Integrand size = 19, antiderivative size = 136 \[ \int \frac {(c+d x)^{5/4}}{(a+b x)^{25/4}} \, dx=-\frac {4 (c+d x)^{9/4}}{21 (b c-a d) (a+b x)^{21/4}}+\frac {16 d (c+d x)^{9/4}}{119 (b c-a d)^2 (a+b x)^{17/4}}-\frac {128 d^2 (c+d x)^{9/4}}{1547 (b c-a d)^3 (a+b x)^{13/4}}+\frac {512 d^3 (c+d x)^{9/4}}{13923 (b c-a d)^4 (a+b x)^{9/4}} \] Output:
-4/21*(d*x+c)^(9/4)/(-a*d+b*c)/(b*x+a)^(21/4)+16/119*d*(d*x+c)^(9/4)/(-a*d +b*c)^2/(b*x+a)^(17/4)-128/1547*d^2*(d*x+c)^(9/4)/(-a*d+b*c)^3/(b*x+a)^(13 /4)+512/13923*d^3*(d*x+c)^(9/4)/(-a*d+b*c)^4/(b*x+a)^(9/4)
Time = 1.69 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.87 \[ \int \frac {(c+d x)^{5/4}}{(a+b x)^{25/4}} \, dx=\frac {4 (c+d x)^{9/4} \left (1547 a^3 d^3+357 a^2 b d^2 (-9 c+4 d x)+21 a b^2 d \left (117 c^2-72 c d x+32 d^2 x^2\right )+b^3 \left (-663 c^3+468 c^2 d x-288 c d^2 x^2+128 d^3 x^3\right )\right )}{13923 (b c-a d)^4 (a+b x)^{21/4}} \] Input:
Integrate[(c + d*x)^(5/4)/(a + b*x)^(25/4),x]
Output:
(4*(c + d*x)^(9/4)*(1547*a^3*d^3 + 357*a^2*b*d^2*(-9*c + 4*d*x) + 21*a*b^2 *d*(117*c^2 - 72*c*d*x + 32*d^2*x^2) + b^3*(-663*c^3 + 468*c^2*d*x - 288*c *d^2*x^2 + 128*d^3*x^3)))/(13923*(b*c - a*d)^4*(a + b*x)^(21/4))
Time = 0.20 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.19, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {55, 55, 55, 48}
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.
| \(\displaystyle \int \frac {(c+d x)^{5/4}}{(a+b x)^{25/4}} \, dx\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle -\frac {4 d \int \frac {(c+d x)^{5/4}}{(a+b x)^{21/4}}dx}{7 (b c-a d)}-\frac {4 (c+d x)^{9/4}}{21 (a+b x)^{21/4} (b c-a d)}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle -\frac {4 d \left (-\frac {8 d \int \frac {(c+d x)^{5/4}}{(a+b x)^{17/4}}dx}{17 (b c-a d)}-\frac {4 (c+d x)^{9/4}}{17 (a+b x)^{17/4} (b c-a d)}\right )}{7 (b c-a d)}-\frac {4 (c+d x)^{9/4}}{21 (a+b x)^{21/4} (b c-a d)}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle -\frac {4 d \left (-\frac {8 d \left (-\frac {4 d \int \frac {(c+d x)^{5/4}}{(a+b x)^{13/4}}dx}{13 (b c-a d)}-\frac {4 (c+d x)^{9/4}}{13 (a+b x)^{13/4} (b c-a d)}\right )}{17 (b c-a d)}-\frac {4 (c+d x)^{9/4}}{17 (a+b x)^{17/4} (b c-a d)}\right )}{7 (b c-a d)}-\frac {4 (c+d x)^{9/4}}{21 (a+b x)^{21/4} (b c-a d)}\) |
| \(\Big \downarrow \) 48 |
| \(\displaystyle -\frac {4 (c+d x)^{9/4}}{21 (a+b x)^{21/4} (b c-a d)}-\frac {4 d \left (-\frac {4 (c+d x)^{9/4}}{17 (a+b x)^{17/4} (b c-a d)}-\frac {8 d \left (\frac {16 d (c+d x)^{9/4}}{117 (a+b x)^{9/4} (b c-a d)^2}-\frac {4 (c+d x)^{9/4}}{13 (a+b x)^{13/4} (b c-a d)}\right )}{17 (b c-a d)}\right )}{7 (b c-a d)}\) |
Input:
Int[(c + d*x)^(5/4)/(a + b*x)^(25/4),x]
Output:
(-4*(c + d*x)^(9/4))/(21*(b*c - a*d)*(a + b*x)^(21/4)) - (4*d*((-4*(c + d* x)^(9/4))/(17*(b*c - a*d)*(a + b*x)^(17/4)) - (8*d*((-4*(c + d*x)^(9/4))/( 13*(b*c - a*d)*(a + b*x)^(13/4)) + (16*d*(c + d*x)^(9/4))/(117*(b*c - a*d) ^2*(a + b*x)^(9/4))))/(17*(b*c - a*d))))/(7*(b*c - a*d))
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp [(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{ a, b, c, d, m, n}, x] && EqQ[m + n + 2, 0] && NeQ[m, -1]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(S implify[m + n + 2]/((b*c - a*d)*(m + 1))) Int[(a + b*x)^Simplify[m + 1]*( c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && ILtQ[Simplify[m + n + 2], 0] && NeQ[m, -1] && !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[ c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (SumSimplerQ[m, 1] || !SumSimp lerQ[n, 1])
Time = 0.18 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.26
| method | result | size |
| gosper | \(\frac {4 \left (x d +c \right )^{\frac {9}{4}} \left (128 d^{3} x^{3} b^{3}+672 x^{2} a \,b^{2} d^{3}-288 x^{2} b^{3} c \,d^{2}+1428 x \,a^{2} b \,d^{3}-1512 x a \,b^{2} c \,d^{2}+468 x \,b^{3} c^{2} d +1547 a^{3} d^{3}-3213 a^{2} b c \,d^{2}+2457 a \,b^{2} c^{2} d -663 b^{3} c^{3}\right )}{13923 \left (b x +a \right )^{\frac {21}{4}} \left (d^{4} a^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +c^{4} b^{4}\right )}\) | \(171\) |
| orering | \(\frac {4 \left (x d +c \right )^{\frac {9}{4}} \left (128 d^{3} x^{3} b^{3}+672 x^{2} a \,b^{2} d^{3}-288 x^{2} b^{3} c \,d^{2}+1428 x \,a^{2} b \,d^{3}-1512 x a \,b^{2} c \,d^{2}+468 x \,b^{3} c^{2} d +1547 a^{3} d^{3}-3213 a^{2} b c \,d^{2}+2457 a \,b^{2} c^{2} d -663 b^{3} c^{3}\right )}{13923 \left (b x +a \right )^{\frac {21}{4}} \left (d^{4} a^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +c^{4} b^{4}\right )}\) | \(171\) |
Input:
int((d*x+c)^(5/4)/(b*x+a)^(25/4),x,method=_RETURNVERBOSE)
Output:
4/13923*(d*x+c)^(9/4)*(128*b^3*d^3*x^3+672*a*b^2*d^3*x^2-288*b^3*c*d^2*x^2 +1428*a^2*b*d^3*x-1512*a*b^2*c*d^2*x+468*b^3*c^2*d*x+1547*a^3*d^3-3213*a^2 *b*c*d^2+2457*a*b^2*c^2*d-663*b^3*c^3)/(b*x+a)^(21/4)/(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)
Leaf count of result is larger than twice the leaf count of optimal. 649 vs. \(2 (112) = 224\).
Time = 0.23 (sec) , antiderivative size = 649, normalized size of antiderivative = 4.77 \[ \int \frac {(c+d x)^{5/4}}{(a+b x)^{25/4}} \, dx=\frac {4 \, {\left (128 \, b^{3} d^{5} x^{5} - 663 \, b^{3} c^{5} + 2457 \, a b^{2} c^{4} d - 3213 \, a^{2} b c^{3} d^{2} + 1547 \, a^{3} c^{2} d^{3} - 32 \, {\left (b^{3} c d^{4} - 21 \, a b^{2} d^{5}\right )} x^{4} + 4 \, {\left (5 \, b^{3} c^{2} d^{3} - 42 \, a b^{2} c d^{4} + 357 \, a^{2} b d^{5}\right )} x^{3} - {\left (15 \, b^{3} c^{3} d^{2} - 105 \, a b^{2} c^{2} d^{3} + 357 \, a^{2} b c d^{4} - 1547 \, a^{3} d^{5}\right )} x^{2} - 2 \, {\left (429 \, b^{3} c^{4} d - 1701 \, a b^{2} c^{3} d^{2} + 2499 \, a^{2} b c^{2} d^{3} - 1547 \, a^{3} c d^{4}\right )} x\right )} {\left (b x + a\right )}^{\frac {3}{4}} {\left (d x + c\right )}^{\frac {1}{4}}}{13923 \, {\left (a^{6} b^{4} c^{4} - 4 \, a^{7} b^{3} c^{3} d + 6 \, a^{8} b^{2} c^{2} d^{2} - 4 \, a^{9} b c d^{3} + a^{10} d^{4} + {\left (b^{10} c^{4} - 4 \, a b^{9} c^{3} d + 6 \, a^{2} b^{8} c^{2} d^{2} - 4 \, a^{3} b^{7} c d^{3} + a^{4} b^{6} d^{4}\right )} x^{6} + 6 \, {\left (a b^{9} c^{4} - 4 \, a^{2} b^{8} c^{3} d + 6 \, a^{3} b^{7} c^{2} d^{2} - 4 \, a^{4} b^{6} c d^{3} + a^{5} b^{5} d^{4}\right )} x^{5} + 15 \, {\left (a^{2} b^{8} c^{4} - 4 \, a^{3} b^{7} c^{3} d + 6 \, a^{4} b^{6} c^{2} d^{2} - 4 \, a^{5} b^{5} c d^{3} + a^{6} b^{4} d^{4}\right )} x^{4} + 20 \, {\left (a^{3} b^{7} c^{4} - 4 \, a^{4} b^{6} c^{3} d + 6 \, a^{5} b^{5} c^{2} d^{2} - 4 \, a^{6} b^{4} c d^{3} + a^{7} b^{3} d^{4}\right )} x^{3} + 15 \, {\left (a^{4} b^{6} c^{4} - 4 \, a^{5} b^{5} c^{3} d + 6 \, a^{6} b^{4} c^{2} d^{2} - 4 \, a^{7} b^{3} c d^{3} + a^{8} b^{2} d^{4}\right )} x^{2} + 6 \, {\left (a^{5} b^{5} c^{4} - 4 \, a^{6} b^{4} c^{3} d + 6 \, a^{7} b^{3} c^{2} d^{2} - 4 \, a^{8} b^{2} c d^{3} + a^{9} b d^{4}\right )} x\right )}} \] Input:
integrate((d*x+c)^(5/4)/(b*x+a)^(25/4),x, algorithm="fricas")
Output:
4/13923*(128*b^3*d^5*x^5 - 663*b^3*c^5 + 2457*a*b^2*c^4*d - 3213*a^2*b*c^3 *d^2 + 1547*a^3*c^2*d^3 - 32*(b^3*c*d^4 - 21*a*b^2*d^5)*x^4 + 4*(5*b^3*c^2 *d^3 - 42*a*b^2*c*d^4 + 357*a^2*b*d^5)*x^3 - (15*b^3*c^3*d^2 - 105*a*b^2*c ^2*d^3 + 357*a^2*b*c*d^4 - 1547*a^3*d^5)*x^2 - 2*(429*b^3*c^4*d - 1701*a*b ^2*c^3*d^2 + 2499*a^2*b*c^2*d^3 - 1547*a^3*c*d^4)*x)*(b*x + a)^(3/4)*(d*x + c)^(1/4)/(a^6*b^4*c^4 - 4*a^7*b^3*c^3*d + 6*a^8*b^2*c^2*d^2 - 4*a^9*b*c* d^3 + a^10*d^4 + (b^10*c^4 - 4*a*b^9*c^3*d + 6*a^2*b^8*c^2*d^2 - 4*a^3*b^7 *c*d^3 + a^4*b^6*d^4)*x^6 + 6*(a*b^9*c^4 - 4*a^2*b^8*c^3*d + 6*a^3*b^7*c^2 *d^2 - 4*a^4*b^6*c*d^3 + a^5*b^5*d^4)*x^5 + 15*(a^2*b^8*c^4 - 4*a^3*b^7*c^ 3*d + 6*a^4*b^6*c^2*d^2 - 4*a^5*b^5*c*d^3 + a^6*b^4*d^4)*x^4 + 20*(a^3*b^7 *c^4 - 4*a^4*b^6*c^3*d + 6*a^5*b^5*c^2*d^2 - 4*a^6*b^4*c*d^3 + a^7*b^3*d^4 )*x^3 + 15*(a^4*b^6*c^4 - 4*a^5*b^5*c^3*d + 6*a^6*b^4*c^2*d^2 - 4*a^7*b^3* c*d^3 + a^8*b^2*d^4)*x^2 + 6*(a^5*b^5*c^4 - 4*a^6*b^4*c^3*d + 6*a^7*b^3*c^ 2*d^2 - 4*a^8*b^2*c*d^3 + a^9*b*d^4)*x)
Timed out. \[ \int \frac {(c+d x)^{5/4}}{(a+b x)^{25/4}} \, dx=\text {Timed out} \] Input:
integrate((d*x+c)**(5/4)/(b*x+a)**(25/4),x)
Output:
Timed out
\[ \int \frac {(c+d x)^{5/4}}{(a+b x)^{25/4}} \, dx=\int { \frac {{\left (d x + c\right )}^{\frac {5}{4}}}{{\left (b x + a\right )}^{\frac {25}{4}}} \,d x } \] Input:
integrate((d*x+c)^(5/4)/(b*x+a)^(25/4),x, algorithm="maxima")
Output:
integrate((d*x + c)^(5/4)/(b*x + a)^(25/4), x)
\[ \int \frac {(c+d x)^{5/4}}{(a+b x)^{25/4}} \, dx=\int { \frac {{\left (d x + c\right )}^{\frac {5}{4}}}{{\left (b x + a\right )}^{\frac {25}{4}}} \,d x } \] Input:
integrate((d*x+c)^(5/4)/(b*x+a)^(25/4),x, algorithm="giac")
Output:
integrate((d*x + c)^(5/4)/(b*x + a)^(25/4), x)
Time = 0.77 (sec) , antiderivative size = 376, normalized size of antiderivative = 2.76 \[ \int \frac {(c+d x)^{5/4}}{(a+b x)^{25/4}} \, dx=\frac {{\left (c+d\,x\right )}^{1/4}\,\left (\frac {x^2\,\left (6188\,a^3\,d^5-1428\,a^2\,b\,c\,d^4+420\,a\,b^2\,c^2\,d^3-60\,b^3\,c^3\,d^2\right )}{13923\,b^5\,{\left (a\,d-b\,c\right )}^4}-\frac {-6188\,a^3\,c^2\,d^3+12852\,a^2\,b\,c^3\,d^2-9828\,a\,b^2\,c^4\,d+2652\,b^3\,c^5}{13923\,b^5\,{\left (a\,d-b\,c\right )}^4}+\frac {x\,\left (12376\,a^3\,c\,d^4-19992\,a^2\,b\,c^2\,d^3+13608\,a\,b^2\,c^3\,d^2-3432\,b^3\,c^4\,d\right )}{13923\,b^5\,{\left (a\,d-b\,c\right )}^4}+\frac {512\,d^5\,x^5}{13923\,b^2\,{\left (a\,d-b\,c\right )}^4}+\frac {128\,d^4\,x^4\,\left (21\,a\,d-b\,c\right )}{13923\,b^3\,{\left (a\,d-b\,c\right )}^4}+\frac {16\,d^3\,x^3\,\left (357\,a^2\,d^2-42\,a\,b\,c\,d+5\,b^2\,c^2\right )}{13923\,b^4\,{\left (a\,d-b\,c\right )}^4}\right )}{x^5\,{\left (a+b\,x\right )}^{1/4}+\frac {a^5\,{\left (a+b\,x\right )}^{1/4}}{b^5}+\frac {10\,a^2\,x^3\,{\left (a+b\,x\right )}^{1/4}}{b^2}+\frac {10\,a^3\,x^2\,{\left (a+b\,x\right )}^{1/4}}{b^3}+\frac {5\,a\,x^4\,{\left (a+b\,x\right )}^{1/4}}{b}+\frac {5\,a^4\,x\,{\left (a+b\,x\right )}^{1/4}}{b^4}} \] Input:
int((c + d*x)^(5/4)/(a + b*x)^(25/4),x)
Output:
((c + d*x)^(1/4)*((x^2*(6188*a^3*d^5 - 60*b^3*c^3*d^2 + 420*a*b^2*c^2*d^3 - 1428*a^2*b*c*d^4))/(13923*b^5*(a*d - b*c)^4) - (2652*b^3*c^5 - 6188*a^3* c^2*d^3 + 12852*a^2*b*c^3*d^2 - 9828*a*b^2*c^4*d)/(13923*b^5*(a*d - b*c)^4 ) + (x*(12376*a^3*c*d^4 - 3432*b^3*c^4*d + 13608*a*b^2*c^3*d^2 - 19992*a^2 *b*c^2*d^3))/(13923*b^5*(a*d - b*c)^4) + (512*d^5*x^5)/(13923*b^2*(a*d - b *c)^4) + (128*d^4*x^4*(21*a*d - b*c))/(13923*b^3*(a*d - b*c)^4) + (16*d^3* x^3*(357*a^2*d^2 + 5*b^2*c^2 - 42*a*b*c*d))/(13923*b^4*(a*d - b*c)^4)))/(x ^5*(a + b*x)^(1/4) + (a^5*(a + b*x)^(1/4))/b^5 + (10*a^2*x^3*(a + b*x)^(1/ 4))/b^2 + (10*a^3*x^2*(a + b*x)^(1/4))/b^3 + (5*a*x^4*(a + b*x)^(1/4))/b + (5*a^4*x*(a + b*x)^(1/4))/b^4)
Time = 0.31 (sec) , antiderivative size = 641, normalized size of antiderivative = 4.71 \[ \int \frac {(c+d x)^{5/4}}{(a+b x)^{25/4}} \, dx=\frac {4 \left (d x +c \right )^{\frac {1}{4}} \left (128 b^{3} d^{5} x^{5}+672 a \,b^{2} d^{5} x^{4}-32 b^{3} c \,d^{4} x^{4}+1428 a^{2} b \,d^{5} x^{3}-168 a \,b^{2} c \,d^{4} x^{3}+20 b^{3} c^{2} d^{3} x^{3}+1547 a^{3} d^{5} x^{2}-357 a^{2} b c \,d^{4} x^{2}+105 a \,b^{2} c^{2} d^{3} x^{2}-15 b^{3} c^{3} d^{2} x^{2}+3094 a^{3} c \,d^{4} x -4998 a^{2} b \,c^{2} d^{3} x +3402 a \,b^{2} c^{3} d^{2} x -858 b^{3} c^{4} d x +1547 a^{3} c^{2} d^{3}-3213 a^{2} b \,c^{3} d^{2}+2457 a \,b^{2} c^{4} d -663 b^{3} c^{5}\right )}{13923 \left (b x +a \right )^{\frac {1}{4}} \left (a^{4} b^{5} d^{4} x^{5}-4 a^{3} b^{6} c \,d^{3} x^{5}+6 a^{2} b^{7} c^{2} d^{2} x^{5}-4 a \,b^{8} c^{3} d \,x^{5}+b^{9} c^{4} x^{5}+5 a^{5} b^{4} d^{4} x^{4}-20 a^{4} b^{5} c \,d^{3} x^{4}+30 a^{3} b^{6} c^{2} d^{2} x^{4}-20 a^{2} b^{7} c^{3} d \,x^{4}+5 a \,b^{8} c^{4} x^{4}+10 a^{6} b^{3} d^{4} x^{3}-40 a^{5} b^{4} c \,d^{3} x^{3}+60 a^{4} b^{5} c^{2} d^{2} x^{3}-40 a^{3} b^{6} c^{3} d \,x^{3}+10 a^{2} b^{7} c^{4} x^{3}+10 a^{7} b^{2} d^{4} x^{2}-40 a^{6} b^{3} c \,d^{3} x^{2}+60 a^{5} b^{4} c^{2} d^{2} x^{2}-40 a^{4} b^{5} c^{3} d \,x^{2}+10 a^{3} b^{6} c^{4} x^{2}+5 a^{8} b \,d^{4} x -20 a^{7} b^{2} c \,d^{3} x +30 a^{6} b^{3} c^{2} d^{2} x -20 a^{5} b^{4} c^{3} d x +5 a^{4} b^{5} c^{4} x +a^{9} d^{4}-4 a^{8} b c \,d^{3}+6 a^{7} b^{2} c^{2} d^{2}-4 a^{6} b^{3} c^{3} d +a^{5} b^{4} c^{4}\right )} \] Input:
int((d*x+c)^(5/4)/(b*x+a)^(25/4),x)
Output:
(4*(c + d*x)**(1/4)*(1547*a**3*c**2*d**3 + 3094*a**3*c*d**4*x + 1547*a**3* d**5*x**2 - 3213*a**2*b*c**3*d**2 - 4998*a**2*b*c**2*d**3*x - 357*a**2*b*c *d**4*x**2 + 1428*a**2*b*d**5*x**3 + 2457*a*b**2*c**4*d + 3402*a*b**2*c**3 *d**2*x + 105*a*b**2*c**2*d**3*x**2 - 168*a*b**2*c*d**4*x**3 + 672*a*b**2* d**5*x**4 - 663*b**3*c**5 - 858*b**3*c**4*d*x - 15*b**3*c**3*d**2*x**2 + 2 0*b**3*c**2*d**3*x**3 - 32*b**3*c*d**4*x**4 + 128*b**3*d**5*x**5))/(13923* (a + b*x)**(1/4)*(a**9*d**4 - 4*a**8*b*c*d**3 + 5*a**8*b*d**4*x + 6*a**7*b **2*c**2*d**2 - 20*a**7*b**2*c*d**3*x + 10*a**7*b**2*d**4*x**2 - 4*a**6*b* *3*c**3*d + 30*a**6*b**3*c**2*d**2*x - 40*a**6*b**3*c*d**3*x**2 + 10*a**6* b**3*d**4*x**3 + a**5*b**4*c**4 - 20*a**5*b**4*c**3*d*x + 60*a**5*b**4*c** 2*d**2*x**2 - 40*a**5*b**4*c*d**3*x**3 + 5*a**5*b**4*d**4*x**4 + 5*a**4*b* *5*c**4*x - 40*a**4*b**5*c**3*d*x**2 + 60*a**4*b**5*c**2*d**2*x**3 - 20*a* *4*b**5*c*d**3*x**4 + a**4*b**5*d**4*x**5 + 10*a**3*b**6*c**4*x**2 - 40*a* *3*b**6*c**3*d*x**3 + 30*a**3*b**6*c**2*d**2*x**4 - 4*a**3*b**6*c*d**3*x** 5 + 10*a**2*b**7*c**4*x**3 - 20*a**2*b**7*c**3*d*x**4 + 6*a**2*b**7*c**2*d **2*x**5 + 5*a*b**8*c**4*x**4 - 4*a*b**8*c**3*d*x**5 + b**9*c**4*x**5))