Integrand size = 23, antiderivative size = 106 \[ \int \left (a-c x^6\right )^3 \left (A+B x^6+C x^{12}\right ) \, dx=a^3 A x+\frac {1}{7} a^2 (a B-3 A c) x^7-\frac {1}{13} a \left (3 a B c-3 A c^2-a^2 C\right ) x^{13}+\frac {1}{19} c \left (3 a B c-A c^2-3 a^2 C\right ) x^{19}-\frac {1}{25} c^2 (B c-3 a C) x^{25}-\frac {1}{31} c^3 C x^{31} \] Output:
a^3*A*x+1/7*a^2*(-3*A*c+B*a)*x^7-1/13*a*(-3*A*c^2+3*B*a*c-C*a^2)*x^13+1/19 *c*(-A*c^2+3*B*a*c-3*C*a^2)*x^19-1/25*c^2*(B*c-3*C*a)*x^25-1/31*c^3*C*x^31
Time = 0.02 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.98 \[ \int \left (a-c x^6\right )^3 \left (A+B x^6+C x^{12}\right ) \, dx=a^3 A x+\frac {1}{7} a^2 (a B-3 A c) x^7+\frac {1}{13} a \left (-3 a B c+3 A c^2+a^2 C\right ) x^{13}-\frac {1}{19} c \left (-3 a B c+A c^2+3 a^2 C\right ) x^{19}-\frac {1}{25} c^2 (B c-3 a C) x^{25}-\frac {1}{31} c^3 C x^{31} \] Input:
Integrate[(a - c*x^6)^3*(A + B*x^6 + C*x^12),x]
Output:
a^3*A*x + (a^2*(a*B - 3*A*c)*x^7)/7 + (a*(-3*a*B*c + 3*A*c^2 + a^2*C)*x^13 )/13 - (c*(-3*a*B*c + A*c^2 + 3*a^2*C)*x^19)/19 - (c^2*(B*c - 3*a*C)*x^25) /25 - (c^3*C*x^31)/31
Time = 0.50 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, 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.
\(\displaystyle \int \left (a-c x^6\right )^3 \left (A+B x^6+C x^{12}\right ) \, dx\) |
\(\Big \downarrow \) 1737 |
\(\displaystyle \int \left (a^3 A-c x^{18} \left (3 a^2 C-3 a B c+A c^2\right )+a x^{12} \left (a^2 C-3 a B c+3 A c^2\right )+a^2 x^6 (a B-3 A c)-c^2 x^{24} (B c-3 a C)-c^3 C x^{30}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle a^3 A x+\frac {1}{19} c x^{19} \left (-3 a^2 C+3 a B c-A c^2\right )-\frac {1}{13} a x^{13} \left (a^2 (-C)+3 a B c-3 A c^2\right )+\frac {1}{7} a^2 x^7 (a B-3 A c)-\frac {1}{25} c^2 x^{25} (B c-3 a C)-\frac {1}{31} c^3 C x^{31}\) |
Input:
Int[(a - c*x^6)^3*(A + B*x^6 + C*x^12),x]
Output:
a^3*A*x + (a^2*(a*B - 3*A*c)*x^7)/7 - (a*(3*a*B*c - 3*A*c^2 - a^2*C)*x^13) /13 + (c*(3*a*B*c - A*c^2 - 3*a^2*C)*x^19)/19 - (c^2*(B*c - 3*a*C)*x^25)/2 5 - (c^3*C*x^31)/31
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 = 103, normalized size of antiderivative = 0.97
method | result | size |
norman | \(-\frac {c^{3} C \,x^{31}}{31}+\left (-\frac {1}{25} c^{3} B +\frac {3}{25} a \,c^{2} C \right ) x^{25}+\left (\frac {3}{13} A a \,c^{2}-\frac {3}{13} a^{2} c B +\frac {1}{13} a^{3} C \right ) x^{13}+\left (-\frac {1}{19} A \,c^{3}+\frac {3}{19} a \,c^{2} B -\frac {3}{19} a^{2} c C \right ) x^{19}+\left (-\frac {3}{7} a^{2} A c +\frac {1}{7} B \,a^{3}\right ) x^{7}+a^{3} A x\) | \(103\) |
default | \(-\frac {c^{3} C \,x^{31}}{31}+\frac {\left (-c^{3} B +3 a \,c^{2} C \right ) x^{25}}{25}+\frac {\left (-A \,c^{3}+3 a \,c^{2} B -3 a^{2} c C \right ) x^{19}}{19}+\frac {\left (3 A a \,c^{2}-3 a^{2} c B +a^{3} C \right ) x^{13}}{13}+\frac {\left (-3 a^{2} A c +B \,a^{3}\right ) x^{7}}{7}+a^{3} A x\) | \(105\) |
gosper | \(-\frac {1}{31} c^{3} C \,x^{31}-\frac {1}{25} x^{25} c^{3} B +\frac {3}{25} x^{25} a \,c^{2} C +\frac {3}{13} x^{13} A a \,c^{2}-\frac {3}{13} x^{13} a^{2} c B +\frac {1}{13} x^{13} a^{3} C -\frac {1}{19} x^{19} A \,c^{3}+\frac {3}{19} x^{19} a \,c^{2} B -\frac {3}{19} x^{19} a^{2} c C -\frac {3}{7} x^{7} a^{2} A c +\frac {1}{7} x^{7} B \,a^{3}+a^{3} A x\) | \(113\) |
risch | \(-\frac {1}{31} c^{3} C \,x^{31}-\frac {1}{25} x^{25} c^{3} B +\frac {3}{25} x^{25} a \,c^{2} C +\frac {3}{13} x^{13} A a \,c^{2}-\frac {3}{13} x^{13} a^{2} c B +\frac {1}{13} x^{13} a^{3} C -\frac {1}{19} x^{19} A \,c^{3}+\frac {3}{19} x^{19} a \,c^{2} B -\frac {3}{19} x^{19} a^{2} c C -\frac {3}{7} x^{7} a^{2} A c +\frac {1}{7} x^{7} B \,a^{3}+a^{3} A x\) | \(113\) |
parallelrisch | \(-\frac {1}{31} c^{3} C \,x^{31}-\frac {1}{25} x^{25} c^{3} B +\frac {3}{25} x^{25} a \,c^{2} C +\frac {3}{13} x^{13} A a \,c^{2}-\frac {3}{13} x^{13} a^{2} c B +\frac {1}{13} x^{13} a^{3} C -\frac {1}{19} x^{19} A \,c^{3}+\frac {3}{19} x^{19} a \,c^{2} B -\frac {3}{19} x^{19} a^{2} c C -\frac {3}{7} x^{7} a^{2} A c +\frac {1}{7} x^{7} B \,a^{3}+a^{3} A x\) | \(113\) |
orering | \(\frac {x \left (-43225 c^{3} C \,x^{30}-53599 B \,c^{3} x^{24}+160797 C a \,c^{2} x^{24}-70525 A \,c^{3} x^{18}+211575 B a \,c^{2} x^{18}-211575 C \,a^{2} c \,x^{18}+309225 A a \,c^{2} x^{12}-309225 B \,a^{2} c \,x^{12}+103075 C \,a^{3} x^{12}-574275 A \,a^{2} c \,x^{6}+191425 B \,a^{3} x^{6}+1339975 a^{3} A \right )}{1339975}\) | \(116\) |
Input:
int((-c*x^6+a)^3*(C*x^12+B*x^6+A),x,method=_RETURNVERBOSE)
Output:
-1/31*c^3*C*x^31+(-1/25*c^3*B+3/25*a*c^2*C)*x^25+(3/13*A*a*c^2-3/13*a^2*c* B+1/13*a^3*C)*x^13+(-1/19*A*c^3+3/19*a*c^2*B-3/19*a^2*c*C)*x^19+(-3/7*a^2* A*c+1/7*B*a^3)*x^7+a^3*A*x
Time = 0.07 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.97 \[ \int \left (a-c x^6\right )^3 \left (A+B x^6+C x^{12}\right ) \, dx=-\frac {1}{31} \, C c^{3} x^{31} + \frac {1}{25} \, {\left (3 \, C a c^{2} - B c^{3}\right )} x^{25} - \frac {1}{19} \, {\left (3 \, C a^{2} c - 3 \, B a c^{2} + A c^{3}\right )} x^{19} + \frac {1}{13} \, {\left (C a^{3} - 3 \, B a^{2} c + 3 \, A a c^{2}\right )} x^{13} + \frac {1}{7} \, {\left (B a^{3} - 3 \, A a^{2} c\right )} x^{7} + A a^{3} x \] Input:
integrate((-c*x^6+a)^3*(C*x^12+B*x^6+A),x, algorithm="fricas")
Output:
-1/31*C*c^3*x^31 + 1/25*(3*C*a*c^2 - B*c^3)*x^25 - 1/19*(3*C*a^2*c - 3*B*a *c^2 + A*c^3)*x^19 + 1/13*(C*a^3 - 3*B*a^2*c + 3*A*a*c^2)*x^13 + 1/7*(B*a^ 3 - 3*A*a^2*c)*x^7 + A*a^3*x
Time = 0.04 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.10 \[ \int \left (a-c x^6\right )^3 \left (A+B x^6+C x^{12}\right ) \, dx=A a^{3} x - \frac {C c^{3} x^{31}}{31} + x^{25} \left (- \frac {B c^{3}}{25} + \frac {3 C a c^{2}}{25}\right ) + x^{19} \left (- \frac {A c^{3}}{19} + \frac {3 B a c^{2}}{19} - \frac {3 C a^{2} c}{19}\right ) + x^{13} \cdot \left (\frac {3 A a c^{2}}{13} - \frac {3 B a^{2} c}{13} + \frac {C a^{3}}{13}\right ) + x^{7} \left (- \frac {3 A a^{2} c}{7} + \frac {B a^{3}}{7}\right ) \] Input:
integrate((-c*x**6+a)**3*(C*x**12+B*x**6+A),x)
Output:
A*a**3*x - C*c**3*x**31/31 + x**25*(-B*c**3/25 + 3*C*a*c**2/25) + x**19*(- A*c**3/19 + 3*B*a*c**2/19 - 3*C*a**2*c/19) + x**13*(3*A*a*c**2/13 - 3*B*a* *2*c/13 + C*a**3/13) + x**7*(-3*A*a**2*c/7 + B*a**3/7)
Time = 0.03 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.97 \[ \int \left (a-c x^6\right )^3 \left (A+B x^6+C x^{12}\right ) \, dx=-\frac {1}{31} \, C c^{3} x^{31} + \frac {1}{25} \, {\left (3 \, C a c^{2} - B c^{3}\right )} x^{25} - \frac {1}{19} \, {\left (3 \, C a^{2} c - 3 \, B a c^{2} + A c^{3}\right )} x^{19} + \frac {1}{13} \, {\left (C a^{3} - 3 \, B a^{2} c + 3 \, A a c^{2}\right )} x^{13} + \frac {1}{7} \, {\left (B a^{3} - 3 \, A a^{2} c\right )} x^{7} + A a^{3} x \] Input:
integrate((-c*x^6+a)^3*(C*x^12+B*x^6+A),x, algorithm="maxima")
Output:
-1/31*C*c^3*x^31 + 1/25*(3*C*a*c^2 - B*c^3)*x^25 - 1/19*(3*C*a^2*c - 3*B*a *c^2 + A*c^3)*x^19 + 1/13*(C*a^3 - 3*B*a^2*c + 3*A*a*c^2)*x^13 + 1/7*(B*a^ 3 - 3*A*a^2*c)*x^7 + A*a^3*x
Time = 0.13 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.06 \[ \int \left (a-c x^6\right )^3 \left (A+B x^6+C x^{12}\right ) \, dx=-\frac {1}{31} \, C c^{3} x^{31} + \frac {3}{25} \, C a c^{2} x^{25} - \frac {1}{25} \, B c^{3} x^{25} - \frac {3}{19} \, C a^{2} c x^{19} + \frac {3}{19} \, B a c^{2} x^{19} - \frac {1}{19} \, A c^{3} x^{19} + \frac {1}{13} \, C a^{3} x^{13} - \frac {3}{13} \, B a^{2} c x^{13} + \frac {3}{13} \, A a c^{2} x^{13} + \frac {1}{7} \, B a^{3} x^{7} - \frac {3}{7} \, A a^{2} c x^{7} + A a^{3} x \] Input:
integrate((-c*x^6+a)^3*(C*x^12+B*x^6+A),x, algorithm="giac")
Output:
-1/31*C*c^3*x^31 + 3/25*C*a*c^2*x^25 - 1/25*B*c^3*x^25 - 3/19*C*a^2*c*x^19 + 3/19*B*a*c^2*x^19 - 1/19*A*c^3*x^19 + 1/13*C*a^3*x^13 - 3/13*B*a^2*c*x^ 13 + 3/13*A*a*c^2*x^13 + 1/7*B*a^3*x^7 - 3/7*A*a^2*c*x^7 + A*a^3*x
Time = 5.92 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.98 \[ \int \left (a-c x^6\right )^3 \left (A+B x^6+C x^{12}\right ) \, dx=x^{13}\,\left (\frac {C\,a^3}{13}-\frac {3\,B\,a^2\,c}{13}+\frac {3\,A\,a\,c^2}{13}\right )-x^{19}\,\left (\frac {3\,C\,a^2\,c}{19}-\frac {3\,B\,a\,c^2}{19}+\frac {A\,c^3}{19}\right )+x^7\,\left (\frac {B\,a^3}{7}-\frac {3\,A\,a^2\,c}{7}\right )-x^{25}\,\left (\frac {B\,c^3}{25}-\frac {3\,C\,a\,c^2}{25}\right )-\frac {C\,c^3\,x^{31}}{31}+A\,a^3\,x \] Input:
int((a - c*x^6)^3*(A + B*x^6 + C*x^12),x)
Output:
x^13*((C*a^3)/13 + (3*A*a*c^2)/13 - (3*B*a^2*c)/13) - x^19*((A*c^3)/19 - ( 3*B*a*c^2)/19 + (3*C*a^2*c)/19) + x^7*((B*a^3)/7 - (3*A*a^2*c)/7) - x^25*( (B*c^3)/25 - (3*C*a*c^2)/25) - (C*c^3*x^31)/31 + A*a^3*x
Time = 0.15 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.07 \[ \int \left (a-c x^6\right )^3 \left (A+B x^6+C x^{12}\right ) \, dx=\frac {x \left (-43225 c^{4} x^{30}+160797 a \,c^{3} x^{24}-53599 b \,c^{3} x^{24}-211575 a^{2} c^{2} x^{18}+211575 a b \,c^{2} x^{18}-70525 a \,c^{3} x^{18}+103075 a^{3} c \,x^{12}-309225 a^{2} b c \,x^{12}+309225 a^{2} c^{2} x^{12}+191425 a^{3} b \,x^{6}-574275 a^{3} c \,x^{6}+1339975 a^{4}\right )}{1339975} \] Input:
int((-c*x^6+a)^3*(C*x^12+B*x^6+A),x)
Output:
(x*(1339975*a**4 + 191425*a**3*b*x**6 + 103075*a**3*c*x**12 - 574275*a**3* c*x**6 - 309225*a**2*b*c*x**12 - 211575*a**2*c**2*x**18 + 309225*a**2*c**2 *x**12 + 211575*a*b*c**2*x**18 + 160797*a*c**3*x**24 - 70525*a*c**3*x**18 - 53599*b*c**3*x**24 - 43225*c**4*x**30))/1339975