Integrand size = 19, antiderivative size = 122 \[ \int \left (a+b x^3\right )^2 \left (c+d x^3\right )^3 \, dx=a^2 c^3 x+\frac {1}{4} a c^2 (2 b c+3 a d) x^4+\frac {1}{7} c \left (b^2 c^2+6 a b c d+3 a^2 d^2\right ) x^7+\frac {1}{10} d \left (3 b^2 c^2+6 a b c d+a^2 d^2\right ) x^{10}+\frac {1}{13} b d^2 (3 b c+2 a d) x^{13}+\frac {1}{16} b^2 d^3 x^{16} \] Output:
a^2*c^3*x+1/4*a*c^2*(3*a*d+2*b*c)*x^4+1/7*c*(3*a^2*d^2+6*a*b*c*d+b^2*c^2)* x^7+1/10*d*(a^2*d^2+6*a*b*c*d+3*b^2*c^2)*x^10+1/13*b*d^2*(2*a*d+3*b*c)*x^1 3+1/16*b^2*d^3*x^16
Time = 0.01 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.00 \[ \int \left (a+b x^3\right )^2 \left (c+d x^3\right )^3 \, dx=a^2 c^3 x+\frac {1}{4} a c^2 (2 b c+3 a d) x^4+\frac {1}{7} c \left (b^2 c^2+6 a b c d+3 a^2 d^2\right ) x^7+\frac {1}{10} d \left (3 b^2 c^2+6 a b c d+a^2 d^2\right ) x^{10}+\frac {1}{13} b d^2 (3 b c+2 a d) x^{13}+\frac {1}{16} b^2 d^3 x^{16} \] Input:
Integrate[(a + b*x^3)^2*(c + d*x^3)^3,x]
Output:
a^2*c^3*x + (a*c^2*(2*b*c + 3*a*d)*x^4)/4 + (c*(b^2*c^2 + 6*a*b*c*d + 3*a^ 2*d^2)*x^7)/7 + (d*(3*b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^10)/10 + (b*d^2*(3* b*c + 2*a*d)*x^13)/13 + (b^2*d^3*x^16)/16
Time = 0.48 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {897, 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.
\(\displaystyle \int \left (a+b x^3\right )^2 \left (c+d x^3\right )^3 \, dx\) |
\(\Big \downarrow \) 897 |
\(\displaystyle \int \left (d x^9 \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+c x^6 \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )+a^2 c^3+a c^2 x^3 (3 a d+2 b c)+b d^2 x^{12} (2 a d+3 b c)+b^2 d^3 x^{15}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{10} d x^{10} \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac {1}{7} c x^7 \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )+a^2 c^3 x+\frac {1}{4} a c^2 x^4 (3 a d+2 b c)+\frac {1}{13} b d^2 x^{13} (2 a d+3 b c)+\frac {1}{16} b^2 d^3 x^{16}\) |
Input:
Int[(a + b*x^3)^2*(c + d*x^3)^3,x]
Output:
a^2*c^3*x + (a*c^2*(2*b*c + 3*a*d)*x^4)/4 + (c*(b^2*c^2 + 6*a*b*c*d + 3*a^ 2*d^2)*x^7)/7 + (d*(3*b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^10)/10 + (b*d^2*(3* b*c + 2*a*d)*x^13)/13 + (b^2*d^3*x^16)/16
Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol ] :> Int[ExpandIntegrand[(a + b*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b , c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[p, 0] && IGtQ[q, 0]
Time = 0.76 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.01
method | result | size |
norman | \(a^{2} c^{3} x +\left (\frac {3}{4} a^{2} c^{2} d +\frac {1}{2} a b \,c^{3}\right ) x^{4}+\left (\frac {3}{7} c \,a^{2} d^{2}+\frac {6}{7} a b \,c^{2} d +\frac {1}{7} b^{2} c^{3}\right ) x^{7}+\left (\frac {1}{10} a^{2} d^{3}+\frac {3}{5} a c \,d^{2} b +\frac {3}{10} b^{2} c^{2} d \right ) x^{10}+\left (\frac {2}{13} a \,d^{3} b +\frac {3}{13} b^{2} c \,d^{2}\right ) x^{13}+\frac {b^{2} d^{3} x^{16}}{16}\) | \(123\) |
default | \(\frac {b^{2} d^{3} x^{16}}{16}+\frac {\left (2 a \,d^{3} b +3 b^{2} c \,d^{2}\right ) x^{13}}{13}+\frac {\left (a^{2} d^{3}+6 a c \,d^{2} b +3 b^{2} c^{2} d \right ) x^{10}}{10}+\frac {\left (3 c \,a^{2} d^{2}+6 a b \,c^{2} d +b^{2} c^{3}\right ) x^{7}}{7}+\frac {\left (3 a^{2} c^{2} d +2 a b \,c^{3}\right ) x^{4}}{4}+a^{2} c^{3} x\) | \(125\) |
gosper | \(a^{2} c^{3} x +\frac {3}{4} x^{4} a^{2} c^{2} d +\frac {1}{2} x^{4} a b \,c^{3}+\frac {3}{7} x^{7} c \,a^{2} d^{2}+\frac {6}{7} x^{7} a b \,c^{2} d +\frac {1}{7} x^{7} b^{2} c^{3}+\frac {1}{10} x^{10} a^{2} d^{3}+\frac {3}{5} x^{10} a c \,d^{2} b +\frac {3}{10} x^{10} b^{2} c^{2} d +\frac {2}{13} x^{13} a \,d^{3} b +\frac {3}{13} x^{13} b^{2} c \,d^{2}+\frac {1}{16} b^{2} d^{3} x^{16}\) | \(133\) |
risch | \(a^{2} c^{3} x +\frac {3}{4} x^{4} a^{2} c^{2} d +\frac {1}{2} x^{4} a b \,c^{3}+\frac {3}{7} x^{7} c \,a^{2} d^{2}+\frac {6}{7} x^{7} a b \,c^{2} d +\frac {1}{7} x^{7} b^{2} c^{3}+\frac {1}{10} x^{10} a^{2} d^{3}+\frac {3}{5} x^{10} a c \,d^{2} b +\frac {3}{10} x^{10} b^{2} c^{2} d +\frac {2}{13} x^{13} a \,d^{3} b +\frac {3}{13} x^{13} b^{2} c \,d^{2}+\frac {1}{16} b^{2} d^{3} x^{16}\) | \(133\) |
parallelrisch | \(a^{2} c^{3} x +\frac {3}{4} x^{4} a^{2} c^{2} d +\frac {1}{2} x^{4} a b \,c^{3}+\frac {3}{7} x^{7} c \,a^{2} d^{2}+\frac {6}{7} x^{7} a b \,c^{2} d +\frac {1}{7} x^{7} b^{2} c^{3}+\frac {1}{10} x^{10} a^{2} d^{3}+\frac {3}{5} x^{10} a c \,d^{2} b +\frac {3}{10} x^{10} b^{2} c^{2} d +\frac {2}{13} x^{13} a \,d^{3} b +\frac {3}{13} x^{13} b^{2} c \,d^{2}+\frac {1}{16} b^{2} d^{3} x^{16}\) | \(133\) |
orering | \(\frac {x \left (455 b^{2} d^{3} x^{15}+1120 a b \,d^{3} x^{12}+1680 b^{2} c \,d^{2} x^{12}+728 a^{2} d^{3} x^{9}+4368 a b c \,d^{2} x^{9}+2184 b^{2} c^{2} d \,x^{9}+3120 a^{2} c \,d^{2} x^{6}+6240 a b \,c^{2} d \,x^{6}+1040 b^{2} c^{3} x^{6}+5460 a^{2} c^{2} d \,x^{3}+3640 a b \,c^{3} x^{3}+7280 a^{2} c^{3}\right )}{7280}\) | \(136\) |
Input:
int((b*x^3+a)^2*(d*x^3+c)^3,x,method=_RETURNVERBOSE)
Output:
a^2*c^3*x+(3/4*a^2*c^2*d+1/2*a*b*c^3)*x^4+(3/7*c*a^2*d^2+6/7*a*b*c^2*d+1/7 *b^2*c^3)*x^7+(1/10*a^2*d^3+3/5*a*c*d^2*b+3/10*b^2*c^2*d)*x^10+(2/13*a*d^3 *b+3/13*b^2*c*d^2)*x^13+1/16*b^2*d^3*x^16
Time = 0.07 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.02 \[ \int \left (a+b x^3\right )^2 \left (c+d x^3\right )^3 \, dx=\frac {1}{16} \, b^{2} d^{3} x^{16} + \frac {1}{13} \, {\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} x^{13} + \frac {1}{10} \, {\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} x^{10} + \frac {1}{7} \, {\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x^{7} + a^{2} c^{3} x + \frac {1}{4} \, {\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} x^{4} \] Input:
integrate((b*x^3+a)^2*(d*x^3+c)^3,x, algorithm="fricas")
Output:
1/16*b^2*d^3*x^16 + 1/13*(3*b^2*c*d^2 + 2*a*b*d^3)*x^13 + 1/10*(3*b^2*c^2* d + 6*a*b*c*d^2 + a^2*d^3)*x^10 + 1/7*(b^2*c^3 + 6*a*b*c^2*d + 3*a^2*c*d^2 )*x^7 + a^2*c^3*x + 1/4*(2*a*b*c^3 + 3*a^2*c^2*d)*x^4
Time = 0.04 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.14 \[ \int \left (a+b x^3\right )^2 \left (c+d x^3\right )^3 \, dx=a^{2} c^{3} x + \frac {b^{2} d^{3} x^{16}}{16} + x^{13} \cdot \left (\frac {2 a b d^{3}}{13} + \frac {3 b^{2} c d^{2}}{13}\right ) + x^{10} \left (\frac {a^{2} d^{3}}{10} + \frac {3 a b c d^{2}}{5} + \frac {3 b^{2} c^{2} d}{10}\right ) + x^{7} \cdot \left (\frac {3 a^{2} c d^{2}}{7} + \frac {6 a b c^{2} d}{7} + \frac {b^{2} c^{3}}{7}\right ) + x^{4} \cdot \left (\frac {3 a^{2} c^{2} d}{4} + \frac {a b c^{3}}{2}\right ) \] Input:
integrate((b*x**3+a)**2*(d*x**3+c)**3,x)
Output:
a**2*c**3*x + b**2*d**3*x**16/16 + x**13*(2*a*b*d**3/13 + 3*b**2*c*d**2/13 ) + x**10*(a**2*d**3/10 + 3*a*b*c*d**2/5 + 3*b**2*c**2*d/10) + x**7*(3*a** 2*c*d**2/7 + 6*a*b*c**2*d/7 + b**2*c**3/7) + x**4*(3*a**2*c**2*d/4 + a*b*c **3/2)
Time = 0.03 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.02 \[ \int \left (a+b x^3\right )^2 \left (c+d x^3\right )^3 \, dx=\frac {1}{16} \, b^{2} d^{3} x^{16} + \frac {1}{13} \, {\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} x^{13} + \frac {1}{10} \, {\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} x^{10} + \frac {1}{7} \, {\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x^{7} + a^{2} c^{3} x + \frac {1}{4} \, {\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} x^{4} \] Input:
integrate((b*x^3+a)^2*(d*x^3+c)^3,x, algorithm="maxima")
Output:
1/16*b^2*d^3*x^16 + 1/13*(3*b^2*c*d^2 + 2*a*b*d^3)*x^13 + 1/10*(3*b^2*c^2* d + 6*a*b*c*d^2 + a^2*d^3)*x^10 + 1/7*(b^2*c^3 + 6*a*b*c^2*d + 3*a^2*c*d^2 )*x^7 + a^2*c^3*x + 1/4*(2*a*b*c^3 + 3*a^2*c^2*d)*x^4
Time = 0.12 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.08 \[ \int \left (a+b x^3\right )^2 \left (c+d x^3\right )^3 \, dx=\frac {1}{16} \, b^{2} d^{3} x^{16} + \frac {3}{13} \, b^{2} c d^{2} x^{13} + \frac {2}{13} \, a b d^{3} x^{13} + \frac {3}{10} \, b^{2} c^{2} d x^{10} + \frac {3}{5} \, a b c d^{2} x^{10} + \frac {1}{10} \, a^{2} d^{3} x^{10} + \frac {1}{7} \, b^{2} c^{3} x^{7} + \frac {6}{7} \, a b c^{2} d x^{7} + \frac {3}{7} \, a^{2} c d^{2} x^{7} + \frac {1}{2} \, a b c^{3} x^{4} + \frac {3}{4} \, a^{2} c^{2} d x^{4} + a^{2} c^{3} x \] Input:
integrate((b*x^3+a)^2*(d*x^3+c)^3,x, algorithm="giac")
Output:
1/16*b^2*d^3*x^16 + 3/13*b^2*c*d^2*x^13 + 2/13*a*b*d^3*x^13 + 3/10*b^2*c^2 *d*x^10 + 3/5*a*b*c*d^2*x^10 + 1/10*a^2*d^3*x^10 + 1/7*b^2*c^3*x^7 + 6/7*a *b*c^2*d*x^7 + 3/7*a^2*c*d^2*x^7 + 1/2*a*b*c^3*x^4 + 3/4*a^2*c^2*d*x^4 + a ^2*c^3*x
Time = 0.07 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.95 \[ \int \left (a+b x^3\right )^2 \left (c+d x^3\right )^3 \, dx=x^7\,\left (\frac {3\,a^2\,c\,d^2}{7}+\frac {6\,a\,b\,c^2\,d}{7}+\frac {b^2\,c^3}{7}\right )+x^{10}\,\left (\frac {a^2\,d^3}{10}+\frac {3\,a\,b\,c\,d^2}{5}+\frac {3\,b^2\,c^2\,d}{10}\right )+a^2\,c^3\,x+\frac {b^2\,d^3\,x^{16}}{16}+\frac {a\,c^2\,x^4\,\left (3\,a\,d+2\,b\,c\right )}{4}+\frac {b\,d^2\,x^{13}\,\left (2\,a\,d+3\,b\,c\right )}{13} \] Input:
int((a + b*x^3)^2*(c + d*x^3)^3,x)
Output:
x^7*((b^2*c^3)/7 + (3*a^2*c*d^2)/7 + (6*a*b*c^2*d)/7) + x^10*((a^2*d^3)/10 + (3*b^2*c^2*d)/10 + (3*a*b*c*d^2)/5) + a^2*c^3*x + (b^2*d^3*x^16)/16 + ( a*c^2*x^4*(3*a*d + 2*b*c))/4 + (b*d^2*x^13*(2*a*d + 3*b*c))/13
Time = 0.25 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.11 \[ \int \left (a+b x^3\right )^2 \left (c+d x^3\right )^3 \, dx=\frac {x \left (455 b^{2} d^{3} x^{15}+1120 a b \,d^{3} x^{12}+1680 b^{2} c \,d^{2} x^{12}+728 a^{2} d^{3} x^{9}+4368 a b c \,d^{2} x^{9}+2184 b^{2} c^{2} d \,x^{9}+3120 a^{2} c \,d^{2} x^{6}+6240 a b \,c^{2} d \,x^{6}+1040 b^{2} c^{3} x^{6}+5460 a^{2} c^{2} d \,x^{3}+3640 a b \,c^{3} x^{3}+7280 a^{2} c^{3}\right )}{7280} \] Input:
int((b*x^3+a)^2*(d*x^3+c)^3,x)
Output:
(x*(7280*a**2*c**3 + 5460*a**2*c**2*d*x**3 + 3120*a**2*c*d**2*x**6 + 728*a **2*d**3*x**9 + 3640*a*b*c**3*x**3 + 6240*a*b*c**2*d*x**6 + 4368*a*b*c*d** 2*x**9 + 1120*a*b*d**3*x**12 + 1040*b**2*c**3*x**6 + 2184*b**2*c**2*d*x**9 + 1680*b**2*c*d**2*x**12 + 455*b**2*d**3*x**15))/7280