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

Optimal. Leaf size=183 \[ \frac{\left (a d (b d-4 c e)+b c e^2\right ) \tanh ^{-1}\left (\frac{2 a x+b}{\sqrt{b^2-4 a c}}\right )}{\sqrt{b^2-4 a c} \left (a d^2-e (b d-c e)\right )^2}+\frac{\left (a d^2-c e^2\right ) \log \left (a x^2+b x+c\right )}{2 \left (a d^2-e (b d-c e)\right )^2}+\frac{d}{(d+e x) \left (a d^2-b d e+c e^2\right )}-\frac{\left (a d^2-c e^2\right ) \log (d+e x)}{\left (a d^2-e (b d-c e)\right )^2} \]

[Out]

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

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Rubi [A]  time = 0.507696, antiderivative size = 183, 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 \[ \frac{\left (a d (b d-4 c e)+b c e^2\right ) \tanh ^{-1}\left (\frac{2 a x+b}{\sqrt{b^2-4 a c}}\right )}{\sqrt{b^2-4 a c} \left (a d^2-e (b d-c e)\right )^2}+\frac{\left (a d^2-c e^2\right ) \log \left (a x^2+b x+c\right )}{2 \left (a d^2-e (b d-c e)\right )^2}+\frac{d}{(d+e x) \left (a d^2-b d e+c e^2\right )}-\frac{\left (a d^2-c e^2\right ) \log (d+e x)}{\left (a d^2-e (b d-c e)\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 103.401, size = 167, normalized size = 0.91 \[ \frac{d}{\left (d + e x\right ) \left (a d^{2} - b d e + c e^{2}\right )} - \frac{\left (a d^{2} - c e^{2}\right ) \log{\left (d + e x \right )}}{\left (a d^{2} - b d e + c e^{2}\right )^{2}} + \frac{\left (a d^{2} - c e^{2}\right ) \log{\left (a x^{2} + b x + c \right )}}{2 \left (a d^{2} - b d e + c e^{2}\right )^{2}} + \frac{\left (a b d^{2} - 4 a c d e + b c e^{2}\right ) \operatorname{atanh}{\left (\frac{2 a x + b}{\sqrt{- 4 a c + b^{2}}} \right )}}{\sqrt{- 4 a c + b^{2}} \left (a d^{2} - b d e + c e^{2}\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.440807, size = 148, normalized size = 0.81 \[ \frac{-\frac{2 \left (a d (b d-4 c e)+b c e^2\right ) \tan ^{-1}\left (\frac{2 a x+b}{\sqrt{4 a c-b^2}}\right )}{\sqrt{4 a c-b^2}}+\left (a d^2-c e^2\right ) \log (x (a x+b)+c)+\frac{2 d \left (a d^2+e (c e-b d)\right )}{d+e x}+\left (2 c e^2-2 a d^2\right ) \log (d+e x)}{2 \left (a d^2+e (c e-b d)\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.012, size = 328, normalized size = 1.8 \[{\frac{d}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ) \left ( ex+d \right ) }}-{\frac{\ln \left ( ex+d \right ) a{d}^{2}}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ) ^{2}}}+{\frac{\ln \left ( ex+d \right ){e}^{2}c}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ) ^{2}}}+{\frac{a\ln \left ( a{x}^{2}+bx+c \right ){d}^{2}}{2\, \left ( a{d}^{2}-bde+{e}^{2}c \right ) ^{2}}}-{\frac{\ln \left ( a{x}^{2}+bx+c \right ) c{e}^{2}}{2\, \left ( a{d}^{2}-bde+{e}^{2}c \right ) ^{2}}}-{\frac{ab{d}^{2}}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ) ^{2}}\arctan \left ({(2\,ax+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}+4\,{\frac{acde}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ) ^{2}\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,ax+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-{\frac{bc{e}^{2}}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ) ^{2}}\arctan \left ({(2\,ax+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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)^2*(a + b/x + c/x^2)*x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 5.22996, size = 1, normalized size = 0.01 \[ \left [\frac{{\left (a b d^{3} - 4 \, a c d^{2} e + b c d e^{2} +{\left (a b d^{2} e - 4 \, a c d e^{2} + b c e^{3}\right )} x\right )} \log \left (\frac{b^{3} - 4 \, a b c + 2 \,{\left (a b^{2} - 4 \, a^{2} c\right )} x +{\left (2 \, a^{2} x^{2} + 2 \, a b x + b^{2} - 2 \, a c\right )} \sqrt{b^{2} - 4 \, a c}}{a x^{2} + b x + c}\right ) +{\left (2 \, a d^{3} - 2 \, b d^{2} e + 2 \, c d e^{2} +{\left (a d^{3} - c d e^{2} +{\left (a d^{2} e - c e^{3}\right )} x\right )} \log \left (a x^{2} + b x + c\right ) - 2 \,{\left (a d^{3} - c d e^{2} +{\left (a d^{2} e - c e^{3}\right )} x\right )} \log \left (e x + d\right )\right )} \sqrt{b^{2} - 4 \, a c}}{2 \,{\left (a^{2} d^{5} - 2 \, a b d^{4} e - 2 \, b c d^{2} e^{3} + c^{2} d e^{4} +{\left (b^{2} + 2 \, a c\right )} d^{3} e^{2} +{\left (a^{2} d^{4} e - 2 \, a b d^{3} e^{2} - 2 \, b c d e^{4} + c^{2} e^{5} +{\left (b^{2} + 2 \, a c\right )} d^{2} e^{3}\right )} x\right )} \sqrt{b^{2} - 4 \, a c}}, -\frac{2 \,{\left (a b d^{3} - 4 \, a c d^{2} e + b c d e^{2} +{\left (a b d^{2} e - 4 \, a c d e^{2} + b c e^{3}\right )} x\right )} \arctan \left (-\frac{\sqrt{-b^{2} + 4 \, a c}{\left (2 \, a x + b\right )}}{b^{2} - 4 \, a c}\right ) -{\left (2 \, a d^{3} - 2 \, b d^{2} e + 2 \, c d e^{2} +{\left (a d^{3} - c d e^{2} +{\left (a d^{2} e - c e^{3}\right )} x\right )} \log \left (a x^{2} + b x + c\right ) - 2 \,{\left (a d^{3} - c d e^{2} +{\left (a d^{2} e - c e^{3}\right )} x\right )} \log \left (e x + d\right )\right )} \sqrt{-b^{2} + 4 \, a c}}{2 \,{\left (a^{2} d^{5} - 2 \, a b d^{4} e - 2 \, b c d^{2} e^{3} + c^{2} d e^{4} +{\left (b^{2} + 2 \, a c\right )} d^{3} e^{2} +{\left (a^{2} d^{4} e - 2 \, a b d^{3} e^{2} - 2 \, b c d e^{4} + c^{2} e^{5} +{\left (b^{2} + 2 \, a c\right )} d^{2} e^{3}\right )} x\right )} \sqrt{-b^{2} + 4 \, a c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/2*((a*b*d^3 - 4*a*c*d^2*e + b*c*d*e^2 + (a*b*d^2*e - 4*a*c*d*e^2 + b*c*e^3)*x
)*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^3 - 2*b*d^2*e + 2*c*d*e^2 + (a
*d^3 - c*d*e^2 + (a*d^2*e - c*e^3)*x)*log(a*x^2 + b*x + c) - 2*(a*d^3 - c*d*e^2
+ (a*d^2*e - c*e^3)*x)*log(e*x + d))*sqrt(b^2 - 4*a*c))/((a^2*d^5 - 2*a*b*d^4*e
- 2*b*c*d^2*e^3 + c^2*d*e^4 + (b^2 + 2*a*c)*d^3*e^2 + (a^2*d^4*e - 2*a*b*d^3*e^2
- 2*b*c*d*e^4 + c^2*e^5 + (b^2 + 2*a*c)*d^2*e^3)*x)*sqrt(b^2 - 4*a*c)), -1/2*(2
*(a*b*d^3 - 4*a*c*d^2*e + b*c*d*e^2 + (a*b*d^2*e - 4*a*c*d*e^2 + b*c*e^3)*x)*arc
tan(-sqrt(-b^2 + 4*a*c)*(2*a*x + b)/(b^2 - 4*a*c)) - (2*a*d^3 - 2*b*d^2*e + 2*c*
d*e^2 + (a*d^3 - c*d*e^2 + (a*d^2*e - c*e^3)*x)*log(a*x^2 + b*x + c) - 2*(a*d^3
- c*d*e^2 + (a*d^2*e - c*e^3)*x)*log(e*x + d))*sqrt(-b^2 + 4*a*c))/((a^2*d^5 - 2
*a*b*d^4*e - 2*b*c*d^2*e^3 + c^2*d*e^4 + (b^2 + 2*a*c)*d^3*e^2 + (a^2*d^4*e - 2*
a*b*d^3*e^2 - 2*b*c*d*e^4 + c^2*e^5 + (b^2 + 2*a*c)*d^2*e^3)*x)*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/(e*x+d)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.276847, size = 441, normalized size = 2.41 \[ \frac{1}{2} \,{\left (\frac{2 \,{\left (a b d^{2} e - 4 \, a c d e^{2} + b c e^{3}\right )} \arctan \left (-\frac{{\left (2 \, a d - \frac{2 \, a d^{2}}{x e + d} - b e + \frac{2 \, b d e}{x e + d} - \frac{2 \, c e^{2}}{x e + d}\right )} e^{\left (-1\right )}}{\sqrt{-b^{2} + 4 \, a c}}\right ) e^{\left (-2\right )}}{{\left (a^{2} d^{4} - 2 \, a b d^{3} e + b^{2} d^{2} e^{2} + 2 \, a c d^{2} e^{2} - 2 \, b c d e^{3} + c^{2} e^{4}\right )} \sqrt{-b^{2} + 4 \, a c}} + \frac{{\left (a d^{2} - c e^{2}\right )}{\rm ln}\left (-a + \frac{2 \, a d}{x e + d} - \frac{a d^{2}}{{\left (x e + d\right )}^{2}} - \frac{b e}{x e + d} + \frac{b d e}{{\left (x e + d\right )}^{2}} - \frac{c e^{2}}{{\left (x e + d\right )}^{2}}\right )}{a^{2} d^{4} e - 2 \, a b d^{3} e^{2} + b^{2} d^{2} e^{3} + 2 \, a c d^{2} e^{3} - 2 \, b c d e^{4} + c^{2} e^{5}} + \frac{2 \, d e}{{\left (a d^{2} e^{2} - b d e^{3} + c e^{4}\right )}{\left (x e + d\right )}}\right )} e \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(2*(a*b*d^2*e - 4*a*c*d*e^2 + b*c*e^3)*arctan(-(2*a*d - 2*a*d^2/(x*e + d) -
b*e + 2*b*d*e/(x*e + d) - 2*c*e^2/(x*e + d))*e^(-1)/sqrt(-b^2 + 4*a*c))*e^(-2)/(
(a^2*d^4 - 2*a*b*d^3*e + b^2*d^2*e^2 + 2*a*c*d^2*e^2 - 2*b*c*d*e^3 + c^2*e^4)*sq
rt(-b^2 + 4*a*c)) + (a*d^2 - c*e^2)*ln(-a + 2*a*d/(x*e + d) - a*d^2/(x*e + d)^2
- b*e/(x*e + d) + b*d*e/(x*e + d)^2 - c*e^2/(x*e + d)^2)/(a^2*d^4*e - 2*a*b*d^3*
e^2 + b^2*d^2*e^3 + 2*a*c*d^2*e^3 - 2*b*c*d*e^4 + c^2*e^5) + 2*d*e/((a*d^2*e^2 -
b*d*e^3 + c*e^4)*(x*e + d)))*e