Integrand size = 18, antiderivative size = 102 \[ \int x (c+d x)^2 \left (a+b x^2\right )^3 \, dx=\frac {2}{3} a^3 c d x^3+\frac {6}{5} a^2 b c d x^5+\frac {6}{7} a b^2 c d x^7+\frac {2}{9} b^3 c d x^9+\frac {\left (5 b c^2-a d^2\right ) \left (a+b x^2\right )^4}{40 b^2}+\frac {d^2 x^2 \left (a+b x^2\right )^4}{10 b} \] Output:
2/3*a^3*c*d*x^3+6/5*a^2*b*c*d*x^5+6/7*a*b^2*c*d*x^7+2/9*b^3*c*d*x^9+1/40*( -a*d^2+5*b*c^2)*(b*x^2+a)^4/b^2+1/10*d^2*x^2*(b*x^2+a)^4/b
Time = 0.03 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.13 \[ \int x (c+d x)^2 \left (a+b x^2\right )^3 \, dx=\frac {x^2 \left (210 a^3 \left (6 c^2+8 c d x+3 d^2 x^2\right )+126 a^2 b x^2 \left (15 c^2+24 c d x+10 d^2 x^2\right )+45 a b^2 x^4 \left (28 c^2+48 c d x+21 d^2 x^2\right )+7 b^3 x^6 \left (45 c^2+80 c d x+36 d^2 x^2\right )\right )}{2520} \] Input:
Integrate[x*(c + d*x)^2*(a + b*x^2)^3,x]
Output:
(x^2*(210*a^3*(6*c^2 + 8*c*d*x + 3*d^2*x^2) + 126*a^2*b*x^2*(15*c^2 + 24*c *d*x + 10*d^2*x^2) + 45*a*b^2*x^4*(28*c^2 + 48*c*d*x + 21*d^2*x^2) + 7*b^3 *x^6*(45*c^2 + 80*c*d*x + 36*d^2*x^2)))/2520
Time = 0.50 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.38, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {522, 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.
\(\displaystyle \int x \left (a+b x^2\right )^3 (c+d x)^2 \, dx\) |
\(\Big \downarrow \) 522 |
\(\displaystyle \int \left (a^3 c^2 x+2 a^3 c d x^2+a^2 x^3 \left (a d^2+3 b c^2\right )+6 a^2 b c d x^4+b^2 x^7 \left (3 a d^2+b c^2\right )+6 a b^2 c d x^6+3 a b x^5 \left (a d^2+b c^2\right )+2 b^3 c d x^8+b^3 d^2 x^9\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} a^3 c^2 x^2+\frac {2}{3} a^3 c d x^3+\frac {1}{4} a^2 x^4 \left (a d^2+3 b c^2\right )+\frac {6}{5} a^2 b c d x^5+\frac {1}{8} b^2 x^8 \left (3 a d^2+b c^2\right )+\frac {6}{7} a b^2 c d x^7+\frac {1}{2} a b x^6 \left (a d^2+b c^2\right )+\frac {2}{9} b^3 c d x^9+\frac {1}{10} b^3 d^2 x^{10}\) |
Input:
Int[x*(c + d*x)^2*(a + b*x^2)^3,x]
Output:
(a^3*c^2*x^2)/2 + (2*a^3*c*d*x^3)/3 + (a^2*(3*b*c^2 + a*d^2)*x^4)/4 + (6*a ^2*b*c*d*x^5)/5 + (a*b*(b*c^2 + a*d^2)*x^6)/2 + (6*a*b^2*c*d*x^7)/7 + (b^2 *(b*c^2 + 3*a*d^2)*x^8)/8 + (2*b^3*c*d*x^9)/9 + (b^3*d^2*x^10)/10
Int[((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_. ), x_Symbol] :> Int[ExpandIntegrand[(e*x)^m*(c + d*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 0]
Time = 0.24 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.30
method | result | size |
norman | \(\frac {d^{2} b^{3} x^{10}}{10}+\frac {2 b^{3} c d \,x^{9}}{9}+\left (\frac {3}{8} a \,b^{2} d^{2}+\frac {1}{8} b^{3} c^{2}\right ) x^{8}+\frac {6 a \,b^{2} c d \,x^{7}}{7}+\left (\frac {1}{2} a^{2} b \,d^{2}+\frac {1}{2} a \,c^{2} b^{2}\right ) x^{6}+\frac {6 a^{2} b c d \,x^{5}}{5}+\left (\frac {1}{4} a^{3} d^{2}+\frac {3}{4} a^{2} b \,c^{2}\right ) x^{4}+\frac {2 a^{3} c d \,x^{3}}{3}+\frac {c^{2} a^{3} x^{2}}{2}\) | \(133\) |
default | \(\frac {d^{2} b^{3} x^{10}}{10}+\frac {2 b^{3} c d \,x^{9}}{9}+\frac {\left (3 a \,b^{2} d^{2}+b^{3} c^{2}\right ) x^{8}}{8}+\frac {6 a \,b^{2} c d \,x^{7}}{7}+\frac {\left (3 a^{2} b \,d^{2}+3 a \,c^{2} b^{2}\right ) x^{6}}{6}+\frac {6 a^{2} b c d \,x^{5}}{5}+\frac {\left (a^{3} d^{2}+3 a^{2} b \,c^{2}\right ) x^{4}}{4}+\frac {2 a^{3} c d \,x^{3}}{3}+\frac {c^{2} a^{3} x^{2}}{2}\) | \(134\) |
gosper | \(\frac {1}{10} d^{2} b^{3} x^{10}+\frac {2}{9} b^{3} c d \,x^{9}+\frac {3}{8} x^{8} a \,b^{2} d^{2}+\frac {1}{8} x^{8} b^{3} c^{2}+\frac {6}{7} a \,b^{2} c d \,x^{7}+\frac {1}{2} x^{6} a^{2} b \,d^{2}+\frac {1}{2} x^{6} a \,c^{2} b^{2}+\frac {6}{5} a^{2} b c d \,x^{5}+\frac {1}{4} x^{4} a^{3} d^{2}+\frac {3}{4} x^{4} a^{2} b \,c^{2}+\frac {2}{3} a^{3} c d \,x^{3}+\frac {1}{2} c^{2} a^{3} x^{2}\) | \(136\) |
risch | \(\frac {1}{10} d^{2} b^{3} x^{10}+\frac {2}{9} b^{3} c d \,x^{9}+\frac {3}{8} x^{8} a \,b^{2} d^{2}+\frac {1}{8} x^{8} b^{3} c^{2}+\frac {6}{7} a \,b^{2} c d \,x^{7}+\frac {1}{2} x^{6} a^{2} b \,d^{2}+\frac {1}{2} x^{6} a \,c^{2} b^{2}+\frac {6}{5} a^{2} b c d \,x^{5}+\frac {1}{4} x^{4} a^{3} d^{2}+\frac {3}{4} x^{4} a^{2} b \,c^{2}+\frac {2}{3} a^{3} c d \,x^{3}+\frac {1}{2} c^{2} a^{3} x^{2}\) | \(136\) |
parallelrisch | \(\frac {1}{10} d^{2} b^{3} x^{10}+\frac {2}{9} b^{3} c d \,x^{9}+\frac {3}{8} x^{8} a \,b^{2} d^{2}+\frac {1}{8} x^{8} b^{3} c^{2}+\frac {6}{7} a \,b^{2} c d \,x^{7}+\frac {1}{2} x^{6} a^{2} b \,d^{2}+\frac {1}{2} x^{6} a \,c^{2} b^{2}+\frac {6}{5} a^{2} b c d \,x^{5}+\frac {1}{4} x^{4} a^{3} d^{2}+\frac {3}{4} x^{4} a^{2} b \,c^{2}+\frac {2}{3} a^{3} c d \,x^{3}+\frac {1}{2} c^{2} a^{3} x^{2}\) | \(136\) |
orering | \(\frac {x^{2} \left (252 b^{3} d^{2} x^{8}+560 d \,x^{7} c \,b^{3}+945 a \,b^{2} d^{2} x^{6}+315 b^{3} c^{2} x^{6}+2160 a \,b^{2} c d \,x^{5}+1260 a^{2} b \,d^{2} x^{4}+1260 a \,b^{2} c^{2} x^{4}+3024 a^{2} b c d \,x^{3}+630 a^{3} d^{2} x^{2}+1890 a^{2} b \,c^{2} x^{2}+1680 a^{3} c d x +1260 c^{2} a^{3}\right )}{2520}\) | \(136\) |
Input:
int(x*(d*x+c)^2*(b*x^2+a)^3,x,method=_RETURNVERBOSE)
Output:
1/10*d^2*b^3*x^10+2/9*b^3*c*d*x^9+(3/8*a*b^2*d^2+1/8*b^3*c^2)*x^8+6/7*a*b^ 2*c*d*x^7+(1/2*a^2*b*d^2+1/2*a*c^2*b^2)*x^6+6/5*a^2*b*c*d*x^5+(1/4*a^3*d^2 +3/4*a^2*b*c^2)*x^4+2/3*a^3*c*d*x^3+1/2*c^2*a^3*x^2
Time = 0.08 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.28 \[ \int x (c+d x)^2 \left (a+b x^2\right )^3 \, dx=\frac {1}{10} \, b^{3} d^{2} x^{10} + \frac {2}{9} \, b^{3} c d x^{9} + \frac {6}{7} \, a b^{2} c d x^{7} + \frac {6}{5} \, a^{2} b c d x^{5} + \frac {1}{8} \, {\left (b^{3} c^{2} + 3 \, a b^{2} d^{2}\right )} x^{8} + \frac {2}{3} \, a^{3} c d x^{3} + \frac {1}{2} \, a^{3} c^{2} x^{2} + \frac {1}{2} \, {\left (a b^{2} c^{2} + a^{2} b d^{2}\right )} x^{6} + \frac {1}{4} \, {\left (3 \, a^{2} b c^{2} + a^{3} d^{2}\right )} x^{4} \] Input:
integrate(x*(d*x+c)^2*(b*x^2+a)^3,x, algorithm="fricas")
Output:
1/10*b^3*d^2*x^10 + 2/9*b^3*c*d*x^9 + 6/7*a*b^2*c*d*x^7 + 6/5*a^2*b*c*d*x^ 5 + 1/8*(b^3*c^2 + 3*a*b^2*d^2)*x^8 + 2/3*a^3*c*d*x^3 + 1/2*a^3*c^2*x^2 + 1/2*(a*b^2*c^2 + a^2*b*d^2)*x^6 + 1/4*(3*a^2*b*c^2 + a^3*d^2)*x^4
Time = 0.03 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.48 \[ \int x (c+d x)^2 \left (a+b x^2\right )^3 \, dx=\frac {a^{3} c^{2} x^{2}}{2} + \frac {2 a^{3} c d x^{3}}{3} + \frac {6 a^{2} b c d x^{5}}{5} + \frac {6 a b^{2} c d x^{7}}{7} + \frac {2 b^{3} c d x^{9}}{9} + \frac {b^{3} d^{2} x^{10}}{10} + x^{8} \cdot \left (\frac {3 a b^{2} d^{2}}{8} + \frac {b^{3} c^{2}}{8}\right ) + x^{6} \left (\frac {a^{2} b d^{2}}{2} + \frac {a b^{2} c^{2}}{2}\right ) + x^{4} \left (\frac {a^{3} d^{2}}{4} + \frac {3 a^{2} b c^{2}}{4}\right ) \] Input:
integrate(x*(d*x+c)**2*(b*x**2+a)**3,x)
Output:
a**3*c**2*x**2/2 + 2*a**3*c*d*x**3/3 + 6*a**2*b*c*d*x**5/5 + 6*a*b**2*c*d* x**7/7 + 2*b**3*c*d*x**9/9 + b**3*d**2*x**10/10 + x**8*(3*a*b**2*d**2/8 + b**3*c**2/8) + x**6*(a**2*b*d**2/2 + a*b**2*c**2/2) + x**4*(a**3*d**2/4 + 3*a**2*b*c**2/4)
Time = 0.03 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.28 \[ \int x (c+d x)^2 \left (a+b x^2\right )^3 \, dx=\frac {1}{10} \, b^{3} d^{2} x^{10} + \frac {2}{9} \, b^{3} c d x^{9} + \frac {6}{7} \, a b^{2} c d x^{7} + \frac {6}{5} \, a^{2} b c d x^{5} + \frac {1}{8} \, {\left (b^{3} c^{2} + 3 \, a b^{2} d^{2}\right )} x^{8} + \frac {2}{3} \, a^{3} c d x^{3} + \frac {1}{2} \, a^{3} c^{2} x^{2} + \frac {1}{2} \, {\left (a b^{2} c^{2} + a^{2} b d^{2}\right )} x^{6} + \frac {1}{4} \, {\left (3 \, a^{2} b c^{2} + a^{3} d^{2}\right )} x^{4} \] Input:
integrate(x*(d*x+c)^2*(b*x^2+a)^3,x, algorithm="maxima")
Output:
1/10*b^3*d^2*x^10 + 2/9*b^3*c*d*x^9 + 6/7*a*b^2*c*d*x^7 + 6/5*a^2*b*c*d*x^ 5 + 1/8*(b^3*c^2 + 3*a*b^2*d^2)*x^8 + 2/3*a^3*c*d*x^3 + 1/2*a^3*c^2*x^2 + 1/2*(a*b^2*c^2 + a^2*b*d^2)*x^6 + 1/4*(3*a^2*b*c^2 + a^3*d^2)*x^4
Time = 0.12 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.32 \[ \int x (c+d x)^2 \left (a+b x^2\right )^3 \, dx=\frac {1}{10} \, b^{3} d^{2} x^{10} + \frac {2}{9} \, b^{3} c d x^{9} + \frac {1}{8} \, b^{3} c^{2} x^{8} + \frac {3}{8} \, a b^{2} d^{2} x^{8} + \frac {6}{7} \, a b^{2} c d x^{7} + \frac {1}{2} \, a b^{2} c^{2} x^{6} + \frac {1}{2} \, a^{2} b d^{2} x^{6} + \frac {6}{5} \, a^{2} b c d x^{5} + \frac {3}{4} \, a^{2} b c^{2} x^{4} + \frac {1}{4} \, a^{3} d^{2} x^{4} + \frac {2}{3} \, a^{3} c d x^{3} + \frac {1}{2} \, a^{3} c^{2} x^{2} \] Input:
integrate(x*(d*x+c)^2*(b*x^2+a)^3,x, algorithm="giac")
Output:
1/10*b^3*d^2*x^10 + 2/9*b^3*c*d*x^9 + 1/8*b^3*c^2*x^8 + 3/8*a*b^2*d^2*x^8 + 6/7*a*b^2*c*d*x^7 + 1/2*a*b^2*c^2*x^6 + 1/2*a^2*b*d^2*x^6 + 6/5*a^2*b*c* d*x^5 + 3/4*a^2*b*c^2*x^4 + 1/4*a^3*d^2*x^4 + 2/3*a^3*c*d*x^3 + 1/2*a^3*c^ 2*x^2
Time = 0.05 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.25 \[ \int x (c+d x)^2 \left (a+b x^2\right )^3 \, dx=x^4\,\left (\frac {a^3\,d^2}{4}+\frac {3\,b\,a^2\,c^2}{4}\right )+x^8\,\left (\frac {b^3\,c^2}{8}+\frac {3\,a\,b^2\,d^2}{8}\right )+\frac {a^3\,c^2\,x^2}{2}+\frac {b^3\,d^2\,x^{10}}{10}+\frac {a\,b\,x^6\,\left (b\,c^2+a\,d^2\right )}{2}+\frac {2\,a^3\,c\,d\,x^3}{3}+\frac {2\,b^3\,c\,d\,x^9}{9}+\frac {6\,a^2\,b\,c\,d\,x^5}{5}+\frac {6\,a\,b^2\,c\,d\,x^7}{7} \] Input:
int(x*(a + b*x^2)^3*(c + d*x)^2,x)
Output:
x^4*((a^3*d^2)/4 + (3*a^2*b*c^2)/4) + x^8*((b^3*c^2)/8 + (3*a*b^2*d^2)/8) + (a^3*c^2*x^2)/2 + (b^3*d^2*x^10)/10 + (a*b*x^6*(a*d^2 + b*c^2))/2 + (2*a ^3*c*d*x^3)/3 + (2*b^3*c*d*x^9)/9 + (6*a^2*b*c*d*x^5)/5 + (6*a*b^2*c*d*x^7 )/7
Time = 0.18 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.32 \[ \int x (c+d x)^2 \left (a+b x^2\right )^3 \, dx=\frac {x^{2} \left (252 b^{3} d^{2} x^{8}+560 b^{3} c d \,x^{7}+945 a \,b^{2} d^{2} x^{6}+315 b^{3} c^{2} x^{6}+2160 a \,b^{2} c d \,x^{5}+1260 a^{2} b \,d^{2} x^{4}+1260 a \,b^{2} c^{2} x^{4}+3024 a^{2} b c d \,x^{3}+630 a^{3} d^{2} x^{2}+1890 a^{2} b \,c^{2} x^{2}+1680 a^{3} c d x +1260 a^{3} c^{2}\right )}{2520} \] Input:
int(x*(d*x+c)^2*(b*x^2+a)^3,x)
Output:
(x**2*(1260*a**3*c**2 + 1680*a**3*c*d*x + 630*a**3*d**2*x**2 + 1890*a**2*b *c**2*x**2 + 3024*a**2*b*c*d*x**3 + 1260*a**2*b*d**2*x**4 + 1260*a*b**2*c* *2*x**4 + 2160*a*b**2*c*d*x**5 + 945*a*b**2*d**2*x**6 + 315*b**3*c**2*x**6 + 560*b**3*c*d*x**7 + 252*b**3*d**2*x**8))/2520