3.1500 \(\int \frac{1}{(d+e x) \left (a^2+2 a b x+b^2 x^2\right )} \, dx\)

Optimal. Leaf size=57 \[ -\frac{1}{(a+b x) (b d-a e)}-\frac{e \log (a+b x)}{(b d-a e)^2}+\frac{e \log (d+e x)}{(b d-a e)^2} \]

[Out]

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

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Rubi [A]  time = 0.0902307, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ -\frac{1}{(a+b x) (b d-a e)}-\frac{e \log (a+b x)}{(b d-a e)^2}+\frac{e \log (d+e x)}{(b d-a e)^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 29.7691, size = 46, normalized size = 0.81 \[ - \frac{e \log{\left (a + b x \right )}}{\left (a e - b d\right )^{2}} + \frac{e \log{\left (d + e x \right )}}{\left (a e - b d\right )^{2}} + \frac{1}{\left (a + b x\right ) \left (a e - b d\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-e*log(a + b*x)/(a*e - b*d)**2 + e*log(d + e*x)/(a*e - b*d)**2 + 1/((a + b*x)*(a
*e - b*d))

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Mathematica [A]  time = 0.0460507, size = 53, normalized size = 0.93 \[ \frac{e (a+b x) \log (d+e x)-e (a+b x) \log (a+b x)+a e-b d}{(a+b x) (b d-a e)^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.023, size = 57, normalized size = 1. \[{\frac{1}{ \left ( ae-bd \right ) \left ( bx+a \right ) }}-{\frac{e\ln \left ( bx+a \right ) }{ \left ( ae-bd \right ) ^{2}}}+{\frac{e\ln \left ( ex+d \right ) }{ \left ( ae-bd \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.680468, size = 124, normalized size = 2.18 \[ -\frac{e \log \left (b x + a\right )}{b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}} + \frac{e \log \left (e x + d\right )}{b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}} - \frac{1}{a b d - a^{2} e +{\left (b^{2} d - a b e\right )} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-e*log(b*x + a)/(b^2*d^2 - 2*a*b*d*e + a^2*e^2) + e*log(e*x + d)/(b^2*d^2 - 2*a*
b*d*e + a^2*e^2) - 1/(a*b*d - a^2*e + (b^2*d - a*b*e)*x)

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Fricas [A]  time = 0.235039, size = 126, normalized size = 2.21 \[ -\frac{b d - a e +{\left (b e x + a e\right )} \log \left (b x + a\right ) -{\left (b e x + a e\right )} \log \left (e x + d\right )}{a b^{2} d^{2} - 2 \, a^{2} b d e + a^{3} e^{2} +{\left (b^{3} d^{2} - 2 \, a b^{2} d e + a^{2} b e^{2}\right )} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(b*d - a*e + (b*e*x + a*e)*log(b*x + a) - (b*e*x + a*e)*log(e*x + d))/(a*b^2*d^
2 - 2*a^2*b*d*e + a^3*e^2 + (b^3*d^2 - 2*a*b^2*d*e + a^2*b*e^2)*x)

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Sympy [A]  time = 3.03263, size = 233, normalized size = 4.09 \[ \frac{e \log{\left (x + \frac{- \frac{a^{3} e^{4}}{\left (a e - b d\right )^{2}} + \frac{3 a^{2} b d e^{3}}{\left (a e - b d\right )^{2}} - \frac{3 a b^{2} d^{2} e^{2}}{\left (a e - b d\right )^{2}} + a e^{2} + \frac{b^{3} d^{3} e}{\left (a e - b d\right )^{2}} + b d e}{2 b e^{2}} \right )}}{\left (a e - b d\right )^{2}} - \frac{e \log{\left (x + \frac{\frac{a^{3} e^{4}}{\left (a e - b d\right )^{2}} - \frac{3 a^{2} b d e^{3}}{\left (a e - b d\right )^{2}} + \frac{3 a b^{2} d^{2} e^{2}}{\left (a e - b d\right )^{2}} + a e^{2} - \frac{b^{3} d^{3} e}{\left (a e - b d\right )^{2}} + b d e}{2 b e^{2}} \right )}}{\left (a e - b d\right )^{2}} + \frac{1}{a^{2} e - a b d + x \left (a b e - b^{2} d\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.211767, size = 128, normalized size = 2.25 \[ -\frac{b e{\rm ln}\left ({\left | b x + a \right |}\right )}{b^{3} d^{2} - 2 \, a b^{2} d e + a^{2} b e^{2}} + \frac{e^{2}{\rm ln}\left ({\left | x e + d \right |}\right )}{b^{2} d^{2} e - 2 \, a b d e^{2} + a^{2} e^{3}} - \frac{1}{{\left (b d - a e\right )}{\left (b x + a\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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