3.4.17 \(\int \frac {1}{x (a+b x)^3 (c+d x)^3} \, dx\) [317]

Optimal. Leaf size=221 \[ \frac {b^3}{2 a (b c-a d)^3 (a+b x)^2}+\frac {b^3 (b c-4 a d)}{a^2 (b c-a d)^4 (a+b x)}-\frac {d^3}{2 c (b c-a d)^3 (c+d x)^2}-\frac {d^3 (4 b c-a d)}{c^2 (b c-a d)^4 (c+d x)}+\frac {\log (x)}{a^3 c^3}-\frac {b^3 \left (b^2 c^2-5 a b c d+10 a^2 d^2\right ) \log (a+b x)}{a^3 (b c-a d)^5}+\frac {d^3 \left (10 b^2 c^2-5 a b c d+a^2 d^2\right ) \log (c+d x)}{c^3 (b c-a d)^5} \]

[Out]

1/2*b^3/a/(-a*d+b*c)^3/(b*x+a)^2+b^3*(-4*a*d+b*c)/a^2/(-a*d+b*c)^4/(b*x+a)-1/2*d^3/c/(-a*d+b*c)^3/(d*x+c)^2-d^
3*(-a*d+4*b*c)/c^2/(-a*d+b*c)^4/(d*x+c)+ln(x)/a^3/c^3-b^3*(10*a^2*d^2-5*a*b*c*d+b^2*c^2)*ln(b*x+a)/a^3/(-a*d+b
*c)^5+d^3*(a^2*d^2-5*a*b*c*d+10*b^2*c^2)*ln(d*x+c)/c^3/(-a*d+b*c)^5

________________________________________________________________________________________

Rubi [A]
time = 0.17, antiderivative size = 221, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {90} \begin {gather*} \frac {\log (x)}{a^3 c^3}+\frac {b^3 (b c-4 a d)}{a^2 (a+b x) (b c-a d)^4}+\frac {d^3 \left (a^2 d^2-5 a b c d+10 b^2 c^2\right ) \log (c+d x)}{c^3 (b c-a d)^5}-\frac {b^3 \left (10 a^2 d^2-5 a b c d+b^2 c^2\right ) \log (a+b x)}{a^3 (b c-a d)^5}+\frac {b^3}{2 a (a+b x)^2 (b c-a d)^3}-\frac {d^3 (4 b c-a d)}{c^2 (c+d x) (b c-a d)^4}-\frac {d^3}{2 c (c+d x)^2 (b c-a d)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x*(a + b*x)^3*(c + d*x)^3),x]

[Out]

b^3/(2*a*(b*c - a*d)^3*(a + b*x)^2) + (b^3*(b*c - 4*a*d))/(a^2*(b*c - a*d)^4*(a + b*x)) - d^3/(2*c*(b*c - a*d)
^3*(c + d*x)^2) - (d^3*(4*b*c - a*d))/(c^2*(b*c - a*d)^4*(c + d*x)) + Log[x]/(a^3*c^3) - (b^3*(b^2*c^2 - 5*a*b
*c*d + 10*a^2*d^2)*Log[a + b*x])/(a^3*(b*c - a*d)^5) + (d^3*(10*b^2*c^2 - 5*a*b*c*d + a^2*d^2)*Log[c + d*x])/(
c^3*(b*c - a*d)^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}{x (a+b x)^3 (c+d x)^3} \, dx &=\int \left (\frac {1}{a^3 c^3 x}+\frac {b^4}{a (-b c+a d)^3 (a+b x)^3}+\frac {b^4 (-b c+4 a d)}{a^2 (-b c+a d)^4 (a+b x)^2}+\frac {b^4 \left (b^2 c^2-5 a b c d+10 a^2 d^2\right )}{a^3 (-b c+a d)^5 (a+b x)}+\frac {d^4}{c (b c-a d)^3 (c+d x)^3}+\frac {d^4 (4 b c-a d)}{c^2 (b c-a d)^4 (c+d x)^2}+\frac {d^4 \left (10 b^2 c^2-5 a b c d+a^2 d^2\right )}{c^3 (b c-a d)^5 (c+d x)}\right ) \, dx\\ &=\frac {b^3}{2 a (b c-a d)^3 (a+b x)^2}+\frac {b^3 (b c-4 a d)}{a^2 (b c-a d)^4 (a+b x)}-\frac {d^3}{2 c (b c-a d)^3 (c+d x)^2}-\frac {d^3 (4 b c-a d)}{c^2 (b c-a d)^4 (c+d x)}+\frac {\log (x)}{a^3 c^3}-\frac {b^3 \left (b^2 c^2-5 a b c d+10 a^2 d^2\right ) \log (a+b x)}{a^3 (b c-a d)^5}+\frac {d^3 \left (10 b^2 c^2-5 a b c d+a^2 d^2\right ) \log (c+d x)}{c^3 (b c-a d)^5}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.26, size = 218, normalized size = 0.99 \begin {gather*} -\frac {b^3}{2 a (-b c+a d)^3 (a+b x)^2}+\frac {b^3 (b c-4 a d)}{a^2 (b c-a d)^4 (a+b x)}-\frac {d^3}{2 c (b c-a d)^3 (c+d x)^2}+\frac {d^3 (-4 b c+a d)}{c^2 (b c-a d)^4 (c+d x)}+\frac {\log (x)}{a^3 c^3}+\frac {b^3 \left (b^2 c^2-5 a b c d+10 a^2 d^2\right ) \log (a+b x)}{a^3 (-b c+a d)^5}+\frac {d^3 \left (10 b^2 c^2-5 a b c d+a^2 d^2\right ) \log (c+d x)}{c^3 (b c-a d)^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*b^3/(a*(-(b*c) + a*d)^3*(a + b*x)^2) + (b^3*(b*c - 4*a*d))/(a^2*(b*c - a*d)^4*(a + b*x)) - d^3/(2*c*(b*c
- a*d)^3*(c + d*x)^2) + (d^3*(-4*b*c + a*d))/(c^2*(b*c - a*d)^4*(c + d*x)) + Log[x]/(a^3*c^3) + (b^3*(b^2*c^2
- 5*a*b*c*d + 10*a^2*d^2)*Log[a + b*x])/(a^3*(-(b*c) + a*d)^5) + (d^3*(10*b^2*c^2 - 5*a*b*c*d + a^2*d^2)*Log[c
+ d*x])/(c^3*(b*c - a*d)^5)

________________________________________________________________________________________

Maple [A]
time = 0.12, size = 218, normalized size = 0.99

method	result	size
default	\(-\frac {b^{3}}{2 a \left (a d -b c \right )^{3} \left (b x +a \right )^{2}}-\frac {b^{3} \left (4 a d -b c \right )}{a^{2} \left (a d -b c \right )^{4} \left (b x +a \right )}+\frac {b^{3} \left (10 a^{2} d^{2}-5 a b c d +b^{2} c^{2}\right ) \ln \left (b x +a \right )}{a^{3} \left (a d -b c \right )^{5}}+\frac {d^{3}}{2 c \left (a d -b c \right )^{3} \left (d x +c \right )^{2}}+\frac {d^{3} \left (a d -4 b c \right )}{c^{2} \left (a d -b c \right )^{4} \left (d x +c \right )}-\frac {d^{3} \left (a^{2} d^{2}-5 a b c d +10 b^{2} c^{2}\right ) \ln \left (d x +c \right )}{c^{3} \left (a d -b c \right )^{5}}+\frac {\ln \left (x \right )}{a^{3} c^{3}}\)	\(218\)
norman	\(\frac {\frac {\left (-2 a^{5} d^{5}+5 a^{4} b c \,d^{4}+5 a \,b^{4} c^{4} d -2 b^{5} c^{5}\right ) x}{c^{2} a^{2} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}+\frac {d b \left (-3 a^{5} d^{5}+7 a^{4} b c \,d^{4}+5 a^{3} b^{2} c^{2} d^{3}+5 a^{2} b^{3} c^{3} d^{2}+7 a \,b^{4} c^{4} d -3 b^{5} c^{5}\right ) x^{3}}{c^{3} a^{3} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}+\frac {\left (-3 a^{6} d^{6}+a^{5} b c \,d^{5}+20 a^{4} b^{2} c^{2} d^{4}+20 a^{2} b^{4} c^{4} d^{2}+a \,b^{5} c^{5} d -3 b^{6} c^{6}\right ) x^{2}}{2 c^{3} a^{3} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}+\frac {b^{2} d^{2} \left (-3 a^{4} d^{4}+9 a^{3} b c \,d^{3}+9 a \,b^{3} c^{3} d -3 b^{4} c^{4}\right ) x^{4}}{2 c^{3} a^{3} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}}{\left (b x +a \right )^{2} \left (d x +c \right )^{2}}+\frac {\ln \left (x \right )}{a^{3} c^{3}}+\frac {b^{3} \left (10 a^{2} d^{2}-5 a b c d +b^{2} c^{2}\right ) \ln \left (b x +a \right )}{a^{3} \left (a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}\right )}-\frac {d^{3} \left (a^{2} d^{2}-5 a b c d +10 b^{2} c^{2}\right ) \ln \left (d x +c \right )}{c^{3} \left (a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}\right )}\)	\(684\)
risch	\(\frac {\frac {b^{2} d^{2} \left (a^{3} d^{3}-4 a^{2} b c \,d^{2}-4 a \,b^{2} c^{2} d +b^{3} c^{3}\right ) x^{3}}{a^{2} c^{2} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}+\frac {b d \left (4 a^{4} d^{4}-13 a^{3} b c \,d^{3}-18 a^{2} b^{2} c^{2} d^{2}-13 a \,b^{3} c^{3} d +4 b^{4} c^{4}\right ) x^{2}}{2 a^{2} c^{2} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}+\frac {\left (a^{5} d^{5}-a^{4} b c \,d^{4}-9 a^{3} b^{2} c^{2} d^{3}-9 a^{2} b^{3} c^{3} d^{2}-a \,b^{4} c^{4} d +b^{5} c^{5}\right ) x}{a^{2} c^{2} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}+\frac {\frac {3}{2} a^{4} d^{4}-\frac {9}{2} a^{3} b c \,d^{3}-\frac {9}{2} a \,b^{3} c^{3} d +\frac {3}{2} b^{4} c^{4}}{a c \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}}{\left (b x +a \right )^{2} \left (d x +c \right )^{2}}-\frac {d^{5} \ln \left (-d x -c \right ) a^{2}}{c^{3} \left (a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}\right )}+\frac {5 d^{4} \ln \left (-d x -c \right ) a b}{c^{2} \left (a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}\right )}-\frac {10 d^{3} \ln \left (-d x -c \right ) b^{2}}{c \left (a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}\right )}+\frac {\ln \left (-x \right )}{a^{3} c^{3}}+\frac {10 b^{3} \ln \left (b x +a \right ) d^{2}}{a \left (a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}\right )}-\frac {5 b^{4} \ln \left (b x +a \right ) c d}{a^{2} \left (a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}\right )}+\frac {b^{5} \ln \left (b x +a \right ) c^{2}}{a^{3} \left (a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}\right )}\)	\(966\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*b^3/a/(a*d-b*c)^3/(b*x+a)^2-b^3*(4*a*d-b*c)/a^2/(a*d-b*c)^4/(b*x+a)+b^3*(10*a^2*d^2-5*a*b*c*d+b^2*c^2)/a^
3/(a*d-b*c)^5*ln(b*x+a)+1/2*d^3/c/(a*d-b*c)^3/(d*x+c)^2+d^3*(a*d-4*b*c)/c^2/(a*d-b*c)^4/(d*x+c)-d^3*(a^2*d^2-5
*a*b*c*d+10*b^2*c^2)/c^3/(a*d-b*c)^5*ln(d*x+c)+ln(x)/a^3/c^3

________________________________________________________________________________________

Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 804 vs. \(2 (217) = 434\).
time = 0.32, size = 804, normalized size = 3.64 \begin {gather*} -\frac {{\left (b^{5} c^{2} - 5 \, a b^{4} c d + 10 \, a^{2} b^{3} d^{2}\right )} \log \left (b x + a\right )}{a^{3} b^{5} c^{5} - 5 \, a^{4} b^{4} c^{4} d + 10 \, a^{5} b^{3} c^{3} d^{2} - 10 \, a^{6} b^{2} c^{2} d^{3} + 5 \, a^{7} b c d^{4} - a^{8} d^{5}} + \frac {{\left (10 \, b^{2} c^{2} d^{3} - 5 \, a b c d^{4} + a^{2} d^{5}\right )} \log \left (d x + c\right )}{b^{5} c^{8} - 5 \, a b^{4} c^{7} d + 10 \, a^{2} b^{3} c^{6} d^{2} - 10 \, a^{3} b^{2} c^{5} d^{3} + 5 \, a^{4} b c^{4} d^{4} - a^{5} c^{3} d^{5}} + \frac {3 \, a b^{4} c^{5} - 9 \, a^{2} b^{3} c^{4} d - 9 \, a^{4} b c^{2} d^{3} + 3 \, a^{5} c d^{4} + 2 \, {\left (b^{5} c^{3} d^{2} - 4 \, a b^{4} c^{2} d^{3} - 4 \, a^{2} b^{3} c d^{4} + a^{3} b^{2} d^{5}\right )} x^{3} + {\left (4 \, b^{5} c^{4} d - 13 \, a b^{4} c^{3} d^{2} - 18 \, a^{2} b^{3} c^{2} d^{3} - 13 \, a^{3} b^{2} c d^{4} + 4 \, a^{4} b d^{5}\right )} x^{2} + 2 \, {\left (b^{5} c^{5} - a b^{4} c^{4} d - 9 \, a^{2} b^{3} c^{3} d^{2} - 9 \, a^{3} b^{2} c^{2} d^{3} - a^{4} b c d^{4} + a^{5} d^{5}\right )} x}{2 \, {\left (a^{4} b^{4} c^{8} - 4 \, a^{5} b^{3} c^{7} d + 6 \, a^{6} b^{2} c^{6} d^{2} - 4 \, a^{7} b c^{5} d^{3} + a^{8} c^{4} d^{4} + {\left (a^{2} b^{6} c^{6} d^{2} - 4 \, a^{3} b^{5} c^{5} d^{3} + 6 \, a^{4} b^{4} c^{4} d^{4} - 4 \, a^{5} b^{3} c^{3} d^{5} + a^{6} b^{2} c^{2} d^{6}\right )} x^{4} + 2 \, {\left (a^{2} b^{6} c^{7} d - 3 \, a^{3} b^{5} c^{6} d^{2} + 2 \, a^{4} b^{4} c^{5} d^{3} + 2 \, a^{5} b^{3} c^{4} d^{4} - 3 \, a^{6} b^{2} c^{3} d^{5} + a^{7} b c^{2} d^{6}\right )} x^{3} + {\left (a^{2} b^{6} c^{8} - 9 \, a^{4} b^{4} c^{6} d^{2} + 16 \, a^{5} b^{3} c^{5} d^{3} - 9 \, a^{6} b^{2} c^{4} d^{4} + a^{8} c^{2} d^{6}\right )} x^{2} + 2 \, {\left (a^{3} b^{5} c^{8} - 3 \, a^{4} b^{4} c^{7} d + 2 \, a^{5} b^{3} c^{6} d^{2} + 2 \, a^{6} b^{2} c^{5} d^{3} - 3 \, a^{7} b c^{4} d^{4} + a^{8} c^{3} d^{5}\right )} x\right )}} + \frac {\log \left (x\right )}{a^{3} c^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(b*x+a)^3/(d*x+c)^3,x, algorithm="maxima")

[Out]

-(b^5*c^2 - 5*a*b^4*c*d + 10*a^2*b^3*d^2)*log(b*x + a)/(a^3*b^5*c^5 - 5*a^4*b^4*c^4*d + 10*a^5*b^3*c^3*d^2 - 1
0*a^6*b^2*c^2*d^3 + 5*a^7*b*c*d^4 - a^8*d^5) + (10*b^2*c^2*d^3 - 5*a*b*c*d^4 + a^2*d^5)*log(d*x + c)/(b^5*c^8
- 5*a*b^4*c^7*d + 10*a^2*b^3*c^6*d^2 - 10*a^3*b^2*c^5*d^3 + 5*a^4*b*c^4*d^4 - a^5*c^3*d^5) + 1/2*(3*a*b^4*c^5
- 9*a^2*b^3*c^4*d - 9*a^4*b*c^2*d^3 + 3*a^5*c*d^4 + 2*(b^5*c^3*d^2 - 4*a*b^4*c^2*d^3 - 4*a^2*b^3*c*d^4 + a^3*b
^2*d^5)*x^3 + (4*b^5*c^4*d - 13*a*b^4*c^3*d^2 - 18*a^2*b^3*c^2*d^3 - 13*a^3*b^2*c*d^4 + 4*a^4*b*d^5)*x^2 + 2*(
b^5*c^5 - a*b^4*c^4*d - 9*a^2*b^3*c^3*d^2 - 9*a^3*b^2*c^2*d^3 - a^4*b*c*d^4 + a^5*d^5)*x)/(a^4*b^4*c^8 - 4*a^5
*b^3*c^7*d + 6*a^6*b^2*c^6*d^2 - 4*a^7*b*c^5*d^3 + a^8*c^4*d^4 + (a^2*b^6*c^6*d^2 - 4*a^3*b^5*c^5*d^3 + 6*a^4*
b^4*c^4*d^4 - 4*a^5*b^3*c^3*d^5 + a^6*b^2*c^2*d^6)*x^4 + 2*(a^2*b^6*c^7*d - 3*a^3*b^5*c^6*d^2 + 2*a^4*b^4*c^5*
d^3 + 2*a^5*b^3*c^4*d^4 - 3*a^6*b^2*c^3*d^5 + a^7*b*c^2*d^6)*x^3 + (a^2*b^6*c^8 - 9*a^4*b^4*c^6*d^2 + 16*a^5*b
^3*c^5*d^3 - 9*a^6*b^2*c^4*d^4 + a^8*c^2*d^6)*x^2 + 2*(a^3*b^5*c^8 - 3*a^4*b^4*c^7*d + 2*a^5*b^3*c^6*d^2 + 2*a
^6*b^2*c^5*d^3 - 3*a^7*b*c^4*d^4 + a^8*c^3*d^5)*x) + log(x)/(a^3*c^3)

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1630 vs. \(2 (217) = 434\).
time = 47.92, size = 1630, normalized size = 7.38 \begin {gather*} \frac {3 \, a^{2} b^{5} c^{7} - 12 \, a^{3} b^{4} c^{6} d + 9 \, a^{4} b^{3} c^{5} d^{2} - 9 \, a^{5} b^{2} c^{4} d^{3} + 12 \, a^{6} b c^{3} d^{4} - 3 \, a^{7} c^{2} d^{5} + 2 \, {\left (a b^{6} c^{5} d^{2} - 5 \, a^{2} b^{5} c^{4} d^{3} + 5 \, a^{4} b^{3} c^{2} d^{5} - a^{5} b^{2} c d^{6}\right )} x^{3} + {\left (4 \, a b^{6} c^{6} d - 17 \, a^{2} b^{5} c^{5} d^{2} - 5 \, a^{3} b^{4} c^{4} d^{3} + 5 \, a^{4} b^{3} c^{3} d^{4} + 17 \, a^{5} b^{2} c^{2} d^{5} - 4 \, a^{6} b c d^{6}\right )} x^{2} + 2 \, {\left (a b^{6} c^{7} - 2 \, a^{2} b^{5} c^{6} d - 8 \, a^{3} b^{4} c^{5} d^{2} + 8 \, a^{5} b^{2} c^{3} d^{4} + 2 \, a^{6} b c^{2} d^{5} - a^{7} c d^{6}\right )} x - 2 \, {\left (a^{2} b^{5} c^{7} - 5 \, a^{3} b^{4} c^{6} d + 10 \, a^{4} b^{3} c^{5} d^{2} + {\left (b^{7} c^{5} d^{2} - 5 \, a b^{6} c^{4} d^{3} + 10 \, a^{2} b^{5} c^{3} d^{4}\right )} x^{4} + 2 \, {\left (b^{7} c^{6} d - 4 \, a b^{6} c^{5} d^{2} + 5 \, a^{2} b^{5} c^{4} d^{3} + 10 \, a^{3} b^{4} c^{3} d^{4}\right )} x^{3} + {\left (b^{7} c^{7} - a b^{6} c^{6} d - 9 \, a^{2} b^{5} c^{5} d^{2} + 35 \, a^{3} b^{4} c^{4} d^{3} + 10 \, a^{4} b^{3} c^{3} d^{4}\right )} x^{2} + 2 \, {\left (a b^{6} c^{7} - 4 \, a^{2} b^{5} c^{6} d + 5 \, a^{3} b^{4} c^{5} d^{2} + 10 \, a^{4} b^{3} c^{4} d^{3}\right )} x\right )} \log \left (b x + a\right ) + 2 \, {\left (10 \, a^{5} b^{2} c^{4} d^{3} - 5 \, a^{6} b c^{3} d^{4} + a^{7} c^{2} d^{5} + {\left (10 \, a^{3} b^{4} c^{2} d^{5} - 5 \, a^{4} b^{3} c d^{6} + a^{5} b^{2} d^{7}\right )} x^{4} + 2 \, {\left (10 \, a^{3} b^{4} c^{3} d^{4} + 5 \, a^{4} b^{3} c^{2} d^{5} - 4 \, a^{5} b^{2} c d^{6} + a^{6} b d^{7}\right )} x^{3} + {\left (10 \, a^{3} b^{4} c^{4} d^{3} + 35 \, a^{4} b^{3} c^{3} d^{4} - 9 \, a^{5} b^{2} c^{2} d^{5} - a^{6} b c d^{6} + a^{7} d^{7}\right )} x^{2} + 2 \, {\left (10 \, a^{4} b^{3} c^{4} d^{3} + 5 \, a^{5} b^{2} c^{3} d^{4} - 4 \, a^{6} b c^{2} d^{5} + a^{7} c d^{6}\right )} x\right )} \log \left (d x + c\right ) + 2 \, {\left (a^{2} b^{5} c^{7} - 5 \, a^{3} b^{4} c^{6} d + 10 \, a^{4} b^{3} c^{5} d^{2} - 10 \, a^{5} b^{2} c^{4} d^{3} + 5 \, a^{6} b c^{3} d^{4} - a^{7} c^{2} d^{5} + {\left (b^{7} c^{5} d^{2} - 5 \, a b^{6} c^{4} d^{3} + 10 \, a^{2} b^{5} c^{3} d^{4} - 10 \, a^{3} b^{4} c^{2} d^{5} + 5 \, a^{4} b^{3} c d^{6} - a^{5} b^{2} d^{7}\right )} x^{4} + 2 \, {\left (b^{7} c^{6} d - 4 \, a b^{6} c^{5} d^{2} + 5 \, a^{2} b^{5} c^{4} d^{3} - 5 \, a^{4} b^{3} c^{2} d^{5} + 4 \, a^{5} b^{2} c d^{6} - a^{6} b d^{7}\right )} x^{3} + {\left (b^{7} c^{7} - a b^{6} c^{6} d - 9 \, a^{2} b^{5} c^{5} d^{2} + 25 \, a^{3} b^{4} c^{4} d^{3} - 25 \, a^{4} b^{3} c^{3} d^{4} + 9 \, a^{5} b^{2} c^{2} d^{5} + a^{6} b c d^{6} - a^{7} d^{7}\right )} x^{2} + 2 \, {\left (a b^{6} c^{7} - 4 \, a^{2} b^{5} c^{6} d + 5 \, a^{3} b^{4} c^{5} d^{2} - 5 \, a^{5} b^{2} c^{3} d^{4} + 4 \, a^{6} b c^{2} d^{5} - a^{7} c d^{6}\right )} x\right )} \log \left (x\right )}{2 \, {\left (a^{5} b^{5} c^{10} - 5 \, a^{6} b^{4} c^{9} d + 10 \, a^{7} b^{3} c^{8} d^{2} - 10 \, a^{8} b^{2} c^{7} d^{3} + 5 \, a^{9} b c^{6} d^{4} - a^{10} c^{5} d^{5} + {\left (a^{3} b^{7} c^{8} d^{2} - 5 \, a^{4} b^{6} c^{7} d^{3} + 10 \, a^{5} b^{5} c^{6} d^{4} - 10 \, a^{6} b^{4} c^{5} d^{5} + 5 \, a^{7} b^{3} c^{4} d^{6} - a^{8} b^{2} c^{3} d^{7}\right )} x^{4} + 2 \, {\left (a^{3} b^{7} c^{9} d - 4 \, a^{4} b^{6} c^{8} d^{2} + 5 \, a^{5} b^{5} c^{7} d^{3} - 5 \, a^{7} b^{3} c^{5} d^{5} + 4 \, a^{8} b^{2} c^{4} d^{6} - a^{9} b c^{3} d^{7}\right )} x^{3} + {\left (a^{3} b^{7} c^{10} - a^{4} b^{6} c^{9} d - 9 \, a^{5} b^{5} c^{8} d^{2} + 25 \, a^{6} b^{4} c^{7} d^{3} - 25 \, a^{7} b^{3} c^{6} d^{4} + 9 \, a^{8} b^{2} c^{5} d^{5} + a^{9} b c^{4} d^{6} - a^{10} c^{3} d^{7}\right )} x^{2} + 2 \, {\left (a^{4} b^{6} c^{10} - 4 \, a^{5} b^{5} c^{9} d + 5 \, a^{6} b^{4} c^{8} d^{2} - 5 \, a^{8} b^{2} c^{6} d^{4} + 4 \, a^{9} b c^{5} d^{5} - a^{10} c^{4} d^{6}\right )} x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(b*x+a)^3/(d*x+c)^3,x, algorithm="fricas")

[Out]

1/2*(3*a^2*b^5*c^7 - 12*a^3*b^4*c^6*d + 9*a^4*b^3*c^5*d^2 - 9*a^5*b^2*c^4*d^3 + 12*a^6*b*c^3*d^4 - 3*a^7*c^2*d
^5 + 2*(a*b^6*c^5*d^2 - 5*a^2*b^5*c^4*d^3 + 5*a^4*b^3*c^2*d^5 - a^5*b^2*c*d^6)*x^3 + (4*a*b^6*c^6*d - 17*a^2*b
^5*c^5*d^2 - 5*a^3*b^4*c^4*d^3 + 5*a^4*b^3*c^3*d^4 + 17*a^5*b^2*c^2*d^5 - 4*a^6*b*c*d^6)*x^2 + 2*(a*b^6*c^7 -
2*a^2*b^5*c^6*d - 8*a^3*b^4*c^5*d^2 + 8*a^5*b^2*c^3*d^4 + 2*a^6*b*c^2*d^5 - a^7*c*d^6)*x - 2*(a^2*b^5*c^7 - 5*
a^3*b^4*c^6*d + 10*a^4*b^3*c^5*d^2 + (b^7*c^5*d^2 - 5*a*b^6*c^4*d^3 + 10*a^2*b^5*c^3*d^4)*x^4 + 2*(b^7*c^6*d -
4*a*b^6*c^5*d^2 + 5*a^2*b^5*c^4*d^3 + 10*a^3*b^4*c^3*d^4)*x^3 + (b^7*c^7 - a*b^6*c^6*d - 9*a^2*b^5*c^5*d^2 +
35*a^3*b^4*c^4*d^3 + 10*a^4*b^3*c^3*d^4)*x^2 + 2*(a*b^6*c^7 - 4*a^2*b^5*c^6*d + 5*a^3*b^4*c^5*d^2 + 10*a^4*b^3
*c^4*d^3)*x)*log(b*x + a) + 2*(10*a^5*b^2*c^4*d^3 - 5*a^6*b*c^3*d^4 + a^7*c^2*d^5 + (10*a^3*b^4*c^2*d^5 - 5*a^
4*b^3*c*d^6 + a^5*b^2*d^7)*x^4 + 2*(10*a^3*b^4*c^3*d^4 + 5*a^4*b^3*c^2*d^5 - 4*a^5*b^2*c*d^6 + a^6*b*d^7)*x^3
+ (10*a^3*b^4*c^4*d^3 + 35*a^4*b^3*c^3*d^4 - 9*a^5*b^2*c^2*d^5 - a^6*b*c*d^6 + a^7*d^7)*x^2 + 2*(10*a^4*b^3*c^
4*d^3 + 5*a^5*b^2*c^3*d^4 - 4*a^6*b*c^2*d^5 + a^7*c*d^6)*x)*log(d*x + c) + 2*(a^2*b^5*c^7 - 5*a^3*b^4*c^6*d +
10*a^4*b^3*c^5*d^2 - 10*a^5*b^2*c^4*d^3 + 5*a^6*b*c^3*d^4 - a^7*c^2*d^5 + (b^7*c^5*d^2 - 5*a*b^6*c^4*d^3 + 10*
a^2*b^5*c^3*d^4 - 10*a^3*b^4*c^2*d^5 + 5*a^4*b^3*c*d^6 - a^5*b^2*d^7)*x^4 + 2*(b^7*c^6*d - 4*a*b^6*c^5*d^2 + 5
*a^2*b^5*c^4*d^3 - 5*a^4*b^3*c^2*d^5 + 4*a^5*b^2*c*d^6 - a^6*b*d^7)*x^3 + (b^7*c^7 - a*b^6*c^6*d - 9*a^2*b^5*c
^5*d^2 + 25*a^3*b^4*c^4*d^3 - 25*a^4*b^3*c^3*d^4 + 9*a^5*b^2*c^2*d^5 + a^6*b*c*d^6 - a^7*d^7)*x^2 + 2*(a*b^6*c
^7 - 4*a^2*b^5*c^6*d + 5*a^3*b^4*c^5*d^2 - 5*a^5*b^2*c^3*d^4 + 4*a^6*b*c^2*d^5 - a^7*c*d^6)*x)*log(x))/(a^5*b^
5*c^10 - 5*a^6*b^4*c^9*d + 10*a^7*b^3*c^8*d^2 - 10*a^8*b^2*c^7*d^3 + 5*a^9*b*c^6*d^4 - a^10*c^5*d^5 + (a^3*b^7
*c^8*d^2 - 5*a^4*b^6*c^7*d^3 + 10*a^5*b^5*c^6*d^4 - 10*a^6*b^4*c^5*d^5 + 5*a^7*b^3*c^4*d^6 - a^8*b^2*c^3*d^7)*
x^4 + 2*(a^3*b^7*c^9*d - 4*a^4*b^6*c^8*d^2 + 5*a^5*b^5*c^7*d^3 - 5*a^7*b^3*c^5*d^5 + 4*a^8*b^2*c^4*d^6 - a^9*b
*c^3*d^7)*x^3 + (a^3*b^7*c^10 - a^4*b^6*c^9*d - 9*a^5*b^5*c^8*d^2 + 25*a^6*b^4*c^7*d^3 - 25*a^7*b^3*c^6*d^4 +
9*a^8*b^2*c^5*d^5 + a^9*b*c^4*d^6 - a^10*c^3*d^7)*x^2 + 2*(a^4*b^6*c^10 - 4*a^5*b^5*c^9*d + 5*a^6*b^4*c^8*d^2
- 5*a^8*b^2*c^6*d^4 + 4*a^9*b*c^5*d^5 - a^10*c^4*d^6)*x)

________________________________________________________________________________________

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/x/(b*x+a)**3/(d*x+c)**3,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 504 vs. \(2 (217) = 434\).
time = 1.04, size = 504, normalized size = 2.28 \begin {gather*} -\frac {{\left (b^{6} c^{2} - 5 \, a b^{5} c d + 10 \, a^{2} b^{4} d^{2}\right )} \log \left ({\left | b x + a \right |}\right )}{a^{3} b^{6} c^{5} - 5 \, a^{4} b^{5} c^{4} d + 10 \, a^{5} b^{4} c^{3} d^{2} - 10 \, a^{6} b^{3} c^{2} d^{3} + 5 \, a^{7} b^{2} c d^{4} - a^{8} b d^{5}} + \frac {{\left (10 \, b^{2} c^{2} d^{4} - 5 \, a b c d^{5} + a^{2} d^{6}\right )} \log \left ({\left | d x + c \right |}\right )}{b^{5} c^{8} d - 5 \, a b^{4} c^{7} d^{2} + 10 \, a^{2} b^{3} c^{6} d^{3} - 10 \, a^{3} b^{2} c^{5} d^{4} + 5 \, a^{4} b c^{4} d^{5} - a^{5} c^{3} d^{6}} + \frac {\log \left ({\left | x \right |}\right )}{a^{3} c^{3}} + \frac {3 \, a^{2} b^{4} c^{6} - 9 \, a^{3} b^{3} c^{5} d - 9 \, a^{5} b c^{3} d^{3} + 3 \, a^{6} c^{2} d^{4} + 2 \, {\left (a b^{5} c^{4} d^{2} - 4 \, a^{2} b^{4} c^{3} d^{3} - 4 \, a^{3} b^{3} c^{2} d^{4} + a^{4} b^{2} c d^{5}\right )} x^{3} + {\left (4 \, a b^{5} c^{5} d - 13 \, a^{2} b^{4} c^{4} d^{2} - 18 \, a^{3} b^{3} c^{3} d^{3} - 13 \, a^{4} b^{2} c^{2} d^{4} + 4 \, a^{5} b c d^{5}\right )} x^{2} + 2 \, {\left (a b^{5} c^{6} - a^{2} b^{4} c^{5} d - 9 \, a^{3} b^{3} c^{4} d^{2} - 9 \, a^{4} b^{2} c^{3} d^{3} - a^{5} b c^{2} d^{4} + a^{6} c d^{5}\right )} x}{2 \, {\left (b c - a d\right )}^{4} {\left (b x + a\right )}^{2} {\left (d x + c\right )}^{2} a^{3} c^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(b*x+a)^3/(d*x+c)^3,x, algorithm="giac")

[Out]

-(b^6*c^2 - 5*a*b^5*c*d + 10*a^2*b^4*d^2)*log(abs(b*x + a))/(a^3*b^6*c^5 - 5*a^4*b^5*c^4*d + 10*a^5*b^4*c^3*d^
2 - 10*a^6*b^3*c^2*d^3 + 5*a^7*b^2*c*d^4 - a^8*b*d^5) + (10*b^2*c^2*d^4 - 5*a*b*c*d^5 + a^2*d^6)*log(abs(d*x +
c))/(b^5*c^8*d - 5*a*b^4*c^7*d^2 + 10*a^2*b^3*c^6*d^3 - 10*a^3*b^2*c^5*d^4 + 5*a^4*b*c^4*d^5 - a^5*c^3*d^6) +
log(abs(x))/(a^3*c^3) + 1/2*(3*a^2*b^4*c^6 - 9*a^3*b^3*c^5*d - 9*a^5*b*c^3*d^3 + 3*a^6*c^2*d^4 + 2*(a*b^5*c^4
*d^2 - 4*a^2*b^4*c^3*d^3 - 4*a^3*b^3*c^2*d^4 + a^4*b^2*c*d^5)*x^3 + (4*a*b^5*c^5*d - 13*a^2*b^4*c^4*d^2 - 18*a
^3*b^3*c^3*d^3 - 13*a^4*b^2*c^2*d^4 + 4*a^5*b*c*d^5)*x^2 + 2*(a*b^5*c^6 - a^2*b^4*c^5*d - 9*a^3*b^3*c^4*d^2 -
9*a^4*b^2*c^3*d^3 - a^5*b*c^2*d^4 + a^6*c*d^5)*x)/((b*c - a*d)^4*(b*x + a)^2*(d*x + c)^2*a^3*c^3)

________________________________________________________________________________________

Mupad [B]
time = 1.44, size = 624, normalized size = 2.82 \begin {gather*} \frac {\frac {3\,\left (a^4\,d^4-3\,a^3\,b\,c\,d^3-3\,a\,b^3\,c^3\,d+b^4\,c^4\right )}{2\,a\,c\,\left (a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}-\frac {x^2\,\left (-4\,a^4\,b\,d^5+13\,a^3\,b^2\,c\,d^4+18\,a^2\,b^3\,c^2\,d^3+13\,a\,b^4\,c^3\,d^2-4\,b^5\,c^4\,d\right )}{2\,a^2\,c^2\,\left (a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}-\frac {x\,\left (-a^5\,d^5+a^4\,b\,c\,d^4+9\,a^3\,b^2\,c^2\,d^3+9\,a^2\,b^3\,c^3\,d^2+a\,b^4\,c^4\,d-b^5\,c^5\right )}{a^2\,c^2\,\left (a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}+\frac {b\,d\,x^3\,\left (a^3\,b\,d^4-4\,a^2\,b^2\,c\,d^3-4\,a\,b^3\,c^2\,d^2+b^4\,c^3\,d\right )}{a^2\,c^2\,\left (a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}}{x\,\left (2\,d\,a^2\,c+2\,b\,a\,c^2\right )+x^2\,\left (a^2\,d^2+4\,a\,b\,c\,d+b^2\,c^2\right )+x^3\,\left (2\,c\,b^2\,d+2\,a\,b\,d^2\right )+a^2\,c^2+b^2\,d^2\,x^4}+\frac {\ln \left (x\right )}{a^3\,c^3}+\frac {b^3\,\ln \left (a+b\,x\right )\,\left (10\,a^2\,d^2-5\,a\,b\,c\,d+b^2\,c^2\right )}{a^3\,{\left (a\,d-b\,c\right )}^5}-\frac {d^3\,\ln \left (c+d\,x\right )\,\left (a^2\,d^2-5\,a\,b\,c\,d+10\,b^2\,c^2\right )}{c^3\,{\left (a\,d-b\,c\right )}^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*(a + b*x)^3*(c + d*x)^3),x)

[Out]

((3*(a^4*d^4 + b^4*c^4 - 3*a*b^3*c^3*d - 3*a^3*b*c*d^3))/(2*a*c*(a^4*d^4 + b^4*c^4 + 6*a^2*b^2*c^2*d^2 - 4*a*b
^3*c^3*d - 4*a^3*b*c*d^3)) - (x^2*(13*a*b^4*c^3*d^2 - 4*b^5*c^4*d - 4*a^4*b*d^5 + 13*a^3*b^2*c*d^4 + 18*a^2*b^
3*c^2*d^3))/(2*a^2*c^2*(a^4*d^4 + b^4*c^4 + 6*a^2*b^2*c^2*d^2 - 4*a*b^3*c^3*d - 4*a^3*b*c*d^3)) - (x*(9*a^2*b^
3*c^3*d^2 - b^5*c^5 - a^5*d^5 + 9*a^3*b^2*c^2*d^3 + a*b^4*c^4*d + a^4*b*c*d^4))/(a^2*c^2*(a^4*d^4 + b^4*c^4 +
6*a^2*b^2*c^2*d^2 - 4*a*b^3*c^3*d - 4*a^3*b*c*d^3)) + (b*d*x^3*(a^3*b*d^4 + b^4*c^3*d - 4*a*b^3*c^2*d^2 - 4*a^
2*b^2*c*d^3))/(a^2*c^2*(a^4*d^4 + b^4*c^4 + 6*a^2*b^2*c^2*d^2 - 4*a*b^3*c^3*d - 4*a^3*b*c*d^3)))/(x*(2*a*b*c^2
+ 2*a^2*c*d) + x^2*(a^2*d^2 + b^2*c^2 + 4*a*b*c*d) + x^3*(2*a*b*d^2 + 2*b^2*c*d) + a^2*c^2 + b^2*d^2*x^4) + l
og(x)/(a^3*c^3) + (b^3*log(a + b*x)*(10*a^2*d^2 + b^2*c^2 - 5*a*b*c*d))/(a^3*(a*d - b*c)^5) - (d^3*log(c + d*x
)*(a^2*d^2 + 10*b^2*c^2 - 5*a*b*c*d))/(c^3*(a*d - b*c)^5)

________________________________________________________________________________________