3.2.73 \(\int \frac {x}{(a+b x)^2} \, dx\)

Optimal. Leaf size=23 \[ \frac {a}{b^2 (a+b x)}+\frac {\log (a+b x)}{b^2} \]

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Rubi [A]  time = 0.01, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {43} \begin {gather*} \frac {a}{b^2 (a+b x)}+\frac {\log (a+b x)}{b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x/(a + b*x)^2,x]

[Out]

a/(b^2*(a + b*x)) + Log[a + b*x]/b^2

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 20, normalized size = 0.87 \begin {gather*} \frac {\frac {a}{a+b x}+\log (a+b x)}{b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x/(a + b*x)^2,x]

[Out]

(a/(a + b*x) + Log[a + b*x])/b^2

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{(a+b x)^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[x/(a + b*x)^2,x]

[Out]

IntegrateAlgebraic[x/(a + b*x)^2, x]

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fricas [A]  time = 1.22, size = 28, normalized size = 1.22 \begin {gather*} \frac {{\left (b x + a\right )} \log \left (b x + a\right ) + a}{b^{3} x + a b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((b*x + a)*log(b*x + a) + a)/(b^3*x + a*b^2)

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giac [A]  time = 1.05, size = 42, normalized size = 1.83 \begin {gather*} -\frac {\frac {\log \left (\frac {{\left | b x + a \right |}}{{\left (b x + a\right )}^{2} {\left | b \right |}}\right )}{b} - \frac {a}{{\left (b x + a\right )} b}}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(log(abs(b*x + a)/((b*x + a)^2*abs(b)))/b - a/((b*x + a)*b))/b

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maple [A]  time = 0.01, size = 24, normalized size = 1.04 \begin {gather*} \frac {a}{\left (b x +a \right ) b^{2}}+\frac {\ln \left (b x +a \right )}{b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(b*x+a)^2,x)

[Out]

a/b^2/(b*x+a)+ln(b*x+a)/b^2

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maxima [A]  time = 1.32, size = 26, normalized size = 1.13 \begin {gather*} \frac {a}{b^{3} x + a b^{2}} + \frac {\log \left (b x + a\right )}{b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a/(b^3*x + a*b^2) + log(b*x + a)/b^2

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mupad [B]  time = 0.04, size = 23, normalized size = 1.00 \begin {gather*} \frac {\ln \left (a+b\,x\right )}{b^2}+\frac {a}{b^2\,\left (a+b\,x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(a + b*x)^2,x)

[Out]

log(a + b*x)/b^2 + a/(b^2*(a + b*x))

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sympy [A]  time = 0.17, size = 20, normalized size = 0.87 \begin {gather*} \frac {a}{a b^{2} + b^{3} x} + \frac {\log {\left (a + b x \right )}}{b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a/(a*b**2 + b**3*x) + log(a + b*x)/b**2

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