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

Optimal. Leaf size=123 \[ -\frac{(2 a d-b e) \tanh ^{-1}\left (\frac{2 a x+b}{\sqrt{b^2-4 a c}}\right )}{\sqrt{b^2-4 a c} \left (a d^2-e (b d-c e)\right )}-\frac{e \log \left (a x^2+b x+c\right )}{2 \left (a d^2-b d e+c e^2\right )}+\frac{e \log (d+e x)}{a d^2-b d e+c e^2} \]

[Out]

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

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Rubi [A]  time = 0.279885, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.28 \[ -\frac{(2 a d-b e) \tanh ^{-1}\left (\frac{2 a x+b}{\sqrt{b^2-4 a c}}\right )}{\sqrt{b^2-4 a c} \left (a d^2-e (b d-c e)\right )}-\frac{e \log \left (a x^2+b x+c\right )}{2 \left (a d^2-b d e+c e^2\right )}+\frac{e \log (d+e x)}{a d^2-b d e+c e^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 60.69, size = 112, normalized size = 0.91 \[ \frac{e \log{\left (d + e x \right )}}{a d^{2} - b d e + c e^{2}} - \frac{e \log{\left (a x^{2} + b x + c \right )}}{2 \left (a d^{2} - b d e + c e^{2}\right )} - \frac{\left (2 a d - b e\right ) \operatorname{atanh}{\left (\frac{2 a x + b}{\sqrt{- 4 a c + b^{2}}} \right )}}{\sqrt{- 4 a c + b^{2}} \left (a d^{2} - b d e + c e^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.135244, size = 105, normalized size = 0.85 \[ \frac{e \sqrt{4 a c-b^2} (\log (x (a x+b)+c)-2 \log (d+e x))+(2 b e-4 a d) \tan ^{-1}\left (\frac{2 a x+b}{\sqrt{4 a c-b^2}}\right )}{2 \sqrt{4 a c-b^2} \left (e (b d-c e)-a d^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.007, size = 168, normalized size = 1.4 \[{\frac{e\ln \left ( ex+d \right ) }{a{d}^{2}-bde+{e}^{2}c}}-{\frac{e\ln \left ( a{x}^{2}+bx+c \right ) }{2\,a{d}^{2}-2\,bde+2\,{e}^{2}c}}+2\,{\frac{ad}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ) \sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,ax+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-{\frac{be}{a{d}^{2}-bde+{e}^{2}c}\arctan \left ({(2\,ax+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.54644, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (2 \, a d - b e\right )} \log \left (\frac{b^{3} - 4 \, a b c + 2 \,{\left (a b^{2} - 4 \, a^{2} c\right )} x +{\left (2 \, a^{2} x^{2} + 2 \, a b x + b^{2} - 2 \, a c\right )} \sqrt{b^{2} - 4 \, a c}}{a x^{2} + b x + c}\right ) + \sqrt{b^{2} - 4 \, a c}{\left (e \log \left (a x^{2} + b x + c\right ) - 2 \, e \log \left (e x + d\right )\right )}}{2 \,{\left (a d^{2} - b d e + c e^{2}\right )} \sqrt{b^{2} - 4 \, a c}}, \frac{2 \,{\left (2 \, a d - b e\right )} \arctan \left (-\frac{\sqrt{-b^{2} + 4 \, a c}{\left (2 \, a x + b\right )}}{b^{2} - 4 \, a c}\right ) - \sqrt{-b^{2} + 4 \, a c}{\left (e \log \left (a x^{2} + b x + c\right ) - 2 \, e \log \left (e x + d\right )\right )}}{2 \,{\left (a d^{2} - b d e + c e^{2}\right )} \sqrt{-b^{2} + 4 \, a c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.296595, size = 170, normalized size = 1.38 \[ -\frac{e{\rm ln}\left (a x^{2} + b x + c\right )}{2 \,{\left (a d^{2} - b d e + c e^{2}\right )}} + \frac{e^{2}{\rm ln}\left ({\left | x e + d \right |}\right )}{a d^{2} e - b d e^{2} + c e^{3}} + \frac{{\left (2 \, a d - b e\right )} \arctan \left (\frac{2 \, a x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{{\left (a d^{2} - b d e + c e^{2}\right )} \sqrt{-b^{2} + 4 \, a c}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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