\(\int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{12}} \, dx\) [1100]

Optimal result

Integrand size = 20, antiderivative size = 321 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{12}} \, dx=-\frac {(B d-A e) (a+b x)^{11}}{11 e (b d-a e) (d+e x)^{11}}-\frac {B (b d-a e)^{10}}{10 e^{12} (d+e x)^{10}}+\frac {10 b B (b d-a e)^9}{9 e^{12} (d+e x)^9}-\frac {45 b^2 B (b d-a e)^8}{8 e^{12} (d+e x)^8}+\frac {120 b^3 B (b d-a e)^7}{7 e^{12} (d+e x)^7}-\frac {35 b^4 B (b d-a e)^6}{e^{12} (d+e x)^6}+\frac {252 b^5 B (b d-a e)^5}{5 e^{12} (d+e x)^5}-\frac {105 b^6 B (b d-a e)^4}{2 e^{12} (d+e x)^4}+\frac {40 b^7 B (b d-a e)^3}{e^{12} (d+e x)^3}-\frac {45 b^8 B (b d-a e)^2}{2 e^{12} (d+e x)^2}+\frac {10 b^9 B (b d-a e)}{e^{12} (d+e x)}+\frac {b^{10} B \log (d+e x)}{e^{12}} \]

[Out]

-1/11*(-A*e+B*d)*(b*x+a)^11/e/(-a*e+b*d)/(e*x+d)^11-1/10*B*(-a*e+b*d)^10/e^12/(e*x+d)^10+10/9*b*B*(-a*e+b*d)^9
/e^12/(e*x+d)^9-45/8*b^2*B*(-a*e+b*d)^8/e^12/(e*x+d)^8+120/7*b^3*B*(-a*e+b*d)^7/e^12/(e*x+d)^7-35*b^4*B*(-a*e+
b*d)^6/e^12/(e*x+d)^6+252/5*b^5*B*(-a*e+b*d)^5/e^12/(e*x+d)^5-105/2*b^6*B*(-a*e+b*d)^4/e^12/(e*x+d)^4+40*b^7*B
*(-a*e+b*d)^3/e^12/(e*x+d)^3-45/2*b^8*B*(-a*e+b*d)^2/e^12/(e*x+d)^2+10*b^9*B*(-a*e+b*d)/e^12/(e*x+d)+b^10*B*ln
(e*x+d)/e^12

Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 321, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {79, 45} \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{12}} \, dx=-\frac {(a+b x)^{11} (B d-A e)}{11 e (d+e x)^{11} (b d-a e)}+\frac {10 b^9 B (b d-a e)}{e^{12} (d+e x)}-\frac {45 b^8 B (b d-a e)^2}{2 e^{12} (d+e x)^2}+\frac {40 b^7 B (b d-a e)^3}{e^{12} (d+e x)^3}-\frac {105 b^6 B (b d-a e)^4}{2 e^{12} (d+e x)^4}+\frac {252 b^5 B (b d-a e)^5}{5 e^{12} (d+e x)^5}-\frac {35 b^4 B (b d-a e)^6}{e^{12} (d+e x)^6}+\frac {120 b^3 B (b d-a e)^7}{7 e^{12} (d+e x)^7}-\frac {45 b^2 B (b d-a e)^8}{8 e^{12} (d+e x)^8}+\frac {10 b B (b d-a e)^9}{9 e^{12} (d+e x)^9}-\frac {B (b d-a e)^{10}}{10 e^{12} (d+e x)^{10}}+\frac {b^{10} B \log (d+e x)}{e^{12}} \]

[In]

Int[((a + b*x)^10*(A + B*x))/(d + e*x)^12,x]

[Out]

-1/11*((B*d - A*e)*(a + b*x)^11)/(e*(b*d - a*e)*(d + e*x)^11) - (B*(b*d - a*e)^10)/(10*e^12*(d + e*x)^10) + (1
0*b*B*(b*d - a*e)^9)/(9*e^12*(d + e*x)^9) - (45*b^2*B*(b*d - a*e)^8)/(8*e^12*(d + e*x)^8) + (120*b^3*B*(b*d -
a*e)^7)/(7*e^12*(d + e*x)^7) - (35*b^4*B*(b*d - a*e)^6)/(e^12*(d + e*x)^6) + (252*b^5*B*(b*d - a*e)^5)/(5*e^12
*(d + e*x)^5) - (105*b^6*B*(b*d - a*e)^4)/(2*e^12*(d + e*x)^4) + (40*b^7*B*(b*d - a*e)^3)/(e^12*(d + e*x)^3) -
(45*b^8*B*(b*d - a*e)^2)/(2*e^12*(d + e*x)^2) + (10*b^9*B*(b*d - a*e))/(e^12*(d + e*x)) + (b^10*B*Log[d + e*x
])/e^12

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

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}}{11 e (b d-a e) (d+e x)^{11}}+\frac {B \int \frac {(a+b x)^{10}}{(d+e x)^{11}} \, dx}{e} \\ & = -\frac {(B d-A e) (a+b x)^{11}}{11 e (b d-a e) (d+e x)^{11}}+\frac {B \int \left (\frac {(-b d+a e)^{10}}{e^{10} (d+e x)^{11}}-\frac {10 b (b d-a e)^9}{e^{10} (d+e x)^{10}}+\frac {45 b^2 (b d-a e)^8}{e^{10} (d+e x)^9}-\frac {120 b^3 (b d-a e)^7}{e^{10} (d+e x)^8}+\frac {210 b^4 (b d-a e)^6}{e^{10} (d+e x)^7}-\frac {252 b^5 (b d-a e)^5}{e^{10} (d+e x)^6}+\frac {210 b^6 (b d-a e)^4}{e^{10} (d+e x)^5}-\frac {120 b^7 (b d-a e)^3}{e^{10} (d+e x)^4}+\frac {45 b^8 (b d-a e)^2}{e^{10} (d+e x)^3}-\frac {10 b^9 (b d-a e)}{e^{10} (d+e x)^2}+\frac {b^{10}}{e^{10} (d+e x)}\right ) \, dx}{e} \\ & = -\frac {(B d-A e) (a+b x)^{11}}{11 e (b d-a e) (d+e x)^{11}}-\frac {B (b d-a e)^{10}}{10 e^{12} (d+e x)^{10}}+\frac {10 b B (b d-a e)^9}{9 e^{12} (d+e x)^9}-\frac {45 b^2 B (b d-a e)^8}{8 e^{12} (d+e x)^8}+\frac {120 b^3 B (b d-a e)^7}{7 e^{12} (d+e x)^7}-\frac {35 b^4 B (b d-a e)^6}{e^{12} (d+e x)^6}+\frac {252 b^5 B (b d-a e)^5}{5 e^{12} (d+e x)^5}-\frac {105 b^6 B (b d-a e)^4}{2 e^{12} (d+e x)^4}+\frac {40 b^7 B (b d-a e)^3}{e^{12} (d+e x)^3}-\frac {45 b^8 B (b d-a e)^2}{2 e^{12} (d+e x)^2}+\frac {10 b^9 B (b d-a e)}{e^{12} (d+e x)}+\frac {b^{10} B \log (d+e x)}{e^{12}} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1443\) vs. \(2(321)=642\).

Time = 1.25 (sec) , antiderivative size = 1443, normalized size of antiderivative = 4.50 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{12}} \, dx=-\frac {252 a^{10} e^{10} (10 A e+B (d+11 e x))+280 a^9 b e^9 \left (9 A e (d+11 e x)+2 B \left (d^2+11 d e x+55 e^2 x^2\right )\right )+315 a^8 b^2 e^8 \left (8 A e \left (d^2+11 d e x+55 e^2 x^2\right )+3 B \left (d^3+11 d^2 e x+55 d e^2 x^2+165 e^3 x^3\right )\right )+360 a^7 b^3 e^7 \left (7 A e \left (d^3+11 d^2 e x+55 d e^2 x^2+165 e^3 x^3\right )+4 B \left (d^4+11 d^3 e x+55 d^2 e^2 x^2+165 d e^3 x^3+330 e^4 x^4\right )\right )+420 a^6 b^4 e^6 \left (6 A e \left (d^4+11 d^3 e x+55 d^2 e^2 x^2+165 d e^3 x^3+330 e^4 x^4\right )+5 B \left (d^5+11 d^4 e x+55 d^3 e^2 x^2+165 d^2 e^3 x^3+330 d e^4 x^4+462 e^5 x^5\right )\right )+504 a^5 b^5 e^5 \left (5 A e \left (d^5+11 d^4 e x+55 d^3 e^2 x^2+165 d^2 e^3 x^3+330 d e^4 x^4+462 e^5 x^5\right )+6 B \left (d^6+11 d^5 e x+55 d^4 e^2 x^2+165 d^3 e^3 x^3+330 d^2 e^4 x^4+462 d e^5 x^5+462 e^6 x^6\right )\right )+630 a^4 b^6 e^4 \left (4 A e \left (d^6+11 d^5 e x+55 d^4 e^2 x^2+165 d^3 e^3 x^3+330 d^2 e^4 x^4+462 d e^5 x^5+462 e^6 x^6\right )+7 B \left (d^7+11 d^6 e x+55 d^5 e^2 x^2+165 d^4 e^3 x^3+330 d^3 e^4 x^4+462 d^2 e^5 x^5+462 d e^6 x^6+330 e^7 x^7\right )\right )+840 a^3 b^7 e^3 \left (3 A e \left (d^7+11 d^6 e x+55 d^5 e^2 x^2+165 d^4 e^3 x^3+330 d^3 e^4 x^4+462 d^2 e^5 x^5+462 d e^6 x^6+330 e^7 x^7\right )+8 B \left (d^8+11 d^7 e x+55 d^6 e^2 x^2+165 d^5 e^3 x^3+330 d^4 e^4 x^4+462 d^3 e^5 x^5+462 d^2 e^6 x^6+330 d e^7 x^7+165 e^8 x^8\right )\right )+1260 a^2 b^8 e^2 \left (2 A e \left (d^8+11 d^7 e x+55 d^6 e^2 x^2+165 d^5 e^3 x^3+330 d^4 e^4 x^4+462 d^3 e^5 x^5+462 d^2 e^6 x^6+330 d e^7 x^7+165 e^8 x^8\right )+9 B \left (d^9+11 d^8 e x+55 d^7 e^2 x^2+165 d^6 e^3 x^3+330 d^5 e^4 x^4+462 d^4 e^5 x^5+462 d^3 e^6 x^6+330 d^2 e^7 x^7+165 d e^8 x^8+55 e^9 x^9\right )\right )+2520 a b^9 e \left (A e \left (d^9+11 d^8 e x+55 d^7 e^2 x^2+165 d^6 e^3 x^3+330 d^5 e^4 x^4+462 d^4 e^5 x^5+462 d^3 e^6 x^6+330 d^2 e^7 x^7+165 d e^8 x^8+55 e^9 x^9\right )+10 B \left (d^{10}+11 d^9 e x+55 d^8 e^2 x^2+165 d^7 e^3 x^3+330 d^6 e^4 x^4+462 d^5 e^5 x^5+462 d^4 e^6 x^6+330 d^3 e^7 x^7+165 d^2 e^8 x^8+55 d e^9 x^9+11 e^{10} x^{10}\right )\right )+b^{10} \left (2520 A e \left (d^{10}+11 d^9 e x+55 d^8 e^2 x^2+165 d^7 e^3 x^3+330 d^6 e^4 x^4+462 d^5 e^5 x^5+462 d^4 e^6 x^6+330 d^3 e^7 x^7+165 d^2 e^8 x^8+55 d e^9 x^9+11 e^{10} x^{10}\right )-B d \left (83711 d^{10}+893101 d^9 e x+4313045 d^8 e^2 x^2+12430935 d^7 e^3 x^3+23718420 d^6 e^4 x^4+31376268 d^5 e^5 x^5+29241828 d^4 e^6 x^6+19057500 d^3 e^7 x^7+8385300 d^2 e^8 x^8+2286900 d e^9 x^9+304920 e^{10} x^{10}\right )\right )-27720 b^{10} B (d+e x)^{11} \log (d+e x)}{27720 e^{12} (d+e x)^{11}} \]

[In]

Integrate[((a + b*x)^10*(A + B*x))/(d + e*x)^12,x]

[Out]

-1/27720*(252*a^10*e^10*(10*A*e + B*(d + 11*e*x)) + 280*a^9*b*e^9*(9*A*e*(d + 11*e*x) + 2*B*(d^2 + 11*d*e*x +
55*e^2*x^2)) + 315*a^8*b^2*e^8*(8*A*e*(d^2 + 11*d*e*x + 55*e^2*x^2) + 3*B*(d^3 + 11*d^2*e*x + 55*d*e^2*x^2 + 1
65*e^3*x^3)) + 360*a^7*b^3*e^7*(7*A*e*(d^3 + 11*d^2*e*x + 55*d*e^2*x^2 + 165*e^3*x^3) + 4*B*(d^4 + 11*d^3*e*x
+ 55*d^2*e^2*x^2 + 165*d*e^3*x^3 + 330*e^4*x^4)) + 420*a^6*b^4*e^6*(6*A*e*(d^4 + 11*d^3*e*x + 55*d^2*e^2*x^2 +
165*d*e^3*x^3 + 330*e^4*x^4) + 5*B*(d^5 + 11*d^4*e*x + 55*d^3*e^2*x^2 + 165*d^2*e^3*x^3 + 330*d*e^4*x^4 + 462
*e^5*x^5)) + 504*a^5*b^5*e^5*(5*A*e*(d^5 + 11*d^4*e*x + 55*d^3*e^2*x^2 + 165*d^2*e^3*x^3 + 330*d*e^4*x^4 + 462
*e^5*x^5) + 6*B*(d^6 + 11*d^5*e*x + 55*d^4*e^2*x^2 + 165*d^3*e^3*x^3 + 330*d^2*e^4*x^4 + 462*d*e^5*x^5 + 462*e
^6*x^6)) + 630*a^4*b^6*e^4*(4*A*e*(d^6 + 11*d^5*e*x + 55*d^4*e^2*x^2 + 165*d^3*e^3*x^3 + 330*d^2*e^4*x^4 + 462
*d*e^5*x^5 + 462*e^6*x^6) + 7*B*(d^7 + 11*d^6*e*x + 55*d^5*e^2*x^2 + 165*d^4*e^3*x^3 + 330*d^3*e^4*x^4 + 462*d
^2*e^5*x^5 + 462*d*e^6*x^6 + 330*e^7*x^7)) + 840*a^3*b^7*e^3*(3*A*e*(d^7 + 11*d^6*e*x + 55*d^5*e^2*x^2 + 165*d
^4*e^3*x^3 + 330*d^3*e^4*x^4 + 462*d^2*e^5*x^5 + 462*d*e^6*x^6 + 330*e^7*x^7) + 8*B*(d^8 + 11*d^7*e*x + 55*d^6
*e^2*x^2 + 165*d^5*e^3*x^3 + 330*d^4*e^4*x^4 + 462*d^3*e^5*x^5 + 462*d^2*e^6*x^6 + 330*d*e^7*x^7 + 165*e^8*x^8
)) + 1260*a^2*b^8*e^2*(2*A*e*(d^8 + 11*d^7*e*x + 55*d^6*e^2*x^2 + 165*d^5*e^3*x^3 + 330*d^4*e^4*x^4 + 462*d^3*
e^5*x^5 + 462*d^2*e^6*x^6 + 330*d*e^7*x^7 + 165*e^8*x^8) + 9*B*(d^9 + 11*d^8*e*x + 55*d^7*e^2*x^2 + 165*d^6*e^
3*x^3 + 330*d^5*e^4*x^4 + 462*d^4*e^5*x^5 + 462*d^3*e^6*x^6 + 330*d^2*e^7*x^7 + 165*d*e^8*x^8 + 55*e^9*x^9)) +
2520*a*b^9*e*(A*e*(d^9 + 11*d^8*e*x + 55*d^7*e^2*x^2 + 165*d^6*e^3*x^3 + 330*d^5*e^4*x^4 + 462*d^4*e^5*x^5 +
462*d^3*e^6*x^6 + 330*d^2*e^7*x^7 + 165*d*e^8*x^8 + 55*e^9*x^9) + 10*B*(d^10 + 11*d^9*e*x + 55*d^8*e^2*x^2 + 1
65*d^7*e^3*x^3 + 330*d^6*e^4*x^4 + 462*d^5*e^5*x^5 + 462*d^4*e^6*x^6 + 330*d^3*e^7*x^7 + 165*d^2*e^8*x^8 + 55*
d*e^9*x^9 + 11*e^10*x^10)) + b^10*(2520*A*e*(d^10 + 11*d^9*e*x + 55*d^8*e^2*x^2 + 165*d^7*e^3*x^3 + 330*d^6*e^
4*x^4 + 462*d^5*e^5*x^5 + 462*d^4*e^6*x^6 + 330*d^3*e^7*x^7 + 165*d^2*e^8*x^8 + 55*d*e^9*x^9 + 11*e^10*x^10) -
B*d*(83711*d^10 + 893101*d^9*e*x + 4313045*d^8*e^2*x^2 + 12430935*d^7*e^3*x^3 + 23718420*d^6*e^4*x^4 + 313762
68*d^5*e^5*x^5 + 29241828*d^4*e^6*x^6 + 19057500*d^3*e^7*x^7 + 8385300*d^2*e^8*x^8 + 2286900*d*e^9*x^9 + 30492
0*e^10*x^10)) - 27720*b^10*B*(d + e*x)^11*Log[d + e*x])/(e^12*(d + e*x)^11)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1902\) vs. \(2(305)=610\).

Time = 2.13 (sec) , antiderivative size = 1903, normalized size of antiderivative = 5.93

method	result	size
risch	\(\text {Expression too large to display}\)	\(1903\)
norman	\(\text {Expression too large to display}\)	\(1939\)
default	\(\text {Expression too large to display}\)	\(1940\)
parallelrisch	\(\text {Expression too large to display}\)	\(2458\)

[In]

int((b*x+a)^10*(B*x+A)/(e*x+d)^12,x,method=_RETURNVERBOSE)

[Out]

(-b^9*(A*b*e+10*B*a*e-11*B*b*d)/e^2*x^10-5/2*b^8*(2*A*a*b*e^2+2*A*b^2*d*e+9*B*a^2*e^2+20*B*a*b*d*e-33*B*b^2*d^
2)/e^3*x^9-5/2*b^7*(6*A*a^2*b*e^3+6*A*a*b^2*d*e^2+6*A*b^3*d^2*e+16*B*a^3*e^3+27*B*a^2*b*d*e^2+60*B*a*b^2*d^2*e
-121*B*b^3*d^3)/e^4*x^8-5/2*b^6*(12*A*a^3*b*e^4+12*A*a^2*b^2*d*e^3+12*A*a*b^3*d^2*e^2+12*A*b^4*d^3*e+21*B*a^4*
e^4+32*B*a^3*b*d*e^3+54*B*a^2*b^2*d^2*e^2+120*B*a*b^3*d^3*e-275*B*b^4*d^4)/e^5*x^7-7/10*b^5*(60*A*a^4*b*e^5+60
*A*a^3*b^2*d*e^4+60*A*a^2*b^3*d^2*e^3+60*A*a*b^4*d^3*e^2+60*A*b^5*d^4*e+72*B*a^5*e^5+105*B*a^4*b*d*e^4+160*B*a
^3*b^2*d^2*e^3+270*B*a^2*b^3*d^3*e^2+600*B*a*b^4*d^4*e-1507*B*b^5*d^5)/e^6*x^6-7/10*b^4*(60*A*a^5*b*e^6+60*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+60*A*a*b^5*d^4*e^2+60*A*b^6*d^5*e+50*B*a^6*e^6+72*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+270*B*a^2*b^4*d^4*e^2+600*B*a*b^5*d^5*e-1617*B*b^6*d^6)/e
^7*x^5-1/14*b^3*(420*A*a^6*b*e^7+420*A*a^5*b^2*d*e^6+420*A*a^4*b^3*d^2*e^5+420*A*a^3*b^4*d^3*e^4+420*A*a^2*b^5
*d^4*e^3+420*A*a*b^6*d^5*e^2+420*A*b^7*d^6*e+240*B*a^7*e^7+350*B*a^6*b*d*e^6+504*B*a^5*b^2*d^2*e^5+735*B*a^4*b
^3*d^3*e^4+1120*B*a^3*b^4*d^4*e^3+1890*B*a^2*b^5*d^5*e^2+4200*B*a*b^6*d^6*e-11979*B*b^7*d^7)/e^8*x^4-1/56*b^2*
(840*A*a^7*b*e^8+840*A*a^6*b^2*d*e^7+840*A*a^5*b^3*d^2*e^6+840*A*a^4*b^4*d^3*e^5+840*A*a^3*b^5*d^4*e^4+840*A*a
^2*b^6*d^5*e^3+840*A*a*b^7*d^6*e^2+840*A*b^8*d^7*e+315*B*a^8*e^8+480*B*a^7*b*d*e^7+700*B*a^6*b^2*d^2*e^6+1008*
B*a^5*b^3*d^3*e^5+1470*B*a^4*b^4*d^4*e^4+2240*B*a^3*b^5*d^5*e^3+3780*B*a^2*b^6*d^6*e^2+8400*B*a*b^7*d^7*e-2511
3*B*b^8*d^8)/e^9*x^3-1/504*b*(2520*A*a^8*b*e^9+2520*A*a^7*b^2*d*e^8+2520*A*a^6*b^3*d^2*e^7+2520*A*a^5*b^4*d^3*
e^6+2520*A*a^4*b^5*d^4*e^5+2520*A*a^3*b^6*d^5*e^4+2520*A*a^2*b^7*d^6*e^3+2520*A*a*b^8*d^7*e^2+2520*A*b^9*d^8*e
+560*B*a^9*e^9+945*B*a^8*b*d*e^8+1440*B*a^7*b^2*d^2*e^7+2100*B*a^6*b^3*d^3*e^6+3024*B*a^5*b^4*d^4*e^5+4410*B*a
^4*b^5*d^5*e^4+6720*B*a^3*b^6*d^6*e^3+11340*B*a^2*b^7*d^7*e^2+25200*B*a*b^8*d^8*e-78419*B*b^9*d^9)/e^10*x^2-1/
2520*(2520*A*a^9*b*e^10+2520*A*a^8*b^2*d*e^9+2520*A*a^7*b^3*d^2*e^8+2520*A*a^6*b^4*d^3*e^7+2520*A*a^5*b^5*d^4*
e^6+2520*A*a^4*b^6*d^5*e^5+2520*A*a^3*b^7*d^6*e^4+2520*A*a^2*b^8*d^7*e^3+2520*A*a*b^9*d^8*e^2+2520*A*b^10*d^9*
e+252*B*a^10*e^10+560*B*a^9*b*d*e^9+945*B*a^8*b^2*d^2*e^8+1440*B*a^7*b^3*d^3*e^7+2100*B*a^6*b^4*d^4*e^6+3024*B
*a^5*b^5*d^5*e^5+4410*B*a^4*b^6*d^6*e^4+6720*B*a^3*b^7*d^7*e^3+11340*B*a^2*b^8*d^8*e^2+25200*B*a*b^9*d^9*e-811
91*B*b^10*d^10)/e^11*x-1/27720*(2520*A*a^10*e^11+2520*A*a^9*b*d*e^10+2520*A*a^8*b^2*d^2*e^9+2520*A*a^7*b^3*d^3
*e^8+2520*A*a^6*b^4*d^4*e^7+2520*A*a^5*b^5*d^5*e^6+2520*A*a^4*b^6*d^6*e^5+2520*A*a^3*b^7*d^7*e^4+2520*A*a^2*b^
8*d^8*e^3+2520*A*a*b^9*d^9*e^2+2520*A*b^10*d^10*e+252*B*a^10*d*e^10+560*B*a^9*b*d^2*e^9+945*B*a^8*b^2*d^3*e^8+
1440*B*a^7*b^3*d^4*e^7+2100*B*a^6*b^4*d^5*e^6+3024*B*a^5*b^5*d^6*e^5+4410*B*a^4*b^6*d^7*e^4+6720*B*a^3*b^7*d^8
*e^3+11340*B*a^2*b^8*d^9*e^2+25200*B*a*b^9*d^10*e-83711*B*b^10*d^11)/e^12)/(e*x+d)^11+b^10*B*ln(e*x+d)/e^12

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2089 vs. \(2 (305) = 610\).

Time = 0.31 (sec) , antiderivative size = 2089, normalized size of antiderivative = 6.51 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{12}} \, dx=\text {Too large to display} \]

[In]

integrate((b*x+a)^10*(B*x+A)/(e*x+d)^12,x, algorithm="fricas")

[Out]

1/27720*(83711*B*b^10*d^11 - 2520*A*a^10*e^11 - 2520*(10*B*a*b^9 + A*b^10)*d^10*e - 1260*(9*B*a^2*b^8 + 2*A*a*
b^9)*d^9*e^2 - 840*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^8*e^3 - 630*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^7*e^4 - 504*(6*B*a^
5*b^5 + 5*A*a^4*b^6)*d^6*e^5 - 420*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^5*e^6 - 360*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^4*e
^7 - 315*(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 - 252*(B*a^10 + 10*A*a^9*
b)*d*e^10 + 27720*(11*B*b^10*d*e^10 - (10*B*a*b^9 + A*b^10)*e^11)*x^10 + 69300*(33*B*b^10*d^2*e^9 - 2*(10*B*a*
b^9 + A*b^10)*d*e^10 - (9*B*a^2*b^8 + 2*A*a*b^9)*e^11)*x^9 + 69300*(121*B*b^10*d^3*e^8 - 6*(10*B*a*b^9 + A*b^1
0)*d^2*e^9 - 3*(9*B*a^2*b^8 + 2*A*a*b^9)*d*e^10 - 2*(8*B*a^3*b^7 + 3*A*a^2*b^8)*e^11)*x^8 + 69300*(275*B*b^10*
d^4*e^7 - 12*(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 - 4*(8*B*a^3*b^7 + 3*A*a^2*b^
8)*d*e^10 - 3*(7*B*a^4*b^6 + 4*A*a^3*b^7)*e^11)*x^7 + 19404*(1507*B*b^10*d^5*e^6 - 60*(10*B*a*b^9 + A*b^10)*d^
4*e^7 - 30*(9*B*a^2*b^8 + 2*A*a*b^9)*d^3*e^8 - 20*(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 - 12*(6*B*a^5*b^5 + 5*A*a^4*b^6)*e^11)*x^6 + 19404*(1617*B*b^10*d^6*e^5 - 60*(10*B*a*b^9 + A*b
^10)*d^5*e^6 - 30*(9*B*a^2*b^8 + 2*A*a*b^9)*d^4*e^7 - 20*(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 - 12*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d*e^10 - 10*(5*B*a^6*b^4 + 6*A*a^5*b^5)*e^11)*x^5 + 1
980*(11979*B*b^10*d^7*e^4 - 420*(10*B*a*b^9 + A*b^10)*d^6*e^5 - 210*(9*B*a^2*b^8 + 2*A*a*b^9)*d^5*e^6 - 140*(8
*B*a^3*b^7 + 3*A*a^2*b^8)*d^4*e^7 - 105*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^3*e^8 - 84*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d
^2*e^9 - 70*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d*e^10 - 60*(4*B*a^7*b^3 + 7*A*a^6*b^4)*e^11)*x^4 + 495*(25113*B*b^10*
d^8*e^3 - 840*(10*B*a*b^9 + A*b^10)*d^7*e^4 - 420*(9*B*a^2*b^8 + 2*A*a*b^9)*d^6*e^5 - 280*(8*B*a^3*b^7 + 3*A*a
^2*b^8)*d^5*e^6 - 210*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^4*e^7 - 168*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^3*e^8 - 140*(5*B
*a^6*b^4 + 6*A*a^5*b^5)*d^2*e^9 - 120*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d*e^10 - 105*(3*B*a^8*b^2 + 8*A*a^7*b^3)*e^1
1)*x^3 + 55*(78419*B*b^10*d^9*e^2 - 2520*(10*B*a*b^9 + A*b^10)*d^8*e^3 - 1260*(9*B*a^2*b^8 + 2*A*a*b^9)*d^7*e^
4 - 840*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^6*e^5 - 630*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^5*e^6 - 504*(6*B*a^5*b^5 + 5*A
*a^4*b^6)*d^4*e^7 - 420*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^3*e^8 - 360*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^2*e^9 - 315*(3
*B*a^8*b^2 + 8*A*a^7*b^3)*d*e^10 - 280*(2*B*a^9*b + 9*A*a^8*b^2)*e^11)*x^2 + 11*(81191*B*b^10*d^10*e - 2520*(1
0*B*a*b^9 + A*b^10)*d^9*e^2 - 1260*(9*B*a^2*b^8 + 2*A*a*b^9)*d^8*e^3 - 840*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^7*e^4
- 630*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^6*e^5 - 504*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^5*e^6 - 420*(5*B*a^6*b^4 + 6*A*
a^5*b^5)*d^4*e^7 - 360*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^3*e^8 - 315*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^2*e^9 - 280*(2*
B*a^9*b + 9*A*a^8*b^2)*d*e^10 - 252*(B*a^10 + 10*A*a^9*b)*e^11)*x + 27720*(B*b^10*e^11*x^11 + 11*B*b^10*d*e^10
*x^10 + 55*B*b^10*d^2*e^9*x^9 + 165*B*b^10*d^3*e^8*x^8 + 330*B*b^10*d^4*e^7*x^7 + 462*B*b^10*d^5*e^6*x^6 + 462
*B*b^10*d^6*e^5*x^5 + 330*B*b^10*d^7*e^4*x^4 + 165*B*b^10*d^8*e^3*x^3 + 55*B*b^10*d^9*e^2*x^2 + 11*B*b^10*d^10
*e*x + B*b^10*d^11)*log(e*x + d))/(e^23*x^11 + 11*d*e^22*x^10 + 55*d^2*e^21*x^9 + 165*d^3*e^20*x^8 + 330*d^4*e
^19*x^7 + 462*d^5*e^18*x^6 + 462*d^6*e^17*x^5 + 330*d^7*e^16*x^4 + 165*d^8*e^15*x^3 + 55*d^9*e^14*x^2 + 11*d^1
0*e^13*x + d^11*e^12)

Sympy [F(-1)]

Timed out. \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{12}} \, dx=\text {Timed out} \]

[In]

integrate((b*x+a)**10*(B*x+A)/(e*x+d)**12,x)

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1932 vs. \(2 (305) = 610\).

Time = 0.28 (sec) , antiderivative size = 1932, normalized size of antiderivative = 6.02 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{12}} \, dx=\text {Too large to display} \]

[In]

integrate((b*x+a)^10*(B*x+A)/(e*x+d)^12,x, algorithm="maxima")

[Out]

1/27720*(83711*B*b^10*d^11 - 2520*A*a^10*e^11 - 2520*(10*B*a*b^9 + A*b^10)*d^10*e - 1260*(9*B*a^2*b^8 + 2*A*a*
b^9)*d^9*e^2 - 840*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^8*e^3 - 630*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^7*e^4 - 504*(6*B*a^
5*b^5 + 5*A*a^4*b^6)*d^6*e^5 - 420*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^5*e^6 - 360*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^4*e
^7 - 315*(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 - 252*(B*a^10 + 10*A*a^9*
b)*d*e^10 + 27720*(11*B*b^10*d*e^10 - (10*B*a*b^9 + A*b^10)*e^11)*x^10 + 69300*(33*B*b^10*d^2*e^9 - 2*(10*B*a*
b^9 + A*b^10)*d*e^10 - (9*B*a^2*b^8 + 2*A*a*b^9)*e^11)*x^9 + 69300*(121*B*b^10*d^3*e^8 - 6*(10*B*a*b^9 + A*b^1
0)*d^2*e^9 - 3*(9*B*a^2*b^8 + 2*A*a*b^9)*d*e^10 - 2*(8*B*a^3*b^7 + 3*A*a^2*b^8)*e^11)*x^8 + 69300*(275*B*b^10*
d^4*e^7 - 12*(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 - 4*(8*B*a^3*b^7 + 3*A*a^2*b^
8)*d*e^10 - 3*(7*B*a^4*b^6 + 4*A*a^3*b^7)*e^11)*x^7 + 19404*(1507*B*b^10*d^5*e^6 - 60*(10*B*a*b^9 + A*b^10)*d^
4*e^7 - 30*(9*B*a^2*b^8 + 2*A*a*b^9)*d^3*e^8 - 20*(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 - 12*(6*B*a^5*b^5 + 5*A*a^4*b^6)*e^11)*x^6 + 19404*(1617*B*b^10*d^6*e^5 - 60*(10*B*a*b^9 + A*b
^10)*d^5*e^6 - 30*(9*B*a^2*b^8 + 2*A*a*b^9)*d^4*e^7 - 20*(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 - 12*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d*e^10 - 10*(5*B*a^6*b^4 + 6*A*a^5*b^5)*e^11)*x^5 + 1
980*(11979*B*b^10*d^7*e^4 - 420*(10*B*a*b^9 + A*b^10)*d^6*e^5 - 210*(9*B*a^2*b^8 + 2*A*a*b^9)*d^5*e^6 - 140*(8
*B*a^3*b^7 + 3*A*a^2*b^8)*d^4*e^7 - 105*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^3*e^8 - 84*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d
^2*e^9 - 70*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d*e^10 - 60*(4*B*a^7*b^3 + 7*A*a^6*b^4)*e^11)*x^4 + 495*(25113*B*b^10*
d^8*e^3 - 840*(10*B*a*b^9 + A*b^10)*d^7*e^4 - 420*(9*B*a^2*b^8 + 2*A*a*b^9)*d^6*e^5 - 280*(8*B*a^3*b^7 + 3*A*a
^2*b^8)*d^5*e^6 - 210*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^4*e^7 - 168*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^3*e^8 - 140*(5*B
*a^6*b^4 + 6*A*a^5*b^5)*d^2*e^9 - 120*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d*e^10 - 105*(3*B*a^8*b^2 + 8*A*a^7*b^3)*e^1
1)*x^3 + 55*(78419*B*b^10*d^9*e^2 - 2520*(10*B*a*b^9 + A*b^10)*d^8*e^3 - 1260*(9*B*a^2*b^8 + 2*A*a*b^9)*d^7*e^
4 - 840*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^6*e^5 - 630*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^5*e^6 - 504*(6*B*a^5*b^5 + 5*A
*a^4*b^6)*d^4*e^7 - 420*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^3*e^8 - 360*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^2*e^9 - 315*(3
*B*a^8*b^2 + 8*A*a^7*b^3)*d*e^10 - 280*(2*B*a^9*b + 9*A*a^8*b^2)*e^11)*x^2 + 11*(81191*B*b^10*d^10*e - 2520*(1
0*B*a*b^9 + A*b^10)*d^9*e^2 - 1260*(9*B*a^2*b^8 + 2*A*a*b^9)*d^8*e^3 - 840*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^7*e^4
- 630*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^6*e^5 - 504*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^5*e^6 - 420*(5*B*a^6*b^4 + 6*A*
a^5*b^5)*d^4*e^7 - 360*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^3*e^8 - 315*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^2*e^9 - 280*(2*
B*a^9*b + 9*A*a^8*b^2)*d*e^10 - 252*(B*a^10 + 10*A*a^9*b)*e^11)*x)/(e^23*x^11 + 11*d*e^22*x^10 + 55*d^2*e^21*x
^9 + 165*d^3*e^20*x^8 + 330*d^4*e^19*x^7 + 462*d^5*e^18*x^6 + 462*d^6*e^17*x^5 + 330*d^7*e^16*x^4 + 165*d^8*e^
15*x^3 + 55*d^9*e^14*x^2 + 11*d^10*e^13*x + d^11*e^12) + B*b^10*log(e*x + d)/e^12

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1974 vs. \(2 (305) = 610\).

Time = 0.30 (sec) , antiderivative size = 1974, normalized size of antiderivative = 6.15 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{12}} \, dx=\text {Too large to display} \]

[In]

integrate((b*x+a)^10*(B*x+A)/(e*x+d)^12,x, algorithm="giac")

[Out]

B*b^10*log(abs(e*x + d))/e^12 + 1/27720*(27720*(11*B*b^10*d*e^9 - 10*B*a*b^9*e^10 - A*b^10*e^10)*x^10 + 69300*
(33*B*b^10*d^2*e^8 - 20*B*a*b^9*d*e^9 - 2*A*b^10*d*e^9 - 9*B*a^2*b^8*e^10 - 2*A*a*b^9*e^10)*x^9 + 69300*(121*B
*b^10*d^3*e^7 - 60*B*a*b^9*d^2*e^8 - 6*A*b^10*d^2*e^8 - 27*B*a^2*b^8*d*e^9 - 6*A*a*b^9*d*e^9 - 16*B*a^3*b^7*e^
10 - 6*A*a^2*b^8*e^10)*x^8 + 69300*(275*B*b^10*d^4*e^6 - 120*B*a*b^9*d^3*e^7 - 12*A*b^10*d^3*e^7 - 54*B*a^2*b^
8*d^2*e^8 - 12*A*a*b^9*d^2*e^8 - 32*B*a^3*b^7*d*e^9 - 12*A*a^2*b^8*d*e^9 - 21*B*a^4*b^6*e^10 - 12*A*a^3*b^7*e^
10)*x^7 + 19404*(1507*B*b^10*d^5*e^5 - 600*B*a*b^9*d^4*e^6 - 60*A*b^10*d^4*e^6 - 270*B*a^2*b^8*d^3*e^7 - 60*A*
a*b^9*d^3*e^7 - 160*B*a^3*b^7*d^2*e^8 - 60*A*a^2*b^8*d^2*e^8 - 105*B*a^4*b^6*d*e^9 - 60*A*a^3*b^7*d*e^9 - 72*B
*a^5*b^5*e^10 - 60*A*a^4*b^6*e^10)*x^6 + 19404*(1617*B*b^10*d^6*e^4 - 600*B*a*b^9*d^5*e^5 - 60*A*b^10*d^5*e^5
- 270*B*a^2*b^8*d^4*e^6 - 60*A*a*b^9*d^4*e^6 - 160*B*a^3*b^7*d^3*e^7 - 60*A*a^2*b^8*d^3*e^7 - 105*B*a^4*b^6*d^
2*e^8 - 60*A*a^3*b^7*d^2*e^8 - 72*B*a^5*b^5*d*e^9 - 60*A*a^4*b^6*d*e^9 - 50*B*a^6*b^4*e^10 - 60*A*a^5*b^5*e^10
)*x^5 + 1980*(11979*B*b^10*d^7*e^3 - 4200*B*a*b^9*d^6*e^4 - 420*A*b^10*d^6*e^4 - 1890*B*a^2*b^8*d^5*e^5 - 420*
A*a*b^9*d^5*e^5 - 1120*B*a^3*b^7*d^4*e^6 - 420*A*a^2*b^8*d^4*e^6 - 735*B*a^4*b^6*d^3*e^7 - 420*A*a^3*b^7*d^3*e
^7 - 504*B*a^5*b^5*d^2*e^8 - 420*A*a^4*b^6*d^2*e^8 - 350*B*a^6*b^4*d*e^9 - 420*A*a^5*b^5*d*e^9 - 240*B*a^7*b^3
*e^10 - 420*A*a^6*b^4*e^10)*x^4 + 495*(25113*B*b^10*d^8*e^2 - 8400*B*a*b^9*d^7*e^3 - 840*A*b^10*d^7*e^3 - 3780
*B*a^2*b^8*d^6*e^4 - 840*A*a*b^9*d^6*e^4 - 2240*B*a^3*b^7*d^5*e^5 - 840*A*a^2*b^8*d^5*e^5 - 1470*B*a^4*b^6*d^4
*e^6 - 840*A*a^3*b^7*d^4*e^6 - 1008*B*a^5*b^5*d^3*e^7 - 840*A*a^4*b^6*d^3*e^7 - 700*B*a^6*b^4*d^2*e^8 - 840*A*
a^5*b^5*d^2*e^8 - 480*B*a^7*b^3*d*e^9 - 840*A*a^6*b^4*d*e^9 - 315*B*a^8*b^2*e^10 - 840*A*a^7*b^3*e^10)*x^3 + 5
5*(78419*B*b^10*d^9*e - 25200*B*a*b^9*d^8*e^2 - 2520*A*b^10*d^8*e^2 - 11340*B*a^2*b^8*d^7*e^3 - 2520*A*a*b^9*d
^7*e^3 - 6720*B*a^3*b^7*d^6*e^4 - 2520*A*a^2*b^8*d^6*e^4 - 4410*B*a^4*b^6*d^5*e^5 - 2520*A*a^3*b^7*d^5*e^5 - 3
024*B*a^5*b^5*d^4*e^6 - 2520*A*a^4*b^6*d^4*e^6 - 2100*B*a^6*b^4*d^3*e^7 - 2520*A*a^5*b^5*d^3*e^7 - 1440*B*a^7*
b^3*d^2*e^8 - 2520*A*a^6*b^4*d^2*e^8 - 945*B*a^8*b^2*d*e^9 - 2520*A*a^7*b^3*d*e^9 - 560*B*a^9*b*e^10 - 2520*A*
a^8*b^2*e^10)*x^2 + 11*(81191*B*b^10*d^10 - 25200*B*a*b^9*d^9*e - 2520*A*b^10*d^9*e - 11340*B*a^2*b^8*d^8*e^2
- 2520*A*a*b^9*d^8*e^2 - 6720*B*a^3*b^7*d^7*e^3 - 2520*A*a^2*b^8*d^7*e^3 - 4410*B*a^4*b^6*d^6*e^4 - 2520*A*a^3
*b^7*d^6*e^4 - 3024*B*a^5*b^5*d^5*e^5 - 2520*A*a^4*b^6*d^5*e^5 - 2100*B*a^6*b^4*d^4*e^6 - 2520*A*a^5*b^5*d^4*e
^6 - 1440*B*a^7*b^3*d^3*e^7 - 2520*A*a^6*b^4*d^3*e^7 - 945*B*a^8*b^2*d^2*e^8 - 2520*A*a^7*b^3*d^2*e^8 - 560*B*
a^9*b*d*e^9 - 2520*A*a^8*b^2*d*e^9 - 252*B*a^10*e^10 - 2520*A*a^9*b*e^10)*x + (83711*B*b^10*d^11 - 25200*B*a*b
^9*d^10*e - 2520*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 - 6720*B*a^3*b^7*d^8*e^3 - 252
0*A*a^2*b^8*d^8*e^3 - 4410*B*a^4*b^6*d^7*e^4 - 2520*A*a^3*b^7*d^7*e^4 - 3024*B*a^5*b^5*d^6*e^5 - 2520*A*a^4*b^
6*d^6*e^5 - 2100*B*a^6*b^4*d^5*e^6 - 2520*A*a^5*b^5*d^5*e^6 - 1440*B*a^7*b^3*d^4*e^7 - 2520*A*a^6*b^4*d^4*e^7
- 945*B*a^8*b^2*d^3*e^8 - 2520*A*a^7*b^3*d^3*e^8 - 560*B*a^9*b*d^2*e^9 - 2520*A*a^8*b^2*d^2*e^9 - 252*B*a^10*d
*e^10 - 2520*A*a^9*b*d*e^10 - 2520*A*a^10*e^11)/e)/((e*x + d)^11*e^11)

Mupad [B] (verification not implemented)

Time = 2.05 (sec) , antiderivative size = 2446, normalized size of antiderivative = 7.62 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{12}} \, dx=\text {Too large to display} \]

[In]

int(((A + B*x)*(a + b*x)^10)/(d + e*x)^12,x)

[Out]

-((A*a^10*e^11)/11 - (83711*B*b^10*d^11)/27720 + (A*b^10*d^10*e)/11 + (B*a^10*d*e^10)/110 - B*b^10*d^11*log(d
+ e*x) + (B*a^10*e^11*x)/10 + A*b^10*e^11*x^10 - (81191*B*b^10*d^10*e*x)/2520 + (A*a*b^9*d^9*e^2)/11 + (2*B*a^
9*b*d^2*e^9)/99 + 5*A*a*b^9*e^11*x^9 + (10*B*a^9*b*e^11*x^2)/9 + 10*B*a*b^9*e^11*x^10 + A*b^10*d^9*e^2*x + 5*A
*b^10*d*e^10*x^9 - 11*B*b^10*d*e^10*x^10 - B*b^10*e^11*x^11*log(d + e*x) + (A*a^2*b^8*d^8*e^3)/11 + (A*a^3*b^7
*d^7*e^4)/11 + (A*a^4*b^6*d^6*e^5)/11 + (A*a^5*b^5*d^5*e^6)/11 + (A*a^6*b^4*d^4*e^7)/11 + (A*a^7*b^3*d^3*e^8)/
11 + (A*a^8*b^2*d^2*e^9)/11 + (9*B*a^2*b^8*d^9*e^2)/22 + (8*B*a^3*b^7*d^8*e^3)/33 + (7*B*a^4*b^6*d^7*e^4)/44 +
(6*B*a^5*b^5*d^6*e^5)/55 + (5*B*a^6*b^4*d^5*e^6)/66 + (4*B*a^7*b^3*d^4*e^7)/77 + (3*B*a^8*b^2*d^3*e^8)/88 + 5
*A*a^8*b^2*e^11*x^2 + 15*A*a^7*b^3*e^11*x^3 + 30*A*a^6*b^4*e^11*x^4 + 42*A*a^5*b^5*e^11*x^5 + 42*A*a^4*b^6*e^1
1*x^6 + 30*A*a^3*b^7*e^11*x^7 + 15*A*a^2*b^8*e^11*x^8 + (45*B*a^8*b^2*e^11*x^3)/8 + (120*B*a^7*b^3*e^11*x^4)/7
+ 35*B*a^6*b^4*e^11*x^5 + (252*B*a^5*b^5*e^11*x^6)/5 + (105*B*a^4*b^6*e^11*x^7)/2 + 40*B*a^3*b^7*e^11*x^8 + (
45*B*a^2*b^8*e^11*x^9)/2 + 5*A*b^10*d^8*e^3*x^2 + 15*A*b^10*d^7*e^4*x^3 + 30*A*b^10*d^6*e^5*x^4 + 42*A*b^10*d^
5*e^6*x^5 + 42*A*b^10*d^4*e^7*x^6 + 30*A*b^10*d^3*e^8*x^7 + 15*A*b^10*d^2*e^9*x^8 - (78419*B*b^10*d^9*e^2*x^2)
/504 - (25113*B*b^10*d^8*e^3*x^3)/56 - (11979*B*b^10*d^7*e^4*x^4)/14 - (11319*B*b^10*d^6*e^5*x^5)/10 - (10549*
B*b^10*d^5*e^6*x^6)/10 - (1375*B*b^10*d^4*e^7*x^7)/2 - (605*B*b^10*d^3*e^8*x^8)/2 - (165*B*b^10*d^2*e^9*x^9)/2
+ (A*a^9*b*d*e^10)/11 + (10*B*a*b^9*d^10*e)/11 + A*a^9*b*e^11*x + A*a^2*b^8*d^7*e^4*x + A*a^3*b^7*d^6*e^5*x +
A*a^4*b^6*d^5*e^6*x + A*a^5*b^5*d^4*e^7*x + A*a^6*b^4*d^3*e^8*x + A*a^7*b^3*d^2*e^9*x + 5*A*a*b^9*d^7*e^4*x^2
+ 5*A*a^7*b^3*d*e^10*x^2 + 15*A*a*b^9*d^6*e^5*x^3 + 15*A*a^6*b^4*d*e^10*x^3 + 30*A*a*b^9*d^5*e^6*x^4 + 30*A*a
^5*b^5*d*e^10*x^4 + 42*A*a*b^9*d^4*e^7*x^5 + 42*A*a^4*b^6*d*e^10*x^5 + 42*A*a*b^9*d^3*e^8*x^6 + 42*A*a^3*b^7*d
*e^10*x^6 + 30*A*a*b^9*d^2*e^9*x^7 + 30*A*a^2*b^8*d*e^10*x^7 + (9*B*a^2*b^8*d^8*e^3*x)/2 + (8*B*a^3*b^7*d^7*e^
4*x)/3 + (7*B*a^4*b^6*d^6*e^5*x)/4 + (6*B*a^5*b^5*d^5*e^6*x)/5 + (5*B*a^6*b^4*d^4*e^7*x)/6 + (4*B*a^7*b^3*d^3*
e^8*x)/7 + (3*B*a^8*b^2*d^2*e^9*x)/8 + 50*B*a*b^9*d^8*e^3*x^2 + (15*B*a^8*b^2*d*e^10*x^2)/8 + 150*B*a*b^9*d^7*
e^4*x^3 + (60*B*a^7*b^3*d*e^10*x^3)/7 + 300*B*a*b^9*d^6*e^5*x^4 + 25*B*a^6*b^4*d*e^10*x^4 + 420*B*a*b^9*d^5*e^
6*x^5 + (252*B*a^5*b^5*d*e^10*x^5)/5 + 420*B*a*b^9*d^4*e^7*x^6 + (147*B*a^4*b^6*d*e^10*x^6)/2 + 300*B*a*b^9*d^
3*e^8*x^7 + 80*B*a^3*b^7*d*e^10*x^7 + 150*B*a*b^9*d^2*e^9*x^8 + (135*B*a^2*b^8*d*e^10*x^8)/2 - 55*B*b^10*d^9*e
^2*x^2*log(d + e*x) - 165*B*b^10*d^8*e^3*x^3*log(d + e*x) - 330*B*b^10*d^7*e^4*x^4*log(d + e*x) - 462*B*b^10*d
^6*e^5*x^5*log(d + e*x) - 462*B*b^10*d^5*e^6*x^6*log(d + e*x) - 330*B*b^10*d^4*e^7*x^7*log(d + e*x) - 165*B*b^
10*d^3*e^8*x^8*log(d + e*x) - 55*B*b^10*d^2*e^9*x^9*log(d + e*x) + (2*B*a^9*b*d*e^10*x)/9 - 11*B*b^10*d^10*e*x
*log(d + e*x) + 5*A*a^2*b^8*d^6*e^5*x^2 + 5*A*a^3*b^7*d^5*e^6*x^2 + 5*A*a^4*b^6*d^4*e^7*x^2 + 5*A*a^5*b^5*d^3*
e^8*x^2 + 5*A*a^6*b^4*d^2*e^9*x^2 + 15*A*a^2*b^8*d^5*e^6*x^3 + 15*A*a^3*b^7*d^4*e^7*x^3 + 15*A*a^4*b^6*d^3*e^8
*x^3 + 15*A*a^5*b^5*d^2*e^9*x^3 + 30*A*a^2*b^8*d^4*e^7*x^4 + 30*A*a^3*b^7*d^3*e^8*x^4 + 30*A*a^4*b^6*d^2*e^9*x
^4 + 42*A*a^2*b^8*d^3*e^8*x^5 + 42*A*a^3*b^7*d^2*e^9*x^5 + 42*A*a^2*b^8*d^2*e^9*x^6 + (45*B*a^2*b^8*d^7*e^4*x^
2)/2 + (40*B*a^3*b^7*d^6*e^5*x^2)/3 + (35*B*a^4*b^6*d^5*e^6*x^2)/4 + 6*B*a^5*b^5*d^4*e^7*x^2 + (25*B*a^6*b^4*d
^3*e^8*x^2)/6 + (20*B*a^7*b^3*d^2*e^9*x^2)/7 + (135*B*a^2*b^8*d^6*e^5*x^3)/2 + 40*B*a^3*b^7*d^5*e^6*x^3 + (105
*B*a^4*b^6*d^4*e^7*x^3)/4 + 18*B*a^5*b^5*d^3*e^8*x^3 + (25*B*a^6*b^4*d^2*e^9*x^3)/2 + 135*B*a^2*b^8*d^5*e^6*x^
4 + 80*B*a^3*b^7*d^4*e^7*x^4 + (105*B*a^4*b^6*d^3*e^8*x^4)/2 + 36*B*a^5*b^5*d^2*e^9*x^4 + 189*B*a^2*b^8*d^4*e^
7*x^5 + 112*B*a^3*b^7*d^3*e^8*x^5 + (147*B*a^4*b^6*d^2*e^9*x^5)/2 + 189*B*a^2*b^8*d^3*e^8*x^6 + 112*B*a^3*b^7*
d^2*e^9*x^6 + 135*B*a^2*b^8*d^2*e^9*x^7 + A*a*b^9*d^8*e^3*x + A*a^8*b^2*d*e^10*x + 15*A*a*b^9*d*e^10*x^8 + 10*
B*a*b^9*d^9*e^2*x + 50*B*a*b^9*d*e^10*x^9 - 11*B*b^10*d*e^10*x^10*log(d + e*x))/(e^12*(d + e*x)^11)