\(\int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{18}} \, dx\) [96]

Optimal result

Integrand size = 20, antiderivative size = 335 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{18}} \, dx=-\frac {(B d-A e) (a+b x)^{11}}{17 e (b d-a e) (d+e x)^{17}}+\frac {(11 b B d+6 A b e-17 a B e) (a+b x)^{11}}{272 e (b d-a e)^2 (d+e x)^{16}}+\frac {b (11 b B d+6 A b e-17 a B e) (a+b x)^{11}}{816 e (b d-a e)^3 (d+e x)^{15}}+\frac {b^2 (11 b B d+6 A b e-17 a B e) (a+b x)^{11}}{2856 e (b d-a e)^4 (d+e x)^{14}}+\frac {b^3 (11 b B d+6 A b e-17 a B e) (a+b x)^{11}}{12376 e (b d-a e)^5 (d+e x)^{13}}+\frac {b^4 (11 b B d+6 A b e-17 a B e) (a+b x)^{11}}{74256 e (b d-a e)^6 (d+e x)^{12}}+\frac {b^5 (11 b B d+6 A b e-17 a B e) (a+b x)^{11}}{816816 e (b d-a e)^7 (d+e x)^{11}} \] Output:

-1/17*(-A*e+B*d)*(b*x+a)^11/e/(-a*e+b*d)/(e*x+d)^17+1/272*(6*A*b*e-17*B*a* 
e+11*B*b*d)*(b*x+a)^11/e/(-a*e+b*d)^2/(e*x+d)^16+1/816*b*(6*A*b*e-17*B*a*e 
+11*B*b*d)*(b*x+a)^11/e/(-a*e+b*d)^3/(e*x+d)^15+1/2856*b^2*(6*A*b*e-17*B*a 
*e+11*B*b*d)*(b*x+a)^11/e/(-a*e+b*d)^4/(e*x+d)^14+1/12376*b^3*(6*A*b*e-17* 
B*a*e+11*B*b*d)*(b*x+a)^11/e/(-a*e+b*d)^5/(e*x+d)^13+1/74256*b^4*(6*A*b*e- 
17*B*a*e+11*B*b*d)*(b*x+a)^11/e/(-a*e+b*d)^6/(e*x+d)^12+1/816816*b^5*(6*A* 
b*e-17*B*a*e+11*B*b*d)*(b*x+a)^11/e/(-a*e+b*d)^7/(e*x+d)^11
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1433\) vs. \(2(335)=670\).

Time = 0.52 (sec) , antiderivative size = 1433, normalized size of antiderivative = 4.28 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{18}} \, dx =\text {Too large to display} \] Input:

Integrate[((a + b*x)^10*(A + B*x))/(d + e*x)^18,x]
 

Output:

-1/816816*(3003*a^10*e^10*(16*A*e + B*(d + 17*e*x)) + 2002*a^9*b*e^9*(15*A 
*e*(d + 17*e*x) + 2*B*(d^2 + 17*d*e*x + 136*e^2*x^2)) + 1287*a^8*b^2*e^8*( 
14*A*e*(d^2 + 17*d*e*x + 136*e^2*x^2) + 3*B*(d^3 + 17*d^2*e*x + 136*d*e^2* 
x^2 + 680*e^3*x^3)) + 792*a^7*b^3*e^7*(13*A*e*(d^3 + 17*d^2*e*x + 136*d*e^ 
2*x^2 + 680*e^3*x^3) + 4*B*(d^4 + 17*d^3*e*x + 136*d^2*e^2*x^2 + 680*d*e^3 
*x^3 + 2380*e^4*x^4)) + 462*a^6*b^4*e^6*(12*A*e*(d^4 + 17*d^3*e*x + 136*d^ 
2*e^2*x^2 + 680*d*e^3*x^3 + 2380*e^4*x^4) + 5*B*(d^5 + 17*d^4*e*x + 136*d^ 
3*e^2*x^2 + 680*d^2*e^3*x^3 + 2380*d*e^4*x^4 + 6188*e^5*x^5)) + 252*a^5*b^ 
5*e^5*(11*A*e*(d^5 + 17*d^4*e*x + 136*d^3*e^2*x^2 + 680*d^2*e^3*x^3 + 2380 
*d*e^4*x^4 + 6188*e^5*x^5) + 6*B*(d^6 + 17*d^5*e*x + 136*d^4*e^2*x^2 + 680 
*d^3*e^3*x^3 + 2380*d^2*e^4*x^4 + 6188*d*e^5*x^5 + 12376*e^6*x^6)) + 126*a 
^4*b^6*e^4*(10*A*e*(d^6 + 17*d^5*e*x + 136*d^4*e^2*x^2 + 680*d^3*e^3*x^3 + 
 2380*d^2*e^4*x^4 + 6188*d*e^5*x^5 + 12376*e^6*x^6) + 7*B*(d^7 + 17*d^6*e* 
x + 136*d^5*e^2*x^2 + 680*d^4*e^3*x^3 + 2380*d^3*e^4*x^4 + 6188*d^2*e^5*x^ 
5 + 12376*d*e^6*x^6 + 19448*e^7*x^7)) + 56*a^3*b^7*e^3*(9*A*e*(d^7 + 17*d^ 
6*e*x + 136*d^5*e^2*x^2 + 680*d^4*e^3*x^3 + 2380*d^3*e^4*x^4 + 6188*d^2*e^ 
5*x^5 + 12376*d*e^6*x^6 + 19448*e^7*x^7) + 8*B*(d^8 + 17*d^7*e*x + 136*d^6 
*e^2*x^2 + 680*d^5*e^3*x^3 + 2380*d^4*e^4*x^4 + 6188*d^3*e^5*x^5 + 12376*d 
^2*e^6*x^6 + 19448*d*e^7*x^7 + 24310*e^8*x^8)) + 21*a^2*b^8*e^2*(8*A*e*(d^ 
8 + 17*d^7*e*x + 136*d^6*e^2*x^2 + 680*d^5*e^3*x^3 + 2380*d^4*e^4*x^4 +...
 

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 307, normalized size of antiderivative = 0.92, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {87, 55, 55, 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.

\(\displaystyle \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{18}} \, dx\)

\(\Big \downarrow \) 87

\(\displaystyle \frac {(-17 a B e+6 A b e+11 b B d) \int \frac {(a+b x)^{10}}{(d+e x)^{17}}dx}{17 e (b d-a e)}-\frac {(a+b x)^{11} (B d-A e)}{17 e (d+e x)^{17} (b d-a e)}\)

\(\Big \downarrow \) 55

\(\displaystyle \frac {(-17 a B e+6 A b e+11 b B d) \left (\frac {5 b \int \frac {(a+b x)^{10}}{(d+e x)^{16}}dx}{16 (b d-a e)}+\frac {(a+b x)^{11}}{16 (d+e x)^{16} (b d-a e)}\right )}{17 e (b d-a e)}-\frac {(a+b x)^{11} (B d-A e)}{17 e (d+e x)^{17} (b d-a e)}\)

\(\Big \downarrow \) 55

\(\displaystyle \frac {(-17 a B e+6 A b e+11 b B d) \left (\frac {5 b \left (\frac {4 b \int \frac {(a+b x)^{10}}{(d+e x)^{15}}dx}{15 (b d-a e)}+\frac {(a+b x)^{11}}{15 (d+e x)^{15} (b d-a e)}\right )}{16 (b d-a e)}+\frac {(a+b x)^{11}}{16 (d+e x)^{16} (b d-a e)}\right )}{17 e (b d-a e)}-\frac {(a+b x)^{11} (B d-A e)}{17 e (d+e x)^{17} (b d-a e)}\)

\(\Big \downarrow \) 55

\(\displaystyle \frac {(-17 a B e+6 A b e+11 b B d) \left (\frac {5 b \left (\frac {4 b \left (\frac {3 b \int \frac {(a+b x)^{10}}{(d+e x)^{14}}dx}{14 (b d-a e)}+\frac {(a+b x)^{11}}{14 (d+e x)^{14} (b d-a e)}\right )}{15 (b d-a e)}+\frac {(a+b x)^{11}}{15 (d+e x)^{15} (b d-a e)}\right )}{16 (b d-a e)}+\frac {(a+b x)^{11}}{16 (d+e x)^{16} (b d-a e)}\right )}{17 e (b d-a e)}-\frac {(a+b x)^{11} (B d-A e)}{17 e (d+e x)^{17} (b d-a e)}\)

\(\Big \downarrow \) 55

\(\displaystyle \frac {(-17 a B e+6 A b e+11 b B d) \left (\frac {5 b \left (\frac {4 b \left (\frac {3 b \left (\frac {2 b \int \frac {(a+b x)^{10}}{(d+e x)^{13}}dx}{13 (b d-a e)}+\frac {(a+b x)^{11}}{13 (d+e x)^{13} (b d-a e)}\right )}{14 (b d-a e)}+\frac {(a+b x)^{11}}{14 (d+e x)^{14} (b d-a e)}\right )}{15 (b d-a e)}+\frac {(a+b x)^{11}}{15 (d+e x)^{15} (b d-a e)}\right )}{16 (b d-a e)}+\frac {(a+b x)^{11}}{16 (d+e x)^{16} (b d-a e)}\right )}{17 e (b d-a e)}-\frac {(a+b x)^{11} (B d-A e)}{17 e (d+e x)^{17} (b d-a e)}\)

\(\Big \downarrow \) 55

\(\displaystyle \frac {(-17 a B e+6 A b e+11 b B d) \left (\frac {5 b \left (\frac {4 b \left (\frac {3 b \left (\frac {2 b \left (\frac {b \int \frac {(a+b x)^{10}}{(d+e x)^{12}}dx}{12 (b d-a e)}+\frac {(a+b x)^{11}}{12 (d+e x)^{12} (b d-a e)}\right )}{13 (b d-a e)}+\frac {(a+b x)^{11}}{13 (d+e x)^{13} (b d-a e)}\right )}{14 (b d-a e)}+\frac {(a+b x)^{11}}{14 (d+e x)^{14} (b d-a e)}\right )}{15 (b d-a e)}+\frac {(a+b x)^{11}}{15 (d+e x)^{15} (b d-a e)}\right )}{16 (b d-a e)}+\frac {(a+b x)^{11}}{16 (d+e x)^{16} (b d-a e)}\right )}{17 e (b d-a e)}-\frac {(a+b x)^{11} (B d-A e)}{17 e (d+e x)^{17} (b d-a e)}\)

\(\Big \downarrow \) 48

\(\displaystyle \frac {\left (\frac {(a+b x)^{11}}{16 (d+e x)^{16} (b d-a e)}+\frac {5 b \left (\frac {(a+b x)^{11}}{15 (d+e x)^{15} (b d-a e)}+\frac {4 b \left (\frac {(a+b x)^{11}}{14 (d+e x)^{14} (b d-a e)}+\frac {3 b \left (\frac {(a+b x)^{11}}{13 (d+e x)^{13} (b d-a e)}+\frac {2 b \left (\frac {b (a+b x)^{11}}{132 (d+e x)^{11} (b d-a e)^2}+\frac {(a+b x)^{11}}{12 (d+e x)^{12} (b d-a e)}\right )}{13 (b d-a e)}\right )}{14 (b d-a e)}\right )}{15 (b d-a e)}\right )}{16 (b d-a e)}\right ) (-17 a B e+6 A b e+11 b B d)}{17 e (b d-a e)}-\frac {(a+b x)^{11} (B d-A e)}{17 e (d+e x)^{17} (b d-a e)}\)

Input:

Int[((a + b*x)^10*(A + B*x))/(d + e*x)^18,x]
 

Output:

-1/17*((B*d - A*e)*(a + b*x)^11)/(e*(b*d - a*e)*(d + e*x)^17) + ((11*b*B*d 
 + 6*A*b*e - 17*a*B*e)*((a + b*x)^11/(16*(b*d - a*e)*(d + e*x)^16) + (5*b* 
((a + b*x)^11/(15*(b*d - a*e)*(d + e*x)^15) + (4*b*((a + b*x)^11/(14*(b*d 
- a*e)*(d + e*x)^14) + (3*b*((a + b*x)^11/(13*(b*d - a*e)*(d + e*x)^13) + 
(2*b*((a + b*x)^11/(12*(b*d - a*e)*(d + e*x)^12) + (b*(a + b*x)^11)/(132*( 
b*d - a*e)^2*(d + e*x)^11)))/(13*(b*d - a*e))))/(14*(b*d - a*e))))/(15*(b* 
d - a*e))))/(16*(b*d - a*e))))/(17*e*(b*d - a*e))
 

Defintions of rubi rules used

rule 48

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]
 

rule 55

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])
 

rule 87

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p 
+ 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p 
+ 1)))/(f*(p + 1)*(c*f - d*e))   Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] 
/; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege 
rQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ[p, n]))))
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1900\) vs. \(2(321)=642\).

Time = 0.32 (sec) , antiderivative size = 1901, normalized size of antiderivative = 5.67

method	result	size
risch	\(\text {Expression too large to display}\)	\(1901\)
default	\(\text {Expression too large to display}\)	\(1942\)
norman	\(\text {Expression too large to display}\)	\(2014\)
gosper	\(\text {Expression too large to display}\)	\(2233\)
orering	\(\text {Expression too large to display}\)	\(2233\)
parallelrisch	\(\text {Expression too large to display}\)	\(2242\)

Input:

int((b*x+a)^10*(B*x+A)/(e*x+d)^18,x,method=_RETURNVERBOSE)
 

Output:

(-1/816816/e^12*(48048*A*a^10*e^11+30030*A*a^9*b*d*e^10+18018*A*a^8*b^2*d^ 
2*e^9+10296*A*a^7*b^3*d^3*e^8+5544*A*a^6*b^4*d^4*e^7+2772*A*a^5*b^5*d^5*e^ 
6+1260*A*a^4*b^6*d^6*e^5+504*A*a^3*b^7*d^7*e^4+168*A*a^2*b^8*d^8*e^3+42*A* 
a*b^9*d^9*e^2+6*A*b^10*d^10*e+3003*B*a^10*d*e^10+4004*B*a^9*b*d^2*e^9+3861 
*B*a^8*b^2*d^3*e^8+3168*B*a^7*b^3*d^4*e^7+2310*B*a^6*b^4*d^5*e^6+1512*B*a^ 
5*b^5*d^6*e^5+882*B*a^4*b^6*d^7*e^4+448*B*a^3*b^7*d^8*e^3+189*B*a^2*b^8*d^ 
9*e^2+60*B*a*b^9*d^10*e+11*B*b^10*d^11)-1/48048/e^11*(30030*A*a^9*b*e^10+1 
8018*A*a^8*b^2*d*e^9+10296*A*a^7*b^3*d^2*e^8+5544*A*a^6*b^4*d^3*e^7+2772*A 
*a^5*b^5*d^4*e^6+1260*A*a^4*b^6*d^5*e^5+504*A*a^3*b^7*d^6*e^4+168*A*a^2*b^ 
8*d^7*e^3+42*A*a*b^9*d^8*e^2+6*A*b^10*d^9*e+3003*B*a^10*e^10+4004*B*a^9*b* 
d*e^9+3861*B*a^8*b^2*d^2*e^8+3168*B*a^7*b^3*d^3*e^7+2310*B*a^6*b^4*d^4*e^6 
+1512*B*a^5*b^5*d^5*e^5+882*B*a^4*b^6*d^6*e^4+448*B*a^3*b^7*d^7*e^3+189*B* 
a^2*b^8*d^8*e^2+60*B*a*b^9*d^9*e+11*B*b^10*d^10)*x-1/6006*b/e^10*(18018*A* 
a^8*b*e^9+10296*A*a^7*b^2*d*e^8+5544*A*a^6*b^3*d^2*e^7+2772*A*a^5*b^4*d^3* 
e^6+1260*A*a^4*b^5*d^4*e^5+504*A*a^3*b^6*d^5*e^4+168*A*a^2*b^7*d^6*e^3+42* 
A*a*b^8*d^7*e^2+6*A*b^9*d^8*e+4004*B*a^9*e^9+3861*B*a^8*b*d*e^8+3168*B*a^7 
*b^2*d^2*e^7+2310*B*a^6*b^3*d^3*e^6+1512*B*a^5*b^4*d^4*e^5+882*B*a^4*b^5*d 
^5*e^4+448*B*a^3*b^6*d^6*e^3+189*B*a^2*b^7*d^7*e^2+60*B*a*b^8*d^8*e+11*B*b 
^9*d^9)*x^2-5/6006*b^2/e^9*(10296*A*a^7*b*e^8+5544*A*a^6*b^2*d*e^7+2772*A* 
a^5*b^3*d^2*e^6+1260*A*a^4*b^4*d^3*e^5+504*A*a^3*b^5*d^4*e^4+168*A*a^2*...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1995 vs. \(2 (321) = 642\).

Time = 0.17 (sec) , antiderivative size = 1995, normalized size of antiderivative = 5.96 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{18}} \, dx=\text {Too large to display} \] Input:

integrate((b*x+a)^10*(B*x+A)/(e*x+d)^18,x, algorithm="fricas")
 

Output:

-1/816816*(136136*B*b^10*e^11*x^11 + 11*B*b^10*d^11 + 48048*A*a^10*e^11 + 
6*(10*B*a*b^9 + A*b^10)*d^10*e + 21*(9*B*a^2*b^8 + 2*A*a*b^9)*d^9*e^2 + 56 
*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^8*e^3 + 126*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^7 
*e^4 + 252*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^6*e^5 + 462*(5*B*a^6*b^4 + 6*A*a^ 
5*b^5)*d^5*e^6 + 792*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^4*e^7 + 1287*(3*B*a^8*b 
^2 + 8*A*a^7*b^3)*d^3*e^8 + 2002*(2*B*a^9*b + 9*A*a^8*b^2)*d^2*e^9 + 3003* 
(B*a^10 + 10*A*a^9*b)*d*e^10 + 19448*(11*B*b^10*d*e^10 + 6*(10*B*a*b^9 + A 
*b^10)*e^11)*x^10 + 24310*(11*B*b^10*d^2*e^9 + 6*(10*B*a*b^9 + A*b^10)*d*e 
^10 + 21*(9*B*a^2*b^8 + 2*A*a*b^9)*e^11)*x^9 + 24310*(11*B*b^10*d^3*e^8 + 
6*(10*B*a*b^9 + A*b^10)*d^2*e^9 + 21*(9*B*a^2*b^8 + 2*A*a*b^9)*d*e^10 + 56 
*(8*B*a^3*b^7 + 3*A*a^2*b^8)*e^11)*x^8 + 19448*(11*B*b^10*d^4*e^7 + 6*(10* 
B*a*b^9 + A*b^10)*d^3*e^8 + 21*(9*B*a^2*b^8 + 2*A*a*b^9)*d^2*e^9 + 56*(8*B 
*a^3*b^7 + 3*A*a^2*b^8)*d*e^10 + 126*(7*B*a^4*b^6 + 4*A*a^3*b^7)*e^11)*x^7 
 + 12376*(11*B*b^10*d^5*e^6 + 6*(10*B*a*b^9 + A*b^10)*d^4*e^7 + 21*(9*B*a^ 
2*b^8 + 2*A*a*b^9)*d^3*e^8 + 56*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^2*e^9 + 126* 
(7*B*a^4*b^6 + 4*A*a^3*b^7)*d*e^10 + 252*(6*B*a^5*b^5 + 5*A*a^4*b^6)*e^11) 
*x^6 + 6188*(11*B*b^10*d^6*e^5 + 6*(10*B*a*b^9 + A*b^10)*d^5*e^6 + 21*(9*B 
*a^2*b^8 + 2*A*a*b^9)*d^4*e^7 + 56*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^3*e^8 + 1 
26*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^2*e^9 + 252*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d 
*e^10 + 462*(5*B*a^6*b^4 + 6*A*a^5*b^5)*e^11)*x^5 + 2380*(11*B*b^10*d^7...
                                                                                    
                                                                                    
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{18}} \, dx=\text {Timed out} \] Input:

integrate((b*x+a)**10*(B*x+A)/(e*x+d)**18,x)
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1995 vs. \(2 (321) = 642\).

Time = 0.16 (sec) , antiderivative size = 1995, normalized size of antiderivative = 5.96 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{18}} \, dx=\text {Too large to display} \] Input:

integrate((b*x+a)^10*(B*x+A)/(e*x+d)^18,x, algorithm="maxima")
 

Output:

-1/816816*(136136*B*b^10*e^11*x^11 + 11*B*b^10*d^11 + 48048*A*a^10*e^11 + 
6*(10*B*a*b^9 + A*b^10)*d^10*e + 21*(9*B*a^2*b^8 + 2*A*a*b^9)*d^9*e^2 + 56 
*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^8*e^3 + 126*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^7 
*e^4 + 252*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^6*e^5 + 462*(5*B*a^6*b^4 + 6*A*a^ 
5*b^5)*d^5*e^6 + 792*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^4*e^7 + 1287*(3*B*a^8*b 
^2 + 8*A*a^7*b^3)*d^3*e^8 + 2002*(2*B*a^9*b + 9*A*a^8*b^2)*d^2*e^9 + 3003* 
(B*a^10 + 10*A*a^9*b)*d*e^10 + 19448*(11*B*b^10*d*e^10 + 6*(10*B*a*b^9 + A 
*b^10)*e^11)*x^10 + 24310*(11*B*b^10*d^2*e^9 + 6*(10*B*a*b^9 + A*b^10)*d*e 
^10 + 21*(9*B*a^2*b^8 + 2*A*a*b^9)*e^11)*x^9 + 24310*(11*B*b^10*d^3*e^8 + 
6*(10*B*a*b^9 + A*b^10)*d^2*e^9 + 21*(9*B*a^2*b^8 + 2*A*a*b^9)*d*e^10 + 56 
*(8*B*a^3*b^7 + 3*A*a^2*b^8)*e^11)*x^8 + 19448*(11*B*b^10*d^4*e^7 + 6*(10* 
B*a*b^9 + A*b^10)*d^3*e^8 + 21*(9*B*a^2*b^8 + 2*A*a*b^9)*d^2*e^9 + 56*(8*B 
*a^3*b^7 + 3*A*a^2*b^8)*d*e^10 + 126*(7*B*a^4*b^6 + 4*A*a^3*b^7)*e^11)*x^7 
 + 12376*(11*B*b^10*d^5*e^6 + 6*(10*B*a*b^9 + A*b^10)*d^4*e^7 + 21*(9*B*a^ 
2*b^8 + 2*A*a*b^9)*d^3*e^8 + 56*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^2*e^9 + 126* 
(7*B*a^4*b^6 + 4*A*a^3*b^7)*d*e^10 + 252*(6*B*a^5*b^5 + 5*A*a^4*b^6)*e^11) 
*x^6 + 6188*(11*B*b^10*d^6*e^5 + 6*(10*B*a*b^9 + A*b^10)*d^5*e^6 + 21*(9*B 
*a^2*b^8 + 2*A*a*b^9)*d^4*e^7 + 56*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^3*e^8 + 1 
26*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^2*e^9 + 252*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d 
*e^10 + 462*(5*B*a^6*b^4 + 6*A*a^5*b^5)*e^11)*x^5 + 2380*(11*B*b^10*d^7...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2232 vs. \(2 (321) = 642\).

Time = 0.13 (sec) , antiderivative size = 2232, normalized size of antiderivative = 6.66 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{18}} \, dx=\text {Too large to display} \] Input:

integrate((b*x+a)^10*(B*x+A)/(e*x+d)^18,x, algorithm="giac")
 

Output:

-1/816816*(136136*B*b^10*e^11*x^11 + 213928*B*b^10*d*e^10*x^10 + 1166880*B 
*a*b^9*e^11*x^10 + 116688*A*b^10*e^11*x^10 + 267410*B*b^10*d^2*e^9*x^9 + 1 
458600*B*a*b^9*d*e^10*x^9 + 145860*A*b^10*d*e^10*x^9 + 4594590*B*a^2*b^8*e 
^11*x^9 + 1021020*A*a*b^9*e^11*x^9 + 267410*B*b^10*d^3*e^8*x^8 + 1458600*B 
*a*b^9*d^2*e^9*x^8 + 145860*A*b^10*d^2*e^9*x^8 + 4594590*B*a^2*b^8*d*e^10* 
x^8 + 1021020*A*a*b^9*d*e^10*x^8 + 10890880*B*a^3*b^7*e^11*x^8 + 4084080*A 
*a^2*b^8*e^11*x^8 + 213928*B*b^10*d^4*e^7*x^7 + 1166880*B*a*b^9*d^3*e^8*x^ 
7 + 116688*A*b^10*d^3*e^8*x^7 + 3675672*B*a^2*b^8*d^2*e^9*x^7 + 816816*A*a 
*b^9*d^2*e^9*x^7 + 8712704*B*a^3*b^7*d*e^10*x^7 + 3267264*A*a^2*b^8*d*e^10 
*x^7 + 17153136*B*a^4*b^6*e^11*x^7 + 9801792*A*a^3*b^7*e^11*x^7 + 136136*B 
*b^10*d^5*e^6*x^6 + 742560*B*a*b^9*d^4*e^7*x^6 + 74256*A*b^10*d^4*e^7*x^6 
+ 2339064*B*a^2*b^8*d^3*e^8*x^6 + 519792*A*a*b^9*d^3*e^8*x^6 + 5544448*B*a 
^3*b^7*d^2*e^9*x^6 + 2079168*A*a^2*b^8*d^2*e^9*x^6 + 10915632*B*a^4*b^6*d* 
e^10*x^6 + 6237504*A*a^3*b^7*d*e^10*x^6 + 18712512*B*a^5*b^5*e^11*x^6 + 15 
593760*A*a^4*b^6*e^11*x^6 + 68068*B*b^10*d^6*e^5*x^5 + 371280*B*a*b^9*d^5* 
e^6*x^5 + 37128*A*b^10*d^5*e^6*x^5 + 1169532*B*a^2*b^8*d^4*e^7*x^5 + 25989 
6*A*a*b^9*d^4*e^7*x^5 + 2772224*B*a^3*b^7*d^3*e^8*x^5 + 1039584*A*a^2*b^8* 
d^3*e^8*x^5 + 5457816*B*a^4*b^6*d^2*e^9*x^5 + 3118752*A*a^3*b^7*d^2*e^9*x^ 
5 + 9356256*B*a^5*b^5*d*e^10*x^5 + 7796880*A*a^4*b^6*d*e^10*x^5 + 14294280 
*B*a^6*b^4*e^11*x^5 + 17153136*A*a^5*b^5*e^11*x^5 + 26180*B*b^10*d^7*e^...
 

Mupad [B] (verification not implemented)

Time = 1.40 (sec) , antiderivative size = 2077, normalized size of antiderivative = 6.20 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{18}} \, dx=\text {Too large to display} \] Input:

int(((A + B*x)*(a + b*x)^10)/(d + e*x)^18,x)
 

Output:

-((48048*A*a^10*e^11 + 11*B*b^10*d^11 + 6*A*b^10*d^10*e + 3003*B*a^10*d*e^ 
10 + 42*A*a*b^9*d^9*e^2 + 4004*B*a^9*b*d^2*e^9 + 168*A*a^2*b^8*d^8*e^3 + 5 
04*A*a^3*b^7*d^7*e^4 + 1260*A*a^4*b^6*d^6*e^5 + 2772*A*a^5*b^5*d^5*e^6 + 5 
544*A*a^6*b^4*d^4*e^7 + 10296*A*a^7*b^3*d^3*e^8 + 18018*A*a^8*b^2*d^2*e^9 
+ 189*B*a^2*b^8*d^9*e^2 + 448*B*a^3*b^7*d^8*e^3 + 882*B*a^4*b^6*d^7*e^4 + 
1512*B*a^5*b^5*d^6*e^5 + 2310*B*a^6*b^4*d^5*e^6 + 3168*B*a^7*b^3*d^4*e^7 + 
 3861*B*a^8*b^2*d^3*e^8 + 30030*A*a^9*b*d*e^10 + 60*B*a*b^9*d^10*e)/(81681 
6*e^12) + (x*(3003*B*a^10*e^10 + 11*B*b^10*d^10 + 30030*A*a^9*b*e^10 + 6*A 
*b^10*d^9*e + 42*A*a*b^9*d^8*e^2 + 18018*A*a^8*b^2*d*e^9 + 168*A*a^2*b^8*d 
^7*e^3 + 504*A*a^3*b^7*d^6*e^4 + 1260*A*a^4*b^6*d^5*e^5 + 2772*A*a^5*b^5*d 
^4*e^6 + 5544*A*a^6*b^4*d^3*e^7 + 10296*A*a^7*b^3*d^2*e^8 + 189*B*a^2*b^8* 
d^8*e^2 + 448*B*a^3*b^7*d^7*e^3 + 882*B*a^4*b^6*d^6*e^4 + 1512*B*a^5*b^5*d 
^5*e^5 + 2310*B*a^6*b^4*d^4*e^6 + 3168*B*a^7*b^3*d^3*e^7 + 3861*B*a^8*b^2* 
d^2*e^8 + 60*B*a*b^9*d^9*e + 4004*B*a^9*b*d*e^9))/(48048*e^11) + (5*b^7*x^ 
8*(448*B*a^3*e^3 + 11*B*b^3*d^3 + 168*A*a^2*b*e^3 + 6*A*b^3*d^2*e + 42*A*a 
*b^2*d*e^2 + 60*B*a*b^2*d^2*e + 189*B*a^2*b*d*e^2))/(168*e^4) + (b^4*x^5*( 
2310*B*a^6*e^6 + 11*B*b^6*d^6 + 2772*A*a^5*b*e^6 + 6*A*b^6*d^5*e + 42*A*a* 
b^5*d^4*e^2 + 1260*A*a^4*b^2*d*e^5 + 168*A*a^2*b^4*d^3*e^3 + 504*A*a^3*b^3 
*d^2*e^4 + 189*B*a^2*b^4*d^4*e^2 + 448*B*a^3*b^3*d^3*e^3 + 882*B*a^4*b^2*d 
^2*e^4 + 60*B*a*b^5*d^5*e + 1512*B*a^5*b*d*e^5))/(132*e^7) + (b^9*x^10*...
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 1327, normalized size of antiderivative = 3.96 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{18}} \, dx =\text {Too large to display} \] Input:

int((b*x+a)^10*(B*x+A)/(e*x+d)^18,x)
 

Output:

( - 4368*a**11*e**11 - 3003*a**10*b*d*e**10 - 51051*a**10*b*e**11*x - 2002 
*a**9*b**2*d**2*e**9 - 34034*a**9*b**2*d*e**10*x - 272272*a**9*b**2*e**11* 
x**2 - 1287*a**8*b**3*d**3*e**8 - 21879*a**8*b**3*d**2*e**9*x - 175032*a** 
8*b**3*d*e**10*x**2 - 875160*a**8*b**3*e**11*x**3 - 792*a**7*b**4*d**4*e** 
7 - 13464*a**7*b**4*d**3*e**8*x - 107712*a**7*b**4*d**2*e**9*x**2 - 538560 
*a**7*b**4*d*e**10*x**3 - 1884960*a**7*b**4*e**11*x**4 - 462*a**6*b**5*d** 
5*e**6 - 7854*a**6*b**5*d**4*e**7*x - 62832*a**6*b**5*d**3*e**8*x**2 - 314 
160*a**6*b**5*d**2*e**9*x**3 - 1099560*a**6*b**5*d*e**10*x**4 - 2858856*a* 
*6*b**5*e**11*x**5 - 252*a**5*b**6*d**6*e**5 - 4284*a**5*b**6*d**5*e**6*x 
- 34272*a**5*b**6*d**4*e**7*x**2 - 171360*a**5*b**6*d**3*e**8*x**3 - 59976 
0*a**5*b**6*d**2*e**9*x**4 - 1559376*a**5*b**6*d*e**10*x**5 - 3118752*a**5 
*b**6*e**11*x**6 - 126*a**4*b**7*d**7*e**4 - 2142*a**4*b**7*d**6*e**5*x - 
17136*a**4*b**7*d**5*e**6*x**2 - 85680*a**4*b**7*d**4*e**7*x**3 - 299880*a 
**4*b**7*d**3*e**8*x**4 - 779688*a**4*b**7*d**2*e**9*x**5 - 1559376*a**4*b 
**7*d*e**10*x**6 - 2450448*a**4*b**7*e**11*x**7 - 56*a**3*b**8*d**8*e**3 - 
 952*a**3*b**8*d**7*e**4*x - 7616*a**3*b**8*d**6*e**5*x**2 - 38080*a**3*b* 
*8*d**5*e**6*x**3 - 133280*a**3*b**8*d**4*e**7*x**4 - 346528*a**3*b**8*d** 
3*e**8*x**5 - 693056*a**3*b**8*d**2*e**9*x**6 - 1089088*a**3*b**8*d*e**10* 
x**7 - 1361360*a**3*b**8*e**11*x**8 - 21*a**2*b**9*d**9*e**2 - 357*a**2*b* 
*9*d**8*e**3*x - 2856*a**2*b**9*d**7*e**4*x**2 - 14280*a**2*b**9*d**6*e...