3.67 \(\int \frac{1}{\left (a+\frac{c}{x^2}+\frac{b}{x}\right ) x^3 (d+e x)} \, dx\)

Optimal. Leaf size=158 \[ \frac{\left (a b d+2 a c e+b^2 (-e)\right ) \tanh ^{-1}\left (\frac{2 a x+b}{\sqrt{b^2-4 a c}}\right )}{c \sqrt{b^2-4 a c} \left (a d^2-e (b d-c e)\right )}-\frac{e^2 \log (d+e x)}{d \left (a d^2-b d e+c e^2\right )}-\frac{(a d-b e) \log \left (a x^2+b x+c\right )}{2 c \left (a d^2-e (b d-c e)\right )}+\frac{\log (x)}{c d} \]

[Out]

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

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Rubi [A]  time = 0.552871, antiderivative size = 159, normalized size of antiderivative = 1.01, number of steps used = 7, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24 \[ \frac{\left (a b d+2 a c e+b^2 (-e)\right ) \tanh ^{-1}\left (\frac{2 a x+b}{\sqrt{b^2-4 a c}}\right )}{c \sqrt{b^2-4 a c} \left (a d^2-e (b d-c e)\right )}-\frac{e^2 \log (d+e x)}{d \left (a d^2-e (b d-c e)\right )}-\frac{(a d-b e) \log \left (a x^2+b x+c\right )}{2 c \left (a d^2-e (b d-c e)\right )}+\frac{\log (x)}{c d} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 99.8035, size = 136, normalized size = 0.86 \[ - \frac{e^{2} \log{\left (d + e x \right )}}{d \left (a d^{2} - b d e + c e^{2}\right )} - \frac{\left (a d - b e\right ) \log{\left (a x^{2} + b x + c \right )}}{2 c \left (a d^{2} - b d e + c e^{2}\right )} - \frac{\left (- a b d - 2 a c e + b^{2} e\right ) \operatorname{atanh}{\left (\frac{2 a x + b}{\sqrt{- 4 a c + b^{2}}} \right )}}{c \sqrt{- 4 a c + b^{2}} \left (a d^{2} - b d e + c e^{2}\right )} + \frac{\log{\left (x \right )}}{c d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.34048, size = 152, normalized size = 0.96 \[ -\frac{\sqrt{4 a c-b^2} \left (-2 \log (x) \left (a d^2+e (c e-b d)\right )+d (a d-b e) \log (x (a x+b)+c)+2 c e^2 \log (d+e x)\right )+2 d \left (a b d+2 a c e+b^2 (-e)\right ) \tan ^{-1}\left (\frac{2 a x+b}{\sqrt{4 a c-b^2}}\right )}{2 c d \sqrt{4 a c-b^2} \left (a d^2+e (c e-b d)\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.011, size = 285, normalized size = 1.8 \[{\frac{\ln \left ( x \right ) }{cd}}-{\frac{{e}^{2}\ln \left ( ex+d \right ) }{d \left ( a{d}^{2}-bde+{e}^{2}c \right ) }}-{\frac{a\ln \left ( a{x}^{2}+bx+c \right ) d}{ \left ( 2\,a{d}^{2}-2\,bde+2\,{e}^{2}c \right ) c}}+{\frac{\ln \left ( a{x}^{2}+bx+c \right ) be}{ \left ( 2\,a{d}^{2}-2\,bde+2\,{e}^{2}c \right ) c}}-{\frac{abd}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ) c}\arctan \left ({(2\,ax+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}-2\,{\frac{ae}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ) \sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,ax+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+{\frac{{b}^{2}e}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ) c}\arctan \left ({(2\,ax+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+c/x^2+b/x)/x^3/(e*x+d),x)

[Out]

ln(x)/c/d-e^2*ln(e*x+d)/d/(a*d^2-b*d*e+c*e^2)-1/2/(a*d^2-b*d*e+c*e^2)/c*a*ln(a*x
^2+b*x+c)*d+1/2/(a*d^2-b*d*e+c*e^2)/c*ln(a*x^2+b*x+c)*b*e-1/(a*d^2-b*d*e+c*e^2)/
c/(4*a*c-b^2)^(1/2)*arctan((2*a*x+b)/(4*a*c-b^2)^(1/2))*a*b*d-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*e+1/(a*d^2-b*d*e+c*e
^2)/c/(4*a*c-b^2)^(1/2)*arctan((2*a*x+b)/(4*a*c-b^2)^(1/2))*b^2*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^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [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/((e*x + d)*(a + b/x + c/x^2)*x^3),x, algorithm="fricas")

[Out]

Timed out

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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**3/(e*x+d),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.293829, size = 221, normalized size = 1.4 \[ -\frac{{\left (a d - b e\right )}{\rm ln}\left (a x^{2} + b x + c\right )}{2 \,{\left (a c d^{2} - b c d e + c^{2} e^{2}\right )}} - \frac{e^{3}{\rm ln}\left ({\left | x e + d \right |}\right )}{a d^{3} e - b d^{2} e^{2} + c d e^{3}} - \frac{{\left (a b d - b^{2} e + 2 \, a c e\right )} \arctan \left (\frac{2 \, a x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{{\left (a c d^{2} - b c d e + c^{2} e^{2}\right )} \sqrt{-b^{2} + 4 \, a c}} + \frac{{\rm ln}\left ({\left | x \right |}\right )}{c d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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