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

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

[Out]

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

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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*}

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Mathematica [A]
time = 0.00, 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)

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Mathics [A]
time = 1.85, size = 30, normalized size = 1.76 \begin {gather*} \frac {-\frac {a}{2}-b x}{b^2 \left (a^2+2 a b x+b^2 x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[x^1/(a + b*x)^3,x]')

[Out]

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

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Maple [A]
time = 0.08, size = 27, normalized size = 1.59

method	result	size
gosper	\(-\frac {2 b x +a}{2 \left (b x +a \right )^{2} b^{2}}\)	\(19\)
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	\(-\frac {1}{b^{2} \left (b x +a \right )}+\frac {a}{2 b^{2} \left (b x +a \right )^{2}}\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 32 vs. \(2 (15) = 30\).
time = 0.26, 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)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 32 vs. \(2 (15) = 30\).
time = 0.30, 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)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 32 vs. \(2 (12) = 24\).
time = 0.10, 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)

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Giac [A]
time = 0.00, size = 22, normalized size = 1.29 \begin {gather*} \frac {-2 x b-a}{2 b^{2} \left (x b+a\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(2*b*x + a)/((b*x + a)^2*b^2)

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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)

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