3.1371 \(\int \frac{1}{(c+d x)^8} \, dx\)

Optimal. Leaf size=14 \[ -\frac{1}{7 d (c+d x)^7} \]

[Out]

-1/(7*d*(c + d*x)^7)

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Rubi [A]  time = 0.00701691, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ -\frac{1}{7 d (c+d x)^7} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x)^(-8),x]

[Out]

-1/(7*d*(c + d*x)^7)

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Rubi in Sympy [A]  time = 1.4395, size = 12, normalized size = 0.86 \[ - \frac{1}{7 d \left (c + d x\right )^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(d*x+c)**8,x)

[Out]

-1/(7*d*(c + d*x)**7)

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Mathematica [A]  time = 0.00436649, size = 14, normalized size = 1. \[ -\frac{1}{7 d (c+d x)^7} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x)^(-8),x]

[Out]

-1/(7*d*(c + d*x)^7)

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Maple [A]  time = 0.002, size = 13, normalized size = 0.9 \[ -{\frac{1}{7\,d \left ( dx+c \right ) ^{7}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(d*x+c)^8,x)

[Out]

-1/7/d/(d*x+c)^7

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Maxima [A]  time = 1.40858, size = 16, normalized size = 1.14 \[ -\frac{1}{7 \,{\left (d x + c\right )}^{7} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^(-8),x, algorithm="maxima")

[Out]

-1/7/((d*x + c)^7*d)

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Fricas [A]  time = 0.219808, size = 107, normalized size = 7.64 \[ -\frac{1}{7 \,{\left (d^{8} x^{7} + 7 \, c d^{7} x^{6} + 21 \, c^{2} d^{6} x^{5} + 35 \, c^{3} d^{5} x^{4} + 35 \, c^{4} d^{4} x^{3} + 21 \, c^{5} d^{3} x^{2} + 7 \, c^{6} d^{2} x + c^{7} d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^(-8),x, algorithm="fricas")

[Out]

-1/7/(d^8*x^7 + 7*c*d^7*x^6 + 21*c^2*d^6*x^5 + 35*c^3*d^5*x^4 + 35*c^4*d^4*x^3 +
21*c^5*d^3*x^2 + 7*c^6*d^2*x + c^7*d)

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Sympy [A]  time = 1.11586, size = 85, normalized size = 6.07 \[ - \frac{1}{7 c^{7} d + 49 c^{6} d^{2} x + 147 c^{5} d^{3} x^{2} + 245 c^{4} d^{4} x^{3} + 245 c^{3} d^{5} x^{4} + 147 c^{2} d^{6} x^{5} + 49 c d^{7} x^{6} + 7 d^{8} x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(d*x+c)**8,x)

[Out]

-1/(7*c**7*d + 49*c**6*d**2*x + 147*c**5*d**3*x**2 + 245*c**4*d**4*x**3 + 245*c*
*3*d**5*x**4 + 147*c**2*d**6*x**5 + 49*c*d**7*x**6 + 7*d**8*x**7)

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GIAC/XCAS [A]  time = 0.220764, size = 16, normalized size = 1.14 \[ -\frac{1}{7 \,{\left (d x + c\right )}^{7} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^(-8),x, algorithm="giac")

[Out]

-1/7/((d*x + c)^7*d)