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

Optimal. Leaf size=69 \[ -\frac{a-b x}{2 \left (a^2+b^2\right ) \left (2 a x-b x^2+b\right )}-\frac{b \tanh ^{-1}\left (\frac{a-b x}{\sqrt{a^2+b^2}}\right )}{2 \left (a^2+b^2\right )^{3/2}} \]

[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 = 0.0336722, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {614, 618, 206} \[ -\frac{a-b x}{2 \left (a^2+b^2\right ) \left (2 a x-b x^2+b\right )}-\frac{b \tanh ^{-1}\left (\frac{a-b x}{\sqrt{a^2+b^2}}\right )}{2 \left (a^2+b^2\right )^{3/2}} \]

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 = 0.0612156, size = 78, normalized size = 1.13 \[ \frac{\frac{b x-a}{2 a x-b x^2+b}-\frac{b \tan ^{-1}\left (\frac{b x-a}{\sqrt{-a^2-b^2}}\right )}{\sqrt{-a^2-b^2}}}{2 \left (a^2+b^2\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[(b + 2*a*x - b*x^2)^(-2),x]

[Out]

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

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Maple [A]  time = 0.149, size = 84, 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}}}{\it Artanh} \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)*arctanh(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 = 2.50362, size = 360, normalized size = 5.22 \begin{align*} -\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 )}} \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*sqr
t(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)

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Sympy [B]  time = 1.03575, size = 218, normalized size = 3.16 \begin{align*} - \frac{b \sqrt{\frac{1}{\left (a^{2} + b^{2}\right )^{3}}} \log{\left (x + \frac{- a^{4} b \sqrt{\frac{1}{\left (a^{2} + b^{2}\right )^{3}}} - 2 a^{2} b^{3} \sqrt{\frac{1}{\left (a^{2} + b^{2}\right )^{3}}} - a b - b^{5} \sqrt{\frac{1}{\left (a^{2} + b^{2}\right )^{3}}}}{b^{2}} \right )}}{4} + \frac{b \sqrt{\frac{1}{\left (a^{2} + b^{2}\right )^{3}}} \log{\left (x + \frac{a^{4} b \sqrt{\frac{1}{\left (a^{2} + b^{2}\right )^{3}}} + 2 a^{2} b^{3} \sqrt{\frac{1}{\left (a^{2} + b^{2}\right )^{3}}} - a b + b^{5} \sqrt{\frac{1}{\left (a^{2} + b^{2}\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((a**2 + b**2)**(-3))*log(x + (-a**4*b*sqrt((a**2 + b**2)**(-3)) - 2*a**2*b**3*sqrt((a**2 + b**2)**(-3)
) - a*b - b**5*sqrt((a**2 + b**2)**(-3)))/b**2)/4 + b*sqrt((a**2 + b**2)**(-3))*log(x + (a**4*b*sqrt((a**2 + b
**2)**(-3)) + 2*a**2*b**3*sqrt((a**2 + b**2)**(-3)) - a*b + b**5*sqrt((a**2 + b**2)**(-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 = 1.21631, size = 122, normalized size = 1.77 \begin{align*} -\frac{b \log \left (\frac{{\left | 2 \, b x - 2 \, a - 2 \, \sqrt{a^{2} + b^{2}} \right |}}{{\left | 2 \, b x - 2 \, a + 2 \, \sqrt{a^{2} + b^{2}} \right |}}\right )}{4 \,{\left (a^{2} + b^{2}\right )}^{\frac{3}{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]

-1/4*b*log(abs(2*b*x - 2*a - 2*sqrt(a^2 + b^2))/abs(2*b*x - 2*a + 2*sqrt(a^2 + b^2)))/(a^2 + b^2)^(3/2) - 1/2*
(b*x - a)/((b*x^2 - 2*a*x - b)*(a^2 + b^2))