3.18.15 \(\int \frac {1}{(a+b x)^2 (c+d x)^2 (e+f x)^2} \, dx\) [1715]

Optimal. Leaf size=234 \[ -\frac {b^3}{(b c-a d)^2 (b e-a f)^2 (a+b x)}-\frac {d^3}{(b c-a d)^2 (d e-c f)^2 (c+d x)}-\frac {f^3}{(b e-a f)^2 (d e-c f)^2 (e+f x)}-\frac {2 b^3 (b d e+b c f-2 a d f) \log (a+b x)}{(b c-a d)^3 (b e-a f)^3}+\frac {2 d^3 (b d e-2 b c f+a d f) \log (c+d x)}{(b c-a d)^3 (d e-c f)^3}+\frac {2 f^3 (2 b d e-b c f-a d f) \log (e+f x)}{(b e-a f)^3 (d e-c f)^3} \]

[Out]

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

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Rubi [A]
time = 0.29, antiderivative size = 234, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {90} \begin {gather*} -\frac {b^3}{(a+b x) (b c-a d)^2 (b e-a f)^2}-\frac {2 b^3 \log (a+b x) (-2 a d f+b c f+b d e)}{(b c-a d)^3 (b e-a f)^3}-\frac {d^3}{(c+d x) (b c-a d)^2 (d e-c f)^2}+\frac {2 d^3 \log (c+d x) (a d f-2 b c f+b d e)}{(b c-a d)^3 (d e-c f)^3}-\frac {f^3}{(e+f x) (b e-a f)^2 (d e-c f)^2}+\frac {2 f^3 \log (e+f x) (-a d f-b c f+2 b d e)}{(b e-a f)^3 (d e-c f)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((a + b*x)^2*(c + d*x)^2*(e + f*x)^2),x]

[Out]

-(b^3/((b*c - a*d)^2*(b*e - a*f)^2*(a + b*x))) - d^3/((b*c - a*d)^2*(d*e - c*f)^2*(c + d*x)) - f^3/((b*e - a*f
)^2*(d*e - c*f)^2*(e + f*x)) - (2*b^3*(b*d*e + b*c*f - 2*a*d*f)*Log[a + b*x])/((b*c - a*d)^3*(b*e - a*f)^3) +
(2*d^3*(b*d*e - 2*b*c*f + a*d*f)*Log[c + d*x])/((b*c - a*d)^3*(d*e - c*f)^3) + (2*f^3*(2*b*d*e - b*c*f - a*d*f
)*Log[e + f*x])/((b*e - a*f)^3*(d*e - c*f)^3)

Rule 90

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rubi steps

\begin {align*} \int \frac {1}{(a+b x)^2 (c+d x)^2 (e+f x)^2} \, dx &=\int \left (\frac {b^4}{(b c-a d)^2 (b e-a f)^2 (a+b x)^2}-\frac {2 b^4 (b d e+b c f-2 a d f)}{(b c-a d)^3 (b e-a f)^3 (a+b x)}+\frac {d^4}{(b c-a d)^2 (-d e+c f)^2 (c+d x)^2}-\frac {2 d^4 (b d e-2 b c f+a d f)}{(b c-a d)^3 (-d e+c f)^3 (c+d x)}+\frac {f^4}{(b e-a f)^2 (d e-c f)^2 (e+f x)^2}-\frac {2 f^4 (-2 b d e+b c f+a d f)}{(b e-a f)^3 (d e-c f)^3 (e+f x)}\right ) \, dx\\ &=-\frac {b^3}{(b c-a d)^2 (b e-a f)^2 (a+b x)}-\frac {d^3}{(b c-a d)^2 (d e-c f)^2 (c+d x)}-\frac {f^3}{(b e-a f)^2 (d e-c f)^2 (e+f x)}-\frac {2 b^3 (b d e+b c f-2 a d f) \log (a+b x)}{(b c-a d)^3 (b e-a f)^3}+\frac {2 d^3 (b d e-2 b c f+a d f) \log (c+d x)}{(b c-a d)^3 (d e-c f)^3}+\frac {2 f^3 (2 b d e-b c f-a d f) \log (e+f x)}{(b e-a f)^3 (d e-c f)^3}\\ \end {align*}

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Mathematica [A]
time = 0.43, size = 232, normalized size = 0.99 \begin {gather*} -\frac {b^3}{(b c-a d)^2 (b e-a f)^2 (a+b x)}-\frac {d^3}{(b c-a d)^2 (d e-c f)^2 (c+d x)}-\frac {f^3}{(b e-a f)^2 (d e-c f)^2 (e+f x)}-\frac {2 b^3 (b d e+b c f-2 a d f) \log (a+b x)}{(b c-a d)^3 (b e-a f)^3}-\frac {2 d^3 (b d e-2 b c f+a d f) \log (c+d x)}{(b c-a d)^3 (-d e+c f)^3}-\frac {2 f^3 (-2 b d e+b c f+a d f) \log (e+f x)}{(b e-a f)^3 (d e-c f)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b*x)^2*(c + d*x)^2*(e + f*x)^2),x]

[Out]

-(b^3/((b*c - a*d)^2*(b*e - a*f)^2*(a + b*x))) - d^3/((b*c - a*d)^2*(d*e - c*f)^2*(c + d*x)) - f^3/((b*e - a*f
)^2*(d*e - c*f)^2*(e + f*x)) - (2*b^3*(b*d*e + b*c*f - 2*a*d*f)*Log[a + b*x])/((b*c - a*d)^3*(b*e - a*f)^3) -
(2*d^3*(b*d*e - 2*b*c*f + a*d*f)*Log[c + d*x])/((b*c - a*d)^3*(-(d*e) + c*f)^3) - (2*f^3*(-2*b*d*e + b*c*f + a
*d*f)*Log[e + f*x])/((b*e - a*f)^3*(d*e - c*f)^3)

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Maple [A]
time = 0.27, size = 235, normalized size = 1.00

method	result	size
default	\(-\frac {b^{3}}{\left (a d -b c \right )^{2} \left (a f -b e \right )^{2} \left (b x +a \right )}+\frac {2 b^{3} \left (2 a d f -b c f -b d e \right ) \ln \left (b x +a \right )}{\left (a d -b c \right )^{3} \left (a f -b e \right )^{3}}-\frac {d^{3}}{\left (a d -b c \right )^{2} \left (c f -d e \right )^{2} \left (d x +c \right )}+\frac {2 d^{3} \left (a d f -2 b c f +b d e \right ) \ln \left (d x +c \right )}{\left (a d -b c \right )^{3} \left (c f -d e \right )^{3}}-\frac {f^{3}}{\left (a f -b e \right )^{2} \left (c f -d e \right )^{2} \left (f x +e \right )}-\frac {2 f^{3} \left (a d f +b c f -2 b d e \right ) \ln \left (f x +e \right )}{\left (a f -b e \right )^{3} \left (c f -d e \right )^{3}}\)	\(235\)
norman	\(\frac {\frac {-a^{3} b c \,d^{3} f^{4}-a^{3} b \,d^{4} e \,f^{3}+2 a^{2} b^{2} c^{2} d^{2} f^{4}+2 a^{2} b^{2} d^{4} e^{2} f^{2}-a \,b^{3} c^{3} d \,f^{4}-a \,b^{3} d^{4} e^{3} f -b^{4} c^{3} d e \,f^{3}+2 b^{4} c^{2} d^{2} e^{2} f^{2}-b^{4} c \,d^{3} e^{3} f}{f d b \left (a^{2} c^{2} f^{4}-2 a^{2} c d e \,f^{3}+a^{2} d^{2} e^{2} f^{2}-2 a b \,c^{2} e \,f^{3}+4 a b c d \,e^{2} f^{2}-2 a b \,d^{2} e^{3} f +b^{2} c^{2} e^{2} f^{2}-2 b^{2} c d \,e^{3} f +b^{2} d^{2} e^{4}\right ) \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {\left (-2 a^{2} b^{2} d^{4} f^{4}+2 a \,b^{3} c \,d^{3} f^{4}+2 a \,b^{3} d^{4} e \,f^{3}-2 b^{4} c^{2} d^{2} f^{4}+2 b^{4} c \,d^{3} e \,f^{3}-2 b^{4} d^{4} e^{2} f^{2}\right ) x^{2}}{f d b \left (a^{2} c^{2} f^{4}-2 a^{2} c d e \,f^{3}+a^{2} d^{2} e^{2} f^{2}-2 a b \,c^{2} e \,f^{3}+4 a b c d \,e^{2} f^{2}-2 a b \,d^{2} e^{3} f +b^{2} c^{2} e^{2} f^{2}-2 b^{2} c d \,e^{3} f +b^{2} d^{2} e^{4}\right ) \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {\left (-2 a^{3} b \,d^{4} f^{4}+a^{2} b^{2} c \,d^{3} f^{4}+a^{2} b^{2} d^{4} e \,f^{3}+a \,b^{3} c^{2} d^{2} f^{4}+a \,b^{3} d^{4} e^{2} f^{2}-2 b^{4} c^{3} d \,f^{4}+b^{4} c^{2} d^{2} e \,f^{3}+b^{4} c \,d^{3} e^{2} f^{2}-2 b^{4} d^{4} e^{3} f \right ) x}{f d b \left (a^{2} c^{2} f^{4}-2 a^{2} c d e \,f^{3}+a^{2} d^{2} e^{2} f^{2}-2 a b \,c^{2} e \,f^{3}+4 a b c d \,e^{2} f^{2}-2 a b \,d^{2} e^{3} f +b^{2} c^{2} e^{2} f^{2}-2 b^{2} c d \,e^{3} f +b^{2} d^{2} e^{4}\right ) \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}}{\left (b x +a \right ) \left (d x +c \right ) \left (f x +e \right )}-\frac {2 f^{3} \left (a d f +b c f -2 b d e \right ) \ln \left (f x +e \right )}{a^{3} c^{3} f^{6}-3 a^{3} c^{2} d e \,f^{5}+3 a^{3} c \,d^{2} e^{2} f^{4}-a^{3} d^{3} e^{3} f^{3}-3 a^{2} b \,c^{3} e \,f^{5}+9 a^{2} b \,c^{2} d \,e^{2} f^{4}-9 a^{2} b c \,d^{2} e^{3} f^{3}+3 a^{2} b \,d^{3} e^{4} f^{2}+3 a \,b^{2} c^{3} e^{2} f^{4}-9 a \,b^{2} c^{2} d \,e^{3} f^{3}+9 a \,b^{2} c \,d^{2} e^{4} f^{2}-3 a \,b^{2} d^{3} e^{5} f -b^{3} c^{3} e^{3} f^{3}+3 b^{3} c^{2} d \,e^{4} f^{2}-3 b^{3} c \,d^{2} e^{5} f +b^{3} d^{3} e^{6}}+\frac {2 b^{3} \left (2 a d f -b c f -b d e \right ) \ln \left (b x +a \right )}{\left (f^{3} a^{3}-3 b e \,f^{2} a^{2}+3 e^{2} b^{2} f a -e^{3} b^{3}\right ) \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {2 d^{3} \left (a d f -2 b c f +b d e \right ) \ln \left (d x +c \right )}{\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \left (c^{3} f^{3}-3 c^{2} d e \,f^{2}+3 c \,d^{2} e^{2} f -d^{3} e^{3}\right )}\)	\(1232\)
risch	\(\text {Expression too large to display}\)	\(3320\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x+a)^2/(d*x+c)^2/(f*x+e)^2,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 2261 vs. \(2 (247) = 494\).
time = 0.45, size = 2261, normalized size = 9.66 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)^2/(d*x+c)^2/(f*x+e)^2,x, algorithm="maxima")

[Out]

-2*(b^4*d*e + (b^4*c - 2*a*b^3*d)*f)*log(b*x + a)/(b^6*c^3*e^3 - 3*a*b^5*c^2*d*e^3 + 3*a^2*b^4*c*d^2*e^3 - a^3
*b^3*d^3*e^3 - (a^3*b^3*c^3 - 3*a^4*b^2*c^2*d + 3*a^5*b*c*d^2 - a^6*d^3)*f^3 + 3*(a^2*b^4*c^3*e - 3*a^3*b^3*c^
2*d*e + 3*a^4*b^2*c*d^2*e - a^5*b*d^3*e)*f^2 - 3*(a*b^5*c^3*e^2 - 3*a^2*b^4*c^2*d*e^2 + 3*a^3*b^3*c*d^2*e^2 -
a^4*b^2*d^3*e^2)*f) + 2*(b*d^4*e - (2*b*c*d^3 - a*d^4)*f)*log(d*x + c)/(b^3*c^3*d^3*e^3 - 3*a*b^2*c^2*d^4*e^3
+ 3*a^2*b*c*d^5*e^3 - a^3*d^6*e^3 - (b^3*c^6 - 3*a*b^2*c^5*d + 3*a^2*b*c^4*d^2 - a^3*c^3*d^3)*f^3 + 3*(b^3*c^5
*d*e - 3*a*b^2*c^4*d^2*e + 3*a^2*b*c^3*d^3*e - a^3*c^2*d^4*e)*f^2 - 3*(b^3*c^4*d^2*e^2 - 3*a*b^2*c^3*d^3*e^2 +
3*a^2*b*c^2*d^4*e^2 - a^3*c*d^5*e^2)*f) + 2*(2*b*d*f^3*e - (b*c + a*d)*f^4)*log(f*x + e)/(a^3*c^3*f^6 + b^3*d
^3*e^6 - 3*(a^2*b*c^3*e + a^3*c^2*d*e)*f^5 + 3*(a*b^2*c^3*e^2 + 3*a^2*b*c^2*d*e^2 + a^3*c*d^2*e^2)*f^4 - (b^3*
c^3*e^3 + 9*a*b^2*c^2*d*e^3 + 9*a^2*b*c*d^2*e^3 + a^3*d^3*e^3)*f^3 + 3*(b^3*c^2*d*e^4 + 3*a*b^2*c*d^2*e^4 + a^
2*b*d^3*e^4)*f^2 - 3*(b^3*c*d^2*e^5 + a*b^2*d^3*e^5)*f) - (b^3*c*d^2*e^3 + a*b^2*d^3*e^3 + (a*b^2*c^3 - 2*a^2*
b*c^2*d + a^3*c*d^2)*f^3 + (b^3*c^3*e + a^3*d^3*e)*f^2 + 2*(b^3*d^3*f*e^2 + (b^3*c^2*d - a*b^2*c*d^2 + a^2*b*d
^3)*f^3 - (b^3*c*d^2*e + a*b^2*d^3*e)*f^2)*x^2 - 2*(b^3*c^2*d*e^2 + a^2*b*d^3*e^2)*f + (2*b^3*d^3*e^3 + (2*b^3
*c^3 - a*b^2*c^2*d - a^2*b*c*d^2 + 2*a^3*d^3)*f^3 - (b^3*c^2*d*e + a^2*b*d^3*e)*f^2 - (b^3*c*d^2*e^2 + a*b^2*d
^3*e^2)*f)*x)/(a*b^4*c^3*d^2*e^5 - 2*a^2*b^3*c^2*d^3*e^5 + a^3*b^2*c*d^4*e^5 + (a^3*b^2*c^5*e - 2*a^4*b*c^4*d*
e + a^5*c^3*d^2*e)*f^4 - 2*(a^2*b^3*c^5*e^2 - a^3*b^2*c^4*d*e^2 - a^4*b*c^3*d^2*e^2 + a^5*c^2*d^3*e^2)*f^3 + (
(a^2*b^3*c^4*d - 2*a^3*b^2*c^3*d^2 + a^4*b*c^2*d^3)*f^5 - 2*(a*b^4*c^4*d*e - a^2*b^3*c^3*d^2*e - a^3*b^2*c^2*d
^3*e + a^4*b*c*d^4*e)*f^4 + (b^5*c^4*d*e^2 + 2*a*b^4*c^3*d^2*e^2 - 6*a^2*b^3*c^2*d^3*e^2 + 2*a^3*b^2*c*d^4*e^2
+ a^4*b*d^5*e^2)*f^3 - 2*(b^5*c^3*d^2*e^3 - a*b^4*c^2*d^3*e^3 - a^2*b^3*c*d^4*e^3 + a^3*b^2*d^5*e^3)*f^2 + (b
^5*c^2*d^3*e^4 - 2*a*b^4*c*d^4*e^4 + a^2*b^3*d^5*e^4)*f)*x^3 + (a*b^4*c^5*e^3 + 2*a^2*b^3*c^4*d*e^3 - 6*a^3*b^
2*c^3*d^2*e^3 + 2*a^4*b*c^2*d^3*e^3 + a^5*c*d^4*e^3)*f^2 + (b^5*c^2*d^3*e^5 - 2*a*b^4*c*d^4*e^5 + a^2*b^3*d^5*
e^5 + (a^2*b^3*c^5 - a^3*b^2*c^4*d - a^4*b*c^3*d^2 + a^5*c^2*d^3)*f^5 - (2*a*b^4*c^5*e - a^2*b^3*c^4*d*e - 2*a
^3*b^2*c^3*d^2*e - a^4*b*c^2*d^3*e + 2*a^5*c*d^4*e)*f^4 + (b^5*c^5*e^2 + a*b^4*c^4*d*e^2 - 2*a^2*b^3*c^3*d^2*e
^2 - 2*a^3*b^2*c^2*d^3*e^2 + a^4*b*c*d^4*e^2 + a^5*d^5*e^2)*f^3 - (b^5*c^4*d*e^3 - 2*a*b^4*c^3*d^2*e^3 + 2*a^2
*b^3*c^2*d^3*e^3 - 2*a^3*b^2*c*d^4*e^3 + a^4*b*d^5*e^3)*f^2 - (b^5*c^3*d^2*e^4 - a*b^4*c^2*d^3*e^4 - a^2*b^3*c
*d^4*e^4 + a^3*b^2*d^5*e^4)*f)*x^2 - 2*(a*b^4*c^4*d*e^4 - a^2*b^3*c^3*d^2*e^4 - a^3*b^2*c^2*d^3*e^4 + a^4*b*c*
d^4*e^4)*f + (b^5*c^3*d^2*e^5 - a*b^4*c^2*d^3*e^5 - a^2*b^3*c*d^4*e^5 + a^3*b^2*d^5*e^5 + (a^3*b^2*c^5 - 2*a^4
*b*c^4*d + a^5*c^3*d^2)*f^5 - (a^2*b^3*c^5*e - a^3*b^2*c^4*d*e - a^4*b*c^3*d^2*e + a^5*c^2*d^3*e)*f^4 - (a*b^4
*c^5*e^2 - 2*a^2*b^3*c^4*d*e^2 + 2*a^3*b^2*c^3*d^2*e^2 - 2*a^4*b*c^2*d^3*e^2 + a^5*c*d^4*e^2)*f^3 + (b^5*c^5*e
^3 + a*b^4*c^4*d*e^3 - 2*a^2*b^3*c^3*d^2*e^3 - 2*a^3*b^2*c^2*d^3*e^3 + a^4*b*c*d^4*e^3 + a^5*d^5*e^3)*f^2 - (2
*b^5*c^4*d*e^4 - a*b^4*c^3*d^2*e^4 - 2*a^2*b^3*c^2*d^3*e^4 - a^3*b^2*c*d^4*e^4 + 2*a^4*b*d^5*e^4)*f)*x)

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Fricas [F(-1)] Timed out
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)^2/(d*x+c)^2/(f*x+e)^2,x, algorithm="fricas")

[Out]

Timed out

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Sympy [F(-1)] Timed out
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)**2/(d*x+c)**2/(f*x+e)**2,x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1738 vs. \(2 (247) = 494\).
time = 0.72, size = 1738, normalized size = 7.43 \begin {gather*} -\frac {b^{7}}{{\left (a^{2} b^{6} c^{2} f^{2} - 2 \, a^{3} b^{5} c d f^{2} + a^{4} b^{4} d^{2} f^{2} - 2 \, a b^{7} c^{2} f e + 4 \, a^{2} b^{6} c d f e - 2 \, a^{3} b^{5} d^{2} f e + b^{8} c^{2} e^{2} - 2 \, a b^{7} c d e^{2} + a^{2} b^{6} d^{2} e^{2}\right )} {\left (b x + a\right )}} - \frac {{\left (b^{4} c f - 2 \, a b^{3} d f + b^{4} d e\right )} \log \left ({\left | \frac {b c f}{b x + a} - \frac {a b c f}{{\left (b x + a\right )}^{2}} - \frac {2 \, a d f}{b x + a} + \frac {a^{2} d f}{{\left (b x + a\right )}^{2}} + d f + \frac {b^{2} c e}{{\left (b x + a\right )}^{2}} + \frac {b d e}{b x + a} - \frac {a b d e}{{\left (b x + a\right )}^{2}} \right |}\right )}{a^{3} b^{3} c^{3} f^{3} - 3 \, a^{4} b^{2} c^{2} d f^{3} + 3 \, a^{5} b c d^{2} f^{3} - a^{6} d^{3} f^{3} - 3 \, a^{2} b^{4} c^{3} f^{2} e + 9 \, a^{3} b^{3} c^{2} d f^{2} e - 9 \, a^{4} b^{2} c d^{2} f^{2} e + 3 \, a^{5} b d^{3} f^{2} e + 3 \, a b^{5} c^{3} f e^{2} - 9 \, a^{2} b^{4} c^{2} d f e^{2} + 9 \, a^{3} b^{3} c d^{2} f e^{2} - 3 \, a^{4} b^{2} d^{3} f e^{2} - b^{6} c^{3} e^{3} + 3 \, a b^{5} c^{2} d e^{3} - 3 \, a^{2} b^{4} c d^{2} e^{3} + a^{3} b^{3} d^{3} e^{3}} - \frac {{\left (b^{6} c^{4} f^{4} - 2 \, a b^{5} c^{3} d f^{4} + 4 \, a^{3} b^{3} c d^{3} f^{4} - 2 \, a^{4} b^{2} d^{4} f^{4} - 2 \, b^{6} c^{3} d f^{3} e + 6 \, a b^{5} c^{2} d^{2} f^{3} e - 12 \, a^{2} b^{4} c d^{3} f^{3} e + 4 \, a^{3} b^{3} d^{4} f^{3} e + 6 \, a b^{5} c d^{3} f^{2} e^{2} - 2 \, b^{6} c d^{3} f e^{3} - 2 \, a b^{5} d^{4} f e^{3} + b^{6} d^{4} e^{4}\right )} \log \left (\frac {{\left | \frac {2 \, a b^{2} c f}{b x + a} - b^{2} c f + 2 \, a b d f - \frac {2 \, a^{2} b d f}{b x + a} - \frac {2 \, b^{3} c e}{b x + a} + \frac {2 \, a b^{2} d e}{b x + a} - b^{2} d e - {\left | -b^{2} c f + b^{2} d e \right |} \right |}}{{\left | \frac {2 \, a b^{2} c f}{b x + a} - b^{2} c f + 2 \, a b d f - \frac {2 \, a^{2} b d f}{b x + a} - \frac {2 \, b^{3} c e}{b x + a} + \frac {2 \, a b^{2} d e}{b x + a} - b^{2} d e + {\left | -b^{2} c f + b^{2} d e \right |} \right |}}\right )}{{\left (a^{3} b^{3} c^{5} f^{5} - 3 \, a^{4} b^{2} c^{4} d f^{5} + 3 \, a^{5} b c^{3} d^{2} f^{5} - a^{6} c^{2} d^{3} f^{5} - 3 \, a^{2} b^{4} c^{5} f^{4} e + 7 \, a^{3} b^{3} c^{4} d f^{4} e - 3 \, a^{4} b^{2} c^{3} d^{2} f^{4} e - 3 \, a^{5} b c^{2} d^{3} f^{4} e + 2 \, a^{6} c d^{4} f^{4} e + 3 \, a b^{5} c^{5} f^{3} e^{2} - 3 \, a^{2} b^{4} c^{4} d f^{3} e^{2} - 8 \, a^{3} b^{3} c^{3} d^{2} f^{3} e^{2} + 12 \, a^{4} b^{2} c^{2} d^{3} f^{3} e^{2} - 3 \, a^{5} b c d^{4} f^{3} e^{2} - a^{6} d^{5} f^{3} e^{2} - b^{6} c^{5} f^{2} e^{3} - 3 \, a b^{5} c^{4} d f^{2} e^{3} + 12 \, a^{2} b^{4} c^{3} d^{2} f^{2} e^{3} - 8 \, a^{3} b^{3} c^{2} d^{3} f^{2} e^{3} - 3 \, a^{4} b^{2} c d^{4} f^{2} e^{3} + 3 \, a^{5} b d^{5} f^{2} e^{3} + 2 \, b^{6} c^{4} d f e^{4} - 3 \, a b^{5} c^{3} d^{2} f e^{4} - 3 \, a^{2} b^{4} c^{2} d^{3} f e^{4} + 7 \, a^{3} b^{3} c d^{4} f e^{4} - 3 \, a^{4} b^{2} d^{5} f e^{4} - b^{6} c^{3} d^{2} e^{5} + 3 \, a b^{5} c^{2} d^{3} e^{5} - 3 \, a^{2} b^{4} c d^{4} e^{5} + a^{3} b^{3} d^{5} e^{5}\right )} {\left | -b^{2} c f + b^{2} d e \right |}} - \frac {\frac {b^{4} c^{3} d f^{4} - 3 \, a b^{3} c^{2} d^{2} f^{4} + 3 \, a^{2} b^{2} c d^{3} f^{4} - 2 \, a^{3} b d^{4} f^{4} + 3 \, a^{2} b^{2} d^{4} f^{3} e - 3 \, a b^{3} d^{4} f^{2} e^{2} + b^{4} d^{4} f e^{3}}{a b c f - a^{2} d f - b^{2} c e + a b d e} + \frac {b^{6} c^{4} f^{4} - 4 \, a b^{5} c^{3} d f^{4} + 6 \, a^{2} b^{4} c^{2} d^{2} f^{4} - 4 \, a^{3} b^{3} c d^{3} f^{4} + 2 \, a^{4} b^{2} d^{4} f^{4} - 4 \, a^{3} b^{3} d^{4} f^{3} e + 6 \, a^{2} b^{4} d^{4} f^{2} e^{2} - 4 \, a b^{5} d^{4} f e^{3} + b^{6} d^{4} e^{4}}{{\left (a b c f - a^{2} d f - b^{2} c e + a b d e\right )} {\left (b x + a\right )} b}}{{\left (b c - a d\right )}^{2} {\left (a f - b e\right )}^{2} {\left (\frac {b c f}{b x + a} - \frac {a b c f}{{\left (b x + a\right )}^{2}} - \frac {2 \, a d f}{b x + a} + \frac {a^{2} d f}{{\left (b x + a\right )}^{2}} + d f + \frac {b^{2} c e}{{\left (b x + a\right )}^{2}} + \frac {b d e}{b x + a} - \frac {a b d e}{{\left (b x + a\right )}^{2}}\right )} {\left (c f - d e\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)^2/(d*x+c)^2/(f*x+e)^2,x, algorithm="giac")

[Out]

-b^7/((a^2*b^6*c^2*f^2 - 2*a^3*b^5*c*d*f^2 + a^4*b^4*d^2*f^2 - 2*a*b^7*c^2*f*e + 4*a^2*b^6*c*d*f*e - 2*a^3*b^5
*d^2*f*e + b^8*c^2*e^2 - 2*a*b^7*c*d*e^2 + a^2*b^6*d^2*e^2)*(b*x + a)) - (b^4*c*f - 2*a*b^3*d*f + b^4*d*e)*log
(abs(b*c*f/(b*x + a) - a*b*c*f/(b*x + a)^2 - 2*a*d*f/(b*x + a) + a^2*d*f/(b*x + a)^2 + d*f + b^2*c*e/(b*x + a)
^2 + b*d*e/(b*x + a) - a*b*d*e/(b*x + a)^2))/(a^3*b^3*c^3*f^3 - 3*a^4*b^2*c^2*d*f^3 + 3*a^5*b*c*d^2*f^3 - a^6*
d^3*f^3 - 3*a^2*b^4*c^3*f^2*e + 9*a^3*b^3*c^2*d*f^2*e - 9*a^4*b^2*c*d^2*f^2*e + 3*a^5*b*d^3*f^2*e + 3*a*b^5*c^
3*f*e^2 - 9*a^2*b^4*c^2*d*f*e^2 + 9*a^3*b^3*c*d^2*f*e^2 - 3*a^4*b^2*d^3*f*e^2 - b^6*c^3*e^3 + 3*a*b^5*c^2*d*e^
3 - 3*a^2*b^4*c*d^2*e^3 + a^3*b^3*d^3*e^3) - (b^6*c^4*f^4 - 2*a*b^5*c^3*d*f^4 + 4*a^3*b^3*c*d^3*f^4 - 2*a^4*b^
2*d^4*f^4 - 2*b^6*c^3*d*f^3*e + 6*a*b^5*c^2*d^2*f^3*e - 12*a^2*b^4*c*d^3*f^3*e + 4*a^3*b^3*d^4*f^3*e + 6*a*b^5
*c*d^3*f^2*e^2 - 2*b^6*c*d^3*f*e^3 - 2*a*b^5*d^4*f*e^3 + b^6*d^4*e^4)*log(abs(2*a*b^2*c*f/(b*x + a) - b^2*c*f
+ 2*a*b*d*f - 2*a^2*b*d*f/(b*x + a) - 2*b^3*c*e/(b*x + a) + 2*a*b^2*d*e/(b*x + a) - b^2*d*e - abs(-b^2*c*f + b
^2*d*e))/abs(2*a*b^2*c*f/(b*x + a) - b^2*c*f + 2*a*b*d*f - 2*a^2*b*d*f/(b*x + a) - 2*b^3*c*e/(b*x + a) + 2*a*b
^2*d*e/(b*x + a) - b^2*d*e + abs(-b^2*c*f + b^2*d*e)))/((a^3*b^3*c^5*f^5 - 3*a^4*b^2*c^4*d*f^5 + 3*a^5*b*c^3*d
^2*f^5 - a^6*c^2*d^3*f^5 - 3*a^2*b^4*c^5*f^4*e + 7*a^3*b^3*c^4*d*f^4*e - 3*a^4*b^2*c^3*d^2*f^4*e - 3*a^5*b*c^2
*d^3*f^4*e + 2*a^6*c*d^4*f^4*e + 3*a*b^5*c^5*f^3*e^2 - 3*a^2*b^4*c^4*d*f^3*e^2 - 8*a^3*b^3*c^3*d^2*f^3*e^2 + 1
2*a^4*b^2*c^2*d^3*f^3*e^2 - 3*a^5*b*c*d^4*f^3*e^2 - a^6*d^5*f^3*e^2 - b^6*c^5*f^2*e^3 - 3*a*b^5*c^4*d*f^2*e^3
+ 12*a^2*b^4*c^3*d^2*f^2*e^3 - 8*a^3*b^3*c^2*d^3*f^2*e^3 - 3*a^4*b^2*c*d^4*f^2*e^3 + 3*a^5*b*d^5*f^2*e^3 + 2*b
^6*c^4*d*f*e^4 - 3*a*b^5*c^3*d^2*f*e^4 - 3*a^2*b^4*c^2*d^3*f*e^4 + 7*a^3*b^3*c*d^4*f*e^4 - 3*a^4*b^2*d^5*f*e^4
- b^6*c^3*d^2*e^5 + 3*a*b^5*c^2*d^3*e^5 - 3*a^2*b^4*c*d^4*e^5 + a^3*b^3*d^5*e^5)*abs(-b^2*c*f + b^2*d*e)) - (
(b^4*c^3*d*f^4 - 3*a*b^3*c^2*d^2*f^4 + 3*a^2*b^2*c*d^3*f^4 - 2*a^3*b*d^4*f^4 + 3*a^2*b^2*d^4*f^3*e - 3*a*b^3*d
^4*f^2*e^2 + b^4*d^4*f*e^3)/(a*b*c*f - a^2*d*f - b^2*c*e + a*b*d*e) + (b^6*c^4*f^4 - 4*a*b^5*c^3*d*f^4 + 6*a^2
*b^4*c^2*d^2*f^4 - 4*a^3*b^3*c*d^3*f^4 + 2*a^4*b^2*d^4*f^4 - 4*a^3*b^3*d^4*f^3*e + 6*a^2*b^4*d^4*f^2*e^2 - 4*a
*b^5*d^4*f*e^3 + b^6*d^4*e^4)/((a*b*c*f - a^2*d*f - b^2*c*e + a*b*d*e)*(b*x + a)*b))/((b*c - a*d)^2*(a*f - b*e
)^2*(b*c*f/(b*x + a) - a*b*c*f/(b*x + a)^2 - 2*a*d*f/(b*x + a) + a^2*d*f/(b*x + a)^2 + d*f + b^2*c*e/(b*x + a)
^2 + b*d*e/(b*x + a) - a*b*d*e/(b*x + a)^2)*(c*f - d*e)^2)

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Mupad [B]
time = 7.10, size = 1940, normalized size = 8.29 \begin {gather*} -\frac {\frac {a^3\,c\,d^2\,f^3+a^3\,d^3\,e\,f^2-2\,a^2\,b\,c^2\,d\,f^3-2\,a^2\,b\,d^3\,e^2\,f+a\,b^2\,c^3\,f^3+a\,b^2\,d^3\,e^3+b^3\,c^3\,e\,f^2-2\,b^3\,c^2\,d\,e^2\,f+b^3\,c\,d^2\,e^3}{a^4\,c^2\,d^2\,f^4-2\,a^4\,c\,d^3\,e\,f^3+a^4\,d^4\,e^2\,f^2-2\,a^3\,b\,c^3\,d\,f^4+2\,a^3\,b\,c^2\,d^2\,e\,f^3+2\,a^3\,b\,c\,d^3\,e^2\,f^2-2\,a^3\,b\,d^4\,e^3\,f+a^2\,b^2\,c^4\,f^4+2\,a^2\,b^2\,c^3\,d\,e\,f^3-6\,a^2\,b^2\,c^2\,d^2\,e^2\,f^2+2\,a^2\,b^2\,c\,d^3\,e^3\,f+a^2\,b^2\,d^4\,e^4-2\,a\,b^3\,c^4\,e\,f^3+2\,a\,b^3\,c^3\,d\,e^2\,f^2+2\,a\,b^3\,c^2\,d^2\,e^3\,f-2\,a\,b^3\,c\,d^3\,e^4+b^4\,c^4\,e^2\,f^2-2\,b^4\,c^3\,d\,e^3\,f+b^4\,c^2\,d^2\,e^4}+\frac {2\,x^2\,\left (a^2\,b\,d^3\,f^3-a\,b^2\,c\,d^2\,f^3-a\,b^2\,d^3\,e\,f^2+b^3\,c^2\,d\,f^3-b^3\,c\,d^2\,e\,f^2+b^3\,d^3\,e^2\,f\right )}{a^4\,c^2\,d^2\,f^4-2\,a^4\,c\,d^3\,e\,f^3+a^4\,d^4\,e^2\,f^2-2\,a^3\,b\,c^3\,d\,f^4+2\,a^3\,b\,c^2\,d^2\,e\,f^3+2\,a^3\,b\,c\,d^3\,e^2\,f^2-2\,a^3\,b\,d^4\,e^3\,f+a^2\,b^2\,c^4\,f^4+2\,a^2\,b^2\,c^3\,d\,e\,f^3-6\,a^2\,b^2\,c^2\,d^2\,e^2\,f^2+2\,a^2\,b^2\,c\,d^3\,e^3\,f+a^2\,b^2\,d^4\,e^4-2\,a\,b^3\,c^4\,e\,f^3+2\,a\,b^3\,c^3\,d\,e^2\,f^2+2\,a\,b^3\,c^2\,d^2\,e^3\,f-2\,a\,b^3\,c\,d^3\,e^4+b^4\,c^4\,e^2\,f^2-2\,b^4\,c^3\,d\,e^3\,f+b^4\,c^2\,d^2\,e^4}-\frac {x\,\left (-2\,a^3\,d^3\,f^3+a^2\,b\,c\,d^2\,f^3+a^2\,b\,d^3\,e\,f^2+a\,b^2\,c^2\,d\,f^3+a\,b^2\,d^3\,e^2\,f-2\,b^3\,c^3\,f^3+b^3\,c^2\,d\,e\,f^2+b^3\,c\,d^2\,e^2\,f-2\,b^3\,d^3\,e^3\right )}{a^4\,c^2\,d^2\,f^4-2\,a^4\,c\,d^3\,e\,f^3+a^4\,d^4\,e^2\,f^2-2\,a^3\,b\,c^3\,d\,f^4+2\,a^3\,b\,c^2\,d^2\,e\,f^3+2\,a^3\,b\,c\,d^3\,e^2\,f^2-2\,a^3\,b\,d^4\,e^3\,f+a^2\,b^2\,c^4\,f^4+2\,a^2\,b^2\,c^3\,d\,e\,f^3-6\,a^2\,b^2\,c^2\,d^2\,e^2\,f^2+2\,a^2\,b^2\,c\,d^3\,e^3\,f+a^2\,b^2\,d^4\,e^4-2\,a\,b^3\,c^4\,e\,f^3+2\,a\,b^3\,c^3\,d\,e^2\,f^2+2\,a\,b^3\,c^2\,d^2\,e^3\,f-2\,a\,b^3\,c\,d^3\,e^4+b^4\,c^4\,e^2\,f^2-2\,b^4\,c^3\,d\,e^3\,f+b^4\,c^2\,d^2\,e^4}}{b\,d\,f\,x^3+\left (a\,d\,f+b\,c\,f+b\,d\,e\right )\,x^2+\left (a\,c\,f+a\,d\,e+b\,c\,e\right )\,x+a\,c\,e}-\frac {\ln \left (a+b\,x\right )\,\left (b^4\,\left (2\,c\,f+2\,d\,e\right )-4\,a\,b^3\,d\,f\right )}{a^6\,d^3\,f^3-3\,a^5\,b\,c\,d^2\,f^3-3\,a^5\,b\,d^3\,e\,f^2+3\,a^4\,b^2\,c^2\,d\,f^3+9\,a^4\,b^2\,c\,d^2\,e\,f^2+3\,a^4\,b^2\,d^3\,e^2\,f-a^3\,b^3\,c^3\,f^3-9\,a^3\,b^3\,c^2\,d\,e\,f^2-9\,a^3\,b^3\,c\,d^2\,e^2\,f-a^3\,b^3\,d^3\,e^3+3\,a^2\,b^4\,c^3\,e\,f^2+9\,a^2\,b^4\,c^2\,d\,e^2\,f+3\,a^2\,b^4\,c\,d^2\,e^3-3\,a\,b^5\,c^3\,e^2\,f-3\,a\,b^5\,c^2\,d\,e^3+b^6\,c^3\,e^3}-\frac {\ln \left (c+d\,x\right )\,\left (d^4\,\left (2\,a\,f+2\,b\,e\right )-4\,b\,c\,d^3\,f\right )}{-a^3\,c^3\,d^3\,f^3+3\,a^3\,c^2\,d^4\,e\,f^2-3\,a^3\,c\,d^5\,e^2\,f+a^3\,d^6\,e^3+3\,a^2\,b\,c^4\,d^2\,f^3-9\,a^2\,b\,c^3\,d^3\,e\,f^2+9\,a^2\,b\,c^2\,d^4\,e^2\,f-3\,a^2\,b\,c\,d^5\,e^3-3\,a\,b^2\,c^5\,d\,f^3+9\,a\,b^2\,c^4\,d^2\,e\,f^2-9\,a\,b^2\,c^3\,d^3\,e^2\,f+3\,a\,b^2\,c^2\,d^4\,e^3+b^3\,c^6\,f^3-3\,b^3\,c^5\,d\,e\,f^2+3\,b^3\,c^4\,d^2\,e^2\,f-b^3\,c^3\,d^3\,e^3}-\frac {\ln \left (e+f\,x\right )\,\left (f^4\,\left (2\,a\,d+2\,b\,c\right )-4\,b\,d\,e\,f^3\right )}{a^3\,c^3\,f^6-3\,a^3\,c^2\,d\,e\,f^5+3\,a^3\,c\,d^2\,e^2\,f^4-a^3\,d^3\,e^3\,f^3-3\,a^2\,b\,c^3\,e\,f^5+9\,a^2\,b\,c^2\,d\,e^2\,f^4-9\,a^2\,b\,c\,d^2\,e^3\,f^3+3\,a^2\,b\,d^3\,e^4\,f^2+3\,a\,b^2\,c^3\,e^2\,f^4-9\,a\,b^2\,c^2\,d\,e^3\,f^3+9\,a\,b^2\,c\,d^2\,e^4\,f^2-3\,a\,b^2\,d^3\,e^5\,f-b^3\,c^3\,e^3\,f^3+3\,b^3\,c^2\,d\,e^4\,f^2-3\,b^3\,c\,d^2\,e^5\,f+b^3\,d^3\,e^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((e + f*x)^2*(a + b*x)^2*(c + d*x)^2),x)

[Out]

- ((a*b^2*c^3*f^3 + a*b^2*d^3*e^3 + a^3*c*d^2*f^3 + b^3*c*d^2*e^3 + a^3*d^3*e*f^2 + b^3*c^3*e*f^2 - 2*a^2*b*c^
2*d*f^3 - 2*a^2*b*d^3*e^2*f - 2*b^3*c^2*d*e^2*f)/(a^2*b^2*c^4*f^4 + a^2*b^2*d^4*e^4 + a^4*c^2*d^2*f^4 + b^4*c^
2*d^2*e^4 + a^4*d^4*e^2*f^2 + b^4*c^4*e^2*f^2 - 2*a*b^3*c*d^3*e^4 - 2*a^3*b*c^3*d*f^4 - 2*a*b^3*c^4*e*f^3 - 2*
a^3*b*d^4*e^3*f - 2*a^4*c*d^3*e*f^3 - 2*b^4*c^3*d*e^3*f + 2*a*b^3*c^2*d^2*e^3*f + 2*a*b^3*c^3*d*e^2*f^2 + 2*a^
2*b^2*c*d^3*e^3*f + 2*a^2*b^2*c^3*d*e*f^3 + 2*a^3*b*c*d^3*e^2*f^2 + 2*a^3*b*c^2*d^2*e*f^3 - 6*a^2*b^2*c^2*d^2*
e^2*f^2) + (2*x^2*(a^2*b*d^3*f^3 + b^3*c^2*d*f^3 + b^3*d^3*e^2*f - a*b^2*c*d^2*f^3 - a*b^2*d^3*e*f^2 - b^3*c*d
^2*e*f^2))/(a^2*b^2*c^4*f^4 + a^2*b^2*d^4*e^4 + a^4*c^2*d^2*f^4 + b^4*c^2*d^2*e^4 + a^4*d^4*e^2*f^2 + b^4*c^4*
e^2*f^2 - 2*a*b^3*c*d^3*e^4 - 2*a^3*b*c^3*d*f^4 - 2*a*b^3*c^4*e*f^3 - 2*a^3*b*d^4*e^3*f - 2*a^4*c*d^3*e*f^3 -
2*b^4*c^3*d*e^3*f + 2*a*b^3*c^2*d^2*e^3*f + 2*a*b^3*c^3*d*e^2*f^2 + 2*a^2*b^2*c*d^3*e^3*f + 2*a^2*b^2*c^3*d*e*
f^3 + 2*a^3*b*c*d^3*e^2*f^2 + 2*a^3*b*c^2*d^2*e*f^3 - 6*a^2*b^2*c^2*d^2*e^2*f^2) - (x*(a*b^2*c^2*d*f^3 - 2*b^3
*c^3*f^3 - 2*b^3*d^3*e^3 - 2*a^3*d^3*f^3 + a^2*b*c*d^2*f^3 + a*b^2*d^3*e^2*f + a^2*b*d^3*e*f^2 + b^3*c*d^2*e^2
*f + b^3*c^2*d*e*f^2))/(a^2*b^2*c^4*f^4 + a^2*b^2*d^4*e^4 + a^4*c^2*d^2*f^4 + b^4*c^2*d^2*e^4 + a^4*d^4*e^2*f^
2 + b^4*c^4*e^2*f^2 - 2*a*b^3*c*d^3*e^4 - 2*a^3*b*c^3*d*f^4 - 2*a*b^3*c^4*e*f^3 - 2*a^3*b*d^4*e^3*f - 2*a^4*c*
d^3*e*f^3 - 2*b^4*c^3*d*e^3*f + 2*a*b^3*c^2*d^2*e^3*f + 2*a*b^3*c^3*d*e^2*f^2 + 2*a^2*b^2*c*d^3*e^3*f + 2*a^2*
b^2*c^3*d*e*f^3 + 2*a^3*b*c*d^3*e^2*f^2 + 2*a^3*b*c^2*d^2*e*f^3 - 6*a^2*b^2*c^2*d^2*e^2*f^2))/(x^2*(a*d*f + b*
c*f + b*d*e) + x*(a*c*f + a*d*e + b*c*e) + a*c*e + b*d*f*x^3) - (log(a + b*x)*(b^4*(2*c*f + 2*d*e) - 4*a*b^3*d
*f))/(b^6*c^3*e^3 + a^6*d^3*f^3 - a^3*b^3*c^3*f^3 - a^3*b^3*d^3*e^3 - 3*a*b^5*c^2*d*e^3 - 3*a^5*b*c*d^2*f^3 -
3*a*b^5*c^3*e^2*f - 3*a^5*b*d^3*e*f^2 + 3*a^2*b^4*c*d^2*e^3 + 3*a^4*b^2*c^2*d*f^3 + 3*a^2*b^4*c^3*e*f^2 + 3*a^
4*b^2*d^3*e^2*f + 9*a^2*b^4*c^2*d*e^2*f - 9*a^3*b^3*c*d^2*e^2*f - 9*a^3*b^3*c^2*d*e*f^2 + 9*a^4*b^2*c*d^2*e*f^
2) - (log(c + d*x)*(d^4*(2*a*f + 2*b*e) - 4*b*c*d^3*f))/(a^3*d^6*e^3 + b^3*c^6*f^3 - a^3*c^3*d^3*f^3 - b^3*c^3
*d^3*e^3 - 3*a^2*b*c*d^5*e^3 - 3*a*b^2*c^5*d*f^3 - 3*a^3*c*d^5*e^2*f - 3*b^3*c^5*d*e*f^2 + 3*a*b^2*c^2*d^4*e^3
+ 3*a^2*b*c^4*d^2*f^3 + 3*a^3*c^2*d^4*e*f^2 + 3*b^3*c^4*d^2*e^2*f - 9*a*b^2*c^3*d^3*e^2*f + 9*a*b^2*c^4*d^2*e
*f^2 + 9*a^2*b*c^2*d^4*e^2*f - 9*a^2*b*c^3*d^3*e*f^2) - (log(e + f*x)*(f^4*(2*a*d + 2*b*c) - 4*b*d*e*f^3))/(a^
3*c^3*f^6 + b^3*d^3*e^6 - a^3*d^3*e^3*f^3 - b^3*c^3*e^3*f^3 - 3*a^2*b*c^3*e*f^5 - 3*a*b^2*d^3*e^5*f - 3*a^3*c^
2*d*e*f^5 - 3*b^3*c*d^2*e^5*f + 3*a*b^2*c^3*e^2*f^4 + 3*a^2*b*d^3*e^4*f^2 + 3*a^3*c*d^2*e^2*f^4 + 3*b^3*c^2*d*
e^4*f^2 + 9*a*b^2*c*d^2*e^4*f^2 - 9*a*b^2*c^2*d*e^3*f^3 - 9*a^2*b*c*d^2*e^3*f^3 + 9*a^2*b*c^2*d*e^2*f^4)

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