3.909 \(\int \frac{A+B x}{A^2+2 A B x+B^2 x^2} \, dx\)

Optimal. Leaf size=10 \[ \frac{\log (A+B x)}{B} \]

[Out]

Log[A + B*x]/B

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Rubi [A]  time = 0.0083106, antiderivative size = 10, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ \frac{\log (A+B x)}{B} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x)/(A^2 + 2*A*B*x + B^2*x^2),x]

[Out]

Log[A + B*x]/B

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Rubi in Sympy [A]  time = 9.50858, size = 7, normalized size = 0.7 \[ \frac{\log{\left (A + B x \right )}}{B} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)/(B**2*x**2+2*A*B*x+A**2),x)

[Out]

log(A + B*x)/B

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Mathematica [A]  time = 0.00195446, size = 10, normalized size = 1. \[ \frac{\log (A+B x)}{B} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x)/(A^2 + 2*A*B*x + B^2*x^2),x]

[Out]

Log[A + B*x]/B

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Maple [A]  time = 0.001, size = 11, normalized size = 1.1 \[{\frac{\ln \left ( Bx+A \right ) }{B}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)/(B^2*x^2+2*A*B*x+A^2),x)

[Out]

ln(B*x+A)/B

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Maxima [A]  time = 0.688626, size = 30, normalized size = 3. \[ \frac{\log \left (B^{2} x^{2} + 2 \, A B x + A^{2}\right )}{2 \, B} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/(B^2*x^2 + 2*A*B*x + A^2),x, algorithm="maxima")

[Out]

1/2*log(B^2*x^2 + 2*A*B*x + A^2)/B

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Fricas [A]  time = 0.270751, size = 14, normalized size = 1.4 \[ \frac{\log \left (B x + A\right )}{B} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/(B^2*x^2 + 2*A*B*x + A^2),x, algorithm="fricas")

[Out]

log(B*x + A)/B

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Sympy [A]  time = 0.107085, size = 7, normalized size = 0.7 \[ \frac{\log{\left (A + B x \right )}}{B} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)/(B**2*x**2+2*A*B*x+A**2),x)

[Out]

log(A + B*x)/B

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GIAC/XCAS [A]  time = 0.27056, size = 15, normalized size = 1.5 \[ \frac{{\rm ln}\left ({\left | B x + A \right |}\right )}{B} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/(B^2*x^2 + 2*A*B*x + A^2),x, algorithm="giac")

[Out]

ln(abs(B*x + A))/B