Integrand size = 22, antiderivative size = 80 \[ \int \frac {\left (c+d x^2\right )^2}{x^3 \left (a+b x^2\right )^2} \, dx=-\frac {c^2}{2 a^2 x^2}-\frac {(b c-a d)^2}{2 a^2 b \left (a+b x^2\right )}-\frac {2 c (b c-a d) \log (x)}{a^3}+\frac {c (b c-a d) \log \left (a+b x^2\right )}{a^3} \] Output:
-1/2*c^2/a^2/x^2-1/2*(-a*d+b*c)^2/a^2/b/(b*x^2+a)-2*c*(-a*d+b*c)*ln(x)/a^3 +c*(-a*d+b*c)*ln(b*x^2+a)/a^3
Time = 0.06 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.90 \[ \int \frac {\left (c+d x^2\right )^2}{x^3 \left (a+b x^2\right )^2} \, dx=-\frac {\frac {a c^2}{x^2}+\frac {a (b c-a d)^2}{b \left (a+b x^2\right )}+4 c (b c-a d) \log (x)-2 c (b c-a d) \log \left (a+b x^2\right )}{2 a^3} \] Input:
Integrate[(c + d*x^2)^2/(x^3*(a + b*x^2)^2),x]
Output:
-1/2*((a*c^2)/x^2 + (a*(b*c - a*d)^2)/(b*(a + b*x^2)) + 4*c*(b*c - a*d)*Lo g[x] - 2*c*(b*c - a*d)*Log[a + b*x^2])/a^3
Time = 0.25 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.04, 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.
\(\displaystyle \int \frac {\left (c+d x^2\right )^2}{x^3 \left (a+b x^2\right )^2} \, dx\) |
\(\Big \downarrow \) 354 |
\(\displaystyle \frac {1}{2} \int \frac {\left (d x^2+c\right )^2}{x^4 \left (b x^2+a\right )^2}dx^2\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \frac {1}{2} \int \left (\frac {c^2}{a^2 x^4}-\frac {2 b (a d-b c) c}{a^3 \left (b x^2+a\right )}+\frac {2 (a d-b c) c}{a^3 x^2}+\frac {(a d-b c)^2}{a^2 \left (b x^2+a\right )^2}\right )dx^2\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} \left (-\frac {2 c \log \left (x^2\right ) (b c-a d)}{a^3}+\frac {2 c (b c-a d) \log \left (a+b x^2\right )}{a^3}-\frac {(b c-a d)^2}{a^2 b \left (a+b x^2\right )}-\frac {c^2}{a^2 x^2}\right )\) |
Input:
Int[(c + d*x^2)^2/(x^3*(a + b*x^2)^2),x]
Output:
(-(c^2/(a^2*x^2)) - (b*c - a*d)^2/(a^2*b*(a + b*x^2)) - (2*c*(b*c - a*d)*L og[x^2])/a^3 + (2*c*(b*c - a*d)*Log[a + b*x^2])/a^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 = 0.47 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.96
method | result | size |
default | \(\frac {\left (a d -b c \right ) \left (-\frac {\left (a d -b c \right ) a}{b \left (b \,x^{2}+a \right )}-2 c \ln \left (b \,x^{2}+a \right )\right )}{2 a^{3}}-\frac {c^{2}}{2 a^{2} x^{2}}+\frac {2 \left (a d -b c \right ) c \ln \left (x \right )}{a^{3}}\) | \(77\) |
norman | \(\frac {-\frac {c^{2}}{2 a}+\frac {\left (a^{2} d^{2}-2 a b c d +2 b^{2} c^{2}\right ) x^{4}}{2 a^{3}}}{x^{2} \left (b \,x^{2}+a \right )}+\frac {2 \left (a d -b c \right ) c \ln \left (x \right )}{a^{3}}-\frac {\left (a d -b c \right ) c \ln \left (b \,x^{2}+a \right )}{a^{3}}\) | \(92\) |
risch | \(\frac {-\frac {\left (a^{2} d^{2}-2 a b c d +2 b^{2} c^{2}\right ) x^{2}}{2 a^{2} b}-\frac {c^{2}}{2 a}}{x^{2} \left (b \,x^{2}+a \right )}+\frac {2 c \ln \left (x \right ) d}{a^{2}}-\frac {2 c^{2} \ln \left (x \right ) b}{a^{3}}-\frac {c \ln \left (b \,x^{2}+a \right ) d}{a^{2}}+\frac {c^{2} \ln \left (b \,x^{2}+a \right ) b}{a^{3}}\) | \(108\) |
parallelrisch | \(\frac {4 \ln \left (x \right ) x^{4} a b c d -4 \ln \left (x \right ) x^{4} b^{2} c^{2}-2 \ln \left (b \,x^{2}+a \right ) x^{4} a b c d +2 \ln \left (b \,x^{2}+a \right ) x^{4} b^{2} c^{2}+a^{2} d^{2} x^{4}-2 a b c d \,x^{4}+2 b^{2} c^{2} x^{4}+4 \ln \left (x \right ) x^{2} a^{2} c d -4 \ln \left (x \right ) x^{2} a b \,c^{2}-2 \ln \left (b \,x^{2}+a \right ) x^{2} a^{2} c d +2 \ln \left (b \,x^{2}+a \right ) x^{2} a b \,c^{2}-a^{2} c^{2}}{2 a^{3} x^{2} \left (b \,x^{2}+a \right )}\) | \(177\) |
Input:
int((d*x^2+c)^2/x^3/(b*x^2+a)^2,x,method=_RETURNVERBOSE)
Output:
1/2/a^3*(a*d-b*c)*(-(a*d-b*c)*a/b/(b*x^2+a)-2*c*ln(b*x^2+a))-1/2*c^2/a^2/x ^2+2*(a*d-b*c)*c/a^3*ln(x)
Leaf count of result is larger than twice the leaf count of optimal. 159 vs. \(2 (76) = 152\).
Time = 0.07 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.99 \[ \int \frac {\left (c+d x^2\right )^2}{x^3 \left (a+b x^2\right )^2} \, dx=-\frac {a^{2} b c^{2} + {\left (2 \, a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}\right )} x^{2} - 2 \, {\left ({\left (b^{3} c^{2} - a b^{2} c d\right )} x^{4} + {\left (a b^{2} c^{2} - a^{2} b c d\right )} x^{2}\right )} \log \left (b x^{2} + a\right ) + 4 \, {\left ({\left (b^{3} c^{2} - a b^{2} c d\right )} x^{4} + {\left (a b^{2} c^{2} - a^{2} b c d\right )} x^{2}\right )} \log \left (x\right )}{2 \, {\left (a^{3} b^{2} x^{4} + a^{4} b x^{2}\right )}} \] Input:
integrate((d*x^2+c)^2/x^3/(b*x^2+a)^2,x, algorithm="fricas")
Output:
-1/2*(a^2*b*c^2 + (2*a*b^2*c^2 - 2*a^2*b*c*d + a^3*d^2)*x^2 - 2*((b^3*c^2 - a*b^2*c*d)*x^4 + (a*b^2*c^2 - a^2*b*c*d)*x^2)*log(b*x^2 + a) + 4*((b^3*c ^2 - a*b^2*c*d)*x^4 + (a*b^2*c^2 - a^2*b*c*d)*x^2)*log(x))/(a^3*b^2*x^4 + a^4*b*x^2)
Time = 0.68 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.15 \[ \int \frac {\left (c+d x^2\right )^2}{x^3 \left (a+b x^2\right )^2} \, dx=\frac {- a b c^{2} + x^{2} \left (- a^{2} d^{2} + 2 a b c d - 2 b^{2} c^{2}\right )}{2 a^{3} b x^{2} + 2 a^{2} b^{2} x^{4}} + \frac {2 c \left (a d - b c\right ) \log {\left (x \right )}}{a^{3}} - \frac {c \left (a d - b c\right ) \log {\left (\frac {a}{b} + x^{2} \right )}}{a^{3}} \] Input:
integrate((d*x**2+c)**2/x**3/(b*x**2+a)**2,x)
Output:
(-a*b*c**2 + x**2*(-a**2*d**2 + 2*a*b*c*d - 2*b**2*c**2))/(2*a**3*b*x**2 + 2*a**2*b**2*x**4) + 2*c*(a*d - b*c)*log(x)/a**3 - c*(a*d - b*c)*log(a/b + x**2)/a**3
Time = 0.03 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.25 \[ \int \frac {\left (c+d x^2\right )^2}{x^3 \left (a+b x^2\right )^2} \, dx=-\frac {a b c^{2} + {\left (2 \, b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x^{2}}{2 \, {\left (a^{2} b^{2} x^{4} + a^{3} b x^{2}\right )}} + \frac {{\left (b c^{2} - a c d\right )} \log \left (b x^{2} + a\right )}{a^{3}} - \frac {{\left (b c^{2} - a c d\right )} \log \left (x^{2}\right )}{a^{3}} \] Input:
integrate((d*x^2+c)^2/x^3/(b*x^2+a)^2,x, algorithm="maxima")
Output:
-1/2*(a*b*c^2 + (2*b^2*c^2 - 2*a*b*c*d + a^2*d^2)*x^2)/(a^2*b^2*x^4 + a^3* b*x^2) + (b*c^2 - a*c*d)*log(b*x^2 + a)/a^3 - (b*c^2 - a*c*d)*log(x^2)/a^3
Time = 0.12 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.36 \[ \int \frac {\left (c+d x^2\right )^2}{x^3 \left (a+b x^2\right )^2} \, dx=-\frac {{\left (b c^{2} - a c d\right )} \log \left (x^{2}\right )}{a^{3}} + \frac {{\left (b^{2} c^{2} - a b c d\right )} \log \left ({\left | b x^{2} + a \right |}\right )}{a^{3} b} - \frac {2 \, b^{2} c^{2} x^{2} - 2 \, a b c d x^{2} + a^{2} d^{2} x^{2} + a b c^{2}}{2 \, {\left (b x^{4} + a x^{2}\right )} a^{2} b} \] Input:
integrate((d*x^2+c)^2/x^3/(b*x^2+a)^2,x, algorithm="giac")
Output:
-(b*c^2 - a*c*d)*log(x^2)/a^3 + (b^2*c^2 - a*b*c*d)*log(abs(b*x^2 + a))/(a ^3*b) - 1/2*(2*b^2*c^2*x^2 - 2*a*b*c*d*x^2 + a^2*d^2*x^2 + a*b*c^2)/((b*x^ 4 + a*x^2)*a^2*b)
Time = 0.55 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.25 \[ \int \frac {\left (c+d x^2\right )^2}{x^3 \left (a+b x^2\right )^2} \, dx=\frac {\ln \left (b\,x^2+a\right )\,\left (b\,c^2-a\,c\,d\right )}{a^3}-\frac {\frac {c^2}{2\,a}+\frac {x^2\,\left (a^2\,d^2-2\,a\,b\,c\,d+2\,b^2\,c^2\right )}{2\,a^2\,b}}{b\,x^4+a\,x^2}-\frac {\ln \left (x\right )\,\left (2\,b\,c^2-2\,a\,c\,d\right )}{a^3} \] Input:
int((c + d*x^2)^2/(x^3*(a + b*x^2)^2),x)
Output:
(log(a + b*x^2)*(b*c^2 - a*c*d))/a^3 - (c^2/(2*a) + (x^2*(a^2*d^2 + 2*b^2* c^2 - 2*a*b*c*d))/(2*a^2*b))/(a*x^2 + b*x^4) - (log(x)*(2*b*c^2 - 2*a*c*d) )/a^3
Time = 0.21 (sec) , antiderivative size = 176, normalized size of antiderivative = 2.20 \[ \int \frac {\left (c+d x^2\right )^2}{x^3 \left (a+b x^2\right )^2} \, dx=\frac {-2 \,\mathrm {log}\left (b \,x^{2}+a \right ) a^{2} c d \,x^{2}+2 \,\mathrm {log}\left (b \,x^{2}+a \right ) a b \,c^{2} x^{2}-2 \,\mathrm {log}\left (b \,x^{2}+a \right ) a b c d \,x^{4}+2 \,\mathrm {log}\left (b \,x^{2}+a \right ) b^{2} c^{2} x^{4}+4 \,\mathrm {log}\left (x \right ) a^{2} c d \,x^{2}-4 \,\mathrm {log}\left (x \right ) a b \,c^{2} x^{2}+4 \,\mathrm {log}\left (x \right ) a b c d \,x^{4}-4 \,\mathrm {log}\left (x \right ) b^{2} c^{2} x^{4}-a^{2} c^{2}+a^{2} d^{2} x^{4}-2 a b c d \,x^{4}+2 b^{2} c^{2} x^{4}}{2 a^{3} x^{2} \left (b \,x^{2}+a \right )} \] Input:
int((d*x^2+c)^2/x^3/(b*x^2+a)^2,x)
Output:
( - 2*log(a + b*x**2)*a**2*c*d*x**2 + 2*log(a + b*x**2)*a*b*c**2*x**2 - 2* log(a + b*x**2)*a*b*c*d*x**4 + 2*log(a + b*x**2)*b**2*c**2*x**4 + 4*log(x) *a**2*c*d*x**2 - 4*log(x)*a*b*c**2*x**2 + 4*log(x)*a*b*c*d*x**4 - 4*log(x) *b**2*c**2*x**4 - a**2*c**2 + a**2*d**2*x**4 - 2*a*b*c*d*x**4 + 2*b**2*c** 2*x**4)/(2*a**3*x**2*(a + b*x**2))