Integrand size = 22, antiderivative size = 98 \[ \int \frac {\left (c+d x^2\right )^3}{x^3 \left (a+b x^2\right )^2} \, dx=-\frac {c^3}{2 a^2 x^2}-\frac {(b c-a d)^3}{2 a^2 b^2 \left (a+b x^2\right )}-\frac {c^2 (2 b c-3 a d) \log (x)}{a^3}+\frac {(b c-a d)^2 (2 b c+a d) \log \left (a+b x^2\right )}{2 a^3 b^2} \] Output:
-1/2*c^3/a^2/x^2-1/2*(-a*d+b*c)^3/a^2/b^2/(b*x^2+a)-c^2*(-3*a*d+2*b*c)*ln( x)/a^3+1/2*(-a*d+b*c)^2*(a*d+2*b*c)*ln(b*x^2+a)/a^3/b^2
Time = 0.06 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.89 \[ \int \frac {\left (c+d x^2\right )^3}{x^3 \left (a+b x^2\right )^2} \, dx=\frac {-\frac {a c^3}{x^2}+\frac {a (-b c+a d)^3}{b^2 \left (a+b x^2\right )}+2 c^2 (-2 b c+3 a d) \log (x)+\frac {(b c-a d)^2 (2 b c+a d) \log \left (a+b x^2\right )}{b^2}}{2 a^3} \] Input:
Integrate[(c + d*x^2)^3/(x^3*(a + b*x^2)^2),x]
Output:
(-((a*c^3)/x^2) + (a*(-(b*c) + a*d)^3)/(b^2*(a + b*x^2)) + 2*c^2*(-2*b*c + 3*a*d)*Log[x] + ((b*c - a*d)^2*(2*b*c + a*d)*Log[a + b*x^2])/b^2)/(2*a^3)
Time = 0.28 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.99, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, 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.
\(\displaystyle \int \frac {\left (c+d x^2\right )^3}{x^3 \left (a+b x^2\right )^2} \, dx\) |
\(\Big \downarrow \) 354 |
\(\displaystyle \frac {1}{2} \int \frac {\left (d x^2+c\right )^3}{x^4 \left (b x^2+a\right )^2}dx^2\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \frac {1}{2} \int \left (\frac {c^3}{a^2 x^4}+\frac {(3 a d-2 b c) c^2}{a^3 x^2}+\frac {(a d-b c)^2 (2 b c+a d)}{a^3 b \left (b x^2+a\right )}-\frac {(a d-b c)^3}{a^2 b \left (b x^2+a\right )^2}\right )dx^2\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} \left (\frac {(b c-a d)^2 (a d+2 b c) \log \left (a+b x^2\right )}{a^3 b^2}-\frac {c^2 \log \left (x^2\right ) (2 b c-3 a d)}{a^3}-\frac {(b c-a d)^3}{a^2 b^2 \left (a+b x^2\right )}-\frac {c^3}{a^2 x^2}\right )\) |
Input:
Int[(c + d*x^2)^3/(x^3*(a + b*x^2)^2),x]
Output:
(-(c^3/(a^2*x^2)) - (b*c - a*d)^3/(a^2*b^2*(a + b*x^2)) - (c^2*(2*b*c - 3* a*d)*Log[x^2])/a^3 + ((b*c - a*d)^2*(2*b*c + a*d)*Log[a + b*x^2])/(a^3*b^2 ))/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 = 0.46 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.02
method | result | size |
default | \(\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (\frac {\left (a d -b c \right ) a}{b^{2} \left (b \,x^{2}+a \right )}+\frac {\left (a d +2 b c \right ) \ln \left (b \,x^{2}+a \right )}{b^{2}}\right )}{2 a^{3}}-\frac {c^{3}}{2 a^{2} x^{2}}+\frac {c^{2} \left (3 a d -2 b c \right ) \ln \left (x \right )}{a^{3}}\) | \(100\) |
norman | \(\frac {-\frac {c^{3}}{2 a}+\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -2 b^{3} c^{3}\right ) x^{2}}{2 a^{2} b^{2}}}{x^{2} \left (b \,x^{2}+a \right )}+\frac {c^{2} \left (3 a d -2 b c \right ) \ln \left (x \right )}{a^{3}}+\frac {\left (a^{3} d^{3}-3 a \,b^{2} c^{2} d +2 b^{3} c^{3}\right ) \ln \left (b \,x^{2}+a \right )}{2 a^{3} b^{2}}\) | \(131\) |
risch | \(\frac {-\frac {c^{3}}{2 a}+\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -2 b^{3} c^{3}\right ) x^{2}}{2 a^{2} b^{2}}}{x^{2} \left (b \,x^{2}+a \right )}+\frac {3 c^{2} \ln \left (x \right ) d}{a^{2}}-\frac {2 c^{3} \ln \left (x \right ) b}{a^{3}}+\frac {\ln \left (-b \,x^{2}-a \right ) d^{3}}{2 b^{2}}-\frac {3 \ln \left (-b \,x^{2}-a \right ) c^{2} d}{2 a^{2}}+\frac {b \ln \left (-b \,x^{2}-a \right ) c^{3}}{a^{3}}\) | \(151\) |
parallelrisch | \(\frac {6 \ln \left (x \right ) x^{4} a \,b^{3} c^{2} d -4 \ln \left (x \right ) x^{4} b^{4} c^{3}+\ln \left (b \,x^{2}+a \right ) x^{4} a^{3} b \,d^{3}-3 \ln \left (b \,x^{2}+a \right ) x^{4} a \,b^{3} c^{2} d +2 \ln \left (b \,x^{2}+a \right ) x^{4} b^{4} c^{3}+6 \ln \left (x \right ) x^{2} a^{2} b^{2} c^{2} d -4 \ln \left (x \right ) x^{2} a \,b^{3} c^{3}+\ln \left (b \,x^{2}+a \right ) x^{2} a^{4} d^{3}-3 \ln \left (b \,x^{2}+a \right ) x^{2} a^{2} b^{2} c^{2} d +2 \ln \left (b \,x^{2}+a \right ) x^{2} a \,b^{3} c^{3}+x^{2} a^{4} d^{3}-3 x^{2} a^{3} b c \,d^{2}+3 x^{2} a^{2} b^{2} c^{2} d -2 x^{2} a \,b^{3} c^{3}-a^{2} b^{2} c^{3}}{2 a^{3} b^{2} x^{2} \left (b \,x^{2}+a \right )}\) | \(262\) |
Input:
int((d*x^2+c)^3/x^3/(b*x^2+a)^2,x,method=_RETURNVERBOSE)
Output:
1/2/a^3*(a^2*d^2-2*a*b*c*d+b^2*c^2)*((a*d-b*c)*a/b^2/(b*x^2+a)+1/b^2*(a*d+ 2*b*c)*ln(b*x^2+a))-1/2*c^3/a^2/x^2+c^2*(3*a*d-2*b*c)/a^3*ln(x)
Leaf count of result is larger than twice the leaf count of optimal. 209 vs. \(2 (92) = 184\).
Time = 0.09 (sec) , antiderivative size = 209, normalized size of antiderivative = 2.13 \[ \int \frac {\left (c+d x^2\right )^3}{x^3 \left (a+b x^2\right )^2} \, dx=-\frac {a^{2} b^{2} c^{3} + {\left (2 \, a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3}\right )} x^{2} - {\left ({\left (2 \, b^{4} c^{3} - 3 \, a b^{3} c^{2} d + a^{3} b d^{3}\right )} x^{4} + {\left (2 \, a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + a^{4} d^{3}\right )} x^{2}\right )} \log \left (b x^{2} + a\right ) + 2 \, {\left ({\left (2 \, b^{4} c^{3} - 3 \, a b^{3} c^{2} d\right )} x^{4} + {\left (2 \, a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d\right )} x^{2}\right )} \log \left (x\right )}{2 \, {\left (a^{3} b^{3} x^{4} + a^{4} b^{2} x^{2}\right )}} \] Input:
integrate((d*x^2+c)^3/x^3/(b*x^2+a)^2,x, algorithm="fricas")
Output:
-1/2*(a^2*b^2*c^3 + (2*a*b^3*c^3 - 3*a^2*b^2*c^2*d + 3*a^3*b*c*d^2 - a^4*d ^3)*x^2 - ((2*b^4*c^3 - 3*a*b^3*c^2*d + a^3*b*d^3)*x^4 + (2*a*b^3*c^3 - 3* a^2*b^2*c^2*d + a^4*d^3)*x^2)*log(b*x^2 + a) + 2*((2*b^4*c^3 - 3*a*b^3*c^2 *d)*x^4 + (2*a*b^3*c^3 - 3*a^2*b^2*c^2*d)*x^2)*log(x))/(a^3*b^3*x^4 + a^4* b^2*x^2)
Time = 1.63 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.31 \[ \int \frac {\left (c+d x^2\right )^3}{x^3 \left (a+b x^2\right )^2} \, dx=\frac {- a b^{2} c^{3} + x^{2} \left (a^{3} d^{3} - 3 a^{2} b c d^{2} + 3 a b^{2} c^{2} d - 2 b^{3} c^{3}\right )}{2 a^{3} b^{2} x^{2} + 2 a^{2} b^{3} x^{4}} + \frac {c^{2} \cdot \left (3 a d - 2 b c\right ) \log {\left (x \right )}}{a^{3}} + \frac {\left (a d - b c\right )^{2} \left (a d + 2 b c\right ) \log {\left (\frac {a}{b} + x^{2} \right )}}{2 a^{3} b^{2}} \] Input:
integrate((d*x**2+c)**3/x**3/(b*x**2+a)**2,x)
Output:
(-a*b**2*c**3 + x**2*(a**3*d**3 - 3*a**2*b*c*d**2 + 3*a*b**2*c**2*d - 2*b* *3*c**3))/(2*a**3*b**2*x**2 + 2*a**2*b**3*x**4) + c**2*(3*a*d - 2*b*c)*log (x)/a**3 + (a*d - b*c)**2*(a*d + 2*b*c)*log(a/b + x**2)/(2*a**3*b**2)
Time = 0.04 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.44 \[ \int \frac {\left (c+d x^2\right )^3}{x^3 \left (a+b x^2\right )^2} \, dx=-\frac {a b^{2} c^{3} + {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} x^{2}}{2 \, {\left (a^{2} b^{3} x^{4} + a^{3} b^{2} x^{2}\right )}} - \frac {{\left (2 \, b c^{3} - 3 \, a c^{2} d\right )} \log \left (x^{2}\right )}{2 \, a^{3}} + \frac {{\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d + a^{3} d^{3}\right )} \log \left (b x^{2} + a\right )}{2 \, a^{3} b^{2}} \] Input:
integrate((d*x^2+c)^3/x^3/(b*x^2+a)^2,x, algorithm="maxima")
Output:
-1/2*(a*b^2*c^3 + (2*b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*x^ 2)/(a^2*b^3*x^4 + a^3*b^2*x^2) - 1/2*(2*b*c^3 - 3*a*c^2*d)*log(x^2)/a^3 + 1/2*(2*b^3*c^3 - 3*a*b^2*c^2*d + a^3*d^3)*log(b*x^2 + a)/(a^3*b^2)
Time = 0.12 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.60 \[ \int \frac {\left (c+d x^2\right )^3}{x^3 \left (a+b x^2\right )^2} \, dx=-\frac {{\left (2 \, b c^{3} - 3 \, a c^{2} d\right )} \log \left (x^{2}\right )}{2 \, a^{3}} + \frac {{\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d + a^{3} d^{3}\right )} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, a^{3} b^{2}} - \frac {a^{2} b d^{3} x^{4} + 4 \, b^{3} c^{3} x^{2} - 6 \, a b^{2} c^{2} d x^{2} + 6 \, a^{2} b c d^{2} x^{2} - a^{3} d^{3} x^{2} + 2 \, a b^{2} c^{3}}{4 \, {\left (b x^{4} + a x^{2}\right )} a^{2} b^{2}} \] Input:
integrate((d*x^2+c)^3/x^3/(b*x^2+a)^2,x, algorithm="giac")
Output:
-1/2*(2*b*c^3 - 3*a*c^2*d)*log(x^2)/a^3 + 1/2*(2*b^3*c^3 - 3*a*b^2*c^2*d + a^3*d^3)*log(abs(b*x^2 + a))/(a^3*b^2) - 1/4*(a^2*b*d^3*x^4 + 4*b^3*c^3*x ^2 - 6*a*b^2*c^2*d*x^2 + 6*a^2*b*c*d^2*x^2 - a^3*d^3*x^2 + 2*a*b^2*c^3)/(( b*x^4 + a*x^2)*a^2*b^2)
Time = 0.66 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.38 \[ \int \frac {\left (c+d x^2\right )^3}{x^3 \left (a+b x^2\right )^2} \, dx=\frac {\ln \left (b\,x^2+a\right )\,\left (a^3\,d^3-3\,a\,b^2\,c^2\,d+2\,b^3\,c^3\right )}{2\,a^3\,b^2}-\frac {\ln \left (x\right )\,\left (2\,b\,c^3-3\,a\,c^2\,d\right )}{a^3}-\frac {\frac {c^3}{2\,a}-\frac {x^2\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-2\,b^3\,c^3\right )}{2\,a^2\,b^2}}{b\,x^4+a\,x^2} \] Input:
int((c + d*x^2)^3/(x^3*(a + b*x^2)^2),x)
Output:
(log(a + b*x^2)*(a^3*d^3 + 2*b^3*c^3 - 3*a*b^2*c^2*d))/(2*a^3*b^2) - (log( x)*(2*b*c^3 - 3*a*c^2*d))/a^3 - (c^3/(2*a) - (x^2*(a^3*d^3 - 2*b^3*c^3 + 3 *a*b^2*c^2*d - 3*a^2*b*c*d^2))/(2*a^2*b^2))/(a*x^2 + b*x^4)
Time = 0.20 (sec) , antiderivative size = 262, normalized size of antiderivative = 2.67 \[ \int \frac {\left (c+d x^2\right )^3}{x^3 \left (a+b x^2\right )^2} \, dx=\frac {\mathrm {log}\left (b \,x^{2}+a \right ) a^{4} d^{3} x^{2}+\mathrm {log}\left (b \,x^{2}+a \right ) a^{3} b \,d^{3} x^{4}-3 \,\mathrm {log}\left (b \,x^{2}+a \right ) a^{2} b^{2} c^{2} d \,x^{2}+2 \,\mathrm {log}\left (b \,x^{2}+a \right ) a \,b^{3} c^{3} x^{2}-3 \,\mathrm {log}\left (b \,x^{2}+a \right ) a \,b^{3} c^{2} d \,x^{4}+2 \,\mathrm {log}\left (b \,x^{2}+a \right ) b^{4} c^{3} x^{4}+6 \,\mathrm {log}\left (x \right ) a^{2} b^{2} c^{2} d \,x^{2}-4 \,\mathrm {log}\left (x \right ) a \,b^{3} c^{3} x^{2}+6 \,\mathrm {log}\left (x \right ) a \,b^{3} c^{2} d \,x^{4}-4 \,\mathrm {log}\left (x \right ) b^{4} c^{3} x^{4}-a^{3} b \,d^{3} x^{4}-a^{2} b^{2} c^{3}+3 a^{2} b^{2} c \,d^{2} x^{4}-3 a \,b^{3} c^{2} d \,x^{4}+2 b^{4} c^{3} x^{4}}{2 a^{3} b^{2} x^{2} \left (b \,x^{2}+a \right )} \] Input:
int((d*x^2+c)^3/x^3/(b*x^2+a)^2,x)
Output:
(log(a + b*x**2)*a**4*d**3*x**2 + log(a + b*x**2)*a**3*b*d**3*x**4 - 3*log (a + b*x**2)*a**2*b**2*c**2*d*x**2 + 2*log(a + b*x**2)*a*b**3*c**3*x**2 - 3*log(a + b*x**2)*a*b**3*c**2*d*x**4 + 2*log(a + b*x**2)*b**4*c**3*x**4 + 6*log(x)*a**2*b**2*c**2*d*x**2 - 4*log(x)*a*b**3*c**3*x**2 + 6*log(x)*a*b* *3*c**2*d*x**4 - 4*log(x)*b**4*c**3*x**4 - a**3*b*d**3*x**4 - a**2*b**2*c* *3 + 3*a**2*b**2*c*d**2*x**4 - 3*a*b**3*c**2*d*x**4 + 2*b**4*c**3*x**4)/(2 *a**3*b**2*x**2*(a + b*x**2))