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]

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

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Rubi [A]  time = 0.00, antiderivative size = 28, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.01, 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]

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

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fricas [A]  time = 1.13, size = 38, normalized size = 1.36 \[ -\frac {2 \, b e x + b d + a e}{2 \, {\left (b^{4} x^{2} + 2 \, a b^{3} x + a^{2} b^{2}\right )}} \]

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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giac [A]  time = 0.16, size = 26, normalized size = 0.93 \[ -\frac {2 \, b x e + b d + a e}{2 \, {\left (b x + a\right )}^{2} b^{2}} \]

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)

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maple [A]  time = 0.04, size = 35, normalized size = 1.25 \[ -\frac {e}{\left (b x +a \right ) b^{2}}-\frac {-a e +b d}{2 \left (b x +a \right )^{2} b^{2}} \]

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]

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

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maxima [A]  time = 0.50, size = 38, normalized size = 1.36 \[ -\frac {2 \, b e x + b d + a e}{2 \, {\left (b^{4} x^{2} + 2 \, a b^{3} x + a^{2} b^{2}\right )}} \]

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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mupad [B]  time = 2.01, size = 39, normalized size = 1.39 \[ -\frac {\frac {a\,e+b\,d}{2\,b^2}+\frac {e\,x}{b}}{a^2+2\,a\,b\,x+b^2\,x^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.28, size = 39, normalized size = 1.39 \[ \frac {- a e - b d - 2 b e x}{2 a^{2} b^{2} + 4 a b^{3} x + 2 b^{4} x^{2}} \]

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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