Integrand size = 20, antiderivative size = 71 \[ \int x \left (a+b x^2\right )^2 \left (c+d x^2\right )^3 \, dx=\frac {(b c-a d)^2 \left (c+d x^2\right )^4}{8 d^3}-\frac {b (b c-a d) \left (c+d x^2\right )^5}{5 d^3}+\frac {b^2 \left (c+d x^2\right )^6}{12 d^3} \] Output:
1/8*(-a*d+b*c)^2*(d*x^2+c)^4/d^3-1/5*b*(-a*d+b*c)*(d*x^2+c)^5/d^3+1/12*b^2 *(d*x^2+c)^6/d^3
Time = 0.02 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.68 \[ \int x \left (a+b x^2\right )^2 \left (c+d x^2\right )^3 \, dx=\frac {1}{120} x^2 \left (60 a^2 c^3+30 a c^2 (2 b c+3 a d) x^2+20 c \left (b^2 c^2+6 a b c d+3 a^2 d^2\right ) x^4+15 d \left (3 b^2 c^2+6 a b c d+a^2 d^2\right ) x^6+12 b d^2 (3 b c+2 a d) x^8+10 b^2 d^3 x^{10}\right ) \] Input:
Integrate[x*(a + b*x^2)^2*(c + d*x^2)^3,x]
Output:
(x^2*(60*a^2*c^3 + 30*a*c^2*(2*b*c + 3*a*d)*x^2 + 20*c*(b^2*c^2 + 6*a*b*c* d + 3*a^2*d^2)*x^4 + 15*d*(3*b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^6 + 12*b*d^2 *(3*b*c + 2*a*d)*x^8 + 10*b^2*d^3*x^10))/120
Time = 0.24 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.06, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {353, 49, 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.
\(\displaystyle \int x \left (a+b x^2\right )^2 \left (c+d x^2\right )^3 \, dx\) |
\(\Big \downarrow \) 353 |
\(\displaystyle \frac {1}{2} \int \left (b x^2+a\right )^2 \left (d x^2+c\right )^3dx^2\) |
\(\Big \downarrow \) 49 |
\(\displaystyle \frac {1}{2} \int \left (\frac {b^2 \left (d x^2+c\right )^5}{d^2}-\frac {2 b (b c-a d) \left (d x^2+c\right )^4}{d^2}+\frac {(a d-b c)^2 \left (d x^2+c\right )^3}{d^2}\right )dx^2\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} \left (-\frac {2 b \left (c+d x^2\right )^5 (b c-a d)}{5 d^3}+\frac {\left (c+d x^2\right )^4 (b c-a d)^2}{4 d^3}+\frac {b^2 \left (c+d x^2\right )^6}{6 d^3}\right )\) |
Input:
Int[x*(a + b*x^2)^2*(c + d*x^2)^3,x]
Output:
(((b*c - a*d)^2*(c + d*x^2)^4)/(4*d^3) - (2*b*(b*c - a*d)*(c + d*x^2)^5)/( 5*d^3) + (b^2*(c + d*x^2)^6)/(6*d^3))/2
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
Int[(x_)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_Symbol] :> Simp[1/2 Subst[Int[(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]
Time = 0.39 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.76
method | result | size |
norman | \(\frac {b^{2} d^{3} x^{12}}{12}+\left (\frac {1}{5} a \,d^{3} b +\frac {3}{10} b^{2} c \,d^{2}\right ) x^{10}+\left (\frac {1}{8} a^{2} d^{3}+\frac {3}{4} a c \,d^{2} b +\frac {3}{8} b^{2} c^{2} d \right ) x^{8}+\left (\frac {1}{2} c \,a^{2} d^{2}+a b \,c^{2} d +\frac {1}{6} b^{2} c^{3}\right ) x^{6}+\left (\frac {3}{4} a^{2} c^{2} d +\frac {1}{2} a b \,c^{3}\right ) x^{4}+\frac {a^{2} c^{3} x^{2}}{2}\) | \(125\) |
default | \(\frac {b^{2} d^{3} x^{12}}{12}+\frac {\left (2 a \,d^{3} b +3 b^{2} c \,d^{2}\right ) x^{10}}{10}+\frac {\left (a^{2} d^{3}+6 a c \,d^{2} b +3 b^{2} c^{2} d \right ) x^{8}}{8}+\frac {\left (3 c \,a^{2} d^{2}+6 a b \,c^{2} d +b^{2} c^{3}\right ) x^{6}}{6}+\frac {\left (3 a^{2} c^{2} d +2 a b \,c^{3}\right ) x^{4}}{4}+\frac {a^{2} c^{3} x^{2}}{2}\) | \(128\) |
gosper | \(\frac {1}{12} b^{2} d^{3} x^{12}+\frac {1}{5} x^{10} a \,d^{3} b +\frac {3}{10} x^{10} b^{2} c \,d^{2}+\frac {1}{8} x^{8} a^{2} d^{3}+\frac {3}{4} x^{8} a c \,d^{2} b +\frac {3}{8} x^{8} b^{2} c^{2} d +\frac {1}{2} x^{6} c \,a^{2} d^{2}+x^{6} a b \,c^{2} d +\frac {1}{6} x^{6} b^{2} c^{3}+\frac {3}{4} x^{4} a^{2} c^{2} d +\frac {1}{2} x^{4} a b \,c^{3}+\frac {1}{2} a^{2} c^{3} x^{2}\) | \(135\) |
risch | \(\frac {1}{12} b^{2} d^{3} x^{12}+\frac {1}{5} x^{10} a \,d^{3} b +\frac {3}{10} x^{10} b^{2} c \,d^{2}+\frac {1}{8} x^{8} a^{2} d^{3}+\frac {3}{4} x^{8} a c \,d^{2} b +\frac {3}{8} x^{8} b^{2} c^{2} d +\frac {1}{2} x^{6} c \,a^{2} d^{2}+x^{6} a b \,c^{2} d +\frac {1}{6} x^{6} b^{2} c^{3}+\frac {3}{4} x^{4} a^{2} c^{2} d +\frac {1}{2} x^{4} a b \,c^{3}+\frac {1}{2} a^{2} c^{3} x^{2}\) | \(135\) |
parallelrisch | \(\frac {1}{12} b^{2} d^{3} x^{12}+\frac {1}{5} x^{10} a \,d^{3} b +\frac {3}{10} x^{10} b^{2} c \,d^{2}+\frac {1}{8} x^{8} a^{2} d^{3}+\frac {3}{4} x^{8} a c \,d^{2} b +\frac {3}{8} x^{8} b^{2} c^{2} d +\frac {1}{2} x^{6} c \,a^{2} d^{2}+x^{6} a b \,c^{2} d +\frac {1}{6} x^{6} b^{2} c^{3}+\frac {3}{4} x^{4} a^{2} c^{2} d +\frac {1}{2} x^{4} a b \,c^{3}+\frac {1}{2} a^{2} c^{3} x^{2}\) | \(135\) |
orering | \(\frac {x^{2} \left (10 b^{2} d^{3} x^{10}+24 a b \,d^{3} x^{8}+36 b^{2} c \,d^{2} x^{8}+15 a^{2} d^{3} x^{6}+90 a b c \,d^{2} x^{6}+45 b^{2} c^{2} d \,x^{6}+60 a^{2} c \,d^{2} x^{4}+120 a b \,c^{2} d \,x^{4}+20 b^{2} c^{3} x^{4}+90 a^{2} c^{2} d \,x^{2}+60 a b \,c^{3} x^{2}+60 a^{2} c^{3}\right )}{120}\) | \(138\) |
Input:
int(x*(b*x^2+a)^2*(d*x^2+c)^3,x,method=_RETURNVERBOSE)
Output:
1/12*b^2*d^3*x^12+(1/5*a*d^3*b+3/10*b^2*c*d^2)*x^10+(1/8*a^2*d^3+3/4*a*c*d ^2*b+3/8*b^2*c^2*d)*x^8+(1/2*c*a^2*d^2+a*b*c^2*d+1/6*b^2*c^3)*x^6+(3/4*a^2 *c^2*d+1/2*a*b*c^3)*x^4+1/2*a^2*c^3*x^2
Time = 0.13 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.79 \[ \int x \left (a+b x^2\right )^2 \left (c+d x^2\right )^3 \, dx=\frac {1}{12} \, b^{2} d^{3} x^{12} + \frac {1}{10} \, {\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} x^{10} + \frac {1}{8} \, {\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} x^{8} + \frac {1}{2} \, a^{2} c^{3} x^{2} + \frac {1}{6} \, {\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x^{6} + \frac {1}{4} \, {\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} x^{4} \] Input:
integrate(x*(b*x^2+a)^2*(d*x^2+c)^3,x, algorithm="fricas")
Output:
1/12*b^2*d^3*x^12 + 1/10*(3*b^2*c*d^2 + 2*a*b*d^3)*x^10 + 1/8*(3*b^2*c^2*d + 6*a*b*c*d^2 + a^2*d^3)*x^8 + 1/2*a^2*c^3*x^2 + 1/6*(b^2*c^3 + 6*a*b*c^2 *d + 3*a^2*c*d^2)*x^6 + 1/4*(2*a*b*c^3 + 3*a^2*c^2*d)*x^4
Leaf count of result is larger than twice the leaf count of optimal. 136 vs. \(2 (60) = 120\).
Time = 0.03 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.92 \[ \int x \left (a+b x^2\right )^2 \left (c+d x^2\right )^3 \, dx=\frac {a^{2} c^{3} x^{2}}{2} + \frac {b^{2} d^{3} x^{12}}{12} + x^{10} \left (\frac {a b d^{3}}{5} + \frac {3 b^{2} c d^{2}}{10}\right ) + x^{8} \left (\frac {a^{2} d^{3}}{8} + \frac {3 a b c d^{2}}{4} + \frac {3 b^{2} c^{2} d}{8}\right ) + x^{6} \left (\frac {a^{2} c d^{2}}{2} + a b c^{2} d + \frac {b^{2} c^{3}}{6}\right ) + x^{4} \cdot \left (\frac {3 a^{2} c^{2} d}{4} + \frac {a b c^{3}}{2}\right ) \] Input:
integrate(x*(b*x**2+a)**2*(d*x**2+c)**3,x)
Output:
a**2*c**3*x**2/2 + b**2*d**3*x**12/12 + x**10*(a*b*d**3/5 + 3*b**2*c*d**2/ 10) + x**8*(a**2*d**3/8 + 3*a*b*c*d**2/4 + 3*b**2*c**2*d/8) + x**6*(a**2*c *d**2/2 + a*b*c**2*d + b**2*c**3/6) + x**4*(3*a**2*c**2*d/4 + a*b*c**3/2)
Time = 0.03 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.79 \[ \int x \left (a+b x^2\right )^2 \left (c+d x^2\right )^3 \, dx=\frac {1}{12} \, b^{2} d^{3} x^{12} + \frac {1}{10} \, {\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} x^{10} + \frac {1}{8} \, {\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} x^{8} + \frac {1}{2} \, a^{2} c^{3} x^{2} + \frac {1}{6} \, {\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x^{6} + \frac {1}{4} \, {\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} x^{4} \] Input:
integrate(x*(b*x^2+a)^2*(d*x^2+c)^3,x, algorithm="maxima")
Output:
1/12*b^2*d^3*x^12 + 1/10*(3*b^2*c*d^2 + 2*a*b*d^3)*x^10 + 1/8*(3*b^2*c^2*d + 6*a*b*c*d^2 + a^2*d^3)*x^8 + 1/2*a^2*c^3*x^2 + 1/6*(b^2*c^3 + 6*a*b*c^2 *d + 3*a^2*c*d^2)*x^6 + 1/4*(2*a*b*c^3 + 3*a^2*c^2*d)*x^4
Leaf count of result is larger than twice the leaf count of optimal. 134 vs. \(2 (65) = 130\).
Time = 0.12 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.89 \[ \int x \left (a+b x^2\right )^2 \left (c+d x^2\right )^3 \, dx=\frac {1}{12} \, b^{2} d^{3} x^{12} + \frac {3}{10} \, b^{2} c d^{2} x^{10} + \frac {1}{5} \, a b d^{3} x^{10} + \frac {3}{8} \, b^{2} c^{2} d x^{8} + \frac {3}{4} \, a b c d^{2} x^{8} + \frac {1}{8} \, a^{2} d^{3} x^{8} + \frac {1}{6} \, b^{2} c^{3} x^{6} + a b c^{2} d x^{6} + \frac {1}{2} \, a^{2} c d^{2} x^{6} + \frac {1}{2} \, a b c^{3} x^{4} + \frac {3}{4} \, a^{2} c^{2} d x^{4} + \frac {1}{2} \, a^{2} c^{3} x^{2} \] Input:
integrate(x*(b*x^2+a)^2*(d*x^2+c)^3,x, algorithm="giac")
Output:
1/12*b^2*d^3*x^12 + 3/10*b^2*c*d^2*x^10 + 1/5*a*b*d^3*x^10 + 3/8*b^2*c^2*d *x^8 + 3/4*a*b*c*d^2*x^8 + 1/8*a^2*d^3*x^8 + 1/6*b^2*c^3*x^6 + a*b*c^2*d*x ^6 + 1/2*a^2*c*d^2*x^6 + 1/2*a*b*c^3*x^4 + 3/4*a^2*c^2*d*x^4 + 1/2*a^2*c^3 *x^2
Time = 0.04 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.66 \[ \int x \left (a+b x^2\right )^2 \left (c+d x^2\right )^3 \, dx=x^6\,\left (\frac {a^2\,c\,d^2}{2}+a\,b\,c^2\,d+\frac {b^2\,c^3}{6}\right )+x^8\,\left (\frac {a^2\,d^3}{8}+\frac {3\,a\,b\,c\,d^2}{4}+\frac {3\,b^2\,c^2\,d}{8}\right )+\frac {a^2\,c^3\,x^2}{2}+\frac {b^2\,d^3\,x^{12}}{12}+\frac {a\,c^2\,x^4\,\left (3\,a\,d+2\,b\,c\right )}{4}+\frac {b\,d^2\,x^{10}\,\left (2\,a\,d+3\,b\,c\right )}{10} \] Input:
int(x*(a + b*x^2)^2*(c + d*x^2)^3,x)
Output:
x^6*((b^2*c^3)/6 + (a^2*c*d^2)/2 + a*b*c^2*d) + x^8*((a^2*d^3)/8 + (3*b^2* c^2*d)/8 + (3*a*b*c*d^2)/4) + (a^2*c^3*x^2)/2 + (b^2*d^3*x^12)/12 + (a*c^2 *x^4*(3*a*d + 2*b*c))/4 + (b*d^2*x^10*(2*a*d + 3*b*c))/10
Time = 0.21 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.93 \[ \int x \left (a+b x^2\right )^2 \left (c+d x^2\right )^3 \, dx=\frac {x^{2} \left (10 b^{2} d^{3} x^{10}+24 a b \,d^{3} x^{8}+36 b^{2} c \,d^{2} x^{8}+15 a^{2} d^{3} x^{6}+90 a b c \,d^{2} x^{6}+45 b^{2} c^{2} d \,x^{6}+60 a^{2} c \,d^{2} x^{4}+120 a b \,c^{2} d \,x^{4}+20 b^{2} c^{3} x^{4}+90 a^{2} c^{2} d \,x^{2}+60 a b \,c^{3} x^{2}+60 a^{2} c^{3}\right )}{120} \] Input:
int(x*(b*x^2+a)^2*(d*x^2+c)^3,x)
Output:
(x**2*(60*a**2*c**3 + 90*a**2*c**2*d*x**2 + 60*a**2*c*d**2*x**4 + 15*a**2* d**3*x**6 + 60*a*b*c**3*x**2 + 120*a*b*c**2*d*x**4 + 90*a*b*c*d**2*x**6 + 24*a*b*d**3*x**8 + 20*b**2*c**3*x**4 + 45*b**2*c**2*d*x**6 + 36*b**2*c*d** 2*x**8 + 10*b**2*d**3*x**10))/120