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

Optimal. Leaf size=163 \[ \frac {b^2 \log (a+b x)}{(b c-a d) (b e-a f) (b g-a h)}-\frac {d^2 \log (c+d x)}{(b c-a d) (d e-c f) (d g-c h)}+\frac {f^2 \log (e+f x)}{(b e-a f) (d e-c f) (f g-e h)}-\frac {h^2 \log (g+h x)}{(b g-a h) (d g-c h) (f g-e h)} \]

________________________________________________________________________________________

Rubi [A]  time = 0.21, antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.034, Rules used = {180} \begin {gather*} \frac {b^2 \log (a+b x)}{(b c-a d) (b e-a f) (b g-a h)}-\frac {d^2 \log (c+d x)}{(b c-a d) (d e-c f) (d g-c h)}+\frac {f^2 \log (e+f x)}{(b e-a f) (d e-c f) (f g-e h)}-\frac {h^2 \log (g+h x)}{(b g-a h) (d g-c h) (f g-e h)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 180

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

Rubi steps

\begin {align*} \int \frac {1}{(a+b x) (c+d x) (e+f x) (g+h x)} \, dx &=\int \left (\frac {b^3}{(b c-a d) (b e-a f) (b g-a h) (a+b x)}-\frac {d^3}{(b c-a d) (-d e+c f) (-d g+c h) (c+d x)}-\frac {f^3}{(b e-a f) (d e-c f) (-f g+e h) (e+f x)}-\frac {h^3}{(b g-a h) (d g-c h) (f g-e h) (g+h x)}\right ) \, dx\\ &=\frac {b^2 \log (a+b x)}{(b c-a d) (b e-a f) (b g-a h)}-\frac {d^2 \log (c+d x)}{(b c-a d) (d e-c f) (d g-c h)}+\frac {f^2 \log (e+f x)}{(b e-a f) (d e-c f) (f g-e h)}-\frac {h^2 \log (g+h x)}{(b g-a h) (d g-c h) (f g-e h)}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.20, size = 164, normalized size = 1.01 \begin {gather*} \frac {b^2 \log (a+b x)}{(b c-a d) (b e-a f) (b g-a h)}-\frac {d^2 \log (c+d x)}{(b c-a d) (c f-d e) (c h-d g)}-\frac {f^2 \log (e+f x)}{(b e-a f) (d e-c f) (e h-f g)}-\frac {h^2 \log (g+h x)}{(b g-a h) (d g-c h) (f g-e h)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{(a+b x) (c+d x) (e+f x) (g+h x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[1/((a + b*x)*(c + d*x)*(e + f*x)*(g + h*x)),x]

[Out]

IntegrateAlgebraic[1/((a + b*x)*(c + d*x)*(e + f*x)*(g + h*x)), x]

________________________________________________________________________________________

fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)/(d*x+c)/(f*x+e)/(h*x+g),x, algorithm="fricas")

[Out]

Timed out

________________________________________________________________________________________

giac [B]  time = 1.19, size = 363, normalized size = 2.23 \begin {gather*} -\frac {b^{3} \log \left ({\left | b x + a \right |}\right )}{a b^{3} c f g - a^{2} b^{2} d f g - a^{2} b^{2} c f h + a^{3} b d f h - b^{4} c g e + a b^{3} d g e + a b^{3} c h e - a^{2} b^{2} d h e} + \frac {d^{3} \log \left ({\left | d x + c \right |}\right )}{b c^{2} d^{2} f g - a c d^{3} f g - b c^{3} d f h + a c^{2} d^{2} f h - b c d^{3} g e + a d^{4} g e + b c^{2} d^{2} h e - a c d^{3} h e} + \frac {f^{3} \log \left ({\left | f x + e \right |}\right )}{a c f^{4} g - b c f^{3} g e - a d f^{3} g e - a c f^{3} h e + b d f^{2} g e^{2} + b c f^{2} h e^{2} + a d f^{2} h e^{2} - b d f h e^{3}} - \frac {h^{3} \log \left ({\left | h x + g \right |}\right )}{b d f g^{3} h - b c f g^{2} h^{2} - a d f g^{2} h^{2} + a c f g h^{3} - b d g^{2} h^{2} e + b c g h^{3} e + a d g h^{3} e - a c h^{4} e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)/(d*x+c)/(f*x+e)/(h*x+g),x, algorithm="giac")

[Out]

-b^3*log(abs(b*x + a))/(a*b^3*c*f*g - a^2*b^2*d*f*g - a^2*b^2*c*f*h + a^3*b*d*f*h - b^4*c*g*e + a*b^3*d*g*e +
a*b^3*c*h*e - a^2*b^2*d*h*e) + d^3*log(abs(d*x + c))/(b*c^2*d^2*f*g - a*c*d^3*f*g - b*c^3*d*f*h + a*c^2*d^2*f*
h - b*c*d^3*g*e + a*d^4*g*e + b*c^2*d^2*h*e - a*c*d^3*h*e) + f^3*log(abs(f*x + e))/(a*c*f^4*g - b*c*f^3*g*e -
a*d*f^3*g*e - a*c*f^3*h*e + b*d*f^2*g*e^2 + b*c*f^2*h*e^2 + a*d*f^2*h*e^2 - b*d*f*h*e^3) - h^3*log(abs(h*x + g
))/(b*d*f*g^3*h - b*c*f*g^2*h^2 - a*d*f*g^2*h^2 + a*c*f*g*h^3 - b*d*g^2*h^2*e + b*c*g*h^3*e + a*d*g*h^3*e - a*
c*h^4*e)

________________________________________________________________________________________

maple [A]  time = 0.01, size = 164, normalized size = 1.01 \begin {gather*} -\frac {b^{2} \ln \left (b x +a \right )}{\left (a d -b c \right ) \left (a f -b e \right ) \left (a h -b g \right )}+\frac {d^{2} \ln \left (d x +c \right )}{\left (a d -b c \right ) \left (c f -d e \right ) \left (c h -d g \right )}-\frac {f^{2} \ln \left (f x +e \right )}{\left (c f -d e \right ) \left (e h -f g \right ) \left (a f -b e \right )}+\frac {h^{2} \ln \left (h x +g \right )}{\left (c h -d g \right ) \left (a h -b g \right ) \left (e h -f g \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x+a)/(d*x+c)/(f*x+e)/(h*x+g),x)

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.49, size = 310, normalized size = 1.90 \begin {gather*} \frac {b^{2} \log \left (b x + a\right )}{{\left ({\left (b^{3} c - a b^{2} d\right )} e - {\left (a b^{2} c - a^{2} b d\right )} f\right )} g - {\left ({\left (a b^{2} c - a^{2} b d\right )} e - {\left (a^{2} b c - a^{3} d\right )} f\right )} h} - \frac {d^{2} \log \left (d x + c\right )}{{\left ({\left (b c d^{2} - a d^{3}\right )} e - {\left (b c^{2} d - a c d^{2}\right )} f\right )} g - {\left ({\left (b c^{2} d - a c d^{2}\right )} e - {\left (b c^{3} - a c^{2} d\right )} f\right )} h} + \frac {f^{2} \log \left (f x + e\right )}{{\left (b d e^{2} f + a c f^{3} - {\left (b c + a d\right )} e f^{2}\right )} g - {\left (b d e^{3} + a c e f^{2} - {\left (b c + a d\right )} e^{2} f\right )} h} - \frac {h^{2} \log \left (h x + g\right )}{b d f g^{3} - a c e h^{3} - {\left (b d e + {\left (b c + a d\right )} f\right )} g^{2} h + {\left (a c f + {\left (b c + a d\right )} e\right )} g h^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)/(d*x+c)/(f*x+e)/(h*x+g),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 6.62, size = 317, normalized size = 1.94 \begin {gather*} \frac {b^2\,\ln \left (a+b\,x\right )}{b^3\,c\,e\,g-a^3\,d\,f\,h-a\,b^2\,c\,e\,h-a\,b^2\,c\,f\,g-a\,b^2\,d\,e\,g+a^2\,b\,c\,f\,h+a^2\,b\,d\,e\,h+a^2\,b\,d\,f\,g}+\frac {d^2\,\ln \left (c+d\,x\right )}{a\,d^3\,e\,g-b\,c^3\,f\,h-a\,c\,d^2\,e\,h-a\,c\,d^2\,f\,g-b\,c\,d^2\,e\,g+a\,c^2\,d\,f\,h+b\,c^2\,d\,e\,h+b\,c^2\,d\,f\,g}+\frac {f^2\,\ln \left (e+f\,x\right )}{a\,c\,f^3\,g-b\,d\,e^3\,h-a\,c\,e\,f^2\,h-a\,d\,e\,f^2\,g-b\,c\,e\,f^2\,g+a\,d\,e^2\,f\,h+b\,c\,e^2\,f\,h+b\,d\,e^2\,f\,g}+\frac {h^2\,\ln \left (g+h\,x\right )}{a\,c\,e\,h^3-b\,d\,f\,g^3-a\,c\,f\,g\,h^2-a\,d\,e\,g\,h^2-b\,c\,e\,g\,h^2+a\,d\,f\,g^2\,h+b\,c\,f\,g^2\,h+b\,d\,e\,g^2\,h} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((e + f*x)*(g + h*x)*(a + b*x)*(c + d*x)),x)

[Out]

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

________________________________________________________________________________________

sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________