Integrand size = 17, antiderivative size = 94 \[ \int \left (a+b x^3\right ) \left (c+d x^3\right )^4 \, dx=a c^4 x+\frac {1}{4} c^3 (b c+4 a d) x^4+\frac {2}{7} c^2 d (2 b c+3 a d) x^7+\frac {1}{5} c d^2 (3 b c+2 a d) x^{10}+\frac {1}{13} d^3 (4 b c+a d) x^{13}+\frac {1}{16} b d^4 x^{16} \] Output:
a*c^4*x+1/4*c^3*(4*a*d+b*c)*x^4+2/7*c^2*d*(3*a*d+2*b*c)*x^7+1/5*c*d^2*(2*a *d+3*b*c)*x^10+1/13*d^3*(a*d+4*b*c)*x^13+1/16*b*d^4*x^16
Time = 0.02 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.00 \[ \int \left (a+b x^3\right ) \left (c+d x^3\right )^4 \, dx=a c^4 x+\frac {1}{4} c^3 (b c+4 a d) x^4+\frac {2}{7} c^2 d (2 b c+3 a d) x^7+\frac {1}{5} c d^2 (3 b c+2 a d) x^{10}+\frac {1}{13} d^3 (4 b c+a d) x^{13}+\frac {1}{16} b d^4 x^{16} \] Input:
Integrate[(a + b*x^3)*(c + d*x^3)^4,x]
Output:
a*c^4*x + (c^3*(b*c + 4*a*d)*x^4)/4 + (2*c^2*d*(2*b*c + 3*a*d)*x^7)/7 + (c *d^2*(3*b*c + 2*a*d)*x^10)/5 + (d^3*(4*b*c + a*d)*x^13)/13 + (b*d^4*x^16)/ 16
Time = 0.43 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, 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.
\(\displaystyle \int \left (a+b x^3\right ) \left (c+d x^3\right )^4 \, dx\) |
\(\Big \downarrow \) 897 |
\(\displaystyle \int \left (c^3 x^3 (4 a d+b c)+2 c^2 d x^6 (3 a d+2 b c)+d^3 x^{12} (a d+4 b c)+2 c d^2 x^9 (2 a d+3 b c)+a c^4+b d^4 x^{15}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{4} c^3 x^4 (4 a d+b c)+\frac {2}{7} c^2 d x^7 (3 a d+2 b c)+\frac {1}{13} d^3 x^{13} (a d+4 b c)+\frac {1}{5} c d^2 x^{10} (2 a d+3 b c)+a c^4 x+\frac {1}{16} b d^4 x^{16}\) |
Input:
Int[(a + b*x^3)*(c + d*x^3)^4,x]
Output:
a*c^4*x + (c^3*(b*c + 4*a*d)*x^4)/4 + (2*c^2*d*(2*b*c + 3*a*d)*x^7)/7 + (c *d^2*(3*b*c + 2*a*d)*x^10)/5 + (d^3*(4*b*c + a*d)*x^13)/13 + (b*d^4*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 = 1.27 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.00
method | result | size |
norman | \(a \,c^{4} x +\left (a \,c^{3} d +\frac {1}{4} b \,c^{4}\right ) x^{4}+\left (\frac {6}{7} a \,c^{2} d^{2}+\frac {4}{7} b \,c^{3} d \right ) x^{7}+\left (\frac {2}{5} a c \,d^{3}+\frac {3}{5} b \,c^{2} d^{2}\right ) x^{10}+\left (\frac {1}{13} a \,d^{4}+\frac {4}{13} b c \,d^{3}\right ) x^{13}+\frac {b \,d^{4} x^{16}}{16}\) | \(94\) |
default | \(\frac {b \,d^{4} x^{16}}{16}+\frac {\left (a \,d^{4}+4 b c \,d^{3}\right ) x^{13}}{13}+\frac {\left (4 a c \,d^{3}+6 b \,c^{2} d^{2}\right ) x^{10}}{10}+\frac {\left (6 a \,c^{2} d^{2}+4 b \,c^{3} d \right ) x^{7}}{7}+\frac {\left (4 a \,c^{3} d +b \,c^{4}\right ) x^{4}}{4}+a \,c^{4} x\) | \(97\) |
gosper | \(a \,c^{4} x +x^{4} a \,c^{3} d +\frac {1}{4} x^{4} b \,c^{4}+\frac {6}{7} x^{7} a \,c^{2} d^{2}+\frac {4}{7} x^{7} b \,c^{3} d +\frac {2}{5} x^{10} a c \,d^{3}+\frac {3}{5} x^{10} b \,c^{2} d^{2}+\frac {1}{13} x^{13} a \,d^{4}+\frac {4}{13} x^{13} b c \,d^{3}+\frac {1}{16} b \,d^{4} x^{16}\) | \(98\) |
risch | \(a \,c^{4} x +x^{4} a \,c^{3} d +\frac {1}{4} x^{4} b \,c^{4}+\frac {6}{7} x^{7} a \,c^{2} d^{2}+\frac {4}{7} x^{7} b \,c^{3} d +\frac {2}{5} x^{10} a c \,d^{3}+\frac {3}{5} x^{10} b \,c^{2} d^{2}+\frac {1}{13} x^{13} a \,d^{4}+\frac {4}{13} x^{13} b c \,d^{3}+\frac {1}{16} b \,d^{4} x^{16}\) | \(98\) |
parallelrisch | \(a \,c^{4} x +x^{4} a \,c^{3} d +\frac {1}{4} x^{4} b \,c^{4}+\frac {6}{7} x^{7} a \,c^{2} d^{2}+\frac {4}{7} x^{7} b \,c^{3} d +\frac {2}{5} x^{10} a c \,d^{3}+\frac {3}{5} x^{10} b \,c^{2} d^{2}+\frac {1}{13} x^{13} a \,d^{4}+\frac {4}{13} x^{13} b c \,d^{3}+\frac {1}{16} b \,d^{4} x^{16}\) | \(98\) |
orering | \(\frac {x \left (455 b \,d^{4} x^{15}+560 a \,d^{4} x^{12}+2240 b c \,d^{3} x^{12}+2912 a c \,d^{3} x^{9}+4368 b \,c^{2} d^{2} x^{9}+6240 a \,c^{2} d^{2} x^{6}+4160 b \,c^{3} d \,x^{6}+7280 a \,c^{3} d \,x^{3}+1820 b \,c^{4} x^{3}+7280 a \,c^{4}\right )}{7280}\) | \(102\) |
Input:
int((b*x^3+a)*(d*x^3+c)^4,x,method=_RETURNVERBOSE)
Output:
a*c^4*x+(a*c^3*d+1/4*b*c^4)*x^4+(6/7*a*c^2*d^2+4/7*b*c^3*d)*x^7+(2/5*a*c*d ^3+3/5*b*c^2*d^2)*x^10+(1/13*a*d^4+4/13*b*c*d^3)*x^13+1/16*b*d^4*x^16
Time = 0.07 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.02 \[ \int \left (a+b x^3\right ) \left (c+d x^3\right )^4 \, dx=\frac {1}{16} \, b d^{4} x^{16} + \frac {1}{13} \, {\left (4 \, b c d^{3} + a d^{4}\right )} x^{13} + \frac {1}{5} \, {\left (3 \, b c^{2} d^{2} + 2 \, a c d^{3}\right )} x^{10} + \frac {2}{7} \, {\left (2 \, b c^{3} d + 3 \, a c^{2} d^{2}\right )} x^{7} + a c^{4} x + \frac {1}{4} \, {\left (b c^{4} + 4 \, a c^{3} d\right )} x^{4} \] Input:
integrate((b*x^3+a)*(d*x^3+c)^4,x, algorithm="fricas")
Output:
1/16*b*d^4*x^16 + 1/13*(4*b*c*d^3 + a*d^4)*x^13 + 1/5*(3*b*c^2*d^2 + 2*a*c *d^3)*x^10 + 2/7*(2*b*c^3*d + 3*a*c^2*d^2)*x^7 + a*c^4*x + 1/4*(b*c^4 + 4* a*c^3*d)*x^4
Time = 0.03 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.11 \[ \int \left (a+b x^3\right ) \left (c+d x^3\right )^4 \, dx=a c^{4} x + \frac {b d^{4} x^{16}}{16} + x^{13} \left (\frac {a d^{4}}{13} + \frac {4 b c d^{3}}{13}\right ) + x^{10} \cdot \left (\frac {2 a c d^{3}}{5} + \frac {3 b c^{2} d^{2}}{5}\right ) + x^{7} \cdot \left (\frac {6 a c^{2} d^{2}}{7} + \frac {4 b c^{3} d}{7}\right ) + x^{4} \left (a c^{3} d + \frac {b c^{4}}{4}\right ) \] Input:
integrate((b*x**3+a)*(d*x**3+c)**4,x)
Output:
a*c**4*x + b*d**4*x**16/16 + x**13*(a*d**4/13 + 4*b*c*d**3/13) + x**10*(2* a*c*d**3/5 + 3*b*c**2*d**2/5) + x**7*(6*a*c**2*d**2/7 + 4*b*c**3*d/7) + x* *4*(a*c**3*d + b*c**4/4)
Time = 0.03 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.02 \[ \int \left (a+b x^3\right ) \left (c+d x^3\right )^4 \, dx=\frac {1}{16} \, b d^{4} x^{16} + \frac {1}{13} \, {\left (4 \, b c d^{3} + a d^{4}\right )} x^{13} + \frac {1}{5} \, {\left (3 \, b c^{2} d^{2} + 2 \, a c d^{3}\right )} x^{10} + \frac {2}{7} \, {\left (2 \, b c^{3} d + 3 \, a c^{2} d^{2}\right )} x^{7} + a c^{4} x + \frac {1}{4} \, {\left (b c^{4} + 4 \, a c^{3} d\right )} x^{4} \] Input:
integrate((b*x^3+a)*(d*x^3+c)^4,x, algorithm="maxima")
Output:
1/16*b*d^4*x^16 + 1/13*(4*b*c*d^3 + a*d^4)*x^13 + 1/5*(3*b*c^2*d^2 + 2*a*c *d^3)*x^10 + 2/7*(2*b*c^3*d + 3*a*c^2*d^2)*x^7 + a*c^4*x + 1/4*(b*c^4 + 4* a*c^3*d)*x^4
Time = 0.13 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.03 \[ \int \left (a+b x^3\right ) \left (c+d x^3\right )^4 \, dx=\frac {1}{16} \, b d^{4} x^{16} + \frac {4}{13} \, b c d^{3} x^{13} + \frac {1}{13} \, a d^{4} x^{13} + \frac {3}{5} \, b c^{2} d^{2} x^{10} + \frac {2}{5} \, a c d^{3} x^{10} + \frac {4}{7} \, b c^{3} d x^{7} + \frac {6}{7} \, a c^{2} d^{2} x^{7} + \frac {1}{4} \, b c^{4} x^{4} + a c^{3} d x^{4} + a c^{4} x \] Input:
integrate((b*x^3+a)*(d*x^3+c)^4,x, algorithm="giac")
Output:
1/16*b*d^4*x^16 + 4/13*b*c*d^3*x^13 + 1/13*a*d^4*x^13 + 3/5*b*c^2*d^2*x^10 + 2/5*a*c*d^3*x^10 + 4/7*b*c^3*d*x^7 + 6/7*a*c^2*d^2*x^7 + 1/4*b*c^4*x^4 + a*c^3*d*x^4 + a*c^4*x
Time = 0.47 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.93 \[ \int \left (a+b x^3\right ) \left (c+d x^3\right )^4 \, dx=x^4\,\left (\frac {b\,c^4}{4}+a\,d\,c^3\right )+x^{13}\,\left (\frac {a\,d^4}{13}+\frac {4\,b\,c\,d^3}{13}\right )+\frac {b\,d^4\,x^{16}}{16}+a\,c^4\,x+\frac {2\,c^2\,d\,x^7\,\left (3\,a\,d+2\,b\,c\right )}{7}+\frac {c\,d^2\,x^{10}\,\left (2\,a\,d+3\,b\,c\right )}{5} \] Input:
int((a + b*x^3)*(c + d*x^3)^4,x)
Output:
x^4*((b*c^4)/4 + a*c^3*d) + x^13*((a*d^4)/13 + (4*b*c*d^3)/13) + (b*d^4*x^ 16)/16 + a*c^4*x + (2*c^2*d*x^7*(3*a*d + 2*b*c))/7 + (c*d^2*x^10*(2*a*d + 3*b*c))/5
Time = 0.24 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.07 \[ \int \left (a+b x^3\right ) \left (c+d x^3\right )^4 \, dx=\frac {x \left (455 b \,d^{4} x^{15}+560 a \,d^{4} x^{12}+2240 b c \,d^{3} x^{12}+2912 a c \,d^{3} x^{9}+4368 b \,c^{2} d^{2} x^{9}+6240 a \,c^{2} d^{2} x^{6}+4160 b \,c^{3} d \,x^{6}+7280 a \,c^{3} d \,x^{3}+1820 b \,c^{4} x^{3}+7280 a \,c^{4}\right )}{7280} \] Input:
int((b*x^3+a)*(d*x^3+c)^4,x)
Output:
(x*(7280*a*c**4 + 7280*a*c**3*d*x**3 + 6240*a*c**2*d**2*x**6 + 2912*a*c*d* *3*x**9 + 560*a*d**4*x**12 + 1820*b*c**4*x**3 + 4160*b*c**3*d*x**6 + 4368* b*c**2*d**2*x**9 + 2240*b*c*d**3*x**12 + 455*b*d**4*x**15))/7280