Integrand size = 22, antiderivative size = 163 \[ \int \left (d+e x^3\right )^5 \left (a+b x^3+c x^6\right ) \, dx=a d^5 x+\frac {1}{4} d^4 (b d+5 a e) x^4+\frac {1}{7} d^3 \left (c d^2+5 e (b d+2 a e)\right ) x^7+\frac {1}{2} d^2 e \left (c d^2+2 e (b d+a e)\right ) x^{10}+\frac {5}{13} d e^2 \left (2 c d^2+e (2 b d+a e)\right ) x^{13}+\frac {1}{16} e^3 \left (10 c d^2+e (5 b d+a e)\right ) x^{16}+\frac {1}{19} e^4 (5 c d+b e) x^{19}+\frac {1}{22} c e^5 x^{22} \] Output:
a*d^5*x+1/4*d^4*(5*a*e+b*d)*x^4+1/7*d^3*(c*d^2+5*e*(2*a*e+b*d))*x^7+1/2*d^ 2*e*(c*d^2+2*e*(a*e+b*d))*x^10+5/13*d*e^2*(2*c*d^2+e*(a*e+2*b*d))*x^13+1/1 6*e^3*(10*c*d^2+e*(a*e+5*b*d))*x^16+1/19*e^4*(b*e+5*c*d)*x^19+1/22*c*e^5*x ^22
Time = 0.05 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.01 \[ \int \left (d+e x^3\right )^5 \left (a+b x^3+c x^6\right ) \, dx=a d^5 x+\frac {1}{4} d^4 (b d+5 a e) x^4+\frac {1}{7} d^3 \left (c d^2+5 b d e+10 a e^2\right ) x^7+\frac {1}{2} d^2 e \left (c d^2+2 b d e+2 a e^2\right ) x^{10}+\frac {5}{13} d e^2 \left (2 c d^2+2 b d e+a e^2\right ) x^{13}+\frac {1}{16} e^3 \left (10 c d^2+5 b d e+a e^2\right ) x^{16}+\frac {1}{19} e^4 (5 c d+b e) x^{19}+\frac {1}{22} c e^5 x^{22} \] Input:
Integrate[(d + e*x^3)^5*(a + b*x^3 + c*x^6),x]
Output:
a*d^5*x + (d^4*(b*d + 5*a*e)*x^4)/4 + (d^3*(c*d^2 + 5*b*d*e + 10*a*e^2)*x^ 7)/7 + (d^2*e*(c*d^2 + 2*b*d*e + 2*a*e^2)*x^10)/2 + (5*d*e^2*(2*c*d^2 + 2* b*d*e + a*e^2)*x^13)/13 + (e^3*(10*c*d^2 + 5*b*d*e + a*e^2)*x^16)/16 + (e^ 4*(5*c*d + b*e)*x^19)/19 + (c*e^5*x^22)/22
Time = 0.38 (sec) , antiderivative size = 163, 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.
| \(\displaystyle \int \left (d+e x^3\right )^5 \left (a+b x^3+c x^6\right ) \, dx\) |
| \(\Big \downarrow \) 1737 |
| \(\displaystyle \int \left (e^3 x^{15} \left (e (a e+5 b d)+10 c d^2\right )+5 d e^2 x^{12} \left (e (a e+2 b d)+2 c d^2\right )+5 d^2 e x^9 \left (2 e (a e+b d)+c d^2\right )+d^3 x^6 \left (5 e (2 a e+b d)+c d^2\right )+d^4 x^3 (5 a e+b d)+a d^5+e^4 x^{18} (b e+5 c d)+c e^5 x^{21}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{16} e^3 x^{16} \left (e (a e+5 b d)+10 c d^2\right )+\frac {5}{13} d e^2 x^{13} \left (e (a e+2 b d)+2 c d^2\right )+\frac {1}{2} d^2 e x^{10} \left (2 e (a e+b d)+c d^2\right )+\frac {1}{7} d^3 x^7 \left (5 e (2 a e+b d)+c d^2\right )+\frac {1}{4} d^4 x^4 (5 a e+b d)+a d^5 x+\frac {1}{19} e^4 x^{19} (b e+5 c d)+\frac {1}{22} c e^5 x^{22}\) |
Input:
Int[(d + e*x^3)^5*(a + b*x^3 + c*x^6),x]
Output:
a*d^5*x + (d^4*(b*d + 5*a*e)*x^4)/4 + (d^3*(c*d^2 + 5*e*(b*d + 2*a*e))*x^7 )/7 + (d^2*e*(c*d^2 + 2*e*(b*d + a*e))*x^10)/2 + (5*d*e^2*(2*c*d^2 + e*(2* b*d + a*e))*x^13)/13 + (e^3*(10*c*d^2 + e*(5*b*d + a*e))*x^16)/16 + (e^4*( 5*c*d + b*e)*x^19)/19 + (c*e^5*x^22)/22
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 = 165, normalized size of antiderivative = 1.01
| method | result | size |
| norman | \(a \,d^{5} x +\left (\frac {5}{4} d^{4} e a +\frac {1}{4} d^{5} b \right ) x^{4}+\left (\frac {10}{7} d^{3} e^{2} a +\frac {5}{7} d^{4} e b +\frac {1}{7} d^{5} c \right ) x^{7}+\left (d^{2} e^{3} a +d^{3} e^{2} b +\frac {1}{2} d^{4} e c \right ) x^{10}+\left (\frac {5}{13} d \,e^{4} a +\frac {10}{13} d^{2} e^{3} b +\frac {10}{13} d^{3} e^{2} c \right ) x^{13}+\left (\frac {1}{16} e^{5} a +\frac {5}{16} d \,e^{4} b +\frac {5}{8} d^{2} e^{3} c \right ) x^{16}+\left (\frac {1}{19} e^{5} b +\frac {5}{19} d \,e^{4} c \right ) x^{19}+\frac {c \,e^{5} x^{22}}{22}\) | \(165\) |
| default | \(\frac {c \,e^{5} x^{22}}{22}+\frac {\left (e^{5} b +5 d \,e^{4} c \right ) x^{19}}{19}+\frac {\left (e^{5} a +5 d \,e^{4} b +10 d^{2} e^{3} c \right ) x^{16}}{16}+\frac {\left (5 d \,e^{4} a +10 d^{2} e^{3} b +10 d^{3} e^{2} c \right ) x^{13}}{13}+\frac {\left (10 d^{2} e^{3} a +10 d^{3} e^{2} b +5 d^{4} e c \right ) x^{10}}{10}+\frac {\left (10 d^{3} e^{2} a +5 d^{4} e b +d^{5} c \right ) x^{7}}{7}+\frac {\left (5 d^{4} e a +d^{5} b \right ) x^{4}}{4}+a \,d^{5} x\) | \(169\) |
| gosper | \(a \,d^{5} x +\frac {5}{4} x^{4} d^{4} e a +\frac {1}{4} x^{4} d^{5} b +\frac {10}{7} x^{7} d^{3} e^{2} a +\frac {5}{7} x^{7} d^{4} e b +\frac {1}{7} x^{7} d^{5} c +x^{10} d^{2} e^{3} a +x^{10} d^{3} e^{2} b +\frac {1}{2} x^{10} d^{4} e c +\frac {5}{13} x^{13} d \,e^{4} a +\frac {10}{13} x^{13} d^{2} e^{3} b +\frac {10}{13} x^{13} d^{3} e^{2} c +\frac {1}{16} x^{16} e^{5} a +\frac {5}{16} x^{16} d \,e^{4} b +\frac {5}{8} x^{16} d^{2} e^{3} c +\frac {1}{19} x^{19} e^{5} b +\frac {5}{19} x^{19} d \,e^{4} c +\frac {1}{22} c \,e^{5} x^{22}\) | \(183\) |
| risch | \(a \,d^{5} x +\frac {5}{4} x^{4} d^{4} e a +\frac {1}{4} x^{4} d^{5} b +\frac {10}{7} x^{7} d^{3} e^{2} a +\frac {5}{7} x^{7} d^{4} e b +\frac {1}{7} x^{7} d^{5} c +x^{10} d^{2} e^{3} a +x^{10} d^{3} e^{2} b +\frac {1}{2} x^{10} d^{4} e c +\frac {5}{13} x^{13} d \,e^{4} a +\frac {10}{13} x^{13} d^{2} e^{3} b +\frac {10}{13} x^{13} d^{3} e^{2} c +\frac {1}{16} x^{16} e^{5} a +\frac {5}{16} x^{16} d \,e^{4} b +\frac {5}{8} x^{16} d^{2} e^{3} c +\frac {1}{19} x^{19} e^{5} b +\frac {5}{19} x^{19} d \,e^{4} c +\frac {1}{22} c \,e^{5} x^{22}\) | \(183\) |
| parallelrisch | \(a \,d^{5} x +\frac {5}{4} x^{4} d^{4} e a +\frac {1}{4} x^{4} d^{5} b +\frac {10}{7} x^{7} d^{3} e^{2} a +\frac {5}{7} x^{7} d^{4} e b +\frac {1}{7} x^{7} d^{5} c +x^{10} d^{2} e^{3} a +x^{10} d^{3} e^{2} b +\frac {1}{2} x^{10} d^{4} e c +\frac {5}{13} x^{13} d \,e^{4} a +\frac {10}{13} x^{13} d^{2} e^{3} b +\frac {10}{13} x^{13} d^{3} e^{2} c +\frac {1}{16} x^{16} e^{5} a +\frac {5}{16} x^{16} d \,e^{4} b +\frac {5}{8} x^{16} d^{2} e^{3} c +\frac {1}{19} x^{19} e^{5} b +\frac {5}{19} x^{19} d \,e^{4} c +\frac {1}{22} c \,e^{5} x^{22}\) | \(183\) |
| orering | \(\frac {x \left (13832 e^{5} c \,x^{21}+16016 b \,e^{5} x^{18}+80080 c d \,e^{4} x^{18}+19019 a \,e^{5} x^{15}+95095 b d \,e^{4} x^{15}+190190 c \,d^{2} e^{3} x^{15}+117040 a d \,e^{4} x^{12}+234080 b \,d^{2} e^{3} x^{12}+234080 c \,d^{3} e^{2} x^{12}+304304 a \,d^{2} e^{3} x^{9}+304304 b \,d^{3} e^{2} x^{9}+152152 c \,d^{4} e \,x^{9}+434720 a \,d^{3} e^{2} x^{6}+217360 b \,d^{4} e \,x^{6}+43472 c \,d^{5} x^{6}+380380 a \,d^{4} e \,x^{3}+76076 b \,d^{5} x^{3}+304304 d^{5} a \right )}{304304}\) | \(188\) |
Input:
int((e*x^3+d)^5*(c*x^6+b*x^3+a),x,method=_RETURNVERBOSE)
Output:
a*d^5*x+(5/4*d^4*e*a+1/4*d^5*b)*x^4+(10/7*d^3*e^2*a+5/7*d^4*e*b+1/7*d^5*c) *x^7+(d^2*e^3*a+d^3*e^2*b+1/2*d^4*e*c)*x^10+(5/13*d*e^4*a+10/13*d^2*e^3*b+ 10/13*d^3*e^2*c)*x^13+(1/16*e^5*a+5/16*d*e^4*b+5/8*d^2*e^3*c)*x^16+(1/19*e ^5*b+5/19*d*e^4*c)*x^19+1/22*c*e^5*x^22
Time = 0.06 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.02 \[ \int \left (d+e x^3\right )^5 \left (a+b x^3+c x^6\right ) \, dx=\frac {1}{22} \, c e^{5} x^{22} + \frac {1}{19} \, {\left (5 \, c d e^{4} + b e^{5}\right )} x^{19} + \frac {1}{16} \, {\left (10 \, c d^{2} e^{3} + 5 \, b d e^{4} + a e^{5}\right )} x^{16} + \frac {5}{13} \, {\left (2 \, c d^{3} e^{2} + 2 \, b d^{2} e^{3} + a d e^{4}\right )} x^{13} + \frac {1}{2} \, {\left (c d^{4} e + 2 \, b d^{3} e^{2} + 2 \, a d^{2} e^{3}\right )} x^{10} + \frac {1}{7} \, {\left (c d^{5} + 5 \, b d^{4} e + 10 \, a d^{3} e^{2}\right )} x^{7} + a d^{5} x + \frac {1}{4} \, {\left (b d^{5} + 5 \, a d^{4} e\right )} x^{4} \] Input:
integrate((e*x^3+d)^5*(c*x^6+b*x^3+a),x, algorithm="fricas")
Output:
1/22*c*e^5*x^22 + 1/19*(5*c*d*e^4 + b*e^5)*x^19 + 1/16*(10*c*d^2*e^3 + 5*b *d*e^4 + a*e^5)*x^16 + 5/13*(2*c*d^3*e^2 + 2*b*d^2*e^3 + a*d*e^4)*x^13 + 1 /2*(c*d^4*e + 2*b*d^3*e^2 + 2*a*d^2*e^3)*x^10 + 1/7*(c*d^5 + 5*b*d^4*e + 1 0*a*d^3*e^2)*x^7 + a*d^5*x + 1/4*(b*d^5 + 5*a*d^4*e)*x^4
Time = 0.03 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.15 \[ \int \left (d+e x^3\right )^5 \left (a+b x^3+c x^6\right ) \, dx=a d^{5} x + \frac {c e^{5} x^{22}}{22} + x^{19} \left (\frac {b e^{5}}{19} + \frac {5 c d e^{4}}{19}\right ) + x^{16} \left (\frac {a e^{5}}{16} + \frac {5 b d e^{4}}{16} + \frac {5 c d^{2} e^{3}}{8}\right ) + x^{13} \cdot \left (\frac {5 a d e^{4}}{13} + \frac {10 b d^{2} e^{3}}{13} + \frac {10 c d^{3} e^{2}}{13}\right ) + x^{10} \left (a d^{2} e^{3} + b d^{3} e^{2} + \frac {c d^{4} e}{2}\right ) + x^{7} \cdot \left (\frac {10 a d^{3} e^{2}}{7} + \frac {5 b d^{4} e}{7} + \frac {c d^{5}}{7}\right ) + x^{4} \cdot \left (\frac {5 a d^{4} e}{4} + \frac {b d^{5}}{4}\right ) \] Input:
integrate((e*x**3+d)**5*(c*x**6+b*x**3+a),x)
Output:
a*d**5*x + c*e**5*x**22/22 + x**19*(b*e**5/19 + 5*c*d*e**4/19) + x**16*(a* e**5/16 + 5*b*d*e**4/16 + 5*c*d**2*e**3/8) + x**13*(5*a*d*e**4/13 + 10*b*d **2*e**3/13 + 10*c*d**3*e**2/13) + x**10*(a*d**2*e**3 + b*d**3*e**2 + c*d* *4*e/2) + x**7*(10*a*d**3*e**2/7 + 5*b*d**4*e/7 + c*d**5/7) + x**4*(5*a*d* *4*e/4 + b*d**5/4)
Time = 0.04 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.02 \[ \int \left (d+e x^3\right )^5 \left (a+b x^3+c x^6\right ) \, dx=\frac {1}{22} \, c e^{5} x^{22} + \frac {1}{19} \, {\left (5 \, c d e^{4} + b e^{5}\right )} x^{19} + \frac {1}{16} \, {\left (10 \, c d^{2} e^{3} + 5 \, b d e^{4} + a e^{5}\right )} x^{16} + \frac {5}{13} \, {\left (2 \, c d^{3} e^{2} + 2 \, b d^{2} e^{3} + a d e^{4}\right )} x^{13} + \frac {1}{2} \, {\left (c d^{4} e + 2 \, b d^{3} e^{2} + 2 \, a d^{2} e^{3}\right )} x^{10} + \frac {1}{7} \, {\left (c d^{5} + 5 \, b d^{4} e + 10 \, a d^{3} e^{2}\right )} x^{7} + a d^{5} x + \frac {1}{4} \, {\left (b d^{5} + 5 \, a d^{4} e\right )} x^{4} \] Input:
integrate((e*x^3+d)^5*(c*x^6+b*x^3+a),x, algorithm="maxima")
Output:
1/22*c*e^5*x^22 + 1/19*(5*c*d*e^4 + b*e^5)*x^19 + 1/16*(10*c*d^2*e^3 + 5*b *d*e^4 + a*e^5)*x^16 + 5/13*(2*c*d^3*e^2 + 2*b*d^2*e^3 + a*d*e^4)*x^13 + 1 /2*(c*d^4*e + 2*b*d^3*e^2 + 2*a*d^2*e^3)*x^10 + 1/7*(c*d^5 + 5*b*d^4*e + 1 0*a*d^3*e^2)*x^7 + a*d^5*x + 1/4*(b*d^5 + 5*a*d^4*e)*x^4
Time = 0.11 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.12 \[ \int \left (d+e x^3\right )^5 \left (a+b x^3+c x^6\right ) \, dx=\frac {1}{22} \, c e^{5} x^{22} + \frac {5}{19} \, c d e^{4} x^{19} + \frac {1}{19} \, b e^{5} x^{19} + \frac {5}{8} \, c d^{2} e^{3} x^{16} + \frac {5}{16} \, b d e^{4} x^{16} + \frac {1}{16} \, a e^{5} x^{16} + \frac {10}{13} \, c d^{3} e^{2} x^{13} + \frac {10}{13} \, b d^{2} e^{3} x^{13} + \frac {5}{13} \, a d e^{4} x^{13} + \frac {1}{2} \, c d^{4} e x^{10} + b d^{3} e^{2} x^{10} + a d^{2} e^{3} x^{10} + \frac {1}{7} \, c d^{5} x^{7} + \frac {5}{7} \, b d^{4} e x^{7} + \frac {10}{7} \, a d^{3} e^{2} x^{7} + \frac {1}{4} \, b d^{5} x^{4} + \frac {5}{4} \, a d^{4} e x^{4} + a d^{5} x \] Input:
integrate((e*x^3+d)^5*(c*x^6+b*x^3+a),x, algorithm="giac")
Output:
1/22*c*e^5*x^22 + 5/19*c*d*e^4*x^19 + 1/19*b*e^5*x^19 + 5/8*c*d^2*e^3*x^16 + 5/16*b*d*e^4*x^16 + 1/16*a*e^5*x^16 + 10/13*c*d^3*e^2*x^13 + 10/13*b*d^ 2*e^3*x^13 + 5/13*a*d*e^4*x^13 + 1/2*c*d^4*e*x^10 + b*d^3*e^2*x^10 + a*d^2 *e^3*x^10 + 1/7*c*d^5*x^7 + 5/7*b*d^4*e*x^7 + 10/7*a*d^3*e^2*x^7 + 1/4*b*d ^5*x^4 + 5/4*a*d^4*e*x^4 + a*d^5*x
Time = 0.05 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.97 \[ \int \left (d+e x^3\right )^5 \left (a+b x^3+c x^6\right ) \, dx=x^4\,\left (\frac {b\,d^5}{4}+\frac {5\,a\,e\,d^4}{4}\right )+x^{19}\,\left (\frac {b\,e^5}{19}+\frac {5\,c\,d\,e^4}{19}\right )+x^7\,\left (\frac {c\,d^5}{7}+\frac {5\,b\,d^4\,e}{7}+\frac {10\,a\,d^3\,e^2}{7}\right )+x^{16}\,\left (\frac {5\,c\,d^2\,e^3}{8}+\frac {5\,b\,d\,e^4}{16}+\frac {a\,e^5}{16}\right )+\frac {c\,e^5\,x^{22}}{22}+a\,d^5\,x+\frac {d^2\,e\,x^{10}\,\left (c\,d^2+2\,b\,d\,e+2\,a\,e^2\right )}{2}+\frac {5\,d\,e^2\,x^{13}\,\left (2\,c\,d^2+2\,b\,d\,e+a\,e^2\right )}{13} \] Input:
int((d + e*x^3)^5*(a + b*x^3 + c*x^6),x)
Output:
x^4*((b*d^5)/4 + (5*a*d^4*e)/4) + x^19*((b*e^5)/19 + (5*c*d*e^4)/19) + x^7 *((c*d^5)/7 + (10*a*d^3*e^2)/7 + (5*b*d^4*e)/7) + x^16*((a*e^5)/16 + (5*c* d^2*e^3)/8 + (5*b*d*e^4)/16) + (c*e^5*x^22)/22 + a*d^5*x + (d^2*e*x^10*(2* a*e^2 + c*d^2 + 2*b*d*e))/2 + (5*d*e^2*x^13*(a*e^2 + 2*c*d^2 + 2*b*d*e))/1 3
Time = 0.22 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.15 \[ \int \left (d+e x^3\right )^5 \left (a+b x^3+c x^6\right ) \, dx=\frac {x \left (13832 c \,e^{5} x^{21}+16016 b \,e^{5} x^{18}+80080 c d \,e^{4} x^{18}+19019 a \,e^{5} x^{15}+95095 b d \,e^{4} x^{15}+190190 c \,d^{2} e^{3} x^{15}+117040 a d \,e^{4} x^{12}+234080 b \,d^{2} e^{3} x^{12}+234080 c \,d^{3} e^{2} x^{12}+304304 a \,d^{2} e^{3} x^{9}+304304 b \,d^{3} e^{2} x^{9}+152152 c \,d^{4} e \,x^{9}+434720 a \,d^{3} e^{2} x^{6}+217360 b \,d^{4} e \,x^{6}+43472 c \,d^{5} x^{6}+380380 a \,d^{4} e \,x^{3}+76076 b \,d^{5} x^{3}+304304 a \,d^{5}\right )}{304304} \] Input:
int((e*x^3+d)^5*(c*x^6+b*x^3+a),x)
Output:
(x*(304304*a*d**5 + 380380*a*d**4*e*x**3 + 434720*a*d**3*e**2*x**6 + 30430 4*a*d**2*e**3*x**9 + 117040*a*d*e**4*x**12 + 19019*a*e**5*x**15 + 76076*b* d**5*x**3 + 217360*b*d**4*e*x**6 + 304304*b*d**3*e**2*x**9 + 234080*b*d**2 *e**3*x**12 + 95095*b*d*e**4*x**15 + 16016*b*e**5*x**18 + 43472*c*d**5*x** 6 + 152152*c*d**4*e*x**9 + 234080*c*d**3*e**2*x**12 + 190190*c*d**2*e**3*x **15 + 80080*c*d*e**4*x**18 + 13832*c*e**5*x**21))/304304