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

Optimal. Leaf size=57 \[ -\frac{2 \tanh ^{-1}\left (\frac{b+2 c e (d+e x)}{\sqrt{-4 a c e^2+b^2+4 b c d e}}\right )}{\sqrt{-4 a c e^2+b^2+4 b c d e}} \]

[Out]

(-2*ArcTanh[(b + 2*c*e*(d + e*x))/Sqrt[b^2 + 4*b*c*d*e - 4*a*c*e^2]])/Sqrt[b^2 + 4*b*c*d*e - 4*a*c*e^2]

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Rubi [A]  time = 0.0815414, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {1981, 618, 206} \[ -\frac{2 \tanh ^{-1}\left (\frac{b+2 c e (d+e x)}{\sqrt{-4 a c e^2+b^2+4 b c d e}}\right )}{\sqrt{-4 a c e^2+b^2+4 b c d e}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x + c*(d + e*x)^2)^(-1),x]

[Out]

(-2*ArcTanh[(b + 2*c*e*(d + e*x))/Sqrt[b^2 + 4*b*c*d*e - 4*a*c*e^2]])/Sqrt[b^2 + 4*b*c*d*e - 4*a*c*e^2]

Rule 1981

Int[(u_)^(p_), x_Symbol] :> Int[ExpandToSum[u, x]^p, x] /; FreeQ[p, x] && QuadraticQ[u, x] &&  !QuadraticMatch
Q[u, x]

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

Mathematica [A]  time = 0.0279165, size = 61, normalized size = 1.07 \[ \frac{2 \tan ^{-1}\left (\frac{b+2 c e (d+e x)}{\sqrt{4 a c e^2-b^2-4 b c d e}}\right )}{\sqrt{4 a c e^2-b^2-4 b c d e}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x + c*(d + e*x)^2)^(-1),x]

[Out]

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

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Maple [A]  time = 0.006, size = 61, normalized size = 1.1 \begin{align*} 2\,{\frac{1}{\sqrt{4\,ac{e}^{2}-4\,bcde-{b}^{2}}}\arctan \left ({\frac{2\,c{e}^{2}x+2\,cde+b}{\sqrt{4\,ac{e}^{2}-4\,bcde-{b}^{2}}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/(4*a*c*e^2-4*b*c*d*e-b^2)^(1/2)*arctan((2*c*e^2*x+2*c*d*e+b)/(4*a*c*e^2-4*b*c*d*e-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/(a+b*x+c*(e*x+d)^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.02391, size = 547, normalized size = 9.6 \begin{align*} \left [\frac{\log \left (\frac{2 \, c^{2} e^{4} x^{2} + 4 \, b c d e + 2 \,{\left (c^{2} d^{2} - a c\right )} e^{2} + b^{2} + 2 \,{\left (2 \, c^{2} d e^{3} + b c e^{2}\right )} x - \sqrt{4 \, b c d e - 4 \, a c e^{2} + b^{2}}{\left (2 \, c e^{2} x + 2 \, c d e + b\right )}}{c e^{2} x^{2} + c d^{2} +{\left (2 \, c d e + b\right )} x + a}\right )}{\sqrt{4 \, b c d e - 4 \, a c e^{2} + b^{2}}}, -\frac{2 \, \sqrt{-4 \, b c d e + 4 \, a c e^{2} - b^{2}} \arctan \left (-\frac{\sqrt{-4 \, b c d e + 4 \, a c e^{2} - b^{2}}{\left (2 \, c e^{2} x + 2 \, c d e + b\right )}}{4 \, b c d e - 4 \, a c e^{2} + b^{2}}\right )}{4 \, b c d e - 4 \, a c e^{2} + b^{2}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 0.323344, size = 294, normalized size = 5.16 \begin{align*} - \sqrt{- \frac{1}{4 a c e^{2} - b^{2} - 4 b c d e}} \log{\left (x + \frac{- 4 a c e^{2} \sqrt{- \frac{1}{4 a c e^{2} - b^{2} - 4 b c d e}} + b^{2} \sqrt{- \frac{1}{4 a c e^{2} - b^{2} - 4 b c d e}} + 4 b c d e \sqrt{- \frac{1}{4 a c e^{2} - b^{2} - 4 b c d e}} + b + 2 c d e}{2 c e^{2}} \right )} + \sqrt{- \frac{1}{4 a c e^{2} - b^{2} - 4 b c d e}} \log{\left (x + \frac{4 a c e^{2} \sqrt{- \frac{1}{4 a c e^{2} - b^{2} - 4 b c d e}} - b^{2} \sqrt{- \frac{1}{4 a c e^{2} - b^{2} - 4 b c d e}} - 4 b c d e \sqrt{- \frac{1}{4 a c e^{2} - b^{2} - 4 b c d e}} + b + 2 c d e}{2 c e^{2}} \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(-1/(4*a*c*e**2 - b**2 - 4*b*c*d*e))*log(x + (-4*a*c*e**2*sqrt(-1/(4*a*c*e**2 - b**2 - 4*b*c*d*e)) + b**2
*sqrt(-1/(4*a*c*e**2 - b**2 - 4*b*c*d*e)) + 4*b*c*d*e*sqrt(-1/(4*a*c*e**2 - b**2 - 4*b*c*d*e)) + b + 2*c*d*e)/
(2*c*e**2)) + sqrt(-1/(4*a*c*e**2 - b**2 - 4*b*c*d*e))*log(x + (4*a*c*e**2*sqrt(-1/(4*a*c*e**2 - b**2 - 4*b*c*
d*e)) - b**2*sqrt(-1/(4*a*c*e**2 - b**2 - 4*b*c*d*e)) - 4*b*c*d*e*sqrt(-1/(4*a*c*e**2 - b**2 - 4*b*c*d*e)) + b
+ 2*c*d*e)/(2*c*e**2))

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Giac [A]  time = 1.19572, size = 81, normalized size = 1.42 \begin{align*} \frac{2 \, \arctan \left (\frac{2 \, c x e^{2} + 2 \, c d e + b}{\sqrt{-4 \, b c d e + 4 \, a c e^{2} - b^{2}}}\right )}{\sqrt{-4 \, b c d e + 4 \, a c e^{2} - b^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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