3.12.2 \(\int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{14}} \, dx\) [1102]

Optimal. Leaf size=135 \[ -\frac {(B d-A e) (a+b x)^{11}}{13 e (b d-a e) (d+e x)^{13}}+\frac {(11 b B d+2 A b e-13 a B e) (a+b x)^{11}}{156 e (b d-a e)^2 (d+e x)^{12}}+\frac {b (11 b B d+2 A b e-13 a B e) (a+b x)^{11}}{1716 e (b d-a e)^3 (d+e x)^{11}} \]

[Out]

-1/13*(-A*e+B*d)*(b*x+a)^11/e/(-a*e+b*d)/(e*x+d)^13+1/156*(2*A*b*e-13*B*a*e+11*B*b*d)*(b*x+a)^11/e/(-a*e+b*d)^
2/(e*x+d)^12+1/1716*b*(2*A*b*e-13*B*a*e+11*B*b*d)*(b*x+a)^11/e/(-a*e+b*d)^3/(e*x+d)^11

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Rubi [A]
time = 0.04, antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {79, 47, 37} \begin {gather*} \frac {b (a+b x)^{11} (-13 a B e+2 A b e+11 b B d)}{1716 e (d+e x)^{11} (b d-a e)^3}+\frac {(a+b x)^{11} (-13 a B e+2 A b e+11 b B d)}{156 e (d+e x)^{12} (b d-a e)^2}-\frac {(a+b x)^{11} (B d-A e)}{13 e (d+e x)^{13} (b d-a e)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^10*(A + B*x))/(d + e*x)^14,x]

[Out]

-1/13*((B*d - A*e)*(a + b*x)^11)/(e*(b*d - a*e)*(d + e*x)^13) + ((11*b*B*d + 2*A*b*e - 13*a*B*e)*(a + b*x)^11)
/(156*e*(b*d - a*e)^2*(d + e*x)^12) + (b*(11*b*B*d + 2*A*b*e - 13*a*B*e)*(a + b*x)^11)/(1716*e*(b*d - a*e)^3*(
d + e*x)^11)

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n +
1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rule 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n + 1
)/((b*c - a*d)*(m + 1))), x] - Dist[d*(Simplify[m + n + 2]/((b*c - a*d)*(m + 1))), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

Rule 79

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(-(b*e - a*f
))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1
) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e,
f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || L
tQ[p, n]))))

Rubi steps

\begin {align*} \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{14}} \, dx &=-\frac {(B d-A e) (a+b x)^{11}}{13 e (b d-a e) (d+e x)^{13}}+\frac {(11 b B d+2 A b e-13 a B e) \int \frac {(a+b x)^{10}}{(d+e x)^{13}} \, dx}{13 e (b d-a e)}\\ &=-\frac {(B d-A e) (a+b x)^{11}}{13 e (b d-a e) (d+e x)^{13}}+\frac {(11 b B d+2 A b e-13 a B e) (a+b x)^{11}}{156 e (b d-a e)^2 (d+e x)^{12}}+\frac {(b (11 b B d+2 A b e-13 a B e)) \int \frac {(a+b x)^{10}}{(d+e x)^{12}} \, dx}{156 e (b d-a e)^2}\\ &=-\frac {(B d-A e) (a+b x)^{11}}{13 e (b d-a e) (d+e x)^{13}}+\frac {(11 b B d+2 A b e-13 a B e) (a+b x)^{11}}{156 e (b d-a e)^2 (d+e x)^{12}}+\frac {b (11 b B d+2 A b e-13 a B e) (a+b x)^{11}}{1716 e (b d-a e)^3 (d+e x)^{11}}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(1433\) vs. \(2(135)=270\).
time = 0.52, size = 1433, normalized size = 10.61 \begin {gather*} -\frac {11 a^{10} e^{10} (12 A e+B (d+13 e x))+10 a^9 b e^9 \left (11 A e (d+13 e x)+2 B \left (d^2+13 d e x+78 e^2 x^2\right )\right )+9 a^8 b^2 e^8 \left (10 A e \left (d^2+13 d e x+78 e^2 x^2\right )+3 B \left (d^3+13 d^2 e x+78 d e^2 x^2+286 e^3 x^3\right )\right )+8 a^7 b^3 e^7 \left (9 A e \left (d^3+13 d^2 e x+78 d e^2 x^2+286 e^3 x^3\right )+4 B \left (d^4+13 d^3 e x+78 d^2 e^2 x^2+286 d e^3 x^3+715 e^4 x^4\right )\right )+7 a^6 b^4 e^6 \left (8 A e \left (d^4+13 d^3 e x+78 d^2 e^2 x^2+286 d e^3 x^3+715 e^4 x^4\right )+5 B \left (d^5+13 d^4 e x+78 d^3 e^2 x^2+286 d^2 e^3 x^3+715 d e^4 x^4+1287 e^5 x^5\right )\right )+6 a^5 b^5 e^5 \left (7 A e \left (d^5+13 d^4 e x+78 d^3 e^2 x^2+286 d^2 e^3 x^3+715 d e^4 x^4+1287 e^5 x^5\right )+6 B \left (d^6+13 d^5 e x+78 d^4 e^2 x^2+286 d^3 e^3 x^3+715 d^2 e^4 x^4+1287 d e^5 x^5+1716 e^6 x^6\right )\right )+5 a^4 b^6 e^4 \left (6 A e \left (d^6+13 d^5 e x+78 d^4 e^2 x^2+286 d^3 e^3 x^3+715 d^2 e^4 x^4+1287 d e^5 x^5+1716 e^6 x^6\right )+7 B \left (d^7+13 d^6 e x+78 d^5 e^2 x^2+286 d^4 e^3 x^3+715 d^3 e^4 x^4+1287 d^2 e^5 x^5+1716 d e^6 x^6+1716 e^7 x^7\right )\right )+4 a^3 b^7 e^3 \left (5 A e \left (d^7+13 d^6 e x+78 d^5 e^2 x^2+286 d^4 e^3 x^3+715 d^3 e^4 x^4+1287 d^2 e^5 x^5+1716 d e^6 x^6+1716 e^7 x^7\right )+8 B \left (d^8+13 d^7 e x+78 d^6 e^2 x^2+286 d^5 e^3 x^3+715 d^4 e^4 x^4+1287 d^3 e^5 x^5+1716 d^2 e^6 x^6+1716 d e^7 x^7+1287 e^8 x^8\right )\right )+3 a^2 b^8 e^2 \left (4 A e \left (d^8+13 d^7 e x+78 d^6 e^2 x^2+286 d^5 e^3 x^3+715 d^4 e^4 x^4+1287 d^3 e^5 x^5+1716 d^2 e^6 x^6+1716 d e^7 x^7+1287 e^8 x^8\right )+9 B \left (d^9+13 d^8 e x+78 d^7 e^2 x^2+286 d^6 e^3 x^3+715 d^5 e^4 x^4+1287 d^4 e^5 x^5+1716 d^3 e^6 x^6+1716 d^2 e^7 x^7+1287 d e^8 x^8+715 e^9 x^9\right )\right )+2 a b^9 e \left (3 A e \left (d^9+13 d^8 e x+78 d^7 e^2 x^2+286 d^6 e^3 x^3+715 d^5 e^4 x^4+1287 d^4 e^5 x^5+1716 d^3 e^6 x^6+1716 d^2 e^7 x^7+1287 d e^8 x^8+715 e^9 x^9\right )+10 B \left (d^{10}+13 d^9 e x+78 d^8 e^2 x^2+286 d^7 e^3 x^3+715 d^6 e^4 x^4+1287 d^5 e^5 x^5+1716 d^4 e^6 x^6+1716 d^3 e^7 x^7+1287 d^2 e^8 x^8+715 d e^9 x^9+286 e^{10} x^{10}\right )\right )+b^{10} \left (2 A e \left (d^{10}+13 d^9 e x+78 d^8 e^2 x^2+286 d^7 e^3 x^3+715 d^6 e^4 x^4+1287 d^5 e^5 x^5+1716 d^4 e^6 x^6+1716 d^3 e^7 x^7+1287 d^2 e^8 x^8+715 d e^9 x^9+286 e^{10} x^{10}\right )+11 B \left (d^{11}+13 d^{10} e x+78 d^9 e^2 x^2+286 d^8 e^3 x^3+715 d^7 e^4 x^4+1287 d^6 e^5 x^5+1716 d^5 e^6 x^6+1716 d^4 e^7 x^7+1287 d^3 e^8 x^8+715 d^2 e^9 x^9+286 d e^{10} x^{10}+78 e^{11} x^{11}\right )\right )}{1716 e^{12} (d+e x)^{13}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^10*(A + B*x))/(d + e*x)^14,x]

[Out]

-1/1716*(11*a^10*e^10*(12*A*e + B*(d + 13*e*x)) + 10*a^9*b*e^9*(11*A*e*(d + 13*e*x) + 2*B*(d^2 + 13*d*e*x + 78
*e^2*x^2)) + 9*a^8*b^2*e^8*(10*A*e*(d^2 + 13*d*e*x + 78*e^2*x^2) + 3*B*(d^3 + 13*d^2*e*x + 78*d*e^2*x^2 + 286*
e^3*x^3)) + 8*a^7*b^3*e^7*(9*A*e*(d^3 + 13*d^2*e*x + 78*d*e^2*x^2 + 286*e^3*x^3) + 4*B*(d^4 + 13*d^3*e*x + 78*
d^2*e^2*x^2 + 286*d*e^3*x^3 + 715*e^4*x^4)) + 7*a^6*b^4*e^6*(8*A*e*(d^4 + 13*d^3*e*x + 78*d^2*e^2*x^2 + 286*d*
e^3*x^3 + 715*e^4*x^4) + 5*B*(d^5 + 13*d^4*e*x + 78*d^3*e^2*x^2 + 286*d^2*e^3*x^3 + 715*d*e^4*x^4 + 1287*e^5*x
^5)) + 6*a^5*b^5*e^5*(7*A*e*(d^5 + 13*d^4*e*x + 78*d^3*e^2*x^2 + 286*d^2*e^3*x^3 + 715*d*e^4*x^4 + 1287*e^5*x^
5) + 6*B*(d^6 + 13*d^5*e*x + 78*d^4*e^2*x^2 + 286*d^3*e^3*x^3 + 715*d^2*e^4*x^4 + 1287*d*e^5*x^5 + 1716*e^6*x^
6)) + 5*a^4*b^6*e^4*(6*A*e*(d^6 + 13*d^5*e*x + 78*d^4*e^2*x^2 + 286*d^3*e^3*x^3 + 715*d^2*e^4*x^4 + 1287*d*e^5
*x^5 + 1716*e^6*x^6) + 7*B*(d^7 + 13*d^6*e*x + 78*d^5*e^2*x^2 + 286*d^4*e^3*x^3 + 715*d^3*e^4*x^4 + 1287*d^2*e
^5*x^5 + 1716*d*e^6*x^6 + 1716*e^7*x^7)) + 4*a^3*b^7*e^3*(5*A*e*(d^7 + 13*d^6*e*x + 78*d^5*e^2*x^2 + 286*d^4*e
^3*x^3 + 715*d^3*e^4*x^4 + 1287*d^2*e^5*x^5 + 1716*d*e^6*x^6 + 1716*e^7*x^7) + 8*B*(d^8 + 13*d^7*e*x + 78*d^6*
e^2*x^2 + 286*d^5*e^3*x^3 + 715*d^4*e^4*x^4 + 1287*d^3*e^5*x^5 + 1716*d^2*e^6*x^6 + 1716*d*e^7*x^7 + 1287*e^8*
x^8)) + 3*a^2*b^8*e^2*(4*A*e*(d^8 + 13*d^7*e*x + 78*d^6*e^2*x^2 + 286*d^5*e^3*x^3 + 715*d^4*e^4*x^4 + 1287*d^3
*e^5*x^5 + 1716*d^2*e^6*x^6 + 1716*d*e^7*x^7 + 1287*e^8*x^8) + 9*B*(d^9 + 13*d^8*e*x + 78*d^7*e^2*x^2 + 286*d^
6*e^3*x^3 + 715*d^5*e^4*x^4 + 1287*d^4*e^5*x^5 + 1716*d^3*e^6*x^6 + 1716*d^2*e^7*x^7 + 1287*d*e^8*x^8 + 715*e^
9*x^9)) + 2*a*b^9*e*(3*A*e*(d^9 + 13*d^8*e*x + 78*d^7*e^2*x^2 + 286*d^6*e^3*x^3 + 715*d^5*e^4*x^4 + 1287*d^4*e
^5*x^5 + 1716*d^3*e^6*x^6 + 1716*d^2*e^7*x^7 + 1287*d*e^8*x^8 + 715*e^9*x^9) + 10*B*(d^10 + 13*d^9*e*x + 78*d^
8*e^2*x^2 + 286*d^7*e^3*x^3 + 715*d^6*e^4*x^4 + 1287*d^5*e^5*x^5 + 1716*d^4*e^6*x^6 + 1716*d^3*e^7*x^7 + 1287*
d^2*e^8*x^8 + 715*d*e^9*x^9 + 286*e^10*x^10)) + b^10*(2*A*e*(d^10 + 13*d^9*e*x + 78*d^8*e^2*x^2 + 286*d^7*e^3*
x^3 + 715*d^6*e^4*x^4 + 1287*d^5*e^5*x^5 + 1716*d^4*e^6*x^6 + 1716*d^3*e^7*x^7 + 1287*d^2*e^8*x^8 + 715*d*e^9*
x^9 + 286*e^10*x^10) + 11*B*(d^11 + 13*d^10*e*x + 78*d^9*e^2*x^2 + 286*d^8*e^3*x^3 + 715*d^7*e^4*x^4 + 1287*d^
6*e^5*x^5 + 1716*d^5*e^6*x^6 + 1716*d^4*e^7*x^7 + 1287*d^3*e^8*x^8 + 715*d^2*e^9*x^9 + 286*d*e^10*x^10 + 78*e^
11*x^11)))/(e^12*(d + e*x)^13)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1941\) vs. \(2(129)=258\).
time = 0.09, size = 1942, normalized size = 14.39

method	result	size
risch	\(\text {Expression too large to display}\)	\(1901\)
default	\(\text {Expression too large to display}\)	\(1942\)
norman	\(\text {Expression too large to display}\)	\(1992\)
gosper	\(\text {Expression too large to display}\)	\(2233\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^10*(B*x+A)/(e*x+d)^14,x,method=_RETURNVERBOSE)

[Out]

-1/2*b^10*B/e^12/(e*x+d)^2-5/4*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
)^4-1/13*(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)^13-21/4*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)^8-10/3*b^3/e^12*(7*A*a^6*b*e^7-42*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)^9-6*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)^7-5*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)^6-1/12/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^4-96
0*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)^12-3*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)^5-1/3*b^9
/e^12*(A*b*e+10*B*a*e-11*B*b*d)/(e*x+d)^3-3/2*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)/(e*x+d)^10-5/11*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)^11

________________________________________________________________________________________

Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 1966 vs. \(2 (137) = 274\).
time = 0.51, size = 1966, normalized size = 14.56 \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)^14,x, algorithm="maxima")

[Out]

-1/1716*(858*B*b^10*x^11*e^11 + 11*B*b^10*d^11 + 132*A*a^10*e^11 + 2*(10*B*a*b^9*e + A*b^10*e)*d^10 + 286*(11*
B*b^10*d*e^10 + 20*B*a*b^9*e^11 + 2*A*b^10*e^11)*x^10 + 3*(9*B*a^2*b^8*e^2 + 2*A*a*b^9*e^2)*d^9 + 715*(11*B*b^
10*d^2*e^9 + 27*B*a^2*b^8*e^11 + 6*A*a*b^9*e^11 + 2*(10*B*a*b^9*e^10 + A*b^10*e^10)*d)*x^9 + 4*(8*B*a^3*b^7*e^
3 + 3*A*a^2*b^8*e^3)*d^8 + 1287*(11*B*b^10*d^3*e^8 + 32*B*a^3*b^7*e^11 + 12*A*a^2*b^8*e^11 + 2*(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 + 5*(7*B*a^4*b^6*e^4 + 4*A*a^3*b^7*e^4)*d^7
+ 1716*(11*B*b^10*d^4*e^7 + 35*B*a^4*b^6*e^11 + 20*A*a^3*b^7*e^11 + 2*(10*B*a*b^9*e^8 + A*b^10*e^8)*d^3 + 3*(9
*B*a^2*b^8*e^9 + 2*A*a*b^9*e^9)*d^2 + 4*(8*B*a^3*b^7*e^10 + 3*A*a^2*b^8*e^10)*d)*x^7 + 6*(6*B*a^5*b^5*e^5 + 5*
A*a^4*b^6*e^5)*d^6 + 1716*(11*B*b^10*d^5*e^6 + 36*B*a^5*b^5*e^11 + 30*A*a^4*b^6*e^11 + 2*(10*B*a*b^9*e^7 + A*b
^10*e^7)*d^4 + 3*(9*B*a^2*b^8*e^8 + 2*A*a*b^9*e^8)*d^3 + 4*(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 + 7*(5*B*a^6*b^4*e^6 + 6*A*a^5*b^5*e^6)*d^5 + 1287*(11*B*b^10*d^6*e^5 +
35*B*a^6*b^4*e^11 + 42*A*a^5*b^5*e^11 + 2*(10*B*a*b^9*e^6 + A*b^10*e^6)*d^5 + 3*(9*B*a^2*b^8*e^7 + 2*A*a*b^9*e
^7)*d^4 + 4*(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 + 6*(6*B*a^5*b
^5*e^10 + 5*A*a^4*b^6*e^10)*d)*x^5 + 8*(4*B*a^7*b^3*e^7 + 7*A*a^6*b^4*e^7)*d^4 + 715*(11*B*b^10*d^7*e^4 + 32*B
*a^7*b^3*e^11 + 56*A*a^6*b^4*e^11 + 2*(10*B*a*b^9*e^5 + A*b^10*e^5)*d^6 + 3*(9*B*a^2*b^8*e^6 + 2*A*a*b^9*e^6)*
d^5 + 4*(8*B*a^3*b^7*e^7 + 3*A*a^2*b^8*e^7)*d^4 + 5*(7*B*a^4*b^6*e^8 + 4*A*a^3*b^7*e^8)*d^3 + 6*(6*B*a^5*b^5*e
^9 + 5*A*a^4*b^6*e^9)*d^2 + 7*(5*B*a^6*b^4*e^10 + 6*A*a^5*b^5*e^10)*d)*x^4 + 9*(3*B*a^8*b^2*e^8 + 8*A*a^7*b^3*
e^8)*d^3 + 286*(11*B*b^10*d^8*e^3 + 27*B*a^8*b^2*e^11 + 72*A*a^7*b^3*e^11 + 2*(10*B*a*b^9*e^4 + A*b^10*e^4)*d^
7 + 3*(9*B*a^2*b^8*e^5 + 2*A*a*b^9*e^5)*d^6 + 4*(8*B*a^3*b^7*e^6 + 3*A*a^2*b^8*e^6)*d^5 + 5*(7*B*a^4*b^6*e^7 +
4*A*a^3*b^7*e^7)*d^4 + 6*(6*B*a^5*b^5*e^8 + 5*A*a^4*b^6*e^8)*d^3 + 7*(5*B*a^6*b^4*e^9 + 6*A*a^5*b^5*e^9)*d^2
+ 8*(4*B*a^7*b^3*e^10 + 7*A*a^6*b^4*e^10)*d)*x^3 + 10*(2*B*a^9*b*e^9 + 9*A*a^8*b^2*e^9)*d^2 + 78*(11*B*b^10*d^
9*e^2 + 20*B*a^9*b*e^11 + 90*A*a^8*b^2*e^11 + 2*(10*B*a*b^9*e^3 + A*b^10*e^3)*d^8 + 3*(9*B*a^2*b^8*e^4 + 2*A*a
*b^9*e^4)*d^7 + 4*(8*B*a^3*b^7*e^5 + 3*A*a^2*b^8*e^5)*d^6 + 5*(7*B*a^4*b^6*e^6 + 4*A*a^3*b^7*e^6)*d^5 + 6*(6*B
*a^5*b^5*e^7 + 5*A*a^4*b^6*e^7)*d^4 + 7*(5*B*a^6*b^4*e^8 + 6*A*a^5*b^5*e^8)*d^3 + 8*(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 + 11*(B*a^10*e^10 + 10*A*a^9*b*e^10)*d + 13*(11
*B*b^10*d^10*e + 11*B*a^10*e^11 + 110*A*a^9*b*e^11 + 2*(10*B*a*b^9*e^2 + A*b^10*e^2)*d^9 + 3*(9*B*a^2*b^8*e^3
+ 2*A*a*b^9*e^3)*d^8 + 4*(8*B*a^3*b^7*e^4 + 3*A*a^2*b^8*e^4)*d^7 + 5*(7*B*a^4*b^6*e^5 + 4*A*a^3*b^7*e^5)*d^6 +
6*(6*B*a^5*b^5*e^6 + 5*A*a^4*b^6*e^6)*d^5 + 7*(5*B*a^6*b^4*e^7 + 6*A*a^5*b^5*e^7)*d^4 + 8*(4*B*a^7*b^3*e^8 +
7*A*a^6*b^4*e^8)*d^3 + 9*(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^13*e^25 + 13*d*x^12*e^24 + 78*d^2*x^11*e^23 + 286*d^3*x^10*e^22 + 715*d^4*x^9*e^21 + 1287*d^5*x^8*e^20 +
1716*d^6*x^7*e^19 + 1716*d^7*x^6*e^18 + 1287*d^8*x^5*e^17 + 715*d^9*x^4*e^16 + 286*d^10*x^3*e^15 + 78*d^11*x^2
*e^14 + 13*d^12*x*e^13 + d^13*e^12)

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1869 vs. \(2 (137) = 274\).
time = 1.24, size = 1869, normalized size = 13.84 \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)^14,x, algorithm="fricas")

[Out]

-1/1716*(11*B*b^10*d^11 + (858*B*b^10*x^11 + 132*A*a^10 + 572*(10*B*a*b^9 + A*b^10)*x^10 + 2145*(9*B*a^2*b^8 +
2*A*a*b^9)*x^9 + 5148*(8*B*a^3*b^7 + 3*A*a^2*b^8)*x^8 + 8580*(7*B*a^4*b^6 + 4*A*a^3*b^7)*x^7 + 10296*(6*B*a^5
*b^5 + 5*A*a^4*b^6)*x^6 + 9009*(5*B*a^6*b^4 + 6*A*a^5*b^5)*x^5 + 5720*(4*B*a^7*b^3 + 7*A*a^6*b^4)*x^4 + 2574*(
3*B*a^8*b^2 + 8*A*a^7*b^3)*x^3 + 780*(2*B*a^9*b + 9*A*a^8*b^2)*x^2 + 143*(B*a^10 + 10*A*a^9*b)*x)*e^11 + (3146
*B*b^10*d*x^10 + 1430*(10*B*a*b^9 + A*b^10)*d*x^9 + 3861*(9*B*a^2*b^8 + 2*A*a*b^9)*d*x^8 + 6864*(8*B*a^3*b^7 +
3*A*a^2*b^8)*d*x^7 + 8580*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d*x^6 + 7722*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d*x^5 + 5005*(
5*B*a^6*b^4 + 6*A*a^5*b^5)*d*x^4 + 2288*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d*x^3 + 702*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d*
x^2 + 130*(2*B*a^9*b + 9*A*a^8*b^2)*d*x + 11*(B*a^10 + 10*A*a^9*b)*d)*e^10 + (7865*B*b^10*d^2*x^9 + 2574*(10*B
*a*b^9 + A*b^10)*d^2*x^8 + 5148*(9*B*a^2*b^8 + 2*A*a*b^9)*d^2*x^7 + 6864*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^2*x^6 +
6435*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^2*x^5 + 4290*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^2*x^4 + 2002*(5*B*a^6*b^4 + 6*A
*a^5*b^5)*d^2*x^3 + 624*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^2*x^2 + 117*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^2*x + 10*(2*B*
a^9*b + 9*A*a^8*b^2)*d^2)*e^9 + (14157*B*b^10*d^3*x^8 + 3432*(10*B*a*b^9 + A*b^10)*d^3*x^7 + 5148*(9*B*a^2*b^8
+ 2*A*a*b^9)*d^3*x^6 + 5148*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^3*x^5 + 3575*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^3*x^4 +
1716*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^3*x^3 + 546*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^3*x^2 + 104*(4*B*a^7*b^3 + 7*A*a^
6*b^4)*d^3*x + 9*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^3)*e^8 + (18876*B*b^10*d^4*x^7 + 3432*(10*B*a*b^9 + A*b^10)*d^4
*x^6 + 3861*(9*B*a^2*b^8 + 2*A*a*b^9)*d^4*x^5 + 2860*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^4*x^4 + 1430*(7*B*a^4*b^6 +
4*A*a^3*b^7)*d^4*x^3 + 468*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^4*x^2 + 91*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^4*x + 8*(4*
B*a^7*b^3 + 7*A*a^6*b^4)*d^4)*e^7 + (18876*B*b^10*d^5*x^6 + 2574*(10*B*a*b^9 + A*b^10)*d^5*x^5 + 2145*(9*B*a^2
*b^8 + 2*A*a*b^9)*d^5*x^4 + 1144*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^5*x^3 + 390*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^5*x^2
+ 78*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^5*x + 7*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^5)*e^6 + (14157*B*b^10*d^6*x^5 + 143
0*(10*B*a*b^9 + A*b^10)*d^6*x^4 + 858*(9*B*a^2*b^8 + 2*A*a*b^9)*d^6*x^3 + 312*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^6*
x^2 + 65*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^6*x + 6*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^6)*e^5 + (7865*B*b^10*d^7*x^4 + 5
72*(10*B*a*b^9 + A*b^10)*d^7*x^3 + 234*(9*B*a^2*b^8 + 2*A*a*b^9)*d^7*x^2 + 52*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^7*
x + 5*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^7)*e^4 + (3146*B*b^10*d^8*x^3 + 156*(10*B*a*b^9 + A*b^10)*d^8*x^2 + 39*(9*
B*a^2*b^8 + 2*A*a*b^9)*d^8*x + 4*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^8)*e^3 + (858*B*b^10*d^9*x^2 + 26*(10*B*a*b^9 +
A*b^10)*d^9*x + 3*(9*B*a^2*b^8 + 2*A*a*b^9)*d^9)*e^2 + (143*B*b^10*d^10*x + 2*(10*B*a*b^9 + A*b^10)*d^10)*e)/
(x^13*e^25 + 13*d*x^12*e^24 + 78*d^2*x^11*e^23 + 286*d^3*x^10*e^22 + 715*d^4*x^9*e^21 + 1287*d^5*x^8*e^20 + 17
16*d^6*x^7*e^19 + 1716*d^7*x^6*e^18 + 1287*d^8*x^5*e^17 + 715*d^9*x^4*e^16 + 286*d^10*x^3*e^15 + 78*d^11*x^2*e
^14 + 13*d^12*x*e^13 + d^13*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)**14,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 2096 vs. \(2 (137) = 274\).
time = 1.00, size = 2096, normalized size = 15.53 \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)^14,x, algorithm="giac")

[Out]

-1/1716*(858*B*b^10*x^11*e^11 + 3146*B*b^10*d*x^10*e^10 + 7865*B*b^10*d^2*x^9*e^9 + 14157*B*b^10*d^3*x^8*e^8 +
18876*B*b^10*d^4*x^7*e^7 + 18876*B*b^10*d^5*x^6*e^6 + 14157*B*b^10*d^6*x^5*e^5 + 7865*B*b^10*d^7*x^4*e^4 + 31
46*B*b^10*d^8*x^3*e^3 + 858*B*b^10*d^9*x^2*e^2 + 143*B*b^10*d^10*x*e + 11*B*b^10*d^11 + 5720*B*a*b^9*x^10*e^11
+ 572*A*b^10*x^10*e^11 + 14300*B*a*b^9*d*x^9*e^10 + 1430*A*b^10*d*x^9*e^10 + 25740*B*a*b^9*d^2*x^8*e^9 + 2574
*A*b^10*d^2*x^8*e^9 + 34320*B*a*b^9*d^3*x^7*e^8 + 3432*A*b^10*d^3*x^7*e^8 + 34320*B*a*b^9*d^4*x^6*e^7 + 3432*A
*b^10*d^4*x^6*e^7 + 25740*B*a*b^9*d^5*x^5*e^6 + 2574*A*b^10*d^5*x^5*e^6 + 14300*B*a*b^9*d^6*x^4*e^5 + 1430*A*b
^10*d^6*x^4*e^5 + 5720*B*a*b^9*d^7*x^3*e^4 + 572*A*b^10*d^7*x^3*e^4 + 1560*B*a*b^9*d^8*x^2*e^3 + 156*A*b^10*d^
8*x^2*e^3 + 260*B*a*b^9*d^9*x*e^2 + 26*A*b^10*d^9*x*e^2 + 20*B*a*b^9*d^10*e + 2*A*b^10*d^10*e + 19305*B*a^2*b^
8*x^9*e^11 + 4290*A*a*b^9*x^9*e^11 + 34749*B*a^2*b^8*d*x^8*e^10 + 7722*A*a*b^9*d*x^8*e^10 + 46332*B*a^2*b^8*d^
2*x^7*e^9 + 10296*A*a*b^9*d^2*x^7*e^9 + 46332*B*a^2*b^8*d^3*x^6*e^8 + 10296*A*a*b^9*d^3*x^6*e^8 + 34749*B*a^2*
b^8*d^4*x^5*e^7 + 7722*A*a*b^9*d^4*x^5*e^7 + 19305*B*a^2*b^8*d^5*x^4*e^6 + 4290*A*a*b^9*d^5*x^4*e^6 + 7722*B*a
^2*b^8*d^6*x^3*e^5 + 1716*A*a*b^9*d^6*x^3*e^5 + 2106*B*a^2*b^8*d^7*x^2*e^4 + 468*A*a*b^9*d^7*x^2*e^4 + 351*B*a
^2*b^8*d^8*x*e^3 + 78*A*a*b^9*d^8*x*e^3 + 27*B*a^2*b^8*d^9*e^2 + 6*A*a*b^9*d^9*e^2 + 41184*B*a^3*b^7*x^8*e^11
+ 15444*A*a^2*b^8*x^8*e^11 + 54912*B*a^3*b^7*d*x^7*e^10 + 20592*A*a^2*b^8*d*x^7*e^10 + 54912*B*a^3*b^7*d^2*x^6
*e^9 + 20592*A*a^2*b^8*d^2*x^6*e^9 + 41184*B*a^3*b^7*d^3*x^5*e^8 + 15444*A*a^2*b^8*d^3*x^5*e^8 + 22880*B*a^3*b
^7*d^4*x^4*e^7 + 8580*A*a^2*b^8*d^4*x^4*e^7 + 9152*B*a^3*b^7*d^5*x^3*e^6 + 3432*A*a^2*b^8*d^5*x^3*e^6 + 2496*B
*a^3*b^7*d^6*x^2*e^5 + 936*A*a^2*b^8*d^6*x^2*e^5 + 416*B*a^3*b^7*d^7*x*e^4 + 156*A*a^2*b^8*d^7*x*e^4 + 32*B*a^
3*b^7*d^8*e^3 + 12*A*a^2*b^8*d^8*e^3 + 60060*B*a^4*b^6*x^7*e^11 + 34320*A*a^3*b^7*x^7*e^11 + 60060*B*a^4*b^6*d
*x^6*e^10 + 34320*A*a^3*b^7*d*x^6*e^10 + 45045*B*a^4*b^6*d^2*x^5*e^9 + 25740*A*a^3*b^7*d^2*x^5*e^9 + 25025*B*a
^4*b^6*d^3*x^4*e^8 + 14300*A*a^3*b^7*d^3*x^4*e^8 + 10010*B*a^4*b^6*d^4*x^3*e^7 + 5720*A*a^3*b^7*d^4*x^3*e^7 +
2730*B*a^4*b^6*d^5*x^2*e^6 + 1560*A*a^3*b^7*d^5*x^2*e^6 + 455*B*a^4*b^6*d^6*x*e^5 + 260*A*a^3*b^7*d^6*x*e^5 +
35*B*a^4*b^6*d^7*e^4 + 20*A*a^3*b^7*d^7*e^4 + 61776*B*a^5*b^5*x^6*e^11 + 51480*A*a^4*b^6*x^6*e^11 + 46332*B*a^
5*b^5*d*x^5*e^10 + 38610*A*a^4*b^6*d*x^5*e^10 + 25740*B*a^5*b^5*d^2*x^4*e^9 + 21450*A*a^4*b^6*d^2*x^4*e^9 + 10
296*B*a^5*b^5*d^3*x^3*e^8 + 8580*A*a^4*b^6*d^3*x^3*e^8 + 2808*B*a^5*b^5*d^4*x^2*e^7 + 2340*A*a^4*b^6*d^4*x^2*e
^7 + 468*B*a^5*b^5*d^5*x*e^6 + 390*A*a^4*b^6*d^5*x*e^6 + 36*B*a^5*b^5*d^6*e^5 + 30*A*a^4*b^6*d^6*e^5 + 45045*B
*a^6*b^4*x^5*e^11 + 54054*A*a^5*b^5*x^5*e^11 + 25025*B*a^6*b^4*d*x^4*e^10 + 30030*A*a^5*b^5*d*x^4*e^10 + 10010
*B*a^6*b^4*d^2*x^3*e^9 + 12012*A*a^5*b^5*d^2*x^3*e^9 + 2730*B*a^6*b^4*d^3*x^2*e^8 + 3276*A*a^5*b^5*d^3*x^2*e^8
+ 455*B*a^6*b^4*d^4*x*e^7 + 546*A*a^5*b^5*d^4*x*e^7 + 35*B*a^6*b^4*d^5*e^6 + 42*A*a^5*b^5*d^5*e^6 + 22880*B*a
^7*b^3*x^4*e^11 + 40040*A*a^6*b^4*x^4*e^11 + 9152*B*a^7*b^3*d*x^3*e^10 + 16016*A*a^6*b^4*d*x^3*e^10 + 2496*B*a
^7*b^3*d^2*x^2*e^9 + 4368*A*a^6*b^4*d^2*x^2*e^9 + 416*B*a^7*b^3*d^3*x*e^8 + 728*A*a^6*b^4*d^3*x*e^8 + 32*B*a^7
*b^3*d^4*e^7 + 56*A*a^6*b^4*d^4*e^7 + 7722*B*a^8*b^2*x^3*e^11 + 20592*A*a^7*b^3*x^3*e^11 + 2106*B*a^8*b^2*d*x^
2*e^10 + 5616*A*a^7*b^3*d*x^2*e^10 + 351*B*a^8*b^2*d^2*x*e^9 + 936*A*a^7*b^3*d^2*x*e^9 + 27*B*a^8*b^2*d^3*e^8
+ 72*A*a^7*b^3*d^3*e^8 + 1560*B*a^9*b*x^2*e^11 + 7020*A*a^8*b^2*x^2*e^11 + 260*B*a^9*b*d*x*e^10 + 1170*A*a^8*b
^2*d*x*e^10 + 20*B*a^9*b*d^2*e^9 + 90*A*a^8*b^2*d^2*e^9 + 143*B*a^10*x*e^11 + 1430*A*a^9*b*x*e^11 + 11*B*a^10*
d*e^10 + 110*A*a^9*b*d*e^10 + 132*A*a^10*e^11)*e^(-12)/(x*e + d)^13

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Mupad [B]
time = 2.03, size = 2031, normalized size = 15.04 \begin {gather*} -\frac {\frac {11\,B\,a^{10}\,d\,e^{10}+132\,A\,a^{10}\,e^{11}+20\,B\,a^9\,b\,d^2\,e^9+110\,A\,a^9\,b\,d\,e^{10}+27\,B\,a^8\,b^2\,d^3\,e^8+90\,A\,a^8\,b^2\,d^2\,e^9+32\,B\,a^7\,b^3\,d^4\,e^7+72\,A\,a^7\,b^3\,d^3\,e^8+35\,B\,a^6\,b^4\,d^5\,e^6+56\,A\,a^6\,b^4\,d^4\,e^7+36\,B\,a^5\,b^5\,d^6\,e^5+42\,A\,a^5\,b^5\,d^5\,e^6+35\,B\,a^4\,b^6\,d^7\,e^4+30\,A\,a^4\,b^6\,d^6\,e^5+32\,B\,a^3\,b^7\,d^8\,e^3+20\,A\,a^3\,b^7\,d^7\,e^4+27\,B\,a^2\,b^8\,d^9\,e^2+12\,A\,a^2\,b^8\,d^8\,e^3+20\,B\,a\,b^9\,d^{10}\,e+6\,A\,a\,b^9\,d^9\,e^2+11\,B\,b^{10}\,d^{11}+2\,A\,b^{10}\,d^{10}\,e}{1716\,e^{12}}+\frac {x\,\left (11\,B\,a^{10}\,e^{10}+20\,B\,a^9\,b\,d\,e^9+110\,A\,a^9\,b\,e^{10}+27\,B\,a^8\,b^2\,d^2\,e^8+90\,A\,a^8\,b^2\,d\,e^9+32\,B\,a^7\,b^3\,d^3\,e^7+72\,A\,a^7\,b^3\,d^2\,e^8+35\,B\,a^6\,b^4\,d^4\,e^6+56\,A\,a^6\,b^4\,d^3\,e^7+36\,B\,a^5\,b^5\,d^5\,e^5+42\,A\,a^5\,b^5\,d^4\,e^6+35\,B\,a^4\,b^6\,d^6\,e^4+30\,A\,a^4\,b^6\,d^5\,e^5+32\,B\,a^3\,b^7\,d^7\,e^3+20\,A\,a^3\,b^7\,d^6\,e^4+27\,B\,a^2\,b^8\,d^8\,e^2+12\,A\,a^2\,b^8\,d^7\,e^3+20\,B\,a\,b^9\,d^9\,e+6\,A\,a\,b^9\,d^8\,e^2+11\,B\,b^{10}\,d^{10}+2\,A\,b^{10}\,d^9\,e\right )}{132\,e^{11}}+\frac {3\,b^7\,x^8\,\left (32\,B\,a^3\,e^3+27\,B\,a^2\,b\,d\,e^2+12\,A\,a^2\,b\,e^3+20\,B\,a\,b^2\,d^2\,e+6\,A\,a\,b^2\,d\,e^2+11\,B\,b^3\,d^3+2\,A\,b^3\,d^2\,e\right )}{4\,e^4}+\frac {3\,b^4\,x^5\,\left (35\,B\,a^6\,e^6+36\,B\,a^5\,b\,d\,e^5+42\,A\,a^5\,b\,e^6+35\,B\,a^4\,b^2\,d^2\,e^4+30\,A\,a^4\,b^2\,d\,e^5+32\,B\,a^3\,b^3\,d^3\,e^3+20\,A\,a^3\,b^3\,d^2\,e^4+27\,B\,a^2\,b^4\,d^4\,e^2+12\,A\,a^2\,b^4\,d^3\,e^3+20\,B\,a\,b^5\,d^5\,e+6\,A\,a\,b^5\,d^4\,e^2+11\,B\,b^6\,d^6+2\,A\,b^6\,d^5\,e\right )}{4\,e^7}+\frac {b^9\,x^{10}\,\left (2\,A\,b\,e+20\,B\,a\,e+11\,B\,b\,d\right )}{6\,e^2}+\frac {b^6\,x^7\,\left (35\,B\,a^4\,e^4+32\,B\,a^3\,b\,d\,e^3+20\,A\,a^3\,b\,e^4+27\,B\,a^2\,b^2\,d^2\,e^2+12\,A\,a^2\,b^2\,d\,e^3+20\,B\,a\,b^3\,d^3\,e+6\,A\,a\,b^3\,d^2\,e^2+11\,B\,b^4\,d^4+2\,A\,b^4\,d^3\,e\right )}{e^5}+\frac {5\,b^3\,x^4\,\left (32\,B\,a^7\,e^7+35\,B\,a^6\,b\,d\,e^6+56\,A\,a^6\,b\,e^7+36\,B\,a^5\,b^2\,d^2\,e^5+42\,A\,a^5\,b^2\,d\,e^6+35\,B\,a^4\,b^3\,d^3\,e^4+30\,A\,a^4\,b^3\,d^2\,e^5+32\,B\,a^3\,b^4\,d^4\,e^3+20\,A\,a^3\,b^4\,d^3\,e^4+27\,B\,a^2\,b^5\,d^5\,e^2+12\,A\,a^2\,b^5\,d^4\,e^3+20\,B\,a\,b^6\,d^6\,e+6\,A\,a\,b^6\,d^5\,e^2+11\,B\,b^7\,d^7+2\,A\,b^7\,d^6\,e\right )}{12\,e^8}+\frac {b\,x^2\,\left (20\,B\,a^9\,e^9+27\,B\,a^8\,b\,d\,e^8+90\,A\,a^8\,b\,e^9+32\,B\,a^7\,b^2\,d^2\,e^7+72\,A\,a^7\,b^2\,d\,e^8+35\,B\,a^6\,b^3\,d^3\,e^6+56\,A\,a^6\,b^3\,d^2\,e^7+36\,B\,a^5\,b^4\,d^4\,e^5+42\,A\,a^5\,b^4\,d^3\,e^6+35\,B\,a^4\,b^5\,d^5\,e^4+30\,A\,a^4\,b^5\,d^4\,e^5+32\,B\,a^3\,b^6\,d^6\,e^3+20\,A\,a^3\,b^6\,d^5\,e^4+27\,B\,a^2\,b^7\,d^7\,e^2+12\,A\,a^2\,b^7\,d^6\,e^3+20\,B\,a\,b^8\,d^8\,e+6\,A\,a\,b^8\,d^7\,e^2+11\,B\,b^9\,d^9+2\,A\,b^9\,d^8\,e\right )}{22\,e^{10}}+\frac {5\,b^8\,x^9\,\left (27\,B\,a^2\,e^2+20\,B\,a\,b\,d\,e+6\,A\,a\,b\,e^2+11\,B\,b^2\,d^2+2\,A\,b^2\,d\,e\right )}{12\,e^3}+\frac {b^5\,x^6\,\left (36\,B\,a^5\,e^5+35\,B\,a^4\,b\,d\,e^4+30\,A\,a^4\,b\,e^5+32\,B\,a^3\,b^2\,d^2\,e^3+20\,A\,a^3\,b^2\,d\,e^4+27\,B\,a^2\,b^3\,d^3\,e^2+12\,A\,a^2\,b^3\,d^2\,e^3+20\,B\,a\,b^4\,d^4\,e+6\,A\,a\,b^4\,d^3\,e^2+11\,B\,b^5\,d^5+2\,A\,b^5\,d^4\,e\right )}{e^6}+\frac {b^2\,x^3\,\left (27\,B\,a^8\,e^8+32\,B\,a^7\,b\,d\,e^7+72\,A\,a^7\,b\,e^8+35\,B\,a^6\,b^2\,d^2\,e^6+56\,A\,a^6\,b^2\,d\,e^7+36\,B\,a^5\,b^3\,d^3\,e^5+42\,A\,a^5\,b^3\,d^2\,e^6+35\,B\,a^4\,b^4\,d^4\,e^4+30\,A\,a^4\,b^4\,d^3\,e^5+32\,B\,a^3\,b^5\,d^5\,e^3+20\,A\,a^3\,b^5\,d^4\,e^4+27\,B\,a^2\,b^6\,d^6\,e^2+12\,A\,a^2\,b^6\,d^5\,e^3+20\,B\,a\,b^7\,d^7\,e+6\,A\,a\,b^7\,d^6\,e^2+11\,B\,b^8\,d^8+2\,A\,b^8\,d^7\,e\right )}{6\,e^9}+\frac {B\,b^{10}\,x^{11}}{2\,e}}{d^{13}+13\,d^{12}\,e\,x+78\,d^{11}\,e^2\,x^2+286\,d^{10}\,e^3\,x^3+715\,d^9\,e^4\,x^4+1287\,d^8\,e^5\,x^5+1716\,d^7\,e^6\,x^6+1716\,d^6\,e^7\,x^7+1287\,d^5\,e^8\,x^8+715\,d^4\,e^9\,x^9+286\,d^3\,e^{10}\,x^{10}+78\,d^2\,e^{11}\,x^{11}+13\,d\,e^{12}\,x^{12}+e^{13}\,x^{13}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(a + b*x)^10)/(d + e*x)^14,x)

[Out]

-((132*A*a^10*e^11 + 11*B*b^10*d^11 + 2*A*b^10*d^10*e + 11*B*a^10*d*e^10 + 6*A*a*b^9*d^9*e^2 + 20*B*a^9*b*d^2*
e^9 + 12*A*a^2*b^8*d^8*e^3 + 20*A*a^3*b^7*d^7*e^4 + 30*A*a^4*b^6*d^6*e^5 + 42*A*a^5*b^5*d^5*e^6 + 56*A*a^6*b^4
*d^4*e^7 + 72*A*a^7*b^3*d^3*e^8 + 90*A*a^8*b^2*d^2*e^9 + 27*B*a^2*b^8*d^9*e^2 + 32*B*a^3*b^7*d^8*e^3 + 35*B*a^
4*b^6*d^7*e^4 + 36*B*a^5*b^5*d^6*e^5 + 35*B*a^6*b^4*d^5*e^6 + 32*B*a^7*b^3*d^4*e^7 + 27*B*a^8*b^2*d^3*e^8 + 11
0*A*a^9*b*d*e^10 + 20*B*a*b^9*d^10*e)/(1716*e^12) + (x*(11*B*a^10*e^10 + 11*B*b^10*d^10 + 110*A*a^9*b*e^10 + 2
*A*b^10*d^9*e + 6*A*a*b^9*d^8*e^2 + 90*A*a^8*b^2*d*e^9 + 12*A*a^2*b^8*d^7*e^3 + 20*A*a^3*b^7*d^6*e^4 + 30*A*a^
4*b^6*d^5*e^5 + 42*A*a^5*b^5*d^4*e^6 + 56*A*a^6*b^4*d^3*e^7 + 72*A*a^7*b^3*d^2*e^8 + 27*B*a^2*b^8*d^8*e^2 + 32
*B*a^3*b^7*d^7*e^3 + 35*B*a^4*b^6*d^6*e^4 + 36*B*a^5*b^5*d^5*e^5 + 35*B*a^6*b^4*d^4*e^6 + 32*B*a^7*b^3*d^3*e^7
+ 27*B*a^8*b^2*d^2*e^8 + 20*B*a*b^9*d^9*e + 20*B*a^9*b*d*e^9))/(132*e^11) + (3*b^7*x^8*(32*B*a^3*e^3 + 11*B*b
^3*d^3 + 12*A*a^2*b*e^3 + 2*A*b^3*d^2*e + 6*A*a*b^2*d*e^2 + 20*B*a*b^2*d^2*e + 27*B*a^2*b*d*e^2))/(4*e^4) + (3
*b^4*x^5*(35*B*a^6*e^6 + 11*B*b^6*d^6 + 42*A*a^5*b*e^6 + 2*A*b^6*d^5*e + 6*A*a*b^5*d^4*e^2 + 30*A*a^4*b^2*d*e^
5 + 12*A*a^2*b^4*d^3*e^3 + 20*A*a^3*b^3*d^2*e^4 + 27*B*a^2*b^4*d^4*e^2 + 32*B*a^3*b^3*d^3*e^3 + 35*B*a^4*b^2*d
^2*e^4 + 20*B*a*b^5*d^5*e + 36*B*a^5*b*d*e^5))/(4*e^7) + (b^9*x^10*(2*A*b*e + 20*B*a*e + 11*B*b*d))/(6*e^2) +
(b^6*x^7*(35*B*a^4*e^4 + 11*B*b^4*d^4 + 20*A*a^3*b*e^4 + 2*A*b^4*d^3*e + 6*A*a*b^3*d^2*e^2 + 12*A*a^2*b^2*d*e^
3 + 27*B*a^2*b^2*d^2*e^2 + 20*B*a*b^3*d^3*e + 32*B*a^3*b*d*e^3))/e^5 + (5*b^3*x^4*(32*B*a^7*e^7 + 11*B*b^7*d^7
+ 56*A*a^6*b*e^7 + 2*A*b^7*d^6*e + 6*A*a*b^6*d^5*e^2 + 42*A*a^5*b^2*d*e^6 + 12*A*a^2*b^5*d^4*e^3 + 20*A*a^3*b
^4*d^3*e^4 + 30*A*a^4*b^3*d^2*e^5 + 27*B*a^2*b^5*d^5*e^2 + 32*B*a^3*b^4*d^4*e^3 + 35*B*a^4*b^3*d^3*e^4 + 36*B*
a^5*b^2*d^2*e^5 + 20*B*a*b^6*d^6*e + 35*B*a^6*b*d*e^6))/(12*e^8) + (b*x^2*(20*B*a^9*e^9 + 11*B*b^9*d^9 + 90*A*
a^8*b*e^9 + 2*A*b^9*d^8*e + 6*A*a*b^8*d^7*e^2 + 72*A*a^7*b^2*d*e^8 + 12*A*a^2*b^7*d^6*e^3 + 20*A*a^3*b^6*d^5*e
^4 + 30*A*a^4*b^5*d^4*e^5 + 42*A*a^5*b^4*d^3*e^6 + 56*A*a^6*b^3*d^2*e^7 + 27*B*a^2*b^7*d^7*e^2 + 32*B*a^3*b^6*
d^6*e^3 + 35*B*a^4*b^5*d^5*e^4 + 36*B*a^5*b^4*d^4*e^5 + 35*B*a^6*b^3*d^3*e^6 + 32*B*a^7*b^2*d^2*e^7 + 20*B*a*b
^8*d^8*e + 27*B*a^8*b*d*e^8))/(22*e^10) + (5*b^8*x^9*(27*B*a^2*e^2 + 11*B*b^2*d^2 + 6*A*a*b*e^2 + 2*A*b^2*d*e
+ 20*B*a*b*d*e))/(12*e^3) + (b^5*x^6*(36*B*a^5*e^5 + 11*B*b^5*d^5 + 30*A*a^4*b*e^5 + 2*A*b^5*d^4*e + 6*A*a*b^4
*d^3*e^2 + 20*A*a^3*b^2*d*e^4 + 12*A*a^2*b^3*d^2*e^3 + 27*B*a^2*b^3*d^3*e^2 + 32*B*a^3*b^2*d^2*e^3 + 20*B*a*b^
4*d^4*e + 35*B*a^4*b*d*e^4))/e^6 + (b^2*x^3*(27*B*a^8*e^8 + 11*B*b^8*d^8 + 72*A*a^7*b*e^8 + 2*A*b^8*d^7*e + 6*
A*a*b^7*d^6*e^2 + 56*A*a^6*b^2*d*e^7 + 12*A*a^2*b^6*d^5*e^3 + 20*A*a^3*b^5*d^4*e^4 + 30*A*a^4*b^4*d^3*e^5 + 42
*A*a^5*b^3*d^2*e^6 + 27*B*a^2*b^6*d^6*e^2 + 32*B*a^3*b^5*d^5*e^3 + 35*B*a^4*b^4*d^4*e^4 + 36*B*a^5*b^3*d^3*e^5
+ 35*B*a^6*b^2*d^2*e^6 + 20*B*a*b^7*d^7*e + 32*B*a^7*b*d*e^7))/(6*e^9) + (B*b^10*x^11)/(2*e))/(d^13 + e^13*x^
13 + 13*d*e^12*x^12 + 78*d^11*e^2*x^2 + 286*d^10*e^3*x^3 + 715*d^9*e^4*x^4 + 1287*d^8*e^5*x^5 + 1716*d^7*e^6*x
^6 + 1716*d^6*e^7*x^7 + 1287*d^5*e^8*x^8 + 715*d^4*e^9*x^9 + 286*d^3*e^10*x^10 + 78*d^2*e^11*x^11 + 13*d^12*e*
x)

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