^{2}-4 \ln \left (x \right ) x^{4} a b c d -2 \ln \left (x^{2} d +c \right ) x^{4} a^{2} d^{2}+2 \ln \left (x^{2} d +c \right ) x^{4} a b c d -2 a^{2} d^{2} x^{4}+2 a b c d \,x^{4}-b^{2} c^{2} x^{4}+4 \ln \left (x \right ) x^{2} a^{2} c d -4 \ln \left (x \right ) x^{2} a b \,c^{2}-2 \ln \left (x^{2} d +c \right ) x^{2} a^{2} c d +2 \ln \left (x^{2} d +c \right ) x^{2} a b \,c^{2}+a^{2} c^{2}}{2 c^{3} x^{2} \left (x^{2} d +c \right )}\) | \(177\) |
Input:
int((b*x^2+a)^2/x^3/(d*x^2+c)^2,x,method=_RETURNVERBOSE)
Output:
-1/2*a^2/c^2/x^2-2*(a*d-b*c)*a/c^3*ln(x)+1/2/c^3*(a*d-b*c)*(-(a*d-b*c)*c/d /(d*x^2+c)+2*a*ln(d*x^2+c))
Leaf count of result is larger than twice the leaf count of optimal. 159 vs. \(2 (77) = 154\).
Time = 0.09 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.96 \[ \int \frac {\left (a+b x^2\right )^2}{x^3 \left (c+d x^2\right )^2} \, dx=-\frac {a^{2} c^{2} d + {\left (b^{2} c^{3} - 2 \, a b c^{2} d + 2 \, a^{2} c d^{2}\right )} x^{2} + 2 \, {\left ({\left (a b c d^{2} - a^{2} d^{3}\right )} x^{4} + {\left (a b c^{2} d - a^{2} c d^{2}\right )} x^{2}\right )} \log \left (d x^{2} + c\right ) - 4 \, {\left ({\left (a b c d^{2} - a^{2} d^{3}\right )} x^{4} + {\left (a b c^{2} d - a^{2} c d^{2}\right )} x^{2}\right )} \log \left (x\right )}{2 \, {\left (c^{3} d^{2} x^{4} + c^{4} d x^{2}\right )}} \] Input:
integrate((b*x^2+a)^2/x^3/(d*x^2+c)^2,x, algorithm="fricas")
Output:
-1/2*(a^2*c^2*d + (b^2*c^3 - 2*a*b*c^2*d + 2*a^2*c*d^2)*x^2 + 2*((a*b*c*d^ 2 - a^2*d^3)*x^4 + (a*b*c^2*d - a^2*c*d^2)*x^2)*log(d*x^2 + c) - 4*((a*b*c *d^2 - a^2*d^3)*x^4 + (a*b*c^2*d - a^2*c*d^2)*x^2)*log(x))/(c^3*d^2*x^4 + c^4*d*x^2)
Time = 0.68 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.14 \[ \int \frac {\left (a+b x^2\right )^2}{x^3 \left (c+d x^2\right )^2} \, dx=- \frac {2 a \left (a d - b c\right ) \log {\left (x \right )}}{c^{3}} + \frac {a \left (a d - b c\right ) \log {\left (\frac {c}{d} + x^{2} \right )}}{c^{3}} + \frac {- a^{2} c d + x^{2} \left (- 2 a^{2} d^{2} + 2 a b c d - b^{2} c^{2}\right )}{2 c^{3} d x^{2} + 2 c^{2} d^{2} x^{4}} \] Input:
integrate((b*x**2+a)**2/x**3/(d*x**2+c)**2,x)
Output:
-2*a*(a*d - b*c)*log(x)/c**3 + a*(a*d - b*c)*log(c/d + x**2)/c**3 + (-a**2 *c*d + x**2*(-2*a**2*d**2 + 2*a*b*c*d - b**2*c**2))/(2*c**3*d*x**2 + 2*c** 2*d**2*x**4)
Time = 0.03 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.23 \[ \int \frac {\left (a+b x^2\right )^2}{x^3 \left (c+d x^2\right )^2} \, dx=-\frac {a^{2} c d + {\left (b^{2} c^{2} - 2 \, a b c d + 2 \, a^{2} d^{2}\right )} x^{2}}{2 \, {\left (c^{2} d^{2} x^{4} + c^{3} d x^{2}\right )}} - \frac {{\left (a b c - a^{2} d\right )} \log \left (d x^{2} + c\right )}{c^{3}} + \frac {{\left (a b c - a^{2} d\right )} \log \left (x^{2}\right )}{c^{3}} \] Input:
integrate((b*x^2+a)^2/x^3/(d*x^2+c)^2,x, algorithm="maxima")
Output:
-1/2*(a^2*c*d + (b^2*c^2 - 2*a*b*c*d + 2*a^2*d^2)*x^2)/(c^2*d^2*x^4 + c^3* d*x^2) - (a*b*c - a^2*d)*log(d*x^2 + c)/c^3 + (a*b*c - a^2*d)*log(x^2)/c^3
Time = 0.12 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.35 \[ \int \frac {\left (a+b x^2\right )^2}{x^3 \left (c+d x^2\right )^2} \, dx=\frac {{\left (a b c - a^{2} d\right )} \log \left (x^{2}\right )}{c^{3}} - \frac {{\left (a b c d - a^{2} d^{2}\right )} \log \left ({\left | d x^{2} + c \right |}\right )}{c^{3} d} - \frac {b^{2} c^{2} x^{2} - 2 \, a b c d x^{2} + 2 \, a^{2} d^{2} x^{2} + a^{2} c d}{2 \, {\left (d x^{4} + c x^{2}\right )} c^{2} d} \] Input:
integrate((b*x^2+a)^2/x^3/(d*x^2+c)^2,x, algorithm="giac")
Output:
(a*b*c - a^2*d)*log(x^2)/c^3 - (a*b*c*d - a^2*d^2)*log(abs(d*x^2 + c))/(c^ 3*d) - 1/2*(b^2*c^2*x^2 - 2*a*b*c*d*x^2 + 2*a^2*d^2*x^2 + a^2*c*d)/((d*x^4 + c*x^2)*c^2*d)
Time = 0.10 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.23 \[ \int \frac {\left (a+b x^2\right )^2}{x^3 \left (c+d x^2\right )^2} \, dx=\frac {\ln \left (d\,x^2+c\right )\,\left (a^2\,d-a\,b\,c\right )}{c^3}-\frac {\frac {a^2}{2\,c}+\frac {x^2\,\left (2\,a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}{2\,c^2\,d}}{d\,x^4+c\,x^2}-\frac {\ln \left (x\right )\,\left (2\,a^2\,d-2\,a\,b\,c\right )}{c^3} \] Input:
int((a + b*x^2)^2/(x^3*(c + d*x^2)^2),x)
Output:
(log(c + d*x^2)*(a^2*d - a*b*c))/c^3 - (a^2/(2*c) + (x^2*(2*a^2*d^2 + b^2* c^2 - 2*a*b*c*d))/(2*c^2*d))/(c*x^2 + d*x^4) - (log(x)*(2*a^2*d - 2*a*b*c) )/c^3
Time = 0.25 (sec) , antiderivative size = 176, normalized size of antiderivative = 2.17 \[ \int \frac {\left (a+b x^2\right )^2}{x^3 \left (c+d x^2\right )^2} \, dx=\frac {2 \,\mathrm {log}\left (d \,x^{2}+c \right ) a^{2} c d \,x^{2}+2 \,\mathrm {log}\left (d \,x^{2}+c \right ) a^{2} d^{2} x^{4}-2 \,\mathrm {log}\left (d \,x^{2}+c \right ) a b \,c^{2} x^{2}-2 \,\mathrm {log}\left (d \,x^{2}+c \right ) a b c d \,x^{4}-4 \,\mathrm {log}\left (x \right ) a^{2} c d \,x^{2}-4 \,\mathrm {log}\left (x \right ) a^{2} d^{2} x^{4}+4 \,\mathrm {log}\left (x \right ) a b \,c^{2} x^{2}+4 \,\mathrm {log}\left (x \right ) a b c d \,x^{4}-a^{2} c^{2}+2 a^{2} d^{2} x^{4}-2 a b c d \,x^{4}+b^{2} c^{2} x^{4}}{2 c^{3} x^{2} \left (d \,x^{2}+c \right )} \] Input:
int((b*x^2+a)^2/x^3/(d*x^2+c)^2,x)
Output:
(2*log(c + d*x**2)*a**2*c*d*x**2 + 2*log(c + d*x**2)*a**2*d**2*x**4 - 2*lo g(c + d*x**2)*a*b*c**2*x**2 - 2*log(c + d*x**2)*a*b*c*d*x**4 - 4*log(x)*a* *2*c*d*x**2 - 4*log(x)*a**2*d**2*x**4 + 4*log(x)*a*b*c**2*x**2 + 4*log(x)* a*b*c*d*x**4 - a**2*c**2 + 2*a**2*d**2*x**4 - 2*a*b*c*d*x**4 + b**2*c**2*x **4)/(2*c**3*x**2*(c + d*x**2))