3.192 \(\int \left (b+2 c x+3 d x^2\right ) \left (a+b x+c x^2+d x^3\right )^7 \, dx\)

Optimal. Leaf size=21 \[ \frac{1}{8} \left (a+b x+c x^2+d x^3\right )^8 \]

[Out]

(a + b*x + c*x^2 + d*x^3)^8/8

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Rubi [A]  time = 0.0432703, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.033 \[ \frac{1}{8} \left (a+b x+c x^2+d x^3\right )^8 \]

Antiderivative was successfully verified.

[In]  Int[(b + 2*c*x + 3*d*x^2)*(a + b*x + c*x^2 + d*x^3)^7,x]

[Out]

(a + b*x + c*x^2 + d*x^3)^8/8

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Rubi in Sympy [A]  time = 14.4395, size = 17, normalized size = 0.81 \[ \frac{\left (a + b x + c x^{2} + d x^{3}\right )^{8}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((3*d*x**2+2*c*x+b)*(d*x**3+c*x**2+b*x+a)**7,x)

[Out]

(a + b*x + c*x**2 + d*x**3)**8/8

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Mathematica [B]  time = 0.242511, size = 143, normalized size = 6.81 \[ \frac{1}{8} x (b+x (c+d x)) \left (8 a^7+28 a^6 x (b+x (c+d x))+56 a^5 x^2 (b+x (c+d x))^2+70 a^4 x^3 (b+x (c+d x))^3+56 a^3 x^4 (b+x (c+d x))^4+28 a^2 x^5 (b+x (c+d x))^5+8 a x^6 (b+x (c+d x))^6+x^7 (b+x (c+d x))^7\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(b + 2*c*x + 3*d*x^2)*(a + b*x + c*x^2 + d*x^3)^7,x]

[Out]

(x*(b + x*(c + d*x))*(8*a^7 + 28*a^6*x*(b + x*(c + d*x)) + 56*a^5*x^2*(b + x*(c
+ d*x))^2 + 70*a^4*x^3*(b + x*(c + d*x))^3 + 56*a^3*x^4*(b + x*(c + d*x))^4 + 28
*a^2*x^5*(b + x*(c + d*x))^5 + 8*a*x^6*(b + x*(c + d*x))^6 + x^7*(b + x*(c + d*x
))^7))/8

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Maple [B]  time = 0.006, size = 25686, normalized size = 1223.1 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((3*d*x^2+2*c*x+b)*(d*x^3+c*x^2+b*x+a)^7,x)

[Out]

result too large to display

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Maxima [A]  time = 0.788848, size = 26, normalized size = 1.24 \[ \frac{1}{8} \,{\left (d x^{3} + c x^{2} + b x + a\right )}^{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^3 + c*x^2 + b*x + a)^7*(3*d*x^2 + 2*c*x + b),x, algorithm="maxima")

[Out]

1/8*(d*x^3 + c*x^2 + b*x + a)^8

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Fricas [A]  time = 0.235679, size = 1, normalized size = 0.05 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^3 + c*x^2 + b*x + a)^7*(3*d*x^2 + 2*c*x + b),x, algorithm="fricas")

[Out]

1/8*x^24*d^8 + x^23*d^7*c + 7/2*x^22*d^6*c^2 + x^22*d^7*b + 7*x^21*d^5*c^3 + 7*x
^21*d^6*c*b + x^21*d^7*a + 35/4*x^20*d^4*c^4 + 21*x^20*d^5*c^2*b + 7/2*x^20*d^6*
b^2 + 7*x^20*d^6*c*a + 7*x^19*d^3*c^5 + 35*x^19*d^4*c^3*b + 21*x^19*d^5*c*b^2 +
21*x^19*d^5*c^2*a + 7*x^19*d^6*b*a + 7/2*x^18*d^2*c^6 + 35*x^18*d^3*c^4*b + 105/
2*x^18*d^4*c^2*b^2 + 7*x^18*d^5*b^3 + 35*x^18*d^4*c^3*a + 42*x^18*d^5*c*b*a + 7/
2*x^18*d^6*a^2 + x^17*d*c^7 + 21*x^17*d^2*c^5*b + 70*x^17*d^3*c^3*b^2 + 35*x^17*
d^4*c*b^3 + 35*x^17*d^3*c^4*a + 105*x^17*d^4*c^2*b*a + 21*x^17*d^5*b^2*a + 21*x^
17*d^5*c*a^2 + 1/8*x^16*c^8 + 7*x^16*d*c^6*b + 105/2*x^16*d^2*c^4*b^2 + 70*x^16*
d^3*c^2*b^3 + 35/4*x^16*d^4*b^4 + 21*x^16*d^2*c^5*a + 140*x^16*d^3*c^3*b*a + 105
*x^16*d^4*c*b^2*a + 105/2*x^16*d^4*c^2*a^2 + 21*x^16*d^5*b*a^2 + x^15*c^7*b + 21
*x^15*d*c^5*b^2 + 70*x^15*d^2*c^3*b^3 + 35*x^15*d^3*c*b^4 + 7*x^15*d*c^6*a + 105
*x^15*d^2*c^4*b*a + 210*x^15*d^3*c^2*b^2*a + 35*x^15*d^4*b^3*a + 70*x^15*d^3*c^3
*a^2 + 105*x^15*d^4*c*b*a^2 + 7*x^15*d^5*a^3 + 7/2*x^14*c^6*b^2 + 35*x^14*d*c^4*
b^3 + 105/2*x^14*d^2*c^2*b^4 + 7*x^14*d^3*b^5 + x^14*c^7*a + 42*x^14*d*c^5*b*a +
210*x^14*d^2*c^3*b^2*a + 140*x^14*d^3*c*b^3*a + 105/2*x^14*d^2*c^4*a^2 + 210*x^
14*d^3*c^2*b*a^2 + 105/2*x^14*d^4*b^2*a^2 + 35*x^14*d^4*c*a^3 + 7*x^13*c^5*b^3 +
35*x^13*d*c^3*b^4 + 21*x^13*d^2*c*b^5 + 7*x^13*c^6*b*a + 105*x^13*d*c^4*b^2*a +
210*x^13*d^2*c^2*b^3*a + 35*x^13*d^3*b^4*a + 21*x^13*d*c^5*a^2 + 210*x^13*d^2*c
^3*b*a^2 + 210*x^13*d^3*c*b^2*a^2 + 70*x^13*d^3*c^2*a^3 + 35*x^13*d^4*b*a^3 + 35
/4*x^12*c^4*b^4 + 21*x^12*d*c^2*b^5 + 7/2*x^12*d^2*b^6 + 21*x^12*c^5*b^2*a + 140
*x^12*d*c^3*b^3*a + 105*x^12*d^2*c*b^4*a + 7/2*x^12*c^6*a^2 + 105*x^12*d*c^4*b*a
^2 + 315*x^12*d^2*c^2*b^2*a^2 + 70*x^12*d^3*b^3*a^2 + 70*x^12*d^2*c^3*a^3 + 140*
x^12*d^3*c*b*a^3 + 35/4*x^12*d^4*a^4 + 7*x^11*c^3*b^5 + 7*x^11*d*c*b^6 + 35*x^11
*c^4*b^3*a + 105*x^11*d*c^2*b^4*a + 21*x^11*d^2*b^5*a + 21*x^11*c^5*b*a^2 + 210*
x^11*d*c^3*b^2*a^2 + 210*x^11*d^2*c*b^3*a^2 + 35*x^11*d*c^4*a^3 + 210*x^11*d^2*c
^2*b*a^3 + 70*x^11*d^3*b^2*a^3 + 35*x^11*d^3*c*a^4 + 7/2*x^10*c^2*b^6 + x^10*d*b
^7 + 35*x^10*c^3*b^4*a + 42*x^10*d*c*b^5*a + 105/2*x^10*c^4*b^2*a^2 + 210*x^10*d
*c^2*b^3*a^2 + 105/2*x^10*d^2*b^4*a^2 + 7*x^10*c^5*a^3 + 140*x^10*d*c^3*b*a^3 +
210*x^10*d^2*c*b^2*a^3 + 105/2*x^10*d^2*c^2*a^4 + 35*x^10*d^3*b*a^4 + x^9*c*b^7
+ 21*x^9*c^2*b^5*a + 7*x^9*d*b^6*a + 70*x^9*c^3*b^3*a^2 + 105*x^9*d*c*b^4*a^2 +
35*x^9*c^4*b*a^3 + 210*x^9*d*c^2*b^2*a^3 + 70*x^9*d^2*b^3*a^3 + 35*x^9*d*c^3*a^4
+ 105*x^9*d^2*c*b*a^4 + 7*x^9*d^3*a^5 + 1/8*x^8*b^8 + 7*x^8*c*b^6*a + 105/2*x^8
*c^2*b^4*a^2 + 21*x^8*d*b^5*a^2 + 70*x^8*c^3*b^2*a^3 + 140*x^8*d*c*b^3*a^3 + 35/
4*x^8*c^4*a^4 + 105*x^8*d*c^2*b*a^4 + 105/2*x^8*d^2*b^2*a^4 + 21*x^8*d^2*c*a^5 +
x^7*b^7*a + 21*x^7*c*b^5*a^2 + 70*x^7*c^2*b^3*a^3 + 35*x^7*d*b^4*a^3 + 35*x^7*c
^3*b*a^4 + 105*x^7*d*c*b^2*a^4 + 21*x^7*d*c^2*a^5 + 21*x^7*d^2*b*a^5 + 7/2*x^6*b
^6*a^2 + 35*x^6*c*b^4*a^3 + 105/2*x^6*c^2*b^2*a^4 + 35*x^6*d*b^3*a^4 + 7*x^6*c^3
*a^5 + 42*x^6*d*c*b*a^5 + 7/2*x^6*d^2*a^6 + 7*x^5*b^5*a^3 + 35*x^5*c*b^3*a^4 + 2
1*x^5*c^2*b*a^5 + 21*x^5*d*b^2*a^5 + 7*x^5*d*c*a^6 + 35/4*x^4*b^4*a^4 + 21*x^4*c
*b^2*a^5 + 7/2*x^4*c^2*a^6 + 7*x^4*d*b*a^6 + 7*x^3*b^3*a^5 + 7*x^3*c*b*a^6 + x^3
*d*a^7 + 7/2*x^2*b^2*a^6 + x^2*c*a^7 + x*b*a^7

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Sympy [A]  time = 1.17226, size = 1771, normalized size = 84.33 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*d*x**2+2*c*x+b)*(d*x**3+c*x**2+b*x+a)**7,x)

[Out]

a**7*b*x + c*d**7*x**23 + d**8*x**24/8 + x**22*(b*d**7 + 7*c**2*d**6/2) + x**21*
(a*d**7 + 7*b*c*d**6 + 7*c**3*d**5) + x**20*(7*a*c*d**6 + 7*b**2*d**6/2 + 21*b*c
**2*d**5 + 35*c**4*d**4/4) + x**19*(7*a*b*d**6 + 21*a*c**2*d**5 + 21*b**2*c*d**5
+ 35*b*c**3*d**4 + 7*c**5*d**3) + x**18*(7*a**2*d**6/2 + 42*a*b*c*d**5 + 35*a*c
**3*d**4 + 7*b**3*d**5 + 105*b**2*c**2*d**4/2 + 35*b*c**4*d**3 + 7*c**6*d**2/2)
+ x**17*(21*a**2*c*d**5 + 21*a*b**2*d**5 + 105*a*b*c**2*d**4 + 35*a*c**4*d**3 +
35*b**3*c*d**4 + 70*b**2*c**3*d**3 + 21*b*c**5*d**2 + c**7*d) + x**16*(21*a**2*b
*d**5 + 105*a**2*c**2*d**4/2 + 105*a*b**2*c*d**4 + 140*a*b*c**3*d**3 + 21*a*c**5
*d**2 + 35*b**4*d**4/4 + 70*b**3*c**2*d**3 + 105*b**2*c**4*d**2/2 + 7*b*c**6*d +
c**8/8) + x**15*(7*a**3*d**5 + 105*a**2*b*c*d**4 + 70*a**2*c**3*d**3 + 35*a*b**
3*d**4 + 210*a*b**2*c**2*d**3 + 105*a*b*c**4*d**2 + 7*a*c**6*d + 35*b**4*c*d**3
+ 70*b**3*c**3*d**2 + 21*b**2*c**5*d + b*c**7) + x**14*(35*a**3*c*d**4 + 105*a**
2*b**2*d**4/2 + 210*a**2*b*c**2*d**3 + 105*a**2*c**4*d**2/2 + 140*a*b**3*c*d**3
+ 210*a*b**2*c**3*d**2 + 42*a*b*c**5*d + a*c**7 + 7*b**5*d**3 + 105*b**4*c**2*d*
*2/2 + 35*b**3*c**4*d + 7*b**2*c**6/2) + x**13*(35*a**3*b*d**4 + 70*a**3*c**2*d*
*3 + 210*a**2*b**2*c*d**3 + 210*a**2*b*c**3*d**2 + 21*a**2*c**5*d + 35*a*b**4*d*
*3 + 210*a*b**3*c**2*d**2 + 105*a*b**2*c**4*d + 7*a*b*c**6 + 21*b**5*c*d**2 + 35
*b**4*c**3*d + 7*b**3*c**5) + x**12*(35*a**4*d**4/4 + 140*a**3*b*c*d**3 + 70*a**
3*c**3*d**2 + 70*a**2*b**3*d**3 + 315*a**2*b**2*c**2*d**2 + 105*a**2*b*c**4*d +
7*a**2*c**6/2 + 105*a*b**4*c*d**2 + 140*a*b**3*c**3*d + 21*a*b**2*c**5 + 7*b**6*
d**2/2 + 21*b**5*c**2*d + 35*b**4*c**4/4) + x**11*(35*a**4*c*d**3 + 70*a**3*b**2
*d**3 + 210*a**3*b*c**2*d**2 + 35*a**3*c**4*d + 210*a**2*b**3*c*d**2 + 210*a**2*
b**2*c**3*d + 21*a**2*b*c**5 + 21*a*b**5*d**2 + 105*a*b**4*c**2*d + 35*a*b**3*c*
*4 + 7*b**6*c*d + 7*b**5*c**3) + x**10*(35*a**4*b*d**3 + 105*a**4*c**2*d**2/2 +
210*a**3*b**2*c*d**2 + 140*a**3*b*c**3*d + 7*a**3*c**5 + 105*a**2*b**4*d**2/2 +
210*a**2*b**3*c**2*d + 105*a**2*b**2*c**4/2 + 42*a*b**5*c*d + 35*a*b**4*c**3 + b
**7*d + 7*b**6*c**2/2) + x**9*(7*a**5*d**3 + 105*a**4*b*c*d**2 + 35*a**4*c**3*d
+ 70*a**3*b**3*d**2 + 210*a**3*b**2*c**2*d + 35*a**3*b*c**4 + 105*a**2*b**4*c*d
+ 70*a**2*b**3*c**3 + 7*a*b**6*d + 21*a*b**5*c**2 + b**7*c) + x**8*(21*a**5*c*d*
*2 + 105*a**4*b**2*d**2/2 + 105*a**4*b*c**2*d + 35*a**4*c**4/4 + 140*a**3*b**3*c
*d + 70*a**3*b**2*c**3 + 21*a**2*b**5*d + 105*a**2*b**4*c**2/2 + 7*a*b**6*c + b*
*8/8) + x**7*(21*a**5*b*d**2 + 21*a**5*c**2*d + 105*a**4*b**2*c*d + 35*a**4*b*c*
*3 + 35*a**3*b**4*d + 70*a**3*b**3*c**2 + 21*a**2*b**5*c + a*b**7) + x**6*(7*a**
6*d**2/2 + 42*a**5*b*c*d + 7*a**5*c**3 + 35*a**4*b**3*d + 105*a**4*b**2*c**2/2 +
35*a**3*b**4*c + 7*a**2*b**6/2) + x**5*(7*a**6*c*d + 21*a**5*b**2*d + 21*a**5*b
*c**2 + 35*a**4*b**3*c + 7*a**3*b**5) + x**4*(7*a**6*b*d + 7*a**6*c**2/2 + 21*a*
*5*b**2*c + 35*a**4*b**4/4) + x**3*(a**7*d + 7*a**6*b*c + 7*a**5*b**3) + x**2*(a
**7*c + 7*a**6*b**2/2)

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GIAC/XCAS [A]  time = 0.26797, size = 1, normalized size = 0.05 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^3 + c*x^2 + b*x + a)^7*(3*d*x^2 + 2*c*x + b),x, algorithm="giac")

[Out]

Done