Integrand size = 22, antiderivative size = 103 \[ \int \left (d+e x^3\right )^3 \left (a+b x^3+c x^6\right ) \, dx=a d^3 x+\frac {1}{4} d^2 (b d+3 a e) x^4+\frac {1}{7} d \left (c d^2+3 e (b d+a e)\right ) x^7+\frac {1}{10} e \left (3 c d^2+e (3 b d+a e)\right ) x^{10}+\frac {1}{13} e^2 (3 c d+b e) x^{13}+\frac {1}{16} c e^3 x^{16} \] Output:
a*d^3*x+1/4*d^2*(3*a*e+b*d)*x^4+1/7*d*(c*d^2+3*e*(a*e+b*d))*x^7+1/10*e*(3* c*d^2+e*(a*e+3*b*d))*x^10+1/13*e^2*(b*e+3*c*d)*x^13+1/16*c*e^3*x^16
Time = 0.03 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.01 \[ \int \left (d+e x^3\right )^3 \left (a+b x^3+c x^6\right ) \, dx=a d^3 x+\frac {1}{4} d^2 (b d+3 a e) x^4+\frac {1}{7} d \left (c d^2+3 b d e+3 a e^2\right ) x^7+\frac {1}{10} e \left (3 c d^2+3 b d e+a e^2\right ) x^{10}+\frac {1}{13} e^2 (3 c d+b e) x^{13}+\frac {1}{16} c e^3 x^{16} \] Input:
Integrate[(d + e*x^3)^3*(a + b*x^3 + c*x^6),x]
Output:
a*d^3*x + (d^2*(b*d + 3*a*e)*x^4)/4 + (d*(c*d^2 + 3*b*d*e + 3*a*e^2)*x^7)/ 7 + (e*(3*c*d^2 + 3*b*d*e + a*e^2)*x^10)/10 + (e^2*(3*c*d + b*e)*x^13)/13 + (c*e^3*x^16)/16
Time = 0.29 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {1737, 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.
\(\displaystyle \int \left (d+e x^3\right )^3 \left (a+b x^3+c x^6\right ) \, dx\) |
\(\Big \downarrow \) 1737 |
\(\displaystyle \int \left (e x^9 \left (e (a e+3 b d)+3 c d^2\right )+d x^6 \left (3 e (a e+b d)+c d^2\right )+d^2 x^3 (3 a e+b d)+a d^3+e^2 x^{12} (b e+3 c d)+c e^3 x^{15}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{10} e x^{10} \left (e (a e+3 b d)+3 c d^2\right )+\frac {1}{7} d x^7 \left (3 e (a e+b d)+c d^2\right )+\frac {1}{4} d^2 x^4 (3 a e+b d)+a d^3 x+\frac {1}{13} e^2 x^{13} (b e+3 c d)+\frac {1}{16} c e^3 x^{16}\) |
Input:
Int[(d + e*x^3)^3*(a + b*x^3 + c*x^6),x]
Output:
a*d^3*x + (d^2*(b*d + 3*a*e)*x^4)/4 + (d*(c*d^2 + 3*e*(b*d + a*e))*x^7)/7 + (e*(3*c*d^2 + e*(3*b*d + a*e))*x^10)/10 + (e^2*(3*c*d + b*e)*x^13)/13 + (c*e^3*x^16)/16
Int[((d_) + (e_.)*(x_)^(n_))^(q_.)*((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2 _)), x_Symbol] :> Int[ExpandIntegrand[(d + e*x^n)^q*(a + b*x^n + c*x^(2*n)) , x], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c , 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[q, 0]
Time = 0.14 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.00
method | result | size |
default | \(\frac {c \,e^{3} x^{16}}{16}+\frac {\left (b \,e^{3}+3 c d \,e^{2}\right ) x^{13}}{13}+\frac {\left (e^{3} a +3 b d \,e^{2}+3 d^{2} e c \right ) x^{10}}{10}+\frac {\left (3 a d \,e^{2}+3 b \,d^{2} e +c \,d^{3}\right ) x^{7}}{7}+\frac {\left (3 d^{2} e a +b \,d^{3}\right ) x^{4}}{4}+a \,d^{3} x\) | \(103\) |
norman | \(a \,d^{3} x +\left (\frac {3}{4} d^{2} e a +\frac {1}{4} b \,d^{3}\right ) x^{4}+\left (\frac {3}{7} a d \,e^{2}+\frac {3}{7} b \,d^{2} e +\frac {1}{7} c \,d^{3}\right ) x^{7}+\left (\frac {1}{10} e^{3} a +\frac {3}{10} b d \,e^{2}+\frac {3}{10} d^{2} e c \right ) x^{10}+\left (\frac {1}{13} b \,e^{3}+\frac {3}{13} c d \,e^{2}\right ) x^{13}+\frac {c \,e^{3} x^{16}}{16}\) | \(103\) |
gosper | \(a \,d^{3} x +\frac {3}{4} x^{4} d^{2} e a +\frac {1}{4} x^{4} b \,d^{3}+\frac {3}{7} x^{7} a d \,e^{2}+\frac {3}{7} x^{7} b \,d^{2} e +\frac {1}{7} x^{7} c \,d^{3}+\frac {1}{10} x^{10} e^{3} a +\frac {3}{10} x^{10} b d \,e^{2}+\frac {3}{10} x^{10} d^{2} e c +\frac {1}{13} x^{13} b \,e^{3}+\frac {3}{13} x^{13} c d \,e^{2}+\frac {1}{16} c \,e^{3} x^{16}\) | \(113\) |
risch | \(a \,d^{3} x +\frac {3}{4} x^{4} d^{2} e a +\frac {1}{4} x^{4} b \,d^{3}+\frac {3}{7} x^{7} a d \,e^{2}+\frac {3}{7} x^{7} b \,d^{2} e +\frac {1}{7} x^{7} c \,d^{3}+\frac {1}{10} x^{10} e^{3} a +\frac {3}{10} x^{10} b d \,e^{2}+\frac {3}{10} x^{10} d^{2} e c +\frac {1}{13} x^{13} b \,e^{3}+\frac {3}{13} x^{13} c d \,e^{2}+\frac {1}{16} c \,e^{3} x^{16}\) | \(113\) |
parallelrisch | \(a \,d^{3} x +\frac {3}{4} x^{4} d^{2} e a +\frac {1}{4} x^{4} b \,d^{3}+\frac {3}{7} x^{7} a d \,e^{2}+\frac {3}{7} x^{7} b \,d^{2} e +\frac {1}{7} x^{7} c \,d^{3}+\frac {1}{10} x^{10} e^{3} a +\frac {3}{10} x^{10} b d \,e^{2}+\frac {3}{10} x^{10} d^{2} e c +\frac {1}{13} x^{13} b \,e^{3}+\frac {3}{13} x^{13} c d \,e^{2}+\frac {1}{16} c \,e^{3} x^{16}\) | \(113\) |
orering | \(\frac {x \left (455 c \,e^{3} x^{15}+560 b \,e^{3} x^{12}+1680 c d \,e^{2} x^{12}+728 a \,e^{3} x^{9}+2184 b d \,e^{2} x^{9}+2184 c \,d^{2} e \,x^{9}+3120 a d \,e^{2} x^{6}+3120 b \,d^{2} e \,x^{6}+1040 c \,d^{3} x^{6}+5460 a \,d^{2} e \,x^{3}+1820 b \,d^{3} x^{3}+7280 d^{3} a \right )}{7280}\) | \(116\) |
Input:
int((e*x^3+d)^3*(c*x^6+b*x^3+a),x,method=_RETURNVERBOSE)
Output:
1/16*c*e^3*x^16+1/13*(b*e^3+3*c*d*e^2)*x^13+1/10*(a*e^3+3*b*d*e^2+3*c*d^2* e)*x^10+1/7*(3*a*d*e^2+3*b*d^2*e+c*d^3)*x^7+1/4*(3*a*d^2*e+b*d^3)*x^4+a*d^ 3*x
Time = 0.06 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.99 \[ \int \left (d+e x^3\right )^3 \left (a+b x^3+c x^6\right ) \, dx=\frac {1}{16} \, c e^{3} x^{16} + \frac {1}{13} \, {\left (3 \, c d e^{2} + b e^{3}\right )} x^{13} + \frac {1}{10} \, {\left (3 \, c d^{2} e + 3 \, b d e^{2} + a e^{3}\right )} x^{10} + \frac {1}{7} \, {\left (c d^{3} + 3 \, b d^{2} e + 3 \, a d e^{2}\right )} x^{7} + a d^{3} x + \frac {1}{4} \, {\left (b d^{3} + 3 \, a d^{2} e\right )} x^{4} \] Input:
integrate((e*x^3+d)^3*(c*x^6+b*x^3+a),x, algorithm="fricas")
Output:
1/16*c*e^3*x^16 + 1/13*(3*c*d*e^2 + b*e^3)*x^13 + 1/10*(3*c*d^2*e + 3*b*d* e^2 + a*e^3)*x^10 + 1/7*(c*d^3 + 3*b*d^2*e + 3*a*d*e^2)*x^7 + a*d^3*x + 1/ 4*(b*d^3 + 3*a*d^2*e)*x^4
Time = 0.03 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.14 \[ \int \left (d+e x^3\right )^3 \left (a+b x^3+c x^6\right ) \, dx=a d^{3} x + \frac {c e^{3} x^{16}}{16} + x^{13} \left (\frac {b e^{3}}{13} + \frac {3 c d e^{2}}{13}\right ) + x^{10} \left (\frac {a e^{3}}{10} + \frac {3 b d e^{2}}{10} + \frac {3 c d^{2} e}{10}\right ) + x^{7} \cdot \left (\frac {3 a d e^{2}}{7} + \frac {3 b d^{2} e}{7} + \frac {c d^{3}}{7}\right ) + x^{4} \cdot \left (\frac {3 a d^{2} e}{4} + \frac {b d^{3}}{4}\right ) \] Input:
integrate((e*x**3+d)**3*(c*x**6+b*x**3+a),x)
Output:
a*d**3*x + c*e**3*x**16/16 + x**13*(b*e**3/13 + 3*c*d*e**2/13) + x**10*(a* e**3/10 + 3*b*d*e**2/10 + 3*c*d**2*e/10) + x**7*(3*a*d*e**2/7 + 3*b*d**2*e /7 + c*d**3/7) + x**4*(3*a*d**2*e/4 + b*d**3/4)
Time = 0.03 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.99 \[ \int \left (d+e x^3\right )^3 \left (a+b x^3+c x^6\right ) \, dx=\frac {1}{16} \, c e^{3} x^{16} + \frac {1}{13} \, {\left (3 \, c d e^{2} + b e^{3}\right )} x^{13} + \frac {1}{10} \, {\left (3 \, c d^{2} e + 3 \, b d e^{2} + a e^{3}\right )} x^{10} + \frac {1}{7} \, {\left (c d^{3} + 3 \, b d^{2} e + 3 \, a d e^{2}\right )} x^{7} + a d^{3} x + \frac {1}{4} \, {\left (b d^{3} + 3 \, a d^{2} e\right )} x^{4} \] Input:
integrate((e*x^3+d)^3*(c*x^6+b*x^3+a),x, algorithm="maxima")
Output:
1/16*c*e^3*x^16 + 1/13*(3*c*d*e^2 + b*e^3)*x^13 + 1/10*(3*c*d^2*e + 3*b*d* e^2 + a*e^3)*x^10 + 1/7*(c*d^3 + 3*b*d^2*e + 3*a*d*e^2)*x^7 + a*d^3*x + 1/ 4*(b*d^3 + 3*a*d^2*e)*x^4
Time = 0.13 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.09 \[ \int \left (d+e x^3\right )^3 \left (a+b x^3+c x^6\right ) \, dx=\frac {1}{16} \, c e^{3} x^{16} + \frac {3}{13} \, c d e^{2} x^{13} + \frac {1}{13} \, b e^{3} x^{13} + \frac {3}{10} \, c d^{2} e x^{10} + \frac {3}{10} \, b d e^{2} x^{10} + \frac {1}{10} \, a e^{3} x^{10} + \frac {1}{7} \, c d^{3} x^{7} + \frac {3}{7} \, b d^{2} e x^{7} + \frac {3}{7} \, a d e^{2} x^{7} + \frac {1}{4} \, b d^{3} x^{4} + \frac {3}{4} \, a d^{2} e x^{4} + a d^{3} x \] Input:
integrate((e*x^3+d)^3*(c*x^6+b*x^3+a),x, algorithm="giac")
Output:
1/16*c*e^3*x^16 + 3/13*c*d*e^2*x^13 + 1/13*b*e^3*x^13 + 3/10*c*d^2*e*x^10 + 3/10*b*d*e^2*x^10 + 1/10*a*e^3*x^10 + 1/7*c*d^3*x^7 + 3/7*b*d^2*e*x^7 + 3/7*a*d*e^2*x^7 + 1/4*b*d^3*x^4 + 3/4*a*d^2*e*x^4 + a*d^3*x
Time = 10.60 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.99 \[ \int \left (d+e x^3\right )^3 \left (a+b x^3+c x^6\right ) \, dx=x^4\,\left (\frac {b\,d^3}{4}+\frac {3\,a\,e\,d^2}{4}\right )+x^{13}\,\left (\frac {b\,e^3}{13}+\frac {3\,c\,d\,e^2}{13}\right )+x^7\,\left (\frac {c\,d^3}{7}+\frac {3\,b\,d^2\,e}{7}+\frac {3\,a\,d\,e^2}{7}\right )+x^{10}\,\left (\frac {3\,c\,d^2\,e}{10}+\frac {3\,b\,d\,e^2}{10}+\frac {a\,e^3}{10}\right )+\frac {c\,e^3\,x^{16}}{16}+a\,d^3\,x \] Input:
int((d + e*x^3)^3*(a + b*x^3 + c*x^6),x)
Output:
x^4*((b*d^3)/4 + (3*a*d^2*e)/4) + x^13*((b*e^3)/13 + (3*c*d*e^2)/13) + x^7 *((c*d^3)/7 + (3*a*d*e^2)/7 + (3*b*d^2*e)/7) + x^10*((a*e^3)/10 + (3*b*d*e ^2)/10 + (3*c*d^2*e)/10) + (c*e^3*x^16)/16 + a*d^3*x
Time = 0.21 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.12 \[ \int \left (d+e x^3\right )^3 \left (a+b x^3+c x^6\right ) \, dx=\frac {x \left (455 c \,e^{3} x^{15}+560 b \,e^{3} x^{12}+1680 c d \,e^{2} x^{12}+728 a \,e^{3} x^{9}+2184 b d \,e^{2} x^{9}+2184 c \,d^{2} e \,x^{9}+3120 a d \,e^{2} x^{6}+3120 b \,d^{2} e \,x^{6}+1040 c \,d^{3} x^{6}+5460 a \,d^{2} e \,x^{3}+1820 b \,d^{3} x^{3}+7280 a \,d^{3}\right )}{7280} \] Input:
int((e*x^3+d)^3*(c*x^6+b*x^3+a),x)
Output:
(x*(7280*a*d**3 + 5460*a*d**2*e*x**3 + 3120*a*d*e**2*x**6 + 728*a*e**3*x** 9 + 1820*b*d**3*x**3 + 3120*b*d**2*e*x**6 + 2184*b*d*e**2*x**9 + 560*b*e** 3*x**12 + 1040*c*d**3*x**6 + 2184*c*d**2*e*x**9 + 1680*c*d*e**2*x**12 + 45 5*c*e**3*x**15))/7280