3.1141 \(\int \frac{A+B x}{b x+c x^2} \, dx\)

Optimal. Leaf size=29 \[ \frac{(b B-A c) \log (b+c x)}{b c}+\frac{A \log (x)}{b} \]

[Out]

(A*Log[x])/b + ((b*B - A*c)*Log[b + c*x])/(b*c)

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Rubi [A]  time = 0.0588647, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \frac{(b B-A c) \log (b+c x)}{b c}+\frac{A \log (x)}{b} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x)/(b*x + c*x^2),x]

[Out]

(A*Log[x])/b + ((b*B - A*c)*Log[b + c*x])/(b*c)

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Rubi in Sympy [A]  time = 9.2071, size = 22, normalized size = 0.76 \[ \frac{A \log{\left (x \right )}}{b} - \frac{\left (A c - B b\right ) \log{\left (b + c x \right )}}{b c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*log(x)/b - (A*c - B*b)*log(b + c*x)/(b*c)

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Mathematica [A]  time = 0.0147871, size = 29, normalized size = 1. \[ \frac{(b B-A c) \log (b+c x)}{b c}+\frac{A \log (x)}{b} \]

Antiderivative was successfully verified.

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

[Out]

(A*Log[x])/b + ((b*B - A*c)*Log[b + c*x])/(b*c)

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Maple [A]  time = 0.009, size = 32, normalized size = 1.1 \[{\frac{A\ln \left ( x \right ) }{b}}-{\frac{\ln \left ( cx+b \right ) A}{b}}+{\frac{\ln \left ( cx+b \right ) B}{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)/(c*x^2+b*x),x)

[Out]

A*ln(x)/b-1/b*ln(c*x+b)*A+1/c*ln(c*x+b)*B

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Maxima [A]  time = 0.71865, size = 39, normalized size = 1.34 \[ \frac{A \log \left (x\right )}{b} + \frac{{\left (B b - A c\right )} \log \left (c x + b\right )}{b c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*log(x)/b + (B*b - A*c)*log(c*x + b)/(b*c)

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Fricas [A]  time = 0.291278, size = 38, normalized size = 1.31 \[ \frac{A c \log \left (x\right ) +{\left (B b - A c\right )} \log \left (c x + b\right )}{b c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(A*c*log(x) + (B*b - A*c)*log(c*x + b))/(b*c)

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Sympy [A]  time = 1.32054, size = 41, normalized size = 1.41 \[ \frac{A \log{\left (x \right )}}{b} + \frac{\left (- A c + B b\right ) \log{\left (x + \frac{- A b + \frac{b \left (- A c + B b\right )}{c}}{- 2 A c + B b} \right )}}{b c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*log(x)/b + (-A*c + B*b)*log(x + (-A*b + b*(-A*c + B*b)/c)/(-2*A*c + B*b))/(b*c
)

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GIAC/XCAS [A]  time = 0.280528, size = 42, normalized size = 1.45 \[ \frac{A{\rm ln}\left ({\left | x \right |}\right )}{b} + \frac{{\left (B b - A c\right )}{\rm ln}\left ({\left | c x + b \right |}\right )}{b c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*ln(abs(x))/b + (B*b - A*c)*ln(abs(c*x + b))/(b*c)