3.94 \(\int x \left (b+2 c x^2\right ) \left (a+b x^2+c x^4\right )^{13} \, dx\)

Optimal. Leaf size=18 \[ \frac{1}{28} \left (a+b x^2+c x^4\right )^{14} \]

[Out]

(a + b*x^2 + c*x^4)^14/28

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Rubi [A]  time = 0.0659763, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042 \[ \frac{1}{28} \left (a+b x^2+c x^4\right )^{14} \]

Antiderivative was successfully verified.

[In]  Int[x*(b + 2*c*x^2)*(a + b*x^2 + c*x^4)^13,x]

[Out]

(a + b*x^2 + c*x^4)^14/28

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Rubi in Sympy [A]  time = 4.94578, size = 14, normalized size = 0.78 \[ \frac{\left (a + b x^{2} + c x^{4}\right )^{14}}{28} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x*(2*c*x**2+b)*(c*x**4+b*x**2+a)**13,x)

[Out]

(a + b*x**2 + c*x**4)**14/28

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Mathematica [B]  time = 0.286706, size = 233, normalized size = 12.94 \[ \frac{1}{28} x^2 \left (b+c x^2\right ) \left (14 a^{13}+91 a^{12} x^2 \left (b+c x^2\right )+364 a^{11} x^4 \left (b+c x^2\right )^2+1001 a^{10} x^6 \left (b+c x^2\right )^3+2002 a^9 x^8 \left (b+c x^2\right )^4+3003 a^8 x^{10} \left (b+c x^2\right )^5+3432 a^7 x^{12} \left (b+c x^2\right )^6+3003 a^6 x^{14} \left (b+c x^2\right )^7+2002 a^5 x^{16} \left (b+c x^2\right )^8+1001 a^4 x^{18} \left (b+c x^2\right )^9+364 a^3 x^{20} \left (b+c x^2\right )^{10}+91 a^2 x^{22} \left (b+c x^2\right )^{11}+14 a x^{24} \left (b+c x^2\right )^{12}+x^{26} \left (b+c x^2\right )^{13}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x*(b + 2*c*x^2)*(a + b*x^2 + c*x^4)^13,x]

[Out]

(x^2*(b + c*x^2)*(14*a^13 + 91*a^12*x^2*(b + c*x^2) + 364*a^11*x^4*(b + c*x^2)^2
+ 1001*a^10*x^6*(b + c*x^2)^3 + 2002*a^9*x^8*(b + c*x^2)^4 + 3003*a^8*x^10*(b +
c*x^2)^5 + 3432*a^7*x^12*(b + c*x^2)^6 + 3003*a^6*x^14*(b + c*x^2)^7 + 2002*a^5
*x^16*(b + c*x^2)^8 + 1001*a^4*x^18*(b + c*x^2)^9 + 364*a^3*x^20*(b + c*x^2)^10
+ 91*a^2*x^22*(b + c*x^2)^11 + 14*a*x^24*(b + c*x^2)^12 + x^26*(b + c*x^2)^13))/
28

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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*c*x^2+b)*(c*x^4+b*x^2+a)^13,x)

[Out]

result too large to display

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Maxima [A]  time = 0.793231, 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^4 + b*x^2 + a)^13*(2*c*x^2 + b)*x,x, algorithm="maxima")

[Out]

1/28*c^14*x^56 + 1/2*b*c^13*x^54 + 1/4*(13*b^2*c^12 + 2*a*c^13)*x^52 + 13/2*(2*b
^3*c^11 + a*b*c^12)*x^50 + 13/4*(11*b^4*c^10 + 12*a*b^2*c^11 + a^2*c^12)*x^48 +
13/2*(11*b^5*c^9 + 22*a*b^3*c^10 + 6*a^2*b*c^11)*x^46 + 13/4*(33*b^6*c^8 + 110*a
*b^4*c^9 + 66*a^2*b^2*c^10 + 4*a^3*c^11)*x^44 + 143/14*(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^42 + 143/4*(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^40 + 143/2*(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^38 + 143/4*(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^36 + 13/
2*(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^34 + 13/4*(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^32 + 1/2*(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^30 + 1/28*(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^28 + 1/2*(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^
26 + 13/4*(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^24 + 13/2*(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^22 +
143/4*(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^20 + 143/2*(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^18 + 143/4*(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^16 + 1/2*a^13*b*x^2 + 143/14*(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^14 + 13/4*(33*a^8*b^6 + 110*a^9*b^4*c +
66*a^10*b^2*c^2 + 4*a^11*c^3)*x^12 + 13/2*(11*a^9*b^5 + 22*a^10*b^3*c + 6*a^11*
b*c^2)*x^10 + 13/4*(11*a^10*b^4 + 12*a^11*b^2*c + a^12*c^2)*x^8 + 13/2*(2*a^11*b
^3 + a^12*b*c)*x^6 + 1/4*(13*a^12*b^2 + 2*a^13*c)*x^4

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Fricas [A]  time = 0.277022, 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^4 + b*x^2 + a)^13*(2*c*x^2 + b)*x,x, algorithm="fricas")

[Out]

1/28*x^56*c^14 + 1/2*x^54*c^13*b + 13/4*x^52*c^12*b^2 + 1/2*x^52*c^13*a + 13*x^5
0*c^11*b^3 + 13/2*x^50*c^12*b*a + 143/4*x^48*c^10*b^4 + 39*x^48*c^11*b^2*a + 13/
4*x^48*c^12*a^2 + 143/2*x^46*c^9*b^5 + 143*x^46*c^10*b^3*a + 39*x^46*c^11*b*a^2
+ 429/4*x^44*c^8*b^6 + 715/2*x^44*c^9*b^4*a + 429/2*x^44*c^10*b^2*a^2 + 13*x^44*
c^11*a^3 + 858/7*x^42*c^7*b^7 + 1287/2*x^42*c^8*b^5*a + 715*x^42*c^9*b^3*a^2 + 1
43*x^42*c^10*b*a^3 + 429/4*x^40*c^6*b^8 + 858*x^40*c^7*b^6*a + 6435/4*x^40*c^8*b
^4*a^2 + 715*x^40*c^9*b^2*a^3 + 143/4*x^40*c^10*a^4 + 143/2*x^38*c^5*b^9 + 858*x
^38*c^6*b^7*a + 2574*x^38*c^7*b^5*a^2 + 2145*x^38*c^8*b^3*a^3 + 715/2*x^38*c^9*b
*a^4 + 143/4*x^36*c^4*b^10 + 1287/2*x^36*c^5*b^8*a + 3003*x^36*c^6*b^6*a^2 + 429
0*x^36*c^7*b^4*a^3 + 6435/4*x^36*c^8*b^2*a^4 + 143/2*x^36*c^9*a^5 + 13*x^34*c^3*
b^11 + 715/2*x^34*c^4*b^9*a + 2574*x^34*c^5*b^7*a^2 + 6006*x^34*c^6*b^5*a^3 + 42
90*x^34*c^7*b^3*a^4 + 1287/2*x^34*c^8*b*a^5 + 13/4*x^32*c^2*b^12 + 143*x^32*c^3*
b^10*a + 6435/4*x^32*c^4*b^8*a^2 + 6006*x^32*c^5*b^6*a^3 + 15015/2*x^32*c^6*b^4*
a^4 + 2574*x^32*c^7*b^2*a^5 + 429/4*x^32*c^8*a^6 + 1/2*x^30*c*b^13 + 39*x^30*c^2
*b^11*a + 715*x^30*c^3*b^9*a^2 + 4290*x^30*c^4*b^7*a^3 + 9009*x^30*c^5*b^5*a^4 +
6006*x^30*c^6*b^3*a^5 + 858*x^30*c^7*b*a^6 + 1/28*x^28*b^14 + 13/2*x^28*c*b^12*
a + 429/2*x^28*c^2*b^10*a^2 + 2145*x^28*c^3*b^8*a^3 + 15015/2*x^28*c^4*b^6*a^4 +
9009*x^28*c^5*b^4*a^5 + 3003*x^28*c^6*b^2*a^6 + 858/7*x^28*c^7*a^7 + 1/2*x^26*b
^13*a + 39*x^26*c*b^11*a^2 + 715*x^26*c^2*b^9*a^3 + 4290*x^26*c^3*b^7*a^4 + 9009
*x^26*c^4*b^5*a^5 + 6006*x^26*c^5*b^3*a^6 + 858*x^26*c^6*b*a^7 + 13/4*x^24*b^12*
a^2 + 143*x^24*c*b^10*a^3 + 6435/4*x^24*c^2*b^8*a^4 + 6006*x^24*c^3*b^6*a^5 + 15
015/2*x^24*c^4*b^4*a^6 + 2574*x^24*c^5*b^2*a^7 + 429/4*x^24*c^6*a^8 + 13*x^22*b^
11*a^3 + 715/2*x^22*c*b^9*a^4 + 2574*x^22*c^2*b^7*a^5 + 6006*x^22*c^3*b^5*a^6 +
4290*x^22*c^4*b^3*a^7 + 1287/2*x^22*c^5*b*a^8 + 143/4*x^20*b^10*a^4 + 1287/2*x^2
0*c*b^8*a^5 + 3003*x^20*c^2*b^6*a^6 + 4290*x^20*c^3*b^4*a^7 + 6435/4*x^20*c^4*b^
2*a^8 + 143/2*x^20*c^5*a^9 + 143/2*x^18*b^9*a^5 + 858*x^18*c*b^7*a^6 + 2574*x^18
*c^2*b^5*a^7 + 2145*x^18*c^3*b^3*a^8 + 715/2*x^18*c^4*b*a^9 + 429/4*x^16*b^8*a^6
+ 858*x^16*c*b^6*a^7 + 6435/4*x^16*c^2*b^4*a^8 + 715*x^16*c^3*b^2*a^9 + 143/4*x
^16*c^4*a^10 + 858/7*x^14*b^7*a^7 + 1287/2*x^14*c*b^5*a^8 + 715*x^14*c^2*b^3*a^9
+ 143*x^14*c^3*b*a^10 + 429/4*x^12*b^6*a^8 + 715/2*x^12*c*b^4*a^9 + 429/2*x^12*
c^2*b^2*a^10 + 13*x^12*c^3*a^11 + 143/2*x^10*b^5*a^9 + 143*x^10*c*b^3*a^10 + 39*
x^10*c^2*b*a^11 + 143/4*x^8*b^4*a^10 + 39*x^8*c*b^2*a^11 + 13/4*x^8*c^2*a^12 + 1
3*x^6*b^3*a^11 + 13/2*x^6*c*b*a^12 + 13/4*x^4*b^2*a^12 + 1/2*x^4*c*a^13 + 1/2*x^
2*b*a^13

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Sympy [A]  time = 0.916498, size = 1384, normalized size = 76.89 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*(2*c*x**2+b)*(c*x**4+b*x**2+a)**13,x)

[Out]

a**13*b*x**2/2 + b*c**13*x**54/2 + c**14*x**56/28 + x**52*(a*c**13/2 + 13*b**2*c
**12/4) + x**50*(13*a*b*c**12/2 + 13*b**3*c**11) + x**48*(13*a**2*c**12/4 + 39*a
*b**2*c**11 + 143*b**4*c**10/4) + x**46*(39*a**2*b*c**11 + 143*a*b**3*c**10 + 14
3*b**5*c**9/2) + x**44*(13*a**3*c**11 + 429*a**2*b**2*c**10/2 + 715*a*b**4*c**9/
2 + 429*b**6*c**8/4) + x**42*(143*a**3*b*c**10 + 715*a**2*b**3*c**9 + 1287*a*b**
5*c**8/2 + 858*b**7*c**7/7) + x**40*(143*a**4*c**10/4 + 715*a**3*b**2*c**9 + 643
5*a**2*b**4*c**8/4 + 858*a*b**6*c**7 + 429*b**8*c**6/4) + x**38*(715*a**4*b*c**9
/2 + 2145*a**3*b**3*c**8 + 2574*a**2*b**5*c**7 + 858*a*b**7*c**6 + 143*b**9*c**5
/2) + x**36*(143*a**5*c**9/2 + 6435*a**4*b**2*c**8/4 + 4290*a**3*b**4*c**7 + 300
3*a**2*b**6*c**6 + 1287*a*b**8*c**5/2 + 143*b**10*c**4/4) + x**34*(1287*a**5*b*c
**8/2 + 4290*a**4*b**3*c**7 + 6006*a**3*b**5*c**6 + 2574*a**2*b**7*c**5 + 715*a*
b**9*c**4/2 + 13*b**11*c**3) + x**32*(429*a**6*c**8/4 + 2574*a**5*b**2*c**7 + 15
015*a**4*b**4*c**6/2 + 6006*a**3*b**6*c**5 + 6435*a**2*b**8*c**4/4 + 143*a*b**10
*c**3 + 13*b**12*c**2/4) + x**30*(858*a**6*b*c**7 + 6006*a**5*b**3*c**6 + 9009*a
**4*b**5*c**5 + 4290*a**3*b**7*c**4 + 715*a**2*b**9*c**3 + 39*a*b**11*c**2 + b**
13*c/2) + x**28*(858*a**7*c**7/7 + 3003*a**6*b**2*c**6 + 9009*a**5*b**4*c**5 + 1
5015*a**4*b**6*c**4/2 + 2145*a**3*b**8*c**3 + 429*a**2*b**10*c**2/2 + 13*a*b**12
*c/2 + b**14/28) + x**26*(858*a**7*b*c**6 + 6006*a**6*b**3*c**5 + 9009*a**5*b**5
*c**4 + 4290*a**4*b**7*c**3 + 715*a**3*b**9*c**2 + 39*a**2*b**11*c + a*b**13/2)
+ x**24*(429*a**8*c**6/4 + 2574*a**7*b**2*c**5 + 15015*a**6*b**4*c**4/2 + 6006*a
**5*b**6*c**3 + 6435*a**4*b**8*c**2/4 + 143*a**3*b**10*c + 13*a**2*b**12/4) + x*
*22*(1287*a**8*b*c**5/2 + 4290*a**7*b**3*c**4 + 6006*a**6*b**5*c**3 + 2574*a**5*
b**7*c**2 + 715*a**4*b**9*c/2 + 13*a**3*b**11) + x**20*(143*a**9*c**5/2 + 6435*a
**8*b**2*c**4/4 + 4290*a**7*b**4*c**3 + 3003*a**6*b**6*c**2 + 1287*a**5*b**8*c/2
+ 143*a**4*b**10/4) + x**18*(715*a**9*b*c**4/2 + 2145*a**8*b**3*c**3 + 2574*a**
7*b**5*c**2 + 858*a**6*b**7*c + 143*a**5*b**9/2) + x**16*(143*a**10*c**4/4 + 715
*a**9*b**2*c**3 + 6435*a**8*b**4*c**2/4 + 858*a**7*b**6*c + 429*a**6*b**8/4) + x
**14*(143*a**10*b*c**3 + 715*a**9*b**3*c**2 + 1287*a**8*b**5*c/2 + 858*a**7*b**7
/7) + x**12*(13*a**11*c**3 + 429*a**10*b**2*c**2/2 + 715*a**9*b**4*c/2 + 429*a**
8*b**6/4) + x**10*(39*a**11*b*c**2 + 143*a**10*b**3*c + 143*a**9*b**5/2) + x**8*
(13*a**12*c**2/4 + 39*a**11*b**2*c + 143*a**10*b**4/4) + x**6*(13*a**12*b*c/2 +
13*a**11*b**3) + x**4*(a**13*c/2 + 13*a**12*b**2/4)

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GIAC/XCAS [A]  time = 0.275046, size = 1, normalized size = 0.06 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2 + a)^13*(2*c*x^2 + b)*x,x, algorithm="giac")

[Out]

Done