Integrand size = 24, antiderivative size = 97 \[ \int \sqrt {x} \left (a+b x^2\right )^2 \left (c+d x^2\right )^2 \, dx=\frac {2}{3} a^2 c^2 x^{3/2}+\frac {4}{7} a c (b c+a d) x^{7/2}+\frac {2}{11} \left (b^2 c^2+4 a b c d+a^2 d^2\right ) x^{11/2}+\frac {4}{15} b d (b c+a d) x^{15/2}+\frac {2}{19} b^2 d^2 x^{19/2} \] Output:
2/3*a^2*c^2*x^(3/2)+4/7*a*c*(a*d+b*c)*x^(7/2)+2/11*(a^2*d^2+4*a*b*c*d+b^2* c^2)*x^(11/2)+4/15*b*d*(a*d+b*c)*x^(15/2)+2/19*b^2*d^2*x^(19/2)
Time = 0.05 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.96 \[ \int \sqrt {x} \left (a+b x^2\right )^2 \left (c+d x^2\right )^2 \, dx=\frac {2 x^{3/2} \left (95 a^2 \left (77 c^2+66 c d x^2+21 d^2 x^4\right )+38 a b x^2 \left (165 c^2+210 c d x^2+77 d^2 x^4\right )+7 b^2 x^4 \left (285 c^2+418 c d x^2+165 d^2 x^4\right )\right )}{21945} \] Input:
Integrate[Sqrt[x]*(a + b*x^2)^2*(c + d*x^2)^2,x]
Output:
(2*x^(3/2)*(95*a^2*(77*c^2 + 66*c*d*x^2 + 21*d^2*x^4) + 38*a*b*x^2*(165*c^ 2 + 210*c*d*x^2 + 77*d^2*x^4) + 7*b^2*x^4*(285*c^2 + 418*c*d*x^2 + 165*d^2 *x^4)))/21945
Time = 0.23 (sec) , antiderivative size = 97, 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.
\(\displaystyle \int \sqrt {x} \left (a+b x^2\right )^2 \left (c+d x^2\right )^2 \, dx\) |
\(\Big \downarrow \) 355 |
\(\displaystyle \int \left (x^{9/2} \left (a^2 d^2+4 a b c d+b^2 c^2\right )+a^2 c^2 \sqrt {x}+2 b d x^{13/2} (a d+b c)+2 a c x^{5/2} (a d+b c)+b^2 d^2 x^{17/2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {2}{11} x^{11/2} \left (a^2 d^2+4 a b c d+b^2 c^2\right )+\frac {2}{3} a^2 c^2 x^{3/2}+\frac {4}{15} b d x^{15/2} (a d+b c)+\frac {4}{7} a c x^{7/2} (a d+b c)+\frac {2}{19} b^2 d^2 x^{19/2}\) |
Input:
Int[Sqrt[x]*(a + b*x^2)^2*(c + d*x^2)^2,x]
Output:
(2*a^2*c^2*x^(3/2))/3 + (4*a*c*(b*c + a*d)*x^(7/2))/7 + (2*(b^2*c^2 + 4*a* b*c*d + a^2*d^2)*x^(11/2))/11 + (4*b*d*(b*c + a*d)*x^(15/2))/15 + (2*b^2*d ^2*x^(19/2))/19
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 = 0.40 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.93
method | result | size |
derivativedivides | \(\frac {2 b^{2} d^{2} x^{\frac {19}{2}}}{19}+\frac {2 \left (2 a b \,d^{2}+2 b^{2} c d \right ) x^{\frac {15}{2}}}{15}+\frac {2 \left (a^{2} d^{2}+4 a b c d +b^{2} c^{2}\right ) x^{\frac {11}{2}}}{11}+\frac {2 \left (2 a^{2} c d +2 b \,c^{2} a \right ) x^{\frac {7}{2}}}{7}+\frac {2 a^{2} c^{2} x^{\frac {3}{2}}}{3}\) | \(90\) |
default | \(\frac {2 b^{2} d^{2} x^{\frac {19}{2}}}{19}+\frac {2 \left (2 a b \,d^{2}+2 b^{2} c d \right ) x^{\frac {15}{2}}}{15}+\frac {2 \left (a^{2} d^{2}+4 a b c d +b^{2} c^{2}\right ) x^{\frac {11}{2}}}{11}+\frac {2 \left (2 a^{2} c d +2 b \,c^{2} a \right ) x^{\frac {7}{2}}}{7}+\frac {2 a^{2} c^{2} x^{\frac {3}{2}}}{3}\) | \(90\) |
gosper | \(\frac {2 x^{\frac {3}{2}} \left (1155 b^{2} d^{2} x^{8}+2926 a b \,d^{2} x^{6}+2926 b^{2} c d \,x^{6}+1995 a^{2} d^{2} x^{4}+7980 a b c d \,x^{4}+1995 b^{2} c^{2} x^{4}+6270 a^{2} c d \,x^{2}+6270 a b \,c^{2} x^{2}+7315 a^{2} c^{2}\right )}{21945}\) | \(97\) |
trager | \(\frac {2 x^{\frac {3}{2}} \left (1155 b^{2} d^{2} x^{8}+2926 a b \,d^{2} x^{6}+2926 b^{2} c d \,x^{6}+1995 a^{2} d^{2} x^{4}+7980 a b c d \,x^{4}+1995 b^{2} c^{2} x^{4}+6270 a^{2} c d \,x^{2}+6270 a b \,c^{2} x^{2}+7315 a^{2} c^{2}\right )}{21945}\) | \(97\) |
risch | \(\frac {2 x^{\frac {3}{2}} \left (1155 b^{2} d^{2} x^{8}+2926 a b \,d^{2} x^{6}+2926 b^{2} c d \,x^{6}+1995 a^{2} d^{2} x^{4}+7980 a b c d \,x^{4}+1995 b^{2} c^{2} x^{4}+6270 a^{2} c d \,x^{2}+6270 a b \,c^{2} x^{2}+7315 a^{2} c^{2}\right )}{21945}\) | \(97\) |
orering | \(\frac {2 x^{\frac {3}{2}} \left (1155 b^{2} d^{2} x^{8}+2926 a b \,d^{2} x^{6}+2926 b^{2} c d \,x^{6}+1995 a^{2} d^{2} x^{4}+7980 a b c d \,x^{4}+1995 b^{2} c^{2} x^{4}+6270 a^{2} c d \,x^{2}+6270 a b \,c^{2} x^{2}+7315 a^{2} c^{2}\right )}{21945}\) | \(97\) |
Input:
int(x^(1/2)*(b*x^2+a)^2*(d*x^2+c)^2,x,method=_RETURNVERBOSE)
Output:
2/19*b^2*d^2*x^(19/2)+2/15*(2*a*b*d^2+2*b^2*c*d)*x^(15/2)+2/11*(a^2*d^2+4* a*b*c*d+b^2*c^2)*x^(11/2)+2/7*(2*a^2*c*d+2*a*b*c^2)*x^(7/2)+2/3*a^2*c^2*x^ (3/2)
Time = 0.08 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.91 \[ \int \sqrt {x} \left (a+b x^2\right )^2 \left (c+d x^2\right )^2 \, dx=\frac {2}{21945} \, {\left (1155 \, b^{2} d^{2} x^{9} + 2926 \, {\left (b^{2} c d + a b d^{2}\right )} x^{7} + 1995 \, {\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{5} + 7315 \, a^{2} c^{2} x + 6270 \, {\left (a b c^{2} + a^{2} c d\right )} x^{3}\right )} \sqrt {x} \] Input:
integrate(x^(1/2)*(b*x^2+a)^2*(d*x^2+c)^2,x, algorithm="fricas")
Output:
2/21945*(1155*b^2*d^2*x^9 + 2926*(b^2*c*d + a*b*d^2)*x^7 + 1995*(b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^5 + 7315*a^2*c^2*x + 6270*(a*b*c^2 + a^2*c*d)*x^3) *sqrt(x)
Time = 0.67 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.13 \[ \int \sqrt {x} \left (a+b x^2\right )^2 \left (c+d x^2\right )^2 \, dx=\frac {2 a^{2} c^{2} x^{\frac {3}{2}}}{3} + \frac {2 b^{2} d^{2} x^{\frac {19}{2}}}{19} + \frac {2 x^{\frac {15}{2}} \cdot \left (2 a b d^{2} + 2 b^{2} c d\right )}{15} + \frac {2 x^{\frac {11}{2}} \left (a^{2} d^{2} + 4 a b c d + b^{2} c^{2}\right )}{11} + \frac {2 x^{\frac {7}{2}} \cdot \left (2 a^{2} c d + 2 a b c^{2}\right )}{7} \] Input:
integrate(x**(1/2)*(b*x**2+a)**2*(d*x**2+c)**2,x)
Output:
2*a**2*c**2*x**(3/2)/3 + 2*b**2*d**2*x**(19/2)/19 + 2*x**(15/2)*(2*a*b*d** 2 + 2*b**2*c*d)/15 + 2*x**(11/2)*(a**2*d**2 + 4*a*b*c*d + b**2*c**2)/11 + 2*x**(7/2)*(2*a**2*c*d + 2*a*b*c**2)/7
Time = 0.03 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.88 \[ \int \sqrt {x} \left (a+b x^2\right )^2 \left (c+d x^2\right )^2 \, dx=\frac {2}{19} \, b^{2} d^{2} x^{\frac {19}{2}} + \frac {4}{15} \, {\left (b^{2} c d + a b d^{2}\right )} x^{\frac {15}{2}} + \frac {2}{11} \, {\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{\frac {11}{2}} + \frac {2}{3} \, a^{2} c^{2} x^{\frac {3}{2}} + \frac {4}{7} \, {\left (a b c^{2} + a^{2} c d\right )} x^{\frac {7}{2}} \] Input:
integrate(x^(1/2)*(b*x^2+a)^2*(d*x^2+c)^2,x, algorithm="maxima")
Output:
2/19*b^2*d^2*x^(19/2) + 4/15*(b^2*c*d + a*b*d^2)*x^(15/2) + 2/11*(b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^(11/2) + 2/3*a^2*c^2*x^(3/2) + 4/7*(a*b*c^2 + a^2 *c*d)*x^(7/2)
Time = 0.12 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.97 \[ \int \sqrt {x} \left (a+b x^2\right )^2 \left (c+d x^2\right )^2 \, dx=\frac {2}{19} \, b^{2} d^{2} x^{\frac {19}{2}} + \frac {4}{15} \, b^{2} c d x^{\frac {15}{2}} + \frac {4}{15} \, a b d^{2} x^{\frac {15}{2}} + \frac {2}{11} \, b^{2} c^{2} x^{\frac {11}{2}} + \frac {8}{11} \, a b c d x^{\frac {11}{2}} + \frac {2}{11} \, a^{2} d^{2} x^{\frac {11}{2}} + \frac {4}{7} \, a b c^{2} x^{\frac {7}{2}} + \frac {4}{7} \, a^{2} c d x^{\frac {7}{2}} + \frac {2}{3} \, a^{2} c^{2} x^{\frac {3}{2}} \] Input:
integrate(x^(1/2)*(b*x^2+a)^2*(d*x^2+c)^2,x, algorithm="giac")
Output:
2/19*b^2*d^2*x^(19/2) + 4/15*b^2*c*d*x^(15/2) + 4/15*a*b*d^2*x^(15/2) + 2/ 11*b^2*c^2*x^(11/2) + 8/11*a*b*c*d*x^(11/2) + 2/11*a^2*d^2*x^(11/2) + 4/7* a*b*c^2*x^(7/2) + 4/7*a^2*c*d*x^(7/2) + 2/3*a^2*c^2*x^(3/2)
Time = 0.03 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.80 \[ \int \sqrt {x} \left (a+b x^2\right )^2 \left (c+d x^2\right )^2 \, dx=x^{11/2}\,\left (\frac {2\,a^2\,d^2}{11}+\frac {8\,a\,b\,c\,d}{11}+\frac {2\,b^2\,c^2}{11}\right )+\frac {2\,a^2\,c^2\,x^{3/2}}{3}+\frac {2\,b^2\,d^2\,x^{19/2}}{19}+\frac {4\,a\,c\,x^{7/2}\,\left (a\,d+b\,c\right )}{7}+\frac {4\,b\,d\,x^{15/2}\,\left (a\,d+b\,c\right )}{15} \] Input:
int(x^(1/2)*(a + b*x^2)^2*(c + d*x^2)^2,x)
Output:
x^(11/2)*((2*a^2*d^2)/11 + (2*b^2*c^2)/11 + (8*a*b*c*d)/11) + (2*a^2*c^2*x ^(3/2))/3 + (2*b^2*d^2*x^(19/2))/19 + (4*a*c*x^(7/2)*(a*d + b*c))/7 + (4*b *d*x^(15/2)*(a*d + b*c))/15
Time = 0.19 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.99 \[ \int \sqrt {x} \left (a+b x^2\right )^2 \left (c+d x^2\right )^2 \, dx=\frac {2 \sqrt {x}\, x \left (1155 b^{2} d^{2} x^{8}+2926 a b \,d^{2} x^{6}+2926 b^{2} c d \,x^{6}+1995 a^{2} d^{2} x^{4}+7980 a b c d \,x^{4}+1995 b^{2} c^{2} x^{4}+6270 a^{2} c d \,x^{2}+6270 a b \,c^{2} x^{2}+7315 a^{2} c^{2}\right )}{21945} \] Input:
int(x^(1/2)*(b*x^2+a)^2*(d*x^2+c)^2,x)
Output:
(2*sqrt(x)*x*(7315*a**2*c**2 + 6270*a**2*c*d*x**2 + 1995*a**2*d**2*x**4 + 6270*a*b*c**2*x**2 + 7980*a*b*c*d*x**4 + 2926*a*b*d**2*x**6 + 1995*b**2*c* *2*x**4 + 2926*b**2*c*d*x**6 + 1155*b**2*d**2*x**8))/21945