3.95 \(\int x^2 \left (b+2 c x^3\right ) \left (a+b x^3+c x^6\right )^{13} \, dx\)

Optimal. Leaf size=18 \[ \frac{1}{42} \left (a+b x^3+c x^6\right )^{14} \]

[Out]

(a + b*x^3 + c*x^6)^14/42

_______________________________________________________________________________________

Rubi [A]  time = 0.0701099, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.038 \[ \frac{1}{42} \left (a+b x^3+c x^6\right )^{14} \]

Antiderivative was successfully verified.

[In]  Int[x^2*(b + 2*c*x^3)*(a + b*x^3 + c*x^6)^13,x]

[Out]

(a + b*x^3 + c*x^6)^14/42

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 4.9908, size = 14, normalized size = 0.78 \[ \frac{\left (a + b x^{3} + c x^{6}\right )^{14}}{42} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2*(2*c*x**3+b)*(c*x**6+b*x**3+a)**13,x)

[Out]

(a + b*x**3 + c*x**6)**14/42

_______________________________________________________________________________________

Mathematica [B]  time = 0.287124, size = 233, normalized size = 12.94 \[ \frac{1}{42} x^3 \left (b+c x^3\right ) \left (14 a^{13}+91 a^{12} x^3 \left (b+c x^3\right )+364 a^{11} x^6 \left (b+c x^3\right )^2+1001 a^{10} x^9 \left (b+c x^3\right )^3+2002 a^9 x^{12} \left (b+c x^3\right )^4+3003 a^8 x^{15} \left (b+c x^3\right )^5+3432 a^7 x^{18} \left (b+c x^3\right )^6+3003 a^6 x^{21} \left (b+c x^3\right )^7+2002 a^5 x^{24} \left (b+c x^3\right )^8+1001 a^4 x^{27} \left (b+c x^3\right )^9+364 a^3 x^{30} \left (b+c x^3\right )^{10}+91 a^2 x^{33} \left (b+c x^3\right )^{11}+14 a x^{36} \left (b+c x^3\right )^{12}+x^{39} \left (b+c x^3\right )^{13}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^2*(b + 2*c*x^3)*(a + b*x^3 + c*x^6)^13,x]

[Out]

(x^3*(b + c*x^3)*(14*a^13 + 91*a^12*x^3*(b + c*x^3) + 364*a^11*x^6*(b + c*x^3)^2
+ 1001*a^10*x^9*(b + c*x^3)^3 + 2002*a^9*x^12*(b + c*x^3)^4 + 3003*a^8*x^15*(b
+ c*x^3)^5 + 3432*a^7*x^18*(b + c*x^3)^6 + 3003*a^6*x^21*(b + c*x^3)^7 + 2002*a^
5*x^24*(b + c*x^3)^8 + 1001*a^4*x^27*(b + c*x^3)^9 + 364*a^3*x^30*(b + c*x^3)^10
+ 91*a^2*x^33*(b + c*x^3)^11 + 14*a*x^36*(b + c*x^3)^12 + x^39*(b + c*x^3)^13))
/42

_______________________________________________________________________________________

Maple [B]  time = 0.004, size = 46552, normalized size = 2586.2 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(2*c*x^3+b)*(c*x^6+b*x^3+a)^13,x)

[Out]

result too large to display

_______________________________________________________________________________________

Maxima [A]  time = 0.779615, size = 1674, normalized size = 93. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^6 + b*x^3 + a)^13*(2*c*x^3 + b)*x^2,x, algorithm="maxima")

[Out]

1/42*c^14*x^84 + 1/3*b*c^13*x^81 + 1/6*(13*b^2*c^12 + 2*a*c^13)*x^78 + 13/3*(2*b
^3*c^11 + a*b*c^12)*x^75 + 13/6*(11*b^4*c^10 + 12*a*b^2*c^11 + a^2*c^12)*x^72 +
13/3*(11*b^5*c^9 + 22*a*b^3*c^10 + 6*a^2*b*c^11)*x^69 + 13/6*(33*b^6*c^8 + 110*a
*b^4*c^9 + 66*a^2*b^2*c^10 + 4*a^3*c^11)*x^66 + 143/21*(12*b^7*c^7 + 63*a*b^5*c^
8 + 70*a^2*b^3*c^9 + 14*a^3*b*c^10)*x^63 + 143/6*(3*b^8*c^6 + 24*a*b^6*c^7 + 45*
a^2*b^4*c^8 + 20*a^3*b^2*c^9 + a^4*c^10)*x^60 + 143/3*(b^9*c^5 + 12*a*b^7*c^6 +
36*a^2*b^5*c^7 + 30*a^3*b^3*c^8 + 5*a^4*b*c^9)*x^57 + 143/6*(b^10*c^4 + 18*a*b^8
*c^5 + 84*a^2*b^6*c^6 + 120*a^3*b^4*c^7 + 45*a^4*b^2*c^8 + 2*a^5*c^9)*x^54 + 13/
3*(2*b^11*c^3 + 55*a*b^9*c^4 + 396*a^2*b^7*c^5 + 924*a^3*b^5*c^6 + 660*a^4*b^3*c
^7 + 99*a^5*b*c^8)*x^51 + 13/6*(b^12*c^2 + 44*a*b^10*c^3 + 495*a^2*b^8*c^4 + 184
8*a^3*b^6*c^5 + 2310*a^4*b^4*c^6 + 792*a^5*b^2*c^7 + 33*a^6*c^8)*x^48 + 1/3*(b^1
3*c + 78*a*b^11*c^2 + 1430*a^2*b^9*c^3 + 8580*a^3*b^7*c^4 + 18018*a^4*b^5*c^5 +
12012*a^5*b^3*c^6 + 1716*a^6*b*c^7)*x^45 + 1/42*(b^14 + 182*a*b^12*c + 6006*a^2*
b^10*c^2 + 60060*a^3*b^8*c^3 + 210210*a^4*b^6*c^4 + 252252*a^5*b^4*c^5 + 84084*a
^6*b^2*c^6 + 3432*a^7*c^7)*x^42 + 1/3*(a*b^13 + 78*a^2*b^11*c + 1430*a^3*b^9*c^2
+ 8580*a^4*b^7*c^3 + 18018*a^5*b^5*c^4 + 12012*a^6*b^3*c^5 + 1716*a^7*b*c^6)*x^
39 + 13/6*(a^2*b^12 + 44*a^3*b^10*c + 495*a^4*b^8*c^2 + 1848*a^5*b^6*c^3 + 2310*
a^6*b^4*c^4 + 792*a^7*b^2*c^5 + 33*a^8*c^6)*x^36 + 13/3*(2*a^3*b^11 + 55*a^4*b^9
*c + 396*a^5*b^7*c^2 + 924*a^6*b^5*c^3 + 660*a^7*b^3*c^4 + 99*a^8*b*c^5)*x^33 +
143/6*(a^4*b^10 + 18*a^5*b^8*c + 84*a^6*b^6*c^2 + 120*a^7*b^4*c^3 + 45*a^8*b^2*c
^4 + 2*a^9*c^5)*x^30 + 143/3*(a^5*b^9 + 12*a^6*b^7*c + 36*a^7*b^5*c^2 + 30*a^8*b
^3*c^3 + 5*a^9*b*c^4)*x^27 + 143/6*(3*a^6*b^8 + 24*a^7*b^6*c + 45*a^8*b^4*c^2 +
20*a^9*b^2*c^3 + a^10*c^4)*x^24 + 143/21*(12*a^7*b^7 + 63*a^8*b^5*c + 70*a^9*b^3
*c^2 + 14*a^10*b*c^3)*x^21 + 13/6*(33*a^8*b^6 + 110*a^9*b^4*c + 66*a^10*b^2*c^2
+ 4*a^11*c^3)*x^18 + 1/3*a^13*b*x^3 + 13/3*(11*a^9*b^5 + 22*a^10*b^3*c + 6*a^11*
b*c^2)*x^15 + 13/6*(11*a^10*b^4 + 12*a^11*b^2*c + a^12*c^2)*x^12 + 13/3*(2*a^11*
b^3 + a^12*b*c)*x^9 + 1/6*(13*a^12*b^2 + 2*a^13*c)*x^6

_______________________________________________________________________________________

Fricas [A]  time = 0.265948, size = 1, normalized size = 0.06 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^6 + b*x^3 + a)^13*(2*c*x^3 + b)*x^2,x, algorithm="fricas")

[Out]

1/42*x^84*c^14 + 1/3*x^81*c^13*b + 13/6*x^78*c^12*b^2 + 1/3*x^78*c^13*a + 26/3*x
^75*c^11*b^3 + 13/3*x^75*c^12*b*a + 143/6*x^72*c^10*b^4 + 26*x^72*c^11*b^2*a + 1
3/6*x^72*c^12*a^2 + 143/3*x^69*c^9*b^5 + 286/3*x^69*c^10*b^3*a + 26*x^69*c^11*b*
a^2 + 143/2*x^66*c^8*b^6 + 715/3*x^66*c^9*b^4*a + 143*x^66*c^10*b^2*a^2 + 26/3*x
^66*c^11*a^3 + 572/7*x^63*c^7*b^7 + 429*x^63*c^8*b^5*a + 1430/3*x^63*c^9*b^3*a^2
+ 286/3*x^63*c^10*b*a^3 + 143/2*x^60*c^6*b^8 + 572*x^60*c^7*b^6*a + 2145/2*x^60
*c^8*b^4*a^2 + 1430/3*x^60*c^9*b^2*a^3 + 143/6*x^60*c^10*a^4 + 143/3*x^57*c^5*b^
9 + 572*x^57*c^6*b^7*a + 1716*x^57*c^7*b^5*a^2 + 1430*x^57*c^8*b^3*a^3 + 715/3*x
^57*c^9*b*a^4 + 143/6*x^54*c^4*b^10 + 429*x^54*c^5*b^8*a + 2002*x^54*c^6*b^6*a^2
+ 2860*x^54*c^7*b^4*a^3 + 2145/2*x^54*c^8*b^2*a^4 + 143/3*x^54*c^9*a^5 + 26/3*x
^51*c^3*b^11 + 715/3*x^51*c^4*b^9*a + 1716*x^51*c^5*b^7*a^2 + 4004*x^51*c^6*b^5*
a^3 + 2860*x^51*c^7*b^3*a^4 + 429*x^51*c^8*b*a^5 + 13/6*x^48*c^2*b^12 + 286/3*x^
48*c^3*b^10*a + 2145/2*x^48*c^4*b^8*a^2 + 4004*x^48*c^5*b^6*a^3 + 5005*x^48*c^6*
b^4*a^4 + 1716*x^48*c^7*b^2*a^5 + 143/2*x^48*c^8*a^6 + 1/3*x^45*c*b^13 + 26*x^45
*c^2*b^11*a + 1430/3*x^45*c^3*b^9*a^2 + 2860*x^45*c^4*b^7*a^3 + 6006*x^45*c^5*b^
5*a^4 + 4004*x^45*c^6*b^3*a^5 + 572*x^45*c^7*b*a^6 + 1/42*x^42*b^14 + 13/3*x^42*
c*b^12*a + 143*x^42*c^2*b^10*a^2 + 1430*x^42*c^3*b^8*a^3 + 5005*x^42*c^4*b^6*a^4
+ 6006*x^42*c^5*b^4*a^5 + 2002*x^42*c^6*b^2*a^6 + 572/7*x^42*c^7*a^7 + 1/3*x^39
*b^13*a + 26*x^39*c*b^11*a^2 + 1430/3*x^39*c^2*b^9*a^3 + 2860*x^39*c^3*b^7*a^4 +
6006*x^39*c^4*b^5*a^5 + 4004*x^39*c^5*b^3*a^6 + 572*x^39*c^6*b*a^7 + 13/6*x^36*
b^12*a^2 + 286/3*x^36*c*b^10*a^3 + 2145/2*x^36*c^2*b^8*a^4 + 4004*x^36*c^3*b^6*a
^5 + 5005*x^36*c^4*b^4*a^6 + 1716*x^36*c^5*b^2*a^7 + 143/2*x^36*c^6*a^8 + 26/3*x
^33*b^11*a^3 + 715/3*x^33*c*b^9*a^4 + 1716*x^33*c^2*b^7*a^5 + 4004*x^33*c^3*b^5*
a^6 + 2860*x^33*c^4*b^3*a^7 + 429*x^33*c^5*b*a^8 + 143/6*x^30*b^10*a^4 + 429*x^3
0*c*b^8*a^5 + 2002*x^30*c^2*b^6*a^6 + 2860*x^30*c^3*b^4*a^7 + 2145/2*x^30*c^4*b^
2*a^8 + 143/3*x^30*c^5*a^9 + 143/3*x^27*b^9*a^5 + 572*x^27*c*b^7*a^6 + 1716*x^27
*c^2*b^5*a^7 + 1430*x^27*c^3*b^3*a^8 + 715/3*x^27*c^4*b*a^9 + 143/2*x^24*b^8*a^6
+ 572*x^24*c*b^6*a^7 + 2145/2*x^24*c^2*b^4*a^8 + 1430/3*x^24*c^3*b^2*a^9 + 143/
6*x^24*c^4*a^10 + 572/7*x^21*b^7*a^7 + 429*x^21*c*b^5*a^8 + 1430/3*x^21*c^2*b^3*
a^9 + 286/3*x^21*c^3*b*a^10 + 143/2*x^18*b^6*a^8 + 715/3*x^18*c*b^4*a^9 + 143*x^
18*c^2*b^2*a^10 + 26/3*x^18*c^3*a^11 + 143/3*x^15*b^5*a^9 + 286/3*x^15*c*b^3*a^1
0 + 26*x^15*c^2*b*a^11 + 143/6*x^12*b^4*a^10 + 26*x^12*c*b^2*a^11 + 13/6*x^12*c^
2*a^12 + 26/3*x^9*b^3*a^11 + 13/3*x^9*c*b*a^12 + 13/6*x^6*b^2*a^12 + 1/3*x^6*c*a
^13 + 1/3*x^3*b*a^13

_______________________________________________________________________________________

Sympy [A]  time = 0.923851, size = 1394, normalized size = 77.44 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2*(2*c*x**3+b)*(c*x**6+b*x**3+a)**13,x)

[Out]

a**13*b*x**3/3 + b*c**13*x**81/3 + c**14*x**84/42 + x**78*(a*c**13/3 + 13*b**2*c
**12/6) + x**75*(13*a*b*c**12/3 + 26*b**3*c**11/3) + x**72*(13*a**2*c**12/6 + 26
*a*b**2*c**11 + 143*b**4*c**10/6) + x**69*(26*a**2*b*c**11 + 286*a*b**3*c**10/3
+ 143*b**5*c**9/3) + x**66*(26*a**3*c**11/3 + 143*a**2*b**2*c**10 + 715*a*b**4*c
**9/3 + 143*b**6*c**8/2) + x**63*(286*a**3*b*c**10/3 + 1430*a**2*b**3*c**9/3 + 4
29*a*b**5*c**8 + 572*b**7*c**7/7) + x**60*(143*a**4*c**10/6 + 1430*a**3*b**2*c**
9/3 + 2145*a**2*b**4*c**8/2 + 572*a*b**6*c**7 + 143*b**8*c**6/2) + x**57*(715*a*
*4*b*c**9/3 + 1430*a**3*b**3*c**8 + 1716*a**2*b**5*c**7 + 572*a*b**7*c**6 + 143*
b**9*c**5/3) + x**54*(143*a**5*c**9/3 + 2145*a**4*b**2*c**8/2 + 2860*a**3*b**4*c
**7 + 2002*a**2*b**6*c**6 + 429*a*b**8*c**5 + 143*b**10*c**4/6) + x**51*(429*a**
5*b*c**8 + 2860*a**4*b**3*c**7 + 4004*a**3*b**5*c**6 + 1716*a**2*b**7*c**5 + 715
*a*b**9*c**4/3 + 26*b**11*c**3/3) + x**48*(143*a**6*c**8/2 + 1716*a**5*b**2*c**7
+ 5005*a**4*b**4*c**6 + 4004*a**3*b**6*c**5 + 2145*a**2*b**8*c**4/2 + 286*a*b**
10*c**3/3 + 13*b**12*c**2/6) + x**45*(572*a**6*b*c**7 + 4004*a**5*b**3*c**6 + 60
06*a**4*b**5*c**5 + 2860*a**3*b**7*c**4 + 1430*a**2*b**9*c**3/3 + 26*a*b**11*c**
2 + b**13*c/3) + x**42*(572*a**7*c**7/7 + 2002*a**6*b**2*c**6 + 6006*a**5*b**4*c
**5 + 5005*a**4*b**6*c**4 + 1430*a**3*b**8*c**3 + 143*a**2*b**10*c**2 + 13*a*b**
12*c/3 + b**14/42) + x**39*(572*a**7*b*c**6 + 4004*a**6*b**3*c**5 + 6006*a**5*b*
*5*c**4 + 2860*a**4*b**7*c**3 + 1430*a**3*b**9*c**2/3 + 26*a**2*b**11*c + a*b**1
3/3) + x**36*(143*a**8*c**6/2 + 1716*a**7*b**2*c**5 + 5005*a**6*b**4*c**4 + 4004
*a**5*b**6*c**3 + 2145*a**4*b**8*c**2/2 + 286*a**3*b**10*c/3 + 13*a**2*b**12/6)
+ x**33*(429*a**8*b*c**5 + 2860*a**7*b**3*c**4 + 4004*a**6*b**5*c**3 + 1716*a**5
*b**7*c**2 + 715*a**4*b**9*c/3 + 26*a**3*b**11/3) + x**30*(143*a**9*c**5/3 + 214
5*a**8*b**2*c**4/2 + 2860*a**7*b**4*c**3 + 2002*a**6*b**6*c**2 + 429*a**5*b**8*c
+ 143*a**4*b**10/6) + x**27*(715*a**9*b*c**4/3 + 1430*a**8*b**3*c**3 + 1716*a**
7*b**5*c**2 + 572*a**6*b**7*c + 143*a**5*b**9/3) + x**24*(143*a**10*c**4/6 + 143
0*a**9*b**2*c**3/3 + 2145*a**8*b**4*c**2/2 + 572*a**7*b**6*c + 143*a**6*b**8/2)
+ x**21*(286*a**10*b*c**3/3 + 1430*a**9*b**3*c**2/3 + 429*a**8*b**5*c + 572*a**7
*b**7/7) + x**18*(26*a**11*c**3/3 + 143*a**10*b**2*c**2 + 715*a**9*b**4*c/3 + 14
3*a**8*b**6/2) + x**15*(26*a**11*b*c**2 + 286*a**10*b**3*c/3 + 143*a**9*b**5/3)
+ x**12*(13*a**12*c**2/6 + 26*a**11*b**2*c + 143*a**10*b**4/6) + x**9*(13*a**12*
b*c/3 + 26*a**11*b**3/3) + x**6*(a**13*c/3 + 13*a**12*b**2/6)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.277393, size = 1, normalized size = 0.06 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^6 + b*x^3 + a)^13*(2*c*x^3 + b)*x^2,x, algorithm="giac")

[Out]

Done