3.96 \(\int \frac{1}{(b+2 a x+b x^2)^2} \, dx\)

Optimal. Leaf size=72 \[ \frac{b \tanh ^{-1}\left (\frac{a+b x}{\sqrt{a^2-b^2}}\right )}{2 \left (a^2-b^2\right )^{3/2}}-\frac{a+b x}{2 \left (a^2-b^2\right ) \left (2 a x+b x^2+b\right )} \]

[Out]

-(a + b*x)/(2*(a^2 - b^2)*(b + 2*a*x + b*x^2)) + (b*ArcTanh[(a + b*x)/Sqrt[a^2 - b^2]])/(2*(a^2 - b^2)^(3/2))

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Rubi [A]  time = 0.0355411, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {614, 618, 206} \[ \frac{b \tanh ^{-1}\left (\frac{a+b x}{\sqrt{a^2-b^2}}\right )}{2 \left (a^2-b^2\right )^{3/2}}-\frac{a+b x}{2 \left (a^2-b^2\right ) \left (2 a x+b x^2+b\right )} \]

Antiderivative was successfully verified.

[In]

Int[(b + 2*a*x + b*x^2)^(-2),x]

[Out]

-(a + b*x)/(2*(a^2 - b^2)*(b + 2*a*x + b*x^2)) + (b*ArcTanh[(a + b*x)/Sqrt[a^2 - b^2]])/(2*(a^2 - b^2)^(3/2))

Rule 614

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b + 2*c*x)*(a + b*x + c*x^2)^(p + 1))/((p +
1)*(b^2 - 4*a*c)), x] - Dist[(2*c*(2*p + 3))/((p + 1)*(b^2 - 4*a*c)), Int[(a + b*x + c*x^2)^(p + 1), x], x] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[p, -3/2] && IntegerQ[4*p]

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

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

Mathematica [A]  time = 0.0427679, size = 72, normalized size = 1. \[ \frac{a+b x}{2 \left (b^2-a^2\right ) \left (2 a x+b x^2+b\right )}+\frac{b \tan ^{-1}\left (\frac{a+b x}{\sqrt{b^2-a^2}}\right )}{2 \left (b^2-a^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(b + 2*a*x + b*x^2)^(-2),x]

[Out]

(a + b*x)/(2*(-a^2 + b^2)*(b + 2*a*x + b*x^2)) + (b*ArcTan[(a + b*x)/Sqrt[-a^2 + b^2]])/(2*(-a^2 + b^2)^(3/2))

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Maple [A]  time = 0.152, size = 86, normalized size = 1.2 \begin{align*}{\frac{2\,bx+2\,a}{ \left ( -4\,{a}^{2}+4\,{b}^{2} \right ) \left ( b{x}^{2}+2\,ax+b \right ) }}+2\,{\frac{b}{ \left ( -4\,{a}^{2}+4\,{b}^{2} \right ) \sqrt{-{a}^{2}+{b}^{2}}}\arctan \left ( 1/2\,{\frac{2\,bx+2\,a}{\sqrt{-{a}^{2}+{b}^{2}}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*b*x+2*a)/(-4*a^2+4*b^2)/(b*x^2+2*a*x+b)+2*b/(-4*a^2+4*b^2)/(-a^2+b^2)^(1/2)*arctan(1/2*(2*b*x+2*a)/(-a^2+b^
2)^(1/2))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^2+2*a*x+b)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.41407, size = 664, normalized size = 9.22 \begin{align*} \left [-\frac{2 \, a^{3} - 2 \, a b^{2} +{\left (b^{2} x^{2} + 2 \, a b x + b^{2}\right )} \sqrt{a^{2} - b^{2}} \log \left (\frac{b^{2} x^{2} + 2 \, a b x + 2 \, a^{2} - b^{2} - 2 \, \sqrt{a^{2} - b^{2}}{\left (b x + a\right )}}{b x^{2} + 2 \, a x + b}\right ) + 2 \,{\left (a^{2} b - b^{3}\right )} x}{4 \,{\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5} +{\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} x^{2} + 2 \,{\left (a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right )} x\right )}}, -\frac{a^{3} - a b^{2} -{\left (b^{2} x^{2} + 2 \, a b x + b^{2}\right )} \sqrt{-a^{2} + b^{2}} \arctan \left (-\frac{\sqrt{-a^{2} + b^{2}}{\left (b x + a\right )}}{a^{2} - b^{2}}\right ) +{\left (a^{2} b - b^{3}\right )} x}{2 \,{\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5} +{\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} x^{2} + 2 \,{\left (a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right )} x\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^2+2*a*x+b)^2,x, algorithm="fricas")

[Out]

[-1/4*(2*a^3 - 2*a*b^2 + (b^2*x^2 + 2*a*b*x + b^2)*sqrt(a^2 - b^2)*log((b^2*x^2 + 2*a*b*x + 2*a^2 - b^2 - 2*sq
rt(a^2 - b^2)*(b*x + a))/(b*x^2 + 2*a*x + b)) + 2*(a^2*b - b^3)*x)/(a^4*b - 2*a^2*b^3 + b^5 + (a^4*b - 2*a^2*b
^3 + b^5)*x^2 + 2*(a^5 - 2*a^3*b^2 + a*b^4)*x), -1/2*(a^3 - a*b^2 - (b^2*x^2 + 2*a*b*x + b^2)*sqrt(-a^2 + b^2)
*arctan(-sqrt(-a^2 + b^2)*(b*x + a)/(a^2 - b^2)) + (a^2*b - b^3)*x)/(a^4*b - 2*a^2*b^3 + b^5 + (a^4*b - 2*a^2*
b^3 + b^5)*x^2 + 2*(a^5 - 2*a^3*b^2 + a*b^4)*x)]

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Sympy [B]  time = 1.39258, size = 228, normalized size = 3.17 \begin{align*} - \frac{b \sqrt{\frac{1}{\left (a - b\right )^{3} \left (a + b\right )^{3}}} \log{\left (x + \frac{- a^{4} b \sqrt{\frac{1}{\left (a - b\right )^{3} \left (a + b\right )^{3}}} + 2 a^{2} b^{3} \sqrt{\frac{1}{\left (a - b\right )^{3} \left (a + b\right )^{3}}} + a b - b^{5} \sqrt{\frac{1}{\left (a - b\right )^{3} \left (a + b\right )^{3}}}}{b^{2}} \right )}}{4} + \frac{b \sqrt{\frac{1}{\left (a - b\right )^{3} \left (a + b\right )^{3}}} \log{\left (x + \frac{a^{4} b \sqrt{\frac{1}{\left (a - b\right )^{3} \left (a + b\right )^{3}}} - 2 a^{2} b^{3} \sqrt{\frac{1}{\left (a - b\right )^{3} \left (a + b\right )^{3}}} + a b + b^{5} \sqrt{\frac{1}{\left (a - b\right )^{3} \left (a + b\right )^{3}}}}{b^{2}} \right )}}{4} - \frac{a + b x}{2 a^{2} b - 2 b^{3} + x^{2} \left (2 a^{2} b - 2 b^{3}\right ) + x \left (4 a^{3} - 4 a b^{2}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x**2+2*a*x+b)**2,x)

[Out]

-b*sqrt(1/((a - b)**3*(a + b)**3))*log(x + (-a**4*b*sqrt(1/((a - b)**3*(a + b)**3)) + 2*a**2*b**3*sqrt(1/((a -
b)**3*(a + b)**3)) + a*b - b**5*sqrt(1/((a - b)**3*(a + b)**3)))/b**2)/4 + b*sqrt(1/((a - b)**3*(a + b)**3))*
log(x + (a**4*b*sqrt(1/((a - b)**3*(a + b)**3)) - 2*a**2*b**3*sqrt(1/((a - b)**3*(a + b)**3)) + a*b + b**5*sqr
t(1/((a - b)**3*(a + b)**3)))/b**2)/4 - (a + b*x)/(2*a**2*b - 2*b**3 + x**2*(2*a**2*b - 2*b**3) + x*(4*a**3 -
4*a*b**2))

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Giac [A]  time = 1.19746, size = 101, normalized size = 1.4 \begin{align*} -\frac{b \arctan \left (\frac{b x + a}{\sqrt{-a^{2} + b^{2}}}\right )}{2 \,{\left (a^{2} - b^{2}\right )} \sqrt{-a^{2} + b^{2}}} - \frac{b x + a}{2 \,{\left (b x^{2} + 2 \, a x + b\right )}{\left (a^{2} - b^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^2+2*a*x+b)^2,x, algorithm="giac")

[Out]

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