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))

________________________________________________________________________________________

Rubi [A]  time = 0.110342, 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, Rules used = {148} \[ -\frac{(b c-a d) \log (c+d x)}{(d e-c f) (d g-c h)}+\frac{(b e-a f) \log (e+f x)}{(d e-c f) (f g-e h)}-\frac{(b g-a h) \log (g+h x)}{(d g-c h) (f g-e h)} \]

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))

Rule 148

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p*(g + h*x), x], x] /; FreeQ[{a, b, c, d, e, f, g
, h, m}, x] && (IntegersQ[m, n, p] || (IGtQ[n, 0] && IGtQ[p, 0]))

Rubi steps

\begin{align*} \int \frac{a+b x}{(c+d x) (e+f x) (g+h x)} \, dx &=\int \left (\frac{d (-b c+a d)}{(d e-c f) (d g-c h) (c+d x)}+\frac{f (-b e+a f)}{(d e-c f) (-f g+e h) (e+f x)}+\frac{h (-b g+a h)}{(d g-c h) (f g-e h) (g+h x)}\right ) \, dx\\ &=-\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)}\\ \end{align*}

Mathematica [A]  time = 0.0862102, 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))

________________________________________________________________________________________

Maple [A]  time = 0.007, size = 179, normalized size = 1.7 \begin{align*}{\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 ( 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 ( hx+g \right ) ah}{ \left ( ch-dg \right ) \left ( eh-fg \right ) }}-{\frac{\ln \left ( hx+g \right ) bg}{ \left ( ch-dg \right ) \left ( eh-fg \right ) }} \end{align*}

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

________________________________________________________________________________________

Maxima [A]  time = 1.64723, size = 181, normalized size = 1.68 \begin{align*} -\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} \end{align*}

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)

________________________________________________________________________________________

Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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]

Timed out

________________________________________________________________________________________

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

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

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b x + a}{{\left (d x + c\right )}{\left (f x + e\right )}{\left (h x + g\right )}}\,{d x} \end{align*}

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)