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

Optimal. Leaf size=86 \[ -\frac{\left (-2 a c d+b^2 d-b c e\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c^2 \sqrt{b^2-4 a c}}-\frac{(b d-c e) \log \left (a+b x+c x^2\right )}{2 c^2}+\frac{d x}{c} \]

[Out]

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

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Rubi [A]  time = 0.183937, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ -\frac{\left (-2 a c d+b^2 d-b c e\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c^2 \sqrt{b^2-4 a c}}-\frac{(b d-c e) \log \left (a+b x+c x^2\right )}{2 c^2}+\frac{d x}{c} \]

Antiderivative was successfully verified.

[In]  Int[(d + e/x)/(c + a/x^2 + b/x),x]

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{\int d\, dx}{c} - \frac{\left (b d - c e\right ) \log{\left (a + b x + c x^{2} \right )}}{2 c^{2}} - \frac{\left (- 2 a c d + b^{2} d - b c e\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{\sqrt{- 4 a c + b^{2}}} \right )}}{c^{2} \sqrt{- 4 a c + b^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d+e/x)/(c+a/x**2+b/x),x)

[Out]

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

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Mathematica [A]  time = 0.145556, size = 86, normalized size = 1. \[ \frac{\frac{2 \left (-2 a c d+b^2 d-b c e\right ) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\sqrt{4 a c-b^2}}+(c e-b d) \log (a+x (b+c x))+2 c d x}{2 c^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e/x)/(c + a/x^2 + b/x),x]

[Out]

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

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Maple [A]  time = 0.005, size = 161, normalized size = 1.9 \[{\frac{dx}{c}}-{\frac{\ln \left ( c{x}^{2}+bx+a \right ) bd}{2\,{c}^{2}}}+{\frac{\ln \left ( c{x}^{2}+bx+a \right ) e}{2\,c}}-2\,{\frac{ad}{c\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+{\frac{{b}^{2}d}{{c}^{2}}\arctan \left ({(2\,cx+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}-{\frac{be}{c}\arctan \left ({(2\,cx+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d+e/x)/(c+a/x^2+b/x),x)

[Out]

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

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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((d + e/x)/(c + b/x + a/x^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.267912, size = 1, normalized size = 0.01 \[ \left [\frac{{\left (b c e -{\left (b^{2} - 2 \, a c\right )} d\right )} \log \left (\frac{b^{3} - 4 \, a b c + 2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} x +{\left (2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c\right )} \sqrt{b^{2} - 4 \, a c}}{c x^{2} + b x + a}\right ) +{\left (2 \, c d x -{\left (b d - c e\right )} \log \left (c x^{2} + b x + a\right )\right )} \sqrt{b^{2} - 4 \, a c}}{2 \, \sqrt{b^{2} - 4 \, a c} c^{2}}, -\frac{2 \,{\left (b c e -{\left (b^{2} - 2 \, a c\right )} d\right )} \arctan \left (-\frac{\sqrt{-b^{2} + 4 \, a c}{\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) -{\left (2 \, c d x -{\left (b d - c e\right )} \log \left (c x^{2} + b x + a\right )\right )} \sqrt{-b^{2} + 4 \, a c}}{2 \, \sqrt{-b^{2} + 4 \, a c} c^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d + e/x)/(c + b/x + a/x^2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 4.56724, size = 423, normalized size = 4.92 \[ \left (- \frac{\sqrt{- 4 a c + b^{2}} \left (2 a c d - b^{2} d + b c e\right )}{2 c^{2} \left (4 a c - b^{2}\right )} - \frac{b d - c e}{2 c^{2}}\right ) \log{\left (x + \frac{- a b d - 4 a c^{2} \left (- \frac{\sqrt{- 4 a c + b^{2}} \left (2 a c d - b^{2} d + b c e\right )}{2 c^{2} \left (4 a c - b^{2}\right )} - \frac{b d - c e}{2 c^{2}}\right ) + 2 a c e + b^{2} c \left (- \frac{\sqrt{- 4 a c + b^{2}} \left (2 a c d - b^{2} d + b c e\right )}{2 c^{2} \left (4 a c - b^{2}\right )} - \frac{b d - c e}{2 c^{2}}\right )}{2 a c d - b^{2} d + b c e} \right )} + \left (\frac{\sqrt{- 4 a c + b^{2}} \left (2 a c d - b^{2} d + b c e\right )}{2 c^{2} \left (4 a c - b^{2}\right )} - \frac{b d - c e}{2 c^{2}}\right ) \log{\left (x + \frac{- a b d - 4 a c^{2} \left (\frac{\sqrt{- 4 a c + b^{2}} \left (2 a c d - b^{2} d + b c e\right )}{2 c^{2} \left (4 a c - b^{2}\right )} - \frac{b d - c e}{2 c^{2}}\right ) + 2 a c e + b^{2} c \left (\frac{\sqrt{- 4 a c + b^{2}} \left (2 a c d - b^{2} d + b c e\right )}{2 c^{2} \left (4 a c - b^{2}\right )} - \frac{b d - c e}{2 c^{2}}\right )}{2 a c d - b^{2} d + b c e} \right )} + \frac{d x}{c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d+e/x)/(c+a/x**2+b/x),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.267372, size = 115, normalized size = 1.34 \[ \frac{d x}{c} - \frac{{\left (b d - c e\right )}{\rm ln}\left (c x^{2} + b x + a\right )}{2 \, c^{2}} + \frac{{\left (b^{2} d - 2 \, a c d - b c e\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{\sqrt{-b^{2} + 4 \, a c} c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d + e/x)/(c + b/x + a/x^2),x, algorithm="giac")

[Out]

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