Integrand size = 22, antiderivative size = 75 \[ \int \frac {\left (a+b x^2\right )^2}{x^5 \left (c+d x^2\right )} \, dx=-\frac {a^2}{4 c x^4}-\frac {a (2 b c-a d)}{2 c^2 x^2}+\frac {(b c-a d)^2 \log (x)}{c^3}-\frac {(b c-a d)^2 \log \left (c+d x^2\right )}{2 c^3} \] Output:
-1/4*a^2/c/x^4-1/2*a*(-a*d+2*b*c)/c^2/x^2+(-a*d+b*c)^2*ln(x)/c^3-1/2*(-a*d +b*c)^2*ln(d*x^2+c)/c^3
Time = 0.03 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.96 \[ \int \frac {\left (a+b x^2\right )^2}{x^5 \left (c+d x^2\right )} \, dx=-\frac {a c \left (a c+4 b c x^2-2 a d x^2\right )-4 (b c-a d)^2 x^4 \log (x)+2 (b c-a d)^2 x^4 \log \left (c+d x^2\right )}{4 c^3 x^4} \] Input:
Integrate[(a + b*x^2)^2/(x^5*(c + d*x^2)),x]
Output:
-1/4*(a*c*(a*c + 4*b*c*x^2 - 2*a*d*x^2) - 4*(b*c - a*d)^2*x^4*Log[x] + 2*( b*c - a*d)^2*x^4*Log[c + d*x^2])/(c^3*x^4)
Time = 0.24 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.03, 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.
\(\displaystyle \int \frac {\left (a+b x^2\right )^2}{x^5 \left (c+d x^2\right )} \, dx\) |
\(\Big \downarrow \) 354 |
\(\displaystyle \frac {1}{2} \int \frac {\left (b x^2+a\right )^2}{x^6 \left (d x^2+c\right )}dx^2\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \frac {1}{2} \int \left (\frac {a^2}{c x^6}-\frac {(a d-2 b c) a}{c^2 x^4}-\frac {d (b c-a d)^2}{c^3 \left (d x^2+c\right )}+\frac {(b c-a d)^2}{c^3 x^2}\right )dx^2\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} \left (-\frac {a^2}{2 c x^4}+\frac {\log \left (x^2\right ) (b c-a d)^2}{c^3}-\frac {(b c-a d)^2 \log \left (c+d x^2\right )}{c^3}-\frac {a (2 b c-a d)}{c^2 x^2}\right )\) |
Input:
Int[(a + b*x^2)^2/(x^5*(c + d*x^2)),x]
Output:
(-1/2*a^2/(c*x^4) - (a*(2*b*c - a*d))/(c^2*x^2) + ((b*c - a*d)^2*Log[x^2]) /c^3 - ((b*c - a*d)^2*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 = 0.42 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.21
method | result | size |
default | \(-\frac {a^{2}}{4 c \,x^{4}}+\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \ln \left (x \right )}{c^{3}}+\frac {a \left (a d -2 b c \right )}{2 c^{2} x^{2}}-\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \ln \left (x^{2} d +c \right )}{2 c^{3}}\) | \(91\) |
norman | \(\frac {-\frac {a^{2}}{4 c}+\frac {a \left (a d -2 b c \right ) x^{2}}{2 c^{2}}}{x^{4}}+\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \ln \left (x \right )}{c^{3}}-\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \ln \left (x^{2} d +c \right )}{2 c^{3}}\) | \(93\) |
risch | \(\frac {-\frac {a^{2}}{4 c}+\frac {a \left (a d -2 b c \right ) x^{2}}{2 c^{2}}}{x^{4}}+\frac {\ln \left (x \right ) a^{2} d^{2}}{c^{3}}-\frac {2 \ln \left (x \right ) a b d}{c^{2}}+\frac {\ln \left (x \right ) b^{2}}{c}-\frac {\ln \left (x^{2} d +c \right ) a^{2} d^{2}}{2 c^{3}}+\frac {\ln \left (x^{2} d +c \right ) a b d}{c^{2}}-\frac {\ln \left (x^{2} d +c \right ) b^{2}}{2 c}\) | \(113\) |
parallelrisch | \(\frac {4 \ln \left (x \right ) x^{4} a^{2} d^{2}-8 \ln \left (x \right ) x^{4} a b c d +4 \ln \left (x \right ) x^{4} b^{2} c^{2}-2 \ln \left (x^{2} d +c \right ) x^{4} a^{2} d^{2}+4 \ln \left (x^{2} d +c \right ) x^{4} a b c d -2 \ln \left (x^{2} d +c \right ) x^{4} b^{2} c^{2}+2 a^{2} c d \,x^{2}-4 a b \,c^{2} x^{2}-a^{2} c^{2}}{4 c^{3} x^{4}}\) | \(130\) |
Input:
int((b*x^2+a)^2/x^5/(d*x^2+c),x,method=_RETURNVERBOSE)
Output:
-1/4*a^2/c/x^4+(a^2*d^2-2*a*b*c*d+b^2*c^2)/c^3*ln(x)+1/2*a*(a*d-2*b*c)/c^2 /x^2-1/2*(a^2*d^2-2*a*b*c*d+b^2*c^2)/c^3*ln(d*x^2+c)
Time = 0.12 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.31 \[ \int \frac {\left (a+b x^2\right )^2}{x^5 \left (c+d x^2\right )} \, dx=-\frac {2 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x^{4} \log \left (d x^{2} + c\right ) - 4 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x^{4} \log \left (x\right ) + a^{2} c^{2} + 2 \, {\left (2 \, a b c^{2} - a^{2} c d\right )} x^{2}}{4 \, c^{3} x^{4}} \] Input:
integrate((b*x^2+a)^2/x^5/(d*x^2+c),x, algorithm="fricas")
Output:
-1/4*(2*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*x^4*log(d*x^2 + c) - 4*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*x^4*log(x) + a^2*c^2 + 2*(2*a*b*c^2 - a^2*c*d)*x^2)/( c^3*x^4)
Time = 0.68 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.88 \[ \int \frac {\left (a+b x^2\right )^2}{x^5 \left (c+d x^2\right )} \, dx=\frac {- a^{2} c + x^{2} \cdot \left (2 a^{2} d - 4 a b c\right )}{4 c^{2} x^{4}} + \frac {\left (a d - b c\right )^{2} \log {\left (x \right )}}{c^{3}} - \frac {\left (a d - b c\right )^{2} \log {\left (\frac {c}{d} + x^{2} \right )}}{2 c^{3}} \] Input:
integrate((b*x**2+a)**2/x**5/(d*x**2+c),x)
Output:
(-a**2*c + x**2*(2*a**2*d - 4*a*b*c))/(4*c**2*x**4) + (a*d - b*c)**2*log(x )/c**3 - (a*d - b*c)**2*log(c/d + x**2)/(2*c**3)
Time = 0.04 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.28 \[ \int \frac {\left (a+b x^2\right )^2}{x^5 \left (c+d x^2\right )} \, dx=-\frac {{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (d x^{2} + c\right )}{2 \, c^{3}} + \frac {{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (x^{2}\right )}{2 \, c^{3}} - \frac {a^{2} c + 2 \, {\left (2 \, a b c - a^{2} d\right )} x^{2}}{4 \, c^{2} x^{4}} \] Input:
integrate((b*x^2+a)^2/x^5/(d*x^2+c),x, algorithm="maxima")
Output:
-1/2*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*log(d*x^2 + c)/c^3 + 1/2*(b^2*c^2 - 2 *a*b*c*d + a^2*d^2)*log(x^2)/c^3 - 1/4*(a^2*c + 2*(2*a*b*c - a^2*d)*x^2)/( c^2*x^4)
Leaf count of result is larger than twice the leaf count of optimal. 139 vs. \(2 (69) = 138\).
Time = 0.13 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.85 \[ \int \frac {\left (a+b x^2\right )^2}{x^5 \left (c+d x^2\right )} \, dx=\frac {{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (x^{2}\right )}{2 \, c^{3}} - \frac {{\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} \log \left ({\left | d x^{2} + c \right |}\right )}{2 \, c^{3} d} - \frac {3 \, b^{2} c^{2} x^{4} - 6 \, a b c d x^{4} + 3 \, a^{2} d^{2} x^{4} + 4 \, a b c^{2} x^{2} - 2 \, a^{2} c d x^{2} + a^{2} c^{2}}{4 \, c^{3} x^{4}} \] Input:
integrate((b*x^2+a)^2/x^5/(d*x^2+c),x, algorithm="giac")
Output:
1/2*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*log(x^2)/c^3 - 1/2*(b^2*c^2*d - 2*a*b* c*d^2 + a^2*d^3)*log(abs(d*x^2 + c))/(c^3*d) - 1/4*(3*b^2*c^2*x^4 - 6*a*b* c*d*x^4 + 3*a^2*d^2*x^4 + 4*a*b*c^2*x^2 - 2*a^2*c*d*x^2 + a^2*c^2)/(c^3*x^ 4)
Time = 0.57 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.24 \[ \int \frac {\left (a+b x^2\right )^2}{x^5 \left (c+d x^2\right )} \, dx=\frac {\ln \left (x\right )\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}{c^3}-\frac {\frac {a^2}{4\,c}-\frac {a\,x^2\,\left (a\,d-2\,b\,c\right )}{2\,c^2}}{x^4}-\frac {\ln \left (d\,x^2+c\right )\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}{2\,c^3} \] Input:
int((a + b*x^2)^2/(x^5*(c + d*x^2)),x)
Output:
(log(x)*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d))/c^3 - (a^2/(4*c) - (a*x^2*(a*d - 2*b*c))/(2*c^2))/x^4 - (log(c + d*x^2)*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d))/(2 *c^3)
Time = 0.20 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.72 \[ \int \frac {\left (a+b x^2\right )^2}{x^5 \left (c+d x^2\right )} \, dx=\frac {-2 \,\mathrm {log}\left (d \,x^{2}+c \right ) a^{2} d^{2} x^{4}+4 \,\mathrm {log}\left (d \,x^{2}+c \right ) a b c d \,x^{4}-2 \,\mathrm {log}\left (d \,x^{2}+c \right ) b^{2} c^{2} x^{4}+4 \,\mathrm {log}\left (x \right ) a^{2} d^{2} x^{4}-8 \,\mathrm {log}\left (x \right ) a b c d \,x^{4}+4 \,\mathrm {log}\left (x \right ) b^{2} c^{2} x^{4}-a^{2} c^{2}+2 a^{2} c d \,x^{2}-4 a b \,c^{2} x^{2}}{4 c^{3} x^{4}} \] Input:
int((b*x^2+a)^2/x^5/(d*x^2+c),x)
Output:
( - 2*log(c + d*x**2)*a**2*d**2*x**4 + 4*log(c + d*x**2)*a*b*c*d*x**4 - 2* log(c + d*x**2)*b**2*c**2*x**4 + 4*log(x)*a**2*d**2*x**4 - 8*log(x)*a*b*c* d*x**4 + 4*log(x)*b**2*c**2*x**4 - a**2*c**2 + 2*a**2*c*d*x**2 - 4*a*b*c** 2*x**2)/(4*c**3*x**4)