3.1949 \(\int \frac{a+b x}{(a^2+2 a b x+b^2 x^2)^3} \, dx\)

Optimal. Leaf size=14 \[ -\frac{1}{4 b (a+b x)^4} \]

[Out]

-1/(4*b*(a + b*x)^4)

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Rubi [A]  time = 0.002937, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {27, 32} \[ -\frac{1}{4 b (a+b x)^4} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)/(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

-1/(4*b*(a + b*x)^4)

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps

\begin{align*} \int \frac{a+b x}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx &=\int \frac{1}{(a+b x)^5} \, dx\\ &=-\frac{1}{4 b (a+b x)^4}\\ \end{align*}

Mathematica [A]  time = 0.0031205, size = 14, normalized size = 1. \[ -\frac{1}{4 b (a+b x)^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)/(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

-1/(4*b*(a + b*x)^4)

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Maple [A]  time = 0.003, size = 13, normalized size = 0.9 \begin{align*} -{\frac{1}{4\,b \left ( bx+a \right ) ^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)/(b^2*x^2+2*a*b*x+a^2)^3,x)

[Out]

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

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Maxima [A]  time = 0.963375, size = 31, normalized size = 2.21 \begin{align*} -\frac{1}{4 \,{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{2} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="maxima")

[Out]

-1/4/((b^2*x^2 + 2*a*b*x + a^2)^2*b)

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Fricas [B]  time = 1.42221, size = 92, normalized size = 6.57 \begin{align*} -\frac{1}{4 \,{\left (b^{5} x^{4} + 4 \, a b^{4} x^{3} + 6 \, a^{2} b^{3} x^{2} + 4 \, a^{3} b^{2} x + a^{4} b\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="fricas")

[Out]

-1/4/(b^5*x^4 + 4*a*b^4*x^3 + 6*a^2*b^3*x^2 + 4*a^3*b^2*x + a^4*b)

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Sympy [B]  time = 0.43722, size = 49, normalized size = 3.5 \begin{align*} - \frac{1}{4 a^{4} b + 16 a^{3} b^{2} x + 24 a^{2} b^{3} x^{2} + 16 a b^{4} x^{3} + 4 b^{5} x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)/(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

-1/(4*a**4*b + 16*a**3*b**2*x + 24*a**2*b**3*x**2 + 16*a*b**4*x**3 + 4*b**5*x**4)

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Giac [A]  time = 1.13992, size = 16, normalized size = 1.14 \begin{align*} -\frac{1}{4 \,{\left (b x + a\right )}^{4} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="giac")

[Out]

-1/4/((b*x + a)^4*b)