3.18.16 \(\int \frac {1}{(a+b x)^3 (c+d x)^3 (e+f x)^3} \, dx\) [1716]

Optimal. Leaf size=495 \[ -\frac {b^5}{2 (b c-a d)^3 (b e-a f)^3 (a+b x)^2}+\frac {3 b^5 (b d e+b c f-2 a d f)}{(b c-a d)^4 (b e-a f)^4 (a+b x)}+\frac {d^5}{2 (b c-a d)^3 (d e-c f)^3 (c+d x)^2}+\frac {3 d^5 (b d e-2 b c f+a d f)}{(b c-a d)^4 (d e-c f)^4 (c+d x)}-\frac {f^5}{2 (b e-a f)^3 (d e-c f)^3 (e+f x)^2}-\frac {3 f^5 (2 b d e-b c f-a d f)}{(b e-a f)^4 (d e-c f)^4 (e+f x)}+\frac {3 b^5 \left (7 a^2 d^2 f^2-7 a b d f (d e+c f)+b^2 \left (2 d^2 e^2+3 c d e f+2 c^2 f^2\right )\right ) \log (a+b x)}{(b c-a d)^5 (b e-a f)^5}-\frac {3 d^5 \left (2 a^2 d^2 f^2+a b d f (3 d e-7 c f)+b^2 \left (2 d^2 e^2-7 c d e f+7 c^2 f^2\right )\right ) \log (c+d x)}{(b c-a d)^5 (d e-c f)^5}+\frac {3 f^5 \left (2 a^2 d^2 f^2-a b d f (7 d e-3 c f)+b^2 \left (7 d^2 e^2-7 c d e f+2 c^2 f^2\right )\right ) \log (e+f x)}{(b e-a f)^5 (d e-c f)^5} \]

[Out]

-1/2*b^5/(-a*d+b*c)^3/(-a*f+b*e)^3/(b*x+a)^2+3*b^5*(-2*a*d*f+b*c*f+b*d*e)/(-a*d+b*c)^4/(-a*f+b*e)^4/(b*x+a)+1/
2*d^5/(-a*d+b*c)^3/(-c*f+d*e)^3/(d*x+c)^2+3*d^5*(a*d*f-2*b*c*f+b*d*e)/(-a*d+b*c)^4/(-c*f+d*e)^4/(d*x+c)-1/2*f^
5/(-a*f+b*e)^3/(-c*f+d*e)^3/(f*x+e)^2-3*f^5*(-a*d*f-b*c*f+2*b*d*e)/(-a*f+b*e)^4/(-c*f+d*e)^4/(f*x+e)+3*b^5*(7*
a^2*d^2*f^2-7*a*b*d*f*(c*f+d*e)+b^2*(2*c^2*f^2+3*c*d*e*f+2*d^2*e^2))*ln(b*x+a)/(-a*d+b*c)^5/(-a*f+b*e)^5-3*d^5
*(2*a^2*d^2*f^2+a*b*d*f*(-7*c*f+3*d*e)+b^2*(7*c^2*f^2-7*c*d*e*f+2*d^2*e^2))*ln(d*x+c)/(-a*d+b*c)^5/(-c*f+d*e)^
5+3*f^5*(2*a^2*d^2*f^2-a*b*d*f*(-3*c*f+7*d*e)+b^2*(2*c^2*f^2-7*c*d*e*f+7*d^2*e^2))*ln(f*x+e)/(-a*f+b*e)^5/(-c*
f+d*e)^5

________________________________________________________________________________________

Rubi [A]
time = 1.05, antiderivative size = 495, 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 {3 f^5 \log (e+f x) \left (2 a^2 d^2 f^2-a b d f (7 d e-3 c f)+b^2 \left (2 c^2 f^2-7 c d e f+7 d^2 e^2\right )\right )}{(b e-a f)^5 (d e-c f)^5}-\frac {3 d^5 \log (c+d x) \left (2 a^2 d^2 f^2+a b d f (3 d e-7 c f)+b^2 \left (7 c^2 f^2-7 c d e f+2 d^2 e^2\right )\right )}{(b c-a d)^5 (d e-c f)^5}+\frac {3 b^5 \log (a+b x) \left (7 a^2 d^2 f^2-7 a b d f (c f+d e)+b^2 \left (2 c^2 f^2+3 c d e f+2 d^2 e^2\right )\right )}{(b c-a d)^5 (b e-a f)^5}+\frac {3 b^5 (-2 a d f+b c f+b d e)}{(a+b x) (b c-a d)^4 (b e-a f)^4}-\frac {b^5}{2 (a+b x)^2 (b c-a d)^3 (b e-a f)^3}+\frac {3 d^5 (a d f-2 b c f+b d e)}{(c+d x) (b c-a d)^4 (d e-c f)^4}+\frac {d^5}{2 (c+d x)^2 (b c-a d)^3 (d e-c f)^3}-\frac {3 f^5 (-a d f-b c f+2 b d e)}{(e+f x) (b e-a f)^4 (d e-c f)^4}-\frac {f^5}{2 (e+f x)^2 (b e-a f)^3 (d e-c f)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((a + b*x)^3*(c + d*x)^3*(e + f*x)^3),x]

[Out]

-1/2*b^5/((b*c - a*d)^3*(b*e - a*f)^3*(a + b*x)^2) + (3*b^5*(b*d*e + b*c*f - 2*a*d*f))/((b*c - a*d)^4*(b*e - a
*f)^4*(a + b*x)) + d^5/(2*(b*c - a*d)^3*(d*e - c*f)^3*(c + d*x)^2) + (3*d^5*(b*d*e - 2*b*c*f + a*d*f))/((b*c -
a*d)^4*(d*e - c*f)^4*(c + d*x)) - f^5/(2*(b*e - a*f)^3*(d*e - c*f)^3*(e + f*x)^2) - (3*f^5*(2*b*d*e - b*c*f -
a*d*f))/((b*e - a*f)^4*(d*e - c*f)^4*(e + f*x)) + (3*b^5*(7*a^2*d^2*f^2 - 7*a*b*d*f*(d*e + c*f) + b^2*(2*d^2*
e^2 + 3*c*d*e*f + 2*c^2*f^2))*Log[a + b*x])/((b*c - a*d)^5*(b*e - a*f)^5) - (3*d^5*(2*a^2*d^2*f^2 + a*b*d*f*(3
*d*e - 7*c*f) + b^2*(2*d^2*e^2 - 7*c*d*e*f + 7*c^2*f^2))*Log[c + d*x])/((b*c - a*d)^5*(d*e - c*f)^5) + (3*f^5*
(2*a^2*d^2*f^2 - a*b*d*f*(7*d*e - 3*c*f) + b^2*(7*d^2*e^2 - 7*c*d*e*f + 2*c^2*f^2))*Log[e + f*x])/((b*e - a*f)
^5*(d*e - c*f)^5)

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)^3 (c+d x)^3 (e+f x)^3} \, dx &=\int \left (\frac {b^6}{(b c-a d)^3 (b e-a f)^3 (a+b x)^3}-\frac {3 b^6 (b d e+b c f-2 a d f)}{(b c-a d)^4 (b e-a f)^4 (a+b x)^2}+\frac {3 b^6 \left (7 a^2 d^2 f^2-7 a b d f (d e+c f)+b^2 \left (2 d^2 e^2+3 c d e f+2 c^2 f^2\right )\right )}{(b c-a d)^5 (b e-a f)^5 (a+b x)}+\frac {d^6}{(b c-a d)^3 (-d e+c f)^3 (c+d x)^3}-\frac {3 d^6 (b d e-2 b c f+a d f)}{(b c-a d)^4 (-d e+c f)^4 (c+d x)^2}+\frac {3 d^6 \left (-2 a^2 d^2 f^2-a b d f (3 d e-7 c f)-b^2 \left (2 d^2 e^2-7 c d e f+7 c^2 f^2\right )\right )}{(b c-a d)^5 (d e-c f)^5 (c+d x)}+\frac {f^6}{(b e-a f)^3 (d e-c f)^3 (e+f x)^3}-\frac {3 f^6 (-2 b d e+b c f+a d f)}{(b e-a f)^4 (d e-c f)^4 (e+f x)^2}+\frac {3 f^6 \left (2 a^2 d^2 f^2-a b d f (7 d e-3 c f)+b^2 \left (7 d^2 e^2-7 c d e f+2 c^2 f^2\right )\right )}{(b e-a f)^5 (d e-c f)^5 (e+f x)}\right ) \, dx\\ &=-\frac {b^5}{2 (b c-a d)^3 (b e-a f)^3 (a+b x)^2}+\frac {3 b^5 (b d e+b c f-2 a d f)}{(b c-a d)^4 (b e-a f)^4 (a+b x)}+\frac {d^5}{2 (b c-a d)^3 (d e-c f)^3 (c+d x)^2}+\frac {3 d^5 (b d e-2 b c f+a d f)}{(b c-a d)^4 (d e-c f)^4 (c+d x)}-\frac {f^5}{2 (b e-a f)^3 (d e-c f)^3 (e+f x)^2}-\frac {3 f^5 (2 b d e-b c f-a d f)}{(b e-a f)^4 (d e-c f)^4 (e+f x)}+\frac {3 b^5 \left (7 a^2 d^2 f^2-7 a b d f (d e+c f)+b^2 \left (2 d^2 e^2+3 c d e f+2 c^2 f^2\right )\right ) \log (a+b x)}{(b c-a d)^5 (b e-a f)^5}-\frac {3 d^5 \left (2 a^2 d^2 f^2+a b d f (3 d e-7 c f)+b^2 \left (2 d^2 e^2-7 c d e f+7 c^2 f^2\right )\right ) \log (c+d x)}{(b c-a d)^5 (d e-c f)^5}+\frac {3 f^5 \left (2 a^2 d^2 f^2-a b d f (7 d e-3 c f)+b^2 \left (7 d^2 e^2-7 c d e f+2 c^2 f^2\right )\right ) \log (e+f x)}{(b e-a f)^5 (d e-c f)^5}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 1.01, size = 490, normalized size = 0.99 \begin {gather*} \frac {1}{2} \left (-\frac {b^5}{(b c-a d)^3 (b e-a f)^3 (a+b x)^2}+\frac {6 b^5 (b d e+b c f-2 a d f)}{(b c-a d)^4 (b e-a f)^4 (a+b x)}-\frac {d^5}{(b c-a d)^3 (-d e+c f)^3 (c+d x)^2}+\frac {6 d^5 (b d e-2 b c f+a d f)}{(b c-a d)^4 (d e-c f)^4 (c+d x)}-\frac {f^5}{(b e-a f)^3 (d e-c f)^3 (e+f x)^2}+\frac {6 f^5 (-2 b d e+b c f+a d f)}{(b e-a f)^4 (d e-c f)^4 (e+f x)}+\frac {6 b^5 \left (7 a^2 d^2 f^2-7 a b d f (d e+c f)+b^2 \left (2 d^2 e^2+3 c d e f+2 c^2 f^2\right )\right ) \log (a+b x)}{(b c-a d)^5 (b e-a f)^5}+\frac {6 d^5 \left (2 a^2 d^2 f^2+a b d f (3 d e-7 c f)+b^2 \left (2 d^2 e^2-7 c d e f+7 c^2 f^2\right )\right ) \log (c+d x)}{(b c-a d)^5 (-d e+c f)^5}+\frac {6 f^5 \left (2 a^2 d^2 f^2+a b d f (-7 d e+3 c f)+b^2 \left (7 d^2 e^2-7 c d e f+2 c^2 f^2\right )\right ) \log (e+f x)}{(b e-a f)^5 (d e-c f)^5}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b*x)^3*(c + d*x)^3*(e + f*x)^3),x]

[Out]

(-(b^5/((b*c - a*d)^3*(b*e - a*f)^3*(a + b*x)^2)) + (6*b^5*(b*d*e + b*c*f - 2*a*d*f))/((b*c - a*d)^4*(b*e - a*
f)^4*(a + b*x)) - d^5/((b*c - a*d)^3*(-(d*e) + c*f)^3*(c + d*x)^2) + (6*d^5*(b*d*e - 2*b*c*f + a*d*f))/((b*c -
a*d)^4*(d*e - c*f)^4*(c + d*x)) - f^5/((b*e - a*f)^3*(d*e - c*f)^3*(e + f*x)^2) + (6*f^5*(-2*b*d*e + b*c*f +
a*d*f))/((b*e - a*f)^4*(d*e - c*f)^4*(e + f*x)) + (6*b^5*(7*a^2*d^2*f^2 - 7*a*b*d*f*(d*e + c*f) + b^2*(2*d^2*e
^2 + 3*c*d*e*f + 2*c^2*f^2))*Log[a + b*x])/((b*c - a*d)^5*(b*e - a*f)^5) + (6*d^5*(2*a^2*d^2*f^2 + a*b*d*f*(3*
d*e - 7*c*f) + b^2*(2*d^2*e^2 - 7*c*d*e*f + 7*c^2*f^2))*Log[c + d*x])/((b*c - a*d)^5*(-(d*e) + c*f)^5) + (6*f^
5*(2*a^2*d^2*f^2 + a*b*d*f*(-7*d*e + 3*c*f) + b^2*(7*d^2*e^2 - 7*c*d*e*f + 2*c^2*f^2))*Log[e + f*x])/((b*e - a
*f)^5*(d*e - c*f)^5))/2

________________________________________________________________________________________

Maple [A]
time = 0.74, size = 514, normalized size = 1.04

method	result	size
default	\(-\frac {b^{5}}{2 \left (a d -b c \right )^{3} \left (a f -b e \right )^{3} \left (b x +a \right )^{2}}-\frac {3 b^{5} \left (2 a d f -b c f -b d e \right )}{\left (a d -b c \right )^{4} \left (a f -b e \right )^{4} \left (b x +a \right )}+\frac {3 b^{5} \left (7 a^{2} d^{2} f^{2}-7 a b c d \,f^{2}-7 a b \,d^{2} e f +2 b^{2} c^{2} f^{2}+3 b^{2} c d e f +2 b^{2} d^{2} e^{2}\right ) \ln \left (b x +a \right )}{\left (a d -b c \right )^{5} \left (a f -b e \right )^{5}}+\frac {d^{5}}{2 \left (a d -b c \right )^{3} \left (c f -d e \right )^{3} \left (d x +c \right )^{2}}+\frac {3 d^{5} \left (a d f -2 b c f +b d e \right )}{\left (a d -b c \right )^{4} \left (c f -d e \right )^{4} \left (d x +c \right )}-\frac {3 d^{5} \left (2 a^{2} d^{2} f^{2}-7 a b c d \,f^{2}+3 a b \,d^{2} e f +7 b^{2} c^{2} f^{2}-7 b^{2} c d e f +2 b^{2} d^{2} e^{2}\right ) \ln \left (d x +c \right )}{\left (a d -b c \right )^{5} \left (c f -d e \right )^{5}}-\frac {f^{5}}{2 \left (a f -b e \right )^{3} \left (c f -d e \right )^{3} \left (f x +e \right )^{2}}+\frac {3 f^{5} \left (a d f +b c f -2 b d e \right )}{\left (a f -b e \right )^{4} \left (c f -d e \right )^{4} \left (f x +e \right )}+\frac {3 f^{5} \left (2 a^{2} d^{2} f^{2}+3 a b c d \,f^{2}-7 a b \,d^{2} e f +2 b^{2} c^{2} f^{2}-7 b^{2} c d e f +7 b^{2} d^{2} e^{2}\right ) \ln \left (f x +e \right )}{\left (a f -b e \right )^{5} \left (c f -d e \right )^{5}}\)	\(514\)
norman	\(\text {Expression too large to display}\)	\(9946\)
risch	\(\text {Expression too large to display}\)	\(20834\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x+a)^3/(d*x+c)^3/(f*x+e)^3,x,method=_RETURNVERBOSE)

[Out]

-1/2*b^5/(a*d-b*c)^3/(a*f-b*e)^3/(b*x+a)^2-3*b^5*(2*a*d*f-b*c*f-b*d*e)/(a*d-b*c)^4/(a*f-b*e)^4/(b*x+a)+3*b^5*(
7*a^2*d^2*f^2-7*a*b*c*d*f^2-7*a*b*d^2*e*f+2*b^2*c^2*f^2+3*b^2*c*d*e*f+2*b^2*d^2*e^2)/(a*d-b*c)^5/(a*f-b*e)^5*l
n(b*x+a)+1/2*d^5/(a*d-b*c)^3/(c*f-d*e)^3/(d*x+c)^2+3*d^5*(a*d*f-2*b*c*f+b*d*e)/(a*d-b*c)^4/(c*f-d*e)^4/(d*x+c)
-3*d^5*(2*a^2*d^2*f^2-7*a*b*c*d*f^2+3*a*b*d^2*e*f+7*b^2*c^2*f^2-7*b^2*c*d*e*f+2*b^2*d^2*e^2)/(a*d-b*c)^5/(c*f-
d*e)^5*ln(d*x+c)-1/2*f^5/(a*f-b*e)^3/(c*f-d*e)^3/(f*x+e)^2+3*f^5*(a*d*f+b*c*f-2*b*d*e)/(a*f-b*e)^4/(c*f-d*e)^4
/(f*x+e)+3*f^5*(2*a^2*d^2*f^2+3*a*b*c*d*f^2-7*a*b*d^2*e*f+2*b^2*c^2*f^2-7*b^2*c*d*e*f+7*b^2*d^2*e^2)/(a*f-b*e)
^5/(c*f-d*e)^5*ln(f*x+e)

________________________________________________________________________________________

Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 12087 vs. \(2 (510) = 1020\).
time = 1.29, size = 12087, normalized size = 24.42 \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)^3/(d*x+c)^3/(f*x+e)^3,x, algorithm="maxima")

[Out]

3*(2*b^7*d^2*e^2 + (2*b^7*c^2 - 7*a*b^6*c*d + 7*a^2*b^5*d^2)*f^2 + (3*b^7*c*d*e - 7*a*b^6*d^2*e)*f)*log(b*x +
a)/(b^10*c^5*e^5 - 5*a*b^9*c^4*d*e^5 + 10*a^2*b^8*c^3*d^2*e^5 - 10*a^3*b^7*c^2*d^3*e^5 + 5*a^4*b^6*c*d^4*e^5 -
a^5*b^5*d^5*e^5 - (a^5*b^5*c^5 - 5*a^6*b^4*c^4*d + 10*a^7*b^3*c^3*d^2 - 10*a^8*b^2*c^2*d^3 + 5*a^9*b*c*d^4 -
a^10*d^5)*f^5 + 5*(a^4*b^6*c^5*e - 5*a^5*b^5*c^4*d*e + 10*a^6*b^4*c^3*d^2*e - 10*a^7*b^3*c^2*d^3*e + 5*a^8*b^2
*c*d^4*e - a^9*b*d^5*e)*f^4 - 10*(a^3*b^7*c^5*e^2 - 5*a^4*b^6*c^4*d*e^2 + 10*a^5*b^5*c^3*d^2*e^2 - 10*a^6*b^4*
c^2*d^3*e^2 + 5*a^7*b^3*c*d^4*e^2 - a^8*b^2*d^5*e^2)*f^3 + 10*(a^2*b^8*c^5*e^3 - 5*a^3*b^7*c^4*d*e^3 + 10*a^4*
b^6*c^3*d^2*e^3 - 10*a^5*b^5*c^2*d^3*e^3 + 5*a^6*b^4*c*d^4*e^3 - a^7*b^3*d^5*e^3)*f^2 - 5*(a*b^9*c^5*e^4 - 5*a
^2*b^8*c^4*d*e^4 + 10*a^3*b^7*c^3*d^2*e^4 - 10*a^4*b^6*c^2*d^3*e^4 + 5*a^5*b^5*c*d^4*e^4 - a^6*b^4*d^5*e^4)*f)
- 3*(2*b^2*d^7*e^2 + (7*b^2*c^2*d^5 - 7*a*b*c*d^6 + 2*a^2*d^7)*f^2 - (7*b^2*c*d^6*e - 3*a*b*d^7*e)*f)*log(d*x
+ c)/(b^5*c^5*d^5*e^5 - 5*a*b^4*c^4*d^6*e^5 + 10*a^2*b^3*c^3*d^7*e^5 - 10*a^3*b^2*c^2*d^8*e^5 + 5*a^4*b*c*d^9
*e^5 - a^5*d^10*e^5 - (b^5*c^10 - 5*a*b^4*c^9*d + 10*a^2*b^3*c^8*d^2 - 10*a^3*b^2*c^7*d^3 + 5*a^4*b*c^6*d^4 -
a^5*c^5*d^5)*f^5 + 5*(b^5*c^9*d*e - 5*a*b^4*c^8*d^2*e + 10*a^2*b^3*c^7*d^3*e - 10*a^3*b^2*c^6*d^4*e + 5*a^4*b*
c^5*d^5*e - a^5*c^4*d^6*e)*f^4 - 10*(b^5*c^8*d^2*e^2 - 5*a*b^4*c^7*d^3*e^2 + 10*a^2*b^3*c^6*d^4*e^2 - 10*a^3*b
^2*c^5*d^5*e^2 + 5*a^4*b*c^4*d^6*e^2 - a^5*c^3*d^7*e^2)*f^3 + 10*(b^5*c^7*d^3*e^3 - 5*a*b^4*c^6*d^4*e^3 + 10*a
^2*b^3*c^5*d^5*e^3 - 10*a^3*b^2*c^4*d^6*e^3 + 5*a^4*b*c^3*d^7*e^3 - a^5*c^2*d^8*e^3)*f^2 - 5*(b^5*c^6*d^4*e^4
- 5*a*b^4*c^5*d^5*e^4 + 10*a^2*b^3*c^4*d^6*e^4 - 10*a^3*b^2*c^3*d^7*e^4 + 5*a^4*b*c^2*d^8*e^4 - a^5*c*d^9*e^4)
*f) + 3*(7*b^2*d^2*f^5*e^2 + (2*b^2*c^2 + 3*a*b*c*d + 2*a^2*d^2)*f^7 - 7*(b^2*c*d*e + a*b*d^2*e)*f^6)*log(f*x
+ e)/(a^5*c^5*f^10 + b^5*d^5*e^10 - 5*(a^4*b*c^5*e + a^5*c^4*d*e)*f^9 + 5*(2*a^3*b^2*c^5*e^2 + 5*a^4*b*c^4*d*e
^2 + 2*a^5*c^3*d^2*e^2)*f^8 - 10*(a^2*b^3*c^5*e^3 + 5*a^3*b^2*c^4*d*e^3 + 5*a^4*b*c^3*d^2*e^3 + a^5*c^2*d^3*e^
3)*f^7 + 5*(a*b^4*c^5*e^4 + 10*a^2*b^3*c^4*d*e^4 + 20*a^3*b^2*c^3*d^2*e^4 + 10*a^4*b*c^2*d^3*e^4 + a^5*c*d^4*e
^4)*f^6 - (b^5*c^5*e^5 + 25*a*b^4*c^4*d*e^5 + 100*a^2*b^3*c^3*d^2*e^5 + 100*a^3*b^2*c^2*d^3*e^5 + 25*a^4*b*c*d
^4*e^5 + a^5*d^5*e^5)*f^5 + 5*(b^5*c^4*d*e^6 + 10*a*b^4*c^3*d^2*e^6 + 20*a^2*b^3*c^2*d^3*e^6 + 10*a^3*b^2*c*d^
4*e^6 + a^4*b*d^5*e^6)*f^4 - 10*(b^5*c^3*d^2*e^7 + 5*a*b^4*c^2*d^3*e^7 + 5*a^2*b^3*c*d^4*e^7 + a^3*b^2*d^5*e^7
)*f^3 + 5*(2*b^5*c^2*d^3*e^8 + 5*a*b^4*c*d^4*e^8 + 2*a^2*b^3*d^5*e^8)*f^2 - 5*(b^5*c*d^4*e^9 + a*b^4*d^5*e^9)*
f) - 1/2*(b^7*c^3*d^4*e^7 - 7*a*b^6*c^2*d^5*e^7 - 7*a^2*b^5*c*d^6*e^7 + a^3*b^4*d^7*e^7 + (a^3*b^4*c^7 - 4*a^4
*b^3*c^6*d + 6*a^5*b^2*c^5*d^2 - 4*a^6*b*c^4*d^3 + a^7*c^3*d^4)*f^7 - 7*(a^2*b^5*c^7*e - 3*a^3*b^4*c^6*d*e + 2
*a^4*b^3*c^5*d^2*e + 2*a^5*b^2*c^4*d^3*e - 3*a^6*b*c^3*d^4*e + a^7*c^2*d^5*e)*f^6 - (7*a*b^6*c^7*e^2 - 26*a^2*
b^5*c^6*d*e^2 + 52*a^3*b^4*c^5*d^2*e^2 - 78*a^4*b^3*c^4*d^3*e^2 + 52*a^5*b^2*c^3*d^4*e^2 - 26*a^6*b*c^2*d^5*e^
2 + 7*a^7*c*d^6*e^2)*f^5 - 6*(2*b^7*d^7*f^2*e^5 + (2*b^7*c^5*d^2 - 5*a*b^6*c^4*d^3 + 2*a^2*b^5*c^3*d^4 + 2*a^3
*b^4*c^2*d^5 - 5*a^4*b^3*c*d^6 + 2*a^5*b^2*d^7)*f^7 - (5*b^7*c^4*d^3*e - 16*a*b^6*c^3*d^4*e + 12*a^2*b^5*c^2*d
^5*e - 16*a^3*b^4*c*d^6*e + 5*a^4*b^3*d^7*e)*f^6 + 2*(b^7*c^3*d^4*e^2 - 6*a*b^6*c^2*d^5*e^2 - 6*a^2*b^5*c*d^6*
e^2 + a^3*b^4*d^7*e^2)*f^5 + 2*(b^7*c^2*d^5*e^3 + 8*a*b^6*c*d^6*e^3 + a^2*b^5*d^7*e^3)*f^4 - 5*(b^7*c*d^6*e^4
+ a*b^6*d^7*e^4)*f^3)*x^5 + (b^7*c^7*e^3 + 21*a*b^6*c^6*d*e^3 - 52*a^2*b^5*c^5*d^2*e^3 - 52*a^5*b^2*c^2*d^5*e^
3 + 21*a^6*b*c*d^6*e^3 + a^7*d^7*e^3)*f^4 - 3*(8*b^7*d^7*f*e^6 + (8*b^7*c^6*d - 14*a*b^6*c^5*d^2 - 7*a^2*b^5*c
^4*d^3 + 14*a^3*b^4*c^3*d^4 - 7*a^4*b^3*c^2*d^5 - 14*a^5*b^2*c*d^6 + 8*a^6*b*d^7)*f^7 - 2*(7*b^7*c^5*d^2*e - 1
7*a*b^6*c^4*d^3*e - 3*a^2*b^5*c^3*d^4*e - 3*a^3*b^4*c^2*d^5*e - 17*a^4*b^3*c*d^6*e + 7*a^5*b^2*d^7*e)*f^6 - (7
*b^7*c^4*d^3*e^2 - 6*a*b^6*c^3*d^4*e^2 + 78*a^2*b^5*c^2*d^5*e^2 - 6*a^3*b^4*c*d^6*e^2 + 7*a^4*b^3*d^7*e^2)*f^5
+ 2*(7*b^7*c^3*d^4*e^3 + 3*a*b^6*c^2*d^5*e^3 + 3*a^2*b^5*c*d^6*e^3 + 7*a^3*b^4*d^7*e^3)*f^4 - (7*b^7*c^2*d^5*
e^4 - 34*a*b^6*c*d^6*e^4 + 7*a^2*b^5*d^7*e^4)*f^3 - 14*(b^7*c*d^6*e^5 + a*b^6*d^7*e^5)*f^2)*x^4 - 2*(2*b^7*c^6
*d*e^4 + 7*a*b^6*c^5*d^2*e^4 - 39*a^2*b^5*c^4*d^3*e^4 - 39*a^4*b^3*c^2*d^5*e^4 + 7*a^5*b^2*c*d^6*e^4 + 2*a^6*b
*d^7*e^4)*f^3 - 2*(6*b^7*d^7*e^7 + (6*b^7*c^7 + 3*a*b^6*c^6*d - 37*a^2*b^5*c^5*d^2 + 19*a^3*b^4*c^4*d^3 + 19*a
^4*b^3*c^3*d^4 - 37*a^5*b^2*c^2*d^5 + 3*a^6*b*c*d^6 + 6*a^7*d^7)*f^7 + (3*b^7*c^6*d*e - 28*a*b^6*c^5*d^2*e + 8
6*a^2*b^5*c^4*d^3*e - 68*a^3*b^4*c^3*d^4*e + 86*a^4*b^3*c^2*d^5*e - 28*a^5*b^2*c*d^6*e + 3*a^6*b*d^7*e)*f^6 -
(37*b^7*c^5*d^2*e^2 - 86*a*b^6*c^4*d^3*e^2 + 52*a^2*b^5*c^3*d^4*e^2 + 52*a^3*b^4*c^2*d^5*e^2 - 86*a^4*b^3*c*d^
6*e^2 + 37*a^5*b^2*d^7*e^2)*f^5 + (19*b^7*c^4*d^3*e^3 - 68*a*b^6*c^3*d^4*e^3 - 52*a^2*b^5*c^2*d^5*e^3 - 68*a^3
*b^4*c*d^6*e^3 + 19*a^4*b^3*d^7*e^3)*f^4 + (19*...

________________________________________________________________________________________

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)^3/(d*x+c)^3/(f*x+e)^3,x, algorithm="fricas")

[Out]

Timed out

________________________________________________________________________________________

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)**3/(d*x+c)**3/(f*x+e)**3,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 6908 vs. \(2 (510) = 1020\).
time = 0.73, size = 6908, normalized size = 13.96 \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)^3/(d*x+c)^3/(f*x+e)^3,x, algorithm="giac")

[Out]

-3*(2*b^8*c^2*f^2 - 7*a*b^7*c*d*f^2 + 7*a^2*b^6*d^2*f^2 + 3*b^8*c*d*f*e - 7*a*b^7*d^2*f*e + 2*b^8*d^2*e^2)*log
(abs(b*x + a))/(a^5*b^6*c^5*f^5 - 5*a^6*b^5*c^4*d*f^5 + 10*a^7*b^4*c^3*d^2*f^5 - 10*a^8*b^3*c^2*d^3*f^5 + 5*a^
9*b^2*c*d^4*f^5 - a^10*b*d^5*f^5 - 5*a^4*b^7*c^5*f^4*e + 25*a^5*b^6*c^4*d*f^4*e - 50*a^6*b^5*c^3*d^2*f^4*e + 5
0*a^7*b^4*c^2*d^3*f^4*e - 25*a^8*b^3*c*d^4*f^4*e + 5*a^9*b^2*d^5*f^4*e + 10*a^3*b^8*c^5*f^3*e^2 - 50*a^4*b^7*c
^4*d*f^3*e^2 + 100*a^5*b^6*c^3*d^2*f^3*e^2 - 100*a^6*b^5*c^2*d^3*f^3*e^2 + 50*a^7*b^4*c*d^4*f^3*e^2 - 10*a^8*b
^3*d^5*f^3*e^2 - 10*a^2*b^9*c^5*f^2*e^3 + 50*a^3*b^8*c^4*d*f^2*e^3 - 100*a^4*b^7*c^3*d^2*f^2*e^3 + 100*a^5*b^6
*c^2*d^3*f^2*e^3 - 50*a^6*b^5*c*d^4*f^2*e^3 + 10*a^7*b^4*d^5*f^2*e^3 + 5*a*b^10*c^5*f*e^4 - 25*a^2*b^9*c^4*d*f
*e^4 + 50*a^3*b^8*c^3*d^2*f*e^4 - 50*a^4*b^7*c^2*d^3*f*e^4 + 25*a^5*b^6*c*d^4*f*e^4 - 5*a^6*b^5*d^5*f*e^4 - b^
11*c^5*e^5 + 5*a*b^10*c^4*d*e^5 - 10*a^2*b^9*c^3*d^2*e^5 + 10*a^3*b^8*c^2*d^3*e^5 - 5*a^4*b^7*c*d^4*e^5 + a^5*
b^6*d^5*e^5) + 3*(7*b^2*c^2*d^6*f^2 - 7*a*b*c*d^7*f^2 + 2*a^2*d^8*f^2 - 7*b^2*c*d^7*f*e + 3*a*b*d^8*f*e + 2*b^
2*d^8*e^2)*log(abs(d*x + c))/(b^5*c^10*d*f^5 - 5*a*b^4*c^9*d^2*f^5 + 10*a^2*b^3*c^8*d^3*f^5 - 10*a^3*b^2*c^7*d
^4*f^5 + 5*a^4*b*c^6*d^5*f^5 - a^5*c^5*d^6*f^5 - 5*b^5*c^9*d^2*f^4*e + 25*a*b^4*c^8*d^3*f^4*e - 50*a^2*b^3*c^7
*d^4*f^4*e + 50*a^3*b^2*c^6*d^5*f^4*e - 25*a^4*b*c^5*d^6*f^4*e + 5*a^5*c^4*d^7*f^4*e + 10*b^5*c^8*d^3*f^3*e^2
- 50*a*b^4*c^7*d^4*f^3*e^2 + 100*a^2*b^3*c^6*d^5*f^3*e^2 - 100*a^3*b^2*c^5*d^6*f^3*e^2 + 50*a^4*b*c^4*d^7*f^3*
e^2 - 10*a^5*c^3*d^8*f^3*e^2 - 10*b^5*c^7*d^4*f^2*e^3 + 50*a*b^4*c^6*d^5*f^2*e^3 - 100*a^2*b^3*c^5*d^6*f^2*e^3
+ 100*a^3*b^2*c^4*d^7*f^2*e^3 - 50*a^4*b*c^3*d^8*f^2*e^3 + 10*a^5*c^2*d^9*f^2*e^3 + 5*b^5*c^6*d^5*f*e^4 - 25*
a*b^4*c^5*d^6*f*e^4 + 50*a^2*b^3*c^4*d^7*f*e^4 - 50*a^3*b^2*c^3*d^8*f*e^4 + 25*a^4*b*c^2*d^9*f*e^4 - 5*a^5*c*d
^10*f*e^4 - b^5*c^5*d^6*e^5 + 5*a*b^4*c^4*d^7*e^5 - 10*a^2*b^3*c^3*d^8*e^5 + 10*a^3*b^2*c^2*d^9*e^5 - 5*a^4*b*
c*d^10*e^5 + a^5*d^11*e^5) + 3*(2*b^2*c^2*f^8 + 3*a*b*c*d*f^8 + 2*a^2*d^2*f^8 - 7*b^2*c*d*f^7*e - 7*a*b*d^2*f^
7*e + 7*b^2*d^2*f^6*e^2)*log(abs(f*x + e))/(a^5*c^5*f^11 - 5*a^4*b*c^5*f^10*e - 5*a^5*c^4*d*f^10*e + 10*a^3*b^
2*c^5*f^9*e^2 + 25*a^4*b*c^4*d*f^9*e^2 + 10*a^5*c^3*d^2*f^9*e^2 - 10*a^2*b^3*c^5*f^8*e^3 - 50*a^3*b^2*c^4*d*f^
8*e^3 - 50*a^4*b*c^3*d^2*f^8*e^3 - 10*a^5*c^2*d^3*f^8*e^3 + 5*a*b^4*c^5*f^7*e^4 + 50*a^2*b^3*c^4*d*f^7*e^4 + 1
00*a^3*b^2*c^3*d^2*f^7*e^4 + 50*a^4*b*c^2*d^3*f^7*e^4 + 5*a^5*c*d^4*f^7*e^4 - b^5*c^5*f^6*e^5 - 25*a*b^4*c^4*d
*f^6*e^5 - 100*a^2*b^3*c^3*d^2*f^6*e^5 - 100*a^3*b^2*c^2*d^3*f^6*e^5 - 25*a^4*b*c*d^4*f^6*e^5 - a^5*d^5*f^6*e^
5 + 5*b^5*c^4*d*f^5*e^6 + 50*a*b^4*c^3*d^2*f^5*e^6 + 100*a^2*b^3*c^2*d^3*f^5*e^6 + 50*a^3*b^2*c*d^4*f^5*e^6 +
5*a^4*b*d^5*f^5*e^6 - 10*b^5*c^3*d^2*f^4*e^7 - 50*a*b^4*c^2*d^3*f^4*e^7 - 50*a^2*b^3*c*d^4*f^4*e^7 - 10*a^3*b^
2*d^5*f^4*e^7 + 10*b^5*c^2*d^3*f^3*e^8 + 25*a*b^4*c*d^4*f^3*e^8 + 10*a^2*b^3*d^5*f^3*e^8 - 5*b^5*c*d^4*f^2*e^9
- 5*a*b^4*d^5*f^2*e^9 + b^5*d^5*f*e^10) + 1/2*(12*b^7*c^5*d^2*f^7*x^5 - 30*a*b^6*c^4*d^3*f^7*x^5 + 12*a^2*b^5
*c^3*d^4*f^7*x^5 + 12*a^3*b^4*c^2*d^5*f^7*x^5 - 30*a^4*b^3*c*d^6*f^7*x^5 + 12*a^5*b^2*d^7*f^7*x^5 - 30*b^7*c^4
*d^3*f^6*x^5*e + 96*a*b^6*c^3*d^4*f^6*x^5*e - 72*a^2*b^5*c^2*d^5*f^6*x^5*e + 96*a^3*b^4*c*d^6*f^6*x^5*e - 30*a
^4*b^3*d^7*f^6*x^5*e + 24*b^7*c^6*d*f^7*x^4 - 42*a*b^6*c^5*d^2*f^7*x^4 - 21*a^2*b^5*c^4*d^3*f^7*x^4 + 42*a^3*b
^4*c^3*d^4*f^7*x^4 - 21*a^4*b^3*c^2*d^5*f^7*x^4 - 42*a^5*b^2*c*d^6*f^7*x^4 + 24*a^6*b*d^7*f^7*x^4 + 12*b^7*c^3
*d^4*f^5*x^5*e^2 - 72*a*b^6*c^2*d^5*f^5*x^5*e^2 - 72*a^2*b^5*c*d^6*f^5*x^5*e^2 + 12*a^3*b^4*d^7*f^5*x^5*e^2 -
42*b^7*c^5*d^2*f^6*x^4*e + 102*a*b^6*c^4*d^3*f^6*x^4*e + 18*a^2*b^5*c^3*d^4*f^6*x^4*e + 18*a^3*b^4*c^2*d^5*f^6
*x^4*e + 102*a^4*b^3*c*d^6*f^6*x^4*e - 42*a^5*b^2*d^7*f^6*x^4*e + 12*b^7*c^7*f^7*x^3 + 6*a*b^6*c^6*d*f^7*x^3 -
74*a^2*b^5*c^5*d^2*f^7*x^3 + 38*a^3*b^4*c^4*d^3*f^7*x^3 + 38*a^4*b^3*c^3*d^4*f^7*x^3 - 74*a^5*b^2*c^2*d^5*f^7
*x^3 + 6*a^6*b*c*d^6*f^7*x^3 + 12*a^7*d^7*f^7*x^3 + 12*b^7*c^2*d^5*f^4*x^5*e^3 + 96*a*b^6*c*d^6*f^4*x^5*e^3 +
12*a^2*b^5*d^7*f^4*x^5*e^3 - 21*b^7*c^4*d^3*f^5*x^4*e^2 + 18*a*b^6*c^3*d^4*f^5*x^4*e^2 - 234*a^2*b^5*c^2*d^5*f
^5*x^4*e^2 + 18*a^3*b^4*c*d^6*f^5*x^4*e^2 - 21*a^4*b^3*d^7*f^5*x^4*e^2 + 6*b^7*c^6*d*f^6*x^3*e - 56*a*b^6*c^5*
d^2*f^6*x^3*e + 172*a^2*b^5*c^4*d^3*f^6*x^3*e - 136*a^3*b^4*c^3*d^4*f^6*x^3*e + 172*a^4*b^3*c^2*d^5*f^6*x^3*e
- 56*a^5*b^2*c*d^6*f^6*x^3*e + 6*a^6*b*d^7*f^6*x^3*e + 18*a*b^6*c^7*f^7*x^2 - 37*a^2*b^5*c^6*d*f^7*x^2 - 3*a^3
*b^4*c^5*d^2*f^7*x^2 + 32*a^4*b^3*c^4*d^3*f^7*x^2 - 3*a^5*b^2*c^3*d^4*f^7*x^2 - 37*a^6*b*c^2*d^5*f^7*x^2 + 18*
a^7*c*d^6*f^7*x^2 - 30*b^7*c*d^6*f^3*x^5*e^4 - 30*a*b^6*d^7*f^3*x^5*e^4 + 42*b^7*c^3*d^4*f^4*x^4*e^3 + 18*a*b^
6*c^2*d^5*f^4*x^4*e^3 + 18*a^2*b^5*c*d^6*f^4*x^4*e^3 + 42*a^3*b^4*d^7*f^4*x^4*e^3 - 74*b^7*c^5*d^2*f^5*x^3*e^2
+ 172*a*b^6*c^4*d^3*f^5*x^3*e^2 - 104*a^2*b^5*c^3*d^4*f^5*x^3*e^2 - 104*a^3*b^4*c^2*d^5*f^5*x^3*e^2 + 172*a^4
*b^3*c*d^6*f^5*x^3*e^2 - 74*a^5*b^2*d^7*f^5*x^3...

________________________________________________________________________________________

Mupad [B]
time = 20.08, size = 10, normalized size = 0.02 \begin {gather*} \text {too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((e + f*x)^3*(a + b*x)^3*(c + d*x)^3),x)

[Out]

Too large to display

________________________________________________________________________________________