∫ (a+bx)10(A+Bx)___________

3.1107 \(\int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{19}} \, dx\)

Optimal. Leaf size=385 \[ \frac {b^6 (a+b x)^{11} (-18 a B e+7 A b e+11 b B d)}{2450448 e (d+e x)^{11} (b d-a e)^8}+\frac {b^5 (a+b x)^{11} (-18 a B e+7 A b e+11 b B d)}{222768 e (d+e x)^{12} (b d-a e)^7}+\frac {b^4 (a+b x)^{11} (-18 a B e+7 A b e+11 b B d)}{37128 e (d+e x)^{13} (b d-a e)^6}+\frac {b^3 (a+b x)^{11} (-18 a B e+7 A b e+11 b B d)}{8568 e (d+e x)^{14} (b d-a e)^5}+\frac {b^2 (a+b x)^{11} (-18 a B e+7 A b e+11 b B d)}{2448 e (d+e x)^{15} (b d-a e)^4}+\frac {b (a+b x)^{11} (-18 a B e+7 A b e+11 b B d)}{816 e (d+e x)^{16} (b d-a e)^3}+\frac {(a+b x)^{11} (-18 a B e+7 A b e+11 b B d)}{306 e (d+e x)^{17} (b d-a e)^2}-\frac {(a+b x)^{11} (B d-A e)}{18 e (d+e x)^{18} (b d-a e)} \]

[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

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Rubi [A] time = 0.21, antiderivative size = 385, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {78, 45, 37} \[ \frac {b^6 (a+b x)^{11} (-18 a B e+7 A b e+11 b B d)}{2450448 e (d+e x)^{11} (b d-a e)^8}+\frac {b^5 (a+b x)^{11} (-18 a B e+7 A b e+11 b B d)}{222768 e (d+e x)^{12} (b d-a e)^7}+\frac {b^4 (a+b x)^{11} (-18 a B e+7 A b e+11 b B d)}{37128 e (d+e x)^{13} (b d-a e)^6}+\frac {b^3 (a+b x)^{11} (-18 a B e+7 A b e+11 b B d)}{8568 e (d+e x)^{14} (b d-a e)^5}+\frac {b^2 (a+b x)^{11} (-18 a B e+7 A b e+11 b B d)}{2448 e (d+e x)^{15} (b d-a e)^4}+\frac {b (a+b x)^{11} (-18 a B e+7 A b e+11 b B d)}{816 e (d+e x)^{16} (b d-a e)^3}+\frac {(a+b x)^{11} (-18 a B e+7 A b e+11 b B d)}{306 e (d+e x)^{17} (b d-a e)^2}-\frac {(a+b x)^{11} (B d-A e)}{18 e (d+e x)^{18} (b d-a e)} \]

Antiderivative was successfully veriﬁed.

[In]

Int[((a + b*x)^10*(A + B*x))/(d + e*x)^19,x]

[Out]

-((B*d - A*e)*(a + b*x)^11)/(18*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*(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) + (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) + (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) + (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 45

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 78

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] || LtQ

[p, n]))))

Rubi steps

\begin {align*} \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) \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*}

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Mathematica [B] time = 1.00, size = 1428, normalized size = 3.71 \[ -\frac {\left (7 A e \left (d^{10}+18 e x d^9+153 e^2 x^2 d^8+816 e^3 x^3 d^7+3060 e^4 x^4 d^6+8568 e^5 x^5 d^5+18564 e^6 x^6 d^4+31824 e^7 x^7 d^3+43758 e^8 x^8 d^2+48620 e^9 x^9 d+43758 e^{10} x^{10}\right )+11 B \left (d^{11}+18 e x d^{10}+153 e^2 x^2 d^9+816 e^3 x^3 d^8+3060 e^4 x^4 d^7+8568 e^5 x^5 d^6+18564 e^6 x^6 d^5+31824 e^7 x^7 d^4+43758 e^8 x^8 d^3+48620 e^9 x^9 d^2+43758 e^{10} x^{10} d+31824 e^{11} x^{11}\right )\right ) b^{10}+14 a e \left (4 A e \left (d^9+18 e x d^8+153 e^2 x^2 d^7+816 e^3 x^3 d^6+3060 e^4 x^4 d^5+8568 e^5 x^5 d^4+18564 e^6 x^6 d^3+31824 e^7 x^7 d^2+43758 e^8 x^8 d+48620 e^9 x^9\right )+5 B \left (d^{10}+18 e x d^9+153 e^2 x^2 d^8+816 e^3 x^3 d^7+3060 e^4 x^4 d^6+8568 e^5 x^5 d^5+18564 e^6 x^6 d^4+31824 e^7 x^7 d^3+43758 e^8 x^8 d^2+48620 e^9 x^9 d+43758 e^{10} x^{10}\right )\right ) b^9+252 a^2 e^2 \left (A e \left (d^8+18 e x d^7+153 e^2 x^2 d^6+816 e^3 x^3 d^5+3060 e^4 x^4 d^4+8568 e^5 x^5 d^3+18564 e^6 x^6 d^2+31824 e^7 x^7 d+43758 e^8 x^8\right )+B \left (d^9+18 e x d^8+153 e^2 x^2 d^7+816 e^3 x^3 d^6+3060 e^4 x^4 d^5+8568 e^5 x^5 d^4+18564 e^6 x^6 d^3+31824 e^7 x^7 d^2+43758 e^8 x^8 d+48620 e^9 x^9\right )\right ) b^8+168 a^3 e^3 \left (5 A e \left (d^7+18 e x d^6+153 e^2 x^2 d^5+816 e^3 x^3 d^4+3060 e^4 x^4 d^3+8568 e^5 x^5 d^2+18564 e^6 x^6 d+31824 e^7 x^7\right )+4 B \left (d^8+18 e x d^7+153 e^2 x^2 d^6+816 e^3 x^3 d^5+3060 e^4 x^4 d^4+8568 e^5 x^5 d^3+18564 e^6 x^6 d^2+31824 e^7 x^7 d+43758 e^8 x^8\right )\right ) b^7+210 a^4 e^4 \left (11 A e \left (d^6+18 e x d^5+153 e^2 x^2 d^4+816 e^3 x^3 d^3+3060 e^4 x^4 d^2+8568 e^5 x^5 d+18564 e^6 x^6\right )+7 B \left (d^7+18 e x d^6+153 e^2 x^2 d^5+816 e^3 x^3 d^4+3060 e^4 x^4 d^3+8568 e^5 x^5 d^2+18564 e^6 x^6 d+31824 e^7 x^7\right )\right ) b^6+2772 a^5 e^5 \left (2 A e \left (d^5+18 e x d^4+153 e^2 x^2 d^3+816 e^3 x^3 d^2+3060 e^4 x^4 d+8568 e^5 x^5\right )+B \left (d^6+18 e x d^5+153 e^2 x^2 d^4+816 e^3 x^3 d^3+3060 e^4 x^4 d^2+8568 e^5 x^5 d+18564 e^6 x^6\right )\right ) b^5+924 a^6 e^6 \left (13 A e \left (d^4+18 e x d^3+153 e^2 x^2 d^2+816 e^3 x^3 d+3060 e^4 x^4\right )+5 B \left (d^5+18 e x d^4+153 e^2 x^2 d^3+816 e^3 x^3 d^2+3060 e^4 x^4 d+8568 e^5 x^5\right )\right ) b^4+3432 a^7 e^7 \left (7 A e \left (d^3+18 e x d^2+153 e^2 x^2 d+816 e^3 x^3\right )+2 B \left (d^4+18 e x d^3+153 e^2 x^2 d^2+816 e^3 x^3 d+3060 e^4 x^4\right )\right ) b^3+9009 a^8 e^8 \left (5 A e \left (d^2+18 e x d+153 e^2 x^2\right )+B \left (d^3+18 e x d^2+153 e^2 x^2 d+816 e^3 x^3\right )\right ) b^2+10010 a^9 e^9 \left (8 A e (d+18 e x)+B \left (d^2+18 e x d+153 e^2 x^2\right )\right ) b+8008 a^{10} e^{10} (17 A e+B (d+18 e x))}{2450448 e^{12} (d+e x)^{18}} \]

Antiderivative was successfully veriﬁed.

[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)

________________________________________________________________________________________

fricas [B] time = 0.89, size = 2006, normalized size = 5.21 \[ \text {result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

giac [B] time = 1.45, size = 2096, normalized size = 5.44 \[ \text {result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10*(B*x+A)/(e*x+d)^19,x, algorithm="giac")

[Out]

-1/2450448*(350064*B*b^10*x^11*e^11 + 481338*B*b^10*d*x^10*e^10 + 534820*B*b^10*d^2*x^9*e^9 + 481338*B*b^10*d^

3*x^8*e^8 + 350064*B*b^10*d^4*x^7*e^7 + 204204*B*b^10*d^5*x^6*e^6 + 94248*B*b^10*d^6*x^5*e^5 + 33660*B*b^10*d^

7*x^4*e^4 + 8976*B*b^10*d^8*x^3*e^3 + 1683*B*b^10*d^9*x^2*e^2 + 198*B*b^10*d^10*x*e + 11*B*b^10*d^11 + 3063060

*B*a*b^9*x^10*e^11 + 306306*A*b^10*x^10*e^11 + 3403400*B*a*b^9*d*x^9*e^10 + 340340*A*b^10*d*x^9*e^10 + 3063060

*B*a*b^9*d^2*x^8*e^9 + 306306*A*b^10*d^2*x^8*e^9 + 2227680*B*a*b^9*d^3*x^7*e^8 + 222768*A*b^10*d^3*x^7*e^8 + 1

299480*B*a*b^9*d^4*x^6*e^7 + 129948*A*b^10*d^4*x^6*e^7 + 599760*B*a*b^9*d^5*x^5*e^6 + 59976*A*b^10*d^5*x^5*e^6

+ 214200*B*a*b^9*d^6*x^4*e^5 + 21420*A*b^10*d^6*x^4*e^5 + 57120*B*a*b^9*d^7*x^3*e^4 + 5712*A*b^10*d^7*x^3*e^4

+ 10710*B*a*b^9*d^8*x^2*e^3 + 1071*A*b^10*d^8*x^2*e^3 + 1260*B*a*b^9*d^9*x*e^2 + 126*A*b^10*d^9*x*e^2 + 70*B*

a*b^9*d^10*e + 7*A*b^10*d^10*e + 12252240*B*a^2*b^8*x^9*e^11 + 2722720*A*a*b^9*x^9*e^11 + 11027016*B*a^2*b^8*d

*x^8*e^10 + 2450448*A*a*b^9*d*x^8*e^10 + 8019648*B*a^2*b^8*d^2*x^7*e^9 + 1782144*A*a*b^9*d^2*x^7*e^9 + 4678128

*B*a^2*b^8*d^3*x^6*e^8 + 1039584*A*a*b^9*d^3*x^6*e^8 + 2159136*B*a^2*b^8*d^4*x^5*e^7 + 479808*A*a*b^9*d^4*x^5*

e^7 + 771120*B*a^2*b^8*d^5*x^4*e^6 + 171360*A*a*b^9*d^5*x^4*e^6 + 205632*B*a^2*b^8*d^6*x^3*e^5 + 45696*A*a*b^9

*d^6*x^3*e^5 + 38556*B*a^2*b^8*d^7*x^2*e^4 + 8568*A*a*b^9*d^7*x^2*e^4 + 4536*B*a^2*b^8*d^8*x*e^3 + 1008*A*a*b^

9*d^8*x*e^3 + 252*B*a^2*b^8*d^9*e^2 + 56*A*a*b^9*d^9*e^2 + 29405376*B*a^3*b^7*x^8*e^11 + 11027016*A*a^2*b^8*x^

8*e^11 + 21385728*B*a^3*b^7*d*x^7*e^10 + 8019648*A*a^2*b^8*d*x^7*e^10 + 12475008*B*a^3*b^7*d^2*x^6*e^9 + 46781

28*A*a^2*b^8*d^2*x^6*e^9 + 5757696*B*a^3*b^7*d^3*x^5*e^8 + 2159136*A*a^2*b^8*d^3*x^5*e^8 + 2056320*B*a^3*b^7*d

^4*x^4*e^7 + 771120*A*a^2*b^8*d^4*x^4*e^7 + 548352*B*a^3*b^7*d^5*x^3*e^6 + 205632*A*a^2*b^8*d^5*x^3*e^6 + 1028

16*B*a^3*b^7*d^6*x^2*e^5 + 38556*A*a^2*b^8*d^6*x^2*e^5 + 12096*B*a^3*b^7*d^7*x*e^4 + 4536*A*a^2*b^8*d^7*x*e^4

+ 672*B*a^3*b^7*d^8*e^3 + 252*A*a^2*b^8*d^8*e^3 + 46781280*B*a^4*b^6*x^7*e^11 + 26732160*A*a^3*b^7*x^7*e^11 +

27289080*B*a^4*b^6*d*x^6*e^10 + 15593760*A*a^3*b^7*d*x^6*e^10 + 12594960*B*a^4*b^6*d^2*x^5*e^9 + 7197120*A*a^3

*b^7*d^2*x^5*e^9 + 4498200*B*a^4*b^6*d^3*x^4*e^8 + 2570400*A*a^3*b^7*d^3*x^4*e^8 + 1199520*B*a^4*b^6*d^4*x^3*e

^7 + 685440*A*a^3*b^7*d^4*x^3*e^7 + 224910*B*a^4*b^6*d^5*x^2*e^6 + 128520*A*a^3*b^7*d^5*x^2*e^6 + 26460*B*a^4*

b^6*d^6*x*e^5 + 15120*A*a^3*b^7*d^6*x*e^5 + 1470*B*a^4*b^6*d^7*e^4 + 840*A*a^3*b^7*d^7*e^4 + 51459408*B*a^5*b^

5*x^6*e^11 + 42882840*A*a^4*b^6*x^6*e^11 + 23750496*B*a^5*b^5*d*x^5*e^10 + 19792080*A*a^4*b^6*d*x^5*e^10 + 848

2320*B*a^5*b^5*d^2*x^4*e^9 + 7068600*A*a^4*b^6*d^2*x^4*e^9 + 2261952*B*a^5*b^5*d^3*x^3*e^8 + 1884960*A*a^4*b^6

*d^3*x^3*e^8 + 424116*B*a^5*b^5*d^4*x^2*e^7 + 353430*A*a^4*b^6*d^4*x^2*e^7 + 49896*B*a^5*b^5*d^5*x*e^6 + 41580

*A*a^4*b^6*d^5*x*e^6 + 2772*B*a^5*b^5*d^6*e^5 + 2310*A*a^4*b^6*d^6*e^5 + 39584160*B*a^6*b^4*x^5*e^11 + 4750099

2*A*a^5*b^5*x^5*e^11 + 14137200*B*a^6*b^4*d*x^4*e^10 + 16964640*A*a^5*b^5*d*x^4*e^10 + 3769920*B*a^6*b^4*d^2*x

^3*e^9 + 4523904*A*a^5*b^5*d^2*x^3*e^9 + 706860*B*a^6*b^4*d^3*x^2*e^8 + 848232*A*a^5*b^5*d^3*x^2*e^8 + 83160*B

*a^6*b^4*d^4*x*e^7 + 99792*A*a^5*b^5*d^4*x*e^7 + 4620*B*a^6*b^4*d^5*e^6 + 5544*A*a^5*b^5*d^5*e^6 + 21003840*B*

a^7*b^3*x^4*e^11 + 36756720*A*a^6*b^4*x^4*e^11 + 5601024*B*a^7*b^3*d*x^3*e^10 + 9801792*A*a^6*b^4*d*x^3*e^10 +

1050192*B*a^7*b^3*d^2*x^2*e^9 + 1837836*A*a^6*b^4*d^2*x^2*e^9 + 123552*B*a^7*b^3*d^3*x*e^8 + 216216*A*a^6*b^4

*d^3*x*e^8 + 6864*B*a^7*b^3*d^4*e^7 + 12012*A*a^6*b^4*d^4*e^7 + 7351344*B*a^8*b^2*x^3*e^11 + 19603584*A*a^7*b^

3*x^3*e^11 + 1378377*B*a^8*b^2*d*x^2*e^10 + 3675672*A*a^7*b^3*d*x^2*e^10 + 162162*B*a^8*b^2*d^2*x*e^9 + 432432

*A*a^7*b^3*d^2*x*e^9 + 9009*B*a^8*b^2*d^3*e^8 + 24024*A*a^7*b^3*d^3*e^8 + 1531530*B*a^9*b*x^2*e^11 + 6891885*A

*a^8*b^2*x^2*e^11 + 180180*B*a^9*b*d*x*e^10 + 810810*A*a^8*b^2*d*x*e^10 + 10010*B*a^9*b*d^2*e^9 + 45045*A*a^8*

b^2*d^2*e^9 + 144144*B*a^10*x*e^11 + 1441440*A*a^9*b*x*e^11 + 8008*B*a^10*d*e^10 + 80080*A*a^9*b*d*e^10 + 1361

36*A*a^10*e^11)*e^(-12)/(x*e + d)^18

________________________________________________________________________________________

maple [B] time = 0.01, size = 1942, normalized size = 5.04 \[ \text {result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^10*(B*x+A)/(e*x+d)^19,x)

[Out]

-1/17*(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^6-1

260*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^12/(e*x

+d)^17-b^2*(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^12/(e*x+d)^15-42/13*b^4*(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^12/(e*x+d)^13-1/18*(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)^18-15/7*b^3*(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^12/(e*x+d)^14-1/8*b^9*(A*b*e+10*B*a*e-11*B*b*d)/e^12/(e*x+d)^8-5/16*b*(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^12/(e*x+d)^16-7/2*b^5*(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^12/(e*x+d)^12-30/11*b^6*(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^12/(e*x+d)^11

-3/2*b^7*(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^12/(e*x+d)^10-5/9*b^8*(2*A*a*b*e^2-2*A*b^2*d*e+9*B*a^2*e^2-20*B*a*b*d*e+11*B*b^2*d^2)/e^12/(e*x+d)^9-1/7

*B*b^10/e^12/(e*x+d)^7

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maxima [B] time = 1.49, size = 2006, normalized size = 5.21 \[ \text {result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10*(B*x+A)/(e*x+d)^19,x, algorithm="maxima")