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

Optimal. Leaf size=84 \[ \frac {(b e-a f) (d e-c f) \log (e+f x)}{f^2 (f g-e h)}-\frac {(b g-a h) (d g-c h) \log (g+h x)}{h^2 (f g-e h)}+\frac {b d x}{f h} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.09, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {142} \[ \frac {(b e-a f) (d e-c f) \log (e+f x)}{f^2 (f g-e h)}-\frac {(b g-a h) (d g-c h) \log (g+h x)}{h^2 (f g-e h)}+\frac {b d x}{f h} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 142

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

Rubi steps

\begin {align*} \int \frac {(a+b x) (c+d x)}{(e+f x) (g+h x)} \, dx &=\int \left (\frac {b d}{f h}+\frac {(-b e+a f) (-d e+c f)}{f (f g-e h) (e+f x)}+\frac {(-b g+a h) (-d g+c h)}{h (-f g+e h) (g+h x)}\right ) \, dx\\ &=\frac {b d x}{f h}+\frac {(b e-a f) (d e-c f) \log (e+f x)}{f^2 (f g-e h)}-\frac {(b g-a h) (d g-c h) \log (g+h x)}{h^2 (f g-e h)}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.06, size = 85, normalized size = 1.01 \[ \frac {f (b d h x (f g-e h)-f (b g-a h) (d g-c h) \log (g+h x))+h^2 (b e-a f) (d e-c f) \log (e+f x)}{f^2 h^2 (f g-e h)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.79, size = 117, normalized size = 1.39 \[ \frac {{\left (b d e^{2} + a c f^{2} - {\left (b c + a d\right )} e f\right )} h^{2} \log \left (f x + e\right ) + {\left (b d f^{2} g h - b d e f h^{2}\right )} x - {\left (b d f^{2} g^{2} + a c f^{2} h^{2} - {\left (b c + a d\right )} f^{2} g h\right )} \log \left (h x + g\right )}{f^{3} g h^{2} - e f^{2} h^{3}} \]

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*d*e^2 + a*c*f^2 - (b*c + a*d)*e*f)*h^2*log(f*x + e) + (b*d*f^2*g*h - b*d*e*f*h^2)*x - (b*d*f^2*g^2 + a*c*f
^2*h^2 - (b*c + a*d)*f^2*g*h)*log(h*x + g))/(f^3*g*h^2 - e*f^2*h^3)

________________________________________________________________________________________

giac [A]  time = 1.25, size = 112, normalized size = 1.33 \[ \frac {b d x}{f h} + \frac {{\left (a c f^{2} - b c f e - a d f e + b d e^{2}\right )} \log \left ({\left | f x + e \right |}\right )}{f^{3} g - f^{2} h e} - \frac {{\left (b d g^{2} - b c g h - a d g h + a c h^{2}\right )} \log \left ({\left | h x + g \right |}\right )}{f g h^{2} - h^{3} e} \]

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]

b*d*x/(f*h) + (a*c*f^2 - b*c*f*e - a*d*f*e + b*d*e^2)*log(abs(f*x + e))/(f^3*g - f^2*h*e) - (b*d*g^2 - b*c*g*h
- a*d*g*h + a*c*h^2)*log(abs(h*x + g))/(f*g*h^2 - h^3*e)

________________________________________________________________________________________

maple [B]  time = 0.01, size = 196, normalized size = 2.33 \[ -\frac {a c \ln \left (f x +e \right )}{e h -f g}+\frac {a c \ln \left (h x +g \right )}{e h -f g}+\frac {a d e \ln \left (f x +e \right )}{\left (e h -f g \right ) f}-\frac {a d g \ln \left (h x +g \right )}{\left (e h -f g \right ) h}+\frac {b c e \ln \left (f x +e \right )}{\left (e h -f g \right ) f}-\frac {b c g \ln \left (h x +g \right )}{\left (e h -f g \right ) h}-\frac {b d \,e^{2} \ln \left (f x +e \right )}{\left (e h -f g \right ) f^{2}}+\frac {b d \,g^{2} \ln \left (h x +g \right )}{\left (e h -f g \right ) h^{2}}+\frac {b d x}{f h} \]

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]

b*d*x/f/h+1/(e*h-f*g)*ln(h*x+g)*a*c-1/h/(e*h-f*g)*ln(h*x+g)*a*d*g-1/h/(e*h-f*g)*ln(h*x+g)*b*c*g+1/h^2/(e*h-f*g
)*ln(h*x+g)*b*d*g^2-1/(e*h-f*g)*ln(f*x+e)*a*c+1/f/(e*h-f*g)*ln(f*x+e)*a*d*e+1/f/(e*h-f*g)*ln(f*x+e)*b*c*e-1/f^
2/(e*h-f*g)*ln(f*x+e)*b*d*e^2

________________________________________________________________________________________

maxima [A]  time = 0.44, size = 104, normalized size = 1.24 \[ \frac {b d x}{f h} + \frac {{\left (b d e^{2} + a c f^{2} - {\left (b c + a d\right )} e f\right )} \log \left (f x + e\right )}{f^{3} g - e f^{2} h} - \frac {{\left (b d g^{2} + a c h^{2} - {\left (b c + a d\right )} g h\right )} \log \left (h x + g\right )}{f g h^{2} - e h^{3}} \]

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*d*x/(f*h) + (b*d*e^2 + a*c*f^2 - (b*c + a*d)*e*f)*log(f*x + e)/(f^3*g - e*f^2*h) - (b*d*g^2 + a*c*h^2 - (b*c
+ a*d)*g*h)*log(h*x + g)/(f*g*h^2 - e*h^3)

________________________________________________________________________________________

mupad [B]  time = 2.97, size = 105, normalized size = 1.25 \[ \frac {\ln \left (e+f\,x\right )\,\left (a\,c\,f^2-f\,\left (a\,d\,e+b\,c\,e\right )+b\,d\,e^2\right )}{f^3\,g-e\,f^2\,h}+\frac {\ln \left (g+h\,x\right )\,\left (a\,c\,h^2-h\,\left (a\,d\,g+b\,c\,g\right )+b\,d\,g^2\right )}{e\,h^3-f\,g\,h^2}+\frac {b\,d\,x}{f\,h} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(log(e + f*x)*(a*c*f^2 - f*(a*d*e + b*c*e) + b*d*e^2))/(f^3*g - e*f^2*h) + (log(g + h*x)*(a*c*h^2 - h*(a*d*g +
b*c*g) + b*d*g^2))/(e*h^3 - f*g*h^2) + (b*d*x)/(f*h)

________________________________________________________________________________________

sympy [B]  time = 20.49, size = 507, normalized size = 6.04 \[ \frac {b d x}{f h} + \frac {\left (a h - b g\right ) \left (c h - d g\right ) \log {\left (x + \frac {a c e f h^{2} + a c f^{2} g h - 2 a d e f g h - 2 b c e f g h + b d e^{2} g h + b d e f g^{2} - \frac {e^{2} f h \left (a h - b g\right ) \left (c h - d g\right )}{e h - f g} + \frac {2 e f^{2} g \left (a h - b g\right ) \left (c h - d g\right )}{e h - f g} - \frac {f^{3} g^{2} \left (a h - b g\right ) \left (c h - d g\right )}{h \left (e h - f g\right )}}{2 a c f^{2} h^{2} - a d e f h^{2} - a d f^{2} g h - b c e f h^{2} - b c f^{2} g h + b d e^{2} h^{2} + b d f^{2} g^{2}} \right )}}{h^{2} \left (e h - f g\right )} - \frac {\left (a f - b e\right ) \left (c f - d e\right ) \log {\left (x + \frac {a c e f h^{2} + a c f^{2} g h - 2 a d e f g h - 2 b c e f g h + b d e^{2} g h + b d e f g^{2} + \frac {e^{2} h^{3} \left (a f - b e\right ) \left (c f - d e\right )}{f \left (e h - f g\right )} - \frac {2 e g h^{2} \left (a f - b e\right ) \left (c f - d e\right )}{e h - f g} + \frac {f g^{2} h \left (a f - b e\right ) \left (c f - d e\right )}{e h - f g}}{2 a c f^{2} h^{2} - a d e f h^{2} - a d f^{2} g h - b c e f h^{2} - b c f^{2} g h + b d e^{2} h^{2} + b d f^{2} g^{2}} \right )}}{f^{2} \left (e h - f g\right )} \]

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]

b*d*x/(f*h) + (a*h - b*g)*(c*h - d*g)*log(x + (a*c*e*f*h**2 + a*c*f**2*g*h - 2*a*d*e*f*g*h - 2*b*c*e*f*g*h + b
*d*e**2*g*h + b*d*e*f*g**2 - e**2*f*h*(a*h - b*g)*(c*h - d*g)/(e*h - f*g) + 2*e*f**2*g*(a*h - b*g)*(c*h - d*g)
/(e*h - f*g) - f**3*g**2*(a*h - b*g)*(c*h - d*g)/(h*(e*h - f*g)))/(2*a*c*f**2*h**2 - a*d*e*f*h**2 - a*d*f**2*g
*h - b*c*e*f*h**2 - b*c*f**2*g*h + b*d*e**2*h**2 + b*d*f**2*g**2))/(h**2*(e*h - f*g)) - (a*f - b*e)*(c*f - d*e
)*log(x + (a*c*e*f*h**2 + a*c*f**2*g*h - 2*a*d*e*f*g*h - 2*b*c*e*f*g*h + b*d*e**2*g*h + b*d*e*f*g**2 + e**2*h*
*3*(a*f - b*e)*(c*f - d*e)/(f*(e*h - f*g)) - 2*e*g*h**2*(a*f - b*e)*(c*f - d*e)/(e*h - f*g) + f*g**2*h*(a*f -
b*e)*(c*f - d*e)/(e*h - f*g))/(2*a*c*f**2*h**2 - a*d*e*f*h**2 - a*d*f**2*g*h - b*c*e*f*h**2 - b*c*f**2*g*h + b
*d*e**2*h**2 + b*d*f**2*g**2))/(f**2*(e*h - f*g))

________________________________________________________________________________________