\(\int \frac {x}{(a+b x)^3} \, dx\) [186]

Optimal result

Integrand size = 9, antiderivative size = 17 \[ \int \frac {x}{(a+b x)^3} \, dx=\frac {x^2}{2 a (a+b x)^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {37} \[ \int \frac {x}{(a+b x)^3} \, dx=\frac {x^2}{2 a (a+b x)^2} \]

[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*} \text {integral}& = \frac {x^2}{2 a (a+b x)^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.18 \[ \int \frac {x}{(a+b x)^3} \, dx=-\frac {a+2 b x}{2 b^2 (a+b x)^2} \]

[In]

Integrate[x/(a + b*x)^3,x]

[Out]

-1/2*(a + 2*b*x)/(b^2*(a + b*x)^2)

Maple [A] (verified)

Time = 0.18 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.12

method	result	size
gosper	\(-\frac {2 b x +a}{2 b^{2} \left (b x +a \right )^{2}}\)	\(19\)
parallelrisch	\(\frac {-2 b x -a}{2 b^{2} \left (b x +a \right )^{2}}\)	\(21\)
norman	\(\frac {-\frac {x}{b}-\frac {a}{2 b^{2}}}{\left (b x +a \right )^{2}}\)	\(22\)
risch	\(\frac {-\frac {x}{b}-\frac {a}{2 b^{2}}}{\left (b x +a \right )^{2}}\)	\(22\)
default	\(\frac {a}{2 b^{2} \left (b x +a \right )^{2}}-\frac {1}{\left (b x +a \right ) b^{2}}\)	\(27\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 32 vs. \(2 (15) = 30\).

Time = 0.22 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.88 \[ \int \frac {x}{(a+b x)^3} \, dx=-\frac {2 \, b x + a}{2 \, {\left (b^{4} x^{2} + 2 \, a b^{3} x + a^{2} b^{2}\right )}} \]

[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)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 32 vs. \(2 (12) = 24\).

Time = 0.10 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.88 \[ \int \frac {x}{(a+b x)^3} \, dx=\frac {- a - 2 b x}{2 a^{2} b^{2} + 4 a b^{3} x + 2 b^{4} x^{2}} \]

[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)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 32 vs. \(2 (15) = 30\).

Time = 0.20 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.88 \[ \int \frac {x}{(a+b x)^3} \, dx=-\frac {2 \, b x + a}{2 \, {\left (b^{4} x^{2} + 2 \, a b^{3} x + a^{2} b^{2}\right )}} \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.06 \[ \int \frac {x}{(a+b x)^3} \, dx=-\frac {2 \, b x + a}{2 \, {\left (b x + a\right )}^{2} b^{2}} \]

[In]

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

[Out]

-1/2*(2*b*x + a)/((b*x + a)^2*b^2)

Mupad [B] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.88 \[ \int \frac {x}{(a+b x)^3} \, dx=-\frac {\frac {a}{2\,b^2}+\frac {x}{b}}{a^2+2\,a\,b\,x+b^2\,x^2} \]

[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)