3.68 \(\int \frac{1}{\left (a+\frac{c}{x^2}+\frac{b}{x}\right ) x^4 (d+e x)} \, dx\)

Optimal. Leaf size=193 \[ \frac{\left (2 a^2 c d-a b (b d+3 c e)+b^3 e\right ) \tanh ^{-1}\left (\frac{2 a x+b}{\sqrt{b^2-4 a c}}\right )}{c^2 \sqrt{b^2-4 a c} \left (a d^2-e (b d-c e)\right )}+\frac{\left (a b d+a c e+b^2 (-e)\right ) \log \left (a x^2+b x+c\right )}{2 c^2 \left (a d^2-e (b d-c e)\right )}+\frac{e^3 \log (d+e x)}{d^2 \left (a d^2-e (b d-c e)\right )}-\frac{\log (x) (b d+c e)}{c^2 d^2}-\frac{1}{c d x} \]

[Out]

-(1/(c*d*x)) + ((2*a^2*c*d + b^3*e - a*b*(b*d + 3*c*e))*ArcTanh[(b + 2*a*x)/Sqrt
[b^2 - 4*a*c]])/(c^2*Sqrt[b^2 - 4*a*c]*(a*d^2 - e*(b*d - c*e))) - ((b*d + c*e)*L
og[x])/(c^2*d^2) + (e^3*Log[d + e*x])/(d^2*(a*d^2 - e*(b*d - c*e))) + ((a*b*d -
b^2*e + a*c*e)*Log[c + b*x + a*x^2])/(2*c^2*(a*d^2 - e*(b*d - c*e)))

_______________________________________________________________________________________

Rubi [A]  time = 0.713638, antiderivative size = 193, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24 \[ \frac{\left (2 a^2 c d-a b (b d+3 c e)+b^3 e\right ) \tanh ^{-1}\left (\frac{2 a x+b}{\sqrt{b^2-4 a c}}\right )}{c^2 \sqrt{b^2-4 a c} \left (a d^2-e (b d-c e)\right )}+\frac{\left (a b d+a c e+b^2 (-e)\right ) \log \left (a x^2+b x+c\right )}{2 c^2 \left (a d^2-e (b d-c e)\right )}+\frac{e^3 \log (d+e x)}{d^2 \left (a d^2-e (b d-c e)\right )}-\frac{\log (x) (b d+c e)}{c^2 d^2}-\frac{1}{c d x} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + c/x^2 + b/x)*x^4*(d + e*x)),x]

[Out]

-(1/(c*d*x)) + ((2*a^2*c*d + b^3*e - a*b*(b*d + 3*c*e))*ArcTanh[(b + 2*a*x)/Sqrt
[b^2 - 4*a*c]])/(c^2*Sqrt[b^2 - 4*a*c]*(a*d^2 - e*(b*d - c*e))) - ((b*d + c*e)*L
og[x])/(c^2*d^2) + (e^3*Log[d + e*x])/(d^2*(a*d^2 - e*(b*d - c*e))) + ((a*b*d -
b^2*e + a*c*e)*Log[c + b*x + a*x^2])/(2*c^2*(a*d^2 - e*(b*d - c*e)))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 137.446, size = 178, normalized size = 0.92 \[ \frac{e^{3} \log{\left (d + e x \right )}}{d^{2} \left (a d^{2} - b d e + c e^{2}\right )} - \frac{1}{c d x} - \frac{\left (- a b d - a c e + b^{2} e\right ) \log{\left (a x^{2} + b x + c \right )}}{2 c^{2} \left (a d^{2} - b d e + c e^{2}\right )} + \frac{\left (2 a^{2} c d - a b^{2} d - 3 a b c e + b^{3} e\right ) \operatorname{atanh}{\left (\frac{2 a x + b}{\sqrt{- 4 a c + b^{2}}} \right )}}{c^{2} \sqrt{- 4 a c + b^{2}} \left (a d^{2} - b d e + c e^{2}\right )} - \frac{\left (b d + c e\right ) \log{\left (x \right )}}{c^{2} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+c/x**2+b/x)/x**4/(e*x+d),x)

[Out]

e**3*log(d + e*x)/(d**2*(a*d**2 - b*d*e + c*e**2)) - 1/(c*d*x) - (-a*b*d - a*c*e
+ b**2*e)*log(a*x**2 + b*x + c)/(2*c**2*(a*d**2 - b*d*e + c*e**2)) + (2*a**2*c*
d - a*b**2*d - 3*a*b*c*e + b**3*e)*atanh((2*a*x + b)/sqrt(-4*a*c + b**2))/(c**2*
sqrt(-4*a*c + b**2)*(a*d**2 - b*d*e + c*e**2)) - (b*d + c*e)*log(x)/(c**2*d**2)

_______________________________________________________________________________________

Mathematica [A]  time = 0.286311, size = 194, normalized size = 1.01 \[ \frac{\left (2 a^2 c d-a b (b d+3 c e)+b^3 e\right ) \tan ^{-1}\left (\frac{2 a x+b}{\sqrt{4 a c-b^2}}\right )}{c^2 \sqrt{4 a c-b^2} \left (e (b d-c e)-a d^2\right )}+\frac{\left (a b d+a c e+b^2 (-e)\right ) \log (x (a x+b)+c)}{2 c^2 \left (a d^2+e (c e-b d)\right )}+\frac{e^3 \log (d+e x)}{a d^4+d^2 e (c e-b d)}-\frac{\log (x) (b d+c e)}{c^2 d^2}-\frac{1}{c d x} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + c/x^2 + b/x)*x^4*(d + e*x)),x]

[Out]

-(1/(c*d*x)) + ((2*a^2*c*d + b^3*e - a*b*(b*d + 3*c*e))*ArcTan[(b + 2*a*x)/Sqrt[
-b^2 + 4*a*c]])/(c^2*Sqrt[-b^2 + 4*a*c]*(-(a*d^2) + e*(b*d - c*e))) - ((b*d + c*
e)*Log[x])/(c^2*d^2) + (e^3*Log[d + e*x])/(a*d^4 + d^2*e*(-(b*d) + c*e)) + ((a*b
*d - b^2*e + a*c*e)*Log[c + x*(b + a*x)])/(2*c^2*(a*d^2 + e*(-(b*d) + c*e)))

_______________________________________________________________________________________

Maple [B]  time = 0.017, size = 412, normalized size = 2.1 \[ -{\frac{1}{cdx}}-{\frac{\ln \left ( x \right ) b}{{c}^{2}d}}-{\frac{\ln \left ( x \right ) e}{c{d}^{2}}}+{\frac{{e}^{3}\ln \left ( ex+d \right ) }{{d}^{2} \left ( a{d}^{2}-bde+{e}^{2}c \right ) }}+{\frac{a\ln \left ( a{x}^{2}+bx+c \right ) bd}{ \left ( 2\,a{d}^{2}-2\,bde+2\,{e}^{2}c \right ){c}^{2}}}+{\frac{a\ln \left ( a{x}^{2}+bx+c \right ) e}{ \left ( 2\,a{d}^{2}-2\,bde+2\,{e}^{2}c \right ) c}}-{\frac{\ln \left ( a{x}^{2}+bx+c \right ){b}^{2}e}{ \left ( 2\,a{d}^{2}-2\,bde+2\,{e}^{2}c \right ){c}^{2}}}-2\,{\frac{{a}^{2}d}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ) c\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,ax+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+{\frac{a{b}^{2}d}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ){c}^{2}}\arctan \left ({(2\,ax+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}+3\,{\frac{abe}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ) c\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,ax+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-{\frac{{b}^{3}e}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ){c}^{2}}\arctan \left ({(2\,ax+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+c/x^2+b/x)/x^4/(e*x+d),x)

[Out]

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

_______________________________________________________________________________________

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^4),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

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^4),x, algorithm="fricas")

[Out]

Timed out

_______________________________________________________________________________________

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

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.30186, size = 284, normalized size = 1.47 \[ \frac{{\left (a b d - b^{2} e + a c e\right )}{\rm ln}\left (a x^{2} + b x + c\right )}{2 \,{\left (a c^{2} d^{2} - b c^{2} d e + c^{3} e^{2}\right )}} + \frac{e^{4}{\rm ln}\left ({\left | x e + d \right |}\right )}{a d^{4} e - b d^{3} e^{2} + c d^{2} e^{3}} + \frac{{\left (a b^{2} d - 2 \, a^{2} c d - b^{3} e + 3 \, a b c e\right )} \arctan \left (\frac{2 \, a x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{{\left (a c^{2} d^{2} - b c^{2} d e + c^{3} e^{2}\right )} \sqrt{-b^{2} + 4 \, a c}} - \frac{{\left (b d + c e\right )}{\rm ln}\left ({\left | x \right |}\right )}{c^{2} d^{2}} - \frac{1}{c d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((e*x + d)*(a + b/x + c/x^2)*x^4),x, algorithm="giac")

[Out]

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