Integrand size = 30, antiderivative size = 113 \[ \int \left (a+b x^3\right )^3 \left (a c+a d x+b c x^3+b d x^4\right ) \, dx=a^4 c x+\frac {1}{2} a^4 d x^2+a^3 b c x^4+\frac {4}{5} a^3 b d x^5+\frac {6}{7} a^2 b^2 c x^7+\frac {3}{4} a^2 b^2 d x^8+\frac {2}{5} a b^3 c x^{10}+\frac {4}{11} a b^3 d x^{11}+\frac {1}{13} b^4 c x^{13}+\frac {1}{14} b^4 d x^{14} \] Output:
a^4*c*x+1/2*a^4*d*x^2+a^3*b*c*x^4+4/5*a^3*b*d*x^5+6/7*a^2*b^2*c*x^7+3/4*a^ 2*b^2*d*x^8+2/5*a*b^3*c*x^10+4/11*a*b^3*d*x^11+1/13*b^4*c*x^13+1/14*b^4*d* x^14
Time = 0.00 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.00 \[ \int \left (a+b x^3\right )^3 \left (a c+a d x+b c x^3+b d x^4\right ) \, dx=a^4 c x+\frac {1}{2} a^4 d x^2+a^3 b c x^4+\frac {4}{5} a^3 b d x^5+\frac {6}{7} a^2 b^2 c x^7+\frac {3}{4} a^2 b^2 d x^8+\frac {2}{5} a b^3 c x^{10}+\frac {4}{11} a b^3 d x^{11}+\frac {1}{13} b^4 c x^{13}+\frac {1}{14} b^4 d x^{14} \] Input:
Integrate[(a + b*x^3)^3*(a*c + a*d*x + b*c*x^3 + b*d*x^4),x]
Output:
a^4*c*x + (a^4*d*x^2)/2 + a^3*b*c*x^4 + (4*a^3*b*d*x^5)/5 + (6*a^2*b^2*c*x ^7)/7 + (3*a^2*b^2*d*x^8)/4 + (2*a*b^3*c*x^10)/5 + (4*a*b^3*d*x^11)/11 + ( b^4*c*x^13)/13 + (b^4*d*x^14)/14
Time = 0.51 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2389, 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.
\(\displaystyle \int \left (a+b x^3\right )^3 \left (a c+a d x+b c x^3+b d x^4\right ) \, dx\) |
\(\Big \downarrow \) 2389 |
\(\displaystyle \int \left (a^4 c+a^4 d x+4 a^3 b c x^3+4 a^3 b d x^4+6 a^2 b^2 c x^6+6 a^2 b^2 d x^7+4 a b^3 c x^9+4 a b^3 d x^{10}+b^4 c x^{12}+b^4 d x^{13}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle a^4 c x+\frac {1}{2} a^4 d x^2+a^3 b c x^4+\frac {4}{5} a^3 b d x^5+\frac {6}{7} a^2 b^2 c x^7+\frac {3}{4} a^2 b^2 d x^8+\frac {2}{5} a b^3 c x^{10}+\frac {4}{11} a b^3 d x^{11}+\frac {1}{13} b^4 c x^{13}+\frac {1}{14} b^4 d x^{14}\) |
Input:
Int[(a + b*x^3)^3*(a*c + a*d*x + b*c*x^3 + b*d*x^4),x]
Output:
a^4*c*x + (a^4*d*x^2)/2 + a^3*b*c*x^4 + (4*a^3*b*d*x^5)/5 + (6*a^2*b^2*c*x ^7)/7 + (3*a^2*b^2*d*x^8)/4 + (2*a*b^3*c*x^10)/5 + (4*a*b^3*d*x^11)/11 + ( b^4*c*x^13)/13 + (b^4*d*x^14)/14
Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand [Pq*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, n}, x] && PolyQ[Pq, x] && (IGtQ[p , 0] || EqQ[n, 1])
Time = 0.13 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.87
method | result | size |
default | \(a^{4} c x +\frac {1}{2} a^{4} d \,x^{2}+a^{3} b c \,x^{4}+\frac {4}{5} a^{3} b d \,x^{5}+\frac {6}{7} a^{2} b^{2} c \,x^{7}+\frac {3}{4} a^{2} b^{2} d \,x^{8}+\frac {2}{5} a \,b^{3} c \,x^{10}+\frac {4}{11} a \,b^{3} d \,x^{11}+\frac {1}{13} b^{4} c \,x^{13}+\frac {1}{14} b^{4} d \,x^{14}\) | \(98\) |
norman | \(a^{4} c x +\frac {1}{2} a^{4} d \,x^{2}+a^{3} b c \,x^{4}+\frac {4}{5} a^{3} b d \,x^{5}+\frac {6}{7} a^{2} b^{2} c \,x^{7}+\frac {3}{4} a^{2} b^{2} d \,x^{8}+\frac {2}{5} a \,b^{3} c \,x^{10}+\frac {4}{11} a \,b^{3} d \,x^{11}+\frac {1}{13} b^{4} c \,x^{13}+\frac {1}{14} b^{4} d \,x^{14}\) | \(98\) |
risch | \(a^{4} c x +\frac {1}{2} a^{4} d \,x^{2}+a^{3} b c \,x^{4}+\frac {4}{5} a^{3} b d \,x^{5}+\frac {6}{7} a^{2} b^{2} c \,x^{7}+\frac {3}{4} a^{2} b^{2} d \,x^{8}+\frac {2}{5} a \,b^{3} c \,x^{10}+\frac {4}{11} a \,b^{3} d \,x^{11}+\frac {1}{13} b^{4} c \,x^{13}+\frac {1}{14} b^{4} d \,x^{14}\) | \(98\) |
parallelrisch | \(a^{4} c x +\frac {1}{2} a^{4} d \,x^{2}+a^{3} b c \,x^{4}+\frac {4}{5} a^{3} b d \,x^{5}+\frac {6}{7} a^{2} b^{2} c \,x^{7}+\frac {3}{4} a^{2} b^{2} d \,x^{8}+\frac {2}{5} a \,b^{3} c \,x^{10}+\frac {4}{11} a \,b^{3} d \,x^{11}+\frac {1}{13} b^{4} c \,x^{13}+\frac {1}{14} b^{4} d \,x^{14}\) | \(98\) |
gosper | \(\frac {x \left (1430 b^{4} d \,x^{13}+1540 b^{4} c \,x^{12}+7280 a \,b^{3} d \,x^{10}+8008 a \,b^{3} c \,x^{9}+15015 a^{2} b^{2} d \,x^{7}+17160 a^{2} b^{2} c \,x^{6}+16016 a^{3} b d \,x^{4}+20020 a^{3} b c \,x^{3}+10010 a^{4} d x +20020 c \,a^{4}\right )}{20020}\) | \(100\) |
orering | \(\frac {x \left (1430 b^{4} d \,x^{13}+1540 b^{4} c \,x^{12}+7280 a \,b^{3} d \,x^{10}+8008 a \,b^{3} c \,x^{9}+15015 a^{2} b^{2} d \,x^{7}+17160 a^{2} b^{2} c \,x^{6}+16016 a^{3} b d \,x^{4}+20020 a^{3} b c \,x^{3}+10010 a^{4} d x +20020 c \,a^{4}\right ) \left (d b \,x^{4}+b c \,x^{3}+a d x +a c \right )}{20020 \left (d x +c \right ) \left (b \,x^{3}+a \right )}\) | \(136\) |
Input:
int((b*x^3+a)^3*(b*d*x^4+b*c*x^3+a*d*x+a*c),x,method=_RETURNVERBOSE)
Output:
a^4*c*x+1/2*a^4*d*x^2+a^3*b*c*x^4+4/5*a^3*b*d*x^5+6/7*a^2*b^2*c*x^7+3/4*a^ 2*b^2*d*x^8+2/5*a*b^3*c*x^10+4/11*a*b^3*d*x^11+1/13*b^4*c*x^13+1/14*b^4*d* x^14
Time = 0.08 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.86 \[ \int \left (a+b x^3\right )^3 \left (a c+a d x+b c x^3+b d x^4\right ) \, dx=\frac {1}{14} \, b^{4} d x^{14} + \frac {1}{13} \, b^{4} c x^{13} + \frac {4}{11} \, a b^{3} d x^{11} + \frac {2}{5} \, a b^{3} c x^{10} + \frac {3}{4} \, a^{2} b^{2} d x^{8} + \frac {6}{7} \, a^{2} b^{2} c x^{7} + \frac {4}{5} \, a^{3} b d x^{5} + a^{3} b c x^{4} + \frac {1}{2} \, a^{4} d x^{2} + a^{4} c x \] Input:
integrate((b*x^3+a)^3*(b*d*x^4+b*c*x^3+a*d*x+a*c),x, algorithm="fricas")
Output:
1/14*b^4*d*x^14 + 1/13*b^4*c*x^13 + 4/11*a*b^3*d*x^11 + 2/5*a*b^3*c*x^10 + 3/4*a^2*b^2*d*x^8 + 6/7*a^2*b^2*c*x^7 + 4/5*a^3*b*d*x^5 + a^3*b*c*x^4 + 1 /2*a^4*d*x^2 + a^4*c*x
Time = 0.03 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.04 \[ \int \left (a+b x^3\right )^3 \left (a c+a d x+b c x^3+b d x^4\right ) \, dx=a^{4} c x + \frac {a^{4} d x^{2}}{2} + a^{3} b c x^{4} + \frac {4 a^{3} b d x^{5}}{5} + \frac {6 a^{2} b^{2} c x^{7}}{7} + \frac {3 a^{2} b^{2} d x^{8}}{4} + \frac {2 a b^{3} c x^{10}}{5} + \frac {4 a b^{3} d x^{11}}{11} + \frac {b^{4} c x^{13}}{13} + \frac {b^{4} d x^{14}}{14} \] Input:
integrate((b*x**3+a)**3*(b*d*x**4+b*c*x**3+a*d*x+a*c),x)
Output:
a**4*c*x + a**4*d*x**2/2 + a**3*b*c*x**4 + 4*a**3*b*d*x**5/5 + 6*a**2*b**2 *c*x**7/7 + 3*a**2*b**2*d*x**8/4 + 2*a*b**3*c*x**10/5 + 4*a*b**3*d*x**11/1 1 + b**4*c*x**13/13 + b**4*d*x**14/14
Time = 0.03 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.86 \[ \int \left (a+b x^3\right )^3 \left (a c+a d x+b c x^3+b d x^4\right ) \, dx=\frac {1}{14} \, b^{4} d x^{14} + \frac {1}{13} \, b^{4} c x^{13} + \frac {4}{11} \, a b^{3} d x^{11} + \frac {2}{5} \, a b^{3} c x^{10} + \frac {3}{4} \, a^{2} b^{2} d x^{8} + \frac {6}{7} \, a^{2} b^{2} c x^{7} + \frac {4}{5} \, a^{3} b d x^{5} + a^{3} b c x^{4} + \frac {1}{2} \, a^{4} d x^{2} + a^{4} c x \] Input:
integrate((b*x^3+a)^3*(b*d*x^4+b*c*x^3+a*d*x+a*c),x, algorithm="maxima")
Output:
1/14*b^4*d*x^14 + 1/13*b^4*c*x^13 + 4/11*a*b^3*d*x^11 + 2/5*a*b^3*c*x^10 + 3/4*a^2*b^2*d*x^8 + 6/7*a^2*b^2*c*x^7 + 4/5*a^3*b*d*x^5 + a^3*b*c*x^4 + 1 /2*a^4*d*x^2 + a^4*c*x
Time = 0.12 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.86 \[ \int \left (a+b x^3\right )^3 \left (a c+a d x+b c x^3+b d x^4\right ) \, dx=\frac {1}{14} \, b^{4} d x^{14} + \frac {1}{13} \, b^{4} c x^{13} + \frac {4}{11} \, a b^{3} d x^{11} + \frac {2}{5} \, a b^{3} c x^{10} + \frac {3}{4} \, a^{2} b^{2} d x^{8} + \frac {6}{7} \, a^{2} b^{2} c x^{7} + \frac {4}{5} \, a^{3} b d x^{5} + a^{3} b c x^{4} + \frac {1}{2} \, a^{4} d x^{2} + a^{4} c x \] Input:
integrate((b*x^3+a)^3*(b*d*x^4+b*c*x^3+a*d*x+a*c),x, algorithm="giac")
Output:
1/14*b^4*d*x^14 + 1/13*b^4*c*x^13 + 4/11*a*b^3*d*x^11 + 2/5*a*b^3*c*x^10 + 3/4*a^2*b^2*d*x^8 + 6/7*a^2*b^2*c*x^7 + 4/5*a^3*b*d*x^5 + a^3*b*c*x^4 + 1 /2*a^4*d*x^2 + a^4*c*x
Time = 5.67 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.86 \[ \int \left (a+b x^3\right )^3 \left (a c+a d x+b c x^3+b d x^4\right ) \, dx=\frac {d\,a^4\,x^2}{2}+c\,a^4\,x+\frac {4\,d\,a^3\,b\,x^5}{5}+c\,a^3\,b\,x^4+\frac {3\,d\,a^2\,b^2\,x^8}{4}+\frac {6\,c\,a^2\,b^2\,x^7}{7}+\frac {4\,d\,a\,b^3\,x^{11}}{11}+\frac {2\,c\,a\,b^3\,x^{10}}{5}+\frac {d\,b^4\,x^{14}}{14}+\frac {c\,b^4\,x^{13}}{13} \] Input:
int((a + b*x^3)^3*(a*c + a*d*x + b*c*x^3 + b*d*x^4),x)
Output:
(a^4*d*x^2)/2 + (b^4*c*x^13)/13 + (b^4*d*x^14)/14 + a^4*c*x + (6*a^2*b^2*c *x^7)/7 + (3*a^2*b^2*d*x^8)/4 + a^3*b*c*x^4 + (2*a*b^3*c*x^10)/5 + (4*a^3* b*d*x^5)/5 + (4*a*b^3*d*x^11)/11
Time = 0.17 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.88 \[ \int \left (a+b x^3\right )^3 \left (a c+a d x+b c x^3+b d x^4\right ) \, dx=\frac {x \left (1430 b^{4} d \,x^{13}+1540 b^{4} c \,x^{12}+7280 a \,b^{3} d \,x^{10}+8008 a \,b^{3} c \,x^{9}+15015 a^{2} b^{2} d \,x^{7}+17160 a^{2} b^{2} c \,x^{6}+16016 a^{3} b d \,x^{4}+20020 a^{3} b c \,x^{3}+10010 a^{4} d x +20020 a^{4} c \right )}{20020} \] Input:
int((b*x^3+a)^3*(b*d*x^4+b*c*x^3+a*d*x+a*c),x)
Output:
(x*(20020*a**4*c + 10010*a**4*d*x + 20020*a**3*b*c*x**3 + 16016*a**3*b*d*x **4 + 17160*a**2*b**2*c*x**6 + 15015*a**2*b**2*d*x**7 + 8008*a*b**3*c*x**9 + 7280*a*b**3*d*x**10 + 1540*b**4*c*x**12 + 1430*b**4*d*x**13))/20020