3.205 \(\int \frac{x}{(a+b x) (c+d x)} \, dx\)

Optimal. Leaf size=44 \[ \frac{c \log (c+d x)}{d (b c-a d)}-\frac{a \log (a+b x)}{b (b c-a d)} \]

[Out]

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

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Rubi [A]  time = 0.0609283, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062 \[ \frac{c \log (c+d x)}{d (b c-a d)}-\frac{a \log (a+b x)}{b (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[x/((a + b*x)*(c + d*x)),x]

[Out]

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

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Rubi in Sympy [A]  time = 18.2118, size = 32, normalized size = 0.73 \[ \frac{a \log{\left (a + b x \right )}}{b \left (a d - b c\right )} - \frac{c \log{\left (c + d x \right )}}{d \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0260754, size = 38, normalized size = 0.86 \[ -\frac{a d \log (a+b x)-b c \log (c+d x)}{b^2 c d-a b d^2} \]

Antiderivative was successfully verified.

[In]  Integrate[x/((a + b*x)*(c + d*x)),x]

[Out]

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

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Maple [A]  time = 0.009, size = 45, normalized size = 1. \[ -{\frac{c\ln \left ( dx+c \right ) }{d \left ( ad-bc \right ) }}+{\frac{a\ln \left ( bx+a \right ) }{ \left ( ad-bc \right ) b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(b*x+a)/(d*x+c),x)

[Out]

-c/(a*d-b*c)/d*ln(d*x+c)+a/(a*d-b*c)/b*ln(b*x+a)

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Maxima [A]  time = 1.32914, size = 59, normalized size = 1.34 \[ -\frac{a \log \left (b x + a\right )}{b^{2} c - a b d} + \frac{c \log \left (d x + c\right )}{b c d - a d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.215946, size = 51, normalized size = 1.16 \[ -\frac{a d \log \left (b x + a\right ) - b c \log \left (d x + c\right )}{b^{2} c d - a b d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 2.93527, size = 138, normalized size = 3.14 \[ \frac{a \log{\left (x + \frac{\frac{a^{3} d^{2}}{b \left (a d - b c\right )} - \frac{2 a^{2} c d}{a d - b c} + \frac{a b c^{2}}{a d - b c} + 2 a c}{a d + b c} \right )}}{b \left (a d - b c\right )} - \frac{c \log{\left (x + \frac{- \frac{a^{2} c d}{a d - b c} + \frac{2 a b c^{2}}{a d - b c} + 2 a c - \frac{b^{2} c^{3}}{d \left (a d - b c\right )}}{a d + b c} \right )}}{d \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(b*x+a)/(d*x+c),x)

[Out]

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

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: NotImplementedError