\(\int \frac {1}{(a+b x) (c+d x) (e+f x) (g+h x)} \, dx\) [5]

Optimal result

Integrand size = 29, antiderivative size = 163 \[ \int \frac {1}{(a+b x) (c+d x) (e+f x) (g+h x)} \, dx=\frac {b^2 \log (a+b x)}{(b c-a d) (b e-a f) (b g-a h)}-\frac {d^2 \log (c+d x)}{(b c-a d) (d e-c f) (d g-c h)}+\frac {f^2 \log (e+f x)}{(b e-a f) (d e-c f) (f g-e h)}-\frac {h^2 \log (g+h x)}{(b g-a h) (d g-c h) (f g-e h)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.15 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.034, Rules used = {186} \[ \int \frac {1}{(a+b x) (c+d x) (e+f x) (g+h x)} \, dx=\frac {b^2 \log (a+b x)}{(b c-a d) (b e-a f) (b g-a h)}-\frac {d^2 \log (c+d x)}{(b c-a d) (d e-c f) (d g-c h)}+\frac {f^2 \log (e+f x)}{(b e-a f) (d e-c f) (f g-e h)}-\frac {h^2 \log (g+h x)}{(b g-a h) (d g-c h) (f g-e h)} \]

[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 186

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*} \text {integral}& = \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] (verified)

Time = 0.11 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.01 \[ \int \frac {1}{(a+b x) (c+d x) (e+f x) (g+h x)} \, dx=\frac {b^2 \log (a+b x)}{(b c-a d) (b e-a f) (b g-a h)}-\frac {d^2 \log (c+d x)}{(b c-a d) (-d e+c f) (-d g+c h)}-\frac {f^2 \log (e+f x)}{(b e-a f) (d e-c f) (-f g+e h)}-\frac {h^2 \log (g+h x)}{(b g-a h) (d g-c h) (f g-e h)} \]

[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))

Maple [A] (verified)

Time = 1.76 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.01

method	result	size
default	\(\frac {d^{2} \ln \left (d x +c \right )}{\left (a d -b c \right ) \left (c f -d e \right ) \left (c h -d g \right )}-\frac {b^{2} \ln \left (b x +a \right )}{\left (a d -b c \right ) \left (a f -b e \right ) \left (a h -b g \right )}+\frac {h^{2} \ln \left (h x +g \right )}{\left (a h -b g \right ) \left (c h -d g \right ) \left (e h -f g \right )}-\frac {f^{2} \ln \left (f x +e \right )}{\left (a f -b e \right ) \left (c f -d e \right ) \left (e h -f g \right )}\)	\(164\)
norman	\(\frac {h^{2} \ln \left (h x +g \right )}{a c e \,h^{3}-a c f g \,h^{2}-a d e g \,h^{2}+a d f \,g^{2} h -b c e g \,h^{2}+b c f \,g^{2} h +b d e \,g^{2} h -b d f \,g^{3}}+\frac {d^{2} \ln \left (d x +c \right )}{\left (a d -b c \right ) \left (c f -d e \right ) \left (c h -d g \right )}-\frac {f^{2} \ln \left (f x +e \right )}{\left (a c \,f^{2}-a d e f -b c e f +b d \,e^{2}\right ) \left (e h -f g \right )}-\frac {b^{2} \ln \left (b x +a \right )}{\left (a d -b c \right ) \left (a f -b e \right ) \left (a h -b g \right )}\)	\(210\)
risch	\(\frac {d^{2} \ln \left (-d x -c \right )}{a \,c^{2} d f h -a c \,d^{2} e h -a c \,d^{2} f g +a \,d^{3} e g -b \,c^{3} f h +b \,c^{2} d e h +b \,c^{2} d f g -b c \,d^{2} e g}+\frac {h^{2} \ln \left (h x +g \right )}{a c e \,h^{3}-a c f g \,h^{2}-a d e g \,h^{2}+a d f \,g^{2} h -b c e g \,h^{2}+b c f \,g^{2} h +b d e \,g^{2} h -b d f \,g^{3}}-\frac {b^{2} \ln \left (-b x -a \right )}{a^{3} d f h -a^{2} b c f h -a^{2} b d e h -a^{2} b d f g +a \,b^{2} c e h +a \,b^{2} c f g +a \,b^{2} d e g -b^{3} c e g}-\frac {f^{2} \ln \left (f x +e \right )}{a c e \,f^{2} h -a c \,f^{3} g -a d \,e^{2} f h +a d e \,f^{2} g -b c \,e^{2} f h +b c e \,f^{2} g +b d \,e^{3} h -b d \,e^{2} f g}\)	\(326\)
parallelrisch	\(-\frac {\ln \left (b x +a \right ) b^{2} c^{2} e f \,h^{2}-\ln \left (b x +a \right ) b^{2} c^{2} f^{2} g h -\ln \left (b x +a \right ) b^{2} c d \,e^{2} h^{2}+\ln \left (b x +a \right ) b^{2} c d \,f^{2} g^{2}+\ln \left (b x +a \right ) b^{2} d^{2} e^{2} g h -\ln \left (b x +a \right ) b^{2} d^{2} e f \,g^{2}-\ln \left (d x +c \right ) a^{2} d^{2} e f \,h^{2}+\ln \left (d x +c \right ) a^{2} d^{2} f^{2} g h +\ln \left (d x +c \right ) a b \,d^{2} e^{2} h^{2}-\ln \left (d x +c \right ) a b \,d^{2} f^{2} g^{2}-\ln \left (d x +c \right ) b^{2} d^{2} e^{2} g h +\ln \left (d x +c \right ) b^{2} d^{2} e f \,g^{2}+\ln \left (f x +e \right ) a^{2} c d \,f^{2} h^{2}-\ln \left (f x +e \right ) a^{2} d^{2} f^{2} g h -\ln \left (f x +e \right ) a b \,c^{2} f^{2} h^{2}+\ln \left (f x +e \right ) a b \,d^{2} f^{2} g^{2}+\ln \left (f x +e \right ) b^{2} c^{2} f^{2} g h -\ln \left (f x +e \right ) b^{2} c d \,f^{2} g^{2}-\ln \left (h x +g \right ) a^{2} c d \,f^{2} h^{2}+\ln \left (h x +g \right ) a^{2} d^{2} e f \,h^{2}+\ln \left (h x +g \right ) a b \,c^{2} f^{2} h^{2}-\ln \left (h x +g \right ) a b \,d^{2} e^{2} h^{2}-\ln \left (h x +g \right ) b^{2} c^{2} e f \,h^{2}+\ln \left (h x +g \right ) b^{2} c d \,e^{2} h^{2}}{\left (a c e \,h^{3}-a c f g \,h^{2}-a d e g \,h^{2}+a d f \,g^{2} h -b c e g \,h^{2}+b c f \,g^{2} h +b d e \,g^{2} h -b d f \,g^{3}\right ) \left (a c \,f^{2}-a d e f -b c e f +b d \,e^{2}\right ) \left (a d -b c \right )}\)	\(554\)

[In]

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

[Out]

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

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{(a+b x) (c+d x) (e+f x) (g+h x)} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{(a+b x) (c+d x) (e+f x) (g+h x)} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 310, normalized size of antiderivative = 1.90 \[ \int \frac {1}{(a+b x) (c+d x) (e+f x) (g+h x)} \, dx=\frac {b^{2} \log \left (b x + a\right )}{{\left ({\left (b^{3} c - a b^{2} d\right )} e - {\left (a b^{2} c - a^{2} b d\right )} f\right )} g - {\left ({\left (a b^{2} c - a^{2} b d\right )} e - {\left (a^{2} b c - a^{3} d\right )} f\right )} h} - \frac {d^{2} \log \left (d x + c\right )}{{\left ({\left (b c d^{2} - a d^{3}\right )} e - {\left (b c^{2} d - a c d^{2}\right )} f\right )} g - {\left ({\left (b c^{2} d - a c d^{2}\right )} e - {\left (b c^{3} - a c^{2} d\right )} f\right )} h} + \frac {f^{2} \log \left (f x + e\right )}{{\left (b d e^{2} f + a c f^{3} - {\left (b c + a d\right )} e f^{2}\right )} g - {\left (b d e^{3} + a c e f^{2} - {\left (b c + a d\right )} e^{2} f\right )} h} - \frac {h^{2} \log \left (h x + g\right )}{b d f g^{3} - a c e h^{3} - {\left (b d e + {\left (b c + a d\right )} f\right )} g^{2} h + {\left (a c f + {\left (b c + a d\right )} e\right )} g h^{2}} \]

[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)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 351 vs. \(2 (163) = 326\).

Time = 0.29 (sec) , antiderivative size = 351, normalized size of antiderivative = 2.15 \[ \int \frac {1}{(a+b x) (c+d x) (e+f x) (g+h x)} \, dx=\frac {b^{3} \log \left ({\left | b x + a \right |}\right )}{b^{4} c e g - a b^{3} d e g - a b^{3} c f g + a^{2} b^{2} d f g - a b^{3} c e h + a^{2} b^{2} d e h + a^{2} b^{2} c f h - a^{3} b d f h} - \frac {d^{3} \log \left ({\left | d x + c \right |}\right )}{b c d^{3} e g - a d^{4} e g - b c^{2} d^{2} f g + a c d^{3} f g - b c^{2} d^{2} e h + a c d^{3} e h + b c^{3} d f h - a c^{2} d^{2} f h} + \frac {f^{3} \log \left ({\left | f x + e \right |}\right )}{b d e^{2} f^{2} g - b c e f^{3} g - a d e f^{3} g + a c f^{4} g - b d e^{3} f h + b c e^{2} f^{2} h + a d e^{2} f^{2} h - a c e f^{3} h} - \frac {h^{3} \log \left ({\left | h x + g \right |}\right )}{b d f g^{3} h - b d e g^{2} h^{2} - b c f g^{2} h^{2} - a d f g^{2} h^{2} + b c e g h^{3} + a d e g h^{3} + a c f g h^{3} - a c e h^{4}} \]

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

Mupad [B] (verification not implemented)

Time = 7.08 (sec) , antiderivative size = 317, normalized size of antiderivative = 1.94 \[ \int \frac {1}{(a+b x) (c+d x) (e+f x) (g+h x)} \, dx=\frac {b^2\,\ln \left (a+b\,x\right )}{b^3\,c\,e\,g-a^3\,d\,f\,h-a\,b^2\,c\,e\,h-a\,b^2\,c\,f\,g-a\,b^2\,d\,e\,g+a^2\,b\,c\,f\,h+a^2\,b\,d\,e\,h+a^2\,b\,d\,f\,g}+\frac {d^2\,\ln \left (c+d\,x\right )}{a\,d^3\,e\,g-b\,c^3\,f\,h-a\,c\,d^2\,e\,h-a\,c\,d^2\,f\,g-b\,c\,d^2\,e\,g+a\,c^2\,d\,f\,h+b\,c^2\,d\,e\,h+b\,c^2\,d\,f\,g}+\frac {f^2\,\ln \left (e+f\,x\right )}{a\,c\,f^3\,g-b\,d\,e^3\,h-a\,c\,e\,f^2\,h-a\,d\,e\,f^2\,g-b\,c\,e\,f^2\,g+a\,d\,e^2\,f\,h+b\,c\,e^2\,f\,h+b\,d\,e^2\,f\,g}+\frac {h^2\,\ln \left (g+h\,x\right )}{a\,c\,e\,h^3-b\,d\,f\,g^3-a\,c\,f\,g\,h^2-a\,d\,e\,g\,h^2-b\,c\,e\,g\,h^2+a\,d\,f\,g^2\,h+b\,c\,f\,g^2\,h+b\,d\,e\,g^2\,h} \]

[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)