3.12.3 \(\int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{15}} \, dx\) [1103]

Optimal. Leaf size=185 \[ -\frac {(B d-A e) (a+b x)^{11}}{14 e (b d-a e) (d+e x)^{14}}+\frac {(11 b B d+3 A b e-14 a B e) (a+b x)^{11}}{182 e (b d-a e)^2 (d+e x)^{13}}+\frac {b (11 b B d+3 A b e-14 a B e) (a+b x)^{11}}{1092 e (b d-a e)^3 (d+e x)^{12}}+\frac {b^2 (11 b B d+3 A b e-14 a B e) (a+b x)^{11}}{12012 e (b d-a e)^4 (d+e x)^{11}} \]

[Out]

-1/14*(-A*e+B*d)*(b*x+a)^11/e/(-a*e+b*d)/(e*x+d)^14+1/182*(3*A*b*e-14*B*a*e+11*B*b*d)*(b*x+a)^11/e/(-a*e+b*d)^
2/(e*x+d)^13+1/1092*b*(3*A*b*e-14*B*a*e+11*B*b*d)*(b*x+a)^11/e/(-a*e+b*d)^3/(e*x+d)^12+1/12012*b^2*(3*A*b*e-14
*B*a*e+11*B*b*d)*(b*x+a)^11/e/(-a*e+b*d)^4/(e*x+d)^11

________________________________________________________________________________________

Rubi [A]
time = 0.06, antiderivative size = 185, normalized size of antiderivative = 1.00, number of steps used = 4, 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^2 (a+b x)^{11} (-14 a B e+3 A b e+11 b B d)}{12012 e (d+e x)^{11} (b d-a e)^4}+\frac {b (a+b x)^{11} (-14 a B e+3 A b e+11 b B d)}{1092 e (d+e x)^{12} (b d-a e)^3}+\frac {(a+b x)^{11} (-14 a B e+3 A b e+11 b B d)}{182 e (d+e x)^{13} (b d-a e)^2}-\frac {(a+b x)^{11} (B d-A e)}{14 e (d+e x)^{14} (b d-a e)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^10*(A + B*x))/(d + e*x)^15,x]

[Out]

-1/14*((B*d - A*e)*(a + b*x)^11)/(e*(b*d - a*e)*(d + e*x)^14) + ((11*b*B*d + 3*A*b*e - 14*a*B*e)*(a + b*x)^11)
/(182*e*(b*d - a*e)^2*(d + e*x)^13) + (b*(11*b*B*d + 3*A*b*e - 14*a*B*e)*(a + b*x)^11)/(1092*e*(b*d - a*e)^3*(
d + e*x)^12) + (b^2*(11*b*B*d + 3*A*b*e - 14*a*B*e)*(a + b*x)^11)/(12012*e*(b*d - a*e)^4*(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)^{15}} \, dx &=-\frac {(B d-A e) (a+b x)^{11}}{14 e (b d-a e) (d+e x)^{14}}+\frac {(11 b B d+3 A b e-14 a B e) \int \frac {(a+b x)^{10}}{(d+e x)^{14}} \, dx}{14 e (b d-a e)}\\ &=-\frac {(B d-A e) (a+b x)^{11}}{14 e (b d-a e) (d+e x)^{14}}+\frac {(11 b B d+3 A b e-14 a B e) (a+b x)^{11}}{182 e (b d-a e)^2 (d+e x)^{13}}+\frac {(b (11 b B d+3 A b e-14 a B e)) \int \frac {(a+b x)^{10}}{(d+e x)^{13}} \, dx}{91 e (b d-a e)^2}\\ &=-\frac {(B d-A e) (a+b x)^{11}}{14 e (b d-a e) (d+e x)^{14}}+\frac {(11 b B d+3 A b e-14 a B e) (a+b x)^{11}}{182 e (b d-a e)^2 (d+e x)^{13}}+\frac {b (11 b B d+3 A b e-14 a B e) (a+b x)^{11}}{1092 e (b d-a e)^3 (d+e x)^{12}}+\frac {\left (b^2 (11 b B d+3 A b e-14 a B e)\right ) \int \frac {(a+b x)^{10}}{(d+e x)^{12}} \, dx}{1092 e (b d-a e)^3}\\ &=-\frac {(B d-A e) (a+b x)^{11}}{14 e (b d-a e) (d+e x)^{14}}+\frac {(11 b B d+3 A b e-14 a B e) (a+b x)^{11}}{182 e (b d-a e)^2 (d+e x)^{13}}+\frac {b (11 b B d+3 A b e-14 a B e) (a+b x)^{11}}{1092 e (b d-a e)^3 (d+e x)^{12}}+\frac {b^2 (11 b B d+3 A b e-14 a B e) (a+b x)^{11}}{12012 e (b d-a e)^4 (d+e x)^{11}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(1430\) vs. \(2(185)=370\).
time = 0.55, size = 1430, normalized size = 7.73 \begin {gather*} -\frac {66 a^{10} e^{10} (13 A e+B (d+14 e x))+110 a^9 b e^9 \left (6 A e (d+14 e x)+B \left (d^2+14 d e x+91 e^2 x^2\right )\right )+45 a^8 b^2 e^8 \left (11 A e \left (d^2+14 d e x+91 e^2 x^2\right )+3 B \left (d^3+14 d^2 e x+91 d e^2 x^2+364 e^3 x^3\right )\right )+72 a^7 b^3 e^7 \left (5 A e \left (d^3+14 d^2 e x+91 d e^2 x^2+364 e^3 x^3\right )+2 B \left (d^4+14 d^3 e x+91 d^2 e^2 x^2+364 d e^3 x^3+1001 e^4 x^4\right )\right )+28 a^6 b^4 e^6 \left (9 A e \left (d^4+14 d^3 e x+91 d^2 e^2 x^2+364 d e^3 x^3+1001 e^4 x^4\right )+5 B \left (d^5+14 d^4 e x+91 d^3 e^2 x^2+364 d^2 e^3 x^3+1001 d e^4 x^4+2002 e^5 x^5\right )\right )+42 a^5 b^5 e^5 \left (4 A e \left (d^5+14 d^4 e x+91 d^3 e^2 x^2+364 d^2 e^3 x^3+1001 d e^4 x^4+2002 e^5 x^5\right )+3 B \left (d^6+14 d^5 e x+91 d^4 e^2 x^2+364 d^3 e^3 x^3+1001 d^2 e^4 x^4+2002 d e^5 x^5+3003 e^6 x^6\right )\right )+105 a^4 b^6 e^4 \left (A e \left (d^6+14 d^5 e x+91 d^4 e^2 x^2+364 d^3 e^3 x^3+1001 d^2 e^4 x^4+2002 d e^5 x^5+3003 e^6 x^6\right )+B \left (d^7+14 d^6 e x+91 d^5 e^2 x^2+364 d^4 e^3 x^3+1001 d^3 e^4 x^4+2002 d^2 e^5 x^5+3003 d e^6 x^6+3432 e^7 x^7\right )\right )+20 a^3 b^7 e^3 \left (3 A e \left (d^7+14 d^6 e x+91 d^5 e^2 x^2+364 d^4 e^3 x^3+1001 d^3 e^4 x^4+2002 d^2 e^5 x^5+3003 d e^6 x^6+3432 e^7 x^7\right )+4 B \left (d^8+14 d^7 e x+91 d^6 e^2 x^2+364 d^5 e^3 x^3+1001 d^4 e^4 x^4+2002 d^3 e^5 x^5+3003 d^2 e^6 x^6+3432 d e^7 x^7+3003 e^8 x^8\right )\right )+6 a^2 b^8 e^2 \left (5 A e \left (d^8+14 d^7 e x+91 d^6 e^2 x^2+364 d^5 e^3 x^3+1001 d^4 e^4 x^4+2002 d^3 e^5 x^5+3003 d^2 e^6 x^6+3432 d e^7 x^7+3003 e^8 x^8\right )+9 B \left (d^9+14 d^8 e x+91 d^7 e^2 x^2+364 d^6 e^3 x^3+1001 d^5 e^4 x^4+2002 d^4 e^5 x^5+3003 d^3 e^6 x^6+3432 d^2 e^7 x^7+3003 d e^8 x^8+2002 e^9 x^9\right )\right )+6 a b^9 e \left (2 A e \left (d^9+14 d^8 e x+91 d^7 e^2 x^2+364 d^6 e^3 x^3+1001 d^5 e^4 x^4+2002 d^4 e^5 x^5+3003 d^3 e^6 x^6+3432 d^2 e^7 x^7+3003 d e^8 x^8+2002 e^9 x^9\right )+5 B \left (d^{10}+14 d^9 e x+91 d^8 e^2 x^2+364 d^7 e^3 x^3+1001 d^6 e^4 x^4+2002 d^5 e^5 x^5+3003 d^4 e^6 x^6+3432 d^3 e^7 x^7+3003 d^2 e^8 x^8+2002 d e^9 x^9+1001 e^{10} x^{10}\right )\right )+b^{10} \left (3 A e \left (d^{10}+14 d^9 e x+91 d^8 e^2 x^2+364 d^7 e^3 x^3+1001 d^6 e^4 x^4+2002 d^5 e^5 x^5+3003 d^4 e^6 x^6+3432 d^3 e^7 x^7+3003 d^2 e^8 x^8+2002 d e^9 x^9+1001 e^{10} x^{10}\right )+11 B \left (d^{11}+14 d^{10} e x+91 d^9 e^2 x^2+364 d^8 e^3 x^3+1001 d^7 e^4 x^4+2002 d^6 e^5 x^5+3003 d^5 e^6 x^6+3432 d^4 e^7 x^7+3003 d^3 e^8 x^8+2002 d^2 e^9 x^9+1001 d e^{10} x^{10}+364 e^{11} x^{11}\right )\right )}{12012 e^{12} (d+e x)^{14}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^10*(A + B*x))/(d + e*x)^15,x]

[Out]

-1/12012*(66*a^10*e^10*(13*A*e + B*(d + 14*e*x)) + 110*a^9*b*e^9*(6*A*e*(d + 14*e*x) + B*(d^2 + 14*d*e*x + 91*
e^2*x^2)) + 45*a^8*b^2*e^8*(11*A*e*(d^2 + 14*d*e*x + 91*e^2*x^2) + 3*B*(d^3 + 14*d^2*e*x + 91*d*e^2*x^2 + 364*
e^3*x^3)) + 72*a^7*b^3*e^7*(5*A*e*(d^3 + 14*d^2*e*x + 91*d*e^2*x^2 + 364*e^3*x^3) + 2*B*(d^4 + 14*d^3*e*x + 91
*d^2*e^2*x^2 + 364*d*e^3*x^3 + 1001*e^4*x^4)) + 28*a^6*b^4*e^6*(9*A*e*(d^4 + 14*d^3*e*x + 91*d^2*e^2*x^2 + 364
*d*e^3*x^3 + 1001*e^4*x^4) + 5*B*(d^5 + 14*d^4*e*x + 91*d^3*e^2*x^2 + 364*d^2*e^3*x^3 + 1001*d*e^4*x^4 + 2002*
e^5*x^5)) + 42*a^5*b^5*e^5*(4*A*e*(d^5 + 14*d^4*e*x + 91*d^3*e^2*x^2 + 364*d^2*e^3*x^3 + 1001*d*e^4*x^4 + 2002
*e^5*x^5) + 3*B*(d^6 + 14*d^5*e*x + 91*d^4*e^2*x^2 + 364*d^3*e^3*x^3 + 1001*d^2*e^4*x^4 + 2002*d*e^5*x^5 + 300
3*e^6*x^6)) + 105*a^4*b^6*e^4*(A*e*(d^6 + 14*d^5*e*x + 91*d^4*e^2*x^2 + 364*d^3*e^3*x^3 + 1001*d^2*e^4*x^4 + 2
002*d*e^5*x^5 + 3003*e^6*x^6) + B*(d^7 + 14*d^6*e*x + 91*d^5*e^2*x^2 + 364*d^4*e^3*x^3 + 1001*d^3*e^4*x^4 + 20
02*d^2*e^5*x^5 + 3003*d*e^6*x^6 + 3432*e^7*x^7)) + 20*a^3*b^7*e^3*(3*A*e*(d^7 + 14*d^6*e*x + 91*d^5*e^2*x^2 +
364*d^4*e^3*x^3 + 1001*d^3*e^4*x^4 + 2002*d^2*e^5*x^5 + 3003*d*e^6*x^6 + 3432*e^7*x^7) + 4*B*(d^8 + 14*d^7*e*x
+ 91*d^6*e^2*x^2 + 364*d^5*e^3*x^3 + 1001*d^4*e^4*x^4 + 2002*d^3*e^5*x^5 + 3003*d^2*e^6*x^6 + 3432*d*e^7*x^7
+ 3003*e^8*x^8)) + 6*a^2*b^8*e^2*(5*A*e*(d^8 + 14*d^7*e*x + 91*d^6*e^2*x^2 + 364*d^5*e^3*x^3 + 1001*d^4*e^4*x^
4 + 2002*d^3*e^5*x^5 + 3003*d^2*e^6*x^6 + 3432*d*e^7*x^7 + 3003*e^8*x^8) + 9*B*(d^9 + 14*d^8*e*x + 91*d^7*e^2*
x^2 + 364*d^6*e^3*x^3 + 1001*d^5*e^4*x^4 + 2002*d^4*e^5*x^5 + 3003*d^3*e^6*x^6 + 3432*d^2*e^7*x^7 + 3003*d*e^8
*x^8 + 2002*e^9*x^9)) + 6*a*b^9*e*(2*A*e*(d^9 + 14*d^8*e*x + 91*d^7*e^2*x^2 + 364*d^6*e^3*x^3 + 1001*d^5*e^4*x
^4 + 2002*d^4*e^5*x^5 + 3003*d^3*e^6*x^6 + 3432*d^2*e^7*x^7 + 3003*d*e^8*x^8 + 2002*e^9*x^9) + 5*B*(d^10 + 14*
d^9*e*x + 91*d^8*e^2*x^2 + 364*d^7*e^3*x^3 + 1001*d^6*e^4*x^4 + 2002*d^5*e^5*x^5 + 3003*d^4*e^6*x^6 + 3432*d^3
*e^7*x^7 + 3003*d^2*e^8*x^8 + 2002*d*e^9*x^9 + 1001*e^10*x^10)) + b^10*(3*A*e*(d^10 + 14*d^9*e*x + 91*d^8*e^2*
x^2 + 364*d^7*e^3*x^3 + 1001*d^6*e^4*x^4 + 2002*d^5*e^5*x^5 + 3003*d^4*e^6*x^6 + 3432*d^3*e^7*x^7 + 3003*d^2*e
^8*x^8 + 2002*d*e^9*x^9 + 1001*e^10*x^10) + 11*B*(d^11 + 14*d^10*e*x + 91*d^9*e^2*x^2 + 364*d^8*e^3*x^3 + 1001
*d^7*e^4*x^4 + 2002*d^6*e^5*x^5 + 3003*d^5*e^6*x^6 + 3432*d^4*e^7*x^7 + 3003*d^3*e^8*x^8 + 2002*d^2*e^9*x^9 +
1001*d*e^10*x^10 + 364*e^11*x^11)))/(e^12*(d + e*x)^14)

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1941\) vs. \(2(177)=354\).
time = 0.12, size = 1942, normalized size = 10.50

method	result	size
risch	\(\text {Expression too large to display}\)	\(1901\)
default	\(\text {Expression too large to display}\)	\(1942\)
norman	\(\text {Expression too large to display}\)	\(2014\)
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)^15,x,method=_RETURNVERBOSE)

[Out]

-1/4*b^9/e^12*(A*b*e+10*B*a*e-11*B*b*d)/(e*x+d)^4-1/13/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)^13-21/4*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)^8-14/3*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)^9-30/7*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)^7-5/2*b^7/e^12*(3*A*a^2*b*e^3-6*A*a*b^2*d*e^2+3*A*b^3*d^2*e+8*B*a^3*e^3
-27*B*a^2*b*d*e^2+30*B*a*b^2*d^2*e-11*B*b^3*d^3)/(e*x+d)^6-5/12*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)^12-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)^5-1/3*b^10*B/e^12/
(e*x+d)^3-3*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)^10-1/14*(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)^14-15/1
1*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)^11

________________________________________________________________________________________

Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 1976 vs. \(2 (188) = 376\).
time = 0.54, size = 1976, normalized size = 10.68 \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)^15,x, algorithm="maxima")

[Out]

-1/12012*(4004*B*b^10*x^11*e^11 + 11*B*b^10*d^11 + 858*A*a^10*e^11 + 3*(10*B*a*b^9*e + A*b^10*e)*d^10 + 1001*(
11*B*b^10*d*e^10 + 30*B*a*b^9*e^11 + 3*A*b^10*e^11)*x^10 + 6*(9*B*a^2*b^8*e^2 + 2*A*a*b^9*e^2)*d^9 + 2002*(11*
B*b^10*d^2*e^9 + 54*B*a^2*b^8*e^11 + 12*A*a*b^9*e^11 + 3*(10*B*a*b^9*e^10 + A*b^10*e^10)*d)*x^9 + 10*(8*B*a^3*
b^7*e^3 + 3*A*a^2*b^8*e^3)*d^8 + 3003*(11*B*b^10*d^3*e^8 + 80*B*a^3*b^7*e^11 + 30*A*a^2*b^8*e^11 + 3*(10*B*a*b
^9*e^9 + A*b^10*e^9)*d^2 + 6*(9*B*a^2*b^8*e^10 + 2*A*a*b^9*e^10)*d)*x^8 + 15*(7*B*a^4*b^6*e^4 + 4*A*a^3*b^7*e^
4)*d^7 + 3432*(11*B*b^10*d^4*e^7 + 105*B*a^4*b^6*e^11 + 60*A*a^3*b^7*e^11 + 3*(10*B*a*b^9*e^8 + A*b^10*e^8)*d^
3 + 6*(9*B*a^2*b^8*e^9 + 2*A*a*b^9*e^9)*d^2 + 10*(8*B*a^3*b^7*e^10 + 3*A*a^2*b^8*e^10)*d)*x^7 + 21*(6*B*a^5*b^
5*e^5 + 5*A*a^4*b^6*e^5)*d^6 + 3003*(11*B*b^10*d^5*e^6 + 126*B*a^5*b^5*e^11 + 105*A*a^4*b^6*e^11 + 3*(10*B*a*b
^9*e^7 + A*b^10*e^7)*d^4 + 6*(9*B*a^2*b^8*e^8 + 2*A*a*b^9*e^8)*d^3 + 10*(8*B*a^3*b^7*e^9 + 3*A*a^2*b^8*e^9)*d^
2 + 15*(7*B*a^4*b^6*e^10 + 4*A*a^3*b^7*e^10)*d)*x^6 + 28*(5*B*a^6*b^4*e^6 + 6*A*a^5*b^5*e^6)*d^5 + 2002*(11*B*
b^10*d^6*e^5 + 140*B*a^6*b^4*e^11 + 168*A*a^5*b^5*e^11 + 3*(10*B*a*b^9*e^6 + A*b^10*e^6)*d^5 + 6*(9*B*a^2*b^8*
e^7 + 2*A*a*b^9*e^7)*d^4 + 10*(8*B*a^3*b^7*e^8 + 3*A*a^2*b^8*e^8)*d^3 + 15*(7*B*a^4*b^6*e^9 + 4*A*a^3*b^7*e^9)
*d^2 + 21*(6*B*a^5*b^5*e^10 + 5*A*a^4*b^6*e^10)*d)*x^5 + 36*(4*B*a^7*b^3*e^7 + 7*A*a^6*b^4*e^7)*d^4 + 1001*(11
*B*b^10*d^7*e^4 + 144*B*a^7*b^3*e^11 + 252*A*a^6*b^4*e^11 + 3*(10*B*a*b^9*e^5 + A*b^10*e^5)*d^6 + 6*(9*B*a^2*b
^8*e^6 + 2*A*a*b^9*e^6)*d^5 + 10*(8*B*a^3*b^7*e^7 + 3*A*a^2*b^8*e^7)*d^4 + 15*(7*B*a^4*b^6*e^8 + 4*A*a^3*b^7*e
^8)*d^3 + 21*(6*B*a^5*b^5*e^9 + 5*A*a^4*b^6*e^9)*d^2 + 28*(5*B*a^6*b^4*e^10 + 6*A*a^5*b^5*e^10)*d)*x^4 + 45*(3
*B*a^8*b^2*e^8 + 8*A*a^7*b^3*e^8)*d^3 + 364*(11*B*b^10*d^8*e^3 + 135*B*a^8*b^2*e^11 + 360*A*a^7*b^3*e^11 + 3*(
10*B*a*b^9*e^4 + A*b^10*e^4)*d^7 + 6*(9*B*a^2*b^8*e^5 + 2*A*a*b^9*e^5)*d^6 + 10*(8*B*a^3*b^7*e^6 + 3*A*a^2*b^8
*e^6)*d^5 + 15*(7*B*a^4*b^6*e^7 + 4*A*a^3*b^7*e^7)*d^4 + 21*(6*B*a^5*b^5*e^8 + 5*A*a^4*b^6*e^8)*d^3 + 28*(5*B*
a^6*b^4*e^9 + 6*A*a^5*b^5*e^9)*d^2 + 36*(4*B*a^7*b^3*e^10 + 7*A*a^6*b^4*e^10)*d)*x^3 + 55*(2*B*a^9*b*e^9 + 9*A
*a^8*b^2*e^9)*d^2 + 91*(11*B*b^10*d^9*e^2 + 110*B*a^9*b*e^11 + 495*A*a^8*b^2*e^11 + 3*(10*B*a*b^9*e^3 + A*b^10
*e^3)*d^8 + 6*(9*B*a^2*b^8*e^4 + 2*A*a*b^9*e^4)*d^7 + 10*(8*B*a^3*b^7*e^5 + 3*A*a^2*b^8*e^5)*d^6 + 15*(7*B*a^4
*b^6*e^6 + 4*A*a^3*b^7*e^6)*d^5 + 21*(6*B*a^5*b^5*e^7 + 5*A*a^4*b^6*e^7)*d^4 + 28*(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 + 45*(3*B*a^8*b^2*e^10 + 8*A*a^7*b^3*e^10)*d)*x^2 + 6
6*(B*a^10*e^10 + 10*A*a^9*b*e^10)*d + 14*(11*B*b^10*d^10*e + 66*B*a^10*e^11 + 660*A*a^9*b*e^11 + 3*(10*B*a*b^9
*e^2 + A*b^10*e^2)*d^9 + 6*(9*B*a^2*b^8*e^3 + 2*A*a*b^9*e^3)*d^8 + 10*(8*B*a^3*b^7*e^4 + 3*A*a^2*b^8*e^4)*d^7
+ 15*(7*B*a^4*b^6*e^5 + 4*A*a^3*b^7*e^5)*d^6 + 21*(6*B*a^5*b^5*e^6 + 5*A*a^4*b^6*e^6)*d^5 + 28*(5*B*a^6*b^4*e^
7 + 6*A*a^5*b^5*e^7)*d^4 + 36*(4*B*a^7*b^3*e^8 + 7*A*a^6*b^4*e^8)*d^3 + 45*(3*B*a^8*b^2*e^9 + 8*A*a^7*b^3*e^9)
*d^2 + 55*(2*B*a^9*b*e^10 + 9*A*a^8*b^2*e^10)*d)*x)/(x^14*e^26 + 14*d*x^13*e^25 + 91*d^2*x^12*e^24 + 364*d^3*x
^11*e^23 + 1001*d^4*x^10*e^22 + 2002*d^5*x^9*e^21 + 3003*d^6*x^8*e^20 + 3432*d^7*x^7*e^19 + 3003*d^8*x^6*e^18
+ 2002*d^9*x^5*e^17 + 1001*d^10*x^4*e^16 + 364*d^11*x^3*e^15 + 91*d^12*x^2*e^14 + 14*d^13*x*e^13 + d^14*e^12)

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1881 vs. \(2 (188) = 376\).
time = 1.32, size = 1881, normalized size = 10.17 \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)^15,x, algorithm="fricas")

[Out]

-1/12012*(11*B*b^10*d^11 + (4004*B*b^10*x^11 + 858*A*a^10 + 3003*(10*B*a*b^9 + A*b^10)*x^10 + 12012*(9*B*a^2*b
^8 + 2*A*a*b^9)*x^9 + 30030*(8*B*a^3*b^7 + 3*A*a^2*b^8)*x^8 + 51480*(7*B*a^4*b^6 + 4*A*a^3*b^7)*x^7 + 63063*(6
*B*a^5*b^5 + 5*A*a^4*b^6)*x^6 + 56056*(5*B*a^6*b^4 + 6*A*a^5*b^5)*x^5 + 36036*(4*B*a^7*b^3 + 7*A*a^6*b^4)*x^4
+ 16380*(3*B*a^8*b^2 + 8*A*a^7*b^3)*x^3 + 5005*(2*B*a^9*b + 9*A*a^8*b^2)*x^2 + 924*(B*a^10 + 10*A*a^9*b)*x)*e^
11 + (11011*B*b^10*d*x^10 + 6006*(10*B*a*b^9 + A*b^10)*d*x^9 + 18018*(9*B*a^2*b^8 + 2*A*a*b^9)*d*x^8 + 34320*(
8*B*a^3*b^7 + 3*A*a^2*b^8)*d*x^7 + 45045*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d*x^6 + 42042*(6*B*a^5*b^5 + 5*A*a^4*b^6)
*d*x^5 + 28028*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d*x^4 + 13104*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d*x^3 + 4095*(3*B*a^8*b^2
+ 8*A*a^7*b^3)*d*x^2 + 770*(2*B*a^9*b + 9*A*a^8*b^2)*d*x + 66*(B*a^10 + 10*A*a^9*b)*d)*e^10 + (22022*B*b^10*d
^2*x^9 + 9009*(10*B*a*b^9 + A*b^10)*d^2*x^8 + 20592*(9*B*a^2*b^8 + 2*A*a*b^9)*d^2*x^7 + 30030*(8*B*a^3*b^7 + 3
*A*a^2*b^8)*d^2*x^6 + 30030*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^2*x^5 + 21021*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^2*x^4 +
10192*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^2*x^3 + 3276*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^2*x^2 + 630*(3*B*a^8*b^2 + 8*A*
a^7*b^3)*d^2*x + 55*(2*B*a^9*b + 9*A*a^8*b^2)*d^2)*e^9 + (33033*B*b^10*d^3*x^8 + 10296*(10*B*a*b^9 + A*b^10)*d
^3*x^7 + 18018*(9*B*a^2*b^8 + 2*A*a*b^9)*d^3*x^6 + 20020*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^3*x^5 + 15015*(7*B*a^4*
b^6 + 4*A*a^3*b^7)*d^3*x^4 + 7644*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^3*x^3 + 2548*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^3*x
^2 + 504*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^3*x + 45*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^3)*e^8 + (37752*B*b^10*d^4*x^7 +
9009*(10*B*a*b^9 + A*b^10)*d^4*x^6 + 12012*(9*B*a^2*b^8 + 2*A*a*b^9)*d^4*x^5 + 10010*(8*B*a^3*b^7 + 3*A*a^2*b
^8)*d^4*x^4 + 5460*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^4*x^3 + 1911*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^4*x^2 + 392*(5*B*a
^6*b^4 + 6*A*a^5*b^5)*d^4*x + 36*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^4)*e^7 + 7*(4719*B*b^10*d^5*x^6 + 858*(10*B*a*b
^9 + A*b^10)*d^5*x^5 + 858*(9*B*a^2*b^8 + 2*A*a*b^9)*d^5*x^4 + 520*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^5*x^3 + 195*(
7*B*a^4*b^6 + 4*A*a^3*b^7)*d^5*x^2 + 42*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^5*x + 4*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^5)
*e^6 + 7*(3146*B*b^10*d^6*x^5 + 429*(10*B*a*b^9 + A*b^10)*d^6*x^4 + 312*(9*B*a^2*b^8 + 2*A*a*b^9)*d^6*x^3 + 13
0*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^6*x^2 + 30*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^6*x + 3*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d
^6)*e^5 + (11011*B*b^10*d^7*x^4 + 1092*(10*B*a*b^9 + A*b^10)*d^7*x^3 + 546*(9*B*a^2*b^8 + 2*A*a*b^9)*d^7*x^2 +
140*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^7*x + 15*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^7)*e^4 + (4004*B*b^10*d^8*x^3 + 273*
(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 + 10*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^8)*e^3 +
(1001*B*b^10*d^9*x^2 + 42*(10*B*a*b^9 + A*b^10)*d^9*x + 6*(9*B*a^2*b^8 + 2*A*a*b^9)*d^9)*e^2 + (154*B*b^10*d^
10*x + 3*(10*B*a*b^9 + A*b^10)*d^10)*e)/(x^14*e^26 + 14*d*x^13*e^25 + 91*d^2*x^12*e^24 + 364*d^3*x^11*e^23 + 1
001*d^4*x^10*e^22 + 2002*d^5*x^9*e^21 + 3003*d^6*x^8*e^20 + 3432*d^7*x^7*e^19 + 3003*d^8*x^6*e^18 + 2002*d^9*x
^5*e^17 + 1001*d^10*x^4*e^16 + 364*d^11*x^3*e^15 + 91*d^12*x^2*e^14 + 14*d^13*x*e^13 + d^14*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)**15,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 2096 vs. \(2 (188) = 376\).
time = 0.94, size = 2096, normalized size = 11.33 \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)^15,x, algorithm="giac")

[Out]

-1/12012*(4004*B*b^10*x^11*e^11 + 11011*B*b^10*d*x^10*e^10 + 22022*B*b^10*d^2*x^9*e^9 + 33033*B*b^10*d^3*x^8*e
^8 + 37752*B*b^10*d^4*x^7*e^7 + 33033*B*b^10*d^5*x^6*e^6 + 22022*B*b^10*d^6*x^5*e^5 + 11011*B*b^10*d^7*x^4*e^4
+ 4004*B*b^10*d^8*x^3*e^3 + 1001*B*b^10*d^9*x^2*e^2 + 154*B*b^10*d^10*x*e + 11*B*b^10*d^11 + 30030*B*a*b^9*x^
10*e^11 + 3003*A*b^10*x^10*e^11 + 60060*B*a*b^9*d*x^9*e^10 + 6006*A*b^10*d*x^9*e^10 + 90090*B*a*b^9*d^2*x^8*e^
9 + 9009*A*b^10*d^2*x^8*e^9 + 102960*B*a*b^9*d^3*x^7*e^8 + 10296*A*b^10*d^3*x^7*e^8 + 90090*B*a*b^9*d^4*x^6*e^
7 + 9009*A*b^10*d^4*x^6*e^7 + 60060*B*a*b^9*d^5*x^5*e^6 + 6006*A*b^10*d^5*x^5*e^6 + 30030*B*a*b^9*d^6*x^4*e^5
+ 3003*A*b^10*d^6*x^4*e^5 + 10920*B*a*b^9*d^7*x^3*e^4 + 1092*A*b^10*d^7*x^3*e^4 + 2730*B*a*b^9*d^8*x^2*e^3 + 2
73*A*b^10*d^8*x^2*e^3 + 420*B*a*b^9*d^9*x*e^2 + 42*A*b^10*d^9*x*e^2 + 30*B*a*b^9*d^10*e + 3*A*b^10*d^10*e + 10
8108*B*a^2*b^8*x^9*e^11 + 24024*A*a*b^9*x^9*e^11 + 162162*B*a^2*b^8*d*x^8*e^10 + 36036*A*a*b^9*d*x^8*e^10 + 18
5328*B*a^2*b^8*d^2*x^7*e^9 + 41184*A*a*b^9*d^2*x^7*e^9 + 162162*B*a^2*b^8*d^3*x^6*e^8 + 36036*A*a*b^9*d^3*x^6*
e^8 + 108108*B*a^2*b^8*d^4*x^5*e^7 + 24024*A*a*b^9*d^4*x^5*e^7 + 54054*B*a^2*b^8*d^5*x^4*e^6 + 12012*A*a*b^9*d
^5*x^4*e^6 + 19656*B*a^2*b^8*d^6*x^3*e^5 + 4368*A*a*b^9*d^6*x^3*e^5 + 4914*B*a^2*b^8*d^7*x^2*e^4 + 1092*A*a*b^
9*d^7*x^2*e^4 + 756*B*a^2*b^8*d^8*x*e^3 + 168*A*a*b^9*d^8*x*e^3 + 54*B*a^2*b^8*d^9*e^2 + 12*A*a*b^9*d^9*e^2 +
240240*B*a^3*b^7*x^8*e^11 + 90090*A*a^2*b^8*x^8*e^11 + 274560*B*a^3*b^7*d*x^7*e^10 + 102960*A*a^2*b^8*d*x^7*e^
10 + 240240*B*a^3*b^7*d^2*x^6*e^9 + 90090*A*a^2*b^8*d^2*x^6*e^9 + 160160*B*a^3*b^7*d^3*x^5*e^8 + 60060*A*a^2*b
^8*d^3*x^5*e^8 + 80080*B*a^3*b^7*d^4*x^4*e^7 + 30030*A*a^2*b^8*d^4*x^4*e^7 + 29120*B*a^3*b^7*d^5*x^3*e^6 + 109
20*A*a^2*b^8*d^5*x^3*e^6 + 7280*B*a^3*b^7*d^6*x^2*e^5 + 2730*A*a^2*b^8*d^6*x^2*e^5 + 1120*B*a^3*b^7*d^7*x*e^4
+ 420*A*a^2*b^8*d^7*x*e^4 + 80*B*a^3*b^7*d^8*e^3 + 30*A*a^2*b^8*d^8*e^3 + 360360*B*a^4*b^6*x^7*e^11 + 205920*A
*a^3*b^7*x^7*e^11 + 315315*B*a^4*b^6*d*x^6*e^10 + 180180*A*a^3*b^7*d*x^6*e^10 + 210210*B*a^4*b^6*d^2*x^5*e^9 +
120120*A*a^3*b^7*d^2*x^5*e^9 + 105105*B*a^4*b^6*d^3*x^4*e^8 + 60060*A*a^3*b^7*d^3*x^4*e^8 + 38220*B*a^4*b^6*d
^4*x^3*e^7 + 21840*A*a^3*b^7*d^4*x^3*e^7 + 9555*B*a^4*b^6*d^5*x^2*e^6 + 5460*A*a^3*b^7*d^5*x^2*e^6 + 1470*B*a^
4*b^6*d^6*x*e^5 + 840*A*a^3*b^7*d^6*x*e^5 + 105*B*a^4*b^6*d^7*e^4 + 60*A*a^3*b^7*d^7*e^4 + 378378*B*a^5*b^5*x^
6*e^11 + 315315*A*a^4*b^6*x^6*e^11 + 252252*B*a^5*b^5*d*x^5*e^10 + 210210*A*a^4*b^6*d*x^5*e^10 + 126126*B*a^5*
b^5*d^2*x^4*e^9 + 105105*A*a^4*b^6*d^2*x^4*e^9 + 45864*B*a^5*b^5*d^3*x^3*e^8 + 38220*A*a^4*b^6*d^3*x^3*e^8 + 1
1466*B*a^5*b^5*d^4*x^2*e^7 + 9555*A*a^4*b^6*d^4*x^2*e^7 + 1764*B*a^5*b^5*d^5*x*e^6 + 1470*A*a^4*b^6*d^5*x*e^6
+ 126*B*a^5*b^5*d^6*e^5 + 105*A*a^4*b^6*d^6*e^5 + 280280*B*a^6*b^4*x^5*e^11 + 336336*A*a^5*b^5*x^5*e^11 + 1401
40*B*a^6*b^4*d*x^4*e^10 + 168168*A*a^5*b^5*d*x^4*e^10 + 50960*B*a^6*b^4*d^2*x^3*e^9 + 61152*A*a^5*b^5*d^2*x^3*
e^9 + 12740*B*a^6*b^4*d^3*x^2*e^8 + 15288*A*a^5*b^5*d^3*x^2*e^8 + 1960*B*a^6*b^4*d^4*x*e^7 + 2352*A*a^5*b^5*d^
4*x*e^7 + 140*B*a^6*b^4*d^5*e^6 + 168*A*a^5*b^5*d^5*e^6 + 144144*B*a^7*b^3*x^4*e^11 + 252252*A*a^6*b^4*x^4*e^1
1 + 52416*B*a^7*b^3*d*x^3*e^10 + 91728*A*a^6*b^4*d*x^3*e^10 + 13104*B*a^7*b^3*d^2*x^2*e^9 + 22932*A*a^6*b^4*d^
2*x^2*e^9 + 2016*B*a^7*b^3*d^3*x*e^8 + 3528*A*a^6*b^4*d^3*x*e^8 + 144*B*a^7*b^3*d^4*e^7 + 252*A*a^6*b^4*d^4*e^
7 + 49140*B*a^8*b^2*x^3*e^11 + 131040*A*a^7*b^3*x^3*e^11 + 12285*B*a^8*b^2*d*x^2*e^10 + 32760*A*a^7*b^3*d*x^2*
e^10 + 1890*B*a^8*b^2*d^2*x*e^9 + 5040*A*a^7*b^3*d^2*x*e^9 + 135*B*a^8*b^2*d^3*e^8 + 360*A*a^7*b^3*d^3*e^8 + 1
0010*B*a^9*b*x^2*e^11 + 45045*A*a^8*b^2*x^2*e^11 + 1540*B*a^9*b*d*x*e^10 + 6930*A*a^8*b^2*d*x*e^10 + 110*B*a^9
*b*d^2*e^9 + 495*A*a^8*b^2*d^2*e^9 + 924*B*a^10*x*e^11 + 9240*A*a^9*b*x*e^11 + 66*B*a^10*d*e^10 + 660*A*a^9*b*
d*e^10 + 858*A*a^10*e^11)*e^(-12)/(x*e + d)^14

________________________________________________________________________________________

Mupad [B]
time = 2.46, size = 2044, normalized size = 11.05 \begin {gather*} -\frac {\frac {66\,B\,a^{10}\,d\,e^{10}+858\,A\,a^{10}\,e^{11}+110\,B\,a^9\,b\,d^2\,e^9+660\,A\,a^9\,b\,d\,e^{10}+135\,B\,a^8\,b^2\,d^3\,e^8+495\,A\,a^8\,b^2\,d^2\,e^9+144\,B\,a^7\,b^3\,d^4\,e^7+360\,A\,a^7\,b^3\,d^3\,e^8+140\,B\,a^6\,b^4\,d^5\,e^6+252\,A\,a^6\,b^4\,d^4\,e^7+126\,B\,a^5\,b^5\,d^6\,e^5+168\,A\,a^5\,b^5\,d^5\,e^6+105\,B\,a^4\,b^6\,d^7\,e^4+105\,A\,a^4\,b^6\,d^6\,e^5+80\,B\,a^3\,b^7\,d^8\,e^3+60\,A\,a^3\,b^7\,d^7\,e^4+54\,B\,a^2\,b^8\,d^9\,e^2+30\,A\,a^2\,b^8\,d^8\,e^3+30\,B\,a\,b^9\,d^{10}\,e+12\,A\,a\,b^9\,d^9\,e^2+11\,B\,b^{10}\,d^{11}+3\,A\,b^{10}\,d^{10}\,e}{12012\,e^{12}}+\frac {x\,\left (66\,B\,a^{10}\,e^{10}+110\,B\,a^9\,b\,d\,e^9+660\,A\,a^9\,b\,e^{10}+135\,B\,a^8\,b^2\,d^2\,e^8+495\,A\,a^8\,b^2\,d\,e^9+144\,B\,a^7\,b^3\,d^3\,e^7+360\,A\,a^7\,b^3\,d^2\,e^8+140\,B\,a^6\,b^4\,d^4\,e^6+252\,A\,a^6\,b^4\,d^3\,e^7+126\,B\,a^5\,b^5\,d^5\,e^5+168\,A\,a^5\,b^5\,d^4\,e^6+105\,B\,a^4\,b^6\,d^6\,e^4+105\,A\,a^4\,b^6\,d^5\,e^5+80\,B\,a^3\,b^7\,d^7\,e^3+60\,A\,a^3\,b^7\,d^6\,e^4+54\,B\,a^2\,b^8\,d^8\,e^2+30\,A\,a^2\,b^8\,d^7\,e^3+30\,B\,a\,b^9\,d^9\,e+12\,A\,a\,b^9\,d^8\,e^2+11\,B\,b^{10}\,d^{10}+3\,A\,b^{10}\,d^9\,e\right )}{858\,e^{11}}+\frac {b^7\,x^8\,\left (80\,B\,a^3\,e^3+54\,B\,a^2\,b\,d\,e^2+30\,A\,a^2\,b\,e^3+30\,B\,a\,b^2\,d^2\,e+12\,A\,a\,b^2\,d\,e^2+11\,B\,b^3\,d^3+3\,A\,b^3\,d^2\,e\right )}{4\,e^4}+\frac {b^4\,x^5\,\left (140\,B\,a^6\,e^6+126\,B\,a^5\,b\,d\,e^5+168\,A\,a^5\,b\,e^6+105\,B\,a^4\,b^2\,d^2\,e^4+105\,A\,a^4\,b^2\,d\,e^5+80\,B\,a^3\,b^3\,d^3\,e^3+60\,A\,a^3\,b^3\,d^2\,e^4+54\,B\,a^2\,b^4\,d^4\,e^2+30\,A\,a^2\,b^4\,d^3\,e^3+30\,B\,a\,b^5\,d^5\,e+12\,A\,a\,b^5\,d^4\,e^2+11\,B\,b^6\,d^6+3\,A\,b^6\,d^5\,e\right )}{6\,e^7}+\frac {b^9\,x^{10}\,\left (3\,A\,b\,e+30\,B\,a\,e+11\,B\,b\,d\right )}{12\,e^2}+\frac {2\,b^6\,x^7\,\left (105\,B\,a^4\,e^4+80\,B\,a^3\,b\,d\,e^3+60\,A\,a^3\,b\,e^4+54\,B\,a^2\,b^2\,d^2\,e^2+30\,A\,a^2\,b^2\,d\,e^3+30\,B\,a\,b^3\,d^3\,e+12\,A\,a\,b^3\,d^2\,e^2+11\,B\,b^4\,d^4+3\,A\,b^4\,d^3\,e\right )}{7\,e^5}+\frac {b^3\,x^4\,\left (144\,B\,a^7\,e^7+140\,B\,a^6\,b\,d\,e^6+252\,A\,a^6\,b\,e^7+126\,B\,a^5\,b^2\,d^2\,e^5+168\,A\,a^5\,b^2\,d\,e^6+105\,B\,a^4\,b^3\,d^3\,e^4+105\,A\,a^4\,b^3\,d^2\,e^5+80\,B\,a^3\,b^4\,d^4\,e^3+60\,A\,a^3\,b^4\,d^3\,e^4+54\,B\,a^2\,b^5\,d^5\,e^2+30\,A\,a^2\,b^5\,d^4\,e^3+30\,B\,a\,b^6\,d^6\,e+12\,A\,a\,b^6\,d^5\,e^2+11\,B\,b^7\,d^7+3\,A\,b^7\,d^6\,e\right )}{12\,e^8}+\frac {b\,x^2\,\left (110\,B\,a^9\,e^9+135\,B\,a^8\,b\,d\,e^8+495\,A\,a^8\,b\,e^9+144\,B\,a^7\,b^2\,d^2\,e^7+360\,A\,a^7\,b^2\,d\,e^8+140\,B\,a^6\,b^3\,d^3\,e^6+252\,A\,a^6\,b^3\,d^2\,e^7+126\,B\,a^5\,b^4\,d^4\,e^5+168\,A\,a^5\,b^4\,d^3\,e^6+105\,B\,a^4\,b^5\,d^5\,e^4+105\,A\,a^4\,b^5\,d^4\,e^5+80\,B\,a^3\,b^6\,d^6\,e^3+60\,A\,a^3\,b^6\,d^5\,e^4+54\,B\,a^2\,b^7\,d^7\,e^2+30\,A\,a^2\,b^7\,d^6\,e^3+30\,B\,a\,b^8\,d^8\,e+12\,A\,a\,b^8\,d^7\,e^2+11\,B\,b^9\,d^9+3\,A\,b^9\,d^8\,e\right )}{132\,e^{10}}+\frac {b^8\,x^9\,\left (54\,B\,a^2\,e^2+30\,B\,a\,b\,d\,e+12\,A\,a\,b\,e^2+11\,B\,b^2\,d^2+3\,A\,b^2\,d\,e\right )}{6\,e^3}+\frac {b^5\,x^6\,\left (126\,B\,a^5\,e^5+105\,B\,a^4\,b\,d\,e^4+105\,A\,a^4\,b\,e^5+80\,B\,a^3\,b^2\,d^2\,e^3+60\,A\,a^3\,b^2\,d\,e^4+54\,B\,a^2\,b^3\,d^3\,e^2+30\,A\,a^2\,b^3\,d^2\,e^3+30\,B\,a\,b^4\,d^4\,e+12\,A\,a\,b^4\,d^3\,e^2+11\,B\,b^5\,d^5+3\,A\,b^5\,d^4\,e\right )}{4\,e^6}+\frac {b^2\,x^3\,\left (135\,B\,a^8\,e^8+144\,B\,a^7\,b\,d\,e^7+360\,A\,a^7\,b\,e^8+140\,B\,a^6\,b^2\,d^2\,e^6+252\,A\,a^6\,b^2\,d\,e^7+126\,B\,a^5\,b^3\,d^3\,e^5+168\,A\,a^5\,b^3\,d^2\,e^6+105\,B\,a^4\,b^4\,d^4\,e^4+105\,A\,a^4\,b^4\,d^3\,e^5+80\,B\,a^3\,b^5\,d^5\,e^3+60\,A\,a^3\,b^5\,d^4\,e^4+54\,B\,a^2\,b^6\,d^6\,e^2+30\,A\,a^2\,b^6\,d^5\,e^3+30\,B\,a\,b^7\,d^7\,e+12\,A\,a\,b^7\,d^6\,e^2+11\,B\,b^8\,d^8+3\,A\,b^8\,d^7\,e\right )}{33\,e^9}+\frac {B\,b^{10}\,x^{11}}{3\,e}}{d^{14}+14\,d^{13}\,e\,x+91\,d^{12}\,e^2\,x^2+364\,d^{11}\,e^3\,x^3+1001\,d^{10}\,e^4\,x^4+2002\,d^9\,e^5\,x^5+3003\,d^8\,e^6\,x^6+3432\,d^7\,e^7\,x^7+3003\,d^6\,e^8\,x^8+2002\,d^5\,e^9\,x^9+1001\,d^4\,e^{10}\,x^{10}+364\,d^3\,e^{11}\,x^{11}+91\,d^2\,e^{12}\,x^{12}+14\,d\,e^{13}\,x^{13}+e^{14}\,x^{14}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(a + b*x)^10)/(d + e*x)^15,x)

[Out]

-((858*A*a^10*e^11 + 11*B*b^10*d^11 + 3*A*b^10*d^10*e + 66*B*a^10*d*e^10 + 12*A*a*b^9*d^9*e^2 + 110*B*a^9*b*d^
2*e^9 + 30*A*a^2*b^8*d^8*e^3 + 60*A*a^3*b^7*d^7*e^4 + 105*A*a^4*b^6*d^6*e^5 + 168*A*a^5*b^5*d^5*e^6 + 252*A*a^
6*b^4*d^4*e^7 + 360*A*a^7*b^3*d^3*e^8 + 495*A*a^8*b^2*d^2*e^9 + 54*B*a^2*b^8*d^9*e^2 + 80*B*a^3*b^7*d^8*e^3 +
105*B*a^4*b^6*d^7*e^4 + 126*B*a^5*b^5*d^6*e^5 + 140*B*a^6*b^4*d^5*e^6 + 144*B*a^7*b^3*d^4*e^7 + 135*B*a^8*b^2*
d^3*e^8 + 660*A*a^9*b*d*e^10 + 30*B*a*b^9*d^10*e)/(12012*e^12) + (x*(66*B*a^10*e^10 + 11*B*b^10*d^10 + 660*A*a
^9*b*e^10 + 3*A*b^10*d^9*e + 12*A*a*b^9*d^8*e^2 + 495*A*a^8*b^2*d*e^9 + 30*A*a^2*b^8*d^7*e^3 + 60*A*a^3*b^7*d^
6*e^4 + 105*A*a^4*b^6*d^5*e^5 + 168*A*a^5*b^5*d^4*e^6 + 252*A*a^6*b^4*d^3*e^7 + 360*A*a^7*b^3*d^2*e^8 + 54*B*a
^2*b^8*d^8*e^2 + 80*B*a^3*b^7*d^7*e^3 + 105*B*a^4*b^6*d^6*e^4 + 126*B*a^5*b^5*d^5*e^5 + 140*B*a^6*b^4*d^4*e^6
+ 144*B*a^7*b^3*d^3*e^7 + 135*B*a^8*b^2*d^2*e^8 + 30*B*a*b^9*d^9*e + 110*B*a^9*b*d*e^9))/(858*e^11) + (b^7*x^8
*(80*B*a^3*e^3 + 11*B*b^3*d^3 + 30*A*a^2*b*e^3 + 3*A*b^3*d^2*e + 12*A*a*b^2*d*e^2 + 30*B*a*b^2*d^2*e + 54*B*a^
2*b*d*e^2))/(4*e^4) + (b^4*x^5*(140*B*a^6*e^6 + 11*B*b^6*d^6 + 168*A*a^5*b*e^6 + 3*A*b^6*d^5*e + 12*A*a*b^5*d^
4*e^2 + 105*A*a^4*b^2*d*e^5 + 30*A*a^2*b^4*d^3*e^3 + 60*A*a^3*b^3*d^2*e^4 + 54*B*a^2*b^4*d^4*e^2 + 80*B*a^3*b^
3*d^3*e^3 + 105*B*a^4*b^2*d^2*e^4 + 30*B*a*b^5*d^5*e + 126*B*a^5*b*d*e^5))/(6*e^7) + (b^9*x^10*(3*A*b*e + 30*B
*a*e + 11*B*b*d))/(12*e^2) + (2*b^6*x^7*(105*B*a^4*e^4 + 11*B*b^4*d^4 + 60*A*a^3*b*e^4 + 3*A*b^4*d^3*e + 12*A*
a*b^3*d^2*e^2 + 30*A*a^2*b^2*d*e^3 + 54*B*a^2*b^2*d^2*e^2 + 30*B*a*b^3*d^3*e + 80*B*a^3*b*d*e^3))/(7*e^5) + (b
^3*x^4*(144*B*a^7*e^7 + 11*B*b^7*d^7 + 252*A*a^6*b*e^7 + 3*A*b^7*d^6*e + 12*A*a*b^6*d^5*e^2 + 168*A*a^5*b^2*d*
e^6 + 30*A*a^2*b^5*d^4*e^3 + 60*A*a^3*b^4*d^3*e^4 + 105*A*a^4*b^3*d^2*e^5 + 54*B*a^2*b^5*d^5*e^2 + 80*B*a^3*b^
4*d^4*e^3 + 105*B*a^4*b^3*d^3*e^4 + 126*B*a^5*b^2*d^2*e^5 + 30*B*a*b^6*d^6*e + 140*B*a^6*b*d*e^6))/(12*e^8) +
(b*x^2*(110*B*a^9*e^9 + 11*B*b^9*d^9 + 495*A*a^8*b*e^9 + 3*A*b^9*d^8*e + 12*A*a*b^8*d^7*e^2 + 360*A*a^7*b^2*d*
e^8 + 30*A*a^2*b^7*d^6*e^3 + 60*A*a^3*b^6*d^5*e^4 + 105*A*a^4*b^5*d^4*e^5 + 168*A*a^5*b^4*d^3*e^6 + 252*A*a^6*
b^3*d^2*e^7 + 54*B*a^2*b^7*d^7*e^2 + 80*B*a^3*b^6*d^6*e^3 + 105*B*a^4*b^5*d^5*e^4 + 126*B*a^5*b^4*d^4*e^5 + 14
0*B*a^6*b^3*d^3*e^6 + 144*B*a^7*b^2*d^2*e^7 + 30*B*a*b^8*d^8*e + 135*B*a^8*b*d*e^8))/(132*e^10) + (b^8*x^9*(54
*B*a^2*e^2 + 11*B*b^2*d^2 + 12*A*a*b*e^2 + 3*A*b^2*d*e + 30*B*a*b*d*e))/(6*e^3) + (b^5*x^6*(126*B*a^5*e^5 + 11
*B*b^5*d^5 + 105*A*a^4*b*e^5 + 3*A*b^5*d^4*e + 12*A*a*b^4*d^3*e^2 + 60*A*a^3*b^2*d*e^4 + 30*A*a^2*b^3*d^2*e^3
+ 54*B*a^2*b^3*d^3*e^2 + 80*B*a^3*b^2*d^2*e^3 + 30*B*a*b^4*d^4*e + 105*B*a^4*b*d*e^4))/(4*e^6) + (b^2*x^3*(135
*B*a^8*e^8 + 11*B*b^8*d^8 + 360*A*a^7*b*e^8 + 3*A*b^8*d^7*e + 12*A*a*b^7*d^6*e^2 + 252*A*a^6*b^2*d*e^7 + 30*A*
a^2*b^6*d^5*e^3 + 60*A*a^3*b^5*d^4*e^4 + 105*A*a^4*b^4*d^3*e^5 + 168*A*a^5*b^3*d^2*e^6 + 54*B*a^2*b^6*d^6*e^2
+ 80*B*a^3*b^5*d^5*e^3 + 105*B*a^4*b^4*d^4*e^4 + 126*B*a^5*b^3*d^3*e^5 + 140*B*a^6*b^2*d^2*e^6 + 30*B*a*b^7*d^
7*e + 144*B*a^7*b*d*e^7))/(33*e^9) + (B*b^10*x^11)/(3*e))/(d^14 + e^14*x^14 + 14*d*e^13*x^13 + 91*d^12*e^2*x^2
+ 364*d^11*e^3*x^3 + 1001*d^10*e^4*x^4 + 2002*d^9*e^5*x^5 + 3003*d^8*e^6*x^6 + 3432*d^7*e^7*x^7 + 3003*d^6*e^
8*x^8 + 2002*d^5*e^9*x^9 + 1001*d^4*e^10*x^10 + 364*d^3*e^11*x^11 + 91*d^2*e^12*x^12 + 14*d^13*e*x)

________________________________________________________________________________________