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

Optimal. Leaf size=149 \[ -\frac{\left (-2 a c d+b^2 d-b c e\right ) \tanh ^{-1}\left (\frac{2 a x+b}{\sqrt{b^2-4 a c}}\right )}{a \sqrt{b^2-4 a c} \left (a d^2-e (b d-c e)\right )}+\frac{d^2 \log (d+e x)}{e \left (a d^2-b d e+c e^2\right )}-\frac{(b d-c e) \log \left (a x^2+b x+c\right )}{2 a \left (a d^2-e (b d-c e)\right )} \]

[Out]

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

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Rubi [A]  time = 0.399242, antiderivative size = 149, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ -\frac{\left (-2 a c d+b^2 d-b c e\right ) \tanh ^{-1}\left (\frac{2 a x+b}{\sqrt{b^2-4 a c}}\right )}{a \sqrt{b^2-4 a c} \left (a d^2-e (b d-c e)\right )}+\frac{d^2 \log (d+e x)}{e \left (a d^2-b d e+c e^2\right )}-\frac{(b d-c e) \log \left (a x^2+b x+c\right )}{2 a \left (a d^2-e (b d-c e)\right )} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + c/x^2 + b/x)*(d + e*x)),x]

[Out]

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

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Rubi in Sympy [A]  time = 80.2854, size = 129, normalized size = 0.87 \[ \frac{d^{2} \log{\left (d + e x \right )}}{e \left (a d^{2} - b d e + c e^{2}\right )} - \frac{\left (b d - c e\right ) \log{\left (a x^{2} + b x + c \right )}}{2 a \left (a d^{2} - b d e + c e^{2}\right )} - \frac{\left (- 2 a c d + b^{2} d - b c e\right ) \operatorname{atanh}{\left (\frac{2 a x + b}{\sqrt{- 4 a c + b^{2}}} \right )}}{a \sqrt{- 4 a c + b^{2}} \left (a d^{2} - b d e + c e^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.211338, size = 132, normalized size = 0.89 \[ -\frac{\sqrt{4 a c-b^2} \left (e (b d-c e) \log (x (a x+b)+c)-2 a d^2 \log (d+e x)\right )+2 e \left (2 a c d+b^2 (-d)+b c e\right ) \tan ^{-1}\left (\frac{2 a x+b}{\sqrt{4 a c-b^2}}\right )}{2 a e \sqrt{4 a c-b^2} \left (a d^2+e (c e-b d)\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.009, size = 275, normalized size = 1.9 \[{\frac{{d}^{2}\ln \left ( ex+d \right ) }{e \left ( a{d}^{2}-bde+{e}^{2}c \right ) }}-{\frac{\ln \left ( a{x}^{2}+bx+c \right ) bd}{ \left ( 2\,a{d}^{2}-2\,bde+2\,{e}^{2}c \right ) a}}+{\frac{\ln \left ( a{x}^{2}+bx+c \right ) ce}{ \left ( 2\,a{d}^{2}-2\,bde+2\,{e}^{2}c \right ) a}}-2\,{\frac{cd}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ) \sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,ax+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+{\frac{{b}^{2}d}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ) a}\arctan \left ({(2\,ax+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}-{\frac{bce}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ) a}\arctan \left ({(2\,ax+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+c/x^2+b/x)/(e*x+d),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.62713, size = 1, normalized size = 0.01 \[ \left [\frac{{\left (b c e^{2} -{\left (b^{2} - 2 \, a c\right )} d e\right )} \log \left (\frac{b^{3} - 4 \, a b c + 2 \,{\left (a b^{2} - 4 \, a^{2} c\right )} x +{\left (2 \, a^{2} x^{2} + 2 \, a b x + b^{2} - 2 \, a c\right )} \sqrt{b^{2} - 4 \, a c}}{a x^{2} + b x + c}\right ) +{\left (2 \, a d^{2} \log \left (e x + d\right ) -{\left (b d e - c e^{2}\right )} \log \left (a x^{2} + b x + c\right )\right )} \sqrt{b^{2} - 4 \, a c}}{2 \,{\left (a^{2} d^{2} e - a b d e^{2} + a c e^{3}\right )} \sqrt{b^{2} - 4 \, a c}}, -\frac{2 \,{\left (b c e^{2} -{\left (b^{2} - 2 \, a c\right )} d e\right )} \arctan \left (-\frac{\sqrt{-b^{2} + 4 \, a c}{\left (2 \, a x + b\right )}}{b^{2} - 4 \, a c}\right ) -{\left (2 \, a d^{2} \log \left (e x + d\right ) -{\left (b d e - c e^{2}\right )} \log \left (a x^{2} + b x + c\right )\right )} \sqrt{-b^{2} + 4 \, a c}}{2 \,{\left (a^{2} d^{2} e - a b d e^{2} + a c e^{3}\right )} \sqrt{-b^{2} + 4 \, a c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.300595, size = 201, normalized size = 1.35 \[ \frac{d^{2}{\rm ln}\left ({\left | x e + d \right |}\right )}{a d^{2} e - b d e^{2} + c e^{3}} - \frac{{\left (b d - c e\right )}{\rm ln}\left (a x^{2} + b x + c\right )}{2 \,{\left (a^{2} d^{2} - a b d e + a c e^{2}\right )}} + \frac{{\left (b^{2} d - 2 \, a c d - b c e\right )} \arctan \left (\frac{2 \, a x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{{\left (a^{2} d^{2} - a b d e + a c e^{2}\right )} \sqrt{-b^{2} + 4 \, a c}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d^2*ln(abs(x*e + d))/(a*d^2*e - b*d*e^2 + c*e^3) - 1/2*(b*d - c*e)*ln(a*x^2 + b*
x + c)/(a^2*d^2 - a*b*d*e + a*c*e^2) + (b^2*d - 2*a*c*d - b*c*e)*arctan((2*a*x +
b)/sqrt(-b^2 + 4*a*c))/((a^2*d^2 - a*b*d*e + a*c*e^2)*sqrt(-b^2 + 4*a*c))