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

Optimal. Leaf size=71 \[ \frac{2 b x+c}{\left (4 a b-c^2\right ) \left (a+b x^2+c x\right )}+\frac{4 b \tan ^{-1}\left (\frac{2 b x+c}{\sqrt{4 a b-c^2}}\right )}{\left (4 a b-c^2\right )^{3/2}} \]

[Out]

(c + 2*b*x)/((4*a*b - c^2)*(a + c*x + b*x^2)) + (4*b*ArcTan[(c + 2*b*x)/Sqrt[4*a*b - c^2]])/(4*a*b - c^2)^(3/2
)

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Rubi [A]  time = 0.0397878, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {614, 618, 204} \[ \frac{2 b x+c}{\left (4 a b-c^2\right ) \left (a+b x^2+c x\right )}+\frac{4 b \tan ^{-1}\left (\frac{2 b x+c}{\sqrt{4 a b-c^2}}\right )}{\left (4 a b-c^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[(a + c*x + b*x^2)^(-2),x]

[Out]

(c + 2*b*x)/((4*a*b - c^2)*(a + c*x + b*x^2)) + (4*b*ArcTan[(c + 2*b*x)/Sqrt[4*a*b - c^2]])/(4*a*b - c^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 204

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

Rubi steps

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

Mathematica [A]  time = 0.0651828, size = 70, normalized size = 0.99 \[ \frac{2 b x+c}{\left (4 a b-c^2\right ) (a+x (b x+c))}+\frac{4 b \tan ^{-1}\left (\frac{2 b x+c}{\sqrt{4 a b-c^2}}\right )}{\left (4 a b-c^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + c*x + b*x^2)^(-2),x]

[Out]

(c + 2*b*x)/((4*a*b - c^2)*(a + x*(c + b*x))) + (4*b*ArcTan[(c + 2*b*x)/Sqrt[4*a*b - c^2]])/(4*a*b - c^2)^(3/2
)

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Maple [A]  time = 0.191, size = 68, normalized size = 1. \begin{align*}{\frac{2\,bx+c}{ \left ( 4\,ab-{c}^{2} \right ) \left ( b{x}^{2}+cx+a \right ) }}+4\,{\frac{b}{ \left ( 4\,ab-{c}^{2} \right ) ^{3/2}}\arctan \left ({\frac{2\,bx+c}{\sqrt{4\,ab-{c}^{2}}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*b*x+c)/(4*a*b-c^2)/(b*x^2+c*x+a)+4*b*arctan((2*b*x+c)/(4*a*b-c^2)^(1/2))/(4*a*b-c^2)^(3/2)

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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+c*x+a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.19871, size = 724, normalized size = 10.2 \begin{align*} \left [\frac{4 \, a b c - c^{3} + 2 \,{\left (b^{2} x^{2} + b c x + a b\right )} \sqrt{-4 \, a b + c^{2}} \log \left (\frac{2 \, b^{2} x^{2} + 2 \, b c x - 2 \, a b + c^{2} + \sqrt{-4 \, a b + c^{2}}{\left (2 \, b x + c\right )}}{b x^{2} + c x + a}\right ) + 2 \,{\left (4 \, a b^{2} - b c^{2}\right )} x}{16 \, a^{3} b^{2} - 8 \, a^{2} b c^{2} + a c^{4} +{\left (16 \, a^{2} b^{3} - 8 \, a b^{2} c^{2} + b c^{4}\right )} x^{2} +{\left (16 \, a^{2} b^{2} c - 8 \, a b c^{3} + c^{5}\right )} x}, \frac{4 \, a b c - c^{3} - 4 \,{\left (b^{2} x^{2} + b c x + a b\right )} \sqrt{4 \, a b - c^{2}} \arctan \left (-\frac{2 \, b x + c}{\sqrt{4 \, a b - c^{2}}}\right ) + 2 \,{\left (4 \, a b^{2} - b c^{2}\right )} x}{16 \, a^{3} b^{2} - 8 \, a^{2} b c^{2} + a c^{4} +{\left (16 \, a^{2} b^{3} - 8 \, a b^{2} c^{2} + b c^{4}\right )} x^{2} +{\left (16 \, a^{2} b^{2} c - 8 \, a b c^{3} + c^{5}\right )} x}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[(4*a*b*c - c^3 + 2*(b^2*x^2 + b*c*x + a*b)*sqrt(-4*a*b + c^2)*log((2*b^2*x^2 + 2*b*c*x - 2*a*b + c^2 + sqrt(-
4*a*b + c^2)*(2*b*x + c))/(b*x^2 + c*x + a)) + 2*(4*a*b^2 - b*c^2)*x)/(16*a^3*b^2 - 8*a^2*b*c^2 + a*c^4 + (16*
a^2*b^3 - 8*a*b^2*c^2 + b*c^4)*x^2 + (16*a^2*b^2*c - 8*a*b*c^3 + c^5)*x), (4*a*b*c - c^3 - 4*(b^2*x^2 + b*c*x
+ a*b)*sqrt(4*a*b - c^2)*arctan(-(2*b*x + c)/sqrt(4*a*b - c^2)) + 2*(4*a*b^2 - b*c^2)*x)/(16*a^3*b^2 - 8*a^2*b
*c^2 + a*c^4 + (16*a^2*b^3 - 8*a*b^2*c^2 + b*c^4)*x^2 + (16*a^2*b^2*c - 8*a*b*c^3 + c^5)*x)]

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Sympy [B]  time = 1.17108, size = 265, normalized size = 3.73 \begin{align*} - 2 b \sqrt{- \frac{1}{\left (4 a b - c^{2}\right )^{3}}} \log{\left (x + \frac{- 32 a^{2} b^{3} \sqrt{- \frac{1}{\left (4 a b - c^{2}\right )^{3}}} + 16 a b^{2} c^{2} \sqrt{- \frac{1}{\left (4 a b - c^{2}\right )^{3}}} - 2 b c^{4} \sqrt{- \frac{1}{\left (4 a b - c^{2}\right )^{3}}} + 2 b c}{4 b^{2}} \right )} + 2 b \sqrt{- \frac{1}{\left (4 a b - c^{2}\right )^{3}}} \log{\left (x + \frac{32 a^{2} b^{3} \sqrt{- \frac{1}{\left (4 a b - c^{2}\right )^{3}}} - 16 a b^{2} c^{2} \sqrt{- \frac{1}{\left (4 a b - c^{2}\right )^{3}}} + 2 b c^{4} \sqrt{- \frac{1}{\left (4 a b - c^{2}\right )^{3}}} + 2 b c}{4 b^{2}} \right )} + \frac{2 b x + c}{4 a^{2} b - a c^{2} + x^{2} \left (4 a b^{2} - b c^{2}\right ) + x \left (4 a b c - c^{3}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*b*sqrt(-1/(4*a*b - c**2)**3)*log(x + (-32*a**2*b**3*sqrt(-1/(4*a*b - c**2)**3) + 16*a*b**2*c**2*sqrt(-1/(4*
a*b - c**2)**3) - 2*b*c**4*sqrt(-1/(4*a*b - c**2)**3) + 2*b*c)/(4*b**2)) + 2*b*sqrt(-1/(4*a*b - c**2)**3)*log(
x + (32*a**2*b**3*sqrt(-1/(4*a*b - c**2)**3) - 16*a*b**2*c**2*sqrt(-1/(4*a*b - c**2)**3) + 2*b*c**4*sqrt(-1/(4
*a*b - c**2)**3) + 2*b*c)/(4*b**2)) + (2*b*x + c)/(4*a**2*b - a*c**2 + x**2*(4*a*b**2 - b*c**2) + x*(4*a*b*c -
c**3))

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Giac [A]  time = 1.24021, size = 90, normalized size = 1.27 \begin{align*} \frac{4 \, b \arctan \left (\frac{2 \, b x + c}{\sqrt{4 \, a b - c^{2}}}\right )}{{\left (4 \, a b - c^{2}\right )}^{\frac{3}{2}}} + \frac{2 \, b x + c}{{\left (b x^{2} + c x + a\right )}{\left (4 \, a b - c^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4*b*arctan((2*b*x + c)/sqrt(4*a*b - c^2))/(4*a*b - c^2)^(3/2) + (2*b*x + c)/((b*x^2 + c*x + a)*(4*a*b - c^2))