\(\int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{19}} \, dx\) [1107]
Optimal result
Integrand size = 20, antiderivative size = 385 \[
\int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{19}} \, dx=-\frac {(B d-A e) (a+b x)^{11}}{18 e (b d-a e) (d+e x)^{18}}+\frac {(11 b B d+7 A b e-18 a B e) (a+b x)^{11}}{306 e (b d-a e)^2 (d+e x)^{17}}+\frac {b (11 b B d+7 A b e-18 a B e) (a+b x)^{11}}{816 e (b d-a e)^3 (d+e x)^{16}}+\frac {b^2 (11 b B d+7 A b e-18 a B e) (a+b x)^{11}}{2448 e (b d-a e)^4 (d+e x)^{15}}+\frac {b^3 (11 b B d+7 A b e-18 a B e) (a+b x)^{11}}{8568 e (b d-a e)^5 (d+e x)^{14}}+\frac {b^4 (11 b B d+7 A b e-18 a B e) (a+b x)^{11}}{37128 e (b d-a e)^6 (d+e x)^{13}}+\frac {b^5 (11 b B d+7 A b e-18 a B e) (a+b x)^{11}}{222768 e (b d-a e)^7 (d+e x)^{12}}+\frac {b^6 (11 b B d+7 A b e-18 a B e) (a+b x)^{11}}{2450448 e (b d-a e)^8 (d+e x)^{11}}
\]
[Out]
-1/18*(-A*e+B*d)*(b*x+a)^11/e/(-a*e+b*d)/(e*x+d)^18+1/306*(7*A*b*e-18*B*a*e+11*B*b*d)*(b*x+a)^11/e/(-a*e+b*d)^
2/(e*x+d)^17+1/816*b*(7*A*b*e-18*B*a*e+11*B*b*d)*(b*x+a)^11/e/(-a*e+b*d)^3/(e*x+d)^16+1/2448*b^2*(7*A*b*e-18*B
*a*e+11*B*b*d)*(b*x+a)^11/e/(-a*e+b*d)^4/(e*x+d)^15+1/8568*b^3*(7*A*b*e-18*B*a*e+11*B*b*d)*(b*x+a)^11/e/(-a*e+
b*d)^5/(e*x+d)^14+1/37128*b^4*(7*A*b*e-18*B*a*e+11*B*b*d)*(b*x+a)^11/e/(-a*e+b*d)^6/(e*x+d)^13+1/222768*b^5*(7
*A*b*e-18*B*a*e+11*B*b*d)*(b*x+a)^11/e/(-a*e+b*d)^7/(e*x+d)^12+1/2450448*b^6*(7*A*b*e-18*B*a*e+11*B*b*d)*(b*x+
a)^11/e/(-a*e+b*d)^8/(e*x+d)^11
Rubi [A] (verified)
Time = 0.15 (sec) , antiderivative size = 385, normalized size of antiderivative = 1.00, number
of steps used = 8, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {79, 47, 37}
\[
\int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{19}} \, dx=\frac {b^6 (a+b x)^{11} (-18 a B e+7 A b e+11 b B d)}{2450448 e (d+e x)^{11} (b d-a e)^8}+\frac {b^5 (a+b x)^{11} (-18 a B e+7 A b e+11 b B d)}{222768 e (d+e x)^{12} (b d-a e)^7}+\frac {b^4 (a+b x)^{11} (-18 a B e+7 A b e+11 b B d)}{37128 e (d+e x)^{13} (b d-a e)^6}+\frac {b^3 (a+b x)^{11} (-18 a B e+7 A b e+11 b B d)}{8568 e (d+e x)^{14} (b d-a e)^5}+\frac {b^2 (a+b x)^{11} (-18 a B e+7 A b e+11 b B d)}{2448 e (d+e x)^{15} (b d-a e)^4}+\frac {b (a+b x)^{11} (-18 a B e+7 A b e+11 b B d)}{816 e (d+e x)^{16} (b d-a e)^3}+\frac {(a+b x)^{11} (-18 a B e+7 A b e+11 b B d)}{306 e (d+e x)^{17} (b d-a e)^2}-\frac {(a+b x)^{11} (B d-A e)}{18 e (d+e x)^{18} (b d-a e)}
\]
[In]
Int[((a + b*x)^10*(A + B*x))/(d + e*x)^19,x]
[Out]
-1/18*((B*d - A*e)*(a + b*x)^11)/(e*(b*d - a*e)*(d + e*x)^18) + ((11*b*B*d + 7*A*b*e - 18*a*B*e)*(a + b*x)^11)
/(306*e*(b*d - a*e)^2*(d + e*x)^17) + (b*(11*b*B*d + 7*A*b*e - 18*a*B*e)*(a + b*x)^11)/(816*e*(b*d - a*e)^3*(d
+ e*x)^16) + (b^2*(11*b*B*d + 7*A*b*e - 18*a*B*e)*(a + b*x)^11)/(2448*e*(b*d - a*e)^4*(d + e*x)^15) + (b^3*(1
1*b*B*d + 7*A*b*e - 18*a*B*e)*(a + b*x)^11)/(8568*e*(b*d - a*e)^5*(d + e*x)^14) + (b^4*(11*b*B*d + 7*A*b*e - 1
8*a*B*e)*(a + b*x)^11)/(37128*e*(b*d - a*e)^6*(d + e*x)^13) + (b^5*(11*b*B*d + 7*A*b*e - 18*a*B*e)*(a + b*x)^1
1)/(222768*e*(b*d - a*e)^7*(d + e*x)^12) + (b^6*(11*b*B*d + 7*A*b*e - 18*a*B*e)*(a + b*x)^11)/(2450448*e*(b*d
- a*e)^8*(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*}
\text {integral}& = -\frac {(B d-A e) (a+b x)^{11}}{18 e (b d-a e) (d+e x)^{18}}+\frac {(11 b B d+7 A b e-18 a B e) \int \frac {(a+b x)^{10}}{(d+e x)^{18}} \, dx}{18 e (b d-a e)} \\ & = -\frac {(B d-A e) (a+b x)^{11}}{18 e (b d-a e) (d+e x)^{18}}+\frac {(11 b B d+7 A b e-18 a B e) (a+b x)^{11}}{306 e (b d-a e)^2 (d+e x)^{17}}+\frac {(b (11 b B d+7 A b e-18 a B e)) \int \frac {(a+b x)^{10}}{(d+e x)^{17}} \, dx}{51 e (b d-a e)^2} \\ & = -\frac {(B d-A e) (a+b x)^{11}}{18 e (b d-a e) (d+e x)^{18}}+\frac {(11 b B d+7 A b e-18 a B e) (a+b x)^{11}}{306 e (b d-a e)^2 (d+e x)^{17}}+\frac {b (11 b B d+7 A b e-18 a B e) (a+b x)^{11}}{816 e (b d-a e)^3 (d+e x)^{16}}+\frac {\left (5 b^2 (11 b B d+7 A b e-18 a B e)\right ) \int \frac {(a+b x)^{10}}{(d+e x)^{16}} \, dx}{816 e (b d-a e)^3} \\ & = -\frac {(B d-A e) (a+b x)^{11}}{18 e (b d-a e) (d+e x)^{18}}+\frac {(11 b B d+7 A b e-18 a B e) (a+b x)^{11}}{306 e (b d-a e)^2 (d+e x)^{17}}+\frac {b (11 b B d+7 A b e-18 a B e) (a+b x)^{11}}{816 e (b d-a e)^3 (d+e x)^{16}}+\frac {b^2 (11 b B d+7 A b e-18 a B e) (a+b x)^{11}}{2448 e (b d-a e)^4 (d+e x)^{15}}+\frac {\left (b^3 (11 b B d+7 A b e-18 a B e)\right ) \int \frac {(a+b x)^{10}}{(d+e x)^{15}} \, dx}{612 e (b d-a e)^4} \\ & = -\frac {(B d-A e) (a+b x)^{11}}{18 e (b d-a e) (d+e x)^{18}}+\frac {(11 b B d+7 A b e-18 a B e) (a+b x)^{11}}{306 e (b d-a e)^2 (d+e x)^{17}}+\frac {b (11 b B d+7 A b e-18 a B e) (a+b x)^{11}}{816 e (b d-a e)^3 (d+e x)^{16}}+\frac {b^2 (11 b B d+7 A b e-18 a B e) (a+b x)^{11}}{2448 e (b d-a e)^4 (d+e x)^{15}}+\frac {b^3 (11 b B d+7 A b e-18 a B e) (a+b x)^{11}}{8568 e (b d-a e)^5 (d+e x)^{14}}+\frac {\left (b^4 (11 b B d+7 A b e-18 a B e)\right ) \int \frac {(a+b x)^{10}}{(d+e x)^{14}} \, dx}{2856 e (b d-a e)^5} \\ & = -\frac {(B d-A e) (a+b x)^{11}}{18 e (b d-a e) (d+e x)^{18}}+\frac {(11 b B d+7 A b e-18 a B e) (a+b x)^{11}}{306 e (b d-a e)^2 (d+e x)^{17}}+\frac {b (11 b B d+7 A b e-18 a B e) (a+b x)^{11}}{816 e (b d-a e)^3 (d+e x)^{16}}+\frac {b^2 (11 b B d+7 A b e-18 a B e) (a+b x)^{11}}{2448 e (b d-a e)^4 (d+e x)^{15}}+\frac {b^3 (11 b B d+7 A b e-18 a B e) (a+b x)^{11}}{8568 e (b d-a e)^5 (d+e x)^{14}}+\frac {b^4 (11 b B d+7 A b e-18 a B e) (a+b x)^{11}}{37128 e (b d-a e)^6 (d+e x)^{13}}+\frac {\left (b^5 (11 b B d+7 A b e-18 a B e)\right ) \int \frac {(a+b x)^{10}}{(d+e x)^{13}} \, dx}{18564 e (b d-a e)^6} \\ & = -\frac {(B d-A e) (a+b x)^{11}}{18 e (b d-a e) (d+e x)^{18}}+\frac {(11 b B d+7 A b e-18 a B e) (a+b x)^{11}}{306 e (b d-a e)^2 (d+e x)^{17}}+\frac {b (11 b B d+7 A b e-18 a B e) (a+b x)^{11}}{816 e (b d-a e)^3 (d+e x)^{16}}+\frac {b^2 (11 b B d+7 A b e-18 a B e) (a+b x)^{11}}{2448 e (b d-a e)^4 (d+e x)^{15}}+\frac {b^3 (11 b B d+7 A b e-18 a B e) (a+b x)^{11}}{8568 e (b d-a e)^5 (d+e x)^{14}}+\frac {b^4 (11 b B d+7 A b e-18 a B e) (a+b x)^{11}}{37128 e (b d-a e)^6 (d+e x)^{13}}+\frac {b^5 (11 b B d+7 A b e-18 a B e) (a+b x)^{11}}{222768 e (b d-a e)^7 (d+e x)^{12}}+\frac {\left (b^6 (11 b B d+7 A b e-18 a B e)\right ) \int \frac {(a+b x)^{10}}{(d+e x)^{12}} \, dx}{222768 e (b d-a e)^7} \\ & = -\frac {(B d-A e) (a+b x)^{11}}{18 e (b d-a e) (d+e x)^{18}}+\frac {(11 b B d+7 A b e-18 a B e) (a+b x)^{11}}{306 e (b d-a e)^2 (d+e x)^{17}}+\frac {b (11 b B d+7 A b e-18 a B e) (a+b x)^{11}}{816 e (b d-a e)^3 (d+e x)^{16}}+\frac {b^2 (11 b B d+7 A b e-18 a B e) (a+b x)^{11}}{2448 e (b d-a e)^4 (d+e x)^{15}}+\frac {b^3 (11 b B d+7 A b e-18 a B e) (a+b x)^{11}}{8568 e (b d-a e)^5 (d+e x)^{14}}+\frac {b^4 (11 b B d+7 A b e-18 a B e) (a+b x)^{11}}{37128 e (b d-a e)^6 (d+e x)^{13}}+\frac {b^5 (11 b B d+7 A b e-18 a B e) (a+b x)^{11}}{222768 e (b d-a e)^7 (d+e x)^{12}}+\frac {b^6 (11 b B d+7 A b e-18 a B e) (a+b x)^{11}}{2450448 e (b d-a e)^8 (d+e x)^{11}} \\
\end{align*}
Mathematica [B] (verified)
Leaf count is larger than twice the leaf count of optimal. \(1428\) vs. \(2(385)=770\).
Time = 0.49 (sec) , antiderivative size = 1428, normalized size of antiderivative = 3.71
\[
\int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{19}} \, dx=-\frac {8008 a^{10} e^{10} (17 A e+B (d+18 e x))+10010 a^9 b e^9 \left (8 A e (d+18 e x)+B \left (d^2+18 d e x+153 e^2 x^2\right )\right )+9009 a^8 b^2 e^8 \left (5 A e \left (d^2+18 d e x+153 e^2 x^2\right )+B \left (d^3+18 d^2 e x+153 d e^2 x^2+816 e^3 x^3\right )\right )+3432 a^7 b^3 e^7 \left (7 A e \left (d^3+18 d^2 e x+153 d e^2 x^2+816 e^3 x^3\right )+2 B \left (d^4+18 d^3 e x+153 d^2 e^2 x^2+816 d e^3 x^3+3060 e^4 x^4\right )\right )+924 a^6 b^4 e^6 \left (13 A e \left (d^4+18 d^3 e x+153 d^2 e^2 x^2+816 d e^3 x^3+3060 e^4 x^4\right )+5 B \left (d^5+18 d^4 e x+153 d^3 e^2 x^2+816 d^2 e^3 x^3+3060 d e^4 x^4+8568 e^5 x^5\right )\right )+2772 a^5 b^5 e^5 \left (2 A e \left (d^5+18 d^4 e x+153 d^3 e^2 x^2+816 d^2 e^3 x^3+3060 d e^4 x^4+8568 e^5 x^5\right )+B \left (d^6+18 d^5 e x+153 d^4 e^2 x^2+816 d^3 e^3 x^3+3060 d^2 e^4 x^4+8568 d e^5 x^5+18564 e^6 x^6\right )\right )+210 a^4 b^6 e^4 \left (11 A e \left (d^6+18 d^5 e x+153 d^4 e^2 x^2+816 d^3 e^3 x^3+3060 d^2 e^4 x^4+8568 d e^5 x^5+18564 e^6 x^6\right )+7 B \left (d^7+18 d^6 e x+153 d^5 e^2 x^2+816 d^4 e^3 x^3+3060 d^3 e^4 x^4+8568 d^2 e^5 x^5+18564 d e^6 x^6+31824 e^7 x^7\right )\right )+168 a^3 b^7 e^3 \left (5 A e \left (d^7+18 d^6 e x+153 d^5 e^2 x^2+816 d^4 e^3 x^3+3060 d^3 e^4 x^4+8568 d^2 e^5 x^5+18564 d e^6 x^6+31824 e^7 x^7\right )+4 B \left (d^8+18 d^7 e x+153 d^6 e^2 x^2+816 d^5 e^3 x^3+3060 d^4 e^4 x^4+8568 d^3 e^5 x^5+18564 d^2 e^6 x^6+31824 d e^7 x^7+43758 e^8 x^8\right )\right )+252 a^2 b^8 e^2 \left (A e \left (d^8+18 d^7 e x+153 d^6 e^2 x^2+816 d^5 e^3 x^3+3060 d^4 e^4 x^4+8568 d^3 e^5 x^5+18564 d^2 e^6 x^6+31824 d e^7 x^7+43758 e^8 x^8\right )+B \left (d^9+18 d^8 e x+153 d^7 e^2 x^2+816 d^6 e^3 x^3+3060 d^5 e^4 x^4+8568 d^4 e^5 x^5+18564 d^3 e^6 x^6+31824 d^2 e^7 x^7+43758 d e^8 x^8+48620 e^9 x^9\right )\right )+14 a b^9 e \left (4 A e \left (d^9+18 d^8 e x+153 d^7 e^2 x^2+816 d^6 e^3 x^3+3060 d^5 e^4 x^4+8568 d^4 e^5 x^5+18564 d^3 e^6 x^6+31824 d^2 e^7 x^7+43758 d e^8 x^8+48620 e^9 x^9\right )+5 B \left (d^{10}+18 d^9 e x+153 d^8 e^2 x^2+816 d^7 e^3 x^3+3060 d^6 e^4 x^4+8568 d^5 e^5 x^5+18564 d^4 e^6 x^6+31824 d^3 e^7 x^7+43758 d^2 e^8 x^8+48620 d e^9 x^9+43758 e^{10} x^{10}\right )\right )+b^{10} \left (7 A e \left (d^{10}+18 d^9 e x+153 d^8 e^2 x^2+816 d^7 e^3 x^3+3060 d^6 e^4 x^4+8568 d^5 e^5 x^5+18564 d^4 e^6 x^6+31824 d^3 e^7 x^7+43758 d^2 e^8 x^8+48620 d e^9 x^9+43758 e^{10} x^{10}\right )+11 B \left (d^{11}+18 d^{10} e x+153 d^9 e^2 x^2+816 d^8 e^3 x^3+3060 d^7 e^4 x^4+8568 d^6 e^5 x^5+18564 d^5 e^6 x^6+31824 d^4 e^7 x^7+43758 d^3 e^8 x^8+48620 d^2 e^9 x^9+43758 d e^{10} x^{10}+31824 e^{11} x^{11}\right )\right )}{2450448 e^{12} (d+e x)^{18}}
\]
[In]
Integrate[((a + b*x)^10*(A + B*x))/(d + e*x)^19,x]
[Out]
-1/2450448*(8008*a^10*e^10*(17*A*e + B*(d + 18*e*x)) + 10010*a^9*b*e^9*(8*A*e*(d + 18*e*x) + B*(d^2 + 18*d*e*x
+ 153*e^2*x^2)) + 9009*a^8*b^2*e^8*(5*A*e*(d^2 + 18*d*e*x + 153*e^2*x^2) + B*(d^3 + 18*d^2*e*x + 153*d*e^2*x^
2 + 816*e^3*x^3)) + 3432*a^7*b^3*e^7*(7*A*e*(d^3 + 18*d^2*e*x + 153*d*e^2*x^2 + 816*e^3*x^3) + 2*B*(d^4 + 18*d
^3*e*x + 153*d^2*e^2*x^2 + 816*d*e^3*x^3 + 3060*e^4*x^4)) + 924*a^6*b^4*e^6*(13*A*e*(d^4 + 18*d^3*e*x + 153*d^
2*e^2*x^2 + 816*d*e^3*x^3 + 3060*e^4*x^4) + 5*B*(d^5 + 18*d^4*e*x + 153*d^3*e^2*x^2 + 816*d^2*e^3*x^3 + 3060*d
*e^4*x^4 + 8568*e^5*x^5)) + 2772*a^5*b^5*e^5*(2*A*e*(d^5 + 18*d^4*e*x + 153*d^3*e^2*x^2 + 816*d^2*e^3*x^3 + 30
60*d*e^4*x^4 + 8568*e^5*x^5) + B*(d^6 + 18*d^5*e*x + 153*d^4*e^2*x^2 + 816*d^3*e^3*x^3 + 3060*d^2*e^4*x^4 + 85
68*d*e^5*x^5 + 18564*e^6*x^6)) + 210*a^4*b^6*e^4*(11*A*e*(d^6 + 18*d^5*e*x + 153*d^4*e^2*x^2 + 816*d^3*e^3*x^3
+ 3060*d^2*e^4*x^4 + 8568*d*e^5*x^5 + 18564*e^6*x^6) + 7*B*(d^7 + 18*d^6*e*x + 153*d^5*e^2*x^2 + 816*d^4*e^3*
x^3 + 3060*d^3*e^4*x^4 + 8568*d^2*e^5*x^5 + 18564*d*e^6*x^6 + 31824*e^7*x^7)) + 168*a^3*b^7*e^3*(5*A*e*(d^7 +
18*d^6*e*x + 153*d^5*e^2*x^2 + 816*d^4*e^3*x^3 + 3060*d^3*e^4*x^4 + 8568*d^2*e^5*x^5 + 18564*d*e^6*x^6 + 31824
*e^7*x^7) + 4*B*(d^8 + 18*d^7*e*x + 153*d^6*e^2*x^2 + 816*d^5*e^3*x^3 + 3060*d^4*e^4*x^4 + 8568*d^3*e^5*x^5 +
18564*d^2*e^6*x^6 + 31824*d*e^7*x^7 + 43758*e^8*x^8)) + 252*a^2*b^8*e^2*(A*e*(d^8 + 18*d^7*e*x + 153*d^6*e^2*x
^2 + 816*d^5*e^3*x^3 + 3060*d^4*e^4*x^4 + 8568*d^3*e^5*x^5 + 18564*d^2*e^6*x^6 + 31824*d*e^7*x^7 + 43758*e^8*x
^8) + B*(d^9 + 18*d^8*e*x + 153*d^7*e^2*x^2 + 816*d^6*e^3*x^3 + 3060*d^5*e^4*x^4 + 8568*d^4*e^5*x^5 + 18564*d^
3*e^6*x^6 + 31824*d^2*e^7*x^7 + 43758*d*e^8*x^8 + 48620*e^9*x^9)) + 14*a*b^9*e*(4*A*e*(d^9 + 18*d^8*e*x + 153*
d^7*e^2*x^2 + 816*d^6*e^3*x^3 + 3060*d^5*e^4*x^4 + 8568*d^4*e^5*x^5 + 18564*d^3*e^6*x^6 + 31824*d^2*e^7*x^7 +
43758*d*e^8*x^8 + 48620*e^9*x^9) + 5*B*(d^10 + 18*d^9*e*x + 153*d^8*e^2*x^2 + 816*d^7*e^3*x^3 + 3060*d^6*e^4*x
^4 + 8568*d^5*e^5*x^5 + 18564*d^4*e^6*x^6 + 31824*d^3*e^7*x^7 + 43758*d^2*e^8*x^8 + 48620*d*e^9*x^9 + 43758*e^
10*x^10)) + b^10*(7*A*e*(d^10 + 18*d^9*e*x + 153*d^8*e^2*x^2 + 816*d^7*e^3*x^3 + 3060*d^6*e^4*x^4 + 8568*d^5*e
^5*x^5 + 18564*d^4*e^6*x^6 + 31824*d^3*e^7*x^7 + 43758*d^2*e^8*x^8 + 48620*d*e^9*x^9 + 43758*e^10*x^10) + 11*B
*(d^11 + 18*d^10*e*x + 153*d^9*e^2*x^2 + 816*d^8*e^3*x^3 + 3060*d^7*e^4*x^4 + 8568*d^6*e^5*x^5 + 18564*d^5*e^6
*x^6 + 31824*d^4*e^7*x^7 + 43758*d^3*e^8*x^8 + 48620*d^2*e^9*x^9 + 43758*d*e^10*x^10 + 31824*e^11*x^11)))/(e^1
2*(d + e*x)^18)
Maple [B] (verified)
Leaf count of result is larger than twice the leaf count of optimal. \(1900\) vs. \(2(369)=738\).
Time = 2.12 (sec) , antiderivative size = 1901, normalized size of antiderivative =
4.94
| | |
| 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 |
\(\text {Expression too large to display}\) |
\(2233\) |
| parallelrisch |
\(\text {Expression too large to display}\) |
\(2242\) |
| | |
|
|
|
|
[In]
int((b*x+a)^10*(B*x+A)/(e*x+d)^19,x,method=_RETURNVERBOSE)
[Out]
(-1/2450448/e^12*(136136*A*a^10*e^11+80080*A*a^9*b*d*e^10+45045*A*a^8*b^2*d^2*e^9+24024*A*a^7*b^3*d^3*e^8+1201
2*A*a^6*b^4*d^4*e^7+5544*A*a^5*b^5*d^5*e^6+2310*A*a^4*b^6*d^6*e^5+840*A*a^3*b^7*d^7*e^4+252*A*a^2*b^8*d^8*e^3+
56*A*a*b^9*d^9*e^2+7*A*b^10*d^10*e+8008*B*a^10*d*e^10+10010*B*a^9*b*d^2*e^9+9009*B*a^8*b^2*d^3*e^8+6864*B*a^7*
b^3*d^4*e^7+4620*B*a^6*b^4*d^5*e^6+2772*B*a^5*b^5*d^6*e^5+1470*B*a^4*b^6*d^7*e^4+672*B*a^3*b^7*d^8*e^3+252*B*a
^2*b^8*d^9*e^2+70*B*a*b^9*d^10*e+11*B*b^10*d^11)-1/136136/e^11*(80080*A*a^9*b*e^10+45045*A*a^8*b^2*d*e^9+24024
*A*a^7*b^3*d^2*e^8+12012*A*a^6*b^4*d^3*e^7+5544*A*a^5*b^5*d^4*e^6+2310*A*a^4*b^6*d^5*e^5+840*A*a^3*b^7*d^6*e^4
+252*A*a^2*b^8*d^7*e^3+56*A*a*b^9*d^8*e^2+7*A*b^10*d^9*e+8008*B*a^10*e^10+10010*B*a^9*b*d*e^9+9009*B*a^8*b^2*d
^2*e^8+6864*B*a^7*b^3*d^3*e^7+4620*B*a^6*b^4*d^4*e^6+2772*B*a^5*b^5*d^5*e^5+1470*B*a^4*b^6*d^6*e^4+672*B*a^3*b
^7*d^7*e^3+252*B*a^2*b^8*d^8*e^2+70*B*a*b^9*d^9*e+11*B*b^10*d^10)*x-1/16016*b/e^10*(45045*A*a^8*b*e^9+24024*A*
a^7*b^2*d*e^8+12012*A*a^6*b^3*d^2*e^7+5544*A*a^5*b^4*d^3*e^6+2310*A*a^4*b^5*d^4*e^5+840*A*a^3*b^6*d^5*e^4+252*
A*a^2*b^7*d^6*e^3+56*A*a*b^8*d^7*e^2+7*A*b^9*d^8*e+10010*B*a^9*e^9+9009*B*a^8*b*d*e^8+6864*B*a^7*b^2*d^2*e^7+4
620*B*a^6*b^3*d^3*e^6+2772*B*a^5*b^4*d^4*e^5+1470*B*a^4*b^5*d^5*e^4+672*B*a^3*b^6*d^6*e^3+252*B*a^2*b^7*d^7*e^
2+70*B*a*b^8*d^8*e+11*B*b^9*d^9)*x^2-1/3003*b^2/e^9*(24024*A*a^7*b*e^8+12012*A*a^6*b^2*d*e^7+5544*A*a^5*b^3*d^
2*e^6+2310*A*a^4*b^4*d^3*e^5+840*A*a^3*b^5*d^4*e^4+252*A*a^2*b^6*d^5*e^3+56*A*a*b^7*d^6*e^2+7*A*b^8*d^7*e+9009
*B*a^8*e^8+6864*B*a^7*b*d*e^7+4620*B*a^6*b^2*d^2*e^6+2772*B*a^5*b^3*d^3*e^5+1470*B*a^4*b^4*d^4*e^4+672*B*a^3*b
^5*d^5*e^3+252*B*a^2*b^6*d^6*e^2+70*B*a*b^7*d^7*e+11*B*b^8*d^8)*x^3-5/4004*b^3/e^8*(12012*A*a^6*b*e^7+5544*A*a
^5*b^2*d*e^6+2310*A*a^4*b^3*d^2*e^5+840*A*a^3*b^4*d^3*e^4+252*A*a^2*b^5*d^4*e^3+56*A*a*b^6*d^5*e^2+7*A*b^7*d^6
*e+6864*B*a^7*e^7+4620*B*a^6*b*d*e^6+2772*B*a^5*b^2*d^2*e^5+1470*B*a^4*b^3*d^3*e^4+672*B*a^3*b^4*d^4*e^3+252*B
*a^2*b^5*d^5*e^2+70*B*a*b^6*d^6*e+11*B*b^7*d^7)*x^4-1/286*b^4/e^7*(5544*A*a^5*b*e^6+2310*A*a^4*b^2*d*e^5+840*A
*a^3*b^3*d^2*e^4+252*A*a^2*b^4*d^3*e^3+56*A*a*b^5*d^4*e^2+7*A*b^6*d^5*e+4620*B*a^6*e^6+2772*B*a^5*b*d*e^5+1470
*B*a^4*b^2*d^2*e^4+672*B*a^3*b^3*d^3*e^3+252*B*a^2*b^4*d^4*e^2+70*B*a*b^5*d^5*e+11*B*b^6*d^6)*x^5-1/132*b^5/e^
6*(2310*A*a^4*b*e^5+840*A*a^3*b^2*d*e^4+252*A*a^2*b^3*d^2*e^3+56*A*a*b^4*d^3*e^2+7*A*b^5*d^4*e+2772*B*a^5*e^5+
1470*B*a^4*b*d*e^4+672*B*a^3*b^2*d^2*e^3+252*B*a^2*b^3*d^3*e^2+70*B*a*b^4*d^4*e+11*B*b^5*d^5)*x^6-1/77*b^6/e^5
*(840*A*a^3*b*e^4+252*A*a^2*b^2*d*e^3+56*A*a*b^3*d^2*e^2+7*A*b^4*d^3*e+1470*B*a^4*e^4+672*B*a^3*b*d*e^3+252*B*
a^2*b^2*d^2*e^2+70*B*a*b^3*d^3*e+11*B*b^4*d^4)*x^7-1/56*b^7/e^4*(252*A*a^2*b*e^3+56*A*a*b^2*d*e^2+7*A*b^3*d^2*
e+672*B*a^3*e^3+252*B*a^2*b*d*e^2+70*B*a*b^2*d^2*e+11*B*b^3*d^3)*x^8-5/252*b^8/e^3*(56*A*a*b*e^2+7*A*b^2*d*e+2
52*B*a^2*e^2+70*B*a*b*d*e+11*B*b^2*d^2)*x^9-1/56*b^9/e^2*(7*A*b*e+70*B*a*e+11*B*b*d)*x^10-1/7*b^10*B/e*x^11)/(
e*x+d)^18
Fricas [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 2006 vs. \(2 (369) = 738\).
Time = 0.35 (sec) , antiderivative size = 2006, normalized size of antiderivative = 5.21
\[
\int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{19}} \, dx=\text {Too large to display}
\]
[In]
integrate((b*x+a)^10*(B*x+A)/(e*x+d)^19,x, algorithm="fricas")
[Out]
-1/2450448*(350064*B*b^10*e^11*x^11 + 11*B*b^10*d^11 + 136136*A*a^10*e^11 + 7*(10*B*a*b^9 + A*b^10)*d^10*e + 2
8*(9*B*a^2*b^8 + 2*A*a*b^9)*d^9*e^2 + 84*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^8*e^3 + 210*(7*B*a^4*b^6 + 4*A*a^3*b^7)
*d^7*e^4 + 462*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^6*e^5 + 924*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^5*e^6 + 1716*(4*B*a^7*b
^3 + 7*A*a^6*b^4)*d^4*e^7 + 3003*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^3*e^8 + 5005*(2*B*a^9*b + 9*A*a^8*b^2)*d^2*e^9
+ 8008*(B*a^10 + 10*A*a^9*b)*d*e^10 + 43758*(11*B*b^10*d*e^10 + 7*(10*B*a*b^9 + A*b^10)*e^11)*x^10 + 48620*(11
*B*b^10*d^2*e^9 + 7*(10*B*a*b^9 + A*b^10)*d*e^10 + 28*(9*B*a^2*b^8 + 2*A*a*b^9)*e^11)*x^9 + 43758*(11*B*b^10*d
^3*e^8 + 7*(10*B*a*b^9 + A*b^10)*d^2*e^9 + 28*(9*B*a^2*b^8 + 2*A*a*b^9)*d*e^10 + 84*(8*B*a^3*b^7 + 3*A*a^2*b^8
)*e^11)*x^8 + 31824*(11*B*b^10*d^4*e^7 + 7*(10*B*a*b^9 + A*b^10)*d^3*e^8 + 28*(9*B*a^2*b^8 + 2*A*a*b^9)*d^2*e^
9 + 84*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d*e^10 + 210*(7*B*a^4*b^6 + 4*A*a^3*b^7)*e^11)*x^7 + 18564*(11*B*b^10*d^5*e
^6 + 7*(10*B*a*b^9 + A*b^10)*d^4*e^7 + 28*(9*B*a^2*b^8 + 2*A*a*b^9)*d^3*e^8 + 84*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d
^2*e^9 + 210*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d*e^10 + 462*(6*B*a^5*b^5 + 5*A*a^4*b^6)*e^11)*x^6 + 8568*(11*B*b^10*
d^6*e^5 + 7*(10*B*a*b^9 + A*b^10)*d^5*e^6 + 28*(9*B*a^2*b^8 + 2*A*a*b^9)*d^4*e^7 + 84*(8*B*a^3*b^7 + 3*A*a^2*b
^8)*d^3*e^8 + 210*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^2*e^9 + 462*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d*e^10 + 924*(5*B*a^6*
b^4 + 6*A*a^5*b^5)*e^11)*x^5 + 3060*(11*B*b^10*d^7*e^4 + 7*(10*B*a*b^9 + A*b^10)*d^6*e^5 + 28*(9*B*a^2*b^8 + 2
*A*a*b^9)*d^5*e^6 + 84*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^4*e^7 + 210*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^3*e^8 + 462*(6*
B*a^5*b^5 + 5*A*a^4*b^6)*d^2*e^9 + 924*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d*e^10 + 1716*(4*B*a^7*b^3 + 7*A*a^6*b^4)*e
^11)*x^4 + 816*(11*B*b^10*d^8*e^3 + 7*(10*B*a*b^9 + A*b^10)*d^7*e^4 + 28*(9*B*a^2*b^8 + 2*A*a*b^9)*d^6*e^5 + 8
4*(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 + 462*(6*B*a^5*b^5 + 5*A*a^4*b
^6)*d^3*e^8 + 924*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^2*e^9 + 1716*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d*e^10 + 3003*(3*B*a^
8*b^2 + 8*A*a^7*b^3)*e^11)*x^3 + 153*(11*B*b^10*d^9*e^2 + 7*(10*B*a*b^9 + A*b^10)*d^8*e^3 + 28*(9*B*a^2*b^8 +
2*A*a*b^9)*d^7*e^4 + 84*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^6*e^5 + 210*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^5*e^6 + 462*(6
*B*a^5*b^5 + 5*A*a^4*b^6)*d^4*e^7 + 924*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^3*e^8 + 1716*(4*B*a^7*b^3 + 7*A*a^6*b^4)
*d^2*e^9 + 3003*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d*e^10 + 5005*(2*B*a^9*b + 9*A*a^8*b^2)*e^11)*x^2 + 18*(11*B*b^10*
d^10*e + 7*(10*B*a*b^9 + A*b^10)*d^9*e^2 + 28*(9*B*a^2*b^8 + 2*A*a*b^9)*d^8*e^3 + 84*(8*B*a^3*b^7 + 3*A*a^2*b^
8)*d^7*e^4 + 210*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^6*e^5 + 462*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^5*e^6 + 924*(5*B*a^6*
b^4 + 6*A*a^5*b^5)*d^4*e^7 + 1716*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^3*e^8 + 3003*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^2*e
^9 + 5005*(2*B*a^9*b + 9*A*a^8*b^2)*d*e^10 + 8008*(B*a^10 + 10*A*a^9*b)*e^11)*x)/(e^30*x^18 + 18*d*e^29*x^17 +
153*d^2*e^28*x^16 + 816*d^3*e^27*x^15 + 3060*d^4*e^26*x^14 + 8568*d^5*e^25*x^13 + 18564*d^6*e^24*x^12 + 31824
*d^7*e^23*x^11 + 43758*d^8*e^22*x^10 + 48620*d^9*e^21*x^9 + 43758*d^10*e^20*x^8 + 31824*d^11*e^19*x^7 + 18564*
d^12*e^18*x^6 + 8568*d^13*e^17*x^5 + 3060*d^14*e^16*x^4 + 816*d^15*e^15*x^3 + 153*d^16*e^14*x^2 + 18*d^17*e^13
*x + d^18*e^12)
Sympy [F(-1)]
Timed out. \[
\int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{19}} \, dx=\text {Timed out}
\]
[In]
integrate((b*x+a)**10*(B*x+A)/(e*x+d)**19,x)
[Out]
Timed out
Maxima [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 2006 vs. \(2 (369) = 738\).
Time = 0.32 (sec) , antiderivative size = 2006, normalized size of antiderivative = 5.21
\[
\int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{19}} \, dx=\text {Too large to display}
\]
[In]
integrate((b*x+a)^10*(B*x+A)/(e*x+d)^19,x, algorithm="maxima")
[Out]
-1/2450448*(350064*B*b^10*e^11*x^11 + 11*B*b^10*d^11 + 136136*A*a^10*e^11 + 7*(10*B*a*b^9 + A*b^10)*d^10*e + 2
8*(9*B*a^2*b^8 + 2*A*a*b^9)*d^9*e^2 + 84*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^8*e^3 + 210*(7*B*a^4*b^6 + 4*A*a^3*b^7)
*d^7*e^4 + 462*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^6*e^5 + 924*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^5*e^6 + 1716*(4*B*a^7*b
^3 + 7*A*a^6*b^4)*d^4*e^7 + 3003*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^3*e^8 + 5005*(2*B*a^9*b + 9*A*a^8*b^2)*d^2*e^9
+ 8008*(B*a^10 + 10*A*a^9*b)*d*e^10 + 43758*(11*B*b^10*d*e^10 + 7*(10*B*a*b^9 + A*b^10)*e^11)*x^10 + 48620*(11
*B*b^10*d^2*e^9 + 7*(10*B*a*b^9 + A*b^10)*d*e^10 + 28*(9*B*a^2*b^8 + 2*A*a*b^9)*e^11)*x^9 + 43758*(11*B*b^10*d
^3*e^8 + 7*(10*B*a*b^9 + A*b^10)*d^2*e^9 + 28*(9*B*a^2*b^8 + 2*A*a*b^9)*d*e^10 + 84*(8*B*a^3*b^7 + 3*A*a^2*b^8
)*e^11)*x^8 + 31824*(11*B*b^10*d^4*e^7 + 7*(10*B*a*b^9 + A*b^10)*d^3*e^8 + 28*(9*B*a^2*b^8 + 2*A*a*b^9)*d^2*e^
9 + 84*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d*e^10 + 210*(7*B*a^4*b^6 + 4*A*a^3*b^7)*e^11)*x^7 + 18564*(11*B*b^10*d^5*e
^6 + 7*(10*B*a*b^9 + A*b^10)*d^4*e^7 + 28*(9*B*a^2*b^8 + 2*A*a*b^9)*d^3*e^8 + 84*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d
^2*e^9 + 210*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d*e^10 + 462*(6*B*a^5*b^5 + 5*A*a^4*b^6)*e^11)*x^6 + 8568*(11*B*b^10*
d^6*e^5 + 7*(10*B*a*b^9 + A*b^10)*d^5*e^6 + 28*(9*B*a^2*b^8 + 2*A*a*b^9)*d^4*e^7 + 84*(8*B*a^3*b^7 + 3*A*a^2*b
^8)*d^3*e^8 + 210*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^2*e^9 + 462*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d*e^10 + 924*(5*B*a^6*
b^4 + 6*A*a^5*b^5)*e^11)*x^5 + 3060*(11*B*b^10*d^7*e^4 + 7*(10*B*a*b^9 + A*b^10)*d^6*e^5 + 28*(9*B*a^2*b^8 + 2
*A*a*b^9)*d^5*e^6 + 84*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^4*e^7 + 210*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^3*e^8 + 462*(6*
B*a^5*b^5 + 5*A*a^4*b^6)*d^2*e^9 + 924*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d*e^10 + 1716*(4*B*a^7*b^3 + 7*A*a^6*b^4)*e
^11)*x^4 + 816*(11*B*b^10*d^8*e^3 + 7*(10*B*a*b^9 + A*b^10)*d^7*e^4 + 28*(9*B*a^2*b^8 + 2*A*a*b^9)*d^6*e^5 + 8
4*(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 + 462*(6*B*a^5*b^5 + 5*A*a^4*b
^6)*d^3*e^8 + 924*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^2*e^9 + 1716*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d*e^10 + 3003*(3*B*a^
8*b^2 + 8*A*a^7*b^3)*e^11)*x^3 + 153*(11*B*b^10*d^9*e^2 + 7*(10*B*a*b^9 + A*b^10)*d^8*e^3 + 28*(9*B*a^2*b^8 +
2*A*a*b^9)*d^7*e^4 + 84*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^6*e^5 + 210*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^5*e^6 + 462*(6
*B*a^5*b^5 + 5*A*a^4*b^6)*d^4*e^7 + 924*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^3*e^8 + 1716*(4*B*a^7*b^3 + 7*A*a^6*b^4)
*d^2*e^9 + 3003*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d*e^10 + 5005*(2*B*a^9*b + 9*A*a^8*b^2)*e^11)*x^2 + 18*(11*B*b^10*
d^10*e + 7*(10*B*a*b^9 + A*b^10)*d^9*e^2 + 28*(9*B*a^2*b^8 + 2*A*a*b^9)*d^8*e^3 + 84*(8*B*a^3*b^7 + 3*A*a^2*b^
8)*d^7*e^4 + 210*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^6*e^5 + 462*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^5*e^6 + 924*(5*B*a^6*
b^4 + 6*A*a^5*b^5)*d^4*e^7 + 1716*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^3*e^8 + 3003*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^2*e
^9 + 5005*(2*B*a^9*b + 9*A*a^8*b^2)*d*e^10 + 8008*(B*a^10 + 10*A*a^9*b)*e^11)*x)/(e^30*x^18 + 18*d*e^29*x^17 +
153*d^2*e^28*x^16 + 816*d^3*e^27*x^15 + 3060*d^4*e^26*x^14 + 8568*d^5*e^25*x^13 + 18564*d^6*e^24*x^12 + 31824
*d^7*e^23*x^11 + 43758*d^8*e^22*x^10 + 48620*d^9*e^21*x^9 + 43758*d^10*e^20*x^8 + 31824*d^11*e^19*x^7 + 18564*
d^12*e^18*x^6 + 8568*d^13*e^17*x^5 + 3060*d^14*e^16*x^4 + 816*d^15*e^15*x^3 + 153*d^16*e^14*x^2 + 18*d^17*e^13
*x + d^18*e^12)
Giac [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 2232 vs. \(2 (369) = 738\).
Time = 0.29 (sec) , antiderivative size = 2232, normalized size of antiderivative = 5.80
\[
\int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{19}} \, dx=\text {Too large to display}
\]
[In]
integrate((b*x+a)^10*(B*x+A)/(e*x+d)^19,x, algorithm="giac")
[Out]
-1/2450448*(350064*B*b^10*e^11*x^11 + 481338*B*b^10*d*e^10*x^10 + 3063060*B*a*b^9*e^11*x^10 + 306306*A*b^10*e^
11*x^10 + 534820*B*b^10*d^2*e^9*x^9 + 3403400*B*a*b^9*d*e^10*x^9 + 340340*A*b^10*d*e^10*x^9 + 12252240*B*a^2*b
^8*e^11*x^9 + 2722720*A*a*b^9*e^11*x^9 + 481338*B*b^10*d^3*e^8*x^8 + 3063060*B*a*b^9*d^2*e^9*x^8 + 306306*A*b^
10*d^2*e^9*x^8 + 11027016*B*a^2*b^8*d*e^10*x^8 + 2450448*A*a*b^9*d*e^10*x^8 + 29405376*B*a^3*b^7*e^11*x^8 + 11
027016*A*a^2*b^8*e^11*x^8 + 350064*B*b^10*d^4*e^7*x^7 + 2227680*B*a*b^9*d^3*e^8*x^7 + 222768*A*b^10*d^3*e^8*x^
7 + 8019648*B*a^2*b^8*d^2*e^9*x^7 + 1782144*A*a*b^9*d^2*e^9*x^7 + 21385728*B*a^3*b^7*d*e^10*x^7 + 8019648*A*a^
2*b^8*d*e^10*x^7 + 46781280*B*a^4*b^6*e^11*x^7 + 26732160*A*a^3*b^7*e^11*x^7 + 204204*B*b^10*d^5*e^6*x^6 + 129
9480*B*a*b^9*d^4*e^7*x^6 + 129948*A*b^10*d^4*e^7*x^6 + 4678128*B*a^2*b^8*d^3*e^8*x^6 + 1039584*A*a*b^9*d^3*e^8
*x^6 + 12475008*B*a^3*b^7*d^2*e^9*x^6 + 4678128*A*a^2*b^8*d^2*e^9*x^6 + 27289080*B*a^4*b^6*d*e^10*x^6 + 155937
60*A*a^3*b^7*d*e^10*x^6 + 51459408*B*a^5*b^5*e^11*x^6 + 42882840*A*a^4*b^6*e^11*x^6 + 94248*B*b^10*d^6*e^5*x^5
+ 599760*B*a*b^9*d^5*e^6*x^5 + 59976*A*b^10*d^5*e^6*x^5 + 2159136*B*a^2*b^8*d^4*e^7*x^5 + 479808*A*a*b^9*d^4*
e^7*x^5 + 5757696*B*a^3*b^7*d^3*e^8*x^5 + 2159136*A*a^2*b^8*d^3*e^8*x^5 + 12594960*B*a^4*b^6*d^2*e^9*x^5 + 719
7120*A*a^3*b^7*d^2*e^9*x^5 + 23750496*B*a^5*b^5*d*e^10*x^5 + 19792080*A*a^4*b^6*d*e^10*x^5 + 39584160*B*a^6*b^
4*e^11*x^5 + 47500992*A*a^5*b^5*e^11*x^5 + 33660*B*b^10*d^7*e^4*x^4 + 214200*B*a*b^9*d^6*e^5*x^4 + 21420*A*b^1
0*d^6*e^5*x^4 + 771120*B*a^2*b^8*d^5*e^6*x^4 + 171360*A*a*b^9*d^5*e^6*x^4 + 2056320*B*a^3*b^7*d^4*e^7*x^4 + 77
1120*A*a^2*b^8*d^4*e^7*x^4 + 4498200*B*a^4*b^6*d^3*e^8*x^4 + 2570400*A*a^3*b^7*d^3*e^8*x^4 + 8482320*B*a^5*b^5
*d^2*e^9*x^4 + 7068600*A*a^4*b^6*d^2*e^9*x^4 + 14137200*B*a^6*b^4*d*e^10*x^4 + 16964640*A*a^5*b^5*d*e^10*x^4 +
21003840*B*a^7*b^3*e^11*x^4 + 36756720*A*a^6*b^4*e^11*x^4 + 8976*B*b^10*d^8*e^3*x^3 + 57120*B*a*b^9*d^7*e^4*x
^3 + 5712*A*b^10*d^7*e^4*x^3 + 205632*B*a^2*b^8*d^6*e^5*x^3 + 45696*A*a*b^9*d^6*e^5*x^3 + 548352*B*a^3*b^7*d^5
*e^6*x^3 + 205632*A*a^2*b^8*d^5*e^6*x^3 + 1199520*B*a^4*b^6*d^4*e^7*x^3 + 685440*A*a^3*b^7*d^4*e^7*x^3 + 22619
52*B*a^5*b^5*d^3*e^8*x^3 + 1884960*A*a^4*b^6*d^3*e^8*x^3 + 3769920*B*a^6*b^4*d^2*e^9*x^3 + 4523904*A*a^5*b^5*d
^2*e^9*x^3 + 5601024*B*a^7*b^3*d*e^10*x^3 + 9801792*A*a^6*b^4*d*e^10*x^3 + 7351344*B*a^8*b^2*e^11*x^3 + 196035
84*A*a^7*b^3*e^11*x^3 + 1683*B*b^10*d^9*e^2*x^2 + 10710*B*a*b^9*d^8*e^3*x^2 + 1071*A*b^10*d^8*e^3*x^2 + 38556*
B*a^2*b^8*d^7*e^4*x^2 + 8568*A*a*b^9*d^7*e^4*x^2 + 102816*B*a^3*b^7*d^6*e^5*x^2 + 38556*A*a^2*b^8*d^6*e^5*x^2
+ 224910*B*a^4*b^6*d^5*e^6*x^2 + 128520*A*a^3*b^7*d^5*e^6*x^2 + 424116*B*a^5*b^5*d^4*e^7*x^2 + 353430*A*a^4*b^
6*d^4*e^7*x^2 + 706860*B*a^6*b^4*d^3*e^8*x^2 + 848232*A*a^5*b^5*d^3*e^8*x^2 + 1050192*B*a^7*b^3*d^2*e^9*x^2 +
1837836*A*a^6*b^4*d^2*e^9*x^2 + 1378377*B*a^8*b^2*d*e^10*x^2 + 3675672*A*a^7*b^3*d*e^10*x^2 + 1531530*B*a^9*b*
e^11*x^2 + 6891885*A*a^8*b^2*e^11*x^2 + 198*B*b^10*d^10*e*x + 1260*B*a*b^9*d^9*e^2*x + 126*A*b^10*d^9*e^2*x +
4536*B*a^2*b^8*d^8*e^3*x + 1008*A*a*b^9*d^8*e^3*x + 12096*B*a^3*b^7*d^7*e^4*x + 4536*A*a^2*b^8*d^7*e^4*x + 264
60*B*a^4*b^6*d^6*e^5*x + 15120*A*a^3*b^7*d^6*e^5*x + 49896*B*a^5*b^5*d^5*e^6*x + 41580*A*a^4*b^6*d^5*e^6*x + 8
3160*B*a^6*b^4*d^4*e^7*x + 99792*A*a^5*b^5*d^4*e^7*x + 123552*B*a^7*b^3*d^3*e^8*x + 216216*A*a^6*b^4*d^3*e^8*x
+ 162162*B*a^8*b^2*d^2*e^9*x + 432432*A*a^7*b^3*d^2*e^9*x + 180180*B*a^9*b*d*e^10*x + 810810*A*a^8*b^2*d*e^10
*x + 144144*B*a^10*e^11*x + 1441440*A*a^9*b*e^11*x + 11*B*b^10*d^11 + 70*B*a*b^9*d^10*e + 7*A*b^10*d^10*e + 25
2*B*a^2*b^8*d^9*e^2 + 56*A*a*b^9*d^9*e^2 + 672*B*a^3*b^7*d^8*e^3 + 252*A*a^2*b^8*d^8*e^3 + 1470*B*a^4*b^6*d^7*
e^4 + 840*A*a^3*b^7*d^7*e^4 + 2772*B*a^5*b^5*d^6*e^5 + 2310*A*a^4*b^6*d^6*e^5 + 4620*B*a^6*b^4*d^5*e^6 + 5544*
A*a^5*b^5*d^5*e^6 + 6864*B*a^7*b^3*d^4*e^7 + 12012*A*a^6*b^4*d^4*e^7 + 9009*B*a^8*b^2*d^3*e^8 + 24024*A*a^7*b^
3*d^3*e^8 + 10010*B*a^9*b*d^2*e^9 + 45045*A*a^8*b^2*d^2*e^9 + 8008*B*a^10*d*e^10 + 80080*A*a^9*b*d*e^10 + 1361
36*A*a^10*e^11)/((e*x + d)^18*e^12)
Mupad [B] (verification not implemented)
Time = 16.11 (sec) , antiderivative size = 2088, normalized size of antiderivative = 5.42
\[
\int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{19}} \, dx=\text {Too large to display}
\]
[In]
int(((A + B*x)*(a + b*x)^10)/(d + e*x)^19,x)
[Out]
-((136136*A*a^10*e^11 + 11*B*b^10*d^11 + 7*A*b^10*d^10*e + 8008*B*a^10*d*e^10 + 56*A*a*b^9*d^9*e^2 + 10010*B*a
^9*b*d^2*e^9 + 252*A*a^2*b^8*d^8*e^3 + 840*A*a^3*b^7*d^7*e^4 + 2310*A*a^4*b^6*d^6*e^5 + 5544*A*a^5*b^5*d^5*e^6
+ 12012*A*a^6*b^4*d^4*e^7 + 24024*A*a^7*b^3*d^3*e^8 + 45045*A*a^8*b^2*d^2*e^9 + 252*B*a^2*b^8*d^9*e^2 + 672*B
*a^3*b^7*d^8*e^3 + 1470*B*a^4*b^6*d^7*e^4 + 2772*B*a^5*b^5*d^6*e^5 + 4620*B*a^6*b^4*d^5*e^6 + 6864*B*a^7*b^3*d
^4*e^7 + 9009*B*a^8*b^2*d^3*e^8 + 80080*A*a^9*b*d*e^10 + 70*B*a*b^9*d^10*e)/(2450448*e^12) + (x*(8008*B*a^10*e
^10 + 11*B*b^10*d^10 + 80080*A*a^9*b*e^10 + 7*A*b^10*d^9*e + 56*A*a*b^9*d^8*e^2 + 45045*A*a^8*b^2*d*e^9 + 252*
A*a^2*b^8*d^7*e^3 + 840*A*a^3*b^7*d^6*e^4 + 2310*A*a^4*b^6*d^5*e^5 + 5544*A*a^5*b^5*d^4*e^6 + 12012*A*a^6*b^4*
d^3*e^7 + 24024*A*a^7*b^3*d^2*e^8 + 252*B*a^2*b^8*d^8*e^2 + 672*B*a^3*b^7*d^7*e^3 + 1470*B*a^4*b^6*d^6*e^4 + 2
772*B*a^5*b^5*d^5*e^5 + 4620*B*a^6*b^4*d^4*e^6 + 6864*B*a^7*b^3*d^3*e^7 + 9009*B*a^8*b^2*d^2*e^8 + 70*B*a*b^9*
d^9*e + 10010*B*a^9*b*d*e^9))/(136136*e^11) + (b^7*x^8*(672*B*a^3*e^3 + 11*B*b^3*d^3 + 252*A*a^2*b*e^3 + 7*A*b
^3*d^2*e + 56*A*a*b^2*d*e^2 + 70*B*a*b^2*d^2*e + 252*B*a^2*b*d*e^2))/(56*e^4) + (b^4*x^5*(4620*B*a^6*e^6 + 11*
B*b^6*d^6 + 5544*A*a^5*b*e^6 + 7*A*b^6*d^5*e + 56*A*a*b^5*d^4*e^2 + 2310*A*a^4*b^2*d*e^5 + 252*A*a^2*b^4*d^3*e
^3 + 840*A*a^3*b^3*d^2*e^4 + 252*B*a^2*b^4*d^4*e^2 + 672*B*a^3*b^3*d^3*e^3 + 1470*B*a^4*b^2*d^2*e^4 + 70*B*a*b
^5*d^5*e + 2772*B*a^5*b*d*e^5))/(286*e^7) + (b^9*x^10*(7*A*b*e + 70*B*a*e + 11*B*b*d))/(56*e^2) + (b^6*x^7*(14
70*B*a^4*e^4 + 11*B*b^4*d^4 + 840*A*a^3*b*e^4 + 7*A*b^4*d^3*e + 56*A*a*b^3*d^2*e^2 + 252*A*a^2*b^2*d*e^3 + 252
*B*a^2*b^2*d^2*e^2 + 70*B*a*b^3*d^3*e + 672*B*a^3*b*d*e^3))/(77*e^5) + (5*b^3*x^4*(6864*B*a^7*e^7 + 11*B*b^7*d
^7 + 12012*A*a^6*b*e^7 + 7*A*b^7*d^6*e + 56*A*a*b^6*d^5*e^2 + 5544*A*a^5*b^2*d*e^6 + 252*A*a^2*b^5*d^4*e^3 + 8
40*A*a^3*b^4*d^3*e^4 + 2310*A*a^4*b^3*d^2*e^5 + 252*B*a^2*b^5*d^5*e^2 + 672*B*a^3*b^4*d^4*e^3 + 1470*B*a^4*b^3
*d^3*e^4 + 2772*B*a^5*b^2*d^2*e^5 + 70*B*a*b^6*d^6*e + 4620*B*a^6*b*d*e^6))/(4004*e^8) + (b*x^2*(10010*B*a^9*e
^9 + 11*B*b^9*d^9 + 45045*A*a^8*b*e^9 + 7*A*b^9*d^8*e + 56*A*a*b^8*d^7*e^2 + 24024*A*a^7*b^2*d*e^8 + 252*A*a^2
*b^7*d^6*e^3 + 840*A*a^3*b^6*d^5*e^4 + 2310*A*a^4*b^5*d^4*e^5 + 5544*A*a^5*b^4*d^3*e^6 + 12012*A*a^6*b^3*d^2*e
^7 + 252*B*a^2*b^7*d^7*e^2 + 672*B*a^3*b^6*d^6*e^3 + 1470*B*a^4*b^5*d^5*e^4 + 2772*B*a^5*b^4*d^4*e^5 + 4620*B*
a^6*b^3*d^3*e^6 + 6864*B*a^7*b^2*d^2*e^7 + 70*B*a*b^8*d^8*e + 9009*B*a^8*b*d*e^8))/(16016*e^10) + (5*b^8*x^9*(
252*B*a^2*e^2 + 11*B*b^2*d^2 + 56*A*a*b*e^2 + 7*A*b^2*d*e + 70*B*a*b*d*e))/(252*e^3) + (b^5*x^6*(2772*B*a^5*e^
5 + 11*B*b^5*d^5 + 2310*A*a^4*b*e^5 + 7*A*b^5*d^4*e + 56*A*a*b^4*d^3*e^2 + 840*A*a^3*b^2*d*e^4 + 252*A*a^2*b^3
*d^2*e^3 + 252*B*a^2*b^3*d^3*e^2 + 672*B*a^3*b^2*d^2*e^3 + 70*B*a*b^4*d^4*e + 1470*B*a^4*b*d*e^4))/(132*e^6) +
(b^2*x^3*(9009*B*a^8*e^8 + 11*B*b^8*d^8 + 24024*A*a^7*b*e^8 + 7*A*b^8*d^7*e + 56*A*a*b^7*d^6*e^2 + 12012*A*a^
6*b^2*d*e^7 + 252*A*a^2*b^6*d^5*e^3 + 840*A*a^3*b^5*d^4*e^4 + 2310*A*a^4*b^4*d^3*e^5 + 5544*A*a^5*b^3*d^2*e^6
+ 252*B*a^2*b^6*d^6*e^2 + 672*B*a^3*b^5*d^5*e^3 + 1470*B*a^4*b^4*d^4*e^4 + 2772*B*a^5*b^3*d^3*e^5 + 4620*B*a^6
*b^2*d^2*e^6 + 70*B*a*b^7*d^7*e + 6864*B*a^7*b*d*e^7))/(3003*e^9) + (B*b^10*x^11)/(7*e))/(d^18 + e^18*x^18 + 1
8*d*e^17*x^17 + 153*d^16*e^2*x^2 + 816*d^15*e^3*x^3 + 3060*d^14*e^4*x^4 + 8568*d^13*e^5*x^5 + 18564*d^12*e^6*x
^6 + 31824*d^11*e^7*x^7 + 43758*d^10*e^8*x^8 + 48620*d^9*e^9*x^9 + 43758*d^8*e^10*x^10 + 31824*d^7*e^11*x^11 +
18564*d^6*e^12*x^12 + 8568*d^5*e^13*x^13 + 3060*d^4*e^14*x^14 + 816*d^3*e^15*x^15 + 153*d^2*e^16*x^16 + 18*d^
17*e*x)