Integrand size = 22, antiderivative size = 80 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^2}{x} \, dx=a c (b c+a d) x^2+\frac {1}{4} \left (b^2 c^2+4 a b c d+a^2 d^2\right ) x^4+\frac {1}{3} b d (b c+a d) x^6+\frac {1}{8} b^2 d^2 x^8+a^2 c^2 \log (x) \] Output:
a*c*(a*d+b*c)*x^2+1/4*(a^2*d^2+4*a*b*c*d+b^2*c^2)*x^4+1/3*b*d*(a*d+b*c)*x^ 6+1/8*b^2*d^2*x^8+a^2*c^2*ln(x)
Time = 0.02 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^2}{x} \, dx=a c (b c+a d) x^2+\frac {1}{4} \left (b^2 c^2+4 a b c d+a^2 d^2\right ) x^4+\frac {1}{3} b d (b c+a d) x^6+\frac {1}{8} b^2 d^2 x^8+a^2 c^2 \log (x) \] Input:
Integrate[((a + b*x^2)^2*(c + d*x^2)^2)/x,x]
Output:
a*c*(b*c + a*d)*x^2 + ((b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^4)/4 + (b*d*(b*c + a*d)*x^6)/3 + (b^2*d^2*x^8)/8 + a^2*c^2*Log[x]
Time = 0.22 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.09, 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.
\(\displaystyle \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^2}{x} \, dx\) |
\(\Big \downarrow \) 354 |
\(\displaystyle \frac {1}{2} \int \frac {\left (b x^2+a\right )^2 \left (d x^2+c\right )^2}{x^2}dx^2\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \frac {1}{2} \int \left (b^2 d^2 x^6+2 b d (b c+a d) x^4+\left (b^2 c^2+4 a b d c+a^2 d^2\right ) x^2+2 a c (b c+a d)+\frac {a^2 c^2}{x^2}\right )dx^2\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} \left (\frac {1}{2} x^4 \left (a^2 d^2+4 a b c d+b^2 c^2\right )+a^2 c^2 \log \left (x^2\right )+\frac {2}{3} b d x^6 (a d+b c)+2 a c x^2 (a d+b c)+\frac {1}{4} b^2 d^2 x^8\right )\) |
Input:
Int[((a + b*x^2)^2*(c + d*x^2)^2)/x,x]
Output:
(2*a*c*(b*c + a*d)*x^2 + ((b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^4)/2 + (2*b*d* (b*c + a*d)*x^6)/3 + (b^2*d^2*x^8)/4 + a^2*c^2*Log[x^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 = 0.51 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.05
method | result | size |
norman | \(\left (\frac {1}{3} a b \,d^{2}+\frac {1}{3} b^{2} c d \right ) x^{6}+\left (\frac {1}{4} a^{2} d^{2}+a b c d +\frac {1}{4} b^{2} c^{2}\right ) x^{4}+\left (a^{2} c d +b \,c^{2} a \right ) x^{2}+\frac {b^{2} d^{2} x^{8}}{8}+a^{2} c^{2} \ln \left (x \right )\) | \(84\) |
default | \(\frac {b^{2} d^{2} x^{8}}{8}+\frac {a b \,d^{2} x^{6}}{3}+\frac {b^{2} c d \,x^{6}}{3}+\frac {a^{2} d^{2} x^{4}}{4}+a b c d \,x^{4}+\frac {b^{2} c^{2} x^{4}}{4}+a^{2} c d \,x^{2}+a b \,c^{2} x^{2}+a^{2} c^{2} \ln \left (x \right )\) | \(90\) |
parallelrisch | \(\frac {b^{2} d^{2} x^{8}}{8}+\frac {a b \,d^{2} x^{6}}{3}+\frac {b^{2} c d \,x^{6}}{3}+\frac {a^{2} d^{2} x^{4}}{4}+a b c d \,x^{4}+\frac {b^{2} c^{2} x^{4}}{4}+a^{2} c d \,x^{2}+a b \,c^{2} x^{2}+a^{2} c^{2} \ln \left (x \right )\) | \(90\) |
risch | \(\frac {a b \,d^{2} x^{6}}{3}+\frac {b^{2} c d \,x^{6}}{3}+\frac {a^{2} d^{2} x^{4}}{4}+\frac {b^{2} c^{2} x^{4}}{4}-\frac {b^{2} c^{4}}{24 d^{2}}-\frac {d^{2} a^{4}}{24 b^{2}}+a b c d \,x^{4}+\frac {d \,a^{3} c}{3 b}+\frac {3 a^{2} c^{2}}{4}+\frac {b a \,c^{3}}{3 d}+a^{2} c d \,x^{2}+a b \,c^{2} x^{2}+\frac {b^{2} d^{2} x^{8}}{8}+a^{2} c^{2} \ln \left (x \right )\) | \(140\) |
Input:
int((b*x^2+a)^2*(d*x^2+c)^2/x,x,method=_RETURNVERBOSE)
Output:
(1/3*a*b*d^2+1/3*b^2*c*d)*x^6+(1/4*a^2*d^2+a*b*c*d+1/4*b^2*c^2)*x^4+(a^2*c *d+a*b*c^2)*x^2+1/8*b^2*d^2*x^8+a^2*c^2*ln(x)
Time = 0.07 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.02 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^2}{x} \, dx=\frac {1}{8} \, b^{2} d^{2} x^{8} + \frac {1}{3} \, {\left (b^{2} c d + a b d^{2}\right )} x^{6} + \frac {1}{4} \, {\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} \log \left (x\right ) + {\left (a b c^{2} + a^{2} c d\right )} x^{2} \] Input:
integrate((b*x^2+a)^2*(d*x^2+c)^2/x,x, algorithm="fricas")
Output:
1/8*b^2*d^2*x^8 + 1/3*(b^2*c*d + a*b*d^2)*x^6 + 1/4*(b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^4 + a^2*c^2*log(x) + (a*b*c^2 + a^2*c*d)*x^2
Time = 0.07 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.06 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^2}{x} \, dx=a^{2} c^{2} \log {\left (x \right )} + \frac {b^{2} d^{2} x^{8}}{8} + x^{6} \left (\frac {a b d^{2}}{3} + \frac {b^{2} c d}{3}\right ) + x^{4} \left (\frac {a^{2} d^{2}}{4} + a b c d + \frac {b^{2} c^{2}}{4}\right ) + x^{2} \left (a^{2} c d + a b c^{2}\right ) \] Input:
integrate((b*x**2+a)**2*(d*x**2+c)**2/x,x)
Output:
a**2*c**2*log(x) + b**2*d**2*x**8/8 + x**6*(a*b*d**2/3 + b**2*c*d/3) + x** 4*(a**2*d**2/4 + a*b*c*d + b**2*c**2/4) + x**2*(a**2*c*d + a*b*c**2)
Time = 0.04 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.06 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^2}{x} \, dx=\frac {1}{8} \, b^{2} d^{2} x^{8} + \frac {1}{3} \, {\left (b^{2} c d + a b d^{2}\right )} x^{6} + \frac {1}{4} \, {\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{4} + \frac {1}{2} \, a^{2} c^{2} \log \left (x^{2}\right ) + {\left (a b c^{2} + a^{2} c d\right )} x^{2} \] Input:
integrate((b*x^2+a)^2*(d*x^2+c)^2/x,x, algorithm="maxima")
Output:
1/8*b^2*d^2*x^8 + 1/3*(b^2*c*d + a*b*d^2)*x^6 + 1/4*(b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^4 + 1/2*a^2*c^2*log(x^2) + (a*b*c^2 + a^2*c*d)*x^2
Time = 0.13 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.15 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^2}{x} \, dx=\frac {1}{8} \, b^{2} d^{2} x^{8} + \frac {1}{3} \, b^{2} c d x^{6} + \frac {1}{3} \, a b d^{2} x^{6} + \frac {1}{4} \, b^{2} c^{2} x^{4} + a b c d x^{4} + \frac {1}{4} \, a^{2} d^{2} x^{4} + a b c^{2} x^{2} + a^{2} c d x^{2} + \frac {1}{2} \, a^{2} c^{2} \log \left (x^{2}\right ) \] Input:
integrate((b*x^2+a)^2*(d*x^2+c)^2/x,x, algorithm="giac")
Output:
1/8*b^2*d^2*x^8 + 1/3*b^2*c*d*x^6 + 1/3*a*b*d^2*x^6 + 1/4*b^2*c^2*x^4 + a* b*c*d*x^4 + 1/4*a^2*d^2*x^4 + a*b*c^2*x^2 + a^2*c*d*x^2 + 1/2*a^2*c^2*log( x^2)
Time = 0.04 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.92 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^2}{x} \, dx=x^4\,\left (\frac {a^2\,d^2}{4}+a\,b\,c\,d+\frac {b^2\,c^2}{4}\right )+\frac {b^2\,d^2\,x^8}{8}+a^2\,c^2\,\ln \left (x\right )+a\,c\,x^2\,\left (a\,d+b\,c\right )+\frac {b\,d\,x^6\,\left (a\,d+b\,c\right )}{3} \] Input:
int(((a + b*x^2)^2*(c + d*x^2)^2)/x,x)
Output:
x^4*((a^2*d^2)/4 + (b^2*c^2)/4 + a*b*c*d) + (b^2*d^2*x^8)/8 + a^2*c^2*log( x) + a*c*x^2*(a*d + b*c) + (b*d*x^6*(a*d + b*c))/3
Time = 0.24 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.11 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^2}{x} \, dx=\mathrm {log}\left (x \right ) a^{2} c^{2}+a^{2} c d \,x^{2}+\frac {a^{2} d^{2} x^{4}}{4}+a b \,c^{2} x^{2}+a b c d \,x^{4}+\frac {a b \,d^{2} x^{6}}{3}+\frac {b^{2} c^{2} x^{4}}{4}+\frac {b^{2} c d \,x^{6}}{3}+\frac {b^{2} d^{2} x^{8}}{8} \] Input:
int((b*x^2+a)^2*(d*x^2+c)^2/x,x)
Output:
(24*log(x)*a**2*c**2 + 24*a**2*c*d*x**2 + 6*a**2*d**2*x**4 + 24*a*b*c**2*x **2 + 24*a*b*c*d*x**4 + 8*a*b*d**2*x**6 + 6*b**2*c**2*x**4 + 8*b**2*c*d*x* *6 + 3*b**2*d**2*x**8)/24