3.26 \(\int \frac{1}{b x+c x^2+d x^3} \, dx\)

Optimal. Leaf size=62 \[ \frac{c \tanh ^{-1}\left (\frac{c+2 d x}{\sqrt{c^2-4 b d}}\right )}{b \sqrt{c^2-4 b d}}-\frac{\log \left (b+c x+d x^2\right )}{2 b}+\frac{\log (x)}{b} \]

[Out]

(c*ArcTanh[(c + 2*d*x)/Sqrt[c^2 - 4*b*d]])/(b*Sqrt[c^2 - 4*b*d]) + Log[x]/b - Lo
g[b + c*x + d*x^2]/(2*b)

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Rubi [A]  time = 0.112362, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.438 \[ \frac{c \tanh ^{-1}\left (\frac{c+2 d x}{\sqrt{c^2-4 b d}}\right )}{b \sqrt{c^2-4 b d}}-\frac{\log \left (b+c x+d x^2\right )}{2 b}+\frac{\log (x)}{b} \]

Antiderivative was successfully verified.

[In]  Int[(b*x + c*x^2 + d*x^3)^(-1),x]

[Out]

(c*ArcTanh[(c + 2*d*x)/Sqrt[c^2 - 4*b*d]])/(b*Sqrt[c^2 - 4*b*d]) + Log[x]/b - Lo
g[b + c*x + d*x^2]/(2*b)

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Rubi in Sympy [A]  time = 20.222, size = 54, normalized size = 0.87 \[ \frac{c \operatorname{atanh}{\left (\frac{c + 2 d x}{\sqrt{- 4 b d + c^{2}}} \right )}}{b \sqrt{- 4 b d + c^{2}}} + \frac{\log{\left (x \right )}}{b} - \frac{\log{\left (b + c x + d x^{2} \right )}}{2 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(d*x**3+c*x**2+b*x),x)

[Out]

c*atanh((c + 2*d*x)/sqrt(-4*b*d + c**2))/(b*sqrt(-4*b*d + c**2)) + log(x)/b - lo
g(b + c*x + d*x**2)/(2*b)

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Mathematica [A]  time = 0.131815, size = 61, normalized size = 0.98 \[ -\frac{\frac{2 c \tan ^{-1}\left (\frac{c+2 d x}{\sqrt{4 b d-c^2}}\right )}{\sqrt{4 b d-c^2}}+\log (b+x (c+d x))-2 \log (x)}{2 b} \]

Antiderivative was successfully verified.

[In]  Integrate[(b*x + c*x^2 + d*x^3)^(-1),x]

[Out]

-((2*c*ArcTan[(c + 2*d*x)/Sqrt[-c^2 + 4*b*d]])/Sqrt[-c^2 + 4*b*d] - 2*Log[x] + L
og[b + x*(c + d*x)])/(2*b)

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Maple [A]  time = 0.013, size = 62, normalized size = 1. \[ -{\frac{\ln \left ( d{x}^{2}+cx+b \right ) }{2\,b}}-{\frac{c}{b}\arctan \left ({(2\,dx+c){\frac{1}{\sqrt{4\,bd-{c}^{2}}}}} \right ){\frac{1}{\sqrt{4\,bd-{c}^{2}}}}}+{\frac{\ln \left ( x \right ) }{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(d*x^3+c*x^2+b*x),x)

[Out]

-1/2*ln(d*x^2+c*x+b)/b-1/b*c/(4*b*d-c^2)^(1/2)*arctan((2*d*x+c)/(4*b*d-c^2)^(1/2
))+ln(x)/b

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(d*x^3 + c*x^2 + b*x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.304896, size = 1, normalized size = 0.02 \[ \left [\frac{c \log \left (\frac{c^{3} - 4 \, b c d + 2 \,{\left (c^{2} d - 4 \, b d^{2}\right )} x +{\left (2 \, d^{2} x^{2} + 2 \, c d x + c^{2} - 2 \, b d\right )} \sqrt{c^{2} - 4 \, b d}}{d x^{2} + c x + b}\right ) - \sqrt{c^{2} - 4 \, b d}{\left (\log \left (d x^{2} + c x + b\right ) - 2 \, \log \left (x\right )\right )}}{2 \, \sqrt{c^{2} - 4 \, b d} b}, -\frac{2 \, c \arctan \left (-\frac{\sqrt{-c^{2} + 4 \, b d}{\left (2 \, d x + c\right )}}{c^{2} - 4 \, b d}\right ) + \sqrt{-c^{2} + 4 \, b d}{\left (\log \left (d x^{2} + c x + b\right ) - 2 \, \log \left (x\right )\right )}}{2 \, \sqrt{-c^{2} + 4 \, b d} b}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(d*x^3 + c*x^2 + b*x),x, algorithm="fricas")

[Out]

[1/2*(c*log((c^3 - 4*b*c*d + 2*(c^2*d - 4*b*d^2)*x + (2*d^2*x^2 + 2*c*d*x + c^2
- 2*b*d)*sqrt(c^2 - 4*b*d))/(d*x^2 + c*x + b)) - sqrt(c^2 - 4*b*d)*(log(d*x^2 +
c*x + b) - 2*log(x)))/(sqrt(c^2 - 4*b*d)*b), -1/2*(2*c*arctan(-sqrt(-c^2 + 4*b*d
)*(2*d*x + c)/(c^2 - 4*b*d)) + sqrt(-c^2 + 4*b*d)*(log(d*x^2 + c*x + b) - 2*log(
x)))/(sqrt(-c^2 + 4*b*d)*b)]

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Sympy [A]  time = 6.67278, size = 564, normalized size = 9.1 \[ \left (- \frac{c \sqrt{- 4 b d + c^{2}}}{2 b \left (4 b d - c^{2}\right )} - \frac{1}{2 b}\right ) \log{\left (x + \frac{24 b^{4} d^{2} \left (- \frac{c \sqrt{- 4 b d + c^{2}}}{2 b \left (4 b d - c^{2}\right )} - \frac{1}{2 b}\right )^{2} - 14 b^{3} c^{2} d \left (- \frac{c \sqrt{- 4 b d + c^{2}}}{2 b \left (4 b d - c^{2}\right )} - \frac{1}{2 b}\right )^{2} - 12 b^{3} d^{2} \left (- \frac{c \sqrt{- 4 b d + c^{2}}}{2 b \left (4 b d - c^{2}\right )} - \frac{1}{2 b}\right ) + 2 b^{2} c^{4} \left (- \frac{c \sqrt{- 4 b d + c^{2}}}{2 b \left (4 b d - c^{2}\right )} - \frac{1}{2 b}\right )^{2} + 3 b^{2} c^{2} d \left (- \frac{c \sqrt{- 4 b d + c^{2}}}{2 b \left (4 b d - c^{2}\right )} - \frac{1}{2 b}\right ) - 12 b^{2} d^{2} + 11 b c^{2} d - 2 c^{4}}{9 b c d^{2} - 2 c^{3} d} \right )} + \left (\frac{c \sqrt{- 4 b d + c^{2}}}{2 b \left (4 b d - c^{2}\right )} - \frac{1}{2 b}\right ) \log{\left (x + \frac{24 b^{4} d^{2} \left (\frac{c \sqrt{- 4 b d + c^{2}}}{2 b \left (4 b d - c^{2}\right )} - \frac{1}{2 b}\right )^{2} - 14 b^{3} c^{2} d \left (\frac{c \sqrt{- 4 b d + c^{2}}}{2 b \left (4 b d - c^{2}\right )} - \frac{1}{2 b}\right )^{2} - 12 b^{3} d^{2} \left (\frac{c \sqrt{- 4 b d + c^{2}}}{2 b \left (4 b d - c^{2}\right )} - \frac{1}{2 b}\right ) + 2 b^{2} c^{4} \left (\frac{c \sqrt{- 4 b d + c^{2}}}{2 b \left (4 b d - c^{2}\right )} - \frac{1}{2 b}\right )^{2} + 3 b^{2} c^{2} d \left (\frac{c \sqrt{- 4 b d + c^{2}}}{2 b \left (4 b d - c^{2}\right )} - \frac{1}{2 b}\right ) - 12 b^{2} d^{2} + 11 b c^{2} d - 2 c^{4}}{9 b c d^{2} - 2 c^{3} d} \right )} + \frac{\log{\left (x \right )}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(d*x**3+c*x**2+b*x),x)

[Out]

(-c*sqrt(-4*b*d + c**2)/(2*b*(4*b*d - c**2)) - 1/(2*b))*log(x + (24*b**4*d**2*(-
c*sqrt(-4*b*d + c**2)/(2*b*(4*b*d - c**2)) - 1/(2*b))**2 - 14*b**3*c**2*d*(-c*sq
rt(-4*b*d + c**2)/(2*b*(4*b*d - c**2)) - 1/(2*b))**2 - 12*b**3*d**2*(-c*sqrt(-4*
b*d + c**2)/(2*b*(4*b*d - c**2)) - 1/(2*b)) + 2*b**2*c**4*(-c*sqrt(-4*b*d + c**2
)/(2*b*(4*b*d - c**2)) - 1/(2*b))**2 + 3*b**2*c**2*d*(-c*sqrt(-4*b*d + c**2)/(2*
b*(4*b*d - c**2)) - 1/(2*b)) - 12*b**2*d**2 + 11*b*c**2*d - 2*c**4)/(9*b*c*d**2
- 2*c**3*d)) + (c*sqrt(-4*b*d + c**2)/(2*b*(4*b*d - c**2)) - 1/(2*b))*log(x + (2
4*b**4*d**2*(c*sqrt(-4*b*d + c**2)/(2*b*(4*b*d - c**2)) - 1/(2*b))**2 - 14*b**3*
c**2*d*(c*sqrt(-4*b*d + c**2)/(2*b*(4*b*d - c**2)) - 1/(2*b))**2 - 12*b**3*d**2*
(c*sqrt(-4*b*d + c**2)/(2*b*(4*b*d - c**2)) - 1/(2*b)) + 2*b**2*c**4*(c*sqrt(-4*
b*d + c**2)/(2*b*(4*b*d - c**2)) - 1/(2*b))**2 + 3*b**2*c**2*d*(c*sqrt(-4*b*d +
c**2)/(2*b*(4*b*d - c**2)) - 1/(2*b)) - 12*b**2*d**2 + 11*b*c**2*d - 2*c**4)/(9*
b*c*d**2 - 2*c**3*d)) + log(x)/b

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GIAC/XCAS [A]  time = 0.263704, size = 84, normalized size = 1.35 \[ -\frac{c \arctan \left (\frac{2 \, d x + c}{\sqrt{-c^{2} + 4 \, b d}}\right )}{\sqrt{-c^{2} + 4 \, b d} b} - \frac{{\rm ln}\left (d x^{2} + c x + b\right )}{2 \, b} + \frac{{\rm ln}\left ({\left | x \right |}\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(d*x^3 + c*x^2 + b*x),x, algorithm="giac")

[Out]

-c*arctan((2*d*x + c)/sqrt(-c^2 + 4*b*d))/(sqrt(-c^2 + 4*b*d)*b) - 1/2*ln(d*x^2
+ c*x + b)/b + ln(abs(x))/b