Integrand size = 22, antiderivative size = 135 \[ \int \left (d+e x^3\right )^4 \left (a+b x^3+c x^6\right ) \, dx=a d^4 x+\frac {1}{4} d^3 (b d+4 a e) x^4+\frac {1}{7} d^2 \left (c d^2+4 b d e+6 a e^2\right ) x^7+\frac {1}{5} d e \left (2 c d^2+e (3 b d+2 a e)\right ) x^{10}+\frac {1}{13} e^2 \left (6 c d^2+e (4 b d+a e)\right ) x^{13}+\frac {1}{16} e^3 (4 c d+b e) x^{16}+\frac {1}{19} c e^4 x^{19} \] Output:
a*d^4*x+1/4*d^3*(4*a*e+b*d)*x^4+1/7*d^2*(6*a*e^2+4*b*d*e+c*d^2)*x^7+1/5*d* e*(2*c*d^2+e*(2*a*e+3*b*d))*x^10+1/13*e^2*(6*c*d^2+e*(a*e+4*b*d))*x^13+1/1 6*e^3*(b*e+4*c*d)*x^16+1/19*c*e^4*x^19
Time = 0.04 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.00 \[ \int \left (d+e x^3\right )^4 \left (a+b x^3+c x^6\right ) \, dx=a d^4 x+\frac {1}{4} d^3 (b d+4 a e) x^4+\frac {1}{7} d^2 \left (c d^2+4 b d e+6 a e^2\right ) x^7+\frac {1}{5} d e \left (2 c d^2+3 b d e+2 a e^2\right ) x^{10}+\frac {1}{13} e^2 \left (6 c d^2+4 b d e+a e^2\right ) x^{13}+\frac {1}{16} e^3 (4 c d+b e) x^{16}+\frac {1}{19} c e^4 x^{19} \] Input:
Integrate[(d + e*x^3)^4*(a + b*x^3 + c*x^6),x]
Output:
a*d^4*x + (d^3*(b*d + 4*a*e)*x^4)/4 + (d^2*(c*d^2 + 4*b*d*e + 6*a*e^2)*x^7 )/7 + (d*e*(2*c*d^2 + 3*b*d*e + 2*a*e^2)*x^10)/5 + (e^2*(6*c*d^2 + 4*b*d*e + a*e^2)*x^13)/13 + (e^3*(4*c*d + b*e)*x^16)/16 + (c*e^4*x^19)/19
Time = 0.34 (sec) , antiderivative size = 135, 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.
| \(\displaystyle \int \left (d+e x^3\right )^4 \left (a+b x^3+c x^6\right ) \, dx\) |
| \(\Big \downarrow \) 1737 |
| \(\displaystyle \int \left (e^2 x^{12} \left (e (a e+4 b d)+6 c d^2\right )+d^2 x^6 \left (6 a e^2+4 b d e+c d^2\right )+2 d e x^9 \left (e (2 a e+3 b d)+2 c d^2\right )+d^3 x^3 (4 a e+b d)+a d^4+e^3 x^{15} (b e+4 c d)+c e^4 x^{18}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{13} e^2 x^{13} \left (e (a e+4 b d)+6 c d^2\right )+\frac {1}{7} d^2 x^7 \left (6 a e^2+4 b d e+c d^2\right )+\frac {1}{5} d e x^{10} \left (e (2 a e+3 b d)+2 c d^2\right )+\frac {1}{4} d^3 x^4 (4 a e+b d)+a d^4 x+\frac {1}{16} e^3 x^{16} (b e+4 c d)+\frac {1}{19} c e^4 x^{19}\) |
Input:
Int[(d + e*x^3)^4*(a + b*x^3 + c*x^6),x]
Output:
a*d^4*x + (d^3*(b*d + 4*a*e)*x^4)/4 + (d^2*(c*d^2 + 4*b*d*e + 6*a*e^2)*x^7 )/7 + (d*e*(2*c*d^2 + e*(3*b*d + 2*a*e))*x^10)/5 + (e^2*(6*c*d^2 + e*(4*b* d + a*e))*x^13)/13 + (e^3*(4*c*d + b*e)*x^16)/16 + (c*e^4*x^19)/19
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 = 0.13 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.99
| method | result | size |
| norman | \(a \,d^{4} x +\left (d^{3} e a +\frac {1}{4} d^{4} b \right ) x^{4}+\left (\frac {6}{7} d^{2} e^{2} a +\frac {4}{7} d^{3} e b +\frac {1}{7} c \,d^{4}\right ) x^{7}+\left (\frac {2}{5} d \,e^{3} a +\frac {3}{5} d^{2} e^{2} b +\frac {2}{5} c \,d^{3} e \right ) x^{10}+\left (\frac {1}{13} e^{4} a +\frac {4}{13} d \,e^{3} b +\frac {6}{13} d^{2} e^{2} c \right ) x^{13}+\left (\frac {1}{16} e^{4} b +\frac {1}{4} d \,e^{3} c \right ) x^{16}+\frac {c \,e^{4} x^{19}}{19}\) | \(134\) |
| default | \(\frac {c \,e^{4} x^{19}}{19}+\frac {\left (e^{4} b +4 d \,e^{3} c \right ) x^{16}}{16}+\frac {\left (e^{4} a +4 d \,e^{3} b +6 d^{2} e^{2} c \right ) x^{13}}{13}+\frac {\left (4 d \,e^{3} a +6 d^{2} e^{2} b +4 c \,d^{3} e \right ) x^{10}}{10}+\frac {\left (6 d^{2} e^{2} a +4 d^{3} e b +c \,d^{4}\right ) x^{7}}{7}+\frac {\left (4 d^{3} e a +d^{4} b \right ) x^{4}}{4}+a \,d^{4} x\) | \(136\) |
| gosper | \(a \,d^{4} x +x^{4} d^{3} e a +\frac {1}{4} x^{4} d^{4} b +\frac {6}{7} x^{7} d^{2} e^{2} a +\frac {4}{7} x^{7} d^{3} e b +\frac {1}{7} x^{7} c \,d^{4}+\frac {2}{5} x^{10} d \,e^{3} a +\frac {3}{5} x^{10} d^{2} e^{2} b +\frac {2}{5} x^{10} c \,d^{3} e +\frac {1}{13} x^{13} e^{4} a +\frac {4}{13} x^{13} d \,e^{3} b +\frac {6}{13} x^{13} d^{2} e^{2} c +\frac {1}{16} x^{16} e^{4} b +\frac {1}{4} x^{16} d \,e^{3} c +\frac {1}{19} c \,e^{4} x^{19}\) | \(148\) |
| risch | \(a \,d^{4} x +x^{4} d^{3} e a +\frac {1}{4} x^{4} d^{4} b +\frac {6}{7} x^{7} d^{2} e^{2} a +\frac {4}{7} x^{7} d^{3} e b +\frac {1}{7} x^{7} c \,d^{4}+\frac {2}{5} x^{10} d \,e^{3} a +\frac {3}{5} x^{10} d^{2} e^{2} b +\frac {2}{5} x^{10} c \,d^{3} e +\frac {1}{13} x^{13} e^{4} a +\frac {4}{13} x^{13} d \,e^{3} b +\frac {6}{13} x^{13} d^{2} e^{2} c +\frac {1}{16} x^{16} e^{4} b +\frac {1}{4} x^{16} d \,e^{3} c +\frac {1}{19} c \,e^{4} x^{19}\) | \(148\) |
| parallelrisch | \(a \,d^{4} x +x^{4} d^{3} e a +\frac {1}{4} x^{4} d^{4} b +\frac {6}{7} x^{7} d^{2} e^{2} a +\frac {4}{7} x^{7} d^{3} e b +\frac {1}{7} x^{7} c \,d^{4}+\frac {2}{5} x^{10} d \,e^{3} a +\frac {3}{5} x^{10} d^{2} e^{2} b +\frac {2}{5} x^{10} c \,d^{3} e +\frac {1}{13} x^{13} e^{4} a +\frac {4}{13} x^{13} d \,e^{3} b +\frac {6}{13} x^{13} d^{2} e^{2} c +\frac {1}{16} x^{16} e^{4} b +\frac {1}{4} x^{16} d \,e^{3} c +\frac {1}{19} c \,e^{4} x^{19}\) | \(148\) |
| orering | \(\frac {x \left (7280 e^{4} c \,x^{18}+8645 b \,e^{4} x^{15}+34580 c d \,e^{3} x^{15}+10640 a \,e^{4} x^{12}+42560 b d \,e^{3} x^{12}+63840 c \,d^{2} e^{2} x^{12}+55328 a d \,e^{3} x^{9}+82992 b \,d^{2} e^{2} x^{9}+55328 c \,d^{3} e \,x^{9}+118560 a \,d^{2} e^{2} x^{6}+79040 b \,d^{3} e \,x^{6}+19760 c \,d^{4} x^{6}+138320 a \,d^{3} e \,x^{3}+34580 b \,d^{4} x^{3}+138320 d^{4} a \right )}{138320}\) | \(152\) |
Input:
int((e*x^3+d)^4*(c*x^6+b*x^3+a),x,method=_RETURNVERBOSE)
Output:
a*d^4*x+(d^3*e*a+1/4*d^4*b)*x^4+(6/7*d^2*e^2*a+4/7*d^3*e*b+1/7*c*d^4)*x^7+ (2/5*d*e^3*a+3/5*d^2*e^2*b+2/5*c*d^3*e)*x^10+(1/13*e^4*a+4/13*d*e^3*b+6/13 *d^2*e^2*c)*x^13+(1/16*e^4*b+1/4*d*e^3*c)*x^16+1/19*c*e^4*x^19
Time = 0.06 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.00 \[ \int \left (d+e x^3\right )^4 \left (a+b x^3+c x^6\right ) \, dx=\frac {1}{19} \, c e^{4} x^{19} + \frac {1}{16} \, {\left (4 \, c d e^{3} + b e^{4}\right )} x^{16} + \frac {1}{13} \, {\left (6 \, c d^{2} e^{2} + 4 \, b d e^{3} + a e^{4}\right )} x^{13} + \frac {1}{5} \, {\left (2 \, c d^{3} e + 3 \, b d^{2} e^{2} + 2 \, a d e^{3}\right )} x^{10} + \frac {1}{7} \, {\left (c d^{4} + 4 \, b d^{3} e + 6 \, a d^{2} e^{2}\right )} x^{7} + a d^{4} x + \frac {1}{4} \, {\left (b d^{4} + 4 \, a d^{3} e\right )} x^{4} \] Input:
integrate((e*x^3+d)^4*(c*x^6+b*x^3+a),x, algorithm="fricas")
Output:
1/19*c*e^4*x^19 + 1/16*(4*c*d*e^3 + b*e^4)*x^16 + 1/13*(6*c*d^2*e^2 + 4*b* d*e^3 + a*e^4)*x^13 + 1/5*(2*c*d^3*e + 3*b*d^2*e^2 + 2*a*d*e^3)*x^10 + 1/7 *(c*d^4 + 4*b*d^3*e + 6*a*d^2*e^2)*x^7 + a*d^4*x + 1/4*(b*d^4 + 4*a*d^3*e) *x^4
Time = 0.03 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.12 \[ \int \left (d+e x^3\right )^4 \left (a+b x^3+c x^6\right ) \, dx=a d^{4} x + \frac {c e^{4} x^{19}}{19} + x^{16} \left (\frac {b e^{4}}{16} + \frac {c d e^{3}}{4}\right ) + x^{13} \left (\frac {a e^{4}}{13} + \frac {4 b d e^{3}}{13} + \frac {6 c d^{2} e^{2}}{13}\right ) + x^{10} \cdot \left (\frac {2 a d e^{3}}{5} + \frac {3 b d^{2} e^{2}}{5} + \frac {2 c d^{3} e}{5}\right ) + x^{7} \cdot \left (\frac {6 a d^{2} e^{2}}{7} + \frac {4 b d^{3} e}{7} + \frac {c d^{4}}{7}\right ) + x^{4} \left (a d^{3} e + \frac {b d^{4}}{4}\right ) \] Input:
integrate((e*x**3+d)**4*(c*x**6+b*x**3+a),x)
Output:
a*d**4*x + c*e**4*x**19/19 + x**16*(b*e**4/16 + c*d*e**3/4) + x**13*(a*e** 4/13 + 4*b*d*e**3/13 + 6*c*d**2*e**2/13) + x**10*(2*a*d*e**3/5 + 3*b*d**2* e**2/5 + 2*c*d**3*e/5) + x**7*(6*a*d**2*e**2/7 + 4*b*d**3*e/7 + c*d**4/7) + x**4*(a*d**3*e + b*d**4/4)
Time = 0.03 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.00 \[ \int \left (d+e x^3\right )^4 \left (a+b x^3+c x^6\right ) \, dx=\frac {1}{19} \, c e^{4} x^{19} + \frac {1}{16} \, {\left (4 \, c d e^{3} + b e^{4}\right )} x^{16} + \frac {1}{13} \, {\left (6 \, c d^{2} e^{2} + 4 \, b d e^{3} + a e^{4}\right )} x^{13} + \frac {1}{5} \, {\left (2 \, c d^{3} e + 3 \, b d^{2} e^{2} + 2 \, a d e^{3}\right )} x^{10} + \frac {1}{7} \, {\left (c d^{4} + 4 \, b d^{3} e + 6 \, a d^{2} e^{2}\right )} x^{7} + a d^{4} x + \frac {1}{4} \, {\left (b d^{4} + 4 \, a d^{3} e\right )} x^{4} \] Input:
integrate((e*x^3+d)^4*(c*x^6+b*x^3+a),x, algorithm="maxima")
Output:
1/19*c*e^4*x^19 + 1/16*(4*c*d*e^3 + b*e^4)*x^16 + 1/13*(6*c*d^2*e^2 + 4*b* d*e^3 + a*e^4)*x^13 + 1/5*(2*c*d^3*e + 3*b*d^2*e^2 + 2*a*d*e^3)*x^10 + 1/7 *(c*d^4 + 4*b*d^3*e + 6*a*d^2*e^2)*x^7 + a*d^4*x + 1/4*(b*d^4 + 4*a*d^3*e) *x^4
Time = 0.13 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.09 \[ \int \left (d+e x^3\right )^4 \left (a+b x^3+c x^6\right ) \, dx=\frac {1}{19} \, c e^{4} x^{19} + \frac {1}{4} \, c d e^{3} x^{16} + \frac {1}{16} \, b e^{4} x^{16} + \frac {6}{13} \, c d^{2} e^{2} x^{13} + \frac {4}{13} \, b d e^{3} x^{13} + \frac {1}{13} \, a e^{4} x^{13} + \frac {2}{5} \, c d^{3} e x^{10} + \frac {3}{5} \, b d^{2} e^{2} x^{10} + \frac {2}{5} \, a d e^{3} x^{10} + \frac {1}{7} \, c d^{4} x^{7} + \frac {4}{7} \, b d^{3} e x^{7} + \frac {6}{7} \, a d^{2} e^{2} x^{7} + \frac {1}{4} \, b d^{4} x^{4} + a d^{3} e x^{4} + a d^{4} x \] Input:
integrate((e*x^3+d)^4*(c*x^6+b*x^3+a),x, algorithm="giac")
Output:
1/19*c*e^4*x^19 + 1/4*c*d*e^3*x^16 + 1/16*b*e^4*x^16 + 6/13*c*d^2*e^2*x^13 + 4/13*b*d*e^3*x^13 + 1/13*a*e^4*x^13 + 2/5*c*d^3*e*x^10 + 3/5*b*d^2*e^2* x^10 + 2/5*a*d*e^3*x^10 + 1/7*c*d^4*x^7 + 4/7*b*d^3*e*x^7 + 6/7*a*d^2*e^2* x^7 + 1/4*b*d^4*x^4 + a*d^3*e*x^4 + a*d^4*x
Time = 0.04 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.96 \[ \int \left (d+e x^3\right )^4 \left (a+b x^3+c x^6\right ) \, dx=x^4\,\left (\frac {b\,d^4}{4}+a\,e\,d^3\right )+x^{16}\,\left (\frac {b\,e^4}{16}+\frac {c\,d\,e^3}{4}\right )+x^7\,\left (\frac {c\,d^4}{7}+\frac {4\,b\,d^3\,e}{7}+\frac {6\,a\,d^2\,e^2}{7}\right )+x^{13}\,\left (\frac {6\,c\,d^2\,e^2}{13}+\frac {4\,b\,d\,e^3}{13}+\frac {a\,e^4}{13}\right )+\frac {c\,e^4\,x^{19}}{19}+a\,d^4\,x+\frac {d\,e\,x^{10}\,\left (2\,c\,d^2+3\,b\,d\,e+2\,a\,e^2\right )}{5} \] Input:
int((d + e*x^3)^4*(a + b*x^3 + c*x^6),x)
Output:
x^4*((b*d^4)/4 + a*d^3*e) + x^16*((b*e^4)/16 + (c*d*e^3)/4) + x^7*((c*d^4) /7 + (6*a*d^2*e^2)/7 + (4*b*d^3*e)/7) + x^13*((a*e^4)/13 + (6*c*d^2*e^2)/1 3 + (4*b*d*e^3)/13) + (c*e^4*x^19)/19 + a*d^4*x + (d*e*x^10*(2*a*e^2 + 2*c *d^2 + 3*b*d*e))/5
Time = 0.19 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.12 \[ \int \left (d+e x^3\right )^4 \left (a+b x^3+c x^6\right ) \, dx=\frac {x \left (7280 c \,e^{4} x^{18}+8645 b \,e^{4} x^{15}+34580 c d \,e^{3} x^{15}+10640 a \,e^{4} x^{12}+42560 b d \,e^{3} x^{12}+63840 c \,d^{2} e^{2} x^{12}+55328 a d \,e^{3} x^{9}+82992 b \,d^{2} e^{2} x^{9}+55328 c \,d^{3} e \,x^{9}+118560 a \,d^{2} e^{2} x^{6}+79040 b \,d^{3} e \,x^{6}+19760 c \,d^{4} x^{6}+138320 a \,d^{3} e \,x^{3}+34580 b \,d^{4} x^{3}+138320 a \,d^{4}\right )}{138320} \] Input:
int((e*x^3+d)^4*(c*x^6+b*x^3+a),x)
Output:
(x*(138320*a*d**4 + 138320*a*d**3*e*x**3 + 118560*a*d**2*e**2*x**6 + 55328 *a*d*e**3*x**9 + 10640*a*e**4*x**12 + 34580*b*d**4*x**3 + 79040*b*d**3*e*x **6 + 82992*b*d**2*e**2*x**9 + 42560*b*d*e**3*x**12 + 8645*b*e**4*x**15 + 19760*c*d**4*x**6 + 55328*c*d**3*e*x**9 + 63840*c*d**2*e**2*x**12 + 34580* c*d*e**3*x**15 + 7280*c*e**4*x**18))/138320