\(\int \frac {x^6}{(a+b x)^3 (c+d x)^3} \, dx\) [310]

Optimal result

Integrand size = 18, antiderivative size = 222 \[ \int \frac {x^6}{(a+b x)^3 (c+d x)^3} \, dx=\frac {x}{b^3 d^3}-\frac {a^6}{2 b^4 (b c-a d)^3 (a+b x)^2}+\frac {3 a^5 (2 b c-a d)}{b^4 (b c-a d)^4 (a+b x)}+\frac {c^6}{2 d^4 (b c-a d)^3 (c+d x)^2}-\frac {3 c^5 (b c-2 a d)}{d^4 (b c-a d)^4 (c+d x)}+\frac {3 a^4 \left (5 b^2 c^2-4 a b c d+a^2 d^2\right ) \log (a+b x)}{b^4 (b c-a d)^5}-\frac {3 c^4 \left (b^2 c^2-4 a b c d+5 a^2 d^2\right ) \log (c+d x)}{d^4 (b c-a d)^5} \]

[Out]

x/b^3/d^3-1/2*a^6/b^4/(-a*d+b*c)^3/(b*x+a)^2+3*a^5*(-a*d+2*b*c)/b^4/(-a*d+b*c)^4/(b*x+a)+1/2*c^6/d^4/(-a*d+b*c
)^3/(d*x+c)^2-3*c^5*(-2*a*d+b*c)/d^4/(-a*d+b*c)^4/(d*x+c)+3*a^4*(a^2*d^2-4*a*b*c*d+5*b^2*c^2)*ln(b*x+a)/b^4/(-
a*d+b*c)^5-3*c^4*(5*a^2*d^2-4*a*b*c*d+b^2*c^2)*ln(d*x+c)/d^4/(-a*d+b*c)^5

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {90} \[ \int \frac {x^6}{(a+b x)^3 (c+d x)^3} \, dx=-\frac {a^6}{2 b^4 (a+b x)^2 (b c-a d)^3}+\frac {3 a^5 (2 b c-a d)}{b^4 (a+b x) (b c-a d)^4}-\frac {3 c^4 \left (5 a^2 d^2-4 a b c d+b^2 c^2\right ) \log (c+d x)}{d^4 (b c-a d)^5}+\frac {3 a^4 \left (a^2 d^2-4 a b c d+5 b^2 c^2\right ) \log (a+b x)}{b^4 (b c-a d)^5}+\frac {c^6}{2 d^4 (c+d x)^2 (b c-a d)^3}-\frac {3 c^5 (b c-2 a d)}{d^4 (c+d x) (b c-a d)^4}+\frac {x}{b^3 d^3} \]

[In]

Int[x^6/((a + b*x)^3*(c + d*x)^3),x]

[Out]

x/(b^3*d^3) - a^6/(2*b^4*(b*c - a*d)^3*(a + b*x)^2) + (3*a^5*(2*b*c - a*d))/(b^4*(b*c - a*d)^4*(a + b*x)) + c^
6/(2*d^4*(b*c - a*d)^3*(c + d*x)^2) - (3*c^5*(b*c - 2*a*d))/(d^4*(b*c - a*d)^4*(c + d*x)) + (3*a^4*(5*b^2*c^2
- 4*a*b*c*d + a^2*d^2)*Log[a + b*x])/(b^4*(b*c - a*d)^5) - (3*c^4*(b^2*c^2 - 4*a*b*c*d + 5*a^2*d^2)*Log[c + d*
x])/(d^4*(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*} \text {integral}& = \int \left (\frac {1}{b^3 d^3}+\frac {a^6}{b^3 (b c-a d)^3 (a+b x)^3}+\frac {3 a^5 (-2 b c+a d)}{b^3 (b c-a d)^4 (a+b x)^2}+\frac {3 a^4 \left (5 b^2 c^2-4 a b c d+a^2 d^2\right )}{b^3 (b c-a d)^5 (a+b x)}+\frac {c^6}{d^3 (-b c+a d)^3 (c+d x)^3}+\frac {3 c^5 (b c-2 a d)}{d^3 (-b c+a d)^4 (c+d x)^2}+\frac {3 c^4 \left (b^2 c^2-4 a b c d+5 a^2 d^2\right )}{d^3 (-b c+a d)^5 (c+d x)}\right ) \, dx \\ & = \frac {x}{b^3 d^3}-\frac {a^6}{2 b^4 (b c-a d)^3 (a+b x)^2}+\frac {3 a^5 (2 b c-a d)}{b^4 (b c-a d)^4 (a+b x)}+\frac {c^6}{2 d^4 (b c-a d)^3 (c+d x)^2}-\frac {3 c^5 (b c-2 a d)}{d^4 (b c-a d)^4 (c+d x)}+\frac {3 a^4 \left (5 b^2 c^2-4 a b c d+a^2 d^2\right ) \log (a+b x)}{b^4 (b c-a d)^5}-\frac {3 c^4 \left (b^2 c^2-4 a b c d+5 a^2 d^2\right ) \log (c+d x)}{d^4 (b c-a d)^5} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.00 \[ \int \frac {x^6}{(a+b x)^3 (c+d x)^3} \, dx=\frac {x}{b^3 d^3}-\frac {a^6}{2 b^4 (b c-a d)^3 (a+b x)^2}-\frac {3 a^5 (-2 b c+a d)}{b^4 (b c-a d)^4 (a+b x)}-\frac {c^6}{2 d^4 (-b c+a d)^3 (c+d x)^2}-\frac {3 c^5 (b c-2 a d)}{d^4 (b c-a d)^4 (c+d x)}+\frac {3 a^4 \left (5 b^2 c^2-4 a b c d+a^2 d^2\right ) \log (a+b x)}{b^4 (b c-a d)^5}+\frac {3 c^4 \left (b^2 c^2-4 a b c d+5 a^2 d^2\right ) \log (c+d x)}{d^4 (-b c+a d)^5} \]

[In]

Integrate[x^6/((a + b*x)^3*(c + d*x)^3),x]

[Out]

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

Maple [A] (verified)

Time = 0.55 (sec) , antiderivative size = 219, normalized size of antiderivative = 0.99

method	result	size
default	\(\frac {x}{b^{3} d^{3}}-\frac {c^{6}}{2 d^{4} \left (a d -b c \right )^{3} \left (d x +c \right )^{2}}+\frac {3 c^{4} \left (5 a^{2} d^{2}-4 a b c d +b^{2} c^{2}\right ) \ln \left (d x +c \right )}{d^{4} \left (a d -b c \right )^{5}}+\frac {3 c^{5} \left (2 a d -b c \right )}{d^{4} \left (a d -b c \right )^{4} \left (d x +c \right )}+\frac {a^{6}}{2 b^{4} \left (a d -b c \right )^{3} \left (b x +a \right )^{2}}-\frac {3 a^{4} \left (a^{2} d^{2}-4 a b c d +5 b^{2} c^{2}\right ) \ln \left (b x +a \right )}{b^{4} \left (a d -b c \right )^{5}}-\frac {3 a^{5} \left (a d -2 b c \right )}{b^{4} \left (a d -b c \right )^{4} \left (b x +a \right )}\)	\(219\)
norman	\(\frac {\frac {x^{5}}{b d}-\frac {\left (6 a^{6} d^{6}-14 a^{5} b c \,d^{5}+5 a^{4} b^{2} c^{2} d^{4}+5 a^{2} b^{4} c^{4} d^{2}-14 a \,b^{5} c^{5} d +6 b^{6} c^{6}\right ) x^{3}}{d^{3} b^{3} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}-\frac {\left (9 a^{7} d^{7}+a^{6} b c \,d^{6}-48 a^{5} b^{2} c^{2} d^{5}+20 a^{4} b^{3} c^{3} d^{4}+20 a^{3} b^{4} c^{4} d^{3}-48 a^{2} b^{5} c^{5} d^{2}+a \,b^{6} c^{6} d +9 b^{7} c^{7}\right ) x^{2}}{2 d^{4} b^{4} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}-\frac {a^{2} c^{2} \left (9 a^{5} d^{5}-23 a^{4} b c \,d^{4}+8 a^{3} b^{2} c^{2} d^{3}+8 a^{2} b^{3} c^{3} d^{2}-23 a \,b^{4} c^{4} d +9 b^{5} c^{5}\right )}{2 d^{4} b^{4} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}-\frac {c a \left (9 a^{6} d^{6}-17 a^{5} b c \,d^{5}-6 a^{4} b^{2} c^{2} d^{4}+10 a^{3} b^{3} c^{3} d^{3}-6 a^{2} b^{4} c^{4} d^{2}-17 a \,b^{5} c^{5} d +9 b^{6} c^{6}\right ) x}{d^{4} b^{4} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}}{\left (b x +a \right )^{2} \left (d x +c \right )^{2}}-\frac {3 a^{4} \left (a^{2} d^{2}-4 a b c d +5 b^{2} c^{2}\right ) \ln \left (b x +a \right )}{\left (a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}\right ) b^{4}}+\frac {3 c^{4} \left (5 a^{2} d^{2}-4 a b c d +b^{2} c^{2}\right ) \ln \left (d x +c \right )}{d^{4} \left (a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}\right )}\)	\(783\)
risch	\(\frac {x}{b^{3} d^{3}}+\frac {-\frac {3 \left (a^{6} d^{6}-2 a^{5} b c \,d^{5}-2 a \,b^{5} c^{5} d +b^{6} c^{6}\right ) x^{3}}{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}}-\frac {\left (5 a^{7} d^{7}+a^{6} b c \,d^{6}-24 a^{5} b^{2} c^{2} d^{5}-24 a^{2} b^{5} c^{5} d^{2}+a \,b^{6} c^{6} d +5 b^{7} c^{7}\right ) x^{2}}{2 b d \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 (5 a^{6} d^{6}-8 a^{5} b c \,d^{5}-6 a^{4} b^{2} c^{2} d^{4}-6 a^{2} b^{4} c^{4} d^{2}-8 a \,b^{5} c^{5} d +5 b^{6} c^{6}\right ) a c x}{b d \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 {c^{2} a^{2} \left (5 a^{5} d^{5}-11 a^{4} b c \,d^{4}-11 a \,b^{4} c^{4} d +5 b^{5} c^{5}\right )}{2 b d \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 )}}{b^{3} d^{3} \left (b x +a \right )^{2} \left (d x +c \right )^{2}}+\frac {15 c^{4} \ln \left (-d x -c \right ) a^{2}}{d^{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 {12 c^{5} \ln \left (-d x -c \right ) a b}{d^{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 {3 c^{6} \ln \left (-d x -c \right ) b^{2}}{d^{4} \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 {3 a^{6} \ln \left (b x +a \right ) d^{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 ) b^{4}}+\frac {12 a^{5} \ln \left (b x +a \right ) c d}{\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 ) b^{3}}-\frac {15 a^{4} \ln \left (b x +a \right ) 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 ) b^{2}}\)	\(982\)
parallelrisch	\(\text {Expression too large to display}\)	\(1334\)

[In]

int(x^6/(b*x+a)^3/(d*x+c)^3,x,method=_RETURNVERBOSE)

[Out]

x/b^3/d^3-1/2/d^4*c^6/(a*d-b*c)^3/(d*x+c)^2+3/d^4*c^4*(5*a^2*d^2-4*a*b*c*d+b^2*c^2)/(a*d-b*c)^5*ln(d*x+c)+3/d^
4*c^5*(2*a*d-b*c)/(a*d-b*c)^4/(d*x+c)+1/2/b^4*a^6/(a*d-b*c)^3/(b*x+a)^2-3/b^4*a^4*(a^2*d^2-4*a*b*c*d+5*b^2*c^2
)/(a*d-b*c)^5*ln(b*x+a)-3/b^4*a^5*(a*d-2*b*c)/(a*d-b*c)^4/(b*x+a)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1457 vs. \(2 (218) = 436\).

Time = 0.31 (sec) , antiderivative size = 1457, normalized size of antiderivative = 6.56 \[ \int \frac {x^6}{(a+b x)^3 (c+d x)^3} \, dx=\text {Too large to display} \]

[In]

integrate(x^6/(b*x+a)^3/(d*x+c)^3,x, algorithm="fricas")

[Out]

-1/2*(5*a^2*b^6*c^8 - 16*a^3*b^5*c^7*d + 11*a^4*b^4*c^6*d^2 - 11*a^6*b^2*c^4*d^4 + 16*a^7*b*c^3*d^5 - 5*a^8*c^
2*d^6 - 2*(b^8*c^5*d^3 - 5*a*b^7*c^4*d^4 + 10*a^2*b^6*c^3*d^5 - 10*a^3*b^5*c^2*d^6 + 5*a^4*b^4*c*d^7 - a^5*b^3
*d^8)*x^5 - 4*(b^8*c^6*d^2 - 4*a*b^7*c^5*d^3 + 5*a^2*b^6*c^4*d^4 - 5*a^4*b^4*c^2*d^6 + 4*a^5*b^3*c*d^7 - a^6*b
^2*d^8)*x^4 + 2*(2*b^8*c^7*d - 8*a*b^7*c^6*d^2 + 15*a^2*b^6*c^5*d^3 - 25*a^3*b^5*c^4*d^4 + 25*a^4*b^4*c^3*d^5
- 15*a^5*b^3*c^2*d^6 + 8*a^6*b^2*c*d^7 - 2*a^7*b*d^8)*x^3 + (5*b^8*c^8 - 8*a*b^7*c^7*d - 9*a^2*b^6*c^6*d^2 + 4
*a^3*b^5*c^5*d^3 - 4*a^5*b^3*c^3*d^5 + 9*a^6*b^2*c^2*d^6 + 8*a^7*b*c*d^7 - 5*a^8*d^8)*x^2 + 2*(5*a*b^7*c^8 - 1
4*a^2*b^6*c^7*d + 7*a^3*b^5*c^6*d^2 - 4*a^4*b^4*c^5*d^3 + 4*a^5*b^3*c^4*d^4 - 7*a^6*b^2*c^3*d^5 + 14*a^7*b*c^2
*d^6 - 5*a^8*c*d^7)*x - 6*(5*a^6*b^2*c^4*d^4 - 4*a^7*b*c^3*d^5 + a^8*c^2*d^6 + (5*a^4*b^4*c^2*d^6 - 4*a^5*b^3*
c*d^7 + a^6*b^2*d^8)*x^4 + 2*(5*a^4*b^4*c^3*d^5 + a^5*b^3*c^2*d^6 - 3*a^6*b^2*c*d^7 + a^7*b*d^8)*x^3 + (5*a^4*
b^4*c^4*d^4 + 16*a^5*b^3*c^3*d^5 - 10*a^6*b^2*c^2*d^6 + a^8*d^8)*x^2 + 2*(5*a^5*b^3*c^4*d^4 + a^6*b^2*c^3*d^5
- 3*a^7*b*c^2*d^6 + a^8*c*d^7)*x)*log(b*x + a) + 6*(a^2*b^6*c^8 - 4*a^3*b^5*c^7*d + 5*a^4*b^4*c^6*d^2 + (b^8*c
^6*d^2 - 4*a*b^7*c^5*d^3 + 5*a^2*b^6*c^4*d^4)*x^4 + 2*(b^8*c^7*d - 3*a*b^7*c^6*d^2 + a^2*b^6*c^5*d^3 + 5*a^3*b
^5*c^4*d^4)*x^3 + (b^8*c^8 - 10*a^2*b^6*c^6*d^2 + 16*a^3*b^5*c^5*d^3 + 5*a^4*b^4*c^4*d^4)*x^2 + 2*(a*b^7*c^8 -
3*a^2*b^6*c^7*d + a^3*b^5*c^6*d^2 + 5*a^4*b^4*c^5*d^3)*x)*log(d*x + c))/(a^2*b^9*c^7*d^4 - 5*a^3*b^8*c^6*d^5
+ 10*a^4*b^7*c^5*d^6 - 10*a^5*b^6*c^4*d^7 + 5*a^6*b^5*c^3*d^8 - a^7*b^4*c^2*d^9 + (b^11*c^5*d^6 - 5*a*b^10*c^4
*d^7 + 10*a^2*b^9*c^3*d^8 - 10*a^3*b^8*c^2*d^9 + 5*a^4*b^7*c*d^10 - a^5*b^6*d^11)*x^4 + 2*(b^11*c^6*d^5 - 4*a*
b^10*c^5*d^6 + 5*a^2*b^9*c^4*d^7 - 5*a^4*b^7*c^2*d^9 + 4*a^5*b^6*c*d^10 - a^6*b^5*d^11)*x^3 + (b^11*c^7*d^4 -
a*b^10*c^6*d^5 - 9*a^2*b^9*c^5*d^6 + 25*a^3*b^8*c^4*d^7 - 25*a^4*b^7*c^3*d^8 + 9*a^5*b^6*c^2*d^9 + a^6*b^5*c*d
^10 - a^7*b^4*d^11)*x^2 + 2*(a*b^10*c^7*d^4 - 4*a^2*b^9*c^6*d^5 + 5*a^3*b^8*c^5*d^6 - 5*a^5*b^6*c^3*d^8 + 4*a^
6*b^5*c^2*d^9 - a^7*b^4*c*d^10)*x)

Sympy [F(-1)]

Timed out. \[ \int \frac {x^6}{(a+b x)^3 (c+d x)^3} \, dx=\text {Timed out} \]

[In]

integrate(x**6/(b*x+a)**3/(d*x+c)**3,x)

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 818 vs. \(2 (218) = 436\).

Time = 0.30 (sec) , antiderivative size = 818, normalized size of antiderivative = 3.68 \[ \int \frac {x^6}{(a+b x)^3 (c+d x)^3} \, dx=\frac {3 \, {\left (5 \, a^{4} b^{2} c^{2} - 4 \, a^{5} b c d + a^{6} d^{2}\right )} \log \left (b x + a\right )}{b^{9} c^{5} - 5 \, a b^{8} c^{4} d + 10 \, a^{2} b^{7} c^{3} d^{2} - 10 \, a^{3} b^{6} c^{2} d^{3} + 5 \, a^{4} b^{5} c d^{4} - a^{5} b^{4} d^{5}} - \frac {3 \, {\left (b^{2} c^{6} - 4 \, a b c^{5} d + 5 \, a^{2} c^{4} d^{2}\right )} \log \left (d x + c\right )}{b^{5} c^{5} d^{4} - 5 \, a b^{4} c^{4} d^{5} + 10 \, a^{2} b^{3} c^{3} d^{6} - 10 \, a^{3} b^{2} c^{2} d^{7} + 5 \, a^{4} b c d^{8} - a^{5} d^{9}} - \frac {5 \, a^{2} b^{5} c^{7} - 11 \, a^{3} b^{4} c^{6} d - 11 \, a^{6} b c^{3} d^{4} + 5 \, a^{7} c^{2} d^{5} + 6 \, {\left (b^{7} c^{6} d - 2 \, a b^{6} c^{5} d^{2} - 2 \, a^{5} b^{2} c d^{6} + a^{6} b d^{7}\right )} x^{3} + {\left (5 \, b^{7} c^{7} + a b^{6} c^{6} d - 24 \, a^{2} b^{5} c^{5} d^{2} - 24 \, a^{5} b^{2} c^{2} d^{5} + a^{6} b c d^{6} + 5 \, a^{7} d^{7}\right )} x^{2} + 2 \, {\left (5 \, a b^{6} c^{7} - 8 \, a^{2} b^{5} c^{6} d - 6 \, a^{3} b^{4} c^{5} d^{2} - 6 \, a^{5} b^{2} c^{3} d^{4} - 8 \, a^{6} b c^{2} d^{5} + 5 \, a^{7} c d^{6}\right )} x}{2 \, {\left (a^{2} b^{8} c^{6} d^{4} - 4 \, a^{3} b^{7} c^{5} d^{5} + 6 \, a^{4} b^{6} c^{4} d^{6} - 4 \, a^{5} b^{5} c^{3} d^{7} + a^{6} b^{4} c^{2} d^{8} + {\left (b^{10} c^{4} d^{6} - 4 \, a b^{9} c^{3} d^{7} + 6 \, a^{2} b^{8} c^{2} d^{8} - 4 \, a^{3} b^{7} c d^{9} + a^{4} b^{6} d^{10}\right )} x^{4} + 2 \, {\left (b^{10} c^{5} d^{5} - 3 \, a b^{9} c^{4} d^{6} + 2 \, a^{2} b^{8} c^{3} d^{7} + 2 \, a^{3} b^{7} c^{2} d^{8} - 3 \, a^{4} b^{6} c d^{9} + a^{5} b^{5} d^{10}\right )} x^{3} + {\left (b^{10} c^{6} d^{4} - 9 \, a^{2} b^{8} c^{4} d^{6} + 16 \, a^{3} b^{7} c^{3} d^{7} - 9 \, a^{4} b^{6} c^{2} d^{8} + a^{6} b^{4} d^{10}\right )} x^{2} + 2 \, {\left (a b^{9} c^{6} d^{4} - 3 \, a^{2} b^{8} c^{5} d^{5} + 2 \, a^{3} b^{7} c^{4} d^{6} + 2 \, a^{4} b^{6} c^{3} d^{7} - 3 \, a^{5} b^{5} c^{2} d^{8} + a^{6} b^{4} c d^{9}\right )} x\right )}} + \frac {x}{b^{3} d^{3}} \]

[In]

integrate(x^6/(b*x+a)^3/(d*x+c)^3,x, algorithm="maxima")

[Out]

3*(5*a^4*b^2*c^2 - 4*a^5*b*c*d + a^6*d^2)*log(b*x + a)/(b^9*c^5 - 5*a*b^8*c^4*d + 10*a^2*b^7*c^3*d^2 - 10*a^3*
b^6*c^2*d^3 + 5*a^4*b^5*c*d^4 - a^5*b^4*d^5) - 3*(b^2*c^6 - 4*a*b*c^5*d + 5*a^2*c^4*d^2)*log(d*x + c)/(b^5*c^5
*d^4 - 5*a*b^4*c^4*d^5 + 10*a^2*b^3*c^3*d^6 - 10*a^3*b^2*c^2*d^7 + 5*a^4*b*c*d^8 - a^5*d^9) - 1/2*(5*a^2*b^5*c
^7 - 11*a^3*b^4*c^6*d - 11*a^6*b*c^3*d^4 + 5*a^7*c^2*d^5 + 6*(b^7*c^6*d - 2*a*b^6*c^5*d^2 - 2*a^5*b^2*c*d^6 +
a^6*b*d^7)*x^3 + (5*b^7*c^7 + a*b^6*c^6*d - 24*a^2*b^5*c^5*d^2 - 24*a^5*b^2*c^2*d^5 + a^6*b*c*d^6 + 5*a^7*d^7)
*x^2 + 2*(5*a*b^6*c^7 - 8*a^2*b^5*c^6*d - 6*a^3*b^4*c^5*d^2 - 6*a^5*b^2*c^3*d^4 - 8*a^6*b*c^2*d^5 + 5*a^7*c*d^
6)*x)/(a^2*b^8*c^6*d^4 - 4*a^3*b^7*c^5*d^5 + 6*a^4*b^6*c^4*d^6 - 4*a^5*b^5*c^3*d^7 + a^6*b^4*c^2*d^8 + (b^10*c
^4*d^6 - 4*a*b^9*c^3*d^7 + 6*a^2*b^8*c^2*d^8 - 4*a^3*b^7*c*d^9 + a^4*b^6*d^10)*x^4 + 2*(b^10*c^5*d^5 - 3*a*b^9
*c^4*d^6 + 2*a^2*b^8*c^3*d^7 + 2*a^3*b^7*c^2*d^8 - 3*a^4*b^6*c*d^9 + a^5*b^5*d^10)*x^3 + (b^10*c^6*d^4 - 9*a^2
*b^8*c^4*d^6 + 16*a^3*b^7*c^3*d^7 - 9*a^4*b^6*c^2*d^8 + a^6*b^4*d^10)*x^2 + 2*(a*b^9*c^6*d^4 - 3*a^2*b^8*c^5*d
^5 + 2*a^3*b^7*c^4*d^6 + 2*a^4*b^6*c^3*d^7 - 3*a^5*b^5*c^2*d^8 + a^6*b^4*c*d^9)*x) + x/(b^3*d^3)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 489 vs. \(2 (218) = 436\).

Time = 0.30 (sec) , antiderivative size = 489, normalized size of antiderivative = 2.20 \[ \int \frac {x^6}{(a+b x)^3 (c+d x)^3} \, dx=\frac {3 \, {\left (5 \, a^{4} b^{2} c^{2} - 4 \, a^{5} b c d + a^{6} d^{2}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{9} c^{5} - 5 \, a b^{8} c^{4} d + 10 \, a^{2} b^{7} c^{3} d^{2} - 10 \, a^{3} b^{6} c^{2} d^{3} + 5 \, a^{4} b^{5} c d^{4} - a^{5} b^{4} d^{5}} - \frac {3 \, {\left (b^{2} c^{6} - 4 \, a b c^{5} d + 5 \, a^{2} c^{4} d^{2}\right )} \log \left ({\left | d x + c \right |}\right )}{b^{5} c^{5} d^{4} - 5 \, a b^{4} c^{4} d^{5} + 10 \, a^{2} b^{3} c^{3} d^{6} - 10 \, a^{3} b^{2} c^{2} d^{7} + 5 \, a^{4} b c d^{8} - a^{5} d^{9}} + \frac {x}{b^{3} d^{3}} - \frac {5 \, a^{2} b^{5} c^{7} - 11 \, a^{3} b^{4} c^{6} d - 11 \, a^{6} b c^{3} d^{4} + 5 \, a^{7} c^{2} d^{5} + 6 \, {\left (b^{7} c^{6} d - 2 \, a b^{6} c^{5} d^{2} - 2 \, a^{5} b^{2} c d^{6} + a^{6} b d^{7}\right )} x^{3} + {\left (5 \, b^{7} c^{7} + a b^{6} c^{6} d - 24 \, a^{2} b^{5} c^{5} d^{2} - 24 \, a^{5} b^{2} c^{2} d^{5} + a^{6} b c d^{6} + 5 \, a^{7} d^{7}\right )} x^{2} + 2 \, {\left (5 \, a b^{6} c^{7} - 8 \, a^{2} b^{5} c^{6} d - 6 \, a^{3} b^{4} c^{5} d^{2} - 6 \, a^{5} b^{2} c^{3} d^{4} - 8 \, a^{6} b c^{2} d^{5} + 5 \, a^{7} c d^{6}\right )} x}{2 \, {\left (b c - a d\right )}^{4} {\left (b x + a\right )}^{2} {\left (d x + c\right )}^{2} b^{4} d^{4}} \]

[In]

integrate(x^6/(b*x+a)^3/(d*x+c)^3,x, algorithm="giac")

[Out]

3*(5*a^4*b^2*c^2 - 4*a^5*b*c*d + a^6*d^2)*log(abs(b*x + a))/(b^9*c^5 - 5*a*b^8*c^4*d + 10*a^2*b^7*c^3*d^2 - 10
*a^3*b^6*c^2*d^3 + 5*a^4*b^5*c*d^4 - a^5*b^4*d^5) - 3*(b^2*c^6 - 4*a*b*c^5*d + 5*a^2*c^4*d^2)*log(abs(d*x + c)
)/(b^5*c^5*d^4 - 5*a*b^4*c^4*d^5 + 10*a^2*b^3*c^3*d^6 - 10*a^3*b^2*c^2*d^7 + 5*a^4*b*c*d^8 - a^5*d^9) + x/(b^3
*d^3) - 1/2*(5*a^2*b^5*c^7 - 11*a^3*b^4*c^6*d - 11*a^6*b*c^3*d^4 + 5*a^7*c^2*d^5 + 6*(b^7*c^6*d - 2*a*b^6*c^5*
d^2 - 2*a^5*b^2*c*d^6 + a^6*b*d^7)*x^3 + (5*b^7*c^7 + a*b^6*c^6*d - 24*a^2*b^5*c^5*d^2 - 24*a^5*b^2*c^2*d^5 +
a^6*b*c*d^6 + 5*a^7*d^7)*x^2 + 2*(5*a*b^6*c^7 - 8*a^2*b^5*c^6*d - 6*a^3*b^4*c^5*d^2 - 6*a^5*b^2*c^3*d^4 - 8*a^
6*b*c^2*d^5 + 5*a^7*c*d^6)*x)/((b*c - a*d)^4*(b*x + a)^2*(d*x + c)^2*b^4*d^4)

Mupad [B] (verification not implemented)

Time = 0.84 (sec) , antiderivative size = 780, normalized size of antiderivative = 3.51 \[ \int \frac {x^6}{(a+b x)^3 (c+d x)^3} \, dx=\frac {x}{b^3\,d^3}-\frac {\frac {3\,x^3\,\left (a^6\,d^6-2\,a^5\,b\,c\,d^5-2\,a\,b^5\,c^5\,d+b^6\,c^6\right )}{a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4}+\frac {x^2\,\left (5\,a^7\,d^7+a^6\,b\,c\,d^6-24\,a^5\,b^2\,c^2\,d^5-24\,a^2\,b^5\,c^5\,d^2+a\,b^6\,c^6\,d+5\,b^7\,c^7\right )}{2\,b\,d\,\left (a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}+\frac {a^2\,c^2\,\left (5\,a^5\,d^5-11\,a^4\,b\,c\,d^4-11\,a\,b^4\,c^4\,d+5\,b^5\,c^5\right )}{2\,b\,d\,\left (a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}-\frac {a\,c\,x\,\left (-5\,a^6\,d^6+8\,a^5\,b\,c\,d^5+6\,a^4\,b^2\,c^2\,d^4+6\,a^2\,b^4\,c^4\,d^2+8\,a\,b^5\,c^5\,d-5\,b^6\,c^6\right )}{b\,d\,\left (a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}}{x^3\,\left (2\,c\,b^5\,d^4+2\,a\,b^4\,d^5\right )+x\,\left (2\,a^2\,b^3\,c\,d^4+2\,a\,b^4\,c^2\,d^3\right )+x^2\,\left (a^2\,b^3\,d^5+4\,a\,b^4\,c\,d^4+b^5\,c^2\,d^3\right )+b^5\,d^5\,x^4+a^2\,b^3\,c^2\,d^3}+\frac {\ln \left (a+b\,x\right )\,\left (3\,a^6\,d^2-12\,a^5\,b\,c\,d+15\,a^4\,b^2\,c^2\right )}{-a^5\,b^4\,d^5+5\,a^4\,b^5\,c\,d^4-10\,a^3\,b^6\,c^2\,d^3+10\,a^2\,b^7\,c^3\,d^2-5\,a\,b^8\,c^4\,d+b^9\,c^5}+\frac {\ln \left (c+d\,x\right )\,\left (15\,a^2\,c^4\,d^2-12\,a\,b\,c^5\,d+3\,b^2\,c^6\right )}{a^5\,d^9-5\,a^4\,b\,c\,d^8+10\,a^3\,b^2\,c^2\,d^7-10\,a^2\,b^3\,c^3\,d^6+5\,a\,b^4\,c^4\,d^5-b^5\,c^5\,d^4} \]

[In]

int(x^6/((a + b*x)^3*(c + d*x)^3),x)

[Out]

x/(b^3*d^3) - ((3*x^3*(a^6*d^6 + b^6*c^6 - 2*a*b^5*c^5*d - 2*a^5*b*c*d^5))/(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*(5*a^7*d^7 + 5*b^7*c^7 - 24*a^2*b^5*c^5*d^2 - 24*a^5*b^2*c^2*d^5 +
a*b^6*c^6*d + a^6*b*c*d^6))/(2*b*d*(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)) +
(a^2*c^2*(5*a^5*d^5 + 5*b^5*c^5 - 11*a*b^4*c^4*d - 11*a^4*b*c*d^4))/(2*b*d*(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)) - (a*c*x*(6*a^2*b^4*c^4*d^2 - 5*b^6*c^6 - 5*a^6*d^6 + 6*a^4*b^2*c^2*d^4
+ 8*a*b^5*c^5*d + 8*a^5*b*c*d^5))/(b*d*(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^3*(2*a*b^4*d^5 + 2*b^5*c*d^4) + x*(2*a*b^4*c^2*d^3 + 2*a^2*b^3*c*d^4) + x^2*(a^2*b^3*d^5 + b^5*c^2*d^3
+ 4*a*b^4*c*d^4) + b^5*d^5*x^4 + a^2*b^3*c^2*d^3) + (log(a + b*x)*(3*a^6*d^2 + 15*a^4*b^2*c^2 - 12*a^5*b*c*d))
/(b^9*c^5 - a^5*b^4*d^5 + 5*a^4*b^5*c*d^4 + 10*a^2*b^7*c^3*d^2 - 10*a^3*b^6*c^2*d^3 - 5*a*b^8*c^4*d) + (log(c
+ d*x)*(3*b^2*c^6 + 15*a^2*c^4*d^2 - 12*a*b*c^5*d))/(a^5*d^9 - b^5*c^5*d^4 + 5*a*b^4*c^4*d^5 - 10*a^2*b^3*c^3*
d^6 + 10*a^3*b^2*c^2*d^7 - 5*a^4*b*c*d^8)