3.234 \(\int \frac{1}{(a+b x)^{10}} \, dx\)

Optimal. Leaf size=14 \[ -\frac{1}{9 b (a+b x)^9} \]

[Out]

-1/(9*b*(a + b*x)^9)

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Rubi [A]  time = 0.0015978, 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} \[ -\frac{1}{9 b (a+b x)^9} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^(-10),x]

[Out]

-1/(9*b*(a + b*x)^9)

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}{(a+b x)^{10}} \, dx &=-\frac{1}{9 b (a+b x)^9}\\ \end{align*}

Mathematica [A]  time = 0.0033223, size = 14, normalized size = 1. \[ -\frac{1}{9 b (a+b x)^9} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^(-10),x]

[Out]

-1/(9*b*(a + b*x)^9)

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Maple [A]  time = 0., size = 13, normalized size = 0.9 \begin{align*} -{\frac{1}{9\,b \left ( bx+a \right ) ^{9}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x+a)^10,x)

[Out]

-1/9/b/(b*x+a)^9

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Maxima [A]  time = 1.01816, size = 16, normalized size = 1.14 \begin{align*} -\frac{1}{9 \,{\left (b x + a\right )}^{9} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)^10,x, algorithm="maxima")

[Out]

-1/9/((b*x + a)^9*b)

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Fricas [B]  time = 1.45276, size = 212, normalized size = 15.14 \begin{align*} -\frac{1}{9 \,{\left (b^{10} x^{9} + 9 \, a b^{9} x^{8} + 36 \, a^{2} b^{8} x^{7} + 84 \, a^{3} b^{7} x^{6} + 126 \, a^{4} b^{6} x^{5} + 126 \, a^{5} b^{5} x^{4} + 84 \, a^{6} b^{4} x^{3} + 36 \, a^{7} b^{3} x^{2} + 9 \, a^{8} b^{2} x + a^{9} b\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)^10,x, algorithm="fricas")

[Out]

-1/9/(b^10*x^9 + 9*a*b^9*x^8 + 36*a^2*b^8*x^7 + 84*a^3*b^7*x^6 + 126*a^4*b^6*x^5 + 126*a^5*b^5*x^4 + 84*a^6*b^
4*x^3 + 36*a^7*b^3*x^2 + 9*a^8*b^2*x + a^9*b)

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Sympy [B]  time = 1.05076, size = 109, normalized size = 7.79 \begin{align*} - \frac{1}{9 a^{9} b + 81 a^{8} b^{2} x + 324 a^{7} b^{3} x^{2} + 756 a^{6} b^{4} x^{3} + 1134 a^{5} b^{5} x^{4} + 1134 a^{4} b^{6} x^{5} + 756 a^{3} b^{7} x^{6} + 324 a^{2} b^{8} x^{7} + 81 a b^{9} x^{8} + 9 b^{10} x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)**10,x)

[Out]

-1/(9*a**9*b + 81*a**8*b**2*x + 324*a**7*b**3*x**2 + 756*a**6*b**4*x**3 + 1134*a**5*b**5*x**4 + 1134*a**4*b**6
*x**5 + 756*a**3*b**7*x**6 + 324*a**2*b**8*x**7 + 81*a*b**9*x**8 + 9*b**10*x**9)

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Giac [A]  time = 1.19332, size = 16, normalized size = 1.14 \begin{align*} -\frac{1}{9 \,{\left (b x + a\right )}^{9} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)^10,x, algorithm="giac")

[Out]

-1/9/((b*x + a)^9*b)