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 = 0.0016125, 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, Rules used = {32} \[ -\frac{1}{7 d (c+d x)^7} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^(-8),x]

[Out]

-1/(7*d*(c + d*x)^7)

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps

\begin{align*} \int \frac{1}{(c+d x)^8} \, dx &=-\frac{1}{7 d (c+d x)^7}\\ \end{align*}

Mathematica [A]  time = 0.0026768, 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., size = 13, normalized size = 0.9 \begin{align*} -{\frac{1}{7\,d \left ( dx+c \right ) ^{7}}} \end{align*}

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 = 0.961186, size = 16, normalized size = 1.14 \begin{align*} -\frac{1}{7 \,{\left (d x + c\right )}^{7} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x+c)^8,x, algorithm="maxima")

[Out]

-1/7/((d*x + c)^7*d)

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Fricas [B]  time = 1.75918, size = 162, normalized size = 11.57 \begin{align*} -\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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(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 [B]  time = 0.609726, size = 85, normalized size = 6.07 \begin{align*} - \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}} \end{align*}

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 [A]  time = 1.06536, size = 16, normalized size = 1.14 \begin{align*} -\frac{1}{7 \,{\left (d x + c\right )}^{7} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x+c)^8,x, algorithm="giac")

[Out]

-1/7/((d*x + c)^7*d)