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

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

________________________________________________________________________________________

Rubi [A]  time = 0.00, antiderivative size = 17, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {x^2}{2 a (a+b x)^2} \end {gather*}

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 = 0.01, size = 20, normalized size = 1.18 \begin {gather*} -\frac {a+2 b x}{2 b^2 (a+b x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x/(a + b*x)^3,x]

[Out]

-1/2*(a + 2*b*x)/(b^2*(a + b*x)^2)

________________________________________________________________________________________

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

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[x/(a + b*x)^3,x]

[Out]

IntegrateAlgebraic[x/(a + b*x)^3, x]

________________________________________________________________________________________

fricas [B]  time = 1.21, size = 32, normalized size = 1.88 \begin {gather*} -\frac {2 \, b x + a}{2 \, {\left (b^{4} x^{2} + 2 \, a b^{3} x + a^{2} b^{2}\right )}} \end {gather*}

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)

________________________________________________________________________________________

giac [A]  time = 1.14, size = 18, normalized size = 1.06 \begin {gather*} -\frac {2 \, b x + a}{2 \, {\left (b x + a\right )}^{2} b^{2}} \end {gather*}

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)

________________________________________________________________________________________

maple [A]  time = 0.00, size = 27, normalized size = 1.59 \begin {gather*} \frac {a}{2 \left (b x +a \right )^{2} b^{2}}-\frac {1}{\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)^3,x)

[Out]

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

________________________________________________________________________________________

maxima [B]  time = 1.39, size = 32, normalized size = 1.88 \begin {gather*} -\frac {2 \, b x + a}{2 \, {\left (b^{4} x^{2} + 2 \, a b^{3} x + a^{2} b^{2}\right )}} \end {gather*}

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)

________________________________________________________________________________________

mupad [B]  time = 0.07, size = 32, normalized size = 1.88 \begin {gather*} -\frac {\frac {a}{2\,b^2}+\frac {x}{b}}{a^2+2\,a\,b\,x+b^2\,x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(a + b*x)^3,x)

[Out]

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

________________________________________________________________________________________

sympy [B]  time = 0.20, size = 32, normalized size = 1.88 \begin {gather*} \frac {- a - 2 b x}{2 a^{2} b^{2} + 4 a b^{3} x + 2 b^{4} x^{2}} \end {gather*}

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)

________________________________________________________________________________________