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

Optimal. Leaf size=17 \[ \frac{x^2}{2 a (a+b x)^2} \]

[Out]

x^2/(2*a*(a + b*x)^2)

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Rubi [A]  time = 0.0016454, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {37} \[ \frac{x^2}{2 a (a+b x)^2} \]

Antiderivative was successfully verified.

[In]

Int[x/(a + b*x)^3,x]

[Out]

x^2/(2*a*(a + b*x)^2)

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

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

Mathematica [A]  time = 0.0078628, size = 20, normalized size = 1.18 \[ -\frac{a+2 b x}{2 b^2 (a+b x)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x/(a + b*x)^3,x]

[Out]

-(a + 2*b*x)/(2*b^2*(a + b*x)^2)

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Maple [A]  time = 0.004, size = 27, normalized size = 1.6 \begin{align*} -{\frac{1}{{b}^{2} \left ( bx+a \right ) }}+{\frac{a}{2\,{b}^{2} \left ( bx+a \right ) ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 0.989177, size = 43, normalized size = 2.53 \begin{align*} -\frac{2 \, b x + a}{2 \,{\left (b^{4} x^{2} + 2 \, a b^{3} x + a^{2} b^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 1.53172, size = 68, normalized size = 4. \begin{align*} -\frac{2 \, b x + a}{2 \,{\left (b^{4} x^{2} + 2 \, a b^{3} x + a^{2} b^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 0.429561, size = 32, normalized size = 1.88 \begin{align*} - \frac{a + 2 b x}{2 a^{2} b^{2} + 4 a b^{3} x + 2 b^{4} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.2015, size = 24, normalized size = 1.41 \begin{align*} -\frac{2 \, b x + a}{2 \,{\left (b x + a\right )}^{2} b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(2*b*x + a)/((b*x + a)^2*b^2)