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

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Rubi [A]  time = 0.08, antiderivative size = 57, normalized size of antiderivative = 1.00, 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 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])

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 1981

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

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*}

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Mathematica [A]  time = 0.03, 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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fricas [A]  time = 0.63, size = 240, normalized size = 4.21 \[ \left [\frac {\log \left (\frac {2 \, c^{2} e^{4} x^{2} + 4 \, b c d e + 2 \, {\left (c^{2} d^{2} - a c\right )} e^{2} + b^{2} + 2 \, {\left (2 \, c^{2} d e^{3} + b c e^{2}\right )} x - \sqrt {4 \, b c d e - 4 \, a c e^{2} + b^{2}} {\left (2 \, c e^{2} x + 2 \, c d e + b\right )}}{c e^{2} x^{2} + c d^{2} + {\left (2 \, c d e + b\right )} x + a}\right )}{\sqrt {4 \, b c d e - 4 \, a c e^{2} + b^{2}}}, -\frac {2 \, \sqrt {-4 \, b c d e + 4 \, a c e^{2} - b^{2}} \arctan \left (-\frac {\sqrt {-4 \, b c d e + 4 \, a c e^{2} - b^{2}} {\left (2 \, c e^{2} x + 2 \, c d e + b\right )}}{4 \, b c d e - 4 \, a c e^{2} + b^{2}}\right )}{4 \, b c d e - 4 \, a c e^{2} + b^{2}}\right ] \]

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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giac [A]  time = 0.36, size = 60, normalized size = 1.05 \[ \frac {2 \, \arctan \left (\frac {2 \, c x e^{2} + 2 \, c d e + b}{\sqrt {-4 \, b c d e + 4 \, a c e^{2} - b^{2}}}\right )}{\sqrt {-4 \, b c d e + 4 \, a c e^{2} - b^{2}}} \]

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)

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maple [A]  time = 0.00, size = 61, normalized size = 1.07 \[ \frac {2 \arctan \left (\frac {2 c \,e^{2} x +2 c d e +b}{\sqrt {4 a c \,e^{2}-4 b c d e -b^{2}}}\right )}{\sqrt {4 a c \,e^{2}-4 b c d e -b^{2}}} \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

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 >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c*e^2>0)', see `assume?` f
or more details)Is 4*a*c*e^2    -4*b*c*d*e-b^2    positive or negative?

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mupad [B]  time = 2.23, size = 82, normalized size = 1.44 \[ \frac {2\,\mathrm {atan}\left (\frac {b+2\,c\,d\,e}{\sqrt {-b^2-4\,c\,d\,b\,e+4\,a\,c\,e^2}}+\frac {2\,c\,e^2\,x}{\sqrt {-b^2-4\,c\,d\,b\,e+4\,a\,c\,e^2}}\right )}{\sqrt {-b^2-4\,c\,d\,b\,e+4\,a\,c\,e^2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + c*(d + e*x)^2 + b*x),x)

[Out]

(2*atan((b + 2*c*d*e)/(4*a*c*e^2 - b^2 - 4*b*c*d*e)^(1/2) + (2*c*e^2*x)/(4*a*c*e^2 - b^2 - 4*b*c*d*e)^(1/2)))/
(4*a*c*e^2 - b^2 - 4*b*c*d*e)^(1/2)

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sympy [B]  time = 0.35, size = 294, normalized size = 5.16 \[ - \sqrt {- \frac {1}{4 a c e^{2} - b^{2} - 4 b c d e}} \log {\left (x + \frac {- 4 a c e^{2} \sqrt {- \frac {1}{4 a c e^{2} - b^{2} - 4 b c d e}} + b^{2} \sqrt {- \frac {1}{4 a c e^{2} - b^{2} - 4 b c d e}} + 4 b c d e \sqrt {- \frac {1}{4 a c e^{2} - b^{2} - 4 b c d e}} + b + 2 c d e}{2 c e^{2}} \right )} + \sqrt {- \frac {1}{4 a c e^{2} - b^{2} - 4 b c d e}} \log {\left (x + \frac {4 a c e^{2} \sqrt {- \frac {1}{4 a c e^{2} - b^{2} - 4 b c d e}} - b^{2} \sqrt {- \frac {1}{4 a c e^{2} - b^{2} - 4 b c d e}} - 4 b c d e \sqrt {- \frac {1}{4 a c e^{2} - b^{2} - 4 b c d e}} + b + 2 c d e}{2 c e^{2}} \right )} \]

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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