3.1498 \(\int \frac{d+e x}{a^2+2 a b x+b^2 x^2} \, dx\)

Optimal. Leaf size=32 \[ \frac{e \log (a+b x)}{b^2}-\frac{b d-a e}{b^2 (a+b x)} \]

[Out]

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

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Rubi [A]  time = 0.0611504, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ \frac{e \log (a+b x)}{b^2}-\frac{b d-a e}{b^2 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 17.0341, size = 26, normalized size = 0.81 \[ \frac{e \log{\left (a + b x \right )}}{b^{2}} + \frac{a e - b d}{b^{2} \left (a + b x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0203736, size = 31, normalized size = 0.97 \[ \frac{a e-b d}{b^2 (a+b x)}+\frac{e \log (a+b x)}{b^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.01, size = 39, normalized size = 1.2 \[{\frac{e\ln \left ( bx+a \right ) }{{b}^{2}}}+{\frac{ae}{ \left ( bx+a \right ){b}^{2}}}-{\frac{d}{b \left ( bx+a \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.682405, size = 47, normalized size = 1.47 \[ -\frac{b d - a e}{b^{3} x + a b^{2}} + \frac{e \log \left (b x + a\right )}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.204234, size = 53, normalized size = 1.66 \[ -\frac{b d - a e -{\left (b e x + a e\right )} \log \left (b x + a\right )}{b^{3} x + a b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.37061, size = 27, normalized size = 0.84 \[ \frac{a e - b d}{a b^{2} + b^{3} x} + \frac{e \log{\left (a + b x \right )}}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.212371, size = 47, normalized size = 1.47 \[ \frac{e{\rm ln}\left ({\left | b x + a \right |}\right )}{b^{2}} - \frac{b d - a e}{{\left (b x + a\right )} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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