3.1.25 \(\int \frac {1}{(a+b x)^2} \, dx\) [25]

Optimal. Leaf size=12 \[ -\frac {1}{b (a+b x)} \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 12, 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}{b (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^(-2),x]

[Out]

-(1/(b*(a + b*x)))

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

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Mathematica [A]
time = 0.00, size = 12, normalized size = 1.00 \begin {gather*} -\frac {1}{b (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^(-2),x]

[Out]

-(1/(b*(a + b*x)))

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Maple [A]
time = 0.02, size = 13, normalized size = 1.08

method	result	size
gosper	\(-\frac {1}{b \left (b x +a \right )}\)	\(13\)
default	\(-\frac {1}{b \left (b x +a \right )}\)	\(13\)
norman	\(\frac {x}{a \left (b x +a \right )}\)	\(13\)
risch	\(-\frac {1}{b \left (b x +a \right )}\)	\(13\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x+a)^2,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.96, size = 12, normalized size = 1.00 \begin {gather*} -\frac {1}{{\left (b x + a\right )} b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)^2,x, algorithm="maxima")

[Out]

-1/((b*x + a)*b)

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Fricas [A]
time = 0.82, size = 13, normalized size = 1.08 \begin {gather*} -\frac {1}{b^{2} x + a b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)^2,x, algorithm="fricas")

[Out]

-1/(b^2*x + a*b)

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Sympy [A]
time = 0.05, size = 10, normalized size = 0.83 \begin {gather*} - \frac {1}{a b + b^{2} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)**2,x)

[Out]

-1/(a*b + b**2*x)

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Giac [A]
time = 0.47, size = 12, normalized size = 1.00 \begin {gather*} -\frac {1}{{\left (b x + a\right )} b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)^2,x, algorithm="giac")

[Out]

-1/((b*x + a)*b)

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Mupad [B]
time = 0.12, size = 12, normalized size = 1.00 \begin {gather*} -\frac {1}{b\,\left (a+b\,x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + b*x)^2,x)

[Out]

-1/(b*(a + b*x))

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