3.11.92 \(\int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^4} \, dx\) [1092]
Optimal. Leaf size=445 \[ -\frac {30 b^3 (b d-a e)^6 (11 b B d-7 A b e-4 a B e) x}{e^{11}}+\frac {(b d-a e)^{10} (B d-A e)}{3 e^{12} (d+e x)^3}-\frac {(b d-a e)^9 (11 b B d-10 A b e-a B e)}{2 e^{12} (d+e x)^2}+\frac {5 b (b d-a e)^8 (11 b B d-9 A b e-2 a B e)}{e^{12} (d+e x)}+\frac {21 b^4 (b d-a e)^5 (11 b B d-6 A b e-5 a B e) (d+e x)^2}{e^{12}}-\frac {14 b^5 (b d-a e)^4 (11 b B d-5 A b e-6 a B e) (d+e x)^3}{e^{12}}+\frac {15 b^6 (b d-a e)^3 (11 b B d-4 A b e-7 a B e) (d+e x)^4}{2 e^{12}}-\frac {3 b^7 (b d-a e)^2 (11 b B d-3 A b e-8 a B e) (d+e x)^5}{e^{12}}+\frac {5 b^8 (b d-a e) (11 b B d-2 A b e-9 a B e) (d+e x)^6}{6 e^{12}}-\frac {b^9 (11 b B d-A b e-10 a B e) (d+e x)^7}{7 e^{12}}+\frac {b^{10} B (d+e x)^8}{8 e^{12}}+\frac {15 b^2 (b d-a e)^7 (11 b B d-8 A b e-3 a B e) \log (d+e x)}{e^{12}} \]
[Out]
-30*b^3*(-a*e+b*d)^6*(-7*A*b*e-4*B*a*e+11*B*b*d)*x/e^11+1/3*(-a*e+b*d)^10*(-A*e+B*d)/e^12/(e*x+d)^3-1/2*(-a*e+
b*d)^9*(-10*A*b*e-B*a*e+11*B*b*d)/e^12/(e*x+d)^2+5*b*(-a*e+b*d)^8*(-9*A*b*e-2*B*a*e+11*B*b*d)/e^12/(e*x+d)+21*
b^4*(-a*e+b*d)^5*(-6*A*b*e-5*B*a*e+11*B*b*d)*(e*x+d)^2/e^12-14*b^5*(-a*e+b*d)^4*(-5*A*b*e-6*B*a*e+11*B*b*d)*(e
*x+d)^3/e^12+15/2*b^6*(-a*e+b*d)^3*(-4*A*b*e-7*B*a*e+11*B*b*d)*(e*x+d)^4/e^12-3*b^7*(-a*e+b*d)^2*(-3*A*b*e-8*B
*a*e+11*B*b*d)*(e*x+d)^5/e^12+5/6*b^8*(-a*e+b*d)*(-2*A*b*e-9*B*a*e+11*B*b*d)*(e*x+d)^6/e^12-1/7*b^9*(-A*b*e-10
*B*a*e+11*B*b*d)*(e*x+d)^7/e^12+1/8*b^10*B*(e*x+d)^8/e^12+15*b^2*(-a*e+b*d)^7*(-8*A*b*e-3*B*a*e+11*B*b*d)*ln(e
*x+d)/e^12
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Rubi [A]
time = 0.91, antiderivative size = 445, 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 (d+e x)^7 (-10 a B e-A b e+11 b B d)}{7 e^{12}}+\frac {5 b^8 (d+e x)^6 (b d-a e) (-9 a B e-2 A b e+11 b B d)}{6 e^{12}}-\frac {3 b^7 (d+e x)^5 (b d-a e)^2 (-8 a B e-3 A b e+11 b B d)}{e^{12}}+\frac {15 b^6 (d+e x)^4 (b d-a e)^3 (-7 a B e-4 A b e+11 b B d)}{2 e^{12}}-\frac {14 b^5 (d+e x)^3 (b d-a e)^4 (-6 a B e-5 A b e+11 b B d)}{e^{12}}+\frac {21 b^4 (d+e x)^2 (b d-a e)^5 (-5 a B e-6 A b e+11 b B d)}{e^{12}}-\frac {30 b^3 x (b d-a e)^6 (-4 a B e-7 A b e+11 b B d)}{e^{11}}+\frac {15 b^2 (b d-a e)^7 \log (d+e x) (-3 a B e-8 A b e+11 b B d)}{e^{12}}+\frac {5 b (b d-a e)^8 (-2 a B e-9 A b e+11 b B d)}{e^{12} (d+e x)}-\frac {(b d-a e)^9 (-a B e-10 A b e+11 b B d)}{2 e^{12} (d+e x)^2}+\frac {(b d-a e)^{10} (B d-A e)}{3 e^{12} (d+e x)^3}+\frac {b^{10} B (d+e x)^8}{8 e^{12}} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[((a + b*x)^10*(A + B*x))/(d + e*x)^4,x]
[Out]
(-30*b^3*(b*d - a*e)^6*(11*b*B*d - 7*A*b*e - 4*a*B*e)*x)/e^11 + ((b*d - a*e)^10*(B*d - A*e))/(3*e^12*(d + e*x)
^3) - ((b*d - a*e)^9*(11*b*B*d - 10*A*b*e - a*B*e))/(2*e^12*(d + e*x)^2) + (5*b*(b*d - a*e)^8*(11*b*B*d - 9*A*
b*e - 2*a*B*e))/(e^12*(d + e*x)) + (21*b^4*(b*d - a*e)^5*(11*b*B*d - 6*A*b*e - 5*a*B*e)*(d + e*x)^2)/e^12 - (1
4*b^5*(b*d - a*e)^4*(11*b*B*d - 5*A*b*e - 6*a*B*e)*(d + e*x)^3)/e^12 + (15*b^6*(b*d - a*e)^3*(11*b*B*d - 4*A*b
*e - 7*a*B*e)*(d + e*x)^4)/(2*e^12) - (3*b^7*(b*d - a*e)^2*(11*b*B*d - 3*A*b*e - 8*a*B*e)*(d + e*x)^5)/e^12 +
(5*b^8*(b*d - a*e)*(11*b*B*d - 2*A*b*e - 9*a*B*e)*(d + e*x)^6)/(6*e^12) - (b^9*(11*b*B*d - A*b*e - 10*a*B*e)*(
d + e*x)^7)/(7*e^12) + (b^10*B*(d + e*x)^8)/(8*e^12) + (15*b^2*(b*d - a*e)^7*(11*b*B*d - 8*A*b*e - 3*a*B*e)*Lo
g[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)^4} \, dx &=\int \left (\frac {30 b^3 (b d-a e)^6 (-11 b B d+7 A b e+4 a B e)}{e^{11}}+\frac {(-b d+a e)^{10} (-B d+A e)}{e^{11} (d+e x)^4}+\frac {(-b d+a e)^9 (-11 b B d+10 A b e+a B e)}{e^{11} (d+e x)^3}+\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)^2}-\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)}-\frac {42 b^4 (b d-a e)^5 (-11 b B d+6 A b e+5 a B e) (d+e x)}{e^{11}}+\frac {42 b^5 (b d-a e)^4 (-11 b B d+5 A b e+6 a B e) (d+e x)^2}{e^{11}}-\frac {30 b^6 (b d-a e)^3 (-11 b B d+4 A b e+7 a B e) (d+e x)^3}{e^{11}}+\frac {15 b^7 (b d-a e)^2 (-11 b B d+3 A b e+8 a B e) (d+e x)^4}{e^{11}}-\frac {5 b^8 (b d-a e) (-11 b B d+2 A b e+9 a B e) (d+e x)^5}{e^{11}}+\frac {b^9 (-11 b B d+A b e+10 a B e) (d+e x)^6}{e^{11}}+\frac {b^{10} B (d+e x)^7}{e^{11}}\right ) \, dx\\ &=-\frac {30 b^3 (b d-a e)^6 (11 b B d-7 A b e-4 a B e) x}{e^{11}}+\frac {(b d-a e)^{10} (B d-A e)}{3 e^{12} (d+e x)^3}-\frac {(b d-a e)^9 (11 b B d-10 A b e-a B e)}{2 e^{12} (d+e x)^2}+\frac {5 b (b d-a e)^8 (11 b B d-9 A b e-2 a B e)}{e^{12} (d+e x)}+\frac {21 b^4 (b d-a e)^5 (11 b B d-6 A b e-5 a B e) (d+e x)^2}{e^{12}}-\frac {14 b^5 (b d-a e)^4 (11 b B d-5 A b e-6 a B e) (d+e x)^3}{e^{12}}+\frac {15 b^6 (b d-a e)^3 (11 b B d-4 A b e-7 a B e) (d+e x)^4}{2 e^{12}}-\frac {3 b^7 (b d-a e)^2 (11 b B d-3 A b e-8 a B e) (d+e x)^5}{e^{12}}+\frac {5 b^8 (b d-a e) (11 b B d-2 A b e-9 a B e) (d+e x)^6}{6 e^{12}}-\frac {b^9 (11 b B d-A b e-10 a B e) (d+e x)^7}{7 e^{12}}+\frac {b^{10} B (d+e x)^8}{8 e^{12}}+\frac {15 b^2 (b d-a e)^7 (11 b B d-8 A b e-3 a B e) \log (d+e x)}{e^{12}}\\ \end {align*}
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Mathematica [A]
time = 0.32, size = 814, normalized size = 1.83 \begin {gather*} \frac {168 b^3 e \left (120 a^7 B e^7-315 a^2 b^5 d^4 e^2 (8 B d-5 A e)+600 a^3 b^4 d^3 e^3 (7 B d-4 A e)+280 a b^6 d^5 e (3 B d-2 A e)+504 a^5 b^2 d e^5 (5 B d-2 A e)-2100 a^4 b^3 d^2 e^4 (2 B d-A e)+210 a^6 b e^6 (-4 B d+A e)+12 b^7 d^6 (-10 B d+7 A e)\right ) x-84 b^4 e^2 \left (-210 a^6 B e^6+70 a b^5 d^4 e (8 B d-5 A e)-225 a^2 b^4 d^3 e^2 (7 B d-4 A e)-420 a^4 b^2 d e^4 (5 B d-2 A e)+1200 a^3 b^3 d^2 e^3 (2 B d-A e)-252 a^5 b e^5 (-4 B d+A e)+28 b^6 d^5 (-3 B d+2 A e)\right ) x^2+56 b^5 e^3 \left (252 a^5 B e^5-7 b^5 d^4 (8 B d-5 A e)+50 a b^4 d^3 e (7 B d-4 A e)+240 a^3 b^2 d e^3 (5 B d-2 A e)-450 a^2 b^3 d^2 e^2 (2 B d-A e)+210 a^4 b e^4 (-4 B d+A e)\right ) x^3-210 b^6 e^4 \left (-42 a^4 B e^4+20 a b^3 d^2 e (2 B d-A e)-24 a^3 b e^3 (-4 B d+A e)+18 a^2 b^2 d e^2 (-5 B d+2 A e)+b^4 d^3 (-7 B d+4 A e)\right ) x^4+168 b^7 e^5 \left (24 a^3 B e^3+4 a b^2 d e (5 B d-2 A e)+9 a^2 b e^2 (-4 B d+A e)+2 b^3 d^2 (-2 B d+A e)\right ) x^5-28 b^8 e^6 \left (-45 a^2 B e^2-10 a b e (-4 B d+A e)+2 b^2 d (-5 B d+2 A e)\right ) x^6+24 b^9 e^7 (-4 b B d+A b e+10 a B e) x^7+21 b^{10} B e^8 x^8+\frac {56 (b d-a e)^{10} (B d-A e)}{(d+e x)^3}-\frac {84 (b d-a e)^9 (11 b B d-10 A b e-a B e)}{(d+e x)^2}+\frac {840 b (b d-a e)^8 (11 b B d-9 A b e-2 a B e)}{d+e x}+2520 b^2 (b d-a e)^7 (11 b B d-8 A b e-3 a B e) \log (d+e x)}{168 e^{12}} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[((a + b*x)^10*(A + B*x))/(d + e*x)^4,x]
[Out]
(168*b^3*e*(120*a^7*B*e^7 - 315*a^2*b^5*d^4*e^2*(8*B*d - 5*A*e) + 600*a^3*b^4*d^3*e^3*(7*B*d - 4*A*e) + 280*a*
b^6*d^5*e*(3*B*d - 2*A*e) + 504*a^5*b^2*d*e^5*(5*B*d - 2*A*e) - 2100*a^4*b^3*d^2*e^4*(2*B*d - A*e) + 210*a^6*b
*e^6*(-4*B*d + A*e) + 12*b^7*d^6*(-10*B*d + 7*A*e))*x - 84*b^4*e^2*(-210*a^6*B*e^6 + 70*a*b^5*d^4*e*(8*B*d - 5
*A*e) - 225*a^2*b^4*d^3*e^2*(7*B*d - 4*A*e) - 420*a^4*b^2*d*e^4*(5*B*d - 2*A*e) + 1200*a^3*b^3*d^2*e^3*(2*B*d
- A*e) - 252*a^5*b*e^5*(-4*B*d + A*e) + 28*b^6*d^5*(-3*B*d + 2*A*e))*x^2 + 56*b^5*e^3*(252*a^5*B*e^5 - 7*b^5*d
^4*(8*B*d - 5*A*e) + 50*a*b^4*d^3*e*(7*B*d - 4*A*e) + 240*a^3*b^2*d*e^3*(5*B*d - 2*A*e) - 450*a^2*b^3*d^2*e^2*
(2*B*d - A*e) + 210*a^4*b*e^4*(-4*B*d + A*e))*x^3 - 210*b^6*e^4*(-42*a^4*B*e^4 + 20*a*b^3*d^2*e*(2*B*d - A*e)
- 24*a^3*b*e^3*(-4*B*d + A*e) + 18*a^2*b^2*d*e^2*(-5*B*d + 2*A*e) + b^4*d^3*(-7*B*d + 4*A*e))*x^4 + 168*b^7*e^
5*(24*a^3*B*e^3 + 4*a*b^2*d*e*(5*B*d - 2*A*e) + 9*a^2*b*e^2*(-4*B*d + A*e) + 2*b^3*d^2*(-2*B*d + A*e))*x^5 - 2
8*b^8*e^6*(-45*a^2*B*e^2 - 10*a*b*e*(-4*B*d + A*e) + 2*b^2*d*(-5*B*d + 2*A*e))*x^6 + 24*b^9*e^7*(-4*b*B*d + A*
b*e + 10*a*B*e)*x^7 + 21*b^10*B*e^8*x^8 + (56*(b*d - a*e)^10*(B*d - A*e))/(d + e*x)^3 - (84*(b*d - a*e)^9*(11*
b*B*d - 10*A*b*e - a*B*e))/(d + e*x)^2 + (840*b*(b*d - a*e)^8*(11*b*B*d - 9*A*b*e - 2*a*B*e))/(d + e*x) + 2520
*b^2*(b*d - a*e)^7*(11*b*B*d - 8*A*b*e - 3*a*B*e)*Log[d + e*x])/(168*e^12)
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(2072\) vs.
\(2(433)=866\).
time = 0.10, size = 2073, normalized size = 4.66
| | |
| method |
result |
size |
| | |
| norman |
\(\text {Expression too large to display}\) |
\(1907\) |
| default |
\(\text {Expression too large to display}\) |
\(2073\) |
| risch |
\(\text {Expression too large to display}\) |
\(2182\) |
| | |
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Verification of antiderivative is not currently implemented for this CAS.
[In]
int((b*x+a)^10*(B*x+A)/(e*x+d)^4,x,method=_RETURNVERBOSE)
[Out]
b^3/e^11*(120*B*a^7*e^7*x-120*B*b^7*d^7*x+1/8*b^7*B*x^8*e^7+1/7*A*b^7*e^7*x^7+350/3*B*a*b^6*d^4*e^3*x^3-420*A*
a^4*b^3*d*e^6*x^2+600*A*a^3*b^4*d^2*e^5*x^2-450*A*a^2*b^5*d^3*e^4*x^2+175*A*a*b^6*d^4*e^3*x^2-504*B*a^5*b^2*d*
e^6*x^2+1050*B*a^4*b^3*d^2*e^5*x^2-1200*B*a^3*b^4*d^3*e^4*x^2+1575/2*B*a^2*b^5*d^4*e^3*x^2-280*B*a*b^6*d^5*e^2
*x^2-1008*A*a^5*b^2*d*e^6*x+2100*A*a^4*b^3*d^2*e^5*x-2400*A*a^3*b^4*d^3*e^4*x+1575*A*a^2*b^5*d^4*e^3*x-560*A*a
*b^6*d^5*e^2*x-840*B*a^6*b*d*e^6*x+2520*B*a^5*b^2*d^2*e^5*x-4200*B*a^4*b^3*d^3*e^4*x+4200*B*a^3*b^4*d^4*e^3*x-
2520*B*a^2*b^5*d^5*e^2*x+840*B*a*b^6*d^6*e*x-20/3*B*a*b^6*d*e^6*x^6+10/7*B*a*b^6*e^7*x^7-4/7*B*b^7*d*e^6*x^7+5
/3*A*a*b^6*e^7*x^6-2/3*A*b^7*d*e^6*x^6+15/2*B*a^2*b^5*e^7*x^6+5/3*B*b^7*d^2*e^5*x^6+9*A*a^2*b^5*e^7*x^5+2*A*b^
7*d^2*e^5*x^5+24*B*a^3*b^4*e^7*x^5-4*B*b^7*d^3*e^4*x^5+30*A*a^3*b^4*e^7*x^4-5*A*b^7*d^3*e^4*x^4+105/2*B*a^4*b^
3*e^7*x^4+35/4*B*b^7*d^4*e^3*x^4+70*A*a^4*b^3*e^7*x^3+35/3*A*b^7*d^4*e^3*x^3+84*B*a^5*b^2*e^7*x^3-56/3*B*b^7*d
^5*e^2*x^3+126*A*a^5*b^2*e^7*x^2-28*A*b^7*d^5*e^2*x^2+105*B*a^6*b*e^7*x^2+42*B*b^7*d^6*e*x^2+210*A*a^6*b*e^7*x
+84*A*b^7*d^6*e*x-8*A*a*b^6*d*e^6*x^5-36*B*a^2*b^5*d*e^6*x^5+20*B*a*b^6*d^2*e^5*x^5-45*A*a^2*b^5*d*e^6*x^4+25*
A*a*b^6*d^2*e^5*x^4-120*B*a^3*b^4*d*e^6*x^4+225/2*B*a^2*b^5*d^2*e^5*x^4-50*B*a*b^6*d^3*e^4*x^4-160*A*a^3*b^4*d
*e^6*x^3+150*A*a^2*b^5*d^2*e^5*x^3-200/3*A*a*b^6*d^3*e^4*x^3-280*B*a^4*b^3*d*e^6*x^3+400*B*a^3*b^4*d^2*e^5*x^3
-300*B*a^2*b^5*d^3*e^4*x^3)-1/2/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+405*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)^2-5*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)-1/3*(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)^3+15*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^4-448*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)*ln(e
*x+d)
________________________________________________________________________________________
Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 1872 vs.
\(2 (463) = 926\).
time = 0.52, size = 1872, normalized size = 4.21 \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)^4,x, algorithm="maxima")
[Out]
15*(11*B*b^10*d^8 + 3*B*a^8*b^2*e^8 + 8*A*a^7*b^3*e^8 - 8*(10*B*a*b^9*e + A*b^10*e)*d^7 + 28*(9*B*a^2*b^8*e^2
+ 2*A*a*b^9*e^2)*d^6 - 56*(8*B*a^3*b^7*e^3 + 3*A*a^2*b^8*e^3)*d^5 + 70*(7*B*a^4*b^6*e^4 + 4*A*a^3*b^7*e^4)*d^4
- 56*(6*B*a^5*b^5*e^5 + 5*A*a^4*b^6*e^5)*d^3 + 28*(5*B*a^6*b^4*e^6 + 6*A*a^5*b^5*e^6)*d^2 - 8*(4*B*a^7*b^3*e^
7 + 7*A*a^6*b^4*e^7)*d)*e^(-12)*log(x*e + d) + 1/168*(21*B*b^10*x^8*e^7 - 24*(4*B*b^10*d*e^6 - 10*B*a*b^9*e^7
- A*b^10*e^7)*x^7 + 28*(10*B*b^10*d^2*e^5 + 45*B*a^2*b^8*e^7 + 10*A*a*b^9*e^7 - 4*(10*B*a*b^9*e^6 + A*b^10*e^6
)*d)*x^6 - 168*(4*B*b^10*d^3*e^4 - 24*B*a^3*b^7*e^7 - 9*A*a^2*b^8*e^7 - 2*(10*B*a*b^9*e^5 + A*b^10*e^5)*d^2 +
4*(9*B*a^2*b^8*e^6 + 2*A*a*b^9*e^6)*d)*x^5 + 210*(7*B*b^10*d^4*e^3 + 42*B*a^4*b^6*e^7 + 24*A*a^3*b^7*e^7 - 4*(
10*B*a*b^9*e^4 + A*b^10*e^4)*d^3 + 10*(9*B*a^2*b^8*e^5 + 2*A*a*b^9*e^5)*d^2 - 12*(8*B*a^3*b^7*e^6 + 3*A*a^2*b^
8*e^6)*d)*x^4 - 56*(56*B*b^10*d^5*e^2 - 252*B*a^5*b^5*e^7 - 210*A*a^4*b^6*e^7 - 35*(10*B*a*b^9*e^3 + A*b^10*e^
3)*d^4 + 100*(9*B*a^2*b^8*e^4 + 2*A*a*b^9*e^4)*d^3 - 150*(8*B*a^3*b^7*e^5 + 3*A*a^2*b^8*e^5)*d^2 + 120*(7*B*a^
4*b^6*e^6 + 4*A*a^3*b^7*e^6)*d)*x^3 + 84*(84*B*b^10*d^6*e + 210*B*a^6*b^4*e^7 + 252*A*a^5*b^5*e^7 - 56*(10*B*a
*b^9*e^2 + A*b^10*e^2)*d^5 + 175*(9*B*a^2*b^8*e^3 + 2*A*a*b^9*e^3)*d^4 - 300*(8*B*a^3*b^7*e^4 + 3*A*a^2*b^8*e^
4)*d^3 + 300*(7*B*a^4*b^6*e^5 + 4*A*a^3*b^7*e^5)*d^2 - 168*(6*B*a^5*b^5*e^6 + 5*A*a^4*b^6*e^6)*d)*x^2 - 168*(1
20*B*b^10*d^7 - 120*B*a^7*b^3*e^7 - 210*A*a^6*b^4*e^7 - 84*(10*B*a*b^9*e + A*b^10*e)*d^6 + 280*(9*B*a^2*b^8*e^
2 + 2*A*a*b^9*e^2)*d^5 - 525*(8*B*a^3*b^7*e^3 + 3*A*a^2*b^8*e^3)*d^4 + 600*(7*B*a^4*b^6*e^4 + 4*A*a^3*b^7*e^4)
*d^3 - 420*(6*B*a^5*b^5*e^5 + 5*A*a^4*b^6*e^5)*d^2 + 168*(5*B*a^6*b^4*e^6 + 6*A*a^5*b^5*e^6)*d)*x)*e^(-11) + 1
/6*(299*B*b^10*d^11 - 2*A*a^10*e^11 - 242*(10*B*a*b^9*e + A*b^10*e)*d^10 + 955*(9*B*a^2*b^8*e^2 + 2*A*a*b^9*e^
2)*d^9 - 2190*(8*B*a^3*b^7*e^3 + 3*A*a^2*b^8*e^3)*d^8 + 3210*(7*B*a^4*b^6*e^4 + 4*A*a^3*b^7*e^4)*d^7 - 3108*(6
*B*a^5*b^5*e^5 + 5*A*a^4*b^6*e^5)*d^6 + 1974*(5*B*a^6*b^4*e^6 + 6*A*a^5*b^5*e^6)*d^5 - 780*(4*B*a^7*b^3*e^7 +
7*A*a^6*b^4*e^7)*d^4 + 165*(3*B*a^8*b^2*e^8 + 8*A*a^7*b^3*e^8)*d^3 - 10*(2*B*a^9*b*e^9 + 9*A*a^8*b^2*e^9)*d^2
+ 30*(11*B*b^10*d^9*e^2 - 2*B*a^9*b*e^11 - 9*A*a^8*b^2*e^11 - 9*(10*B*a*b^9*e^3 + A*b^10*e^3)*d^8 + 36*(9*B*a^
2*b^8*e^4 + 2*A*a*b^9*e^4)*d^7 - 84*(8*B*a^3*b^7*e^5 + 3*A*a^2*b^8*e^5)*d^6 + 126*(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 - 36*(4
*B*a^7*b^3*e^9 + 7*A*a^6*b^4*e^9)*d^2 + 9*(3*B*a^8*b^2*e^10 + 8*A*a^7*b^3*e^10)*d)*x^2 - (B*a^10*e^10 + 10*A*a
^9*b*e^10)*d + 3*(209*B*b^10*d^10*e - B*a^10*e^11 - 10*A*a^9*b*e^11 - 170*(10*B*a*b^9*e^2 + A*b^10*e^2)*d^9 +
675*(9*B*a^2*b^8*e^3 + 2*A*a*b^9*e^3)*d^8 - 1560*(8*B*a^3*b^7*e^4 + 3*A*a^2*b^8*e^4)*d^7 + 2310*(7*B*a^4*b^6*e
^5 + 4*A*a^3*b^7*e^5)*d^6 - 2268*(6*B*a^5*b^5*e^6 + 5*A*a^4*b^6*e^6)*d^5 + 1470*(5*B*a^6*b^4*e^7 + 6*A*a^5*b^5
*e^7)*d^4 - 600*(4*B*a^7*b^3*e^8 + 7*A*a^6*b^4*e^8)*d^3 + 135*(3*B*a^8*b^2*e^9 + 8*A*a^7*b^3*e^9)*d^2 - 10*(2*
B*a^9*b*e^10 + 9*A*a^8*b^2*e^10)*d)*x)/(x^3*e^15 + 3*d*x^2*e^14 + 3*d^2*x*e^13 + d^3*e^12)
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2639 vs.
\(2 (463) = 926\).
time = 1.05, size = 2639, normalized size = 5.93 \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)^4,x, algorithm="fricas")
[Out]
1/168*(8372*B*b^10*d^11 + (21*B*b^10*x^11 - 56*A*a^10 + 24*(10*B*a*b^9 + A*b^10)*x^10 + 140*(9*B*a^2*b^8 + 2*A
*a*b^9)*x^9 + 504*(8*B*a^3*b^7 + 3*A*a^2*b^8)*x^8 + 1260*(7*B*a^4*b^6 + 4*A*a^3*b^7)*x^7 + 2352*(6*B*a^5*b^5 +
5*A*a^4*b^6)*x^6 + 3528*(5*B*a^6*b^4 + 6*A*a^5*b^5)*x^5 + 5040*(4*B*a^7*b^3 + 7*A*a^6*b^4)*x^4 - 840*(2*B*a^9
*b + 9*A*a^8*b^2)*x^2 - 84*(B*a^10 + 10*A*a^9*b)*x)*e^11 - (33*B*b^10*d*x^10 + 40*(10*B*a*b^9 + A*b^10)*d*x^9
+ 252*(9*B*a^2*b^8 + 2*A*a*b^9)*d*x^8 + 1008*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d*x^7 + 2940*(7*B*a^4*b^6 + 4*A*a^3*b
^7)*d*x^6 + 7056*(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 - 15120*(4*B*a^7*
b^3 + 7*A*a^6*b^4)*d*x^3 - 7560*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d*x^2 + 840*(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 + (55*B*b^10*d^2*x^9 + 72*(10*B*a*b^9 + A*b^10)*d^2*x^8 + 504*(9*B*a^2*b^8 + 2*A*a*
b^9)*d^2*x^7 + 2352*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^2*x^6 + 8820*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^2*x^5 + 35280*(6*
B*a^5*b^5 + 5*A*a^4*b^6)*d^2*x^4 - 74088*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^2*x^3 - 15120*(4*B*a^7*b^3 + 7*A*a^6*b^
4)*d^2*x^2 + 11340*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^2*x - 280*(2*B*a^9*b + 9*A*a^8*b^2)*d^2)*e^9 - 3*(33*B*b^10*d
^3*x^8 + 48*(10*B*a*b^9 + A*b^10)*d^3*x^7 + 392*(9*B*a^2*b^8 + 2*A*a*b^9)*d^3*x^6 + 2352*(8*B*a^3*b^7 + 3*A*a^
2*b^8)*d^3*x^5 + 14700*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^3*x^4 - 57232*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^3*x^3 + 3528*
(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^3*x^2 + 15120*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^3*x - 1540*(3*B*a^8*b^2 + 8*A*a^7*b^
3)*d^3)*e^8 + 6*(33*B*b^10*d^4*x^7 + 56*(10*B*a*b^9 + A*b^10)*d^4*x^6 + 588*(9*B*a^2*b^8 + 2*A*a*b^9)*d^4*x^5
+ 5880*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^4*x^4 - 38920*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^4*x^3 + 15288*(6*B*a^5*b^5 +
5*A*a^4*b^6)*d^4*x^2 + 15876*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^4*x - 3640*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^4)*e^7 - 4
2*(11*B*b^10*d^5*x^6 + 24*(10*B*a*b^9 + A*b^10)*d^5*x^5 + 420*(9*B*a^2*b^8 + 2*A*a*b^9)*d^5*x^4 - 4700*(8*B*a^
3*b^7 + 3*A*a^2*b^8)*d^5*x^3 + 4080*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^5*x^2 + 2856*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^5
*x - 1316*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^5)*e^6 + 14*(99*B*b^10*d^6*x^5 + 360*(10*B*a*b^9 + A*b^10)*d^6*x^4 - 7
330*(9*B*a^2*b^8 + 2*A*a*b^9)*d^6*x^3 + 12060*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^6*x^2 + 6660*(7*B*a^4*b^6 + 4*A*a^
3*b^7)*d^6*x - 6216*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^6)*e^5 - 14*(495*B*b^10*d^7*x^4 - 2156*(10*B*a*b^9 + A*b^10)
*d^7*x^3 + 6870*(9*B*a^2*b^8 + 2*A*a*b^9)*d^7*x^2 + 3060*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^7*x - 6420*(7*B*a^4*b^6
+ 4*A*a^3*b^7)*d^7)*e^4 - 28*(1516*B*b^10*d^8*x^3 - 1074*(10*B*a*b^9 + A*b^10)*d^8*x^2 - 345*(9*B*a^2*b^8 + 2
*A*a*b^9)*d^8*x + 2190*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^8)*e^3 - 28*(1578*B*b^10*d^9*x^2 + 6*(10*B*a*b^9 + A*b^10
)*d^9*x - 955*(9*B*a^2*b^8 + 2*A*a*b^9)*d^9)*e^2 - 28*(93*B*b^10*d^10*x + 242*(10*B*a*b^9 + A*b^10)*d^10)*e +
2520*(11*B*b^10*d^11 + (3*B*a^8*b^2 + 8*A*a^7*b^3)*x^3*e^11 - (8*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d*x^3 - 3*(3*B*a^
8*b^2 + 8*A*a^7*b^3)*d*x^2)*e^10 + (28*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^2*x^3 - 24*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^
2*x^2 + 3*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^2*x)*e^9 - (56*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^3*x^3 - 84*(5*B*a^6*b^4 +
6*A*a^5*b^5)*d^3*x^2 + 24*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^3*x - (3*B*a^8*b^2 + 8*A*a^7*b^3)*d^3)*e^8 + 2*(35*(7
*B*a^4*b^6 + 4*A*a^3*b^7)*d^4*x^3 - 84*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^4*x^2 + 42*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^
4*x - 4*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^4)*e^7 - 14*(4*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^5*x^3 - 15*(7*B*a^4*b^6 + 4
*A*a^3*b^7)*d^5*x^2 + 12*(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 + 14*(2*(9
*B*a^2*b^8 + 2*A*a*b^9)*d^6*x^3 - 12*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^6*x^2 + 15*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^6*
x - 4*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^6)*e^5 - 2*(4*(10*B*a*b^9 + A*b^10)*d^7*x^3 - 42*(9*B*a^2*b^8 + 2*A*a*b^9)
*d^7*x^2 + 84*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^7*x - 35*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^7)*e^4 + (11*B*b^10*d^8*x^3
- 24*(10*B*a*b^9 + A*b^10)*d^8*x^2 + 84*(9*B*a^2*b^8 + 2*A*a*b^9)*d^8*x - 56*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^8)
*e^3 + (33*B*b^10*d^9*x^2 - 24*(10*B*a*b^9 + A*b^10)*d^9*x + 28*(9*B*a^2*b^8 + 2*A*a*b^9)*d^9)*e^2 + (33*B*b^1
0*d^10*x - 8*(10*B*a*b^9 + A*b^10)*d^10)*e)*log(x*e + d))/(x^3*e^15 + 3*d*x^2*e^14 + 3*d^2*x*e^13 + d^3*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)**4,x)
[Out]
Timed out
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1964 vs.
\(2 (463) = 926\).
time = 0.91, size = 1964, normalized size = 4.41 \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)^4,x, algorithm="giac")
[Out]
15*(11*B*b^10*d^8 - 80*B*a*b^9*d^7*e - 8*A*b^10*d^7*e + 252*B*a^2*b^8*d^6*e^2 + 56*A*a*b^9*d^6*e^2 - 448*B*a^3
*b^7*d^5*e^3 - 168*A*a^2*b^8*d^5*e^3 + 490*B*a^4*b^6*d^4*e^4 + 280*A*a^3*b^7*d^4*e^4 - 336*B*a^5*b^5*d^3*e^5 -
280*A*a^4*b^6*d^3*e^5 + 140*B*a^6*b^4*d^2*e^6 + 168*A*a^5*b^5*d^2*e^6 - 32*B*a^7*b^3*d*e^7 - 56*A*a^6*b^4*d*e
^7 + 3*B*a^8*b^2*e^8 + 8*A*a^7*b^3*e^8)*e^(-12)*log(abs(x*e + d)) + 1/168*(21*B*b^10*x^8*e^28 - 96*B*b^10*d*x^
7*e^27 + 280*B*b^10*d^2*x^6*e^26 - 672*B*b^10*d^3*x^5*e^25 + 1470*B*b^10*d^4*x^4*e^24 - 3136*B*b^10*d^5*x^3*e^
23 + 7056*B*b^10*d^6*x^2*e^22 - 20160*B*b^10*d^7*x*e^21 + 240*B*a*b^9*x^7*e^28 + 24*A*b^10*x^7*e^28 - 1120*B*a
*b^9*d*x^6*e^27 - 112*A*b^10*d*x^6*e^27 + 3360*B*a*b^9*d^2*x^5*e^26 + 336*A*b^10*d^2*x^5*e^26 - 8400*B*a*b^9*d
^3*x^4*e^25 - 840*A*b^10*d^3*x^4*e^25 + 19600*B*a*b^9*d^4*x^3*e^24 + 1960*A*b^10*d^4*x^3*e^24 - 47040*B*a*b^9*
d^5*x^2*e^23 - 4704*A*b^10*d^5*x^2*e^23 + 141120*B*a*b^9*d^6*x*e^22 + 14112*A*b^10*d^6*x*e^22 + 1260*B*a^2*b^8
*x^6*e^28 + 280*A*a*b^9*x^6*e^28 - 6048*B*a^2*b^8*d*x^5*e^27 - 1344*A*a*b^9*d*x^5*e^27 + 18900*B*a^2*b^8*d^2*x
^4*e^26 + 4200*A*a*b^9*d^2*x^4*e^26 - 50400*B*a^2*b^8*d^3*x^3*e^25 - 11200*A*a*b^9*d^3*x^3*e^25 + 132300*B*a^2
*b^8*d^4*x^2*e^24 + 29400*A*a*b^9*d^4*x^2*e^24 - 423360*B*a^2*b^8*d^5*x*e^23 - 94080*A*a*b^9*d^5*x*e^23 + 4032
*B*a^3*b^7*x^5*e^28 + 1512*A*a^2*b^8*x^5*e^28 - 20160*B*a^3*b^7*d*x^4*e^27 - 7560*A*a^2*b^8*d*x^4*e^27 + 67200
*B*a^3*b^7*d^2*x^3*e^26 + 25200*A*a^2*b^8*d^2*x^3*e^26 - 201600*B*a^3*b^7*d^3*x^2*e^25 - 75600*A*a^2*b^8*d^3*x
^2*e^25 + 705600*B*a^3*b^7*d^4*x*e^24 + 264600*A*a^2*b^8*d^4*x*e^24 + 8820*B*a^4*b^6*x^4*e^28 + 5040*A*a^3*b^7
*x^4*e^28 - 47040*B*a^4*b^6*d*x^3*e^27 - 26880*A*a^3*b^7*d*x^3*e^27 + 176400*B*a^4*b^6*d^2*x^2*e^26 + 100800*A
*a^3*b^7*d^2*x^2*e^26 - 705600*B*a^4*b^6*d^3*x*e^25 - 403200*A*a^3*b^7*d^3*x*e^25 + 14112*B*a^5*b^5*x^3*e^28 +
11760*A*a^4*b^6*x^3*e^28 - 84672*B*a^5*b^5*d*x^2*e^27 - 70560*A*a^4*b^6*d*x^2*e^27 + 423360*B*a^5*b^5*d^2*x*e
^26 + 352800*A*a^4*b^6*d^2*x*e^26 + 17640*B*a^6*b^4*x^2*e^28 + 21168*A*a^5*b^5*x^2*e^28 - 141120*B*a^6*b^4*d*x
*e^27 - 169344*A*a^5*b^5*d*x*e^27 + 20160*B*a^7*b^3*x*e^28 + 35280*A*a^6*b^4*x*e^28)*e^(-32) + 1/6*(299*B*b^10
*d^11 - 2420*B*a*b^9*d^10*e - 242*A*b^10*d^10*e + 8595*B*a^2*b^8*d^9*e^2 + 1910*A*a*b^9*d^9*e^2 - 17520*B*a^3*
b^7*d^8*e^3 - 6570*A*a^2*b^8*d^8*e^3 + 22470*B*a^4*b^6*d^7*e^4 + 12840*A*a^3*b^7*d^7*e^4 - 18648*B*a^5*b^5*d^6
*e^5 - 15540*A*a^4*b^6*d^6*e^5 + 9870*B*a^6*b^4*d^5*e^6 + 11844*A*a^5*b^5*d^5*e^6 - 3120*B*a^7*b^3*d^4*e^7 - 5
460*A*a^6*b^4*d^4*e^7 + 495*B*a^8*b^2*d^3*e^8 + 1320*A*a^7*b^3*d^3*e^8 - 20*B*a^9*b*d^2*e^9 - 90*A*a^8*b^2*d^2
*e^9 - B*a^10*d*e^10 - 10*A*a^9*b*d*e^10 - 2*A*a^10*e^11 + 30*(11*B*b^10*d^9*e^2 - 90*B*a*b^9*d^8*e^3 - 9*A*b^
10*d^8*e^3 + 324*B*a^2*b^8*d^7*e^4 + 72*A*a*b^9*d^7*e^4 - 672*B*a^3*b^7*d^6*e^5 - 252*A*a^2*b^8*d^6*e^5 + 882*
B*a^4*b^6*d^5*e^6 + 504*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 - 144*B*a^7*b^3*d^2*e^9 - 252*A*a^6*b^4*d^2*e^9 + 27*B*a^8*b^2*d*e^10 + 72*A*a^7*b
^3*d*e^10 - 2*B*a^9*b*e^11 - 9*A*a^8*b^2*e^11)*x^2 + 3*(209*B*b^10*d^10*e - 1700*B*a*b^9*d^9*e^2 - 170*A*b^10*
d^9*e^2 + 6075*B*a^2*b^8*d^8*e^3 + 1350*A*a*b^9*d^8*e^3 - 12480*B*a^3*b^7*d^7*e^4 - 4680*A*a^2*b^8*d^7*e^4 + 1
6170*B*a^4*b^6*d^6*e^5 + 9240*A*a^3*b^7*d^6*e^5 - 13608*B*a^5*b^5*d^5*e^6 - 11340*A*a^4*b^6*d^5*e^6 + 7350*B*a
^6*b^4*d^4*e^7 + 8820*A*a^5*b^5*d^4*e^7 - 2400*B*a^7*b^3*d^3*e^8 - 4200*A*a^6*b^4*d^3*e^8 + 405*B*a^8*b^2*d^2*
e^9 + 1080*A*a^7*b^3*d^2*e^9 - 20*B*a^9*b*d*e^10 - 90*A*a^8*b^2*d*e^10 - B*a^10*e^11 - 10*A*a^9*b*e^11)*x)*e^(
-12)/(x*e + d)^3
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Mupad [B]
time = 1.48, size = 2500, normalized size = 5.62 \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)^4,x)
[Out]
x^2*((2*d^3*((6*d^2*((A*b^10 + 10*B*a*b^9)/e^4 - (4*B*b^10*d)/e^5))/e^2 - (4*d*((4*d*((A*b^10 + 10*B*a*b^9)/e^
4 - (4*B*b^10*d)/e^5))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^4 + (6*B*b^10*d^2)/e^6))/e - (15*a^2*b^7*(3*A*b + 8*B*a
))/e^4 + (4*B*b^10*d^3)/e^7))/e^3 - (3*d^2*((4*d*((6*d^2*((A*b^10 + 10*B*a*b^9)/e^4 - (4*B*b^10*d)/e^5))/e^2 -
(4*d*((4*d*((A*b^10 + 10*B*a*b^9)/e^4 - (4*B*b^10*d)/e^5))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^4 + (6*B*b^10*d^2)
/e^6))/e - (15*a^2*b^7*(3*A*b + 8*B*a))/e^4 + (4*B*b^10*d^3)/e^7))/e - (4*d^3*((A*b^10 + 10*B*a*b^9)/e^4 - (4*
B*b^10*d)/e^5))/e^3 + (6*d^2*((4*d*((A*b^10 + 10*B*a*b^9)/e^4 - (4*B*b^10*d)/e^5))/e - (5*a*b^8*(2*A*b + 9*B*a
))/e^4 + (6*B*b^10*d^2)/e^6))/e^2 + (30*a^3*b^6*(4*A*b + 7*B*a))/e^4 - (B*b^10*d^4)/e^8))/e^2 - (2*d*((6*d^2*(
(6*d^2*((A*b^10 + 10*B*a*b^9)/e^4 - (4*B*b^10*d)/e^5))/e^2 - (4*d*((4*d*((A*b^10 + 10*B*a*b^9)/e^4 - (4*B*b^10
*d)/e^5))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^4 + (6*B*b^10*d^2)/e^6))/e - (15*a^2*b^7*(3*A*b + 8*B*a))/e^4 + (4*B
*b^10*d^3)/e^7))/e^2 - (4*d*((4*d*((6*d^2*((A*b^10 + 10*B*a*b^9)/e^4 - (4*B*b^10*d)/e^5))/e^2 - (4*d*((4*d*((A
*b^10 + 10*B*a*b^9)/e^4 - (4*B*b^10*d)/e^5))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^4 + (6*B*b^10*d^2)/e^6))/e - (15*
a^2*b^7*(3*A*b + 8*B*a))/e^4 + (4*B*b^10*d^3)/e^7))/e - (4*d^3*((A*b^10 + 10*B*a*b^9)/e^4 - (4*B*b^10*d)/e^5))
/e^3 + (6*d^2*((4*d*((A*b^10 + 10*B*a*b^9)/e^4 - (4*B*b^10*d)/e^5))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^4 + (6*B*b
^10*d^2)/e^6))/e^2 + (30*a^3*b^6*(4*A*b + 7*B*a))/e^4 - (B*b^10*d^4)/e^8))/e - (d^4*((A*b^10 + 10*B*a*b^9)/e^4
- (4*B*b^10*d)/e^5))/e^4 + (4*d^3*((4*d*((A*b^10 + 10*B*a*b^9)/e^4 - (4*B*b^10*d)/e^5))/e - (5*a*b^8*(2*A*b +
9*B*a))/e^4 + (6*B*b^10*d^2)/e^6))/e^3 + (42*a^4*b^5*(5*A*b + 6*B*a))/e^4))/e + (d^4*((4*d*((A*b^10 + 10*B*a*
b^9)/e^4 - (4*B*b^10*d)/e^5))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^4 + (6*B*b^10*d^2)/e^6))/(2*e^4) + (21*a^5*b^4*(
6*A*b + 5*B*a))/e^4) - x^6*((2*d*((A*b^10 + 10*B*a*b^9)/e^4 - (4*B*b^10*d)/e^5))/(3*e) - (5*a*b^8*(2*A*b + 9*B
*a))/(6*e^4) + (B*b^10*d^2)/e^6) - ((2*A*a^10*e^11 - 299*B*b^10*d^11 + 242*A*b^10*d^10*e + B*a^10*d*e^10 - 191
0*A*a*b^9*d^9*e^2 + 20*B*a^9*b*d^2*e^9 + 6570*A*a^2*b^8*d^8*e^3 - 12840*A*a^3*b^7*d^7*e^4 + 15540*A*a^4*b^6*d^
6*e^5 - 11844*A*a^5*b^5*d^5*e^6 + 5460*A*a^6*b^4*d^4*e^7 - 1320*A*a^7*b^3*d^3*e^8 + 90*A*a^8*b^2*d^2*e^9 - 859
5*B*a^2*b^8*d^9*e^2 + 17520*B*a^3*b^7*d^8*e^3 - 22470*B*a^4*b^6*d^7*e^4 + 18648*B*a^5*b^5*d^6*e^5 - 9870*B*a^6
*b^4*d^5*e^6 + 3120*B*a^7*b^3*d^4*e^7 - 495*B*a^8*b^2*d^3*e^8 + 10*A*a^9*b*d*e^10 + 2420*B*a*b^9*d^10*e)/(6*e)
+ x*((B*a^10*e^10)/2 - (209*B*b^10*d^10)/2 + 5*A*a^9*b*e^10 + 85*A*b^10*d^9*e - 675*A*a*b^9*d^8*e^2 + 45*A*a^
8*b^2*d*e^9 + 2340*A*a^2*b^8*d^7*e^3 - 4620*A*a^3*b^7*d^6*e^4 + 5670*A*a^4*b^6*d^5*e^5 - 4410*A*a^5*b^5*d^4*e^
6 + 2100*A*a^6*b^4*d^3*e^7 - 540*A*a^7*b^3*d^2*e^8 - (6075*B*a^2*b^8*d^8*e^2)/2 + 6240*B*a^3*b^7*d^7*e^3 - 808
5*B*a^4*b^6*d^6*e^4 + 6804*B*a^5*b^5*d^5*e^5 - 3675*B*a^6*b^4*d^4*e^6 + 1200*B*a^7*b^3*d^3*e^7 - (405*B*a^8*b^
2*d^2*e^8)/2 + 850*B*a*b^9*d^9*e + 10*B*a^9*b*d*e^9) + x^2*(10*B*a^9*b*e^10 - 55*B*b^10*d^9*e + 45*A*a^8*b^2*e
^10 + 45*A*b^10*d^8*e^2 - 360*A*a*b^9*d^7*e^3 - 360*A*a^7*b^3*d*e^9 + 450*B*a*b^9*d^8*e^2 - 135*B*a^8*b^2*d*e^
9 + 1260*A*a^2*b^8*d^6*e^4 - 2520*A*a^3*b^7*d^5*e^5 + 3150*A*a^4*b^6*d^4*e^6 - 2520*A*a^5*b^5*d^3*e^7 + 1260*A
*a^6*b^4*d^2*e^8 - 1620*B*a^2*b^8*d^7*e^3 + 3360*B*a^3*b^7*d^6*e^4 - 4410*B*a^4*b^6*d^5*e^5 + 3780*B*a^5*b^5*d
^4*e^6 - 2100*B*a^6*b^4*d^3*e^7 + 720*B*a^7*b^3*d^2*e^8))/(d^3*e^11 + e^14*x^3 + 3*d^2*e^12*x + 3*d*e^13*x^2)
- x*((6*d^2*((6*d^2*((6*d^2*((A*b^10 + 10*B*a*b^9)/e^4 - (4*B*b^10*d)/e^5))/e^2 - (4*d*((4*d*((A*b^10 + 10*B*a
*b^9)/e^4 - (4*B*b^10*d)/e^5))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^4 + (6*B*b^10*d^2)/e^6))/e - (15*a^2*b^7*(3*A*b
+ 8*B*a))/e^4 + (4*B*b^10*d^3)/e^7))/e^2 - (4*d*((4*d*((6*d^2*((A*b^10 + 10*B*a*b^9)/e^4 - (4*B*b^10*d)/e^5))
/e^2 - (4*d*((4*d*((A*b^10 + 10*B*a*b^9)/e^4 - (4*B*b^10*d)/e^5))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^4 + (6*B*b^1
0*d^2)/e^6))/e - (15*a^2*b^7*(3*A*b + 8*B*a))/e^4 + (4*B*b^10*d^3)/e^7))/e - (4*d^3*((A*b^10 + 10*B*a*b^9)/e^4
- (4*B*b^10*d)/e^5))/e^3 + (6*d^2*((4*d*((A*b^10 + 10*B*a*b^9)/e^4 - (4*B*b^10*d)/e^5))/e - (5*a*b^8*(2*A*b +
9*B*a))/e^4 + (6*B*b^10*d^2)/e^6))/e^2 + (30*a^3*b^6*(4*A*b + 7*B*a))/e^4 - (B*b^10*d^4)/e^8))/e - (d^4*((A*b
^10 + 10*B*a*b^9)/e^4 - (4*B*b^10*d)/e^5))/e^4 + (4*d^3*((4*d*((A*b^10 + 10*B*a*b^9)/e^4 - (4*B*b^10*d)/e^5))/
e - (5*a*b^8*(2*A*b + 9*B*a))/e^4 + (6*B*b^10*d^2)/e^6))/e^3 + (42*a^4*b^5*(5*A*b + 6*B*a))/e^4))/e^2 - (d^4*(
(6*d^2*((A*b^10 + 10*B*a*b^9)/e^4 - (4*B*b^10*d)/e^5))/e^2 - (4*d*((4*d*((A*b^10 + 10*B*a*b^9)/e^4 - (4*B*b^10
*d)/e^5))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^4 + (6*B*b^10*d^2)/e^6))/e - (15*a^2*b^7*(3*A*b + 8*B*a))/e^4 + (4*B
*b^10*d^3)/e^7))/e^4 + (4*d^3*((4*d*((6*d^2*((A*b^10 + 10*B*a*b^9)/e^4 - (4*B*b^10*d)/e^5))/e^2 - (4*d*((4*d*(
(A*b^10 + 10*B*a*b^9)/e^4 - (4*B*b^10*d)/e^5))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^4 + (6*B*b^10*d^2)/e^6))/e - (1
5*a^2*b^7*(3*A*b + 8*B*a))/e^4 + (4*B*b^10*d^3)...
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