3.1697 \(\int \frac{1}{(a+b x) (c+d x) (e+f x)} \, dx\)

Optimal. Leaf size=86 \[ \frac{b \log (a+b x)}{(b c-a d) (b e-a f)}-\frac{d \log (c+d x)}{(b c-a d) (d e-c f)}+\frac{f \log (e+f x)}{(b e-a f) (d e-c f)} \]

[Out]

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

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Rubi [A]  time = 0.161118, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045 \[ \frac{b \log (a+b x)}{(b c-a d) (b e-a f)}-\frac{d \log (c+d x)}{(b c-a d) (d e-c f)}+\frac{f \log (e+f x)}{(b e-a f) (d e-c f)} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b*x)*(c + d*x)*(e + f*x)),x]

[Out]

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

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Rubi in Sympy [A]  time = 25.0541, size = 65, normalized size = 0.76 \[ \frac{b \log{\left (a + b x \right )}}{\left (a d - b c\right ) \left (a f - b e\right )} - \frac{d \log{\left (c + d x \right )}}{\left (a d - b c\right ) \left (c f - d e\right )} + \frac{f \log{\left (e + f x \right )}}{\left (a f - b e\right ) \left (c f - d e\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x+a)/(d*x+c)/(f*x+e),x)

[Out]

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

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Mathematica [A]  time = 0.0852806, size = 80, normalized size = 0.93 \[ \frac{b \log (a+b x) (c f-d e)+d (b e-a f) \log (c+d x)+f (a d-b c) \log (e+f x)}{(b c-a d) (b e-a f) (c f-d e)} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b*x)*(c + d*x)*(e + f*x)),x]

[Out]

(b*(-(d*e) + c*f)*Log[a + b*x] + d*(b*e - a*f)*Log[c + d*x] + (-(b*c) + a*d)*f*L
og[e + f*x])/((b*c - a*d)*(b*e - a*f)*(-(d*e) + c*f))

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Maple [A]  time = 0.011, size = 87, normalized size = 1. \[ -{\frac{d\ln \left ( dx+c \right ) }{ \left ( ad-bc \right ) \left ( cf-de \right ) }}+{\frac{b\ln \left ( bx+a \right ) }{ \left ( ad-bc \right ) \left ( af-be \right ) }}+{\frac{f\ln \left ( fx+e \right ) }{ \left ( af-be \right ) \left ( cf-de \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x+a)/(d*x+c)/(f*x+e),x)

[Out]

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

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Maxima [A]  time = 1.35754, size = 151, normalized size = 1.76 \[ \frac{b \log \left (b x + a\right )}{{\left (b^{2} c - a b d\right )} e -{\left (a b c - a^{2} d\right )} f} - \frac{d \log \left (d x + c\right )}{{\left (b c d - a d^{2}\right )} e -{\left (b c^{2} - a c d\right )} f} + \frac{f \log \left (f x + e\right )}{b d e^{2} + a c f^{2} -{\left (b c + a d\right )} e f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)*(d*x + c)*(f*x + e)),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 5.4974, size = 151, normalized size = 1.76 \[ \frac{{\left (b c - a d\right )} f \log \left (f x + e\right ) +{\left (b d e - b c f\right )} \log \left (b x + a\right ) -{\left (b d e - a d f\right )} \log \left (d x + c\right )}{{\left (b^{2} c d - a b d^{2}\right )} e^{2} -{\left (b^{2} c^{2} - a^{2} d^{2}\right )} e f +{\left (a b c^{2} - a^{2} c d\right )} f^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)*(d*x + c)*(f*x + e)),x, algorithm="fricas")

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b*x+a)/(d*x+c)/(f*x+e),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x + a\right )}{\left (d x + c\right )}{\left (f x + e\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)*(d*x + c)*(f*x + e)),x, algorithm="giac")

[Out]

integrate(1/((b*x + a)*(d*x + c)*(f*x + e)), x)