Integrand size = 18, antiderivative size = 222 \[ \int \frac {x^6}{(a+b x)^3 (c+d x)^3} \, dx=\frac {x}{b^3 d^3}-\frac {a^6}{2 b^4 (b c-a d)^3 (a+b x)^2}+\frac {3 a^5 (2 b c-a d)}{b^4 (b c-a d)^4 (a+b x)}+\frac {c^6}{2 d^4 (b c-a d)^3 (c+d x)^2}-\frac {3 c^5 (b c-2 a d)}{d^4 (b c-a d)^4 (c+d x)}+\frac {3 a^4 \left (5 b^2 c^2-4 a b c d+a^2 d^2\right ) \log (a+b x)}{b^4 (b c-a d)^5}-\frac {3 c^4 \left (b^2 c^2-4 a b c d+5 a^2 d^2\right ) \log (c+d x)}{d^4 (b c-a d)^5} \] Output:
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
Time = 0.19 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.00 \[ \int \frac {x^6}{(a+b x)^3 (c+d x)^3} \, dx=\frac {x}{b^3 d^3}-\frac {a^6}{2 b^4 (b c-a d)^3 (a+b x)^2}-\frac {3 a^5 (-2 b c+a d)}{b^4 (b c-a d)^4 (a+b x)}-\frac {c^6}{2 d^4 (-b c+a d)^3 (c+d x)^2}-\frac {3 c^5 (b c-2 a d)}{d^4 (b c-a d)^4 (c+d x)}+\frac {3 a^4 \left (5 b^2 c^2-4 a b c d+a^2 d^2\right ) \log (a+b x)}{b^4 (b c-a d)^5}+\frac {3 c^4 \left (b^2 c^2-4 a b c d+5 a^2 d^2\right ) \log (c+d x)}{d^4 (-b c+a d)^5} \] Input:
Integrate[x^6/((a + b*x)^3*(c + d*x)^3),x]
Output:
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)
Time = 0.48 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {99, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {x^6}{(a+b x)^3 (c+d x)^3} \, dx\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle \int \left (\frac {a^6}{b^3 (a+b x)^3 (b c-a d)^3}+\frac {3 a^5 (a d-2 b c)}{b^3 (a+b x)^2 (b c-a d)^4}+\frac {3 c^4 \left (5 a^2 d^2-4 a b c d+b^2 c^2\right )}{d^3 (c+d x) (a d-b c)^5}+\frac {3 a^4 \left (a^2 d^2-4 a b c d+5 b^2 c^2\right )}{b^3 (a+b x) (b c-a d)^5}+\frac {c^6}{d^3 (c+d x)^3 (a d-b c)^3}+\frac {3 c^5 (b c-2 a d)}{d^3 (c+d x)^2 (a d-b c)^4}+\frac {1}{b^3 d^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {a^6}{2 b^4 (a+b x)^2 (b c-a d)^3}+\frac {3 a^5 (2 b c-a d)}{b^4 (a+b x) (b c-a d)^4}-\frac {3 c^4 \left (5 a^2 d^2-4 a b c d+b^2 c^2\right ) \log (c+d x)}{d^4 (b c-a d)^5}+\frac {3 a^4 \left (a^2 d^2-4 a b c d+5 b^2 c^2\right ) \log (a+b x)}{b^4 (b c-a d)^5}+\frac {c^6}{2 d^4 (c+d x)^2 (b c-a d)^3}-\frac {3 c^5 (b c-2 a d)}{d^4 (c+d x) (b c-a d)^4}+\frac {x}{b^3 d^3}\) |
Input:
Int[x^6/((a + b*x)^3*(c + d*x)^3),x]
Output:
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)
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Int[ExpandIntegrand[(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]))
Time = 0.31 (sec) , antiderivative size = 219, normalized size of antiderivative = 0.99
| method | result | size |
| default | \(\frac {x}{b^{3} d^{3}}-\frac {c^{6}}{2 d^{4} \left (a d -b c \right )^{3} \left (x d +c \right )^{2}}+\frac {3 c^{4} \left (5 a^{2} d^{2}-4 a b c d +b^{2} c^{2}\right ) \ln \left (x d +c \right )}{d^{4} \left (a d -b c \right )^{5}}+\frac {3 c^{5} \left (2 a d -b c \right )}{d^{4} \left (a d -b c \right )^{4} \left (x d +c \right )}+\frac {a^{6}}{2 b^{4} \left (a d -b c \right )^{3} \left (b x +a \right )^{2}}-\frac {3 a^{4} \left (a^{2} d^{2}-4 a b c d +5 b^{2} c^{2}\right ) \ln \left (b x +a \right )}{b^{4} \left (a d -b c \right )^{5}}-\frac {3 a^{5} \left (a d -2 b c \right )}{b^{4} \left (a d -b c \right )^{4} \left (b x +a \right )}\) | \(219\) |
| norman | \(\frac {\frac {x^{5}}{b d}-\frac {\left (6 a^{6} d^{6}-14 a^{5} b c \,d^{5}+5 a^{4} b^{2} c^{2} d^{4}+5 a^{2} b^{4} c^{4} d^{2}-14 a \,b^{5} c^{5} d +6 c^{6} b^{6}\right ) x^{3}}{d^{3} b^{3} \left (d^{4} a^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +c^{4} b^{4}\right )}-\frac {\left (9 a^{7} d^{7}+a^{6} b c \,d^{6}-48 a^{5} b^{2} c^{2} d^{5}+20 a^{4} b^{3} c^{3} d^{4}+20 a^{3} b^{4} c^{4} d^{3}-48 a^{2} b^{5} c^{5} d^{2}+a \,b^{6} c^{6} d +9 b^{7} c^{7}\right ) x^{2}}{2 d^{4} b^{4} \left (d^{4} a^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +c^{4} b^{4}\right )}-\frac {c^{2} a^{2} \left (9 a^{5} d^{5}-23 a^{4} b c \,d^{4}+8 a^{3} b^{2} c^{2} d^{3}+8 a^{2} b^{3} c^{3} d^{2}-23 a \,b^{4} c^{4} d +9 c^{5} b^{5}\right )}{2 d^{4} b^{4} \left (d^{4} a^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +c^{4} b^{4}\right )}-\frac {c a \left (9 a^{6} d^{6}-17 a^{5} b c \,d^{5}-6 a^{4} b^{2} c^{2} d^{4}+10 a^{3} b^{3} c^{3} d^{3}-6 a^{2} b^{4} c^{4} d^{2}-17 a \,b^{5} c^{5} d +9 c^{6} b^{6}\right ) x}{d^{4} b^{4} \left (d^{4} a^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +c^{4} b^{4}\right )}}{\left (b x +a \right )^{2} \left (x d +c \right )^{2}}-\frac {3 a^{4} \left (a^{2} d^{2}-4 a b c d +5 b^{2} c^{2}\right ) \ln \left (b x +a \right )}{\left (a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -c^{5} b^{5}\right ) b^{4}}+\frac {3 c^{4} \left (5 a^{2} d^{2}-4 a b c d +b^{2} c^{2}\right ) \ln \left (x d +c \right )}{d^{4} \left (a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -c^{5} b^{5}\right )}\) | \(783\) |
| 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 +c^{6} b^{6}\right ) x^{3}}{d^{4} a^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +c^{4} b^{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 d b \left (d^{4} a^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +c^{4} b^{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 c^{6} b^{6}\right ) a c x}{b d \left (d^{4} a^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +c^{4} b^{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 c^{5} b^{5}\right )}{2 d b \left (d^{4} a^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +c^{4} b^{4}\right )}}{b^{3} d^{3} \left (b x +a \right )^{2} \left (x d +c \right )^{2}}-\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 -c^{5} b^{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 -c^{5} b^{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 -c^{5} b^{5}\right ) b^{2}}+\frac {15 c^{4} \ln \left (-x d -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 -c^{5} b^{5}\right )}-\frac {12 c^{5} \ln \left (-x d -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 -c^{5} b^{5}\right )}+\frac {3 c^{6} \ln \left (-x d -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 -c^{5} b^{5}\right )}\) | \(982\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(1334\) |
Input:
int(x^6/(b*x+a)^3/(d*x+c)^3,x,method=_RETURNVERBOSE)
Output:
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)
Leaf count of result is larger than twice the leaf count of optimal. 1457 vs. \(2 (218) = 436\).
Time = 0.18 (sec) , antiderivative size = 1457, normalized size of antiderivative = 6.56 \[ \int \frac {x^6}{(a+b x)^3 (c+d x)^3} \, dx=\text {Too large to display} \] Input:
integrate(x^6/(b*x+a)^3/(d*x+c)^3,x, algorithm="fricas")
Output:
-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 - 14*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*...
Timed out. \[ \int \frac {x^6}{(a+b x)^3 (c+d x)^3} \, dx=\text {Timed out} \] Input:
integrate(x**6/(b*x+a)**3/(d*x+c)**3,x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 818 vs. \(2 (218) = 436\).
Time = 0.07 (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}} \] Input:
integrate(x^6/(b*x+a)^3/(d*x+c)^3,x, algorithm="maxima")
Output:
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)
Leaf count of result is larger than twice the leaf count of optimal. 489 vs. \(2 (218) = 436\).
Time = 0.14 (sec) , antiderivative size = 489, normalized size of antiderivative = 2.20 \[ \int \frac {x^6}{(a+b x)^3 (c+d x)^3} \, dx=\frac {3 \, {\left (5 \, a^{4} b^{2} c^{2} - 4 \, a^{5} b c d + a^{6} d^{2}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{9} c^{5} - 5 \, a b^{8} c^{4} d + 10 \, a^{2} b^{7} c^{3} d^{2} - 10 \, a^{3} b^{6} c^{2} d^{3} + 5 \, a^{4} b^{5} c d^{4} - a^{5} b^{4} d^{5}} - \frac {3 \, {\left (b^{2} c^{6} - 4 \, a b c^{5} d + 5 \, a^{2} c^{4} d^{2}\right )} \log \left ({\left | d x + c \right |}\right )}{b^{5} c^{5} d^{4} - 5 \, a b^{4} c^{4} d^{5} + 10 \, a^{2} b^{3} c^{3} d^{6} - 10 \, a^{3} b^{2} c^{2} d^{7} + 5 \, a^{4} b c d^{8} - a^{5} d^{9}} + \frac {x}{b^{3} d^{3}} - \frac {5 \, a^{2} b^{5} c^{7} - 11 \, a^{3} b^{4} c^{6} d - 11 \, a^{6} b c^{3} d^{4} + 5 \, a^{7} c^{2} d^{5} + 6 \, {\left (b^{7} c^{6} d - 2 \, a b^{6} c^{5} d^{2} - 2 \, a^{5} b^{2} c d^{6} + a^{6} b d^{7}\right )} x^{3} + {\left (5 \, b^{7} c^{7} + a b^{6} c^{6} d - 24 \, a^{2} b^{5} c^{5} d^{2} - 24 \, a^{5} b^{2} c^{2} d^{5} + a^{6} b c d^{6} + 5 \, a^{7} d^{7}\right )} x^{2} + 2 \, {\left (5 \, a b^{6} c^{7} - 8 \, a^{2} b^{5} c^{6} d - 6 \, a^{3} b^{4} c^{5} d^{2} - 6 \, a^{5} b^{2} c^{3} d^{4} - 8 \, a^{6} b c^{2} d^{5} + 5 \, a^{7} c d^{6}\right )} x}{2 \, {\left (b c - a d\right )}^{4} {\left (b x + a\right )}^{2} {\left (d x + c\right )}^{2} b^{4} d^{4}} \] Input:
integrate(x^6/(b*x+a)^3/(d*x+c)^3,x, algorithm="giac")
Output:
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)
Time = 0.78 (sec) , antiderivative size = 780, normalized size of antiderivative = 3.51 \[ \int \frac {x^6}{(a+b x)^3 (c+d x)^3} \, dx=\frac {x}{b^3\,d^3}-\frac {\frac {3\,x^3\,\left (a^6\,d^6-2\,a^5\,b\,c\,d^5-2\,a\,b^5\,c^5\,d+b^6\,c^6\right )}{a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4}+\frac {x^2\,\left (5\,a^7\,d^7+a^6\,b\,c\,d^6-24\,a^5\,b^2\,c^2\,d^5-24\,a^2\,b^5\,c^5\,d^2+a\,b^6\,c^6\,d+5\,b^7\,c^7\right )}{2\,b\,d\,\left (a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}+\frac {a^2\,c^2\,\left (5\,a^5\,d^5-11\,a^4\,b\,c\,d^4-11\,a\,b^4\,c^4\,d+5\,b^5\,c^5\right )}{2\,b\,d\,\left (a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}-\frac {a\,c\,x\,\left (-5\,a^6\,d^6+8\,a^5\,b\,c\,d^5+6\,a^4\,b^2\,c^2\,d^4+6\,a^2\,b^4\,c^4\,d^2+8\,a\,b^5\,c^5\,d-5\,b^6\,c^6\right )}{b\,d\,\left (a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}}{x^3\,\left (2\,c\,b^5\,d^4+2\,a\,b^4\,d^5\right )+x\,\left (2\,a^2\,b^3\,c\,d^4+2\,a\,b^4\,c^2\,d^3\right )+x^2\,\left (a^2\,b^3\,d^5+4\,a\,b^4\,c\,d^4+b^5\,c^2\,d^3\right )+b^5\,d^5\,x^4+a^2\,b^3\,c^2\,d^3}+\frac {\ln \left (a+b\,x\right )\,\left (3\,a^6\,d^2-12\,a^5\,b\,c\,d+15\,a^4\,b^2\,c^2\right )}{-a^5\,b^4\,d^5+5\,a^4\,b^5\,c\,d^4-10\,a^3\,b^6\,c^2\,d^3+10\,a^2\,b^7\,c^3\,d^2-5\,a\,b^8\,c^4\,d+b^9\,c^5}+\frac {\ln \left (c+d\,x\right )\,\left (15\,a^2\,c^4\,d^2-12\,a\,b\,c^5\,d+3\,b^2\,c^6\right )}{a^5\,d^9-5\,a^4\,b\,c\,d^8+10\,a^3\,b^2\,c^2\,d^7-10\,a^2\,b^3\,c^3\,d^6+5\,a\,b^4\,c^4\,d^5-b^5\,c^5\,d^4} \] Input:
int(x^6/((a + b*x)^3*(c + d*x)^3),x)
Output:
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)
Time = 0.19 (sec) , antiderivative size = 2061, normalized size of antiderivative = 9.28 \[ \int \frac {x^6}{(a+b x)^3 (c+d x)^3} \, dx =\text {Too large to display} \] Input:
int(x^6/(b*x+a)^3/(d*x+c)^3,x)
Output:
( - 6*log(a + b*x)*a**9*c**2*d**7 - 12*log(a + b*x)*a**9*c*d**8*x - 6*log( a + b*x)*a**9*d**9*x**2 + 18*log(a + b*x)*a**8*b*c**3*d**6 + 24*log(a + b* x)*a**8*b*c**2*d**7*x - 6*log(a + b*x)*a**8*b*c*d**8*x**2 - 12*log(a + b*x )*a**8*b*d**9*x**3 - 6*log(a + b*x)*a**7*b**2*c**4*d**5 + 24*log(a + b*x)* a**7*b**2*c**3*d**6*x + 60*log(a + b*x)*a**7*b**2*c**2*d**7*x**2 + 24*log( a + b*x)*a**7*b**2*c*d**8*x**3 - 6*log(a + b*x)*a**7*b**2*d**9*x**4 - 30*l og(a + b*x)*a**6*b**3*c**5*d**4 - 72*log(a + b*x)*a**6*b**3*c**4*d**5*x - 36*log(a + b*x)*a**6*b**3*c**3*d**6*x**2 + 24*log(a + b*x)*a**6*b**3*c**2* d**7*x**3 + 18*log(a + b*x)*a**6*b**3*c*d**8*x**4 - 60*log(a + b*x)*a**5*b **4*c**5*d**4*x - 126*log(a + b*x)*a**5*b**4*c**4*d**5*x**2 - 72*log(a + b *x)*a**5*b**4*c**3*d**6*x**3 - 6*log(a + b*x)*a**5*b**4*c**2*d**7*x**4 - 3 0*log(a + b*x)*a**4*b**5*c**5*d**4*x**2 - 60*log(a + b*x)*a**4*b**5*c**4*d **5*x**3 - 30*log(a + b*x)*a**4*b**5*c**3*d**6*x**4 + 30*log(c + d*x)*a**5 *b**4*c**6*d**3 + 60*log(c + d*x)*a**5*b**4*c**5*d**4*x + 30*log(c + d*x)* a**5*b**4*c**4*d**5*x**2 + 6*log(c + d*x)*a**4*b**5*c**7*d**2 + 72*log(c + d*x)*a**4*b**5*c**6*d**3*x + 126*log(c + d*x)*a**4*b**5*c**5*d**4*x**2 + 60*log(c + d*x)*a**4*b**5*c**4*d**5*x**3 - 18*log(c + d*x)*a**3*b**6*c**8* d - 24*log(c + d*x)*a**3*b**6*c**7*d**2*x + 36*log(c + d*x)*a**3*b**6*c**6 *d**3*x**2 + 72*log(c + d*x)*a**3*b**6*c**5*d**4*x**3 + 30*log(c + d*x)*a* *3*b**6*c**4*d**5*x**4 + 6*log(c + d*x)*a**2*b**7*c**9 - 24*log(c + d*x...