3.11.99 \(\int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{11}} \, dx\) [1099]
Optimal. Leaf size=446 \[ \frac {b^{10} B x}{e^{11}}+\frac {(b d-a e)^{10} (B d-A e)}{10 e^{12} (d+e x)^{10}}-\frac {(b d-a e)^9 (11 b B d-10 A b e-a B e)}{9 e^{12} (d+e x)^9}+\frac {5 b (b d-a e)^8 (11 b B d-9 A b e-2 a B e)}{8 e^{12} (d+e x)^8}-\frac {15 b^2 (b d-a e)^7 (11 b B d-8 A b e-3 a B e)}{7 e^{12} (d+e x)^7}+\frac {5 b^3 (b d-a e)^6 (11 b B d-7 A b e-4 a B e)}{e^{12} (d+e x)^6}-\frac {42 b^4 (b d-a e)^5 (11 b B d-6 A b e-5 a B e)}{5 e^{12} (d+e x)^5}+\frac {21 b^5 (b d-a e)^4 (11 b B d-5 A b e-6 a B e)}{2 e^{12} (d+e x)^4}-\frac {10 b^6 (b d-a e)^3 (11 b B d-4 A b e-7 a B e)}{e^{12} (d+e x)^3}+\frac {15 b^7 (b d-a e)^2 (11 b B d-3 A b e-8 a B e)}{2 e^{12} (d+e x)^2}-\frac {5 b^8 (b d-a e) (11 b B d-2 A b e-9 a B e)}{e^{12} (d+e x)}-\frac {b^9 (11 b B d-A b e-10 a B e) \log (d+e x)}{e^{12}} \]
[Out]
b^10*B*x/e^11+1/10*(-a*e+b*d)^10*(-A*e+B*d)/e^12/(e*x+d)^10-1/9*(-a*e+b*d)^9*(-10*A*b*e-B*a*e+11*B*b*d)/e^12/(
e*x+d)^9+5/8*b*(-a*e+b*d)^8*(-9*A*b*e-2*B*a*e+11*B*b*d)/e^12/(e*x+d)^8-15/7*b^2*(-a*e+b*d)^7*(-8*A*b*e-3*B*a*e
+11*B*b*d)/e^12/(e*x+d)^7+5*b^3*(-a*e+b*d)^6*(-7*A*b*e-4*B*a*e+11*B*b*d)/e^12/(e*x+d)^6-42/5*b^4*(-a*e+b*d)^5*
(-6*A*b*e-5*B*a*e+11*B*b*d)/e^12/(e*x+d)^5+21/2*b^5*(-a*e+b*d)^4*(-5*A*b*e-6*B*a*e+11*B*b*d)/e^12/(e*x+d)^4-10
*b^6*(-a*e+b*d)^3*(-4*A*b*e-7*B*a*e+11*B*b*d)/e^12/(e*x+d)^3+15/2*b^7*(-a*e+b*d)^2*(-3*A*b*e-8*B*a*e+11*B*b*d)
/e^12/(e*x+d)^2-5*b^8*(-a*e+b*d)*(-2*A*b*e-9*B*a*e+11*B*b*d)/e^12/(e*x+d)-b^9*(-A*b*e-10*B*a*e+11*B*b*d)*ln(e*
x+d)/e^12
________________________________________________________________________________________
Rubi [A]
time = 0.51, antiderivative size = 446, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {78}
\begin {gather*} -\frac {b^9 \log (d+e x) (-10 a B e-A b e+11 b B d)}{e^{12}}-\frac {5 b^8 (b d-a e) (-9 a B e-2 A b e+11 b B d)}{e^{12} (d+e x)}+\frac {15 b^7 (b d-a e)^2 (-8 a B e-3 A b e+11 b B d)}{2 e^{12} (d+e x)^2}-\frac {10 b^6 (b d-a e)^3 (-7 a B e-4 A b e+11 b B d)}{e^{12} (d+e x)^3}+\frac {21 b^5 (b d-a e)^4 (-6 a B e-5 A b e+11 b B d)}{2 e^{12} (d+e x)^4}-\frac {42 b^4 (b d-a e)^5 (-5 a B e-6 A b e+11 b B d)}{5 e^{12} (d+e x)^5}+\frac {5 b^3 (b d-a e)^6 (-4 a B e-7 A b e+11 b B d)}{e^{12} (d+e x)^6}-\frac {15 b^2 (b d-a e)^7 (-3 a B e-8 A b e+11 b B d)}{7 e^{12} (d+e x)^7}+\frac {5 b (b d-a e)^8 (-2 a B e-9 A b e+11 b B d)}{8 e^{12} (d+e x)^8}-\frac {(b d-a e)^9 (-a B e-10 A b e+11 b B d)}{9 e^{12} (d+e x)^9}+\frac {(b d-a e)^{10} (B d-A e)}{10 e^{12} (d+e x)^{10}}+\frac {b^{10} B x}{e^{11}} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[((a + b*x)^10*(A + B*x))/(d + e*x)^11,x]
[Out]
(b^10*B*x)/e^11 + ((b*d - a*e)^10*(B*d - A*e))/(10*e^12*(d + e*x)^10) - ((b*d - a*e)^9*(11*b*B*d - 10*A*b*e -
a*B*e))/(9*e^12*(d + e*x)^9) + (5*b*(b*d - a*e)^8*(11*b*B*d - 9*A*b*e - 2*a*B*e))/(8*e^12*(d + e*x)^8) - (15*b
^2*(b*d - a*e)^7*(11*b*B*d - 8*A*b*e - 3*a*B*e))/(7*e^12*(d + e*x)^7) + (5*b^3*(b*d - a*e)^6*(11*b*B*d - 7*A*b
*e - 4*a*B*e))/(e^12*(d + e*x)^6) - (42*b^4*(b*d - a*e)^5*(11*b*B*d - 6*A*b*e - 5*a*B*e))/(5*e^12*(d + e*x)^5)
+ (21*b^5*(b*d - a*e)^4*(11*b*B*d - 5*A*b*e - 6*a*B*e))/(2*e^12*(d + e*x)^4) - (10*b^6*(b*d - a*e)^3*(11*b*B*
d - 4*A*b*e - 7*a*B*e))/(e^12*(d + e*x)^3) + (15*b^7*(b*d - a*e)^2*(11*b*B*d - 3*A*b*e - 8*a*B*e))/(2*e^12*(d
+ e*x)^2) - (5*b^8*(b*d - a*e)*(11*b*B*d - 2*A*b*e - 9*a*B*e))/(e^12*(d + e*x)) - (b^9*(11*b*B*d - A*b*e - 10*
a*B*e)*Log[d + e*x])/e^12
Rule 78
Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
Rubi steps
\begin {align*} \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{11}} \, dx &=\int \left (\frac {b^{10} B}{e^{11}}+\frac {(-b d+a e)^{10} (-B d+A e)}{e^{11} (d+e x)^{11}}+\frac {(-b d+a e)^9 (-11 b B d+10 A b e+a B e)}{e^{11} (d+e x)^{10}}+\frac {5 b (b d-a e)^8 (-11 b B d+9 A b e+2 a B e)}{e^{11} (d+e x)^9}-\frac {15 b^2 (b d-a e)^7 (-11 b B d+8 A b e+3 a B e)}{e^{11} (d+e x)^8}+\frac {30 b^3 (b d-a e)^6 (-11 b B d+7 A b e+4 a B e)}{e^{11} (d+e x)^7}-\frac {42 b^4 (b d-a e)^5 (-11 b B d+6 A b e+5 a B e)}{e^{11} (d+e x)^6}+\frac {42 b^5 (b d-a e)^4 (-11 b B d+5 A b e+6 a B e)}{e^{11} (d+e x)^5}-\frac {30 b^6 (b d-a e)^3 (-11 b B d+4 A b e+7 a B e)}{e^{11} (d+e x)^4}+\frac {15 b^7 (b d-a e)^2 (-11 b B d+3 A b e+8 a B e)}{e^{11} (d+e x)^3}-\frac {5 b^8 (b d-a e) (-11 b B d+2 A b e+9 a B e)}{e^{11} (d+e x)^2}+\frac {b^9 (-11 b B d+A b e+10 a B e)}{e^{11} (d+e x)}\right ) \, dx\\ &=\frac {b^{10} B x}{e^{11}}+\frac {(b d-a e)^{10} (B d-A e)}{10 e^{12} (d+e x)^{10}}-\frac {(b d-a e)^9 (11 b B d-10 A b e-a B e)}{9 e^{12} (d+e x)^9}+\frac {5 b (b d-a e)^8 (11 b B d-9 A b e-2 a B e)}{8 e^{12} (d+e x)^8}-\frac {15 b^2 (b d-a e)^7 (11 b B d-8 A b e-3 a B e)}{7 e^{12} (d+e x)^7}+\frac {5 b^3 (b d-a e)^6 (11 b B d-7 A b e-4 a B e)}{e^{12} (d+e x)^6}-\frac {42 b^4 (b d-a e)^5 (11 b B d-6 A b e-5 a B e)}{5 e^{12} (d+e x)^5}+\frac {21 b^5 (b d-a e)^4 (11 b B d-5 A b e-6 a B e)}{2 e^{12} (d+e x)^4}-\frac {10 b^6 (b d-a e)^3 (11 b B d-4 A b e-7 a B e)}{e^{12} (d+e x)^3}+\frac {15 b^7 (b d-a e)^2 (11 b B d-3 A b e-8 a B e)}{2 e^{12} (d+e x)^2}-\frac {5 b^8 (b d-a e) (11 b B d-2 A b e-9 a B e)}{e^{12} (d+e x)}-\frac {b^9 (11 b B d-A b e-10 a B e) \log (d+e x)}{e^{12}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(1447\) vs. \(2(446)=892\).
time = 0.63, size = 1447, normalized size = 3.24 \begin {gather*} -\frac {28 a^{10} e^{10} (9 A e+B (d+10 e x))+70 a^9 b e^9 \left (4 A e (d+10 e x)+B \left (d^2+10 d e x+45 e^2 x^2\right )\right )+45 a^8 b^2 e^8 \left (7 A e \left (d^2+10 d e x+45 e^2 x^2\right )+3 B \left (d^3+10 d^2 e x+45 d e^2 x^2+120 e^3 x^3\right )\right )+120 a^7 b^3 e^7 \left (3 A e \left (d^3+10 d^2 e x+45 d e^2 x^2+120 e^3 x^3\right )+2 B \left (d^4+10 d^3 e x+45 d^2 e^2 x^2+120 d e^3 x^3+210 e^4 x^4\right )\right )+420 a^6 b^4 e^6 \left (A e \left (d^4+10 d^3 e x+45 d^2 e^2 x^2+120 d e^3 x^3+210 e^4 x^4\right )+B \left (d^5+10 d^4 e x+45 d^3 e^2 x^2+120 d^2 e^3 x^3+210 d e^4 x^4+252 e^5 x^5\right )\right )+252 a^5 b^5 e^5 \left (2 A e \left (d^5+10 d^4 e x+45 d^3 e^2 x^2+120 d^2 e^3 x^3+210 d e^4 x^4+252 e^5 x^5\right )+3 B \left (d^6+10 d^5 e x+45 d^4 e^2 x^2+120 d^3 e^3 x^3+210 d^2 e^4 x^4+252 d e^5 x^5+210 e^6 x^6\right )\right )+210 a^4 b^6 e^4 \left (3 A e \left (d^6+10 d^5 e x+45 d^4 e^2 x^2+120 d^3 e^3 x^3+210 d^2 e^4 x^4+252 d e^5 x^5+210 e^6 x^6\right )+7 B \left (d^7+10 d^6 e x+45 d^5 e^2 x^2+120 d^4 e^3 x^3+210 d^3 e^4 x^4+252 d^2 e^5 x^5+210 d e^6 x^6+120 e^7 x^7\right )\right )+840 a^3 b^7 e^3 \left (A e \left (d^7+10 d^6 e x+45 d^5 e^2 x^2+120 d^4 e^3 x^3+210 d^3 e^4 x^4+252 d^2 e^5 x^5+210 d e^6 x^6+120 e^7 x^7\right )+4 B \left (d^8+10 d^7 e x+45 d^6 e^2 x^2+120 d^5 e^3 x^3+210 d^4 e^4 x^4+252 d^3 e^5 x^5+210 d^2 e^6 x^6+120 d e^7 x^7+45 e^8 x^8\right )\right )+1260 a^2 b^8 e^2 \left (A e \left (d^8+10 d^7 e x+45 d^6 e^2 x^2+120 d^5 e^3 x^3+210 d^4 e^4 x^4+252 d^3 e^5 x^5+210 d^2 e^6 x^6+120 d e^7 x^7+45 e^8 x^8\right )+9 B \left (d^9+10 d^8 e x+45 d^7 e^2 x^2+120 d^6 e^3 x^3+210 d^5 e^4 x^4+252 d^4 e^5 x^5+210 d^3 e^6 x^6+120 d^2 e^7 x^7+45 d e^8 x^8+10 e^9 x^9\right )\right )-10 a b^9 e \left (-252 A e \left (d^9+10 d^8 e x+45 d^7 e^2 x^2+120 d^6 e^3 x^3+210 d^5 e^4 x^4+252 d^4 e^5 x^5+210 d^3 e^6 x^6+120 d^2 e^7 x^7+45 d e^8 x^8+10 e^9 x^9\right )+B d \left (7381 d^9+71290 d^8 e x+308205 d^7 e^2 x^2+784080 d^6 e^3 x^3+1296540 d^5 e^4 x^4+1450008 d^4 e^5 x^5+1102500 d^3 e^6 x^6+554400 d^2 e^7 x^7+170100 d e^8 x^8+25200 e^9 x^9\right )\right )-b^{10} \left (A d e \left (7381 d^9+71290 d^8 e x+308205 d^7 e^2 x^2+784080 d^6 e^3 x^3+1296540 d^5 e^4 x^4+1450008 d^4 e^5 x^5+1102500 d^3 e^6 x^6+554400 d^2 e^7 x^7+170100 d e^8 x^8+25200 e^9 x^9\right )-B \left (55991 d^{11}+532190 d^{10} e x+2256255 d^9 e^2 x^2+5600880 d^8 e^3 x^3+8969940 d^7 e^4 x^4+9599688 d^6 e^5 x^5+6835500 d^5 e^6 x^6+3074400 d^4 e^7 x^7+737100 d^3 e^8 x^8+25200 d^2 e^9 x^9-25200 d e^{10} x^{10}-2520 e^{11} x^{11}\right )\right )+2520 b^9 (11 b B d-A b e-10 a B e) (d+e x)^{10} \log (d+e x)}{2520 e^{12} (d+e x)^{10}} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[((a + b*x)^10*(A + B*x))/(d + e*x)^11,x]
[Out]
-1/2520*(28*a^10*e^10*(9*A*e + B*(d + 10*e*x)) + 70*a^9*b*e^9*(4*A*e*(d + 10*e*x) + B*(d^2 + 10*d*e*x + 45*e^2
*x^2)) + 45*a^8*b^2*e^8*(7*A*e*(d^2 + 10*d*e*x + 45*e^2*x^2) + 3*B*(d^3 + 10*d^2*e*x + 45*d*e^2*x^2 + 120*e^3*
x^3)) + 120*a^7*b^3*e^7*(3*A*e*(d^3 + 10*d^2*e*x + 45*d*e^2*x^2 + 120*e^3*x^3) + 2*B*(d^4 + 10*d^3*e*x + 45*d^
2*e^2*x^2 + 120*d*e^3*x^3 + 210*e^4*x^4)) + 420*a^6*b^4*e^6*(A*e*(d^4 + 10*d^3*e*x + 45*d^2*e^2*x^2 + 120*d*e^
3*x^3 + 210*e^4*x^4) + B*(d^5 + 10*d^4*e*x + 45*d^3*e^2*x^2 + 120*d^2*e^3*x^3 + 210*d*e^4*x^4 + 252*e^5*x^5))
+ 252*a^5*b^5*e^5*(2*A*e*(d^5 + 10*d^4*e*x + 45*d^3*e^2*x^2 + 120*d^2*e^3*x^3 + 210*d*e^4*x^4 + 252*e^5*x^5) +
3*B*(d^6 + 10*d^5*e*x + 45*d^4*e^2*x^2 + 120*d^3*e^3*x^3 + 210*d^2*e^4*x^4 + 252*d*e^5*x^5 + 210*e^6*x^6)) +
210*a^4*b^6*e^4*(3*A*e*(d^6 + 10*d^5*e*x + 45*d^4*e^2*x^2 + 120*d^3*e^3*x^3 + 210*d^2*e^4*x^4 + 252*d*e^5*x^5
+ 210*e^6*x^6) + 7*B*(d^7 + 10*d^6*e*x + 45*d^5*e^2*x^2 + 120*d^4*e^3*x^3 + 210*d^3*e^4*x^4 + 252*d^2*e^5*x^5
+ 210*d*e^6*x^6 + 120*e^7*x^7)) + 840*a^3*b^7*e^3*(A*e*(d^7 + 10*d^6*e*x + 45*d^5*e^2*x^2 + 120*d^4*e^3*x^3 +
210*d^3*e^4*x^4 + 252*d^2*e^5*x^5 + 210*d*e^6*x^6 + 120*e^7*x^7) + 4*B*(d^8 + 10*d^7*e*x + 45*d^6*e^2*x^2 + 12
0*d^5*e^3*x^3 + 210*d^4*e^4*x^4 + 252*d^3*e^5*x^5 + 210*d^2*e^6*x^6 + 120*d*e^7*x^7 + 45*e^8*x^8)) + 1260*a^2*
b^8*e^2*(A*e*(d^8 + 10*d^7*e*x + 45*d^6*e^2*x^2 + 120*d^5*e^3*x^3 + 210*d^4*e^4*x^4 + 252*d^3*e^5*x^5 + 210*d^
2*e^6*x^6 + 120*d*e^7*x^7 + 45*e^8*x^8) + 9*B*(d^9 + 10*d^8*e*x + 45*d^7*e^2*x^2 + 120*d^6*e^3*x^3 + 210*d^5*e
^4*x^4 + 252*d^4*e^5*x^5 + 210*d^3*e^6*x^6 + 120*d^2*e^7*x^7 + 45*d*e^8*x^8 + 10*e^9*x^9)) - 10*a*b^9*e*(-252*
A*e*(d^9 + 10*d^8*e*x + 45*d^7*e^2*x^2 + 120*d^6*e^3*x^3 + 210*d^5*e^4*x^4 + 252*d^4*e^5*x^5 + 210*d^3*e^6*x^6
+ 120*d^2*e^7*x^7 + 45*d*e^8*x^8 + 10*e^9*x^9) + B*d*(7381*d^9 + 71290*d^8*e*x + 308205*d^7*e^2*x^2 + 784080*
d^6*e^3*x^3 + 1296540*d^5*e^4*x^4 + 1450008*d^4*e^5*x^5 + 1102500*d^3*e^6*x^6 + 554400*d^2*e^7*x^7 + 170100*d*
e^8*x^8 + 25200*e^9*x^9)) - b^10*(A*d*e*(7381*d^9 + 71290*d^8*e*x + 308205*d^7*e^2*x^2 + 784080*d^6*e^3*x^3 +
1296540*d^5*e^4*x^4 + 1450008*d^4*e^5*x^5 + 1102500*d^3*e^6*x^6 + 554400*d^2*e^7*x^7 + 170100*d*e^8*x^8 + 2520
0*e^9*x^9) - B*(55991*d^11 + 532190*d^10*e*x + 2256255*d^9*e^2*x^2 + 5600880*d^8*e^3*x^3 + 8969940*d^7*e^4*x^4
+ 9599688*d^6*e^5*x^5 + 6835500*d^5*e^6*x^6 + 3074400*d^4*e^7*x^7 + 737100*d^3*e^8*x^8 + 25200*d^2*e^9*x^9 -
25200*d*e^10*x^10 - 2520*e^11*x^11)) + 2520*b^9*(11*b*B*d - A*b*e - 10*a*B*e)*(d + e*x)^10*Log[d + e*x])/(e^12
*(d + e*x)^10)
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1932\) vs.
\(2(432)=864\).
time = 0.08, size = 1933, normalized size = 4.33
| | |
method |
result |
size |
| | |
risch |
\(\text {Expression too large to display}\) |
\(1922\) |
default |
\(\text {Expression too large to display}\) |
\(1933\) |
norman |
\(\text {Expression too large to display}\) |
\(1934\) |
| | |
|
|
|
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((b*x+a)^10*(B*x+A)/(e*x+d)^11,x,method=_RETURNVERBOSE)
[Out]
b^10*B*x/e^11-15/2*b^7/e^12*(3*A*a^2*b*e^3-6*A*a*b^2*d*e^2+3*A*b^3*d^2*e+8*B*a^3*e^3-27*B*a^2*b*d*e^2+30*B*a*b
^2*d^2*e-11*B*b^3*d^3)/(e*x+d)^2-21/2*b^5/e^12*(5*A*a^4*b*e^5-20*A*a^3*b^2*d*e^4+30*A*a^2*b^3*d^2*e^3-20*A*a*b
^4*d^3*e^2+5*A*b^5*d^4*e+6*B*a^5*e^5-35*B*a^4*b*d*e^4+80*B*a^3*b^2*d^2*e^3-90*B*a^2*b^3*d^3*e^2+50*B*a*b^4*d^4
*e-11*B*b^5*d^5)/(e*x+d)^4-5/8*b/e^12*(9*A*a^8*b*e^9-72*A*a^7*b^2*d*e^8+252*A*a^6*b^3*d^2*e^7-504*A*a^5*b^4*d^
3*e^6+630*A*a^4*b^5*d^4*e^5-504*A*a^3*b^6*d^5*e^4+252*A*a^2*b^7*d^6*e^3-72*A*a*b^8*d^7*e^2+9*A*b^9*d^8*e+2*B*a
^9*e^9-27*B*a^8*b*d*e^8+144*B*a^7*b^2*d^2*e^7-420*B*a^6*b^3*d^3*e^6+756*B*a^5*b^4*d^4*e^5-882*B*a^4*b^5*d^5*e^
4+672*B*a^3*b^6*d^6*e^3-324*B*a^2*b^7*d^7*e^2+90*B*a*b^8*d^8*e-11*B*b^9*d^9)/(e*x+d)^8-5*b^8/e^12*(2*A*a*b*e^2
-2*A*b^2*d*e+9*B*a^2*e^2-20*B*a*b*d*e+11*B*b^2*d^2)/(e*x+d)-1/9/e^12*(10*A*a^9*b*e^10-90*A*a^8*b^2*d*e^9+360*A
*a^7*b^3*d^2*e^8-840*A*a^6*b^4*d^3*e^7+1260*A*a^5*b^5*d^4*e^6-1260*A*a^4*b^6*d^5*e^5+840*A*a^3*b^7*d^6*e^4-360
*A*a^2*b^8*d^7*e^3+90*A*a*b^9*d^8*e^2-10*A*b^10*d^9*e+B*a^10*e^10-20*B*a^9*b*d*e^9+135*B*a^8*b^2*d^2*e^8-480*B
*a^7*b^3*d^3*e^7+1050*B*a^6*b^4*d^4*e^6-1512*B*a^5*b^5*d^5*e^5+1470*B*a^4*b^6*d^6*e^4-960*B*a^3*b^7*d^7*e^3+40
5*B*a^2*b^8*d^8*e^2-100*B*a*b^9*d^9*e+11*B*b^10*d^10)/(e*x+d)^9-15/7*b^2/e^12*(8*A*a^7*b*e^8-56*A*a^6*b^2*d*e^
7+168*A*a^5*b^3*d^2*e^6-280*A*a^4*b^4*d^3*e^5+280*A*a^3*b^5*d^4*e^4-168*A*a^2*b^6*d^5*e^3+56*A*a*b^7*d^6*e^2-8
*A*b^8*d^7*e+3*B*a^8*e^8-32*B*a^7*b*d*e^7+140*B*a^6*b^2*d^2*e^6-336*B*a^5*b^3*d^3*e^5+490*B*a^4*b^4*d^4*e^4-44
8*B*a^3*b^5*d^5*e^3+252*B*a^2*b^6*d^6*e^2-80*B*a*b^7*d^7*e+11*B*b^8*d^8)/(e*x+d)^7-5*b^3/e^12*(7*A*a^6*b*e^7-4
2*A*a^5*b^2*d*e^6+105*A*a^4*b^3*d^2*e^5-140*A*a^3*b^4*d^3*e^4+105*A*a^2*b^5*d^4*e^3-42*A*a*b^6*d^5*e^2+7*A*b^7
*d^6*e+4*B*a^7*e^7-35*B*a^6*b*d*e^6+126*B*a^5*b^2*d^2*e^5-245*B*a^4*b^3*d^3*e^4+280*B*a^3*b^4*d^4*e^3-189*B*a^
2*b^5*d^5*e^2+70*B*a*b^6*d^6*e-11*B*b^7*d^7)/(e*x+d)^6-42/5*b^4/e^12*(6*A*a^5*b*e^6-30*A*a^4*b^2*d*e^5+60*A*a^
3*b^3*d^2*e^4-60*A*a^2*b^4*d^3*e^3+30*A*a*b^5*d^4*e^2-6*A*b^6*d^5*e+5*B*a^6*e^6-36*B*a^5*b*d*e^5+105*B*a^4*b^2
*d^2*e^4-160*B*a^3*b^3*d^3*e^3+135*B*a^2*b^4*d^4*e^2-60*B*a*b^5*d^5*e+11*B*b^6*d^6)/(e*x+d)^5-10*b^6/e^12*(4*A
*a^3*b*e^4-12*A*a^2*b^2*d*e^3+12*A*a*b^3*d^2*e^2-4*A*b^4*d^3*e+7*B*a^4*e^4-32*B*a^3*b*d*e^3+54*B*a^2*b^2*d^2*e
^2-40*B*a*b^3*d^3*e+11*B*b^4*d^4)/(e*x+d)^3-1/10*(A*a^10*e^11-10*A*a^9*b*d*e^10+45*A*a^8*b^2*d^2*e^9-120*A*a^7
*b^3*d^3*e^8+210*A*a^6*b^4*d^4*e^7-252*A*a^5*b^5*d^5*e^6+210*A*a^4*b^6*d^6*e^5-120*A*a^3*b^7*d^7*e^4+45*A*a^2*
b^8*d^8*e^3-10*A*a*b^9*d^9*e^2+A*b^10*d^10*e-B*a^10*d*e^10+10*B*a^9*b*d^2*e^9-45*B*a^8*b^2*d^3*e^8+120*B*a^7*b
^3*d^4*e^7-210*B*a^6*b^4*d^5*e^6+252*B*a^5*b^5*d^6*e^5-210*B*a^4*b^6*d^7*e^4+120*B*a^3*b^7*d^8*e^3-45*B*a^2*b^
8*d^9*e^2+10*B*a*b^9*d^10*e-B*b^10*d^11)/e^12/(e*x+d)^10+b^9/e^12*(A*b*e+10*B*a*e-11*B*b*d)*ln(e*x+d)
________________________________________________________________________________________
Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 1938 vs.
\(2 (462) = 924\).
time = 0.41, size = 1938, normalized size = 4.35 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)^10*(B*x+A)/(e*x+d)^11,x, algorithm="maxima")
[Out]
B*b^10*x*e^(-11) - (11*B*b^10*d - 10*B*a*b^9*e - A*b^10*e)*e^(-12)*log(x*e + d) - 1/2520*(55991*B*b^10*d^11 +
252*A*a^10*e^11 - 7381*(10*B*a*b^9*e + A*b^10*e)*d^10 + 1260*(9*B*a^2*b^8*e^2 + 2*A*a*b^9*e^2)*d^9 + 12600*(11
*B*b^10*d^2*e^9 + 9*B*a^2*b^8*e^11 + 2*A*a*b^9*e^11 - 2*(10*B*a*b^9*e^10 + A*b^10*e^10)*d)*x^9 + 420*(8*B*a^3*
b^7*e^3 + 3*A*a^2*b^8*e^3)*d^8 + 18900*(55*B*b^10*d^3*e^8 + 8*B*a^3*b^7*e^11 + 3*A*a^2*b^8*e^11 - 9*(10*B*a*b^
9*e^9 + A*b^10*e^9)*d^2 + 3*(9*B*a^2*b^8*e^10 + 2*A*a*b^9*e^10)*d)*x^8 + 210*(7*B*a^4*b^6*e^4 + 4*A*a^3*b^7*e^
4)*d^7 + 25200*(143*B*b^10*d^4*e^7 + 7*B*a^4*b^6*e^11 + 4*A*a^3*b^7*e^11 - 22*(10*B*a*b^9*e^8 + A*b^10*e^8)*d^
3 + 6*(9*B*a^2*b^8*e^9 + 2*A*a*b^9*e^9)*d^2 + 2*(8*B*a^3*b^7*e^10 + 3*A*a^2*b^8*e^10)*d)*x^7 + 126*(6*B*a^5*b^
5*e^5 + 5*A*a^4*b^6*e^5)*d^6 + 8820*(847*B*b^10*d^5*e^6 + 18*B*a^5*b^5*e^11 + 15*A*a^4*b^6*e^11 - 125*(10*B*a*
b^9*e^7 + A*b^10*e^7)*d^4 + 30*(9*B*a^2*b^8*e^8 + 2*A*a*b^9*e^8)*d^3 + 10*(8*B*a^3*b^7*e^9 + 3*A*a^2*b^8*e^9)*
d^2 + 5*(7*B*a^4*b^6*e^10 + 4*A*a^3*b^7*e^10)*d)*x^6 + 84*(5*B*a^6*b^4*e^6 + 6*A*a^5*b^5*e^6)*d^5 + 10584*(957
*B*b^10*d^6*e^5 + 10*B*a^6*b^4*e^11 + 12*A*a^5*b^5*e^11 - 137*(10*B*a*b^9*e^6 + A*b^10*e^6)*d^5 + 30*(9*B*a^2*
b^8*e^7 + 2*A*a*b^9*e^7)*d^4 + 10*(8*B*a^3*b^7*e^8 + 3*A*a^2*b^8*e^8)*d^3 + 5*(7*B*a^4*b^6*e^9 + 4*A*a^3*b^7*e
^9)*d^2 + 3*(6*B*a^5*b^5*e^10 + 5*A*a^4*b^6*e^10)*d)*x^5 + 60*(4*B*a^7*b^3*e^7 + 7*A*a^6*b^4*e^7)*d^4 + 1260*(
7359*B*b^10*d^7*e^4 + 40*B*a^7*b^3*e^11 + 70*A*a^6*b^4*e^11 - 1029*(10*B*a*b^9*e^5 + A*b^10*e^5)*d^6 + 210*(9*
B*a^2*b^8*e^6 + 2*A*a*b^9*e^6)*d^5 + 70*(8*B*a^3*b^7*e^7 + 3*A*a^2*b^8*e^7)*d^4 + 35*(7*B*a^4*b^6*e^8 + 4*A*a^
3*b^7*e^8)*d^3 + 21*(6*B*a^5*b^5*e^9 + 5*A*a^4*b^6*e^9)*d^2 + 14*(5*B*a^6*b^4*e^10 + 6*A*a^5*b^5*e^10)*d)*x^4
+ 45*(3*B*a^8*b^2*e^8 + 8*A*a^7*b^3*e^8)*d^3 + 360*(15873*B*b^10*d^8*e^3 + 45*B*a^8*b^2*e^11 + 120*A*a^7*b^3*e
^11 - 2178*(10*B*a*b^9*e^4 + A*b^10*e^4)*d^7 + 420*(9*B*a^2*b^8*e^5 + 2*A*a*b^9*e^5)*d^6 + 140*(8*B*a^3*b^7*e^
6 + 3*A*a^2*b^8*e^6)*d^5 + 70*(7*B*a^4*b^6*e^7 + 4*A*a^3*b^7*e^7)*d^4 + 42*(6*B*a^5*b^5*e^8 + 5*A*a^4*b^6*e^8)
*d^3 + 28*(5*B*a^6*b^4*e^9 + 6*A*a^5*b^5*e^9)*d^2 + 20*(4*B*a^7*b^3*e^10 + 7*A*a^6*b^4*e^10)*d)*x^3 + 35*(2*B*
a^9*b*e^9 + 9*A*a^8*b^2*e^9)*d^2 + 45*(50699*B*b^10*d^9*e^2 + 70*B*a^9*b*e^11 + 315*A*a^8*b^2*e^11 - 6849*(10*
B*a*b^9*e^3 + A*b^10*e^3)*d^8 + 1260*(9*B*a^2*b^8*e^4 + 2*A*a*b^9*e^4)*d^7 + 420*(8*B*a^3*b^7*e^5 + 3*A*a^2*b^
8*e^5)*d^6 + 210*(7*B*a^4*b^6*e^6 + 4*A*a^3*b^7*e^6)*d^5 + 126*(6*B*a^5*b^5*e^7 + 5*A*a^4*b^6*e^7)*d^4 + 84*(5
*B*a^6*b^4*e^8 + 6*A*a^5*b^5*e^8)*d^3 + 60*(4*B*a^7*b^3*e^9 + 7*A*a^6*b^4*e^9)*d^2 + 45*(3*B*a^8*b^2*e^10 + 8*
A*a^7*b^3*e^10)*d)*x^2 + 28*(B*a^10*e^10 + 10*A*a^9*b*e^10)*d + 10*(53471*B*b^10*d^10*e + 28*B*a^10*e^11 + 280
*A*a^9*b*e^11 - 7129*(10*B*a*b^9*e^2 + A*b^10*e^2)*d^9 + 1260*(9*B*a^2*b^8*e^3 + 2*A*a*b^9*e^3)*d^8 + 420*(8*B
*a^3*b^7*e^4 + 3*A*a^2*b^8*e^4)*d^7 + 210*(7*B*a^4*b^6*e^5 + 4*A*a^3*b^7*e^5)*d^6 + 126*(6*B*a^5*b^5*e^6 + 5*A
*a^4*b^6*e^6)*d^5 + 84*(5*B*a^6*b^4*e^7 + 6*A*a^5*b^5*e^7)*d^4 + 60*(4*B*a^7*b^3*e^8 + 7*A*a^6*b^4*e^8)*d^3 +
45*(3*B*a^8*b^2*e^9 + 8*A*a^7*b^3*e^9)*d^2 + 35*(2*B*a^9*b*e^10 + 9*A*a^8*b^2*e^10)*d)*x)/(x^10*e^22 + 10*d*x^
9*e^21 + 45*d^2*x^8*e^20 + 120*d^3*x^7*e^19 + 210*d^4*x^6*e^18 + 252*d^5*x^5*e^17 + 210*d^6*x^4*e^16 + 120*d^7
*x^3*e^15 + 45*d^8*x^2*e^14 + 10*d^9*x*e^13 + d^10*e^12)
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2237 vs.
\(2 (462) = 924\).
time = 1.15, size = 2237, normalized size = 5.02 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)^10*(B*x+A)/(e*x+d)^11,x, algorithm="fricas")
[Out]
-1/2520*(55991*B*b^10*d^11 - (2520*B*b^10*x^11 - 252*A*a^10 - 12600*(9*B*a^2*b^8 + 2*A*a*b^9)*x^9 - 18900*(8*B
*a^3*b^7 + 3*A*a^2*b^8)*x^8 - 25200*(7*B*a^4*b^6 + 4*A*a^3*b^7)*x^7 - 26460*(6*B*a^5*b^5 + 5*A*a^4*b^6)*x^6 -
21168*(5*B*a^6*b^4 + 6*A*a^5*b^5)*x^5 - 12600*(4*B*a^7*b^3 + 7*A*a^6*b^4)*x^4 - 5400*(3*B*a^8*b^2 + 8*A*a^7*b^
3)*x^3 - 1575*(2*B*a^9*b + 9*A*a^8*b^2)*x^2 - 280*(B*a^10 + 10*A*a^9*b)*x)*e^11 - (25200*B*b^10*d*x^10 + 25200
*(10*B*a*b^9 + A*b^10)*d*x^9 - 56700*(9*B*a^2*b^8 + 2*A*a*b^9)*d*x^8 - 50400*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d*x^7
- 44100*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d*x^6 - 31752*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d*x^5 - 17640*(5*B*a^6*b^4 + 6*
A*a^5*b^5)*d*x^4 - 7200*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d*x^3 - 2025*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d*x^2 - 350*(2*B*
a^9*b + 9*A*a^8*b^2)*d*x - 28*(B*a^10 + 10*A*a^9*b)*d)*e^10 + 5*(5040*B*b^10*d^2*x^9 - 34020*(10*B*a*b^9 + A*b
^10)*d^2*x^8 + 30240*(9*B*a^2*b^8 + 2*A*a*b^9)*d^2*x^7 + 17640*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^2*x^6 + 10584*(7*
B*a^4*b^6 + 4*A*a^3*b^7)*d^2*x^5 + 5292*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^2*x^4 + 2016*(5*B*a^6*b^4 + 6*A*a^5*b^5)
*d^2*x^3 + 540*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^2*x^2 + 90*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^2*x + 7*(2*B*a^9*b + 9*A
*a^8*b^2)*d^2)*e^9 + 15*(49140*B*b^10*d^3*x^8 - 36960*(10*B*a*b^9 + A*b^10)*d^3*x^7 + 17640*(9*B*a^2*b^8 + 2*A
*a*b^9)*d^3*x^6 + 7056*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^3*x^5 + 2940*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^3*x^4 + 1008*(
6*B*a^5*b^5 + 5*A*a^4*b^6)*d^3*x^3 + 252*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^3*x^2 + 40*(4*B*a^7*b^3 + 7*A*a^6*b^4)*
d^3*x + 3*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^3)*e^8 + 30*(102480*B*b^10*d^4*x^7 - 36750*(10*B*a*b^9 + A*b^10)*d^4*x
^6 + 10584*(9*B*a^2*b^8 + 2*A*a*b^9)*d^4*x^5 + 2940*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^4*x^4 + 840*(7*B*a^4*b^6 + 4
*A*a^3*b^7)*d^4*x^3 + 189*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^4*x^2 + 28*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^4*x + 2*(4*B*
a^7*b^3 + 7*A*a^6*b^4)*d^4)*e^7 + 42*(162750*B*b^10*d^5*x^6 - 34524*(10*B*a*b^9 + A*b^10)*d^5*x^5 + 6300*(9*B*
a^2*b^8 + 2*A*a*b^9)*d^5*x^4 + 1200*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^5*x^3 + 225*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^5*
x^2 + 30*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^5*x + 2*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^5)*e^6 + 42*(228564*B*b^10*d^6*x^
5 - 30870*(10*B*a*b^9 + A*b^10)*d^6*x^4 + 3600*(9*B*a^2*b^8 + 2*A*a*b^9)*d^6*x^3 + 450*(8*B*a^3*b^7 + 3*A*a^2*
b^8)*d^6*x^2 + 50*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^6*x + 3*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^6)*e^5 + 30*(298998*B*b^
10*d^7*x^4 - 26136*(10*B*a*b^9 + A*b^10)*d^7*x^3 + 1890*(9*B*a^2*b^8 + 2*A*a*b^9)*d^7*x^2 + 140*(8*B*a^3*b^7 +
3*A*a^2*b^8)*d^7*x + 7*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^7)*e^4 + 15*(373392*B*b^10*d^8*x^3 - 20547*(10*B*a*b^9 +
A*b^10)*d^8*x^2 + 840*(9*B*a^2*b^8 + 2*A*a*b^9)*d^8*x + 28*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^8)*e^3 + 5*(451251*B
*b^10*d^9*x^2 - 14258*(10*B*a*b^9 + A*b^10)*d^9*x + 252*(9*B*a^2*b^8 + 2*A*a*b^9)*d^9)*e^2 + (532190*B*b^10*d^
10*x - 7381*(10*B*a*b^9 + A*b^10)*d^10)*e + 2520*(11*B*b^10*d^11 - (10*B*a*b^9 + A*b^10)*x^10*e^11 + (11*B*b^1
0*d*x^10 - 10*(10*B*a*b^9 + A*b^10)*d*x^9)*e^10 + 5*(22*B*b^10*d^2*x^9 - 9*(10*B*a*b^9 + A*b^10)*d^2*x^8)*e^9
+ 15*(33*B*b^10*d^3*x^8 - 8*(10*B*a*b^9 + A*b^10)*d^3*x^7)*e^8 + 30*(44*B*b^10*d^4*x^7 - 7*(10*B*a*b^9 + A*b^1
0)*d^4*x^6)*e^7 + 42*(55*B*b^10*d^5*x^6 - 6*(10*B*a*b^9 + A*b^10)*d^5*x^5)*e^6 + 42*(66*B*b^10*d^6*x^5 - 5*(10
*B*a*b^9 + A*b^10)*d^6*x^4)*e^5 + 30*(77*B*b^10*d^7*x^4 - 4*(10*B*a*b^9 + A*b^10)*d^7*x^3)*e^4 + 15*(88*B*b^10
*d^8*x^3 - 3*(10*B*a*b^9 + A*b^10)*d^8*x^2)*e^3 + 5*(99*B*b^10*d^9*x^2 - 2*(10*B*a*b^9 + A*b^10)*d^9*x)*e^2 +
(110*B*b^10*d^10*x - (10*B*a*b^9 + A*b^10)*d^10)*e)*log(x*e + d))/(x^10*e^22 + 10*d*x^9*e^21 + 45*d^2*x^8*e^20
+ 120*d^3*x^7*e^19 + 210*d^4*x^6*e^18 + 252*d^5*x^5*e^17 + 210*d^6*x^4*e^16 + 120*d^7*x^3*e^15 + 45*d^8*x^2*e
^14 + 10*d^9*x*e^13 + d^10*e^12)
________________________________________________________________________________________
Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)**10*(B*x+A)/(e*x+d)**11,x)
[Out]
Timed out
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1839 vs.
\(2 (462) = 924\).
time = 1.82, size = 1839, normalized size = 4.12 \begin {gather*} B b^{10} x e^{\left (-11\right )} - {\left (11 \, B b^{10} d - 10 \, B a b^{9} e - A b^{10} e\right )} e^{\left (-12\right )} \log \left ({\left | x e + d \right |}\right ) - \frac {{\left (55991 \, B b^{10} d^{11} - 73810 \, B a b^{9} d^{10} e - 7381 \, A b^{10} d^{10} e + 11340 \, B a^{2} b^{8} d^{9} e^{2} + 2520 \, A a b^{9} d^{9} e^{2} + 3360 \, B a^{3} b^{7} d^{8} e^{3} + 1260 \, A a^{2} b^{8} d^{8} e^{3} + 1470 \, B a^{4} b^{6} d^{7} e^{4} + 840 \, A a^{3} b^{7} d^{7} e^{4} + 756 \, B a^{5} b^{5} d^{6} e^{5} + 630 \, A a^{4} b^{6} d^{6} e^{5} + 420 \, B a^{6} b^{4} d^{5} e^{6} + 504 \, A a^{5} b^{5} d^{5} e^{6} + 240 \, B a^{7} b^{3} d^{4} e^{7} + 420 \, 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} + 70 \, B a^{9} b d^{2} e^{9} + 315 \, A a^{8} b^{2} d^{2} e^{9} + 28 \, B a^{10} d e^{10} + 280 \, A a^{9} b d e^{10} + 252 \, A a^{10} e^{11} + 12600 \, {\left (11 \, B b^{10} d^{2} e^{9} - 20 \, B a b^{9} d e^{10} - 2 \, A b^{10} d e^{10} + 9 \, B a^{2} b^{8} e^{11} + 2 \, A a b^{9} e^{11}\right )} x^{9} + 18900 \, {\left (55 \, B b^{10} d^{3} e^{8} - 90 \, B a b^{9} d^{2} e^{9} - 9 \, A b^{10} d^{2} e^{9} + 27 \, B a^{2} b^{8} d e^{10} + 6 \, A a b^{9} d e^{10} + 8 \, B a^{3} b^{7} e^{11} + 3 \, A a^{2} b^{8} e^{11}\right )} x^{8} + 25200 \, {\left (143 \, B b^{10} d^{4} e^{7} - 220 \, B a b^{9} d^{3} e^{8} - 22 \, A b^{10} d^{3} e^{8} + 54 \, B a^{2} b^{8} d^{2} e^{9} + 12 \, A a b^{9} d^{2} e^{9} + 16 \, B a^{3} b^{7} d e^{10} + 6 \, A a^{2} b^{8} d e^{10} + 7 \, B a^{4} b^{6} e^{11} + 4 \, A a^{3} b^{7} e^{11}\right )} x^{7} + 8820 \, {\left (847 \, B b^{10} d^{5} e^{6} - 1250 \, B a b^{9} d^{4} e^{7} - 125 \, A b^{10} d^{4} e^{7} + 270 \, B a^{2} b^{8} d^{3} e^{8} + 60 \, A a b^{9} d^{3} e^{8} + 80 \, B a^{3} b^{7} d^{2} e^{9} + 30 \, A a^{2} b^{8} d^{2} e^{9} + 35 \, B a^{4} b^{6} d e^{10} + 20 \, A a^{3} b^{7} d e^{10} + 18 \, B a^{5} b^{5} e^{11} + 15 \, A a^{4} b^{6} e^{11}\right )} x^{6} + 10584 \, {\left (957 \, B b^{10} d^{6} e^{5} - 1370 \, B a b^{9} d^{5} e^{6} - 137 \, A b^{10} d^{5} e^{6} + 270 \, B a^{2} b^{8} d^{4} e^{7} + 60 \, A a b^{9} d^{4} e^{7} + 80 \, B a^{3} b^{7} d^{3} e^{8} + 30 \, A a^{2} b^{8} d^{3} e^{8} + 35 \, B a^{4} b^{6} d^{2} e^{9} + 20 \, A a^{3} b^{7} d^{2} e^{9} + 18 \, B a^{5} b^{5} d e^{10} + 15 \, A a^{4} b^{6} d e^{10} + 10 \, B a^{6} b^{4} e^{11} + 12 \, A a^{5} b^{5} e^{11}\right )} x^{5} + 1260 \, {\left (7359 \, B b^{10} d^{7} e^{4} - 10290 \, B a b^{9} d^{6} e^{5} - 1029 \, A b^{10} d^{6} e^{5} + 1890 \, B a^{2} b^{8} d^{5} e^{6} + 420 \, A a b^{9} d^{5} e^{6} + 560 \, B a^{3} b^{7} d^{4} e^{7} + 210 \, A a^{2} b^{8} d^{4} e^{7} + 245 \, B a^{4} b^{6} d^{3} e^{8} + 140 \, A a^{3} b^{7} d^{3} e^{8} + 126 \, B a^{5} b^{5} d^{2} e^{9} + 105 \, A a^{4} b^{6} d^{2} e^{9} + 70 \, B a^{6} b^{4} d e^{10} + 84 \, A a^{5} b^{5} d e^{10} + 40 \, B a^{7} b^{3} e^{11} + 70 \, A a^{6} b^{4} e^{11}\right )} x^{4} + 360 \, {\left (15873 \, B b^{10} d^{8} e^{3} - 21780 \, B a b^{9} d^{7} e^{4} - 2178 \, A b^{10} d^{7} e^{4} + 3780 \, B a^{2} b^{8} d^{6} e^{5} + 840 \, A a b^{9} d^{6} e^{5} + 1120 \, B a^{3} b^{7} d^{5} e^{6} + 420 \, A a^{2} b^{8} d^{5} e^{6} + 490 \, B a^{4} b^{6} d^{4} e^{7} + 280 \, A a^{3} b^{7} d^{4} e^{7} + 252 \, B a^{5} b^{5} d^{3} e^{8} + 210 \, A a^{4} b^{6} d^{3} e^{8} + 140 \, B a^{6} b^{4} d^{2} e^{9} + 168 \, A a^{5} b^{5} d^{2} e^{9} + 80 \, B a^{7} b^{3} d e^{10} + 140 \, A a^{6} b^{4} d e^{10} + 45 \, B a^{8} b^{2} e^{11} + 120 \, A a^{7} b^{3} e^{11}\right )} x^{3} + 45 \, {\left (50699 \, B b^{10} d^{9} e^{2} - 68490 \, B a b^{9} d^{8} e^{3} - 6849 \, A b^{10} d^{8} e^{3} + 11340 \, B a^{2} b^{8} d^{7} e^{4} + 2520 \, A a b^{9} d^{7} e^{4} + 3360 \, B a^{3} b^{7} d^{6} e^{5} + 1260 \, A a^{2} b^{8} d^{6} e^{5} + 1470 \, B a^{4} b^{6} d^{5} e^{6} + 840 \, A a^{3} b^{7} d^{5} e^{6} + 756 \, B a^{5} b^{5} d^{4} e^{7} + 630 \, A a^{4} b^{6} d^{4} e^{7} + 420 \, B a^{6} b^{4} d^{3} e^{8} + 504 \, A a^{5} b^{5} d^{3} e^{8} + 240 \, B a^{7} b^{3} d^{2} e^{9} + 420 \, A a^{6} b^{4} d^{2} e^{9} + 135 \, B a^{8} b^{2} d e^{10} + 360 \, A a^{7} b^{3} d e^{10} + 70 \, B a^{9} b e^{11} + 315 \, A a^{8} b^{2} e^{11}\right )} x^{2} + 10 \, {\left (53471 \, B b^{10} d^{10} e - 71290 \, B a b^{9} d^{9} e^{2} - 7129 \, A b^{10} d^{9} e^{2} + 11340 \, B a^{2} b^{8} d^{8} e^{3} + 2520 \, A a b^{9} d^{8} e^{3} + 3360 \, B a^{3} b^{7} d^{7} e^{4} + 1260 \, A a^{2} b^{8} d^{7} e^{4} + 1470 \, B a^{4} b^{6} d^{6} e^{5} + 840 \, A a^{3} b^{7} d^{6} e^{5} + 756 \, B a^{5} b^{5} d^{5} e^{6} + 630 \, A a^{4} b^{6} d^{5} e^{6} + 420 \, B a^{6} b^{4} d^{4} e^{7} + 504 \, A a^{5} b^{5} d^{4} e^{7} + 240 \, B a^{7} b^{3} d^{3} e^{8} + 420 \, A a^{6} b^{4} d^{3} e^{8} + 135 \, B a^{8} b^{2} d^{2} e^{9} + 360 \, A a^{7} b^{3} d^{2} e^{9} + 70 \, B a^{9} b d e^{10} + 315 \, A a^{8} b^{2} d e^{10} + 28 \, B a^{10} e^{11} + 280 \, A a^{9} b e^{11}\right )} x\right )} e^{\left (-12\right )}}{2520 \, {\left (x e + d\right )}^{10}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)^10*(B*x+A)/(e*x+d)^11,x, algorithm="giac")
[Out]
B*b^10*x*e^(-11) - (11*B*b^10*d - 10*B*a*b^9*e - A*b^10*e)*e^(-12)*log(abs(x*e + d)) - 1/2520*(55991*B*b^10*d^
11 - 73810*B*a*b^9*d^10*e - 7381*A*b^10*d^10*e + 11340*B*a^2*b^8*d^9*e^2 + 2520*A*a*b^9*d^9*e^2 + 3360*B*a^3*b
^7*d^8*e^3 + 1260*A*a^2*b^8*d^8*e^3 + 1470*B*a^4*b^6*d^7*e^4 + 840*A*a^3*b^7*d^7*e^4 + 756*B*a^5*b^5*d^6*e^5 +
630*A*a^4*b^6*d^6*e^5 + 420*B*a^6*b^4*d^5*e^6 + 504*A*a^5*b^5*d^5*e^6 + 240*B*a^7*b^3*d^4*e^7 + 420*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 + 70*B*a^9*b*d^2*e^9 + 315*A*a^8*b^2*d^2*e^9 + 28*B*a
^10*d*e^10 + 280*A*a^9*b*d*e^10 + 252*A*a^10*e^11 + 12600*(11*B*b^10*d^2*e^9 - 20*B*a*b^9*d*e^10 - 2*A*b^10*d*
e^10 + 9*B*a^2*b^8*e^11 + 2*A*a*b^9*e^11)*x^9 + 18900*(55*B*b^10*d^3*e^8 - 90*B*a*b^9*d^2*e^9 - 9*A*b^10*d^2*e
^9 + 27*B*a^2*b^8*d*e^10 + 6*A*a*b^9*d*e^10 + 8*B*a^3*b^7*e^11 + 3*A*a^2*b^8*e^11)*x^8 + 25200*(143*B*b^10*d^4
*e^7 - 220*B*a*b^9*d^3*e^8 - 22*A*b^10*d^3*e^8 + 54*B*a^2*b^8*d^2*e^9 + 12*A*a*b^9*d^2*e^9 + 16*B*a^3*b^7*d*e^
10 + 6*A*a^2*b^8*d*e^10 + 7*B*a^4*b^6*e^11 + 4*A*a^3*b^7*e^11)*x^7 + 8820*(847*B*b^10*d^5*e^6 - 1250*B*a*b^9*d
^4*e^7 - 125*A*b^10*d^4*e^7 + 270*B*a^2*b^8*d^3*e^8 + 60*A*a*b^9*d^3*e^8 + 80*B*a^3*b^7*d^2*e^9 + 30*A*a^2*b^8
*d^2*e^9 + 35*B*a^4*b^6*d*e^10 + 20*A*a^3*b^7*d*e^10 + 18*B*a^5*b^5*e^11 + 15*A*a^4*b^6*e^11)*x^6 + 10584*(957
*B*b^10*d^6*e^5 - 1370*B*a*b^9*d^5*e^6 - 137*A*b^10*d^5*e^6 + 270*B*a^2*b^8*d^4*e^7 + 60*A*a*b^9*d^4*e^7 + 80*
B*a^3*b^7*d^3*e^8 + 30*A*a^2*b^8*d^3*e^8 + 35*B*a^4*b^6*d^2*e^9 + 20*A*a^3*b^7*d^2*e^9 + 18*B*a^5*b^5*d*e^10 +
15*A*a^4*b^6*d*e^10 + 10*B*a^6*b^4*e^11 + 12*A*a^5*b^5*e^11)*x^5 + 1260*(7359*B*b^10*d^7*e^4 - 10290*B*a*b^9*
d^6*e^5 - 1029*A*b^10*d^6*e^5 + 1890*B*a^2*b^8*d^5*e^6 + 420*A*a*b^9*d^5*e^6 + 560*B*a^3*b^7*d^4*e^7 + 210*A*a
^2*b^8*d^4*e^7 + 245*B*a^4*b^6*d^3*e^8 + 140*A*a^3*b^7*d^3*e^8 + 126*B*a^5*b^5*d^2*e^9 + 105*A*a^4*b^6*d^2*e^9
+ 70*B*a^6*b^4*d*e^10 + 84*A*a^5*b^5*d*e^10 + 40*B*a^7*b^3*e^11 + 70*A*a^6*b^4*e^11)*x^4 + 360*(15873*B*b^10*
d^8*e^3 - 21780*B*a*b^9*d^7*e^4 - 2178*A*b^10*d^7*e^4 + 3780*B*a^2*b^8*d^6*e^5 + 840*A*a*b^9*d^6*e^5 + 1120*B*
a^3*b^7*d^5*e^6 + 420*A*a^2*b^8*d^5*e^6 + 490*B*a^4*b^6*d^4*e^7 + 280*A*a^3*b^7*d^4*e^7 + 252*B*a^5*b^5*d^3*e^
8 + 210*A*a^4*b^6*d^3*e^8 + 140*B*a^6*b^4*d^2*e^9 + 168*A*a^5*b^5*d^2*e^9 + 80*B*a^7*b^3*d*e^10 + 140*A*a^6*b^
4*d*e^10 + 45*B*a^8*b^2*e^11 + 120*A*a^7*b^3*e^11)*x^3 + 45*(50699*B*b^10*d^9*e^2 - 68490*B*a*b^9*d^8*e^3 - 68
49*A*b^10*d^8*e^3 + 11340*B*a^2*b^8*d^7*e^4 + 2520*A*a*b^9*d^7*e^4 + 3360*B*a^3*b^7*d^6*e^5 + 1260*A*a^2*b^8*d
^6*e^5 + 1470*B*a^4*b^6*d^5*e^6 + 840*A*a^3*b^7*d^5*e^6 + 756*B*a^5*b^5*d^4*e^7 + 630*A*a^4*b^6*d^4*e^7 + 420*
B*a^6*b^4*d^3*e^8 + 504*A*a^5*b^5*d^3*e^8 + 240*B*a^7*b^3*d^2*e^9 + 420*A*a^6*b^4*d^2*e^9 + 135*B*a^8*b^2*d*e^
10 + 360*A*a^7*b^3*d*e^10 + 70*B*a^9*b*e^11 + 315*A*a^8*b^2*e^11)*x^2 + 10*(53471*B*b^10*d^10*e - 71290*B*a*b^
9*d^9*e^2 - 7129*A*b^10*d^9*e^2 + 11340*B*a^2*b^8*d^8*e^3 + 2520*A*a*b^9*d^8*e^3 + 3360*B*a^3*b^7*d^7*e^4 + 12
60*A*a^2*b^8*d^7*e^4 + 1470*B*a^4*b^6*d^6*e^5 + 840*A*a^3*b^7*d^6*e^5 + 756*B*a^5*b^5*d^5*e^6 + 630*A*a^4*b^6*
d^5*e^6 + 420*B*a^6*b^4*d^4*e^7 + 504*A*a^5*b^5*d^4*e^7 + 240*B*a^7*b^3*d^3*e^8 + 420*A*a^6*b^4*d^3*e^8 + 135*
B*a^8*b^2*d^2*e^9 + 360*A*a^7*b^3*d^2*e^9 + 70*B*a^9*b*d*e^10 + 315*A*a^8*b^2*d*e^10 + 28*B*a^10*e^11 + 280*A*
a^9*b*e^11)*x)*e^(-12)/(x*e + d)^10
________________________________________________________________________________________
Mupad [B]
time = 1.76, size = 2500, normalized size = 5.61 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(((A + B*x)*(a + b*x)^10)/(d + e*x)^11,x)
[Out]
-((A*a^10*e^11)/10 + (55991*B*b^10*d^11)/2520 - (7381*A*b^10*d^10*e)/2520 + (B*a^10*d*e^10)/90 + 11*B*b^10*d^1
1*log(d + e*x) + (B*a^10*e^11*x)/9 - B*b^10*e^11*x^11 + (53219*B*b^10*d^10*e*x)/252 + A*a*b^9*d^9*e^2 + (B*a^9
*b*d^2*e^9)/36 + 10*A*a*b^9*e^11*x^9 + (5*B*a^9*b*e^11*x^2)/4 - (7129*A*b^10*d^9*e^2*x)/252 - 10*A*b^10*d*e^10
*x^9 - 10*B*b^10*d*e^10*x^10 - A*b^10*e^11*x^10*log(d + e*x) + (A*a^2*b^8*d^8*e^3)/2 + (A*a^3*b^7*d^7*e^4)/3 +
(A*a^4*b^6*d^6*e^5)/4 + (A*a^5*b^5*d^5*e^6)/5 + (A*a^6*b^4*d^4*e^7)/6 + (A*a^7*b^3*d^3*e^8)/7 + (A*a^8*b^2*d^
2*e^9)/8 + (9*B*a^2*b^8*d^9*e^2)/2 + (4*B*a^3*b^7*d^8*e^3)/3 + (7*B*a^4*b^6*d^7*e^4)/12 + (3*B*a^5*b^5*d^6*e^5
)/10 + (B*a^6*b^4*d^5*e^6)/6 + (2*B*a^7*b^3*d^4*e^7)/21 + (3*B*a^8*b^2*d^3*e^8)/56 + (45*A*a^8*b^2*e^11*x^2)/8
+ (120*A*a^7*b^3*e^11*x^3)/7 + 35*A*a^6*b^4*e^11*x^4 + (252*A*a^5*b^5*e^11*x^5)/5 + (105*A*a^4*b^6*e^11*x^6)/
2 + 40*A*a^3*b^7*e^11*x^7 + (45*A*a^2*b^8*e^11*x^8)/2 + (45*B*a^8*b^2*e^11*x^3)/7 + 20*B*a^7*b^3*e^11*x^4 + 42
*B*a^6*b^4*e^11*x^5 + 63*B*a^5*b^5*e^11*x^6 + 70*B*a^4*b^6*e^11*x^7 + 60*B*a^3*b^7*e^11*x^8 + 45*B*a^2*b^8*e^1
1*x^9 - (6849*A*b^10*d^8*e^3*x^2)/56 - (2178*A*b^10*d^7*e^4*x^3)/7 - (1029*A*b^10*d^6*e^5*x^4)/2 - (2877*A*b^1
0*d^5*e^6*x^5)/5 - (875*A*b^10*d^4*e^7*x^6)/2 - 220*A*b^10*d^3*e^8*x^7 - (135*A*b^10*d^2*e^9*x^8)/2 + (50139*B
*b^10*d^9*e^2*x^2)/56 + (15558*B*b^10*d^8*e^3*x^3)/7 + (7119*B*b^10*d^7*e^4*x^4)/2 + (19047*B*b^10*d^6*e^5*x^5
)/5 + (5425*B*b^10*d^5*e^6*x^6)/2 + 1220*B*b^10*d^4*e^7*x^7 + (585*B*b^10*d^3*e^8*x^8)/2 + 10*B*b^10*d^2*e^9*x
^9 + (A*a^9*b*d*e^10)/9 - (7381*B*a*b^9*d^10*e)/252 - A*b^10*d^10*e*log(d + e*x) + (10*A*a^9*b*e^11*x)/9 + 5*A
*a^2*b^8*d^7*e^4*x + (10*A*a^3*b^7*d^6*e^5*x)/3 + (5*A*a^4*b^6*d^5*e^6*x)/2 + 2*A*a^5*b^5*d^4*e^7*x + (5*A*a^6
*b^4*d^3*e^8*x)/3 + (10*A*a^7*b^3*d^2*e^9*x)/7 + 45*A*a*b^9*d^7*e^4*x^2 + (45*A*a^7*b^3*d*e^10*x^2)/7 + 120*A*
a*b^9*d^6*e^5*x^3 + 20*A*a^6*b^4*d*e^10*x^3 + 210*A*a*b^9*d^5*e^6*x^4 + 42*A*a^5*b^5*d*e^10*x^4 + 252*A*a*b^9*
d^4*e^7*x^5 + 63*A*a^4*b^6*d*e^10*x^5 + 210*A*a*b^9*d^3*e^8*x^6 + 70*A*a^3*b^7*d*e^10*x^6 + 120*A*a*b^9*d^2*e^
9*x^7 + 60*A*a^2*b^8*d*e^10*x^7 + 45*B*a^2*b^8*d^8*e^3*x + (40*B*a^3*b^7*d^7*e^4*x)/3 + (35*B*a^4*b^6*d^6*e^5*
x)/6 + 3*B*a^5*b^5*d^5*e^6*x + (5*B*a^6*b^4*d^4*e^7*x)/3 + (20*B*a^7*b^3*d^3*e^8*x)/21 + (15*B*a^8*b^2*d^2*e^9
*x)/28 - (34245*B*a*b^9*d^8*e^3*x^2)/28 + (135*B*a^8*b^2*d*e^10*x^2)/56 - (21780*B*a*b^9*d^7*e^4*x^3)/7 + (80*
B*a^7*b^3*d*e^10*x^3)/7 - 5145*B*a*b^9*d^6*e^5*x^4 + 35*B*a^6*b^4*d*e^10*x^4 - 5754*B*a*b^9*d^5*e^6*x^5 + (378
*B*a^5*b^5*d*e^10*x^5)/5 - 4375*B*a*b^9*d^4*e^7*x^6 + (245*B*a^4*b^6*d*e^10*x^6)/2 - 2200*B*a*b^9*d^3*e^8*x^7
+ 160*B*a^3*b^7*d*e^10*x^7 - 675*B*a*b^9*d^2*e^9*x^8 + (405*B*a^2*b^8*d*e^10*x^8)/2 - 45*A*b^10*d^8*e^3*x^2*lo
g(d + e*x) - 120*A*b^10*d^7*e^4*x^3*log(d + e*x) - 210*A*b^10*d^6*e^5*x^4*log(d + e*x) - 252*A*b^10*d^5*e^6*x^
5*log(d + e*x) - 210*A*b^10*d^4*e^7*x^6*log(d + e*x) - 120*A*b^10*d^3*e^8*x^7*log(d + e*x) - 45*A*b^10*d^2*e^9
*x^8*log(d + e*x) + 495*B*b^10*d^9*e^2*x^2*log(d + e*x) + 1320*B*b^10*d^8*e^3*x^3*log(d + e*x) + 2310*B*b^10*d
^7*e^4*x^4*log(d + e*x) + 2772*B*b^10*d^6*e^5*x^5*log(d + e*x) + 2310*B*b^10*d^5*e^6*x^6*log(d + e*x) + 1320*B
*b^10*d^4*e^7*x^7*log(d + e*x) + 495*B*b^10*d^3*e^8*x^8*log(d + e*x) + 110*B*b^10*d^2*e^9*x^9*log(d + e*x) - 1
0*B*a*b^9*d^10*e*log(d + e*x) + (5*B*a^9*b*d*e^10*x)/18 + 110*B*b^10*d^10*e*x*log(d + e*x) + (45*A*a^2*b^8*d^6
*e^5*x^2)/2 + 15*A*a^3*b^7*d^5*e^6*x^2 + (45*A*a^4*b^6*d^4*e^7*x^2)/4 + 9*A*a^5*b^5*d^3*e^8*x^2 + (15*A*a^6*b^
4*d^2*e^9*x^2)/2 + 60*A*a^2*b^8*d^5*e^6*x^3 + 40*A*a^3*b^7*d^4*e^7*x^3 + 30*A*a^4*b^6*d^3*e^8*x^3 + 24*A*a^5*b
^5*d^2*e^9*x^3 + 105*A*a^2*b^8*d^4*e^7*x^4 + 70*A*a^3*b^7*d^3*e^8*x^4 + (105*A*a^4*b^6*d^2*e^9*x^4)/2 + 126*A*
a^2*b^8*d^3*e^8*x^5 + 84*A*a^3*b^7*d^2*e^9*x^5 + 105*A*a^2*b^8*d^2*e^9*x^6 + (405*B*a^2*b^8*d^7*e^4*x^2)/2 + 6
0*B*a^3*b^7*d^6*e^5*x^2 + (105*B*a^4*b^6*d^5*e^6*x^2)/4 + (27*B*a^5*b^5*d^4*e^7*x^2)/2 + (15*B*a^6*b^4*d^3*e^8
*x^2)/2 + (30*B*a^7*b^3*d^2*e^9*x^2)/7 + 540*B*a^2*b^8*d^6*e^5*x^3 + 160*B*a^3*b^7*d^5*e^6*x^3 + 70*B*a^4*b^6*
d^4*e^7*x^3 + 36*B*a^5*b^5*d^3*e^8*x^3 + 20*B*a^6*b^4*d^2*e^9*x^3 + 945*B*a^2*b^8*d^5*e^6*x^4 + 280*B*a^3*b^7*
d^4*e^7*x^4 + (245*B*a^4*b^6*d^3*e^8*x^4)/2 + 63*B*a^5*b^5*d^2*e^9*x^4 + 1134*B*a^2*b^8*d^4*e^7*x^5 + 336*B*a^
3*b^7*d^3*e^8*x^5 + 147*B*a^4*b^6*d^2*e^9*x^5 + 945*B*a^2*b^8*d^3*e^8*x^6 + 280*B*a^3*b^7*d^2*e^9*x^6 + 540*B*
a^2*b^8*d^2*e^9*x^7 + 10*A*a*b^9*d^8*e^3*x + (5*A*a^8*b^2*d*e^10*x)/4 + 45*A*a*b^9*d*e^10*x^8 - (35645*B*a*b^9
*d^9*e^2*x)/126 - 100*B*a*b^9*d*e^10*x^9 - 10*B*a*b^9*e^11*x^10*log(d + e*x) - 10*A*b^10*d^9*e^2*x*log(d + e*x
) - 10*A*b^10*d*e^10*x^9*log(d + e*x) + 11*B*b^10*d*e^10*x^10*log(d + e*x) - 100*B*a*b^9*d^9*e^2*x*log(d + e*x
) - 100*B*a*b^9*d*e^10*x^9*log(d + e*x) - 450*B*a*b^9*d^8*e^3*x^2*log(d + e*x) - 1200*B*a*b^9*d^7*e^4*x^3*log(
d + e*x) - 2100*B*a*b^9*d^6*e^5*x^4*log(d + e*x) - 2520*B*a*b^9*d^5*e^6*x^5*log(d + e*x) - 2100*B*a*b^9*d^4*e^
7*x^6*log(d + e*x) - 1200*B*a*b^9*d^3*e^8*x^7*l...
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