3.1939 \(\int \frac{(a+b x) (d+e x)}{(a^2+2 a b x+b^2 x^2)^2} \, dx\)

Optimal. Leaf size=28 \[ -\frac{(d+e x)^2}{2 (a+b x)^2 (b d-a e)} \]

[Out]

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

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Rubi [A]  time = 0.0043435, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069, Rules used = {27, 37} \[ -\frac{(d+e x)^2}{2 (a+b x)^2 (b d-a e)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.0092757, size = 26, normalized size = 0.93 \[ -\frac{a e+b (d+2 e x)}{2 b^2 (a+b x)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.006, size = 35, normalized size = 1.3 \begin{align*} -{\frac{e}{{b}^{2} \left ( bx+a \right ) }}-{\frac{-ae+bd}{2\,{b}^{2} \left ( bx+a \right ) ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.973212, size = 51, normalized size = 1.82 \begin{align*} -\frac{2 \, b e x + b d + a e}{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((b*x+a)*(e*x+d)/(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.45195, size = 81, normalized size = 2.89 \begin{align*} -\frac{2 \, b e x + b d + a e}{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((b*x+a)*(e*x+d)/(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.427896, size = 39, normalized size = 1.39 \begin{align*} - \frac{a e + b d + 2 b e 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((b*x+a)*(e*x+d)/(b**2*x**2+2*a*b*x+a**2)**2,x)

[Out]

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

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Giac [A]  time = 1.10995, size = 35, normalized size = 1.25 \begin{align*} -\frac{2 \, b x e + b d + a e}{2 \,{\left (b x + a\right )}^{2} b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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