Integrand size = 24, antiderivative size = 95 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^2}{x^{5/2}} \, dx=-\frac {2 a^2 c^2}{3 x^{3/2}}+4 a c (b c+a d) \sqrt {x}+\frac {2}{5} \left (b^2 c^2+4 a b c d+a^2 d^2\right ) x^{5/2}+\frac {4}{9} b d (b c+a d) x^{9/2}+\frac {2}{13} b^2 d^2 x^{13/2} \] Output:
-2/3*a^2*c^2/x^(3/2)+4*a*c*(a*d+b*c)*x^(1/2)+2/5*(a^2*d^2+4*a*b*c*d+b^2*c^ 2)*x^(5/2)+4/9*b*d*(a*d+b*c)*x^(9/2)+2/13*b^2*d^2*x^(13/2)
Time = 0.04 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.98 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^2}{x^{5/2}} \, dx=\frac {-78 a^2 \left (5 c^2-30 c d x^2-3 d^2 x^4\right )+52 a b x^2 \left (45 c^2+18 c d x^2+5 d^2 x^4\right )+2 b^2 x^4 \left (117 c^2+130 c d x^2+45 d^2 x^4\right )}{585 x^{3/2}} \] Input:
Integrate[((a + b*x^2)^2*(c + d*x^2)^2)/x^(5/2),x]
Output:
(-78*a^2*(5*c^2 - 30*c*d*x^2 - 3*d^2*x^4) + 52*a*b*x^2*(45*c^2 + 18*c*d*x^ 2 + 5*d^2*x^4) + 2*b^2*x^4*(117*c^2 + 130*c*d*x^2 + 45*d^2*x^4))/(585*x^(3 /2))
Time = 0.23 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {355, 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.
\(\displaystyle \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^2}{x^{5/2}} \, dx\) |
\(\Big \downarrow \) 355 |
\(\displaystyle \int \left (x^{3/2} \left (a^2 d^2+4 a b c d+b^2 c^2\right )+\frac {a^2 c^2}{x^{5/2}}+2 b d x^{7/2} (a d+b c)+\frac {2 a c (a d+b c)}{\sqrt {x}}+b^2 d^2 x^{11/2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {2}{5} x^{5/2} \left (a^2 d^2+4 a b c d+b^2 c^2\right )-\frac {2 a^2 c^2}{3 x^{3/2}}+\frac {4}{9} b d x^{9/2} (a d+b c)+4 a c \sqrt {x} (a d+b c)+\frac {2}{13} b^2 d^2 x^{13/2}\) |
Input:
Int[((a + b*x^2)^2*(c + d*x^2)^2)/x^(5/2),x]
Output:
(-2*a^2*c^2)/(3*x^(3/2)) + 4*a*c*(b*c + a*d)*Sqrt[x] + (2*(b^2*c^2 + 4*a*b *c*d + a^2*d^2)*x^(5/2))/5 + (4*b*d*(b*c + a*d)*x^(9/2))/9 + (2*b^2*d^2*x^ (13/2))/13
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q _.), x_Symbol] :> Int[ExpandIntegrand[(e*x)^m*(a + b*x^2)^p*(c + d*x^2)^q, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && IGtQ[p, 0] & & IGtQ[q, 0]
Time = 0.38 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.00
method | result | size |
derivativedivides | \(\frac {2 b^{2} d^{2} x^{\frac {13}{2}}}{13}+\frac {4 a b \,d^{2} x^{\frac {9}{2}}}{9}+\frac {4 b^{2} c d \,x^{\frac {9}{2}}}{9}+\frac {2 a^{2} d^{2} x^{\frac {5}{2}}}{5}+\frac {8 a b c d \,x^{\frac {5}{2}}}{5}+\frac {2 b^{2} c^{2} x^{\frac {5}{2}}}{5}+4 a^{2} c d \sqrt {x}+4 a b \,c^{2} \sqrt {x}-\frac {2 a^{2} c^{2}}{3 x^{\frac {3}{2}}}\) | \(95\) |
default | \(\frac {2 b^{2} d^{2} x^{\frac {13}{2}}}{13}+\frac {4 a b \,d^{2} x^{\frac {9}{2}}}{9}+\frac {4 b^{2} c d \,x^{\frac {9}{2}}}{9}+\frac {2 a^{2} d^{2} x^{\frac {5}{2}}}{5}+\frac {8 a b c d \,x^{\frac {5}{2}}}{5}+\frac {2 b^{2} c^{2} x^{\frac {5}{2}}}{5}+4 a^{2} c d \sqrt {x}+4 a b \,c^{2} \sqrt {x}-\frac {2 a^{2} c^{2}}{3 x^{\frac {3}{2}}}\) | \(95\) |
gosper | \(-\frac {2 \left (-45 b^{2} d^{2} x^{8}-130 a b \,d^{2} x^{6}-130 b^{2} c d \,x^{6}-117 a^{2} d^{2} x^{4}-468 a b c d \,x^{4}-117 b^{2} c^{2} x^{4}-1170 a^{2} c d \,x^{2}-1170 a b \,c^{2} x^{2}+195 a^{2} c^{2}\right )}{585 x^{\frac {3}{2}}}\) | \(97\) |
trager | \(-\frac {2 \left (-45 b^{2} d^{2} x^{8}-130 a b \,d^{2} x^{6}-130 b^{2} c d \,x^{6}-117 a^{2} d^{2} x^{4}-468 a b c d \,x^{4}-117 b^{2} c^{2} x^{4}-1170 a^{2} c d \,x^{2}-1170 a b \,c^{2} x^{2}+195 a^{2} c^{2}\right )}{585 x^{\frac {3}{2}}}\) | \(97\) |
risch | \(-\frac {2 \left (-45 b^{2} d^{2} x^{8}-130 a b \,d^{2} x^{6}-130 b^{2} c d \,x^{6}-117 a^{2} d^{2} x^{4}-468 a b c d \,x^{4}-117 b^{2} c^{2} x^{4}-1170 a^{2} c d \,x^{2}-1170 a b \,c^{2} x^{2}+195 a^{2} c^{2}\right )}{585 x^{\frac {3}{2}}}\) | \(97\) |
orering | \(-\frac {2 \left (-45 b^{2} d^{2} x^{8}-130 a b \,d^{2} x^{6}-130 b^{2} c d \,x^{6}-117 a^{2} d^{2} x^{4}-468 a b c d \,x^{4}-117 b^{2} c^{2} x^{4}-1170 a^{2} c d \,x^{2}-1170 a b \,c^{2} x^{2}+195 a^{2} c^{2}\right )}{585 x^{\frac {3}{2}}}\) | \(97\) |
Input:
int((b*x^2+a)^2*(d*x^2+c)^2/x^(5/2),x,method=_RETURNVERBOSE)
Output:
2/13*b^2*d^2*x^(13/2)+4/9*a*b*d^2*x^(9/2)+4/9*b^2*c*d*x^(9/2)+2/5*a^2*d^2* x^(5/2)+8/5*a*b*c*d*x^(5/2)+2/5*b^2*c^2*x^(5/2)+4*a^2*c*d*x^(1/2)+4*a*b*c^ 2*x^(1/2)-2/3*a^2*c^2/x^(3/2)
Time = 0.08 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.92 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^2}{x^{5/2}} \, dx=\frac {2 \, {\left (45 \, b^{2} d^{2} x^{8} + 130 \, {\left (b^{2} c d + a b d^{2}\right )} x^{6} + 117 \, {\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{4} - 195 \, a^{2} c^{2} + 1170 \, {\left (a b c^{2} + a^{2} c d\right )} x^{2}\right )}}{585 \, x^{\frac {3}{2}}} \] Input:
integrate((b*x^2+a)^2*(d*x^2+c)^2/x^(5/2),x, algorithm="fricas")
Output:
2/585*(45*b^2*d^2*x^8 + 130*(b^2*c*d + a*b*d^2)*x^6 + 117*(b^2*c^2 + 4*a*b *c*d + a^2*d^2)*x^4 - 195*a^2*c^2 + 1170*(a*b*c^2 + a^2*c*d)*x^2)/x^(3/2)
Time = 0.49 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.40 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^2}{x^{5/2}} \, dx=- \frac {2 a^{2} c^{2}}{3 x^{\frac {3}{2}}} + 4 a^{2} c d \sqrt {x} + \frac {2 a^{2} d^{2} x^{\frac {5}{2}}}{5} + 4 a b c^{2} \sqrt {x} + \frac {8 a b c d x^{\frac {5}{2}}}{5} + \frac {4 a b d^{2} x^{\frac {9}{2}}}{9} + \frac {2 b^{2} c^{2} x^{\frac {5}{2}}}{5} + \frac {4 b^{2} c d x^{\frac {9}{2}}}{9} + \frac {2 b^{2} d^{2} x^{\frac {13}{2}}}{13} \] Input:
integrate((b*x**2+a)**2*(d*x**2+c)**2/x**(5/2),x)
Output:
-2*a**2*c**2/(3*x**(3/2)) + 4*a**2*c*d*sqrt(x) + 2*a**2*d**2*x**(5/2)/5 + 4*a*b*c**2*sqrt(x) + 8*a*b*c*d*x**(5/2)/5 + 4*a*b*d**2*x**(9/2)/9 + 2*b**2 *c**2*x**(5/2)/5 + 4*b**2*c*d*x**(9/2)/9 + 2*b**2*d**2*x**(13/2)/13
Time = 0.04 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.89 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^2}{x^{5/2}} \, dx=\frac {2}{13} \, b^{2} d^{2} x^{\frac {13}{2}} + \frac {4}{9} \, {\left (b^{2} c d + a b d^{2}\right )} x^{\frac {9}{2}} + \frac {2}{5} \, {\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{\frac {5}{2}} - \frac {2 \, a^{2} c^{2}}{3 \, x^{\frac {3}{2}}} + 4 \, {\left (a b c^{2} + a^{2} c d\right )} \sqrt {x} \] Input:
integrate((b*x^2+a)^2*(d*x^2+c)^2/x^(5/2),x, algorithm="maxima")
Output:
2/13*b^2*d^2*x^(13/2) + 4/9*(b^2*c*d + a*b*d^2)*x^(9/2) + 2/5*(b^2*c^2 + 4 *a*b*c*d + a^2*d^2)*x^(5/2) - 2/3*a^2*c^2/x^(3/2) + 4*(a*b*c^2 + a^2*c*d)* sqrt(x)
Time = 0.12 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.99 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^2}{x^{5/2}} \, dx=\frac {2}{13} \, b^{2} d^{2} x^{\frac {13}{2}} + \frac {4}{9} \, b^{2} c d x^{\frac {9}{2}} + \frac {4}{9} \, a b d^{2} x^{\frac {9}{2}} + \frac {2}{5} \, b^{2} c^{2} x^{\frac {5}{2}} + \frac {8}{5} \, a b c d x^{\frac {5}{2}} + \frac {2}{5} \, a^{2} d^{2} x^{\frac {5}{2}} + 4 \, a b c^{2} \sqrt {x} + 4 \, a^{2} c d \sqrt {x} - \frac {2 \, a^{2} c^{2}}{3 \, x^{\frac {3}{2}}} \] Input:
integrate((b*x^2+a)^2*(d*x^2+c)^2/x^(5/2),x, algorithm="giac")
Output:
2/13*b^2*d^2*x^(13/2) + 4/9*b^2*c*d*x^(9/2) + 4/9*a*b*d^2*x^(9/2) + 2/5*b^ 2*c^2*x^(5/2) + 8/5*a*b*c*d*x^(5/2) + 2/5*a^2*d^2*x^(5/2) + 4*a*b*c^2*sqrt (x) + 4*a^2*c*d*sqrt(x) - 2/3*a^2*c^2/x^(3/2)
Time = 0.04 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.82 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^2}{x^{5/2}} \, dx=x^{5/2}\,\left (\frac {2\,a^2\,d^2}{5}+\frac {8\,a\,b\,c\,d}{5}+\frac {2\,b^2\,c^2}{5}\right )-\frac {2\,a^2\,c^2}{3\,x^{3/2}}+\frac {2\,b^2\,d^2\,x^{13/2}}{13}+4\,a\,c\,\sqrt {x}\,\left (a\,d+b\,c\right )+\frac {4\,b\,d\,x^{9/2}\,\left (a\,d+b\,c\right )}{9} \] Input:
int(((a + b*x^2)^2*(c + d*x^2)^2)/x^(5/2),x)
Output:
x^(5/2)*((2*a^2*d^2)/5 + (2*b^2*c^2)/5 + (8*a*b*c*d)/5) - (2*a^2*c^2)/(3*x ^(3/2)) + (2*b^2*d^2*x^(13/2))/13 + 4*a*c*x^(1/2)*(a*d + b*c) + (4*b*d*x^( 9/2)*(a*d + b*c))/9
Time = 0.22 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.05 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^2}{x^{5/2}} \, dx=\frac {\frac {2}{13} b^{2} d^{2} x^{8}+\frac {4}{9} a b \,d^{2} x^{6}+\frac {4}{9} b^{2} c d \,x^{6}+\frac {2}{5} a^{2} d^{2} x^{4}+\frac {8}{5} a b c d \,x^{4}+\frac {2}{5} b^{2} c^{2} x^{4}+4 a^{2} c d \,x^{2}+4 a b \,c^{2} x^{2}-\frac {2}{3} a^{2} c^{2}}{\sqrt {x}\, x} \] Input:
int((b*x^2+a)^2*(d*x^2+c)^2/x^(5/2),x)
Output:
(2*( - 195*a**2*c**2 + 1170*a**2*c*d*x**2 + 117*a**2*d**2*x**4 + 1170*a*b* c**2*x**2 + 468*a*b*c*d*x**4 + 130*a*b*d**2*x**6 + 117*b**2*c**2*x**4 + 13 0*b**2*c*d*x**6 + 45*b**2*d**2*x**8))/(585*sqrt(x)*x)