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

Optimal. Leaf size=30 \[ \frac{d \log (x)}{b}-\frac{(c d-b e) \log (b+c x)}{b c} \]

[Out]

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

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Rubi [A]  time = 0.0562437, antiderivative size = 30, 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 \[ \frac{d \log (x)}{b}-\frac{(c d-b e) \log (b+c x)}{b c} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)/(b*x + c*x^2),x]

[Out]

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

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Rubi in Sympy [A]  time = 8.88104, size = 22, normalized size = 0.73 \[ \frac{d \log{\left (x \right )}}{b} + \frac{\left (b e - c d\right ) \log{\left (b + c x \right )}}{b c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d*log(x)/b + (b*e - c*d)*log(b + c*x)/(b*c)

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Mathematica [A]  time = 0.0165687, size = 29, normalized size = 0.97 \[ \frac{(b e-c d) \log (b+c x)}{b c}+\frac{d \log (x)}{b} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.002, size = 32, normalized size = 1.1 \[{\frac{d\ln \left ( x \right ) }{b}}+{\frac{\ln \left ( cx+b \right ) e}{c}}-{\frac{\ln \left ( cx+b \right ) d}{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)/(c*x^2+b*x),x)

[Out]

1/b*d*ln(x)+1/c*ln(c*x+b)*e-1/b*ln(c*x+b)*d

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Maxima [A]  time = 0.71541, size = 41, normalized size = 1.37 \[ \frac{d \log \left (x\right )}{b} - \frac{{\left (c d - b e\right )} \log \left (c x + b\right )}{b c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d*log(x)/b - (c*d - b*e)*log(c*x + b)/(b*c)

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Fricas [A]  time = 0.277731, size = 39, normalized size = 1.3 \[ \frac{c d \log \left (x\right ) -{\left (c d - b e\right )} \log \left (c x + b\right )}{b c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(c*d*log(x) - (c*d - b*e)*log(c*x + b))/(b*c)

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Sympy [A]  time = 1.31038, size = 41, normalized size = 1.37 \[ \frac{d \log{\left (x \right )}}{b} + \frac{\left (b e - c d\right ) \log{\left (x + \frac{- b d + \frac{b \left (b e - c d\right )}{c}}{b e - 2 c d} \right )}}{b c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.266748, size = 45, normalized size = 1.5 \[ \frac{d{\rm ln}\left ({\left | x \right |}\right )}{b} - \frac{{\left (c d - b e\right )}{\rm ln}\left ({\left | c x + b \right |}\right )}{b c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d*ln(abs(x))/b - (c*d - b*e)*ln(abs(c*x + b))/(b*c)