\(\int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{15}} \, dx\) [1103]
Optimal result
Integrand size = 20, antiderivative size = 185 \[
\int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{15}} \, dx=-\frac {(B d-A e) (a+b x)^{11}}{14 e (b d-a e) (d+e x)^{14}}+\frac {(11 b B d+3 A b e-14 a B e) (a+b x)^{11}}{182 e (b d-a e)^2 (d+e x)^{13}}+\frac {b (11 b B d+3 A b e-14 a B e) (a+b x)^{11}}{1092 e (b d-a e)^3 (d+e x)^{12}}+\frac {b^2 (11 b B d+3 A b e-14 a B e) (a+b x)^{11}}{12012 e (b d-a e)^4 (d+e x)^{11}}
\]
[Out]
-1/14*(-A*e+B*d)*(b*x+a)^11/e/(-a*e+b*d)/(e*x+d)^14+1/182*(3*A*b*e-14*B*a*e+11*B*b*d)*(b*x+a)^11/e/(-a*e+b*d)^
2/(e*x+d)^13+1/1092*b*(3*A*b*e-14*B*a*e+11*B*b*d)*(b*x+a)^11/e/(-a*e+b*d)^3/(e*x+d)^12+1/12012*b^2*(3*A*b*e-14
*B*a*e+11*B*b*d)*(b*x+a)^11/e/(-a*e+b*d)^4/(e*x+d)^11
Rubi [A] (verified)
Time = 0.05 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.00, number
of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {79, 47, 37}
\[
\int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{15}} \, dx=\frac {b^2 (a+b x)^{11} (-14 a B e+3 A b e+11 b B d)}{12012 e (d+e x)^{11} (b d-a e)^4}+\frac {b (a+b x)^{11} (-14 a B e+3 A b e+11 b B d)}{1092 e (d+e x)^{12} (b d-a e)^3}+\frac {(a+b x)^{11} (-14 a B e+3 A b e+11 b B d)}{182 e (d+e x)^{13} (b d-a e)^2}-\frac {(a+b x)^{11} (B d-A e)}{14 e (d+e x)^{14} (b d-a e)}
\]
[In]
Int[((a + b*x)^10*(A + B*x))/(d + e*x)^15,x]
[Out]
-1/14*((B*d - A*e)*(a + b*x)^11)/(e*(b*d - a*e)*(d + e*x)^14) + ((11*b*B*d + 3*A*b*e - 14*a*B*e)*(a + b*x)^11)
/(182*e*(b*d - a*e)^2*(d + e*x)^13) + (b*(11*b*B*d + 3*A*b*e - 14*a*B*e)*(a + b*x)^11)/(1092*e*(b*d - a*e)^3*(
d + e*x)^12) + (b^2*(11*b*B*d + 3*A*b*e - 14*a*B*e)*(a + b*x)^11)/(12012*e*(b*d - a*e)^4*(d + e*x)^11)
Rule 37
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] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]
Rule 47
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] - Dist[d*(Simplify[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] && NeQ[b*c - a*d, 0] && 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] || !SumSimplerQ[n, 1])
Rule 79
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(-(b*e - a*f
))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Dist[(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] || IntegerQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || L
tQ[p, n]))))
Rubi steps \begin{align*}
\text {integral}& = -\frac {(B d-A e) (a+b x)^{11}}{14 e (b d-a e) (d+e x)^{14}}+\frac {(11 b B d+3 A b e-14 a B e) \int \frac {(a+b x)^{10}}{(d+e x)^{14}} \, dx}{14 e (b d-a e)} \\ & = -\frac {(B d-A e) (a+b x)^{11}}{14 e (b d-a e) (d+e x)^{14}}+\frac {(11 b B d+3 A b e-14 a B e) (a+b x)^{11}}{182 e (b d-a e)^2 (d+e x)^{13}}+\frac {(b (11 b B d+3 A b e-14 a B e)) \int \frac {(a+b x)^{10}}{(d+e x)^{13}} \, dx}{91 e (b d-a e)^2} \\ & = -\frac {(B d-A e) (a+b x)^{11}}{14 e (b d-a e) (d+e x)^{14}}+\frac {(11 b B d+3 A b e-14 a B e) (a+b x)^{11}}{182 e (b d-a e)^2 (d+e x)^{13}}+\frac {b (11 b B d+3 A b e-14 a B e) (a+b x)^{11}}{1092 e (b d-a e)^3 (d+e x)^{12}}+\frac {\left (b^2 (11 b B d+3 A b e-14 a B e)\right ) \int \frac {(a+b x)^{10}}{(d+e x)^{12}} \, dx}{1092 e (b d-a e)^3} \\ & = -\frac {(B d-A e) (a+b x)^{11}}{14 e (b d-a e) (d+e x)^{14}}+\frac {(11 b B d+3 A b e-14 a B e) (a+b x)^{11}}{182 e (b d-a e)^2 (d+e x)^{13}}+\frac {b (11 b B d+3 A b e-14 a B e) (a+b x)^{11}}{1092 e (b d-a e)^3 (d+e x)^{12}}+\frac {b^2 (11 b B d+3 A b e-14 a B e) (a+b x)^{11}}{12012 e (b d-a e)^4 (d+e x)^{11}} \\
\end{align*}
Mathematica [B] (verified)
Leaf count is larger than twice the leaf count of optimal. \(1430\) vs. \(2(185)=370\).
Time = 0.54 (sec) , antiderivative size = 1430, normalized size of antiderivative = 7.73
\[
\int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{15}} \, dx=-\frac {66 a^{10} e^{10} (13 A e+B (d+14 e x))+110 a^9 b e^9 \left (6 A e (d+14 e x)+B \left (d^2+14 d e x+91 e^2 x^2\right )\right )+45 a^8 b^2 e^8 \left (11 A e \left (d^2+14 d e x+91 e^2 x^2\right )+3 B \left (d^3+14 d^2 e x+91 d e^2 x^2+364 e^3 x^3\right )\right )+72 a^7 b^3 e^7 \left (5 A e \left (d^3+14 d^2 e x+91 d e^2 x^2+364 e^3 x^3\right )+2 B \left (d^4+14 d^3 e x+91 d^2 e^2 x^2+364 d e^3 x^3+1001 e^4 x^4\right )\right )+28 a^6 b^4 e^6 \left (9 A e \left (d^4+14 d^3 e x+91 d^2 e^2 x^2+364 d e^3 x^3+1001 e^4 x^4\right )+5 B \left (d^5+14 d^4 e x+91 d^3 e^2 x^2+364 d^2 e^3 x^3+1001 d e^4 x^4+2002 e^5 x^5\right )\right )+42 a^5 b^5 e^5 \left (4 A e \left (d^5+14 d^4 e x+91 d^3 e^2 x^2+364 d^2 e^3 x^3+1001 d e^4 x^4+2002 e^5 x^5\right )+3 B \left (d^6+14 d^5 e x+91 d^4 e^2 x^2+364 d^3 e^3 x^3+1001 d^2 e^4 x^4+2002 d e^5 x^5+3003 e^6 x^6\right )\right )+105 a^4 b^6 e^4 \left (A e \left (d^6+14 d^5 e x+91 d^4 e^2 x^2+364 d^3 e^3 x^3+1001 d^2 e^4 x^4+2002 d e^5 x^5+3003 e^6 x^6\right )+B \left (d^7+14 d^6 e x+91 d^5 e^2 x^2+364 d^4 e^3 x^3+1001 d^3 e^4 x^4+2002 d^2 e^5 x^5+3003 d e^6 x^6+3432 e^7 x^7\right )\right )+20 a^3 b^7 e^3 \left (3 A e \left (d^7+14 d^6 e x+91 d^5 e^2 x^2+364 d^4 e^3 x^3+1001 d^3 e^4 x^4+2002 d^2 e^5 x^5+3003 d e^6 x^6+3432 e^7 x^7\right )+4 B \left (d^8+14 d^7 e x+91 d^6 e^2 x^2+364 d^5 e^3 x^3+1001 d^4 e^4 x^4+2002 d^3 e^5 x^5+3003 d^2 e^6 x^6+3432 d e^7 x^7+3003 e^8 x^8\right )\right )+6 a^2 b^8 e^2 \left (5 A e \left (d^8+14 d^7 e x+91 d^6 e^2 x^2+364 d^5 e^3 x^3+1001 d^4 e^4 x^4+2002 d^3 e^5 x^5+3003 d^2 e^6 x^6+3432 d e^7 x^7+3003 e^8 x^8\right )+9 B \left (d^9+14 d^8 e x+91 d^7 e^2 x^2+364 d^6 e^3 x^3+1001 d^5 e^4 x^4+2002 d^4 e^5 x^5+3003 d^3 e^6 x^6+3432 d^2 e^7 x^7+3003 d e^8 x^8+2002 e^9 x^9\right )\right )+6 a b^9 e \left (2 A e \left (d^9+14 d^8 e x+91 d^7 e^2 x^2+364 d^6 e^3 x^3+1001 d^5 e^4 x^4+2002 d^4 e^5 x^5+3003 d^3 e^6 x^6+3432 d^2 e^7 x^7+3003 d e^8 x^8+2002 e^9 x^9\right )+5 B \left (d^{10}+14 d^9 e x+91 d^8 e^2 x^2+364 d^7 e^3 x^3+1001 d^6 e^4 x^4+2002 d^5 e^5 x^5+3003 d^4 e^6 x^6+3432 d^3 e^7 x^7+3003 d^2 e^8 x^8+2002 d e^9 x^9+1001 e^{10} x^{10}\right )\right )+b^{10} \left (3 A e \left (d^{10}+14 d^9 e x+91 d^8 e^2 x^2+364 d^7 e^3 x^3+1001 d^6 e^4 x^4+2002 d^5 e^5 x^5+3003 d^4 e^6 x^6+3432 d^3 e^7 x^7+3003 d^2 e^8 x^8+2002 d e^9 x^9+1001 e^{10} x^{10}\right )+11 B \left (d^{11}+14 d^{10} e x+91 d^9 e^2 x^2+364 d^8 e^3 x^3+1001 d^7 e^4 x^4+2002 d^6 e^5 x^5+3003 d^5 e^6 x^6+3432 d^4 e^7 x^7+3003 d^3 e^8 x^8+2002 d^2 e^9 x^9+1001 d e^{10} x^{10}+364 e^{11} x^{11}\right )\right )}{12012 e^{12} (d+e x)^{14}}
\]
[In]
Integrate[((a + b*x)^10*(A + B*x))/(d + e*x)^15,x]
[Out]
-1/12012*(66*a^10*e^10*(13*A*e + B*(d + 14*e*x)) + 110*a^9*b*e^9*(6*A*e*(d + 14*e*x) + B*(d^2 + 14*d*e*x + 91*
e^2*x^2)) + 45*a^8*b^2*e^8*(11*A*e*(d^2 + 14*d*e*x + 91*e^2*x^2) + 3*B*(d^3 + 14*d^2*e*x + 91*d*e^2*x^2 + 364*
e^3*x^3)) + 72*a^7*b^3*e^7*(5*A*e*(d^3 + 14*d^2*e*x + 91*d*e^2*x^2 + 364*e^3*x^3) + 2*B*(d^4 + 14*d^3*e*x + 91
*d^2*e^2*x^2 + 364*d*e^3*x^3 + 1001*e^4*x^4)) + 28*a^6*b^4*e^6*(9*A*e*(d^4 + 14*d^3*e*x + 91*d^2*e^2*x^2 + 364
*d*e^3*x^3 + 1001*e^4*x^4) + 5*B*(d^5 + 14*d^4*e*x + 91*d^3*e^2*x^2 + 364*d^2*e^3*x^3 + 1001*d*e^4*x^4 + 2002*
e^5*x^5)) + 42*a^5*b^5*e^5*(4*A*e*(d^5 + 14*d^4*e*x + 91*d^3*e^2*x^2 + 364*d^2*e^3*x^3 + 1001*d*e^4*x^4 + 2002
*e^5*x^5) + 3*B*(d^6 + 14*d^5*e*x + 91*d^4*e^2*x^2 + 364*d^3*e^3*x^3 + 1001*d^2*e^4*x^4 + 2002*d*e^5*x^5 + 300
3*e^6*x^6)) + 105*a^4*b^6*e^4*(A*e*(d^6 + 14*d^5*e*x + 91*d^4*e^2*x^2 + 364*d^3*e^3*x^3 + 1001*d^2*e^4*x^4 + 2
002*d*e^5*x^5 + 3003*e^6*x^6) + B*(d^7 + 14*d^6*e*x + 91*d^5*e^2*x^2 + 364*d^4*e^3*x^3 + 1001*d^3*e^4*x^4 + 20
02*d^2*e^5*x^5 + 3003*d*e^6*x^6 + 3432*e^7*x^7)) + 20*a^3*b^7*e^3*(3*A*e*(d^7 + 14*d^6*e*x + 91*d^5*e^2*x^2 +
364*d^4*e^3*x^3 + 1001*d^3*e^4*x^4 + 2002*d^2*e^5*x^5 + 3003*d*e^6*x^6 + 3432*e^7*x^7) + 4*B*(d^8 + 14*d^7*e*x
+ 91*d^6*e^2*x^2 + 364*d^5*e^3*x^3 + 1001*d^4*e^4*x^4 + 2002*d^3*e^5*x^5 + 3003*d^2*e^6*x^6 + 3432*d*e^7*x^7
+ 3003*e^8*x^8)) + 6*a^2*b^8*e^2*(5*A*e*(d^8 + 14*d^7*e*x + 91*d^6*e^2*x^2 + 364*d^5*e^3*x^3 + 1001*d^4*e^4*x^
4 + 2002*d^3*e^5*x^5 + 3003*d^2*e^6*x^6 + 3432*d*e^7*x^7 + 3003*e^8*x^8) + 9*B*(d^9 + 14*d^8*e*x + 91*d^7*e^2*
x^2 + 364*d^6*e^3*x^3 + 1001*d^5*e^4*x^4 + 2002*d^4*e^5*x^5 + 3003*d^3*e^6*x^6 + 3432*d^2*e^7*x^7 + 3003*d*e^8
*x^8 + 2002*e^9*x^9)) + 6*a*b^9*e*(2*A*e*(d^9 + 14*d^8*e*x + 91*d^7*e^2*x^2 + 364*d^6*e^3*x^3 + 1001*d^5*e^4*x
^4 + 2002*d^4*e^5*x^5 + 3003*d^3*e^6*x^6 + 3432*d^2*e^7*x^7 + 3003*d*e^8*x^8 + 2002*e^9*x^9) + 5*B*(d^10 + 14*
d^9*e*x + 91*d^8*e^2*x^2 + 364*d^7*e^3*x^3 + 1001*d^6*e^4*x^4 + 2002*d^5*e^5*x^5 + 3003*d^4*e^6*x^6 + 3432*d^3
*e^7*x^7 + 3003*d^2*e^8*x^8 + 2002*d*e^9*x^9 + 1001*e^10*x^10)) + b^10*(3*A*e*(d^10 + 14*d^9*e*x + 91*d^8*e^2*
x^2 + 364*d^7*e^3*x^3 + 1001*d^6*e^4*x^4 + 2002*d^5*e^5*x^5 + 3003*d^4*e^6*x^6 + 3432*d^3*e^7*x^7 + 3003*d^2*e
^8*x^8 + 2002*d*e^9*x^9 + 1001*e^10*x^10) + 11*B*(d^11 + 14*d^10*e*x + 91*d^9*e^2*x^2 + 364*d^8*e^3*x^3 + 1001
*d^7*e^4*x^4 + 2002*d^6*e^5*x^5 + 3003*d^5*e^6*x^6 + 3432*d^4*e^7*x^7 + 3003*d^3*e^8*x^8 + 2002*d^2*e^9*x^9 +
1001*d*e^10*x^10 + 364*e^11*x^11)))/(e^12*(d + e*x)^14)
Maple [B] (verified)
Leaf count of result is larger than twice the leaf count of optimal. \(1900\) vs. \(2(177)=354\).
Time = 0.73 (sec) , antiderivative size = 1901, normalized size of antiderivative =
10.28
| | |
| 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 |
\(\text {Expression too large to display}\) |
\(2233\) |
| parallelrisch |
\(\text {Expression too large to display}\) |
\(2242\) |
| | |
|
|
|
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[In]
int((b*x+a)^10*(B*x+A)/(e*x+d)^15,x,method=_RETURNVERBOSE)
[Out]
(-1/3*b^10*B/e*x^11-1/12*b^9/e^2*(3*A*b*e+30*B*a*e+11*B*b*d)*x^10-1/6*b^8/e^3*(12*A*a*b*e^2+3*A*b^2*d*e+54*B*a
^2*e^2+30*B*a*b*d*e+11*B*b^2*d^2)*x^9-1/4*b^7/e^4*(30*A*a^2*b*e^3+12*A*a*b^2*d*e^2+3*A*b^3*d^2*e+80*B*a^3*e^3+
54*B*a^2*b*d*e^2+30*B*a*b^2*d^2*e+11*B*b^3*d^3)*x^8-2/7*b^6/e^5*(60*A*a^3*b*e^4+30*A*a^2*b^2*d*e^3+12*A*a*b^3*
d^2*e^2+3*A*b^4*d^3*e+105*B*a^4*e^4+80*B*a^3*b*d*e^3+54*B*a^2*b^2*d^2*e^2+30*B*a*b^3*d^3*e+11*B*b^4*d^4)*x^7-1
/4*b^5/e^6*(105*A*a^4*b*e^5+60*A*a^3*b^2*d*e^4+30*A*a^2*b^3*d^2*e^3+12*A*a*b^4*d^3*e^2+3*A*b^5*d^4*e+126*B*a^5
*e^5+105*B*a^4*b*d*e^4+80*B*a^3*b^2*d^2*e^3+54*B*a^2*b^3*d^3*e^2+30*B*a*b^4*d^4*e+11*B*b^5*d^5)*x^6-1/6*b^4/e^
7*(168*A*a^5*b*e^6+105*A*a^4*b^2*d*e^5+60*A*a^3*b^3*d^2*e^4+30*A*a^2*b^4*d^3*e^3+12*A*a*b^5*d^4*e^2+3*A*b^6*d^
5*e+140*B*a^6*e^6+126*B*a^5*b*d*e^5+105*B*a^4*b^2*d^2*e^4+80*B*a^3*b^3*d^3*e^3+54*B*a^2*b^4*d^4*e^2+30*B*a*b^5
*d^5*e+11*B*b^6*d^6)*x^5-1/12*b^3/e^8*(252*A*a^6*b*e^7+168*A*a^5*b^2*d*e^6+105*A*a^4*b^3*d^2*e^5+60*A*a^3*b^4*
d^3*e^4+30*A*a^2*b^5*d^4*e^3+12*A*a*b^6*d^5*e^2+3*A*b^7*d^6*e+144*B*a^7*e^7+140*B*a^6*b*d*e^6+126*B*a^5*b^2*d^
2*e^5+105*B*a^4*b^3*d^3*e^4+80*B*a^3*b^4*d^4*e^3+54*B*a^2*b^5*d^5*e^2+30*B*a*b^6*d^6*e+11*B*b^7*d^7)*x^4-1/33*
b^2/e^9*(360*A*a^7*b*e^8+252*A*a^6*b^2*d*e^7+168*A*a^5*b^3*d^2*e^6+105*A*a^4*b^4*d^3*e^5+60*A*a^3*b^5*d^4*e^4+
30*A*a^2*b^6*d^5*e^3+12*A*a*b^7*d^6*e^2+3*A*b^8*d^7*e+135*B*a^8*e^8+144*B*a^7*b*d*e^7+140*B*a^6*b^2*d^2*e^6+12
6*B*a^5*b^3*d^3*e^5+105*B*a^4*b^4*d^4*e^4+80*B*a^3*b^5*d^5*e^3+54*B*a^2*b^6*d^6*e^2+30*B*a*b^7*d^7*e+11*B*b^8*
d^8)*x^3-1/132*b/e^10*(495*A*a^8*b*e^9+360*A*a^7*b^2*d*e^8+252*A*a^6*b^3*d^2*e^7+168*A*a^5*b^4*d^3*e^6+105*A*a
^4*b^5*d^4*e^5+60*A*a^3*b^6*d^5*e^4+30*A*a^2*b^7*d^6*e^3+12*A*a*b^8*d^7*e^2+3*A*b^9*d^8*e+110*B*a^9*e^9+135*B*
a^8*b*d*e^8+144*B*a^7*b^2*d^2*e^7+140*B*a^6*b^3*d^3*e^6+126*B*a^5*b^4*d^4*e^5+105*B*a^4*b^5*d^5*e^4+80*B*a^3*b
^6*d^6*e^3+54*B*a^2*b^7*d^7*e^2+30*B*a*b^8*d^8*e+11*B*b^9*d^9)*x^2-1/858/e^11*(660*A*a^9*b*e^10+495*A*a^8*b^2*
d*e^9+360*A*a^7*b^3*d^2*e^8+252*A*a^6*b^4*d^3*e^7+168*A*a^5*b^5*d^4*e^6+105*A*a^4*b^6*d^5*e^5+60*A*a^3*b^7*d^6
*e^4+30*A*a^2*b^8*d^7*e^3+12*A*a*b^9*d^8*e^2+3*A*b^10*d^9*e+66*B*a^10*e^10+110*B*a^9*b*d*e^9+135*B*a^8*b^2*d^2
*e^8+144*B*a^7*b^3*d^3*e^7+140*B*a^6*b^4*d^4*e^6+126*B*a^5*b^5*d^5*e^5+105*B*a^4*b^6*d^6*e^4+80*B*a^3*b^7*d^7*
e^3+54*B*a^2*b^8*d^8*e^2+30*B*a*b^9*d^9*e+11*B*b^10*d^10)*x-1/12012/e^12*(858*A*a^10*e^11+660*A*a^9*b*d*e^10+4
95*A*a^8*b^2*d^2*e^9+360*A*a^7*b^3*d^3*e^8+252*A*a^6*b^4*d^4*e^7+168*A*a^5*b^5*d^5*e^6+105*A*a^4*b^6*d^6*e^5+6
0*A*a^3*b^7*d^7*e^4+30*A*a^2*b^8*d^8*e^3+12*A*a*b^9*d^9*e^2+3*A*b^10*d^10*e+66*B*a^10*d*e^10+110*B*a^9*b*d^2*e
^9+135*B*a^8*b^2*d^3*e^8+144*B*a^7*b^3*d^4*e^7+140*B*a^6*b^4*d^5*e^6+126*B*a^5*b^5*d^6*e^5+105*B*a^4*b^6*d^7*e
^4+80*B*a^3*b^7*d^8*e^3+54*B*a^2*b^8*d^9*e^2+30*B*a*b^9*d^10*e+11*B*b^10*d^11))/(e*x+d)^14
Fricas [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 1962 vs. \(2 (177) = 354\).
Time = 0.29 (sec) , antiderivative size = 1962, normalized size of antiderivative = 10.61
\[
\int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{15}} \, dx=\text {Too large to display}
\]
[In]
integrate((b*x+a)^10*(B*x+A)/(e*x+d)^15,x, algorithm="fricas")
[Out]
-1/12012*(4004*B*b^10*e^11*x^11 + 11*B*b^10*d^11 + 858*A*a^10*e^11 + 3*(10*B*a*b^9 + A*b^10)*d^10*e + 6*(9*B*a
^2*b^8 + 2*A*a*b^9)*d^9*e^2 + 10*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^8*e^3 + 15*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^7*e^4
+ 21*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^6*e^5 + 28*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^5*e^6 + 36*(4*B*a^7*b^3 + 7*A*a^6*
b^4)*d^4*e^7 + 45*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^3*e^8 + 55*(2*B*a^9*b + 9*A*a^8*b^2)*d^2*e^9 + 66*(B*a^10 + 10
*A*a^9*b)*d*e^10 + 1001*(11*B*b^10*d*e^10 + 3*(10*B*a*b^9 + A*b^10)*e^11)*x^10 + 2002*(11*B*b^10*d^2*e^9 + 3*(
10*B*a*b^9 + A*b^10)*d*e^10 + 6*(9*B*a^2*b^8 + 2*A*a*b^9)*e^11)*x^9 + 3003*(11*B*b^10*d^3*e^8 + 3*(10*B*a*b^9
+ A*b^10)*d^2*e^9 + 6*(9*B*a^2*b^8 + 2*A*a*b^9)*d*e^10 + 10*(8*B*a^3*b^7 + 3*A*a^2*b^8)*e^11)*x^8 + 3432*(11*B
*b^10*d^4*e^7 + 3*(10*B*a*b^9 + A*b^10)*d^3*e^8 + 6*(9*B*a^2*b^8 + 2*A*a*b^9)*d^2*e^9 + 10*(8*B*a^3*b^7 + 3*A*
a^2*b^8)*d*e^10 + 15*(7*B*a^4*b^6 + 4*A*a^3*b^7)*e^11)*x^7 + 3003*(11*B*b^10*d^5*e^6 + 3*(10*B*a*b^9 + A*b^10)
*d^4*e^7 + 6*(9*B*a^2*b^8 + 2*A*a*b^9)*d^3*e^8 + 10*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^2*e^9 + 15*(7*B*a^4*b^6 + 4*
A*a^3*b^7)*d*e^10 + 21*(6*B*a^5*b^5 + 5*A*a^4*b^6)*e^11)*x^6 + 2002*(11*B*b^10*d^6*e^5 + 3*(10*B*a*b^9 + A*b^1
0)*d^5*e^6 + 6*(9*B*a^2*b^8 + 2*A*a*b^9)*d^4*e^7 + 10*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^3*e^8 + 15*(7*B*a^4*b^6 +
4*A*a^3*b^7)*d^2*e^9 + 21*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d*e^10 + 28*(5*B*a^6*b^4 + 6*A*a^5*b^5)*e^11)*x^5 + 1001
*(11*B*b^10*d^7*e^4 + 3*(10*B*a*b^9 + A*b^10)*d^6*e^5 + 6*(9*B*a^2*b^8 + 2*A*a*b^9)*d^5*e^6 + 10*(8*B*a^3*b^7
+ 3*A*a^2*b^8)*d^4*e^7 + 15*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^3*e^8 + 21*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^2*e^9 + 28*
(5*B*a^6*b^4 + 6*A*a^5*b^5)*d*e^10 + 36*(4*B*a^7*b^3 + 7*A*a^6*b^4)*e^11)*x^4 + 364*(11*B*b^10*d^8*e^3 + 3*(10
*B*a*b^9 + A*b^10)*d^7*e^4 + 6*(9*B*a^2*b^8 + 2*A*a*b^9)*d^6*e^5 + 10*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^5*e^6 + 15
*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^4*e^7 + 21*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^3*e^8 + 28*(5*B*a^6*b^4 + 6*A*a^5*b^5)
*d^2*e^9 + 36*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d*e^10 + 45*(3*B*a^8*b^2 + 8*A*a^7*b^3)*e^11)*x^3 + 91*(11*B*b^10*d^
9*e^2 + 3*(10*B*a*b^9 + A*b^10)*d^8*e^3 + 6*(9*B*a^2*b^8 + 2*A*a*b^9)*d^7*e^4 + 10*(8*B*a^3*b^7 + 3*A*a^2*b^8)
*d^6*e^5 + 15*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^5*e^6 + 21*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^4*e^7 + 28*(5*B*a^6*b^4 +
6*A*a^5*b^5)*d^3*e^8 + 36*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^2*e^9 + 45*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d*e^10 + 55*(2
*B*a^9*b + 9*A*a^8*b^2)*e^11)*x^2 + 14*(11*B*b^10*d^10*e + 3*(10*B*a*b^9 + A*b^10)*d^9*e^2 + 6*(9*B*a^2*b^8 +
2*A*a*b^9)*d^8*e^3 + 10*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^7*e^4 + 15*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^6*e^5 + 21*(6*B
*a^5*b^5 + 5*A*a^4*b^6)*d^5*e^6 + 28*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^4*e^7 + 36*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^3*
e^8 + 45*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^2*e^9 + 55*(2*B*a^9*b + 9*A*a^8*b^2)*d*e^10 + 66*(B*a^10 + 10*A*a^9*b)*
e^11)*x)/(e^26*x^14 + 14*d*e^25*x^13 + 91*d^2*e^24*x^12 + 364*d^3*e^23*x^11 + 1001*d^4*e^22*x^10 + 2002*d^5*e^
21*x^9 + 3003*d^6*e^20*x^8 + 3432*d^7*e^19*x^7 + 3003*d^8*e^18*x^6 + 2002*d^9*e^17*x^5 + 1001*d^10*e^16*x^4 +
364*d^11*e^15*x^3 + 91*d^12*e^14*x^2 + 14*d^13*e^13*x + d^14*e^12)
Sympy [F(-1)]
Timed out. \[
\int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{15}} \, dx=\text {Timed out}
\]
[In]
integrate((b*x+a)**10*(B*x+A)/(e*x+d)**15,x)
[Out]
Timed out
Maxima [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 1962 vs. \(2 (177) = 354\).
Time = 0.30 (sec) , antiderivative size = 1962, normalized size of antiderivative = 10.61
\[
\int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{15}} \, dx=\text {Too large to display}
\]
[In]
integrate((b*x+a)^10*(B*x+A)/(e*x+d)^15,x, algorithm="maxima")
[Out]
-1/12012*(4004*B*b^10*e^11*x^11 + 11*B*b^10*d^11 + 858*A*a^10*e^11 + 3*(10*B*a*b^9 + A*b^10)*d^10*e + 6*(9*B*a
^2*b^8 + 2*A*a*b^9)*d^9*e^2 + 10*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^8*e^3 + 15*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^7*e^4
+ 21*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^6*e^5 + 28*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^5*e^6 + 36*(4*B*a^7*b^3 + 7*A*a^6*
b^4)*d^4*e^7 + 45*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^3*e^8 + 55*(2*B*a^9*b + 9*A*a^8*b^2)*d^2*e^9 + 66*(B*a^10 + 10
*A*a^9*b)*d*e^10 + 1001*(11*B*b^10*d*e^10 + 3*(10*B*a*b^9 + A*b^10)*e^11)*x^10 + 2002*(11*B*b^10*d^2*e^9 + 3*(
10*B*a*b^9 + A*b^10)*d*e^10 + 6*(9*B*a^2*b^8 + 2*A*a*b^9)*e^11)*x^9 + 3003*(11*B*b^10*d^3*e^8 + 3*(10*B*a*b^9
+ A*b^10)*d^2*e^9 + 6*(9*B*a^2*b^8 + 2*A*a*b^9)*d*e^10 + 10*(8*B*a^3*b^7 + 3*A*a^2*b^8)*e^11)*x^8 + 3432*(11*B
*b^10*d^4*e^7 + 3*(10*B*a*b^9 + A*b^10)*d^3*e^8 + 6*(9*B*a^2*b^8 + 2*A*a*b^9)*d^2*e^9 + 10*(8*B*a^3*b^7 + 3*A*
a^2*b^8)*d*e^10 + 15*(7*B*a^4*b^6 + 4*A*a^3*b^7)*e^11)*x^7 + 3003*(11*B*b^10*d^5*e^6 + 3*(10*B*a*b^9 + A*b^10)
*d^4*e^7 + 6*(9*B*a^2*b^8 + 2*A*a*b^9)*d^3*e^8 + 10*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^2*e^9 + 15*(7*B*a^4*b^6 + 4*
A*a^3*b^7)*d*e^10 + 21*(6*B*a^5*b^5 + 5*A*a^4*b^6)*e^11)*x^6 + 2002*(11*B*b^10*d^6*e^5 + 3*(10*B*a*b^9 + A*b^1
0)*d^5*e^6 + 6*(9*B*a^2*b^8 + 2*A*a*b^9)*d^4*e^7 + 10*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^3*e^8 + 15*(7*B*a^4*b^6 +
4*A*a^3*b^7)*d^2*e^9 + 21*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d*e^10 + 28*(5*B*a^6*b^4 + 6*A*a^5*b^5)*e^11)*x^5 + 1001
*(11*B*b^10*d^7*e^4 + 3*(10*B*a*b^9 + A*b^10)*d^6*e^5 + 6*(9*B*a^2*b^8 + 2*A*a*b^9)*d^5*e^6 + 10*(8*B*a^3*b^7
+ 3*A*a^2*b^8)*d^4*e^7 + 15*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^3*e^8 + 21*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^2*e^9 + 28*
(5*B*a^6*b^4 + 6*A*a^5*b^5)*d*e^10 + 36*(4*B*a^7*b^3 + 7*A*a^6*b^4)*e^11)*x^4 + 364*(11*B*b^10*d^8*e^3 + 3*(10
*B*a*b^9 + A*b^10)*d^7*e^4 + 6*(9*B*a^2*b^8 + 2*A*a*b^9)*d^6*e^5 + 10*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^5*e^6 + 15
*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^4*e^7 + 21*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^3*e^8 + 28*(5*B*a^6*b^4 + 6*A*a^5*b^5)
*d^2*e^9 + 36*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d*e^10 + 45*(3*B*a^8*b^2 + 8*A*a^7*b^3)*e^11)*x^3 + 91*(11*B*b^10*d^
9*e^2 + 3*(10*B*a*b^9 + A*b^10)*d^8*e^3 + 6*(9*B*a^2*b^8 + 2*A*a*b^9)*d^7*e^4 + 10*(8*B*a^3*b^7 + 3*A*a^2*b^8)
*d^6*e^5 + 15*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^5*e^6 + 21*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^4*e^7 + 28*(5*B*a^6*b^4 +
6*A*a^5*b^5)*d^3*e^8 + 36*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^2*e^9 + 45*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d*e^10 + 55*(2
*B*a^9*b + 9*A*a^8*b^2)*e^11)*x^2 + 14*(11*B*b^10*d^10*e + 3*(10*B*a*b^9 + A*b^10)*d^9*e^2 + 6*(9*B*a^2*b^8 +
2*A*a*b^9)*d^8*e^3 + 10*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^7*e^4 + 15*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^6*e^5 + 21*(6*B
*a^5*b^5 + 5*A*a^4*b^6)*d^5*e^6 + 28*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^4*e^7 + 36*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^3*
e^8 + 45*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^2*e^9 + 55*(2*B*a^9*b + 9*A*a^8*b^2)*d*e^10 + 66*(B*a^10 + 10*A*a^9*b)*
e^11)*x)/(e^26*x^14 + 14*d*e^25*x^13 + 91*d^2*e^24*x^12 + 364*d^3*e^23*x^11 + 1001*d^4*e^22*x^10 + 2002*d^5*e^
21*x^9 + 3003*d^6*e^20*x^8 + 3432*d^7*e^19*x^7 + 3003*d^8*e^18*x^6 + 2002*d^9*e^17*x^5 + 1001*d^10*e^16*x^4 +
364*d^11*e^15*x^3 + 91*d^12*e^14*x^2 + 14*d^13*e^13*x + d^14*e^12)
Giac [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 2232 vs. \(2 (177) = 354\).
Time = 0.29 (sec) , antiderivative size = 2232, normalized size of antiderivative = 12.06
\[
\int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{15}} \, dx=\text {Too large to display}
\]
[In]
integrate((b*x+a)^10*(B*x+A)/(e*x+d)^15,x, algorithm="giac")
[Out]
-1/12012*(4004*B*b^10*e^11*x^11 + 11011*B*b^10*d*e^10*x^10 + 30030*B*a*b^9*e^11*x^10 + 3003*A*b^10*e^11*x^10 +
22022*B*b^10*d^2*e^9*x^9 + 60060*B*a*b^9*d*e^10*x^9 + 6006*A*b^10*d*e^10*x^9 + 108108*B*a^2*b^8*e^11*x^9 + 24
024*A*a*b^9*e^11*x^9 + 33033*B*b^10*d^3*e^8*x^8 + 90090*B*a*b^9*d^2*e^9*x^8 + 9009*A*b^10*d^2*e^9*x^8 + 162162
*B*a^2*b^8*d*e^10*x^8 + 36036*A*a*b^9*d*e^10*x^8 + 240240*B*a^3*b^7*e^11*x^8 + 90090*A*a^2*b^8*e^11*x^8 + 3775
2*B*b^10*d^4*e^7*x^7 + 102960*B*a*b^9*d^3*e^8*x^7 + 10296*A*b^10*d^3*e^8*x^7 + 185328*B*a^2*b^8*d^2*e^9*x^7 +
41184*A*a*b^9*d^2*e^9*x^7 + 274560*B*a^3*b^7*d*e^10*x^7 + 102960*A*a^2*b^8*d*e^10*x^7 + 360360*B*a^4*b^6*e^11*
x^7 + 205920*A*a^3*b^7*e^11*x^7 + 33033*B*b^10*d^5*e^6*x^6 + 90090*B*a*b^9*d^4*e^7*x^6 + 9009*A*b^10*d^4*e^7*x
^6 + 162162*B*a^2*b^8*d^3*e^8*x^6 + 36036*A*a*b^9*d^3*e^8*x^6 + 240240*B*a^3*b^7*d^2*e^9*x^6 + 90090*A*a^2*b^8
*d^2*e^9*x^6 + 315315*B*a^4*b^6*d*e^10*x^6 + 180180*A*a^3*b^7*d*e^10*x^6 + 378378*B*a^5*b^5*e^11*x^6 + 315315*
A*a^4*b^6*e^11*x^6 + 22022*B*b^10*d^6*e^5*x^5 + 60060*B*a*b^9*d^5*e^6*x^5 + 6006*A*b^10*d^5*e^6*x^5 + 108108*B
*a^2*b^8*d^4*e^7*x^5 + 24024*A*a*b^9*d^4*e^7*x^5 + 160160*B*a^3*b^7*d^3*e^8*x^5 + 60060*A*a^2*b^8*d^3*e^8*x^5
+ 210210*B*a^4*b^6*d^2*e^9*x^5 + 120120*A*a^3*b^7*d^2*e^9*x^5 + 252252*B*a^5*b^5*d*e^10*x^5 + 210210*A*a^4*b^6
*d*e^10*x^5 + 280280*B*a^6*b^4*e^11*x^5 + 336336*A*a^5*b^5*e^11*x^5 + 11011*B*b^10*d^7*e^4*x^4 + 30030*B*a*b^9
*d^6*e^5*x^4 + 3003*A*b^10*d^6*e^5*x^4 + 54054*B*a^2*b^8*d^5*e^6*x^4 + 12012*A*a*b^9*d^5*e^6*x^4 + 80080*B*a^3
*b^7*d^4*e^7*x^4 + 30030*A*a^2*b^8*d^4*e^7*x^4 + 105105*B*a^4*b^6*d^3*e^8*x^4 + 60060*A*a^3*b^7*d^3*e^8*x^4 +
126126*B*a^5*b^5*d^2*e^9*x^4 + 105105*A*a^4*b^6*d^2*e^9*x^4 + 140140*B*a^6*b^4*d*e^10*x^4 + 168168*A*a^5*b^5*d
*e^10*x^4 + 144144*B*a^7*b^3*e^11*x^4 + 252252*A*a^6*b^4*e^11*x^4 + 4004*B*b^10*d^8*e^3*x^3 + 10920*B*a*b^9*d^
7*e^4*x^3 + 1092*A*b^10*d^7*e^4*x^3 + 19656*B*a^2*b^8*d^6*e^5*x^3 + 4368*A*a*b^9*d^6*e^5*x^3 + 29120*B*a^3*b^7
*d^5*e^6*x^3 + 10920*A*a^2*b^8*d^5*e^6*x^3 + 38220*B*a^4*b^6*d^4*e^7*x^3 + 21840*A*a^3*b^7*d^4*e^7*x^3 + 45864
*B*a^5*b^5*d^3*e^8*x^3 + 38220*A*a^4*b^6*d^3*e^8*x^3 + 50960*B*a^6*b^4*d^2*e^9*x^3 + 61152*A*a^5*b^5*d^2*e^9*x
^3 + 52416*B*a^7*b^3*d*e^10*x^3 + 91728*A*a^6*b^4*d*e^10*x^3 + 49140*B*a^8*b^2*e^11*x^3 + 131040*A*a^7*b^3*e^1
1*x^3 + 1001*B*b^10*d^9*e^2*x^2 + 2730*B*a*b^9*d^8*e^3*x^2 + 273*A*b^10*d^8*e^3*x^2 + 4914*B*a^2*b^8*d^7*e^4*x
^2 + 1092*A*a*b^9*d^7*e^4*x^2 + 7280*B*a^3*b^7*d^6*e^5*x^2 + 2730*A*a^2*b^8*d^6*e^5*x^2 + 9555*B*a^4*b^6*d^5*e
^6*x^2 + 5460*A*a^3*b^7*d^5*e^6*x^2 + 11466*B*a^5*b^5*d^4*e^7*x^2 + 9555*A*a^4*b^6*d^4*e^7*x^2 + 12740*B*a^6*b
^4*d^3*e^8*x^2 + 15288*A*a^5*b^5*d^3*e^8*x^2 + 13104*B*a^7*b^3*d^2*e^9*x^2 + 22932*A*a^6*b^4*d^2*e^9*x^2 + 122
85*B*a^8*b^2*d*e^10*x^2 + 32760*A*a^7*b^3*d*e^10*x^2 + 10010*B*a^9*b*e^11*x^2 + 45045*A*a^8*b^2*e^11*x^2 + 154
*B*b^10*d^10*e*x + 420*B*a*b^9*d^9*e^2*x + 42*A*b^10*d^9*e^2*x + 756*B*a^2*b^8*d^8*e^3*x + 168*A*a*b^9*d^8*e^3
*x + 1120*B*a^3*b^7*d^7*e^4*x + 420*A*a^2*b^8*d^7*e^4*x + 1470*B*a^4*b^6*d^6*e^5*x + 840*A*a^3*b^7*d^6*e^5*x +
1764*B*a^5*b^5*d^5*e^6*x + 1470*A*a^4*b^6*d^5*e^6*x + 1960*B*a^6*b^4*d^4*e^7*x + 2352*A*a^5*b^5*d^4*e^7*x + 2
016*B*a^7*b^3*d^3*e^8*x + 3528*A*a^6*b^4*d^3*e^8*x + 1890*B*a^8*b^2*d^2*e^9*x + 5040*A*a^7*b^3*d^2*e^9*x + 154
0*B*a^9*b*d*e^10*x + 6930*A*a^8*b^2*d*e^10*x + 924*B*a^10*e^11*x + 9240*A*a^9*b*e^11*x + 11*B*b^10*d^11 + 30*B
*a*b^9*d^10*e + 3*A*b^10*d^10*e + 54*B*a^2*b^8*d^9*e^2 + 12*A*a*b^9*d^9*e^2 + 80*B*a^3*b^7*d^8*e^3 + 30*A*a^2*
b^8*d^8*e^3 + 105*B*a^4*b^6*d^7*e^4 + 60*A*a^3*b^7*d^7*e^4 + 126*B*a^5*b^5*d^6*e^5 + 105*A*a^4*b^6*d^6*e^5 + 1
40*B*a^6*b^4*d^5*e^6 + 168*A*a^5*b^5*d^5*e^6 + 144*B*a^7*b^3*d^4*e^7 + 252*A*a^6*b^4*d^4*e^7 + 135*B*a^8*b^2*d
^3*e^8 + 360*A*a^7*b^3*d^3*e^8 + 110*B*a^9*b*d^2*e^9 + 495*A*a^8*b^2*d^2*e^9 + 66*B*a^10*d*e^10 + 660*A*a^9*b*
d*e^10 + 858*A*a^10*e^11)/((e*x + d)^14*e^12)
Mupad [B] (verification not implemented)
Time = 2.78 (sec) , antiderivative size = 2044, normalized size of antiderivative = 11.05
\[
\int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{15}} \, dx=\text {Too large to display}
\]
[In]
int(((A + B*x)*(a + b*x)^10)/(d + e*x)^15,x)
[Out]
-((858*A*a^10*e^11 + 11*B*b^10*d^11 + 3*A*b^10*d^10*e + 66*B*a^10*d*e^10 + 12*A*a*b^9*d^9*e^2 + 110*B*a^9*b*d^
2*e^9 + 30*A*a^2*b^8*d^8*e^3 + 60*A*a^3*b^7*d^7*e^4 + 105*A*a^4*b^6*d^6*e^5 + 168*A*a^5*b^5*d^5*e^6 + 252*A*a^
6*b^4*d^4*e^7 + 360*A*a^7*b^3*d^3*e^8 + 495*A*a^8*b^2*d^2*e^9 + 54*B*a^2*b^8*d^9*e^2 + 80*B*a^3*b^7*d^8*e^3 +
105*B*a^4*b^6*d^7*e^4 + 126*B*a^5*b^5*d^6*e^5 + 140*B*a^6*b^4*d^5*e^6 + 144*B*a^7*b^3*d^4*e^7 + 135*B*a^8*b^2*
d^3*e^8 + 660*A*a^9*b*d*e^10 + 30*B*a*b^9*d^10*e)/(12012*e^12) + (x*(66*B*a^10*e^10 + 11*B*b^10*d^10 + 660*A*a
^9*b*e^10 + 3*A*b^10*d^9*e + 12*A*a*b^9*d^8*e^2 + 495*A*a^8*b^2*d*e^9 + 30*A*a^2*b^8*d^7*e^3 + 60*A*a^3*b^7*d^
6*e^4 + 105*A*a^4*b^6*d^5*e^5 + 168*A*a^5*b^5*d^4*e^6 + 252*A*a^6*b^4*d^3*e^7 + 360*A*a^7*b^3*d^2*e^8 + 54*B*a
^2*b^8*d^8*e^2 + 80*B*a^3*b^7*d^7*e^3 + 105*B*a^4*b^6*d^6*e^4 + 126*B*a^5*b^5*d^5*e^5 + 140*B*a^6*b^4*d^4*e^6
+ 144*B*a^7*b^3*d^3*e^7 + 135*B*a^8*b^2*d^2*e^8 + 30*B*a*b^9*d^9*e + 110*B*a^9*b*d*e^9))/(858*e^11) + (b^7*x^8
*(80*B*a^3*e^3 + 11*B*b^3*d^3 + 30*A*a^2*b*e^3 + 3*A*b^3*d^2*e + 12*A*a*b^2*d*e^2 + 30*B*a*b^2*d^2*e + 54*B*a^
2*b*d*e^2))/(4*e^4) + (b^4*x^5*(140*B*a^6*e^6 + 11*B*b^6*d^6 + 168*A*a^5*b*e^6 + 3*A*b^6*d^5*e + 12*A*a*b^5*d^
4*e^2 + 105*A*a^4*b^2*d*e^5 + 30*A*a^2*b^4*d^3*e^3 + 60*A*a^3*b^3*d^2*e^4 + 54*B*a^2*b^4*d^4*e^2 + 80*B*a^3*b^
3*d^3*e^3 + 105*B*a^4*b^2*d^2*e^4 + 30*B*a*b^5*d^5*e + 126*B*a^5*b*d*e^5))/(6*e^7) + (b^9*x^10*(3*A*b*e + 30*B
*a*e + 11*B*b*d))/(12*e^2) + (2*b^6*x^7*(105*B*a^4*e^4 + 11*B*b^4*d^4 + 60*A*a^3*b*e^4 + 3*A*b^4*d^3*e + 12*A*
a*b^3*d^2*e^2 + 30*A*a^2*b^2*d*e^3 + 54*B*a^2*b^2*d^2*e^2 + 30*B*a*b^3*d^3*e + 80*B*a^3*b*d*e^3))/(7*e^5) + (b
^3*x^4*(144*B*a^7*e^7 + 11*B*b^7*d^7 + 252*A*a^6*b*e^7 + 3*A*b^7*d^6*e + 12*A*a*b^6*d^5*e^2 + 168*A*a^5*b^2*d*
e^6 + 30*A*a^2*b^5*d^4*e^3 + 60*A*a^3*b^4*d^3*e^4 + 105*A*a^4*b^3*d^2*e^5 + 54*B*a^2*b^5*d^5*e^2 + 80*B*a^3*b^
4*d^4*e^3 + 105*B*a^4*b^3*d^3*e^4 + 126*B*a^5*b^2*d^2*e^5 + 30*B*a*b^6*d^6*e + 140*B*a^6*b*d*e^6))/(12*e^8) +
(b*x^2*(110*B*a^9*e^9 + 11*B*b^9*d^9 + 495*A*a^8*b*e^9 + 3*A*b^9*d^8*e + 12*A*a*b^8*d^7*e^2 + 360*A*a^7*b^2*d*
e^8 + 30*A*a^2*b^7*d^6*e^3 + 60*A*a^3*b^6*d^5*e^4 + 105*A*a^4*b^5*d^4*e^5 + 168*A*a^5*b^4*d^3*e^6 + 252*A*a^6*
b^3*d^2*e^7 + 54*B*a^2*b^7*d^7*e^2 + 80*B*a^3*b^6*d^6*e^3 + 105*B*a^4*b^5*d^5*e^4 + 126*B*a^5*b^4*d^4*e^5 + 14
0*B*a^6*b^3*d^3*e^6 + 144*B*a^7*b^2*d^2*e^7 + 30*B*a*b^8*d^8*e + 135*B*a^8*b*d*e^8))/(132*e^10) + (b^8*x^9*(54
*B*a^2*e^2 + 11*B*b^2*d^2 + 12*A*a*b*e^2 + 3*A*b^2*d*e + 30*B*a*b*d*e))/(6*e^3) + (b^5*x^6*(126*B*a^5*e^5 + 11
*B*b^5*d^5 + 105*A*a^4*b*e^5 + 3*A*b^5*d^4*e + 12*A*a*b^4*d^3*e^2 + 60*A*a^3*b^2*d*e^4 + 30*A*a^2*b^3*d^2*e^3
+ 54*B*a^2*b^3*d^3*e^2 + 80*B*a^3*b^2*d^2*e^3 + 30*B*a*b^4*d^4*e + 105*B*a^4*b*d*e^4))/(4*e^6) + (b^2*x^3*(135
*B*a^8*e^8 + 11*B*b^8*d^8 + 360*A*a^7*b*e^8 + 3*A*b^8*d^7*e + 12*A*a*b^7*d^6*e^2 + 252*A*a^6*b^2*d*e^7 + 30*A*
a^2*b^6*d^5*e^3 + 60*A*a^3*b^5*d^4*e^4 + 105*A*a^4*b^4*d^3*e^5 + 168*A*a^5*b^3*d^2*e^6 + 54*B*a^2*b^6*d^6*e^2
+ 80*B*a^3*b^5*d^5*e^3 + 105*B*a^4*b^4*d^4*e^4 + 126*B*a^5*b^3*d^3*e^5 + 140*B*a^6*b^2*d^2*e^6 + 30*B*a*b^7*d^
7*e + 144*B*a^7*b*d*e^7))/(33*e^9) + (B*b^10*x^11)/(3*e))/(d^14 + e^14*x^14 + 14*d*e^13*x^13 + 91*d^12*e^2*x^2
+ 364*d^11*e^3*x^3 + 1001*d^10*e^4*x^4 + 2002*d^9*e^5*x^5 + 3003*d^8*e^6*x^6 + 3432*d^7*e^7*x^7 + 3003*d^6*e^
8*x^8 + 2002*d^5*e^9*x^9 + 1001*d^4*e^10*x^10 + 364*d^3*e^11*x^11 + 91*d^2*e^12*x^12 + 14*d^13*e*x)