Integrand size = 19, antiderivative size = 122 \[ \int \left (a+b x^3\right )^3 \left (c+d x^3\right )^2 \, dx=a^3 c^2 x+\frac {1}{4} a^2 c (3 b c+2 a d) x^4+\frac {1}{7} a \left (3 b^2 c^2+6 a b c d+a^2 d^2\right ) x^7+\frac {1}{10} b \left (b^2 c^2+6 a b c d+3 a^2 d^2\right ) x^{10}+\frac {1}{13} b^2 d (2 b c+3 a d) x^{13}+\frac {1}{16} b^3 d^2 x^{16} \] Output:
a^3*c^2*x+1/4*a^2*c*(2*a*d+3*b*c)*x^4+1/7*a*(a^2*d^2+6*a*b*c*d+3*b^2*c^2)* x^7+1/10*b*(3*a^2*d^2+6*a*b*c*d+b^2*c^2)*x^10+1/13*b^2*d*(3*a*d+2*b*c)*x^1 3+1/16*b^3*d^2*x^16
Time = 0.02 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.00 \[ \int \left (a+b x^3\right )^3 \left (c+d x^3\right )^2 \, dx=a^3 c^2 x+\frac {1}{4} a^2 c (3 b c+2 a d) x^4+\frac {1}{7} a \left (3 b^2 c^2+6 a b c d+a^2 d^2\right ) x^7+\frac {1}{10} b \left (b^2 c^2+6 a b c d+3 a^2 d^2\right ) x^{10}+\frac {1}{13} b^2 d (2 b c+3 a d) x^{13}+\frac {1}{16} b^3 d^2 x^{16} \] Input:
Integrate[(a + b*x^3)^3*(c + d*x^3)^2,x]
Output:
a^3*c^2*x + (a^2*c*(3*b*c + 2*a*d)*x^4)/4 + (a*(3*b^2*c^2 + 6*a*b*c*d + a^ 2*d^2)*x^7)/7 + (b*(b^2*c^2 + 6*a*b*c*d + 3*a^2*d^2)*x^10)/10 + (b^2*d*(2* b*c + 3*a*d)*x^13)/13 + (b^3*d^2*x^16)/16
Time = 0.46 (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.
\(\displaystyle \int \left (a+b x^3\right )^3 \left (c+d x^3\right )^2 \, dx\) |
\(\Big \downarrow \) 897 |
\(\displaystyle \int \left (a^3 c^2+b x^9 \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )+a x^6 \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+a^2 c x^3 (2 a d+3 b c)+b^2 d x^{12} (3 a d+2 b c)+b^3 d^2 x^{15}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle a^3 c^2 x+\frac {1}{10} b x^{10} \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )+\frac {1}{7} a x^7 \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac {1}{4} a^2 c x^4 (2 a d+3 b c)+\frac {1}{13} b^2 d x^{13} (3 a d+2 b c)+\frac {1}{16} b^3 d^2 x^{16}\) |
Input:
Int[(a + b*x^3)^3*(c + d*x^3)^2,x]
Output:
a^3*c^2*x + (a^2*c*(3*b*c + 2*a*d)*x^4)/4 + (a*(3*b^2*c^2 + 6*a*b*c*d + a^ 2*d^2)*x^7)/7 + (b*(b^2*c^2 + 6*a*b*c*d + 3*a^2*d^2)*x^10)/10 + (b^2*d*(2* b*c + 3*a*d)*x^13)/13 + (b^3*d^2*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 | \(a^{3} c^{2} x +\left (\frac {1}{2} a^{3} c d +\frac {3}{4} a^{2} b \,c^{2}\right ) x^{4}+\left (\frac {1}{7} a^{3} d^{2}+\frac {6}{7} a^{2} b c d +\frac {3}{7} a \,b^{2} c^{2}\right ) x^{7}+\left (\frac {3}{10} a^{2} b \,d^{2}+\frac {3}{5} a \,b^{2} c d +\frac {1}{10} c^{2} b^{3}\right ) x^{10}+\left (\frac {3}{13} a \,b^{2} d^{2}+\frac {2}{13} d \,b^{3} c \right ) x^{13}+\frac {b^{3} d^{2} x^{16}}{16}\) | \(123\) |
default | \(\frac {b^{3} d^{2} x^{16}}{16}+\frac {\left (3 a \,b^{2} d^{2}+2 d \,b^{3} c \right ) x^{13}}{13}+\frac {\left (3 a^{2} b \,d^{2}+6 a \,b^{2} c d +c^{2} b^{3}\right ) x^{10}}{10}+\frac {\left (a^{3} d^{2}+6 a^{2} b c d +3 a \,b^{2} c^{2}\right ) x^{7}}{7}+\frac {\left (2 a^{3} c d +3 a^{2} b \,c^{2}\right ) x^{4}}{4}+a^{3} c^{2} x\) | \(125\) |
gosper | \(a^{3} c^{2} x +\frac {1}{2} x^{4} a^{3} c d +\frac {3}{4} x^{4} a^{2} b \,c^{2}+\frac {1}{7} x^{7} a^{3} d^{2}+\frac {6}{7} x^{7} a^{2} b c d +\frac {3}{7} x^{7} a \,b^{2} c^{2}+\frac {3}{10} x^{10} a^{2} b \,d^{2}+\frac {3}{5} x^{10} a \,b^{2} c d +\frac {1}{10} x^{10} c^{2} b^{3}+\frac {3}{13} x^{13} a \,b^{2} d^{2}+\frac {2}{13} x^{13} d \,b^{3} c +\frac {1}{16} b^{3} d^{2} x^{16}\) | \(133\) |
risch | \(a^{3} c^{2} x +\frac {1}{2} x^{4} a^{3} c d +\frac {3}{4} x^{4} a^{2} b \,c^{2}+\frac {1}{7} x^{7} a^{3} d^{2}+\frac {6}{7} x^{7} a^{2} b c d +\frac {3}{7} x^{7} a \,b^{2} c^{2}+\frac {3}{10} x^{10} a^{2} b \,d^{2}+\frac {3}{5} x^{10} a \,b^{2} c d +\frac {1}{10} x^{10} c^{2} b^{3}+\frac {3}{13} x^{13} a \,b^{2} d^{2}+\frac {2}{13} x^{13} d \,b^{3} c +\frac {1}{16} b^{3} d^{2} x^{16}\) | \(133\) |
parallelrisch | \(a^{3} c^{2} x +\frac {1}{2} x^{4} a^{3} c d +\frac {3}{4} x^{4} a^{2} b \,c^{2}+\frac {1}{7} x^{7} a^{3} d^{2}+\frac {6}{7} x^{7} a^{2} b c d +\frac {3}{7} x^{7} a \,b^{2} c^{2}+\frac {3}{10} x^{10} a^{2} b \,d^{2}+\frac {3}{5} x^{10} a \,b^{2} c d +\frac {1}{10} x^{10} c^{2} b^{3}+\frac {3}{13} x^{13} a \,b^{2} d^{2}+\frac {2}{13} x^{13} d \,b^{3} c +\frac {1}{16} b^{3} d^{2} x^{16}\) | \(133\) |
orering | \(\frac {x \left (455 b^{3} d^{2} x^{15}+1680 a \,b^{2} d^{2} x^{12}+1120 b^{3} c d \,x^{12}+2184 a^{2} b \,d^{2} x^{9}+4368 a \,b^{2} c d \,x^{9}+728 b^{3} c^{2} x^{9}+1040 a^{3} d^{2} x^{6}+6240 a^{2} b c d \,x^{6}+3120 a \,b^{2} c^{2} x^{6}+3640 a^{3} c d \,x^{3}+5460 a^{2} b \,c^{2} x^{3}+7280 a^{3} c^{2}\right )}{7280}\) | \(136\) |
Input:
int((b*x^3+a)^3*(d*x^3+c)^2,x,method=_RETURNVERBOSE)
Output:
a^3*c^2*x+(1/2*a^3*c*d+3/4*a^2*b*c^2)*x^4+(1/7*a^3*d^2+6/7*a^2*b*c*d+3/7*a *b^2*c^2)*x^7+(3/10*a^2*b*d^2+3/5*a*b^2*c*d+1/10*c^2*b^3)*x^10+(3/13*a*b^2 *d^2+2/13*d*b^3*c)*x^13+1/16*b^3*d^2*x^16
Time = 0.07 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.02 \[ \int \left (a+b x^3\right )^3 \left (c+d x^3\right )^2 \, dx=\frac {1}{16} \, b^{3} d^{2} x^{16} + \frac {1}{13} \, {\left (2 \, b^{3} c d + 3 \, a b^{2} d^{2}\right )} x^{13} + \frac {1}{10} \, {\left (b^{3} c^{2} + 6 \, a b^{2} c d + 3 \, a^{2} b d^{2}\right )} x^{10} + \frac {1}{7} \, {\left (3 \, a b^{2} c^{2} + 6 \, a^{2} b c d + a^{3} d^{2}\right )} x^{7} + a^{3} c^{2} x + \frac {1}{4} \, {\left (3 \, a^{2} b c^{2} + 2 \, a^{3} c d\right )} x^{4} \] Input:
integrate((b*x^3+a)^3*(d*x^3+c)^2,x, algorithm="fricas")
Output:
1/16*b^3*d^2*x^16 + 1/13*(2*b^3*c*d + 3*a*b^2*d^2)*x^13 + 1/10*(b^3*c^2 + 6*a*b^2*c*d + 3*a^2*b*d^2)*x^10 + 1/7*(3*a*b^2*c^2 + 6*a^2*b*c*d + a^3*d^2 )*x^7 + a^3*c^2*x + 1/4*(3*a^2*b*c^2 + 2*a^3*c*d)*x^4
Time = 0.04 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.14 \[ \int \left (a+b x^3\right )^3 \left (c+d x^3\right )^2 \, dx=a^{3} c^{2} x + \frac {b^{3} d^{2} x^{16}}{16} + x^{13} \cdot \left (\frac {3 a b^{2} d^{2}}{13} + \frac {2 b^{3} c d}{13}\right ) + x^{10} \cdot \left (\frac {3 a^{2} b d^{2}}{10} + \frac {3 a b^{2} c d}{5} + \frac {b^{3} c^{2}}{10}\right ) + x^{7} \left (\frac {a^{3} d^{2}}{7} + \frac {6 a^{2} b c d}{7} + \frac {3 a b^{2} c^{2}}{7}\right ) + x^{4} \left (\frac {a^{3} c d}{2} + \frac {3 a^{2} b c^{2}}{4}\right ) \] Input:
integrate((b*x**3+a)**3*(d*x**3+c)**2,x)
Output:
a**3*c**2*x + b**3*d**2*x**16/16 + x**13*(3*a*b**2*d**2/13 + 2*b**3*c*d/13 ) + x**10*(3*a**2*b*d**2/10 + 3*a*b**2*c*d/5 + b**3*c**2/10) + x**7*(a**3* d**2/7 + 6*a**2*b*c*d/7 + 3*a*b**2*c**2/7) + x**4*(a**3*c*d/2 + 3*a**2*b*c **2/4)
Time = 0.03 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.02 \[ \int \left (a+b x^3\right )^3 \left (c+d x^3\right )^2 \, dx=\frac {1}{16} \, b^{3} d^{2} x^{16} + \frac {1}{13} \, {\left (2 \, b^{3} c d + 3 \, a b^{2} d^{2}\right )} x^{13} + \frac {1}{10} \, {\left (b^{3} c^{2} + 6 \, a b^{2} c d + 3 \, a^{2} b d^{2}\right )} x^{10} + \frac {1}{7} \, {\left (3 \, a b^{2} c^{2} + 6 \, a^{2} b c d + a^{3} d^{2}\right )} x^{7} + a^{3} c^{2} x + \frac {1}{4} \, {\left (3 \, a^{2} b c^{2} + 2 \, a^{3} c d\right )} x^{4} \] Input:
integrate((b*x^3+a)^3*(d*x^3+c)^2,x, algorithm="maxima")
Output:
1/16*b^3*d^2*x^16 + 1/13*(2*b^3*c*d + 3*a*b^2*d^2)*x^13 + 1/10*(b^3*c^2 + 6*a*b^2*c*d + 3*a^2*b*d^2)*x^10 + 1/7*(3*a*b^2*c^2 + 6*a^2*b*c*d + a^3*d^2 )*x^7 + a^3*c^2*x + 1/4*(3*a^2*b*c^2 + 2*a^3*c*d)*x^4
Time = 0.12 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.08 \[ \int \left (a+b x^3\right )^3 \left (c+d x^3\right )^2 \, dx=\frac {1}{16} \, b^{3} d^{2} x^{16} + \frac {2}{13} \, b^{3} c d x^{13} + \frac {3}{13} \, a b^{2} d^{2} x^{13} + \frac {1}{10} \, b^{3} c^{2} x^{10} + \frac {3}{5} \, a b^{2} c d x^{10} + \frac {3}{10} \, a^{2} b d^{2} x^{10} + \frac {3}{7} \, a b^{2} c^{2} x^{7} + \frac {6}{7} \, a^{2} b c d x^{7} + \frac {1}{7} \, a^{3} d^{2} x^{7} + \frac {3}{4} \, a^{2} b c^{2} x^{4} + \frac {1}{2} \, a^{3} c d x^{4} + a^{3} c^{2} x \] Input:
integrate((b*x^3+a)^3*(d*x^3+c)^2,x, algorithm="giac")
Output:
1/16*b^3*d^2*x^16 + 2/13*b^3*c*d*x^13 + 3/13*a*b^2*d^2*x^13 + 1/10*b^3*c^2 *x^10 + 3/5*a*b^2*c*d*x^10 + 3/10*a^2*b*d^2*x^10 + 3/7*a*b^2*c^2*x^7 + 6/7 *a^2*b*c*d*x^7 + 1/7*a^3*d^2*x^7 + 3/4*a^2*b*c^2*x^4 + 1/2*a^3*c*d*x^4 + a ^3*c^2*x
Time = 0.06 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.95 \[ \int \left (a+b x^3\right )^3 \left (c+d x^3\right )^2 \, dx=x^7\,\left (\frac {a^3\,d^2}{7}+\frac {6\,a^2\,b\,c\,d}{7}+\frac {3\,a\,b^2\,c^2}{7}\right )+x^{10}\,\left (\frac {3\,a^2\,b\,d^2}{10}+\frac {3\,a\,b^2\,c\,d}{5}+\frac {b^3\,c^2}{10}\right )+a^3\,c^2\,x+\frac {b^3\,d^2\,x^{16}}{16}+\frac {a^2\,c\,x^4\,\left (2\,a\,d+3\,b\,c\right )}{4}+\frac {b^2\,d\,x^{13}\,\left (3\,a\,d+2\,b\,c\right )}{13} \] Input:
int((a + b*x^3)^3*(c + d*x^3)^2,x)
Output:
x^7*((a^3*d^2)/7 + (3*a*b^2*c^2)/7 + (6*a^2*b*c*d)/7) + x^10*((b^3*c^2)/10 + (3*a^2*b*d^2)/10 + (3*a*b^2*c*d)/5) + a^3*c^2*x + (b^3*d^2*x^16)/16 + ( a^2*c*x^4*(2*a*d + 3*b*c))/4 + (b^2*d*x^13*(3*a*d + 2*b*c))/13
Time = 0.20 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.11 \[ \int \left (a+b x^3\right )^3 \left (c+d x^3\right )^2 \, dx=\frac {x \left (455 b^{3} d^{2} x^{15}+1680 a \,b^{2} d^{2} x^{12}+1120 b^{3} c d \,x^{12}+2184 a^{2} b \,d^{2} x^{9}+4368 a \,b^{2} c d \,x^{9}+728 b^{3} c^{2} x^{9}+1040 a^{3} d^{2} x^{6}+6240 a^{2} b c d \,x^{6}+3120 a \,b^{2} c^{2} x^{6}+3640 a^{3} c d \,x^{3}+5460 a^{2} b \,c^{2} x^{3}+7280 a^{3} c^{2}\right )}{7280} \] Input:
int((b*x^3+a)^3*(d*x^3+c)^2,x)
Output:
(x*(7280*a**3*c**2 + 3640*a**3*c*d*x**3 + 1040*a**3*d**2*x**6 + 5460*a**2* b*c**2*x**3 + 6240*a**2*b*c*d*x**6 + 2184*a**2*b*d**2*x**9 + 3120*a*b**2*c **2*x**6 + 4368*a*b**2*c*d*x**9 + 1680*a*b**2*d**2*x**12 + 728*b**3*c**2*x **9 + 1120*b**3*c*d*x**12 + 455*b**3*d**2*x**15))/7280