\(\int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{19}} \, dx\) [97]

Optimal result

Integrand size = 20, antiderivative size = 385 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{19}} \, dx=-\frac {(B d-A e) (a+b x)^{11}}{18 e (b d-a e) (d+e x)^{18}}+\frac {(11 b B d+7 A b e-18 a B e) (a+b x)^{11}}{306 e (b d-a e)^2 (d+e x)^{17}}+\frac {b (11 b B d+7 A b e-18 a B e) (a+b x)^{11}}{816 e (b d-a e)^3 (d+e x)^{16}}+\frac {b^2 (11 b B d+7 A b e-18 a B e) (a+b x)^{11}}{2448 e (b d-a e)^4 (d+e x)^{15}}+\frac {b^3 (11 b B d+7 A b e-18 a B e) (a+b x)^{11}}{8568 e (b d-a e)^5 (d+e x)^{14}}+\frac {b^4 (11 b B d+7 A b e-18 a B e) (a+b x)^{11}}{37128 e (b d-a e)^6 (d+e x)^{13}}+\frac {b^5 (11 b B d+7 A b e-18 a B e) (a+b x)^{11}}{222768 e (b d-a e)^7 (d+e x)^{12}}+\frac {b^6 (11 b B d+7 A b e-18 a B e) (a+b x)^{11}}{2450448 e (b d-a e)^8 (d+e x)^{11}} \] Output:

-1/18*(-A*e+B*d)*(b*x+a)^11/e/(-a*e+b*d)/(e*x+d)^18+1/306*(7*A*b*e-18*B*a* 
e+11*B*b*d)*(b*x+a)^11/e/(-a*e+b*d)^2/(e*x+d)^17+1/816*b*(7*A*b*e-18*B*a*e 
+11*B*b*d)*(b*x+a)^11/e/(-a*e+b*d)^3/(e*x+d)^16+1/2448*b^2*(7*A*b*e-18*B*a 
*e+11*B*b*d)*(b*x+a)^11/e/(-a*e+b*d)^4/(e*x+d)^15+1/8568*b^3*(7*A*b*e-18*B 
*a*e+11*B*b*d)*(b*x+a)^11/e/(-a*e+b*d)^5/(e*x+d)^14+1/37128*b^4*(7*A*b*e-1 
8*B*a*e+11*B*b*d)*(b*x+a)^11/e/(-a*e+b*d)^6/(e*x+d)^13+1/222768*b^5*(7*A*b 
*e-18*B*a*e+11*B*b*d)*(b*x+a)^11/e/(-a*e+b*d)^7/(e*x+d)^12+1/2450448*b^6*( 
7*A*b*e-18*B*a*e+11*B*b*d)*(b*x+a)^11/e/(-a*e+b*d)^8/(e*x+d)^11
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1428\) vs. \(2(385)=770\).

Time = 0.48 (sec) , antiderivative size = 1428, normalized size of antiderivative = 3.71 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{19}} \, dx =\text {Too large to display} \] Input:

Integrate[((a + b*x)^10*(A + B*x))/(d + e*x)^19,x]
 

Output:

-1/2450448*(8008*a^10*e^10*(17*A*e + B*(d + 18*e*x)) + 10010*a^9*b*e^9*(8* 
A*e*(d + 18*e*x) + B*(d^2 + 18*d*e*x + 153*e^2*x^2)) + 9009*a^8*b^2*e^8*(5 
*A*e*(d^2 + 18*d*e*x + 153*e^2*x^2) + B*(d^3 + 18*d^2*e*x + 153*d*e^2*x^2 
+ 816*e^3*x^3)) + 3432*a^7*b^3*e^7*(7*A*e*(d^3 + 18*d^2*e*x + 153*d*e^2*x^ 
2 + 816*e^3*x^3) + 2*B*(d^4 + 18*d^3*e*x + 153*d^2*e^2*x^2 + 816*d*e^3*x^3 
 + 3060*e^4*x^4)) + 924*a^6*b^4*e^6*(13*A*e*(d^4 + 18*d^3*e*x + 153*d^2*e^ 
2*x^2 + 816*d*e^3*x^3 + 3060*e^4*x^4) + 5*B*(d^5 + 18*d^4*e*x + 153*d^3*e^ 
2*x^2 + 816*d^2*e^3*x^3 + 3060*d*e^4*x^4 + 8568*e^5*x^5)) + 2772*a^5*b^5*e 
^5*(2*A*e*(d^5 + 18*d^4*e*x + 153*d^3*e^2*x^2 + 816*d^2*e^3*x^3 + 3060*d*e 
^4*x^4 + 8568*e^5*x^5) + B*(d^6 + 18*d^5*e*x + 153*d^4*e^2*x^2 + 816*d^3*e 
^3*x^3 + 3060*d^2*e^4*x^4 + 8568*d*e^5*x^5 + 18564*e^6*x^6)) + 210*a^4*b^6 
*e^4*(11*A*e*(d^6 + 18*d^5*e*x + 153*d^4*e^2*x^2 + 816*d^3*e^3*x^3 + 3060* 
d^2*e^4*x^4 + 8568*d*e^5*x^5 + 18564*e^6*x^6) + 7*B*(d^7 + 18*d^6*e*x + 15 
3*d^5*e^2*x^2 + 816*d^4*e^3*x^3 + 3060*d^3*e^4*x^4 + 8568*d^2*e^5*x^5 + 18 
564*d*e^6*x^6 + 31824*e^7*x^7)) + 168*a^3*b^7*e^3*(5*A*e*(d^7 + 18*d^6*e*x 
 + 153*d^5*e^2*x^2 + 816*d^4*e^3*x^3 + 3060*d^3*e^4*x^4 + 8568*d^2*e^5*x^5 
 + 18564*d*e^6*x^6 + 31824*e^7*x^7) + 4*B*(d^8 + 18*d^7*e*x + 153*d^6*e^2* 
x^2 + 816*d^5*e^3*x^3 + 3060*d^4*e^4*x^4 + 8568*d^3*e^5*x^5 + 18564*d^2*e^ 
6*x^6 + 31824*d*e^7*x^7 + 43758*e^8*x^8)) + 252*a^2*b^8*e^2*(A*e*(d^8 + 18 
*d^7*e*x + 153*d^6*e^2*x^2 + 816*d^5*e^3*x^3 + 3060*d^4*e^4*x^4 + 8568*...
 

Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 351, normalized size of antiderivative = 0.91, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {87, 55, 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.

\(\displaystyle \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{19}} \, dx\)

\(\Big \downarrow \) 87

\(\displaystyle \frac {(-18 a B e+7 A b e+11 b B d) \int \frac {(a+b x)^{10}}{(d+e x)^{18}}dx}{18 e (b d-a e)}-\frac {(a+b x)^{11} (B d-A e)}{18 e (d+e x)^{18} (b d-a e)}\)

\(\Big \downarrow \) 55

\(\displaystyle \frac {(-18 a B e+7 A b e+11 b B d) \left (\frac {6 b \int \frac {(a+b x)^{10}}{(d+e x)^{17}}dx}{17 (b d-a e)}+\frac {(a+b x)^{11}}{17 (d+e x)^{17} (b d-a e)}\right )}{18 e (b d-a e)}-\frac {(a+b x)^{11} (B d-A e)}{18 e (d+e x)^{18} (b d-a e)}\)

\(\Big \downarrow \) 55

\(\displaystyle \frac {(-18 a B e+7 A b e+11 b B d) \left (\frac {6 b \left (\frac {5 b \int \frac {(a+b x)^{10}}{(d+e x)^{16}}dx}{16 (b d-a e)}+\frac {(a+b x)^{11}}{16 (d+e x)^{16} (b d-a e)}\right )}{17 (b d-a e)}+\frac {(a+b x)^{11}}{17 (d+e x)^{17} (b d-a e)}\right )}{18 e (b d-a e)}-\frac {(a+b x)^{11} (B d-A e)}{18 e (d+e x)^{18} (b d-a e)}\)

\(\Big \downarrow \) 55

\(\displaystyle \frac {(-18 a B e+7 A b e+11 b B d) \left (\frac {6 b \left (\frac {5 b \left (\frac {4 b \int \frac {(a+b x)^{10}}{(d+e x)^{15}}dx}{15 (b d-a e)}+\frac {(a+b x)^{11}}{15 (d+e x)^{15} (b d-a e)}\right )}{16 (b d-a e)}+\frac {(a+b x)^{11}}{16 (d+e x)^{16} (b d-a e)}\right )}{17 (b d-a e)}+\frac {(a+b x)^{11}}{17 (d+e x)^{17} (b d-a e)}\right )}{18 e (b d-a e)}-\frac {(a+b x)^{11} (B d-A e)}{18 e (d+e x)^{18} (b d-a e)}\)

\(\Big \downarrow \) 55

\(\displaystyle \frac {(-18 a B e+7 A b e+11 b B d) \left (\frac {6 b \left (\frac {5 b \left (\frac {4 b \left (\frac {3 b \int \frac {(a+b x)^{10}}{(d+e x)^{14}}dx}{14 (b d-a e)}+\frac {(a+b x)^{11}}{14 (d+e x)^{14} (b d-a e)}\right )}{15 (b d-a e)}+\frac {(a+b x)^{11}}{15 (d+e x)^{15} (b d-a e)}\right )}{16 (b d-a e)}+\frac {(a+b x)^{11}}{16 (d+e x)^{16} (b d-a e)}\right )}{17 (b d-a e)}+\frac {(a+b x)^{11}}{17 (d+e x)^{17} (b d-a e)}\right )}{18 e (b d-a e)}-\frac {(a+b x)^{11} (B d-A e)}{18 e (d+e x)^{18} (b d-a e)}\)

\(\Big \downarrow \) 55

\(\displaystyle \frac {(-18 a B e+7 A b e+11 b B d) \left (\frac {6 b \left (\frac {5 b \left (\frac {4 b \left (\frac {3 b \left (\frac {2 b \int \frac {(a+b x)^{10}}{(d+e x)^{13}}dx}{13 (b d-a e)}+\frac {(a+b x)^{11}}{13 (d+e x)^{13} (b d-a e)}\right )}{14 (b d-a e)}+\frac {(a+b x)^{11}}{14 (d+e x)^{14} (b d-a e)}\right )}{15 (b d-a e)}+\frac {(a+b x)^{11}}{15 (d+e x)^{15} (b d-a e)}\right )}{16 (b d-a e)}+\frac {(a+b x)^{11}}{16 (d+e x)^{16} (b d-a e)}\right )}{17 (b d-a e)}+\frac {(a+b x)^{11}}{17 (d+e x)^{17} (b d-a e)}\right )}{18 e (b d-a e)}-\frac {(a+b x)^{11} (B d-A e)}{18 e (d+e x)^{18} (b d-a e)}\)

\(\Big \downarrow \) 55

\(\displaystyle \frac {(-18 a B e+7 A b e+11 b B d) \left (\frac {6 b \left (\frac {5 b \left (\frac {4 b \left (\frac {3 b \left (\frac {2 b \left (\frac {b \int \frac {(a+b x)^{10}}{(d+e x)^{12}}dx}{12 (b d-a e)}+\frac {(a+b x)^{11}}{12 (d+e x)^{12} (b d-a e)}\right )}{13 (b d-a e)}+\frac {(a+b x)^{11}}{13 (d+e x)^{13} (b d-a e)}\right )}{14 (b d-a e)}+\frac {(a+b x)^{11}}{14 (d+e x)^{14} (b d-a e)}\right )}{15 (b d-a e)}+\frac {(a+b x)^{11}}{15 (d+e x)^{15} (b d-a e)}\right )}{16 (b d-a e)}+\frac {(a+b x)^{11}}{16 (d+e x)^{16} (b d-a e)}\right )}{17 (b d-a e)}+\frac {(a+b x)^{11}}{17 (d+e x)^{17} (b d-a e)}\right )}{18 e (b d-a e)}-\frac {(a+b x)^{11} (B d-A e)}{18 e (d+e x)^{18} (b d-a e)}\)

\(\Big \downarrow \) 48

\(\displaystyle \frac {\left (\frac {(a+b x)^{11}}{17 (d+e x)^{17} (b d-a e)}+\frac {6 b \left (\frac {(a+b x)^{11}}{16 (d+e x)^{16} (b d-a e)}+\frac {5 b \left (\frac {(a+b x)^{11}}{15 (d+e x)^{15} (b d-a e)}+\frac {4 b \left (\frac {(a+b x)^{11}}{14 (d+e x)^{14} (b d-a e)}+\frac {3 b \left (\frac {(a+b x)^{11}}{13 (d+e x)^{13} (b d-a e)}+\frac {2 b \left (\frac {b (a+b x)^{11}}{132 (d+e x)^{11} (b d-a e)^2}+\frac {(a+b x)^{11}}{12 (d+e x)^{12} (b d-a e)}\right )}{13 (b d-a e)}\right )}{14 (b d-a e)}\right )}{15 (b d-a e)}\right )}{16 (b d-a e)}\right )}{17 (b d-a e)}\right ) (-18 a B e+7 A b e+11 b B d)}{18 e (b d-a e)}-\frac {(a+b x)^{11} (B d-A e)}{18 e (d+e x)^{18} (b d-a e)}\)

Input:

Int[((a + b*x)^10*(A + B*x))/(d + e*x)^19,x]
 

Output:

-1/18*((B*d - A*e)*(a + b*x)^11)/(e*(b*d - a*e)*(d + e*x)^18) + ((11*b*B*d 
 + 7*A*b*e - 18*a*B*e)*((a + b*x)^11/(17*(b*d - a*e)*(d + e*x)^17) + (6*b* 
((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*(b*d - a*e))))/(18*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. \(2(369)=738\).

Time = 0.34 (sec) , antiderivative size = 1901, normalized size of antiderivative = 4.94

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)^19,x,method=_RETURNVERBOSE)
 

Output:

(-1/2450448/e^12*(136136*A*a^10*e^11+80080*A*a^9*b*d*e^10+45045*A*a^8*b^2* 
d^2*e^9+24024*A*a^7*b^3*d^3*e^8+12012*A*a^6*b^4*d^4*e^7+5544*A*a^5*b^5*d^5 
*e^6+2310*A*a^4*b^6*d^6*e^5+840*A*a^3*b^7*d^7*e^4+252*A*a^2*b^8*d^8*e^3+56 
*A*a*b^9*d^9*e^2+7*A*b^10*d^10*e+8008*B*a^10*d*e^10+10010*B*a^9*b*d^2*e^9+ 
9009*B*a^8*b^2*d^3*e^8+6864*B*a^7*b^3*d^4*e^7+4620*B*a^6*b^4*d^5*e^6+2772* 
B*a^5*b^5*d^6*e^5+1470*B*a^4*b^6*d^7*e^4+672*B*a^3*b^7*d^8*e^3+252*B*a^2*b 
^8*d^9*e^2+70*B*a*b^9*d^10*e+11*B*b^10*d^11)-1/136136/e^11*(80080*A*a^9*b* 
e^10+45045*A*a^8*b^2*d*e^9+24024*A*a^7*b^3*d^2*e^8+12012*A*a^6*b^4*d^3*e^7 
+5544*A*a^5*b^5*d^4*e^6+2310*A*a^4*b^6*d^5*e^5+840*A*a^3*b^7*d^6*e^4+252*A 
*a^2*b^8*d^7*e^3+56*A*a*b^9*d^8*e^2+7*A*b^10*d^9*e+8008*B*a^10*e^10+10010* 
B*a^9*b*d*e^9+9009*B*a^8*b^2*d^2*e^8+6864*B*a^7*b^3*d^3*e^7+4620*B*a^6*b^4 
*d^4*e^6+2772*B*a^5*b^5*d^5*e^5+1470*B*a^4*b^6*d^6*e^4+672*B*a^3*b^7*d^7*e 
^3+252*B*a^2*b^8*d^8*e^2+70*B*a*b^9*d^9*e+11*B*b^10*d^10)*x-1/16016*b/e^10 
*(45045*A*a^8*b*e^9+24024*A*a^7*b^2*d*e^8+12012*A*a^6*b^3*d^2*e^7+5544*A*a 
^5*b^4*d^3*e^6+2310*A*a^4*b^5*d^4*e^5+840*A*a^3*b^6*d^5*e^4+252*A*a^2*b^7* 
d^6*e^3+56*A*a*b^8*d^7*e^2+7*A*b^9*d^8*e+10010*B*a^9*e^9+9009*B*a^8*b*d*e^ 
8+6864*B*a^7*b^2*d^2*e^7+4620*B*a^6*b^3*d^3*e^6+2772*B*a^5*b^4*d^4*e^5+147 
0*B*a^4*b^5*d^5*e^4+672*B*a^3*b^6*d^6*e^3+252*B*a^2*b^7*d^7*e^2+70*B*a*b^8 
*d^8*e+11*B*b^9*d^9)*x^2-1/3003*b^2/e^9*(24024*A*a^7*b*e^8+12012*A*a^6*b^2 
*d*e^7+5544*A*a^5*b^3*d^2*e^6+2310*A*a^4*b^4*d^3*e^5+840*A*a^3*b^5*d^4*...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2006 vs. \(2 (369) = 738\).

Time = 0.20 (sec) , antiderivative size = 2006, normalized size of antiderivative = 5.21 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{19}} \, dx=\text {Too large to display} \] Input:

integrate((b*x+a)^10*(B*x+A)/(e*x+d)^19,x, algorithm="fricas")
 

Output:

-1/2450448*(350064*B*b^10*e^11*x^11 + 11*B*b^10*d^11 + 136136*A*a^10*e^11 
+ 7*(10*B*a*b^9 + A*b^10)*d^10*e + 28*(9*B*a^2*b^8 + 2*A*a*b^9)*d^9*e^2 + 
84*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^8*e^3 + 210*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d 
^7*e^4 + 462*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^6*e^5 + 924*(5*B*a^6*b^4 + 6*A* 
a^5*b^5)*d^5*e^6 + 1716*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^4*e^7 + 3003*(3*B*a^ 
8*b^2 + 8*A*a^7*b^3)*d^3*e^8 + 5005*(2*B*a^9*b + 9*A*a^8*b^2)*d^2*e^9 + 80 
08*(B*a^10 + 10*A*a^9*b)*d*e^10 + 43758*(11*B*b^10*d*e^10 + 7*(10*B*a*b^9 
+ A*b^10)*e^11)*x^10 + 48620*(11*B*b^10*d^2*e^9 + 7*(10*B*a*b^9 + A*b^10)* 
d*e^10 + 28*(9*B*a^2*b^8 + 2*A*a*b^9)*e^11)*x^9 + 43758*(11*B*b^10*d^3*e^8 
 + 7*(10*B*a*b^9 + A*b^10)*d^2*e^9 + 28*(9*B*a^2*b^8 + 2*A*a*b^9)*d*e^10 + 
 84*(8*B*a^3*b^7 + 3*A*a^2*b^8)*e^11)*x^8 + 31824*(11*B*b^10*d^4*e^7 + 7*( 
10*B*a*b^9 + A*b^10)*d^3*e^8 + 28*(9*B*a^2*b^8 + 2*A*a*b^9)*d^2*e^9 + 84*( 
8*B*a^3*b^7 + 3*A*a^2*b^8)*d*e^10 + 210*(7*B*a^4*b^6 + 4*A*a^3*b^7)*e^11)* 
x^7 + 18564*(11*B*b^10*d^5*e^6 + 7*(10*B*a*b^9 + A*b^10)*d^4*e^7 + 28*(9*B 
*a^2*b^8 + 2*A*a*b^9)*d^3*e^8 + 84*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^2*e^9 + 2 
10*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d*e^10 + 462*(6*B*a^5*b^5 + 5*A*a^4*b^6)*e^ 
11)*x^6 + 8568*(11*B*b^10*d^6*e^5 + 7*(10*B*a*b^9 + A*b^10)*d^5*e^6 + 28*( 
9*B*a^2*b^8 + 2*A*a*b^9)*d^4*e^7 + 84*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^3*e^8 
+ 210*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^2*e^9 + 462*(6*B*a^5*b^5 + 5*A*a^4*b^6 
)*d*e^10 + 924*(5*B*a^6*b^4 + 6*A*a^5*b^5)*e^11)*x^5 + 3060*(11*B*b^10*...
                                                                                    
                                                                                    
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{19}} \, dx=\text {Timed out} \] Input:

integrate((b*x+a)**10*(B*x+A)/(e*x+d)**19,x)
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2006 vs. \(2 (369) = 738\).

Time = 0.13 (sec) , antiderivative size = 2006, normalized size of antiderivative = 5.21 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{19}} \, dx=\text {Too large to display} \] Input:

integrate((b*x+a)^10*(B*x+A)/(e*x+d)^19,x, algorithm="maxima")
 

Output:

-1/2450448*(350064*B*b^10*e^11*x^11 + 11*B*b^10*d^11 + 136136*A*a^10*e^11 
+ 7*(10*B*a*b^9 + A*b^10)*d^10*e + 28*(9*B*a^2*b^8 + 2*A*a*b^9)*d^9*e^2 + 
84*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^8*e^3 + 210*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d 
^7*e^4 + 462*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^6*e^5 + 924*(5*B*a^6*b^4 + 6*A* 
a^5*b^5)*d^5*e^6 + 1716*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^4*e^7 + 3003*(3*B*a^ 
8*b^2 + 8*A*a^7*b^3)*d^3*e^8 + 5005*(2*B*a^9*b + 9*A*a^8*b^2)*d^2*e^9 + 80 
08*(B*a^10 + 10*A*a^9*b)*d*e^10 + 43758*(11*B*b^10*d*e^10 + 7*(10*B*a*b^9 
+ A*b^10)*e^11)*x^10 + 48620*(11*B*b^10*d^2*e^9 + 7*(10*B*a*b^9 + A*b^10)* 
d*e^10 + 28*(9*B*a^2*b^8 + 2*A*a*b^9)*e^11)*x^9 + 43758*(11*B*b^10*d^3*e^8 
 + 7*(10*B*a*b^9 + A*b^10)*d^2*e^9 + 28*(9*B*a^2*b^8 + 2*A*a*b^9)*d*e^10 + 
 84*(8*B*a^3*b^7 + 3*A*a^2*b^8)*e^11)*x^8 + 31824*(11*B*b^10*d^4*e^7 + 7*( 
10*B*a*b^9 + A*b^10)*d^3*e^8 + 28*(9*B*a^2*b^8 + 2*A*a*b^9)*d^2*e^9 + 84*( 
8*B*a^3*b^7 + 3*A*a^2*b^8)*d*e^10 + 210*(7*B*a^4*b^6 + 4*A*a^3*b^7)*e^11)* 
x^7 + 18564*(11*B*b^10*d^5*e^6 + 7*(10*B*a*b^9 + A*b^10)*d^4*e^7 + 28*(9*B 
*a^2*b^8 + 2*A*a*b^9)*d^3*e^8 + 84*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^2*e^9 + 2 
10*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d*e^10 + 462*(6*B*a^5*b^5 + 5*A*a^4*b^6)*e^ 
11)*x^6 + 8568*(11*B*b^10*d^6*e^5 + 7*(10*B*a*b^9 + A*b^10)*d^5*e^6 + 28*( 
9*B*a^2*b^8 + 2*A*a*b^9)*d^4*e^7 + 84*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^3*e^8 
+ 210*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^2*e^9 + 462*(6*B*a^5*b^5 + 5*A*a^4*b^6 
)*d*e^10 + 924*(5*B*a^6*b^4 + 6*A*a^5*b^5)*e^11)*x^5 + 3060*(11*B*b^10*...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2232 vs. \(2 (369) = 738\).

Time = 0.13 (sec) , antiderivative size = 2232, normalized size of antiderivative = 5.80 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{19}} \, dx=\text {Too large to display} \] Input:

integrate((b*x+a)^10*(B*x+A)/(e*x+d)^19,x, algorithm="giac")
 

Output:

-1/2450448*(350064*B*b^10*e^11*x^11 + 481338*B*b^10*d*e^10*x^10 + 3063060* 
B*a*b^9*e^11*x^10 + 306306*A*b^10*e^11*x^10 + 534820*B*b^10*d^2*e^9*x^9 + 
3403400*B*a*b^9*d*e^10*x^9 + 340340*A*b^10*d*e^10*x^9 + 12252240*B*a^2*b^8 
*e^11*x^9 + 2722720*A*a*b^9*e^11*x^9 + 481338*B*b^10*d^3*e^8*x^8 + 3063060 
*B*a*b^9*d^2*e^9*x^8 + 306306*A*b^10*d^2*e^9*x^8 + 11027016*B*a^2*b^8*d*e^ 
10*x^8 + 2450448*A*a*b^9*d*e^10*x^8 + 29405376*B*a^3*b^7*e^11*x^8 + 110270 
16*A*a^2*b^8*e^11*x^8 + 350064*B*b^10*d^4*e^7*x^7 + 2227680*B*a*b^9*d^3*e^ 
8*x^7 + 222768*A*b^10*d^3*e^8*x^7 + 8019648*B*a^2*b^8*d^2*e^9*x^7 + 178214 
4*A*a*b^9*d^2*e^9*x^7 + 21385728*B*a^3*b^7*d*e^10*x^7 + 8019648*A*a^2*b^8* 
d*e^10*x^7 + 46781280*B*a^4*b^6*e^11*x^7 + 26732160*A*a^3*b^7*e^11*x^7 + 2 
04204*B*b^10*d^5*e^6*x^6 + 1299480*B*a*b^9*d^4*e^7*x^6 + 129948*A*b^10*d^4 
*e^7*x^6 + 4678128*B*a^2*b^8*d^3*e^8*x^6 + 1039584*A*a*b^9*d^3*e^8*x^6 + 1 
2475008*B*a^3*b^7*d^2*e^9*x^6 + 4678128*A*a^2*b^8*d^2*e^9*x^6 + 27289080*B 
*a^4*b^6*d*e^10*x^6 + 15593760*A*a^3*b^7*d*e^10*x^6 + 51459408*B*a^5*b^5*e 
^11*x^6 + 42882840*A*a^4*b^6*e^11*x^6 + 94248*B*b^10*d^6*e^5*x^5 + 599760* 
B*a*b^9*d^5*e^6*x^5 + 59976*A*b^10*d^5*e^6*x^5 + 2159136*B*a^2*b^8*d^4*e^7 
*x^5 + 479808*A*a*b^9*d^4*e^7*x^5 + 5757696*B*a^3*b^7*d^3*e^8*x^5 + 215913 
6*A*a^2*b^8*d^3*e^8*x^5 + 12594960*B*a^4*b^6*d^2*e^9*x^5 + 7197120*A*a^3*b 
^7*d^2*e^9*x^5 + 23750496*B*a^5*b^5*d*e^10*x^5 + 19792080*A*a^4*b^6*d*e^10 
*x^5 + 39584160*B*a^6*b^4*e^11*x^5 + 47500992*A*a^5*b^5*e^11*x^5 + 3366...
 

Mupad [B] (verification not implemented)

Time = 9.64 (sec) , antiderivative size = 2088, normalized size of antiderivative = 5.42 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{19}} \, dx=\text {Too large to display} \] Input:

int(((A + B*x)*(a + b*x)^10)/(d + e*x)^19,x)
 

Output:

-((136136*A*a^10*e^11 + 11*B*b^10*d^11 + 7*A*b^10*d^10*e + 8008*B*a^10*d*e 
^10 + 56*A*a*b^9*d^9*e^2 + 10010*B*a^9*b*d^2*e^9 + 252*A*a^2*b^8*d^8*e^3 + 
 840*A*a^3*b^7*d^7*e^4 + 2310*A*a^4*b^6*d^6*e^5 + 5544*A*a^5*b^5*d^5*e^6 + 
 12012*A*a^6*b^4*d^4*e^7 + 24024*A*a^7*b^3*d^3*e^8 + 45045*A*a^8*b^2*d^2*e 
^9 + 252*B*a^2*b^8*d^9*e^2 + 672*B*a^3*b^7*d^8*e^3 + 1470*B*a^4*b^6*d^7*e^ 
4 + 2772*B*a^5*b^5*d^6*e^5 + 4620*B*a^6*b^4*d^5*e^6 + 6864*B*a^7*b^3*d^4*e 
^7 + 9009*B*a^8*b^2*d^3*e^8 + 80080*A*a^9*b*d*e^10 + 70*B*a*b^9*d^10*e)/(2 
450448*e^12) + (x*(8008*B*a^10*e^10 + 11*B*b^10*d^10 + 80080*A*a^9*b*e^10 
+ 7*A*b^10*d^9*e + 56*A*a*b^9*d^8*e^2 + 45045*A*a^8*b^2*d*e^9 + 252*A*a^2* 
b^8*d^7*e^3 + 840*A*a^3*b^7*d^6*e^4 + 2310*A*a^4*b^6*d^5*e^5 + 5544*A*a^5* 
b^5*d^4*e^6 + 12012*A*a^6*b^4*d^3*e^7 + 24024*A*a^7*b^3*d^2*e^8 + 252*B*a^ 
2*b^8*d^8*e^2 + 672*B*a^3*b^7*d^7*e^3 + 1470*B*a^4*b^6*d^6*e^4 + 2772*B*a^ 
5*b^5*d^5*e^5 + 4620*B*a^6*b^4*d^4*e^6 + 6864*B*a^7*b^3*d^3*e^7 + 9009*B*a 
^8*b^2*d^2*e^8 + 70*B*a*b^9*d^9*e + 10010*B*a^9*b*d*e^9))/(136136*e^11) + 
(b^7*x^8*(672*B*a^3*e^3 + 11*B*b^3*d^3 + 252*A*a^2*b*e^3 + 7*A*b^3*d^2*e + 
 56*A*a*b^2*d*e^2 + 70*B*a*b^2*d^2*e + 252*B*a^2*b*d*e^2))/(56*e^4) + (b^4 
*x^5*(4620*B*a^6*e^6 + 11*B*b^6*d^6 + 5544*A*a^5*b*e^6 + 7*A*b^6*d^5*e + 5 
6*A*a*b^5*d^4*e^2 + 2310*A*a^4*b^2*d*e^5 + 252*A*a^2*b^4*d^3*e^3 + 840*A*a 
^3*b^3*d^2*e^4 + 252*B*a^2*b^4*d^4*e^2 + 672*B*a^3*b^3*d^3*e^3 + 1470*B*a^ 
4*b^2*d^2*e^4 + 70*B*a*b^5*d^5*e + 2772*B*a^5*b*d*e^5))/(286*e^7) + (b^...
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 1338, normalized size of antiderivative = 3.48 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{19}} \, dx =\text {Too large to display} \] Input:

int((b*x+a)^10*(B*x+A)/(e*x+d)^19,x)
 

Output:

( - 12376*a**11*e**11 - 8008*a**10*b*d*e**10 - 144144*a**10*b*e**11*x - 50 
05*a**9*b**2*d**2*e**9 - 90090*a**9*b**2*d*e**10*x - 765765*a**9*b**2*e**1 
1*x**2 - 3003*a**8*b**3*d**3*e**8 - 54054*a**8*b**3*d**2*e**9*x - 459459*a 
**8*b**3*d*e**10*x**2 - 2450448*a**8*b**3*e**11*x**3 - 1716*a**7*b**4*d**4 
*e**7 - 30888*a**7*b**4*d**3*e**8*x - 262548*a**7*b**4*d**2*e**9*x**2 - 14 
00256*a**7*b**4*d*e**10*x**3 - 5250960*a**7*b**4*e**11*x**4 - 924*a**6*b** 
5*d**5*e**6 - 16632*a**6*b**5*d**4*e**7*x - 141372*a**6*b**5*d**3*e**8*x** 
2 - 753984*a**6*b**5*d**2*e**9*x**3 - 2827440*a**6*b**5*d*e**10*x**4 - 791 
6832*a**6*b**5*e**11*x**5 - 462*a**5*b**6*d**6*e**5 - 8316*a**5*b**6*d**5* 
e**6*x - 70686*a**5*b**6*d**4*e**7*x**2 - 376992*a**5*b**6*d**3*e**8*x**3 
- 1413720*a**5*b**6*d**2*e**9*x**4 - 3958416*a**5*b**6*d*e**10*x**5 - 8576 
568*a**5*b**6*e**11*x**6 - 210*a**4*b**7*d**7*e**4 - 3780*a**4*b**7*d**6*e 
**5*x - 32130*a**4*b**7*d**5*e**6*x**2 - 171360*a**4*b**7*d**4*e**7*x**3 - 
 642600*a**4*b**7*d**3*e**8*x**4 - 1799280*a**4*b**7*d**2*e**9*x**5 - 3898 
440*a**4*b**7*d*e**10*x**6 - 6683040*a**4*b**7*e**11*x**7 - 84*a**3*b**8*d 
**8*e**3 - 1512*a**3*b**8*d**7*e**4*x - 12852*a**3*b**8*d**6*e**5*x**2 - 6 
8544*a**3*b**8*d**5*e**6*x**3 - 257040*a**3*b**8*d**4*e**7*x**4 - 719712*a 
**3*b**8*d**3*e**8*x**5 - 1559376*a**3*b**8*d**2*e**9*x**6 - 2673216*a**3* 
b**8*d*e**10*x**7 - 3675672*a**3*b**8*e**11*x**8 - 28*a**2*b**9*d**9*e**2 
- 504*a**2*b**9*d**8*e**3*x - 4284*a**2*b**9*d**7*e**4*x**2 - 22848*a**...