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

Optimal. Leaf size=194 \[ -\frac{\left (-2 c \left (a d^2-c e^2\right )+b^2 d^2-2 b c d e\right ) \tanh ^{-1}\left (\frac{2 a x+b}{\sqrt{b^2-4 a c}}\right )}{\sqrt{b^2-4 a c} \left (a d^2-e (b d-c e)\right )^2}-\frac{d^2}{e (d+e x) \left (a d^2-b d e+c e^2\right )}-\frac{d (b d-2 c e) \log \left (a x^2+b x+c\right )}{2 \left (a d^2-e (b d-c e)\right )^2}+\frac{d (b d-2 c e) \log (d+e x)}{\left (a d^2-e (b d-c e)\right )^2} \]

[Out]

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

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Rubi [A]  time = 0.558972, antiderivative size = 194, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ -\frac{\left (-2 c \left (a d^2-c e^2\right )+b^2 d^2-2 b c d e\right ) \tanh ^{-1}\left (\frac{2 a x+b}{\sqrt{b^2-4 a c}}\right )}{\sqrt{b^2-4 a c} \left (a d^2-e (b d-c e)\right )^2}-\frac{d^2}{e (d+e x) \left (a d^2-b d e+c e^2\right )}-\frac{d (b d-2 c e) \log \left (a x^2+b x+c\right )}{2 \left (a d^2-e (b d-c e)\right )^2}+\frac{d (b d-2 c e) \log (d+e x)}{\left (a d^2-e (b d-c e)\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

-(d^2/(e*(a*d^2 - b*d*e + c*e^2)*(d + e*x))) - ((b^2*d^2 - 2*b*c*d*e - 2*c*(a*d^
2 - c*e^2))*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) + (d*(b*d - 2*c*e)*Log[d + e*x])/(a*d^2 - e*(b*d - c*e))^2 - (
d*(b*d - 2*c*e)*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 = 146.522, size = 177, normalized size = 0.91 \[ \frac{d \left (b d - 2 c e\right ) \log{\left (\frac{d}{x} + e \right )}}{\left (a d^{2} - b d e + c e^{2}\right )^{2}} - \frac{d \left (b d - 2 c e\right ) \log{\left (a + \frac{b}{x} + \frac{c}{x^{2}} \right )}}{2 \left (a d^{2} - b d e + c e^{2}\right )^{2}} + \frac{d}{\left (\frac{d}{x} + e\right ) \left (a d^{2} - b d e + c e^{2}\right )} + \frac{\left (- 2 a c d^{2} + b^{2} d^{2} - 2 b c d e + 2 c^{2} e^{2}\right ) \operatorname{atanh}{\left (\frac{b + \frac{2 c}{x}}{\sqrt{- 4 a c + b^{2}}} \right )}}{\sqrt{- 4 a c + b^{2}} \left (a d^{2} - b d e + c e^{2}\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d*(b*d - 2*c*e)*log(d/x + e)/(a*d**2 - b*d*e + c*e**2)**2 - d*(b*d - 2*c*e)*log(
a + b/x + c/x**2)/(2*(a*d**2 - b*d*e + c*e**2)**2) + d/((d/x + e)*(a*d**2 - b*d*
e + c*e**2)) + (-2*a*c*d**2 + b**2*d**2 - 2*b*c*d*e + 2*c**2*e**2)*atanh((b + 2*
c/x)/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 = 0.406467, size = 159, normalized size = 0.82 \[ \frac{\frac{2 \left (2 c \left (c e^2-a d^2\right )+b^2 d^2-2 b c d e\right ) \tan ^{-1}\left (\frac{2 a x+b}{\sqrt{4 a c-b^2}}\right )}{\sqrt{4 a c-b^2}}-\frac{2 d^2 \left (a d^2+e (c e-b d)\right )}{e (d+e x)}-d (b d-2 c e) \log (x (a x+b)+c)+2 d (b d-2 c e) \log (d+e x)}{2 \left (a d^2+e (c e-b d)\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.013, size = 389, normalized size = 2. \[ -{\frac{{d}^{2}}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ) e \left ( ex+d \right ) }}+{\frac{{d}^{2}\ln \left ( ex+d \right ) b}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ) ^{2}}}-2\,{\frac{d\ln \left ( ex+d \right ) ce}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ) ^{2}}}-{\frac{\ln \left ( a{x}^{2}+bx+c \right ) b{d}^{2}}{2\, \left ( a{d}^{2}-bde+{e}^{2}c \right ) ^{2}}}+{\frac{\ln \left ( a{x}^{2}+bx+c \right ) cde}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ) ^{2}}}-2\,{\frac{ac{d}^{2}}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ) ^{2}\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,ax+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+{\frac{{b}^{2}{d}^{2}}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ) ^{2}}\arctan \left ({(2\,ax+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}-2\,{\frac{bcde}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ) ^{2}\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,ax+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+2\,{\frac{{c}^{2}{e}^{2}}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ) ^{2}\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,ax+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 6.04516, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/2*((2*b*c*d^2*e^2 - 2*c^2*d*e^3 - (b^2 - 2*a*c)*d^3*e + (2*b*c*d*e^3 - 2*c^2
*e^4 - (b^2 - 2*a*c)*d^2*e^2)*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^4 - 2*b*d^3*e + 2*c*d^2*e^2 + (b*d^3*e - 2*c*d^2*e^2 + (b*d^2*e^2 - 2*c*d*e^3)
*x)*log(a*x^2 + b*x + c) - 2*(b*d^3*e - 2*c*d^2*e^2 + (b*d^2*e^2 - 2*c*d*e^3)*x)
*log(e*x + d))*sqrt(b^2 - 4*a*c))/((a^2*d^5*e - 2*a*b*d^4*e^2 - 2*b*c*d^2*e^4 +
c^2*d*e^5 + (b^2 + 2*a*c)*d^3*e^3 + (a^2*d^4*e^2 - 2*a*b*d^3*e^3 - 2*b*c*d*e^5 +
c^2*e^6 + (b^2 + 2*a*c)*d^2*e^4)*x)*sqrt(b^2 - 4*a*c)), -1/2*(2*(2*b*c*d^2*e^2
- 2*c^2*d*e^3 - (b^2 - 2*a*c)*d^3*e + (2*b*c*d*e^3 - 2*c^2*e^4 - (b^2 - 2*a*c)*d
^2*e^2)*x)*arctan(-sqrt(-b^2 + 4*a*c)*(2*a*x + b)/(b^2 - 4*a*c)) + (2*a*d^4 - 2*
b*d^3*e + 2*c*d^2*e^2 + (b*d^3*e - 2*c*d^2*e^2 + (b*d^2*e^2 - 2*c*d*e^3)*x)*log(
a*x^2 + b*x + c) - 2*(b*d^3*e - 2*c*d^2*e^2 + (b*d^2*e^2 - 2*c*d*e^3)*x)*log(e*x
+ d))*sqrt(-b^2 + 4*a*c))/((a^2*d^5*e - 2*a*b*d^4*e^2 - 2*b*c*d^2*e^4 + c^2*d*e
^5 + (b^2 + 2*a*c)*d^3*e^3 + (a^2*d^4*e^2 - 2*a*b*d^3*e^3 - 2*b*c*d*e^5 + c^2*e^
6 + (b^2 + 2*a*c)*d^2*e^4)*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)/(e*x+d)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.273423, size = 455, normalized size = 2.35 \[ -\frac{{\left (b^{2} d^{2} e^{2} - 2 \, a c d^{2} e^{2} - 2 \, b c d e^{3} + 2 \, c^{2} e^{4}\right )} \arctan \left (-\frac{{\left (2 \, a d - \frac{2 \, a d^{2}}{x e + d} - b e + \frac{2 \, b d e}{x e + d} - \frac{2 \, c e^{2}}{x e + d}\right )} e^{\left (-1\right )}}{\sqrt{-b^{2} + 4 \, a c}}\right ) e^{\left (-2\right )}}{{\left (a^{2} d^{4} - 2 \, a b d^{3} e + b^{2} d^{2} e^{2} + 2 \, a c d^{2} e^{2} - 2 \, b c d e^{3} + c^{2} e^{4}\right )} \sqrt{-b^{2} + 4 \, a c}} - \frac{d^{2} e}{{\left (a d^{2} e^{2} - b d e^{3} + c e^{4}\right )}{\left (x e + d\right )}} - \frac{{\left (b d^{2} - 2 \, c d e\right )}{\rm ln}\left (-a + \frac{2 \, a d}{x e + d} - \frac{a d^{2}}{{\left (x e + d\right )}^{2}} - \frac{b e}{x e + d} + \frac{b d e}{{\left (x e + d\right )}^{2}} - \frac{c e^{2}}{{\left (x e + d\right )}^{2}}\right )}{2 \,{\left (a^{2} d^{4} - 2 \, a b d^{3} e + b^{2} d^{2} e^{2} + 2 \, a c d^{2} e^{2} - 2 \, b c d e^{3} + c^{2} e^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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