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

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

[Out]

(a - b*x - c*x^2)^14/14

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Rubi [A]  time = 0.0352768, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048 \[ \frac{1}{14} \left (a-b x-c x^2\right )^{14} \]

Antiderivative was successfully verified.

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

[Out]

(a - b*x - c*x^2)^14/14

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Rubi in Sympy [A]  time = 4.02151, size = 12, normalized size = 0.67 \[ \frac{\left (- a + b x + c x^{2}\right )^{14}}{14} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-a + b*x + c*x**2)**14/14

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Mathematica [B]  time = 0.302325, size = 201, normalized size = 11.17 \[ \frac{1}{14} x (b+c x) \left (-14 a^{13}+91 a^{12} x (b+c x)-364 a^{11} x^2 (b+c x)^2+1001 a^{10} x^3 (b+c x)^3-2002 a^9 x^4 (b+c x)^4+3003 a^8 x^5 (b+c x)^5-3432 a^7 x^6 (b+c x)^6+3003 a^6 x^7 (b+c x)^7-2002 a^5 x^8 (b+c x)^8+1001 a^4 x^9 (b+c x)^9-364 a^3 x^{10} (b+c x)^{10}+91 a^2 x^{11} (b+c x)^{11}-14 a x^{12} (b+c x)^{12}+x^{13} (b+c x)^{13}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(x*(b + c*x)*(-14*a^13 + 91*a^12*x*(b + c*x) - 364*a^11*x^2*(b + c*x)^2 + 1001*a
^10*x^3*(b + c*x)^3 - 2002*a^9*x^4*(b + c*x)^4 + 3003*a^8*x^5*(b + c*x)^5 - 3432
*a^7*x^6*(b + c*x)^6 + 3003*a^6*x^7*(b + c*x)^7 - 2002*a^5*x^8*(b + c*x)^8 + 100
1*a^4*x^9*(b + c*x)^9 - 364*a^3*x^10*(b + c*x)^10 + 91*a^2*x^11*(b + c*x)^11 - 1
4*a*x^12*(b + c*x)^12 + x^13*(b + c*x)^13))/14

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Maple [B]  time = 0.007, size = 47685, normalized size = 2649.2 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

result too large to display

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Maxima [A]  time = 0.770251, size = 22, normalized size = 1.22 \[ \frac{1}{14} \,{\left (c x^{2} + b x - a\right )}^{14} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/14*(c*x^2 + b*x - a)^14

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Fricas [A]  time = 0.258895, 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^2 + b*x - a)^13*(2*c*x + b),x, algorithm="fricas")

[Out]

1/14*x^28*c^14 + x^27*c^13*b + 13/2*x^26*c^12*b^2 - x^26*c^13*a + 26*x^25*c^11*b
^3 - 13*x^25*c^12*b*a + 143/2*x^24*c^10*b^4 - 78*x^24*c^11*b^2*a + 13/2*x^24*c^1
2*a^2 + 143*x^23*c^9*b^5 - 286*x^23*c^10*b^3*a + 78*x^23*c^11*b*a^2 + 429/2*x^22
*c^8*b^6 - 715*x^22*c^9*b^4*a + 429*x^22*c^10*b^2*a^2 - 26*x^22*c^11*a^3 + 1716/
7*x^21*c^7*b^7 - 1287*x^21*c^8*b^5*a + 1430*x^21*c^9*b^3*a^2 - 286*x^21*c^10*b*a
^3 + 429/2*x^20*c^6*b^8 - 1716*x^20*c^7*b^6*a + 6435/2*x^20*c^8*b^4*a^2 - 1430*x
^20*c^9*b^2*a^3 + 143/2*x^20*c^10*a^4 + 143*x^19*c^5*b^9 - 1716*x^19*c^6*b^7*a +
5148*x^19*c^7*b^5*a^2 - 4290*x^19*c^8*b^3*a^3 + 715*x^19*c^9*b*a^4 + 143/2*x^18
*c^4*b^10 - 1287*x^18*c^5*b^8*a + 6006*x^18*c^6*b^6*a^2 - 8580*x^18*c^7*b^4*a^3
+ 6435/2*x^18*c^8*b^2*a^4 - 143*x^18*c^9*a^5 + 26*x^17*c^3*b^11 - 715*x^17*c^4*b
^9*a + 5148*x^17*c^5*b^7*a^2 - 12012*x^17*c^6*b^5*a^3 + 8580*x^17*c^7*b^3*a^4 -
1287*x^17*c^8*b*a^5 + 13/2*x^16*c^2*b^12 - 286*x^16*c^3*b^10*a + 6435/2*x^16*c^4
*b^8*a^2 - 12012*x^16*c^5*b^6*a^3 + 15015*x^16*c^6*b^4*a^4 - 5148*x^16*c^7*b^2*a
^5 + 429/2*x^16*c^8*a^6 + x^15*c*b^13 - 78*x^15*c^2*b^11*a + 1430*x^15*c^3*b^9*a
^2 - 8580*x^15*c^4*b^7*a^3 + 18018*x^15*c^5*b^5*a^4 - 12012*x^15*c^6*b^3*a^5 + 1
716*x^15*c^7*b*a^6 + 1/14*x^14*b^14 - 13*x^14*c*b^12*a + 429*x^14*c^2*b^10*a^2 -
4290*x^14*c^3*b^8*a^3 + 15015*x^14*c^4*b^6*a^4 - 18018*x^14*c^5*b^4*a^5 + 6006*
x^14*c^6*b^2*a^6 - 1716/7*x^14*c^7*a^7 - x^13*b^13*a + 78*x^13*c*b^11*a^2 - 1430
*x^13*c^2*b^9*a^3 + 8580*x^13*c^3*b^7*a^4 - 18018*x^13*c^4*b^5*a^5 + 12012*x^13*
c^5*b^3*a^6 - 1716*x^13*c^6*b*a^7 + 13/2*x^12*b^12*a^2 - 286*x^12*c*b^10*a^3 + 6
435/2*x^12*c^2*b^8*a^4 - 12012*x^12*c^3*b^6*a^5 + 15015*x^12*c^4*b^4*a^6 - 5148*
x^12*c^5*b^2*a^7 + 429/2*x^12*c^6*a^8 - 26*x^11*b^11*a^3 + 715*x^11*c*b^9*a^4 -
5148*x^11*c^2*b^7*a^5 + 12012*x^11*c^3*b^5*a^6 - 8580*x^11*c^4*b^3*a^7 + 1287*x^
11*c^5*b*a^8 + 143/2*x^10*b^10*a^4 - 1287*x^10*c*b^8*a^5 + 6006*x^10*c^2*b^6*a^6
- 8580*x^10*c^3*b^4*a^7 + 6435/2*x^10*c^4*b^2*a^8 - 143*x^10*c^5*a^9 - 143*x^9*
b^9*a^5 + 1716*x^9*c*b^7*a^6 - 5148*x^9*c^2*b^5*a^7 + 4290*x^9*c^3*b^3*a^8 - 715
*x^9*c^4*b*a^9 + 429/2*x^8*b^8*a^6 - 1716*x^8*c*b^6*a^7 + 6435/2*x^8*c^2*b^4*a^8
- 1430*x^8*c^3*b^2*a^9 + 143/2*x^8*c^4*a^10 - 1716/7*x^7*b^7*a^7 + 1287*x^7*c*b
^5*a^8 - 1430*x^7*c^2*b^3*a^9 + 286*x^7*c^3*b*a^10 + 429/2*x^6*b^6*a^8 - 715*x^6
*c*b^4*a^9 + 429*x^6*c^2*b^2*a^10 - 26*x^6*c^3*a^11 - 143*x^5*b^5*a^9 + 286*x^5*
c*b^3*a^10 - 78*x^5*c^2*b*a^11 + 143/2*x^4*b^4*a^10 - 78*x^4*c*b^2*a^11 + 13/2*x
^4*c^2*a^12 - 26*x^3*b^3*a^11 + 13*x^3*c*b*a^12 + 13/2*x^2*b^2*a^12 - x^2*c*a^13
- x*b*a^13

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Sympy [A]  time = 0.956125, size = 1326, normalized size = 73.67 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a**13*b*x + b*c**13*x**27 + c**14*x**28/14 + x**26*(-a*c**13 + 13*b**2*c**12/2)
+ x**25*(-13*a*b*c**12 + 26*b**3*c**11) + x**24*(13*a**2*c**12/2 - 78*a*b**2*c*
*11 + 143*b**4*c**10/2) + x**23*(78*a**2*b*c**11 - 286*a*b**3*c**10 + 143*b**5*c
**9) + x**22*(-26*a**3*c**11 + 429*a**2*b**2*c**10 - 715*a*b**4*c**9 + 429*b**6*
c**8/2) + x**21*(-286*a**3*b*c**10 + 1430*a**2*b**3*c**9 - 1287*a*b**5*c**8 + 17
16*b**7*c**7/7) + x**20*(143*a**4*c**10/2 - 1430*a**3*b**2*c**9 + 6435*a**2*b**4
*c**8/2 - 1716*a*b**6*c**7 + 429*b**8*c**6/2) + x**19*(715*a**4*b*c**9 - 4290*a*
*3*b**3*c**8 + 5148*a**2*b**5*c**7 - 1716*a*b**7*c**6 + 143*b**9*c**5) + x**18*(
-143*a**5*c**9 + 6435*a**4*b**2*c**8/2 - 8580*a**3*b**4*c**7 + 6006*a**2*b**6*c*
*6 - 1287*a*b**8*c**5 + 143*b**10*c**4/2) + x**17*(-1287*a**5*b*c**8 + 8580*a**4
*b**3*c**7 - 12012*a**3*b**5*c**6 + 5148*a**2*b**7*c**5 - 715*a*b**9*c**4 + 26*b
**11*c**3) + x**16*(429*a**6*c**8/2 - 5148*a**5*b**2*c**7 + 15015*a**4*b**4*c**6
- 12012*a**3*b**6*c**5 + 6435*a**2*b**8*c**4/2 - 286*a*b**10*c**3 + 13*b**12*c*
*2/2) + x**15*(1716*a**6*b*c**7 - 12012*a**5*b**3*c**6 + 18018*a**4*b**5*c**5 -
8580*a**3*b**7*c**4 + 1430*a**2*b**9*c**3 - 78*a*b**11*c**2 + b**13*c) + x**14*(
-1716*a**7*c**7/7 + 6006*a**6*b**2*c**6 - 18018*a**5*b**4*c**5 + 15015*a**4*b**6
*c**4 - 4290*a**3*b**8*c**3 + 429*a**2*b**10*c**2 - 13*a*b**12*c + b**14/14) + x
**13*(-1716*a**7*b*c**6 + 12012*a**6*b**3*c**5 - 18018*a**5*b**5*c**4 + 8580*a**
4*b**7*c**3 - 1430*a**3*b**9*c**2 + 78*a**2*b**11*c - a*b**13) + x**12*(429*a**8
*c**6/2 - 5148*a**7*b**2*c**5 + 15015*a**6*b**4*c**4 - 12012*a**5*b**6*c**3 + 64
35*a**4*b**8*c**2/2 - 286*a**3*b**10*c + 13*a**2*b**12/2) + x**11*(1287*a**8*b*c
**5 - 8580*a**7*b**3*c**4 + 12012*a**6*b**5*c**3 - 5148*a**5*b**7*c**2 + 715*a**
4*b**9*c - 26*a**3*b**11) + x**10*(-143*a**9*c**5 + 6435*a**8*b**2*c**4/2 - 8580
*a**7*b**4*c**3 + 6006*a**6*b**6*c**2 - 1287*a**5*b**8*c + 143*a**4*b**10/2) + x
**9*(-715*a**9*b*c**4 + 4290*a**8*b**3*c**3 - 5148*a**7*b**5*c**2 + 1716*a**6*b*
*7*c - 143*a**5*b**9) + x**8*(143*a**10*c**4/2 - 1430*a**9*b**2*c**3 + 6435*a**8
*b**4*c**2/2 - 1716*a**7*b**6*c + 429*a**6*b**8/2) + x**7*(286*a**10*b*c**3 - 14
30*a**9*b**3*c**2 + 1287*a**8*b**5*c - 1716*a**7*b**7/7) + x**6*(-26*a**11*c**3
+ 429*a**10*b**2*c**2 - 715*a**9*b**4*c + 429*a**8*b**6/2) + x**5*(-78*a**11*b*c
**2 + 286*a**10*b**3*c - 143*a**9*b**5) + x**4*(13*a**12*c**2/2 - 78*a**11*b**2*
c + 143*a**10*b**4/2) + x**3*(13*a**12*b*c - 26*a**11*b**3) + x**2*(-a**13*c + 1
3*a**12*b**2/2)

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GIAC/XCAS [A]  time = 0.268694, size = 1, normalized size = 0.06 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done