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

Optimal. Leaf size=176 \[ \frac{\left (-a c d+b^2 d-b c e\right ) \log \left (a x^2+b x+c\right )}{2 a^2 \left (a d^2-e (b d-c e)\right )}+\frac{\left (-3 a b c d+2 a c^2 e+b^3 d-b^2 c e\right ) \tanh ^{-1}\left (\frac{2 a x+b}{\sqrt{b^2-4 a c}}\right )}{a^2 \sqrt{b^2-4 a c} \left (a d^2-e (b d-c e)\right )}-\frac{d^3 \log (d+e x)}{e^2 \left (a d^2-e (b d-c e)\right )}+\frac{x}{a e} \]

[Out]

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

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Rubi [A]  time = 0.534576, antiderivative size = 176, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261 \[ \frac{\left (-a c d+b^2 d-b c e\right ) \log \left (a x^2+b x+c\right )}{2 a^2 \left (a d^2-e (b d-c e)\right )}+\frac{\left (-3 a b c d+2 a c^2 e+b^3 d-b^2 c e\right ) \tanh ^{-1}\left (\frac{2 a x+b}{\sqrt{b^2-4 a c}}\right )}{a^2 \sqrt{b^2-4 a c} \left (a d^2-e (b d-c e)\right )}-\frac{d^3 \log (d+e x)}{e^2 \left (a d^2-e (b d-c e)\right )}+\frac{x}{a e} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{d^{3} \log{\left (d + e x \right )}}{e^{2} \left (a d^{2} - b d e + c e^{2}\right )} + \frac{\int \frac{1}{a}\, dx}{e} + \frac{\left (- a c d + b^{2} d - b c e\right ) \log{\left (a x^{2} + b x + c \right )}}{2 a^{2} \left (a d^{2} - b d e + c e^{2}\right )} + \frac{\left (- 3 a b c d + 2 a c^{2} e + b^{3} d - b^{2} c e\right ) \operatorname{atanh}{\left (\frac{2 a x + b}{\sqrt{- 4 a c + b^{2}}} \right )}}{a^{2} \sqrt{- 4 a c + b^{2}} \left (a d^{2} - b d e + c e^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.320269, size = 178, normalized size = 1.01 \[ \frac{\left (-a c d+b^2 d-b c e\right ) \log \left (a x^2+b x+c\right )}{2 a^2 \left (a d^2-b d e+c e^2\right )}+\frac{\left (-3 a b c d+2 a c^2 e+b^3 d-b^2 c e\right ) \tan ^{-1}\left (\frac{2 a x+b}{\sqrt{4 a c-b^2}}\right )}{a^2 \sqrt{4 a c-b^2} \left (-a d^2+b d e-c e^2\right )}-\frac{d^3 \log (d+e x)}{e^2 \left (a d^2-b d e+c e^2\right )}+\frac{x}{a e} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.011, size = 388, normalized size = 2.2 \[{\frac{x}{ae}}-{\frac{{d}^{3}\ln \left ( ex+d \right ) }{{e}^{2} \left ( a{d}^{2}-bde+{e}^{2}c \right ) }}-{\frac{\ln \left ( a{x}^{2}+bx+c \right ) cd}{ \left ( 2\,a{d}^{2}-2\,bde+2\,{e}^{2}c \right ) a}}+{\frac{\ln \left ( a{x}^{2}+bx+c \right ){b}^{2}d}{ \left ( 2\,a{d}^{2}-2\,bde+2\,{e}^{2}c \right ){a}^{2}}}-{\frac{\ln \left ( a{x}^{2}+bx+c \right ) bce}{ \left ( 2\,a{d}^{2}-2\,bde+2\,{e}^{2}c \right ){a}^{2}}}+3\,{\frac{bcd}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ) a\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,ax+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-2\,{\frac{{c}^{2}e}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ) a\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,ax+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-{\frac{{b}^{3}d}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ){a}^{2}}\arctan \left ({(2\,ax+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}+{\frac{{b}^{2}ce}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ){a}^{2}}\arctan \left ({(2\,ax+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(a+c/x^2+b/x)/(e*x+d),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 5.41146, size = 1, normalized size = 0.01 \[ \left [\frac{{\left ({\left (b^{3} - 3 \, a b c\right )} d e^{2} -{\left (b^{2} c - 2 \, a c^{2}\right )} e^{3}\right )} \log \left (\frac{b^{3} - 4 \, a b c + 2 \,{\left (a b^{2} - 4 \, a^{2} c\right )} x +{\left (2 \, a^{2} x^{2} + 2 \, a b x + b^{2} - 2 \, a c\right )} \sqrt{b^{2} - 4 \, a c}}{a x^{2} + b x + c}\right ) -{\left (2 \, a^{2} d^{3} \log \left (e x + d\right ) - 2 \,{\left (a^{2} d^{2} e - a b d e^{2} + a c e^{3}\right )} x +{\left (b c e^{3} -{\left (b^{2} - a c\right )} d e^{2}\right )} \log \left (a x^{2} + b x + c\right )\right )} \sqrt{b^{2} - 4 \, a c}}{2 \,{\left (a^{3} d^{2} e^{2} - a^{2} b d e^{3} + a^{2} c e^{4}\right )} \sqrt{b^{2} - 4 \, a c}}, -\frac{2 \,{\left ({\left (b^{3} - 3 \, a b c\right )} d e^{2} -{\left (b^{2} c - 2 \, a c^{2}\right )} e^{3}\right )} \arctan \left (-\frac{\sqrt{-b^{2} + 4 \, a c}{\left (2 \, a x + b\right )}}{b^{2} - 4 \, a c}\right ) +{\left (2 \, a^{2} d^{3} \log \left (e x + d\right ) - 2 \,{\left (a^{2} d^{2} e - a b d e^{2} + a c e^{3}\right )} x +{\left (b c e^{3} -{\left (b^{2} - a c\right )} d e^{2}\right )} \log \left (a x^{2} + b x + c\right )\right )} \sqrt{-b^{2} + 4 \, a c}}{2 \,{\left (a^{3} d^{2} e^{2} - a^{2} b d e^{3} + a^{2} c e^{4}\right )} \sqrt{-b^{2} + 4 \, a c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.31354, size = 250, normalized size = 1.42 \[ -\frac{d^{3}{\rm ln}\left ({\left | x e + d \right |}\right )}{a d^{2} e^{2} - b d e^{3} + c e^{4}} + \frac{x e^{\left (-1\right )}}{a} + \frac{{\left (b^{2} d - a c d - b c e\right )}{\rm ln}\left (a x^{2} + b x + c\right )}{2 \,{\left (a^{3} d^{2} - a^{2} b d e + a^{2} c e^{2}\right )}} - \frac{{\left (b^{3} d - 3 \, a b c d - b^{2} c e + 2 \, a c^{2} e\right )} \arctan \left (\frac{2 \, a x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{{\left (a^{3} d^{2} - a^{2} b d e + a^{2} c e^{2}\right )} \sqrt{-b^{2} + 4 \, a c}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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