\(\int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^4} \, dx\) [82]

Optimal result

Integrand size = 20, antiderivative size = 445 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^4} \, dx=-\frac {30 b^3 (b d-a e)^6 (11 b B d-7 A b e-4 a B e) x}{e^{11}}+\frac {(b d-a e)^{10} (B d-A e)}{3 e^{12} (d+e x)^3}-\frac {(b d-a e)^9 (11 b B d-10 A b e-a B e)}{2 e^{12} (d+e x)^2}+\frac {5 b (b d-a e)^8 (11 b B d-9 A b e-2 a B e)}{e^{12} (d+e x)}+\frac {21 b^4 (b d-a e)^5 (11 b B d-6 A b e-5 a B e) (d+e x)^2}{e^{12}}-\frac {14 b^5 (b d-a e)^4 (11 b B d-5 A b e-6 a B e) (d+e x)^3}{e^{12}}+\frac {15 b^6 (b d-a e)^3 (11 b B d-4 A b e-7 a B e) (d+e x)^4}{2 e^{12}}-\frac {3 b^7 (b d-a e)^2 (11 b B d-3 A b e-8 a B e) (d+e x)^5}{e^{12}}+\frac {5 b^8 (b d-a e) (11 b B d-2 A b e-9 a B e) (d+e x)^6}{6 e^{12}}-\frac {b^9 (11 b B d-A b e-10 a B e) (d+e x)^7}{7 e^{12}}+\frac {b^{10} B (d+e x)^8}{8 e^{12}}+\frac {15 b^2 (b d-a e)^7 (11 b B d-8 A b e-3 a B e) \log (d+e x)}{e^{12}} \] Output:

-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
 

Mathematica [A] (verified)

Time = 0.26 (sec) , antiderivative size = 814, normalized size of antiderivative = 1.83 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^4} \, dx=\frac {168 b^3 e \left (120 a^7 B e^7-315 a^2 b^5 d^4 e^2 (8 B d-5 A e)+600 a^3 b^4 d^3 e^3 (7 B d-4 A e)+280 a b^6 d^5 e (3 B d-2 A e)+504 a^5 b^2 d e^5 (5 B d-2 A e)-2100 a^4 b^3 d^2 e^4 (2 B d-A e)+210 a^6 b e^6 (-4 B d+A e)+12 b^7 d^6 (-10 B d+7 A e)\right ) x-84 b^4 e^2 \left (-210 a^6 B e^6+70 a b^5 d^4 e (8 B d-5 A e)-225 a^2 b^4 d^3 e^2 (7 B d-4 A e)-420 a^4 b^2 d e^4 (5 B d-2 A e)+1200 a^3 b^3 d^2 e^3 (2 B d-A e)-252 a^5 b e^5 (-4 B d+A e)+28 b^6 d^5 (-3 B d+2 A e)\right ) x^2+56 b^5 e^3 \left (252 a^5 B e^5-7 b^5 d^4 (8 B d-5 A e)+50 a b^4 d^3 e (7 B d-4 A e)+240 a^3 b^2 d e^3 (5 B d-2 A e)-450 a^2 b^3 d^2 e^2 (2 B d-A e)+210 a^4 b e^4 (-4 B d+A e)\right ) x^3-210 b^6 e^4 \left (-42 a^4 B e^4+20 a b^3 d^2 e (2 B d-A e)-24 a^3 b e^3 (-4 B d+A e)+18 a^2 b^2 d e^2 (-5 B d+2 A e)+b^4 d^3 (-7 B d+4 A e)\right ) x^4+168 b^7 e^5 \left (24 a^3 B e^3+4 a b^2 d e (5 B d-2 A e)+9 a^2 b e^2 (-4 B d+A e)+2 b^3 d^2 (-2 B d+A e)\right ) x^5-28 b^8 e^6 \left (-45 a^2 B e^2-10 a b e (-4 B d+A e)+2 b^2 d (-5 B d+2 A e)\right ) x^6+24 b^9 e^7 (-4 b B d+A b e+10 a B e) x^7+21 b^{10} B e^8 x^8+\frac {56 (b d-a e)^{10} (B d-A e)}{(d+e x)^3}-\frac {84 (b d-a e)^9 (11 b B d-10 A b e-a B e)}{(d+e x)^2}+\frac {840 b (b d-a e)^8 (11 b B d-9 A b e-2 a B e)}{d+e x}+2520 b^2 (b d-a e)^7 (11 b B d-8 A b e-3 a B e) \log (d+e x)}{168 e^{12}} \] Input:

Integrate[((a + b*x)^10*(A + B*x))/(d + e*x)^4,x]
 

Output:

(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 - 28*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)
 

Rubi [A] (verified)

Time = 1.53 (sec) , antiderivative size = 445, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {86, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^4} \, dx\)

\(\Big \downarrow \) 86

\(\displaystyle \int \left (\frac {b^9 (d+e x)^6 (10 a B e+A b e-11 b B d)}{e^{11}}-\frac {5 b^8 (d+e x)^5 (b d-a e) (9 a B e+2 A b e-11 b B d)}{e^{11}}+\frac {15 b^7 (d+e x)^4 (b d-a e)^2 (8 a B e+3 A b e-11 b B d)}{e^{11}}-\frac {30 b^6 (d+e x)^3 (b d-a e)^3 (7 a B e+4 A b e-11 b B d)}{e^{11}}+\frac {42 b^5 (d+e x)^2 (b d-a e)^4 (6 a B e+5 A b e-11 b B d)}{e^{11}}-\frac {42 b^4 (d+e x) (b d-a e)^5 (5 a B e+6 A b e-11 b B d)}{e^{11}}+\frac {30 b^3 (b d-a e)^6 (4 a B e+7 A b e-11 b B d)}{e^{11}}-\frac {15 b^2 (b d-a e)^7 (3 a B e+8 A b e-11 b B d)}{e^{11} (d+e x)}+\frac {5 b (b d-a e)^8 (2 a B e+9 A b e-11 b B d)}{e^{11} (d+e x)^2}+\frac {(a e-b d)^9 (a B e+10 A b e-11 b B d)}{e^{11} (d+e x)^3}+\frac {(a e-b d)^{10} (A e-B d)}{e^{11} (d+e x)^4}+\frac {b^{10} B (d+e x)^7}{e^{11}}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {b^9 (d+e x)^7 (-10 a B e-A b e+11 b B d)}{7 e^{12}}+\frac {5 b^8 (d+e x)^6 (b d-a e) (-9 a B e-2 A b e+11 b B d)}{6 e^{12}}-\frac {3 b^7 (d+e x)^5 (b d-a e)^2 (-8 a B e-3 A b e+11 b B d)}{e^{12}}+\frac {15 b^6 (d+e x)^4 (b d-a e)^3 (-7 a B e-4 A b e+11 b B d)}{2 e^{12}}-\frac {14 b^5 (d+e x)^3 (b d-a e)^4 (-6 a B e-5 A b e+11 b B d)}{e^{12}}+\frac {21 b^4 (d+e x)^2 (b d-a e)^5 (-5 a B e-6 A b e+11 b B d)}{e^{12}}-\frac {30 b^3 x (b d-a e)^6 (-4 a B e-7 A b e+11 b B d)}{e^{11}}+\frac {15 b^2 (b d-a e)^7 \log (d+e x) (-3 a B e-8 A b e+11 b B d)}{e^{12}}+\frac {5 b (b d-a e)^8 (-2 a B e-9 A b e+11 b B d)}{e^{12} (d+e x)}-\frac {(b d-a e)^9 (-a B e-10 A b e+11 b B d)}{2 e^{12} (d+e x)^2}+\frac {(b d-a e)^{10} (B d-A e)}{3 e^{12} (d+e x)^3}+\frac {b^{10} B (d+e x)^8}{8 e^{12}}\)

Input:

Int[((a + b*x)^10*(A + B*x))/(d + e*x)^4,x]
 

Output:

(-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 - (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 + (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)*Log[d + e*x])/e^12
 

Defintions of rubi rules used

rule 86

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_ 
.), x_] :> Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; 
 FreeQ[{a, b, c, d, e, f, n}, x] && ((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]))))
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1906\) vs. \(2(433)=866\).

Time = 0.24 (sec) , antiderivative size = 1907, normalized size of antiderivative = 4.29

method	result	size
norman	\(\text {Expression too large to display}\)	\(1907\)
default	\(\text {Expression too large to display}\)	\(2073\)
risch	\(\text {Expression too large to display}\)	\(2182\)
parallelrisch	\(\text {Expression too large to display}\)	\(3431\)

Input:

int((b*x+a)^10*(B*x+A)/(e*x+d)^4,x,method=_RETURNVERBOSE)
 

Output:

(-1/6*(2*A*a^10*e^11+10*A*a^9*b*d*e^10+90*A*a^8*b^2*d^2*e^9-1320*A*a^7*b^3 
*d^3*e^8+9240*A*a^6*b^4*d^4*e^7-27720*A*a^5*b^5*d^5*e^6+46200*A*a^4*b^6*d^ 
6*e^5-46200*A*a^3*b^7*d^7*e^4+27720*A*a^2*b^8*d^8*e^3-9240*A*a*b^9*d^9*e^2 
+1320*A*b^10*d^10*e+B*a^10*d*e^10+20*B*a^9*b*d^2*e^9-495*B*a^8*b^2*d^3*e^8 
+5280*B*a^7*b^3*d^4*e^7-23100*B*a^6*b^4*d^5*e^6+55440*B*a^5*b^5*d^6*e^5-80 
850*B*a^4*b^6*d^7*e^4+73920*B*a^3*b^7*d^8*e^3-41580*B*a^2*b^8*d^9*e^2+1320 
0*B*a*b^9*d^10*e-1815*B*b^10*d^11)/e^12-(45*A*a^8*b^2*e^9-360*A*a^7*b^3*d* 
e^8+2520*A*a^6*b^4*d^2*e^7-7560*A*a^5*b^5*d^3*e^6+12600*A*a^4*b^6*d^4*e^5- 
12600*A*a^3*b^7*d^5*e^4+7560*A*a^2*b^8*d^6*e^3-2520*A*a*b^9*d^7*e^2+360*A* 
b^10*d^8*e+10*B*a^9*b*e^9-135*B*a^8*b^2*d*e^8+1440*B*a^7*b^3*d^2*e^7-6300* 
B*a^6*b^4*d^3*e^6+15120*B*a^5*b^5*d^4*e^5-22050*B*a^4*b^6*d^5*e^4+20160*B* 
a^3*b^7*d^6*e^3-11340*B*a^2*b^8*d^7*e^2+3600*B*a*b^9*d^8*e-495*B*b^10*d^9) 
/e^10*x^2-1/2*(10*A*a^9*b*e^10+90*A*a^8*b^2*d*e^9-1080*A*a^7*b^3*d^2*e^8+7 
560*A*a^6*b^4*d^3*e^7-22680*A*a^5*b^5*d^4*e^6+37800*A*a^4*b^6*d^5*e^5-3780 
0*A*a^3*b^7*d^6*e^4+22680*A*a^2*b^8*d^7*e^3-7560*A*a*b^9*d^8*e^2+1080*A*b^ 
10*d^9*e+B*a^10*e^10+20*B*a^9*b*d*e^9-405*B*a^8*b^2*d^2*e^8+4320*B*a^7*b^3 
*d^3*e^7-18900*B*a^6*b^4*d^4*e^6+45360*B*a^5*b^5*d^5*e^5-66150*B*a^4*b^6*d 
^6*e^4+60480*B*a^3*b^7*d^7*e^3-34020*B*a^2*b^8*d^8*e^2+10800*B*a*b^9*d^9*e 
-1485*B*b^10*d^10)/e^11*x+15/4*b^3*(56*A*a^6*b*e^7-168*A*a^5*b^2*d*e^6+280 
*A*a^4*b^3*d^2*e^5-280*A*a^3*b^4*d^3*e^4+168*A*a^2*b^5*d^4*e^3-56*A*a*b...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2702 vs. \(2 (433) = 866\).

Time = 0.13 (sec) , antiderivative size = 2702, normalized size of antiderivative = 6.07 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^4} \, dx=\text {Too large to display} \] Input:

integrate((b*x+a)^10*(B*x+A)/(e*x+d)^4,x, algorithm="fricas")
 

Output:

1/168*(21*B*b^10*e^11*x^11 + 8372*B*b^10*d^11 - 56*A*a^10*e^11 - 6776*(10* 
B*a*b^9 + A*b^10)*d^10*e + 26740*(9*B*a^2*b^8 + 2*A*a*b^9)*d^9*e^2 - 61320 
*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^8*e^3 + 89880*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d 
^7*e^4 - 87024*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^6*e^5 + 55272*(5*B*a^6*b^4 + 
6*A*a^5*b^5)*d^5*e^6 - 21840*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^4*e^7 + 4620*(3 
*B*a^8*b^2 + 8*A*a^7*b^3)*d^3*e^8 - 280*(2*B*a^9*b + 9*A*a^8*b^2)*d^2*e^9 
- 28*(B*a^10 + 10*A*a^9*b)*d*e^10 - 3*(11*B*b^10*d*e^10 - 8*(10*B*a*b^9 + 
A*b^10)*e^11)*x^10 + 5*(11*B*b^10*d^2*e^9 - 8*(10*B*a*b^9 + A*b^10)*d*e^10 
 + 28*(9*B*a^2*b^8 + 2*A*a*b^9)*e^11)*x^9 - 9*(11*B*b^10*d^3*e^8 - 8*(10*B 
*a*b^9 + A*b^10)*d^2*e^9 + 28*(9*B*a^2*b^8 + 2*A*a*b^9)*d*e^10 - 56*(8*B*a 
^3*b^7 + 3*A*a^2*b^8)*e^11)*x^8 + 18*(11*B*b^10*d^4*e^7 - 8*(10*B*a*b^9 + 
A*b^10)*d^3*e^8 + 28*(9*B*a^2*b^8 + 2*A*a*b^9)*d^2*e^9 - 56*(8*B*a^3*b^7 + 
 3*A*a^2*b^8)*d*e^10 + 70*(7*B*a^4*b^6 + 4*A*a^3*b^7)*e^11)*x^7 - 42*(11*B 
*b^10*d^5*e^6 - 8*(10*B*a*b^9 + A*b^10)*d^4*e^7 + 28*(9*B*a^2*b^8 + 2*A*a* 
b^9)*d^3*e^8 - 56*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^2*e^9 + 70*(7*B*a^4*b^6 + 
4*A*a^3*b^7)*d*e^10 - 56*(6*B*a^5*b^5 + 5*A*a^4*b^6)*e^11)*x^6 + 126*(11*B 
*b^10*d^6*e^5 - 8*(10*B*a*b^9 + A*b^10)*d^5*e^6 + 28*(9*B*a^2*b^8 + 2*A*a* 
b^9)*d^4*e^7 - 56*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^3*e^8 + 70*(7*B*a^4*b^6 + 
4*A*a^3*b^7)*d^2*e^9 - 56*(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 - 630*(11*B*b^10*d^7*e^4 - 8*(10*B*a*b^9 ...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^4} \, dx=\text {Timed out} \] Input:

integrate((b*x+a)**10*(B*x+A)/(e*x+d)**4,x)
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1839 vs. \(2 (433) = 866\).

Time = 0.11 (sec) , antiderivative size = 1839, normalized size of antiderivative = 4.13 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^4} \, dx=\text {Too large to display} \] Input:

integrate((b*x+a)^10*(B*x+A)/(e*x+d)^4,x, algorithm="maxima")
 

Output:

1/6*(299*B*b^10*d^11 - 2*A*a^10*e^11 - 242*(10*B*a*b^9 + A*b^10)*d^10*e + 
955*(9*B*a^2*b^8 + 2*A*a*b^9)*d^9*e^2 - 2190*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d 
^8*e^3 + 3210*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^7*e^4 - 3108*(6*B*a^5*b^5 + 5* 
A*a^4*b^6)*d^6*e^5 + 1974*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^5*e^6 - 780*(4*B*a 
^7*b^3 + 7*A*a^6*b^4)*d^4*e^7 + 165*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^3*e^8 - 
10*(2*B*a^9*b + 9*A*a^8*b^2)*d^2*e^9 - (B*a^10 + 10*A*a^9*b)*d*e^10 + 30*( 
11*B*b^10*d^9*e^2 - 9*(10*B*a*b^9 + A*b^10)*d^8*e^3 + 36*(9*B*a^2*b^8 + 2* 
A*a*b^9)*d^7*e^4 - 84*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^6*e^5 + 126*(7*B*a^4*b 
^6 + 4*A*a^3*b^7)*d^5*e^6 - 126*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^4*e^7 + 84*( 
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 + 9*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d*e^10 - (2*B*a^9*b + 9*A*a^8*b^2)*e^11) 
*x^2 + 3*(209*B*b^10*d^10*e - 170*(10*B*a*b^9 + A*b^10)*d^9*e^2 + 675*(9*B 
*a^2*b^8 + 2*A*a*b^9)*d^8*e^3 - 1560*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^7*e^4 + 
 2310*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^6*e^5 - 2268*(6*B*a^5*b^5 + 5*A*a^4*b^ 
6)*d^5*e^6 + 1470*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^4*e^7 - 600*(4*B*a^7*b^3 + 
 7*A*a^6*b^4)*d^3*e^8 + 135*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^2*e^9 - 10*(2*B* 
a^9*b + 9*A*a^8*b^2)*d*e^10 - (B*a^10 + 10*A*a^9*b)*e^11)*x)/(e^15*x^3 + 3 
*d*e^14*x^2 + 3*d^2*e^13*x + d^3*e^12) + 1/168*(21*B*b^10*e^7*x^8 - 24*(4* 
B*b^10*d*e^6 - (10*B*a*b^9 + A*b^10)*e^7)*x^7 + 28*(10*B*b^10*d^2*e^5 - 4* 
(10*B*a*b^9 + A*b^10)*d*e^6 + 5*(9*B*a^2*b^8 + 2*A*a*b^9)*e^7)*x^6 - 16...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2096 vs. \(2 (433) = 866\).

Time = 0.13 (sec) , antiderivative size = 2096, normalized size of antiderivative = 4.71 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^4} \, dx=\text {Too large to display} \] Input:

integrate((b*x+a)^10*(B*x+A)/(e*x+d)^4,x, algorithm="giac")
 

Output:

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 - 2 
80*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)* 
log(abs(e*x + d))/e^12 + 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 + 987 
0*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 - 2 
0*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...
 

Mupad [B] (verification not implemented)

Time = 1.18 (sec) , antiderivative size = 5544, normalized size of antiderivative = 12.46 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^4} \, dx=\text {Too large to display} \] Input:

int(((A + B*x)*(a + b*x)^10)/(d + e*x)^4,x)
 

Output:

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^1 
0 + 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^1 
0*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...
 

Reduce [B] (verification not implemented)

Time = 0.33 (sec) , antiderivative size = 1786, normalized size of antiderivative = 4.01 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^4} \, dx =\text {Too large to display} \] Input:

int((b*x+a)^10*(B*x+A)/(e*x+d)^4,x)
                                                                                    
                                                                                    
 

Output:

(27720*log(d + e*x)*a**8*b**3*d**4*e**8 + 83160*log(d + e*x)*a**8*b**3*d** 
3*e**9*x + 83160*log(d + e*x)*a**8*b**3*d**2*e**10*x**2 + 27720*log(d + e* 
x)*a**8*b**3*d*e**11*x**3 - 221760*log(d + e*x)*a**7*b**4*d**5*e**7 - 6652 
80*log(d + e*x)*a**7*b**4*d**4*e**8*x - 665280*log(d + e*x)*a**7*b**4*d**3 
*e**9*x**2 - 221760*log(d + e*x)*a**7*b**4*d**2*e**10*x**3 + 776160*log(d 
+ e*x)*a**6*b**5*d**6*e**6 + 2328480*log(d + e*x)*a**6*b**5*d**5*e**7*x + 
2328480*log(d + e*x)*a**6*b**5*d**4*e**8*x**2 + 776160*log(d + e*x)*a**6*b 
**5*d**3*e**9*x**3 - 1552320*log(d + e*x)*a**5*b**6*d**7*e**5 - 4656960*lo 
g(d + e*x)*a**5*b**6*d**6*e**6*x - 4656960*log(d + e*x)*a**5*b**6*d**5*e** 
7*x**2 - 1552320*log(d + e*x)*a**5*b**6*d**4*e**8*x**3 + 1940400*log(d + e 
*x)*a**4*b**7*d**8*e**4 + 5821200*log(d + e*x)*a**4*b**7*d**7*e**5*x + 582 
1200*log(d + e*x)*a**4*b**7*d**6*e**6*x**2 + 1940400*log(d + e*x)*a**4*b** 
7*d**5*e**7*x**3 - 1552320*log(d + e*x)*a**3*b**8*d**9*e**3 - 4656960*log( 
d + e*x)*a**3*b**8*d**8*e**4*x - 4656960*log(d + e*x)*a**3*b**8*d**7*e**5* 
x**2 - 1552320*log(d + e*x)*a**3*b**8*d**6*e**6*x**3 + 776160*log(d + e*x) 
*a**2*b**9*d**10*e**2 + 2328480*log(d + e*x)*a**2*b**9*d**9*e**3*x + 23284 
80*log(d + e*x)*a**2*b**9*d**8*e**4*x**2 + 776160*log(d + e*x)*a**2*b**9*d 
**7*e**5*x**3 - 221760*log(d + e*x)*a*b**10*d**11*e - 665280*log(d + e*x)* 
a*b**10*d**10*e**2*x - 665280*log(d + e*x)*a*b**10*d**9*e**3*x**2 - 221760 
*log(d + e*x)*a*b**10*d**8*e**4*x**3 + 27720*log(d + e*x)*b**11*d**12 +...