\(\int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{17}} \, dx\) [95]

Optimal result

Integrand size = 20, antiderivative size = 285 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{17}} \, dx=-\frac {(B d-A e) (a+b x)^{11}}{16 e (b d-a e) (d+e x)^{16}}+\frac {(11 b B d+5 A b e-16 a B e) (a+b x)^{11}}{240 e (b d-a e)^2 (d+e x)^{15}}+\frac {b (11 b B d+5 A b e-16 a B e) (a+b x)^{11}}{840 e (b d-a e)^3 (d+e x)^{14}}+\frac {b^2 (11 b B d+5 A b e-16 a B e) (a+b x)^{11}}{3640 e (b d-a e)^4 (d+e x)^{13}}+\frac {b^3 (11 b B d+5 A b e-16 a B e) (a+b x)^{11}}{21840 e (b d-a e)^5 (d+e x)^{12}}+\frac {b^4 (11 b B d+5 A b e-16 a B e) (a+b x)^{11}}{240240 e (b d-a e)^6 (d+e x)^{11}} \] Output:

-1/16*(-A*e+B*d)*(b*x+a)^11/e/(-a*e+b*d)/(e*x+d)^16+1/240*(5*A*b*e-16*B*a* 
e+11*B*b*d)*(b*x+a)^11/e/(-a*e+b*d)^2/(e*x+d)^15+1/840*b*(5*A*b*e-16*B*a*e 
+11*B*b*d)*(b*x+a)^11/e/(-a*e+b*d)^3/(e*x+d)^14+1/3640*b^2*(5*A*b*e-16*B*a 
*e+11*B*b*d)*(b*x+a)^11/e/(-a*e+b*d)^4/(e*x+d)^13+1/21840*b^3*(5*A*b*e-16* 
B*a*e+11*B*b*d)*(b*x+a)^11/e/(-a*e+b*d)^5/(e*x+d)^12+1/240240*b^4*(5*A*b*e 
-16*B*a*e+11*B*b*d)*(b*x+a)^11/e/(-a*e+b*d)^6/(e*x+d)^11
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1429\) vs. \(2(285)=570\).

Time = 0.46 (sec) , antiderivative size = 1429, normalized size of antiderivative = 5.01 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{17}} \, dx =\text {Too large to display} \] Input:

Integrate[((a + b*x)^10*(A + B*x))/(d + e*x)^17,x]
 

Output:

-1/240240*(1001*a^10*e^10*(15*A*e + B*(d + 16*e*x)) + 1430*a^9*b*e^9*(7*A* 
e*(d + 16*e*x) + B*(d^2 + 16*d*e*x + 120*e^2*x^2)) + 495*a^8*b^2*e^8*(13*A 
*e*(d^2 + 16*d*e*x + 120*e^2*x^2) + 3*B*(d^3 + 16*d^2*e*x + 120*d*e^2*x^2 
+ 560*e^3*x^3)) + 1320*a^7*b^3*e^7*(3*A*e*(d^3 + 16*d^2*e*x + 120*d*e^2*x^ 
2 + 560*e^3*x^3) + B*(d^4 + 16*d^3*e*x + 120*d^2*e^2*x^2 + 560*d*e^3*x^3 + 
 1820*e^4*x^4)) + 210*a^6*b^4*e^6*(11*A*e*(d^4 + 16*d^3*e*x + 120*d^2*e^2* 
x^2 + 560*d*e^3*x^3 + 1820*e^4*x^4) + 5*B*(d^5 + 16*d^4*e*x + 120*d^3*e^2* 
x^2 + 560*d^2*e^3*x^3 + 1820*d*e^4*x^4 + 4368*e^5*x^5)) + 252*a^5*b^5*e^5* 
(5*A*e*(d^5 + 16*d^4*e*x + 120*d^3*e^2*x^2 + 560*d^2*e^3*x^3 + 1820*d*e^4* 
x^4 + 4368*e^5*x^5) + 3*B*(d^6 + 16*d^5*e*x + 120*d^4*e^2*x^2 + 560*d^3*e^ 
3*x^3 + 1820*d^2*e^4*x^4 + 4368*d*e^5*x^5 + 8008*e^6*x^6)) + 70*a^4*b^6*e^ 
4*(9*A*e*(d^6 + 16*d^5*e*x + 120*d^4*e^2*x^2 + 560*d^3*e^3*x^3 + 1820*d^2* 
e^4*x^4 + 4368*d*e^5*x^5 + 8008*e^6*x^6) + 7*B*(d^7 + 16*d^6*e*x + 120*d^5 
*e^2*x^2 + 560*d^4*e^3*x^3 + 1820*d^3*e^4*x^4 + 4368*d^2*e^5*x^5 + 8008*d* 
e^6*x^6 + 11440*e^7*x^7)) + 280*a^3*b^7*e^3*(A*e*(d^7 + 16*d^6*e*x + 120*d 
^5*e^2*x^2 + 560*d^4*e^3*x^3 + 1820*d^3*e^4*x^4 + 4368*d^2*e^5*x^5 + 8008* 
d*e^6*x^6 + 11440*e^7*x^7) + B*(d^8 + 16*d^7*e*x + 120*d^6*e^2*x^2 + 560*d 
^5*e^3*x^3 + 1820*d^4*e^4*x^4 + 4368*d^3*e^5*x^5 + 8008*d^2*e^6*x^6 + 1144 
0*d*e^7*x^7 + 12870*e^8*x^8)) + 15*a^2*b^8*e^2*(7*A*e*(d^8 + 16*d^7*e*x + 
120*d^6*e^2*x^2 + 560*d^5*e^3*x^3 + 1820*d^4*e^4*x^4 + 4368*d^3*e^5*x^5...
 

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 263, normalized size of antiderivative = 0.92, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {87, 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.

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

\(\Big \downarrow \) 87

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

\(\Big \downarrow \) 55

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

\(\Big \downarrow \) 55

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

\(\Big \downarrow \) 55

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

\(\Big \downarrow \) 55

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

\(\Big \downarrow \) 48

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

Input:

Int[((a + b*x)^10*(A + B*x))/(d + e*x)^17,x]
 

Output:

-1/16*((B*d - A*e)*(a + b*x)^11)/(e*(b*d - a*e)*(d + e*x)^16) + ((11*b*B*d 
 + 5*A*b*e - 16*a*B*e)*((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*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. \(2(273)=546\).

Time = 0.31 (sec) , antiderivative size = 1901, normalized size of antiderivative = 6.67

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)^17,x,method=_RETURNVERBOSE)
 

Output:

(-1/240240/e^12*(15015*A*a^10*e^11+10010*A*a^9*b*d*e^10+6435*A*a^8*b^2*d^2 
*e^9+3960*A*a^7*b^3*d^3*e^8+2310*A*a^6*b^4*d^4*e^7+1260*A*a^5*b^5*d^5*e^6+ 
630*A*a^4*b^6*d^6*e^5+280*A*a^3*b^7*d^7*e^4+105*A*a^2*b^8*d^8*e^3+30*A*a*b 
^9*d^9*e^2+5*A*b^10*d^10*e+1001*B*a^10*d*e^10+1430*B*a^9*b*d^2*e^9+1485*B* 
a^8*b^2*d^3*e^8+1320*B*a^7*b^3*d^4*e^7+1050*B*a^6*b^4*d^5*e^6+756*B*a^5*b^ 
5*d^6*e^5+490*B*a^4*b^6*d^7*e^4+280*B*a^3*b^7*d^8*e^3+135*B*a^2*b^8*d^9*e^ 
2+50*B*a*b^9*d^10*e+11*B*b^10*d^11)-1/15015/e^11*(10010*A*a^9*b*e^10+6435* 
A*a^8*b^2*d*e^9+3960*A*a^7*b^3*d^2*e^8+2310*A*a^6*b^4*d^3*e^7+1260*A*a^5*b 
^5*d^4*e^6+630*A*a^4*b^6*d^5*e^5+280*A*a^3*b^7*d^6*e^4+105*A*a^2*b^8*d^7*e 
^3+30*A*a*b^9*d^8*e^2+5*A*b^10*d^9*e+1001*B*a^10*e^10+1430*B*a^9*b*d*e^9+1 
485*B*a^8*b^2*d^2*e^8+1320*B*a^7*b^3*d^3*e^7+1050*B*a^6*b^4*d^4*e^6+756*B* 
a^5*b^5*d^5*e^5+490*B*a^4*b^6*d^6*e^4+280*B*a^3*b^7*d^7*e^3+135*B*a^2*b^8* 
d^8*e^2+50*B*a*b^9*d^9*e+11*B*b^10*d^10)*x-1/2002*b/e^10*(6435*A*a^8*b*e^9 
+3960*A*a^7*b^2*d*e^8+2310*A*a^6*b^3*d^2*e^7+1260*A*a^5*b^4*d^3*e^6+630*A* 
a^4*b^5*d^4*e^5+280*A*a^3*b^6*d^5*e^4+105*A*a^2*b^7*d^6*e^3+30*A*a*b^8*d^7 
*e^2+5*A*b^9*d^8*e+1430*B*a^9*e^9+1485*B*a^8*b*d*e^8+1320*B*a^7*b^2*d^2*e^ 
7+1050*B*a^6*b^3*d^3*e^6+756*B*a^5*b^4*d^4*e^5+490*B*a^4*b^5*d^5*e^4+280*B 
*a^3*b^6*d^6*e^3+135*B*a^2*b^7*d^7*e^2+50*B*a*b^8*d^8*e+11*B*b^9*d^9)*x^2- 
1/429*b^2/e^9*(3960*A*a^7*b*e^8+2310*A*a^6*b^2*d*e^7+1260*A*a^5*b^3*d^2*e^ 
6+630*A*a^4*b^4*d^3*e^5+280*A*a^3*b^5*d^4*e^4+105*A*a^2*b^6*d^5*e^3+30*...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1984 vs. \(2 (273) = 546\).

Time = 0.18 (sec) , antiderivative size = 1984, normalized size of antiderivative = 6.96 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{17}} \, dx=\text {Too large to display} \] Input:

integrate((b*x+a)^10*(B*x+A)/(e*x+d)^17,x, algorithm="fricas")
 

Output:

-1/240240*(48048*B*b^10*e^11*x^11 + 11*B*b^10*d^11 + 15015*A*a^10*e^11 + 5 
*(10*B*a*b^9 + A*b^10)*d^10*e + 15*(9*B*a^2*b^8 + 2*A*a*b^9)*d^9*e^2 + 35* 
(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^8*e^3 + 70*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^7*e 
^4 + 126*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^6*e^5 + 210*(5*B*a^6*b^4 + 6*A*a^5* 
b^5)*d^5*e^6 + 330*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^4*e^7 + 495*(3*B*a^8*b^2 
+ 8*A*a^7*b^3)*d^3*e^8 + 715*(2*B*a^9*b + 9*A*a^8*b^2)*d^2*e^9 + 1001*(B*a 
^10 + 10*A*a^9*b)*d*e^10 + 8008*(11*B*b^10*d*e^10 + 5*(10*B*a*b^9 + A*b^10 
)*e^11)*x^10 + 11440*(11*B*b^10*d^2*e^9 + 5*(10*B*a*b^9 + A*b^10)*d*e^10 + 
 15*(9*B*a^2*b^8 + 2*A*a*b^9)*e^11)*x^9 + 12870*(11*B*b^10*d^3*e^8 + 5*(10 
*B*a*b^9 + A*b^10)*d^2*e^9 + 15*(9*B*a^2*b^8 + 2*A*a*b^9)*d*e^10 + 35*(8*B 
*a^3*b^7 + 3*A*a^2*b^8)*e^11)*x^8 + 11440*(11*B*b^10*d^4*e^7 + 5*(10*B*a*b 
^9 + A*b^10)*d^3*e^8 + 15*(9*B*a^2*b^8 + 2*A*a*b^9)*d^2*e^9 + 35*(8*B*a^3* 
b^7 + 3*A*a^2*b^8)*d*e^10 + 70*(7*B*a^4*b^6 + 4*A*a^3*b^7)*e^11)*x^7 + 800 
8*(11*B*b^10*d^5*e^6 + 5*(10*B*a*b^9 + A*b^10)*d^4*e^7 + 15*(9*B*a^2*b^8 + 
 2*A*a*b^9)*d^3*e^8 + 35*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^2*e^9 + 70*(7*B*a^4 
*b^6 + 4*A*a^3*b^7)*d*e^10 + 126*(6*B*a^5*b^5 + 5*A*a^4*b^6)*e^11)*x^6 + 4 
368*(11*B*b^10*d^6*e^5 + 5*(10*B*a*b^9 + A*b^10)*d^5*e^6 + 15*(9*B*a^2*b^8 
 + 2*A*a*b^9)*d^4*e^7 + 35*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^3*e^8 + 70*(7*B*a 
^4*b^6 + 4*A*a^3*b^7)*d^2*e^9 + 126*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d*e^10 + 2 
10*(5*B*a^6*b^4 + 6*A*a^5*b^5)*e^11)*x^5 + 1820*(11*B*b^10*d^7*e^4 + 5*...
                                                                                    
                                                                                    
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{17}} \, dx=\text {Timed out} \] Input:

integrate((b*x+a)**10*(B*x+A)/(e*x+d)**17,x)
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1984 vs. \(2 (273) = 546\).

Time = 0.14 (sec) , antiderivative size = 1984, normalized size of antiderivative = 6.96 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{17}} \, dx=\text {Too large to display} \] Input:

integrate((b*x+a)^10*(B*x+A)/(e*x+d)^17,x, algorithm="maxima")
 

Output:

-1/240240*(48048*B*b^10*e^11*x^11 + 11*B*b^10*d^11 + 15015*A*a^10*e^11 + 5 
*(10*B*a*b^9 + A*b^10)*d^10*e + 15*(9*B*a^2*b^8 + 2*A*a*b^9)*d^9*e^2 + 35* 
(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^8*e^3 + 70*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^7*e 
^4 + 126*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^6*e^5 + 210*(5*B*a^6*b^4 + 6*A*a^5* 
b^5)*d^5*e^6 + 330*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^4*e^7 + 495*(3*B*a^8*b^2 
+ 8*A*a^7*b^3)*d^3*e^8 + 715*(2*B*a^9*b + 9*A*a^8*b^2)*d^2*e^9 + 1001*(B*a 
^10 + 10*A*a^9*b)*d*e^10 + 8008*(11*B*b^10*d*e^10 + 5*(10*B*a*b^9 + A*b^10 
)*e^11)*x^10 + 11440*(11*B*b^10*d^2*e^9 + 5*(10*B*a*b^9 + A*b^10)*d*e^10 + 
 15*(9*B*a^2*b^8 + 2*A*a*b^9)*e^11)*x^9 + 12870*(11*B*b^10*d^3*e^8 + 5*(10 
*B*a*b^9 + A*b^10)*d^2*e^9 + 15*(9*B*a^2*b^8 + 2*A*a*b^9)*d*e^10 + 35*(8*B 
*a^3*b^7 + 3*A*a^2*b^8)*e^11)*x^8 + 11440*(11*B*b^10*d^4*e^7 + 5*(10*B*a*b 
^9 + A*b^10)*d^3*e^8 + 15*(9*B*a^2*b^8 + 2*A*a*b^9)*d^2*e^9 + 35*(8*B*a^3* 
b^7 + 3*A*a^2*b^8)*d*e^10 + 70*(7*B*a^4*b^6 + 4*A*a^3*b^7)*e^11)*x^7 + 800 
8*(11*B*b^10*d^5*e^6 + 5*(10*B*a*b^9 + A*b^10)*d^4*e^7 + 15*(9*B*a^2*b^8 + 
 2*A*a*b^9)*d^3*e^8 + 35*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^2*e^9 + 70*(7*B*a^4 
*b^6 + 4*A*a^3*b^7)*d*e^10 + 126*(6*B*a^5*b^5 + 5*A*a^4*b^6)*e^11)*x^6 + 4 
368*(11*B*b^10*d^6*e^5 + 5*(10*B*a*b^9 + A*b^10)*d^5*e^6 + 15*(9*B*a^2*b^8 
 + 2*A*a*b^9)*d^4*e^7 + 35*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^3*e^8 + 70*(7*B*a 
^4*b^6 + 4*A*a^3*b^7)*d^2*e^9 + 126*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d*e^10 + 2 
10*(5*B*a^6*b^4 + 6*A*a^5*b^5)*e^11)*x^5 + 1820*(11*B*b^10*d^7*e^4 + 5*...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2232 vs. \(2 (273) = 546\).

Time = 0.13 (sec) , antiderivative size = 2232, normalized size of antiderivative = 7.83 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{17}} \, dx=\text {Too large to display} \] Input:

integrate((b*x+a)^10*(B*x+A)/(e*x+d)^17,x, algorithm="giac")
 

Output:

-1/240240*(48048*B*b^10*e^11*x^11 + 88088*B*b^10*d*e^10*x^10 + 400400*B*a* 
b^9*e^11*x^10 + 40040*A*b^10*e^11*x^10 + 125840*B*b^10*d^2*e^9*x^9 + 57200 
0*B*a*b^9*d*e^10*x^9 + 57200*A*b^10*d*e^10*x^9 + 1544400*B*a^2*b^8*e^11*x^ 
9 + 343200*A*a*b^9*e^11*x^9 + 141570*B*b^10*d^3*e^8*x^8 + 643500*B*a*b^9*d 
^2*e^9*x^8 + 64350*A*b^10*d^2*e^9*x^8 + 1737450*B*a^2*b^8*d*e^10*x^8 + 386 
100*A*a*b^9*d*e^10*x^8 + 3603600*B*a^3*b^7*e^11*x^8 + 1351350*A*a^2*b^8*e^ 
11*x^8 + 125840*B*b^10*d^4*e^7*x^7 + 572000*B*a*b^9*d^3*e^8*x^7 + 57200*A* 
b^10*d^3*e^8*x^7 + 1544400*B*a^2*b^8*d^2*e^9*x^7 + 343200*A*a*b^9*d^2*e^9* 
x^7 + 3203200*B*a^3*b^7*d*e^10*x^7 + 1201200*A*a^2*b^8*d*e^10*x^7 + 560560 
0*B*a^4*b^6*e^11*x^7 + 3203200*A*a^3*b^7*e^11*x^7 + 88088*B*b^10*d^5*e^6*x 
^6 + 400400*B*a*b^9*d^4*e^7*x^6 + 40040*A*b^10*d^4*e^7*x^6 + 1081080*B*a^2 
*b^8*d^3*e^8*x^6 + 240240*A*a*b^9*d^3*e^8*x^6 + 2242240*B*a^3*b^7*d^2*e^9* 
x^6 + 840840*A*a^2*b^8*d^2*e^9*x^6 + 3923920*B*a^4*b^6*d*e^10*x^6 + 224224 
0*A*a^3*b^7*d*e^10*x^6 + 6054048*B*a^5*b^5*e^11*x^6 + 5045040*A*a^4*b^6*e^ 
11*x^6 + 48048*B*b^10*d^6*e^5*x^5 + 218400*B*a*b^9*d^5*e^6*x^5 + 21840*A*b 
^10*d^5*e^6*x^5 + 589680*B*a^2*b^8*d^4*e^7*x^5 + 131040*A*a*b^9*d^4*e^7*x^ 
5 + 1223040*B*a^3*b^7*d^3*e^8*x^5 + 458640*A*a^2*b^8*d^3*e^8*x^5 + 2140320 
*B*a^4*b^6*d^2*e^9*x^5 + 1223040*A*a^3*b^7*d^2*e^9*x^5 + 3302208*B*a^5*b^5 
*d*e^10*x^5 + 2751840*A*a^4*b^6*d*e^10*x^5 + 4586400*B*a^6*b^4*e^11*x^5 + 
5503680*A*a^5*b^5*e^11*x^5 + 20020*B*b^10*d^7*e^4*x^4 + 91000*B*a*b^9*d...
 

Mupad [B] (verification not implemented)

Time = 1.32 (sec) , antiderivative size = 2066, normalized size of antiderivative = 7.25 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{17}} \, dx=\text {Too large to display} \] Input:

int(((A + B*x)*(a + b*x)^10)/(d + e*x)^17,x)
 

Output:

-((15015*A*a^10*e^11 + 11*B*b^10*d^11 + 5*A*b^10*d^10*e + 1001*B*a^10*d*e^ 
10 + 30*A*a*b^9*d^9*e^2 + 1430*B*a^9*b*d^2*e^9 + 105*A*a^2*b^8*d^8*e^3 + 2 
80*A*a^3*b^7*d^7*e^4 + 630*A*a^4*b^6*d^6*e^5 + 1260*A*a^5*b^5*d^5*e^6 + 23 
10*A*a^6*b^4*d^4*e^7 + 3960*A*a^7*b^3*d^3*e^8 + 6435*A*a^8*b^2*d^2*e^9 + 1 
35*B*a^2*b^8*d^9*e^2 + 280*B*a^3*b^7*d^8*e^3 + 490*B*a^4*b^6*d^7*e^4 + 756 
*B*a^5*b^5*d^6*e^5 + 1050*B*a^6*b^4*d^5*e^6 + 1320*B*a^7*b^3*d^4*e^7 + 148 
5*B*a^8*b^2*d^3*e^8 + 10010*A*a^9*b*d*e^10 + 50*B*a*b^9*d^10*e)/(240240*e^ 
12) + (x*(1001*B*a^10*e^10 + 11*B*b^10*d^10 + 10010*A*a^9*b*e^10 + 5*A*b^1 
0*d^9*e + 30*A*a*b^9*d^8*e^2 + 6435*A*a^8*b^2*d*e^9 + 105*A*a^2*b^8*d^7*e^ 
3 + 280*A*a^3*b^7*d^6*e^4 + 630*A*a^4*b^6*d^5*e^5 + 1260*A*a^5*b^5*d^4*e^6 
 + 2310*A*a^6*b^4*d^3*e^7 + 3960*A*a^7*b^3*d^2*e^8 + 135*B*a^2*b^8*d^8*e^2 
 + 280*B*a^3*b^7*d^7*e^3 + 490*B*a^4*b^6*d^6*e^4 + 756*B*a^5*b^5*d^5*e^5 + 
 1050*B*a^6*b^4*d^4*e^6 + 1320*B*a^7*b^3*d^3*e^7 + 1485*B*a^8*b^2*d^2*e^8 
+ 50*B*a*b^9*d^9*e + 1430*B*a^9*b*d*e^9))/(15015*e^11) + (3*b^7*x^8*(280*B 
*a^3*e^3 + 11*B*b^3*d^3 + 105*A*a^2*b*e^3 + 5*A*b^3*d^2*e + 30*A*a*b^2*d*e 
^2 + 50*B*a*b^2*d^2*e + 135*B*a^2*b*d*e^2))/(56*e^4) + (b^4*x^5*(1050*B*a^ 
6*e^6 + 11*B*b^6*d^6 + 1260*A*a^5*b*e^6 + 5*A*b^6*d^5*e + 30*A*a*b^5*d^4*e 
^2 + 630*A*a^4*b^2*d*e^5 + 105*A*a^2*b^4*d^3*e^3 + 280*A*a^3*b^3*d^2*e^4 + 
 135*B*a^2*b^4*d^4*e^2 + 280*B*a^3*b^3*d^3*e^3 + 490*B*a^4*b^2*d^2*e^4 + 5 
0*B*a*b^5*d^5*e + 756*B*a^5*b*d*e^5))/(55*e^7) + (b^9*x^10*(5*A*b*e + 5...
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 1316, normalized size of antiderivative = 4.62 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{17}} \, dx =\text {Too large to display} \] Input:

int((b*x+a)^10*(B*x+A)/(e*x+d)^17,x)
 

Output:

( - 1365*a**11*e**11 - 1001*a**10*b*d*e**10 - 16016*a**10*b*e**11*x - 715* 
a**9*b**2*d**2*e**9 - 11440*a**9*b**2*d*e**10*x - 85800*a**9*b**2*e**11*x* 
*2 - 495*a**8*b**3*d**3*e**8 - 7920*a**8*b**3*d**2*e**9*x - 59400*a**8*b** 
3*d*e**10*x**2 - 277200*a**8*b**3*e**11*x**3 - 330*a**7*b**4*d**4*e**7 - 5 
280*a**7*b**4*d**3*e**8*x - 39600*a**7*b**4*d**2*e**9*x**2 - 184800*a**7*b 
**4*d*e**10*x**3 - 600600*a**7*b**4*e**11*x**4 - 210*a**6*b**5*d**5*e**6 - 
 3360*a**6*b**5*d**4*e**7*x - 25200*a**6*b**5*d**3*e**8*x**2 - 117600*a**6 
*b**5*d**2*e**9*x**3 - 382200*a**6*b**5*d*e**10*x**4 - 917280*a**6*b**5*e* 
*11*x**5 - 126*a**5*b**6*d**6*e**5 - 2016*a**5*b**6*d**5*e**6*x - 15120*a* 
*5*b**6*d**4*e**7*x**2 - 70560*a**5*b**6*d**3*e**8*x**3 - 229320*a**5*b**6 
*d**2*e**9*x**4 - 550368*a**5*b**6*d*e**10*x**5 - 1009008*a**5*b**6*e**11* 
x**6 - 70*a**4*b**7*d**7*e**4 - 1120*a**4*b**7*d**6*e**5*x - 8400*a**4*b** 
7*d**5*e**6*x**2 - 39200*a**4*b**7*d**4*e**7*x**3 - 127400*a**4*b**7*d**3* 
e**8*x**4 - 305760*a**4*b**7*d**2*e**9*x**5 - 560560*a**4*b**7*d*e**10*x** 
6 - 800800*a**4*b**7*e**11*x**7 - 35*a**3*b**8*d**8*e**3 - 560*a**3*b**8*d 
**7*e**4*x - 4200*a**3*b**8*d**6*e**5*x**2 - 19600*a**3*b**8*d**5*e**6*x** 
3 - 63700*a**3*b**8*d**4*e**7*x**4 - 152880*a**3*b**8*d**3*e**8*x**5 - 280 
280*a**3*b**8*d**2*e**9*x**6 - 400400*a**3*b**8*d*e**10*x**7 - 450450*a**3 
*b**8*e**11*x**8 - 15*a**2*b**9*d**9*e**2 - 240*a**2*b**9*d**8*e**3*x - 18 
00*a**2*b**9*d**7*e**4*x**2 - 8400*a**2*b**9*d**6*e**5*x**3 - 27300*a**...