3.4 \(\int \frac{a+b x}{(c+d x) (e+f x) (g+h x)} \, dx\)

Optimal. Leaf size=108 \[ -\frac{(b c-a d) \log (c+d x)}{(d e-c f) (d g-c h)}+\frac{(b e-a f) \log (e+f x)}{(d e-c f) (f g-e h)}-\frac{(b g-a h) \log (g+h x)}{(d g-c h) (f g-e h)} \]

[Out]

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

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Rubi [A]  time = 0.303578, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.037 \[ -\frac{(b c-a d) \log (c+d x)}{(d e-c f) (d g-c h)}+\frac{(b e-a f) \log (e+f x)}{(d e-c f) (f g-e h)}-\frac{(b g-a h) \log (g+h x)}{(d g-c h) (f g-e h)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 46.0995, size = 80, normalized size = 0.74 \[ \frac{\left (a d - b c\right ) \log{\left (c + d x \right )}}{\left (c f - d e\right ) \left (c h - d g\right )} - \frac{\left (a f - b e\right ) \log{\left (e + f x \right )}}{\left (c f - d e\right ) \left (e h - f g\right )} + \frac{\left (a h - b g\right ) \log{\left (g + h x \right )}}{\left (c h - d g\right ) \left (e h - f g\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.164695, size = 102, normalized size = 0.94 \[ \frac{(b c-a d) \log (c+d x) (f g-e h)-(b e-a f) (d g-c h) \log (e+f x)+(b g-a h) (d e-c f) \log (g+h x)}{(d e-c f) (d g-c h) (e h-f g)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.013, size = 179, normalized size = 1.7 \[ -{\frac{\ln \left ( fx+e \right ) af}{ \left ( cf-de \right ) \left ( eh-fg \right ) }}+{\frac{\ln \left ( fx+e \right ) be}{ \left ( cf-de \right ) \left ( eh-fg \right ) }}+{\frac{\ln \left ( dx+c \right ) ad}{ \left ( cf-de \right ) \left ( ch-dg \right ) }}-{\frac{\ln \left ( dx+c \right ) bc}{ \left ( cf-de \right ) \left ( ch-dg \right ) }}+{\frac{\ln \left ( hx+g \right ) ah}{ \left ( eh-fg \right ) \left ( ch-dg \right ) }}-{\frac{\ln \left ( hx+g \right ) bg}{ \left ( eh-fg \right ) \left ( ch-dg \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/(c*f-d*e)/(e*h-f*g)*ln(f*x+e)*a*f+1/(c*f-d*e)/(e*h-f*g)*ln(f*x+e)*b*e+1/(c*f-
d*e)/(c*h-d*g)*ln(d*x+c)*a*d-1/(c*f-d*e)/(c*h-d*g)*ln(d*x+c)*b*c+1/(e*h-f*g)/(c*
h-d*g)*ln(h*x+g)*a*h-1/(e*h-f*g)/(c*h-d*g)*ln(h*x+g)*b*g

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Maxima [A]  time = 1.35065, size = 181, normalized size = 1.68 \[ -\frac{{\left (b c - a d\right )} \log \left (d x + c\right )}{{\left (d^{2} e - c d f\right )} g -{\left (c d e - c^{2} f\right )} h} + \frac{{\left (b e - a f\right )} \log \left (f x + e\right )}{{\left (d e f - c f^{2}\right )} g -{\left (d e^{2} - c e f\right )} h} - \frac{{\left (b g - a h\right )} \log \left (h x + g\right )}{d f g^{2} + c e h^{2} -{\left (d e + c f\right )} g h} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 74.9794, size = 216, normalized size = 2. \[ -\frac{{\left ({\left (b c - a d\right )} f g -{\left (b c - a d\right )} e h\right )} \log \left (d x + c\right ) -{\left ({\left (b d e - a d f\right )} g -{\left (b c e - a c f\right )} h\right )} \log \left (f x + e\right ) +{\left ({\left (b d e - b c f\right )} g -{\left (a d e - a c f\right )} h\right )} \log \left (h x + g\right )}{{\left (d^{2} e f - c d f^{2}\right )} g^{2} -{\left (d^{2} e^{2} - c^{2} f^{2}\right )} g h +{\left (c d e^{2} - c^{2} e f\right )} h^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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