3.2.87 \(\int \frac {1}{(a+b x)^3} \, dx\)

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

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Rubi [A]  time = 0.00, antiderivative size = 14, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\frac {1}{2 b (a+b x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^(-3),x]

[Out]

-1/(2*b*(a + b*x)^2)

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)^3} \, dx &=-\frac {1}{2 b (a+b x)^2}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 14, normalized size = 1.00 \begin {gather*} -\frac {1}{2 b (a+b x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^(-3),x]

[Out]

-1/2*1/(b*(a + b*x)^2)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{(a+b x)^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a + b*x)^(-3),x]

[Out]

IntegrateAlgebraic[(a + b*x)^(-3), x]

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fricas [A]  time = 1.01, size = 24, normalized size = 1.71 \begin {gather*} -\frac {1}{2 \, {\left (b^{3} x^{2} + 2 \, a b^{2} x + a^{2} b\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)^3,x, algorithm="fricas")

[Out]

-1/2/(b^3*x^2 + 2*a*b^2*x + a^2*b)

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giac [A]  time = 0.95, size = 12, normalized size = 0.86 \begin {gather*} -\frac {1}{2 \, {\left (b x + a\right )}^{2} b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)^3,x, algorithm="giac")

[Out]

-1/2/((b*x + a)^2*b)

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maple [A]  time = 0.00, size = 13, normalized size = 0.93 \begin {gather*} -\frac {1}{2 \left (b x +a \right )^{2} b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x+a)^3,x)

[Out]

-1/2/b/(b*x+a)^2

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maxima [A]  time = 1.29, size = 12, normalized size = 0.86 \begin {gather*} -\frac {1}{2 \, {\left (b x + a\right )}^{2} b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)^3,x, algorithm="maxima")

[Out]

-1/2/((b*x + a)^2*b)

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mupad [B]  time = 0.07, size = 26, normalized size = 1.86 \begin {gather*} -\frac {1}{2\,a^2\,b+4\,a\,b^2\,x+2\,b^3\,x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + b*x)^3,x)

[Out]

-1/(2*a^2*b + 2*b^3*x^2 + 4*a*b^2*x)

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sympy [B]  time = 0.21, size = 26, normalized size = 1.86 \begin {gather*} - \frac {1}{2 a^{2} b + 4 a b^{2} x + 2 b^{3} x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)**3,x)

[Out]

-1/(2*a**2*b + 4*a*b**2*x + 2*b**3*x**2)

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