∫ (a+bx)10(A+Bx)___________

3.10.66 \(\int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{17}} \, dx\)

Optimal. Leaf size=285 \[ \frac {b^4 (a+b x)^{11} (-16 a B e+5 A b e+11 b B d)}{240240 e (d+e x)^{11} (b d-a e)^6}+\frac {b^3 (a+b x)^{11} (-16 a B e+5 A b e+11 b B d)}{21840 e (d+e x)^{12} (b d-a e)^5}+\frac {b^2 (a+b x)^{11} (-16 a B e+5 A b e+11 b B d)}{3640 e (d+e x)^{13} (b d-a e)^4}+\frac {b (a+b x)^{11} (-16 a B e+5 A b e+11 b B d)}{840 e (d+e x)^{14} (b d-a e)^3}+\frac {(a+b x)^{11} (-16 a B e+5 A b e+11 b B d)}{240 e (d+e x)^{15} (b d-a e)^2}-\frac {(a+b x)^{11} (B d-A e)}{16 e (d+e x)^{16} (b d-a e)} \]

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Rubi [A] time = 0.15, antiderivative size = 285, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {78, 45, 37} \begin {gather*} \frac {b^4 (a+b x)^{11} (-16 a B e+5 A b e+11 b B d)}{240240 e (d+e x)^{11} (b d-a e)^6}+\frac {b^3 (a+b x)^{11} (-16 a B e+5 A b e+11 b B d)}{21840 e (d+e x)^{12} (b d-a e)^5}+\frac {b^2 (a+b x)^{11} (-16 a B e+5 A b e+11 b B d)}{3640 e (d+e x)^{13} (b d-a e)^4}+\frac {b (a+b x)^{11} (-16 a B e+5 A b e+11 b B d)}{840 e (d+e x)^{14} (b d-a e)^3}+\frac {(a+b x)^{11} (-16 a B e+5 A b e+11 b B d)}{240 e (d+e x)^{15} (b d-a e)^2}-\frac {(a+b x)^{11} (B d-A e)}{16 e (d+e x)^{16} (b d-a e)} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

Int[((a + b*x)^10*(A + B*x))/(d + e*x)^17,x]

[Out]

-((B*d - A*e)*(a + b*x)^11)/(16*e*(b*d - a*e)*(d + e*x)^16) + ((11*b*B*d + 5*A*b*e - 16*a*B*e)*(a + b*x)^11)/(

240*e*(b*d - a*e)^2*(d + e*x)^15) + (b*(11*b*B*d + 5*A*b*e - 16*a*B*e)*(a + b*x)^11)/(840*e*(b*d - a*e)^3*(d +

e*x)^14) + (b^2*(11*b*B*d + 5*A*b*e - 16*a*B*e)*(a + b*x)^11)/(3640*e*(b*d - a*e)^4*(d + e*x)^13) + (b^3*(11*

b*B*d + 5*A*b*e - 16*a*B*e)*(a + b*x)^11)/(21840*e*(b*d - a*e)^5*(d + e*x)^12) + (b^4*(11*b*B*d + 5*A*b*e - 16

*a*B*e)*(a + b*x)^11)/(240240*e*(b*d - a*e)^6*(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)^{17}} \, dx &=-\frac {(B d-A e) (a+b x)^{11}}{16 e (b d-a e) (d+e x)^{16}}+\frac {(11 b B d+5 A b e-16 a B e) \int \frac {(a+b x)^{10}}{(d+e x)^{16}} \, dx}{16 e (b d-a e)}\\ &=-\frac {(B d-A e) (a+b x)^{11}}{16 e (b d-a e) (d+e x)^{16}}+\frac {(11 b B d+5 A b e-16 a B e) (a+b x)^{11}}{240 e (b d-a e)^2 (d+e x)^{15}}+\frac {(b (11 b B d+5 A b e-16 a B e)) \int \frac {(a+b x)^{10}}{(d+e x)^{15}} \, dx}{60 e (b d-a e)^2}\\ &=-\frac {(B d-A e) (a+b x)^{11}}{16 e (b d-a e) (d+e x)^{16}}+\frac {(11 b B d+5 A b e-16 a B e) (a+b x)^{11}}{240 e (b d-a e)^2 (d+e x)^{15}}+\frac {b (11 b B d+5 A b e-16 a B e) (a+b x)^{11}}{840 e (b d-a e)^3 (d+e x)^{14}}+\frac {\left (b^2 (11 b B d+5 A b e-16 a B e)\right ) \int \frac {(a+b x)^{10}}{(d+e x)^{14}} \, dx}{280 e (b d-a e)^3}\\ &=-\frac {(B d-A e) (a+b x)^{11}}{16 e (b d-a e) (d+e x)^{16}}+\frac {(11 b B d+5 A b e-16 a B e) (a+b x)^{11}}{240 e (b d-a e)^2 (d+e x)^{15}}+\frac {b (11 b B d+5 A b e-16 a B e) (a+b x)^{11}}{840 e (b d-a e)^3 (d+e x)^{14}}+\frac {b^2 (11 b B d+5 A b e-16 a B e) (a+b x)^{11}}{3640 e (b d-a e)^4 (d+e x)^{13}}+\frac {\left (b^3 (11 b B d+5 A b e-16 a B e)\right ) \int \frac {(a+b x)^{10}}{(d+e x)^{13}} \, dx}{1820 e (b d-a e)^4}\\ &=-\frac {(B d-A e) (a+b x)^{11}}{16 e (b d-a e) (d+e x)^{16}}+\frac {(11 b B d+5 A b e-16 a B e) (a+b x)^{11}}{240 e (b d-a e)^2 (d+e x)^{15}}+\frac {b (11 b B d+5 A b e-16 a B e) (a+b x)^{11}}{840 e (b d-a e)^3 (d+e x)^{14}}+\frac {b^2 (11 b B d+5 A b e-16 a B e) (a+b x)^{11}}{3640 e (b d-a e)^4 (d+e x)^{13}}+\frac {b^3 (11 b B d+5 A b e-16 a B e) (a+b x)^{11}}{21840 e (b d-a e)^5 (d+e x)^{12}}+\frac {\left (b^4 (11 b B d+5 A b e-16 a B e)\right ) \int \frac {(a+b x)^{10}}{(d+e x)^{12}} \, dx}{21840 e (b d-a e)^5}\\ &=-\frac {(B d-A e) (a+b x)^{11}}{16 e (b d-a e) (d+e x)^{16}}+\frac {(11 b B d+5 A b e-16 a B e) (a+b x)^{11}}{240 e (b d-a e)^2 (d+e x)^{15}}+\frac {b (11 b B d+5 A b e-16 a B e) (a+b x)^{11}}{840 e (b d-a e)^3 (d+e x)^{14}}+\frac {b^2 (11 b B d+5 A b e-16 a B e) (a+b x)^{11}}{3640 e (b d-a e)^4 (d+e x)^{13}}+\frac {b^3 (11 b B d+5 A b e-16 a B e) (a+b x)^{11}}{21840 e (b d-a e)^5 (d+e x)^{12}}+\frac {b^4 (11 b B d+5 A b e-16 a B e) (a+b x)^{11}}{240240 e (b d-a e)^6 (d+e x)^{11}}\\ \end {align*}

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Mathematica [B] time = 0.97, size = 1429, normalized size = 5.01 \begin {gather*} -\frac {\left (5 A e \left (d^{10}+16 e x d^9+120 e^2 x^2 d^8+560 e^3 x^3 d^7+1820 e^4 x^4 d^6+4368 e^5 x^5 d^5+8008 e^6 x^6 d^4+11440 e^7 x^7 d^3+12870 e^8 x^8 d^2+11440 e^9 x^9 d+8008 e^{10} x^{10}\right )+11 B \left (d^{11}+16 e x d^{10}+120 e^2 x^2 d^9+560 e^3 x^3 d^8+1820 e^4 x^4 d^7+4368 e^5 x^5 d^6+8008 e^6 x^6 d^5+11440 e^7 x^7 d^4+12870 e^8 x^8 d^3+11440 e^9 x^9 d^2+8008 e^{10} x^{10} d+4368 e^{11} x^{11}\right )\right ) b^{10}+10 a e \left (3 A e \left (d^9+16 e x d^8+120 e^2 x^2 d^7+560 e^3 x^3 d^6+1820 e^4 x^4 d^5+4368 e^5 x^5 d^4+8008 e^6 x^6 d^3+11440 e^7 x^7 d^2+12870 e^8 x^8 d+11440 e^9 x^9\right )+5 B \left (d^{10}+16 e x d^9+120 e^2 x^2 d^8+560 e^3 x^3 d^7+1820 e^4 x^4 d^6+4368 e^5 x^5 d^5+8008 e^6 x^6 d^4+11440 e^7 x^7 d^3+12870 e^8 x^8 d^2+11440 e^9 x^9 d+8008 e^{10} x^{10}\right )\right ) b^9+15 a^2 e^2 \left (7 A e \left (d^8+16 e x d^7+120 e^2 x^2 d^6+560 e^3 x^3 d^5+1820 e^4 x^4 d^4+4368 e^5 x^5 d^3+8008 e^6 x^6 d^2+11440 e^7 x^7 d+12870 e^8 x^8\right )+9 B \left (d^9+16 e x d^8+120 e^2 x^2 d^7+560 e^3 x^3 d^6+1820 e^4 x^4 d^5+4368 e^5 x^5 d^4+8008 e^6 x^6 d^3+11440 e^7 x^7 d^2+12870 e^8 x^8 d+11440 e^9 x^9\right )\right ) b^8+280 a^3 e^3 \left (A e \left (d^7+16 e x d^6+120 e^2 x^2 d^5+560 e^3 x^3 d^4+1820 e^4 x^4 d^3+4368 e^5 x^5 d^2+8008 e^6 x^6 d+11440 e^7 x^7\right )+B \left (d^8+16 e x d^7+120 e^2 x^2 d^6+560 e^3 x^3 d^5+1820 e^4 x^4 d^4+4368 e^5 x^5 d^3+8008 e^6 x^6 d^2+11440 e^7 x^7 d+12870 e^8 x^8\right )\right ) b^7+70 a^4 e^4 \left (9 A e \left (d^6+16 e x d^5+120 e^2 x^2 d^4+560 e^3 x^3 d^3+1820 e^4 x^4 d^2+4368 e^5 x^5 d+8008 e^6 x^6\right )+7 B \left (d^7+16 e x d^6+120 e^2 x^2 d^5+560 e^3 x^3 d^4+1820 e^4 x^4 d^3+4368 e^5 x^5 d^2+8008 e^6 x^6 d+11440 e^7 x^7\right )\right ) b^6+252 a^5 e^5 \left (5 A e \left (d^5+16 e x d^4+120 e^2 x^2 d^3+560 e^3 x^3 d^2+1820 e^4 x^4 d+4368 e^5 x^5\right )+3 B \left (d^6+16 e x d^5+120 e^2 x^2 d^4+560 e^3 x^3 d^3+1820 e^4 x^4 d^2+4368 e^5 x^5 d+8008 e^6 x^6\right )\right ) b^5+210 a^6 e^6 \left (11 A e \left (d^4+16 e x d^3+120 e^2 x^2 d^2+560 e^3 x^3 d+1820 e^4 x^4\right )+5 B \left (d^5+16 e x d^4+120 e^2 x^2 d^3+560 e^3 x^3 d^2+1820 e^4 x^4 d+4368 e^5 x^5\right )\right ) b^4+1320 a^7 e^7 \left (3 A e \left (d^3+16 e x d^2+120 e^2 x^2 d+560 e^3 x^3\right )+B \left (d^4+16 e x d^3+120 e^2 x^2 d^2+560 e^3 x^3 d+1820 e^4 x^4\right )\right ) b^3+495 a^8 e^8 \left (13 A e \left (d^2+16 e x d+120 e^2 x^2\right )+3 B \left (d^3+16 e x d^2+120 e^2 x^2 d+560 e^3 x^3\right )\right ) b^2+1430 a^9 e^9 \left (7 A e (d+16 e x)+B \left (d^2+16 e x d+120 e^2 x^2\right )\right ) b+1001 a^{10} e^{10} (15 A e+B (d+16 e x))}{240240 e^{12} (d+e x)^{16}} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

Integrate[((a + b*x)^10*(A + B*x))/(d + e*x)^17,x]

[Out]

-1/240240*(1001*a^10*e^10*(15*A*e + B*(d + 16*e*x)) + 1430*a^9*b*e^9*(7*A*e*(d + 16*e*x) + B*(d^2 + 16*d*e*x +

120*e^2*x^2)) + 495*a^8*b^2*e^8*(13*A*e*(d^2 + 16*d*e*x + 120*e^2*x^2) + 3*B*(d^3 + 16*d^2*e*x + 120*d*e^2*x^

2 + 560*e^3*x^3)) + 1320*a^7*b^3*e^7*(3*A*e*(d^3 + 16*d^2*e*x + 120*d*e^2*x^2 + 560*e^3*x^3) + B*(d^4 + 16*d^3

*e*x + 120*d^2*e^2*x^2 + 560*d*e^3*x^3 + 1820*e^4*x^4)) + 210*a^6*b^4*e^6*(11*A*e*(d^4 + 16*d^3*e*x + 120*d^2*

e^2*x^2 + 560*d*e^3*x^3 + 1820*e^4*x^4) + 5*B*(d^5 + 16*d^4*e*x + 120*d^3*e^2*x^2 + 560*d^2*e^3*x^3 + 1820*d*e

^4*x^4 + 4368*e^5*x^5)) + 252*a^5*b^5*e^5*(5*A*e*(d^5 + 16*d^4*e*x + 120*d^3*e^2*x^2 + 560*d^2*e^3*x^3 + 1820*

d*e^4*x^4 + 4368*e^5*x^5) + 3*B*(d^6 + 16*d^5*e*x + 120*d^4*e^2*x^2 + 560*d^3*e^3*x^3 + 1820*d^2*e^4*x^4 + 436

8*d*e^5*x^5 + 8008*e^6*x^6)) + 70*a^4*b^6*e^4*(9*A*e*(d^6 + 16*d^5*e*x + 120*d^4*e^2*x^2 + 560*d^3*e^3*x^3 + 1

820*d^2*e^4*x^4 + 4368*d*e^5*x^5 + 8008*e^6*x^6) + 7*B*(d^7 + 16*d^6*e*x + 120*d^5*e^2*x^2 + 560*d^4*e^3*x^3 +

1820*d^3*e^4*x^4 + 4368*d^2*e^5*x^5 + 8008*d*e^6*x^6 + 11440*e^7*x^7)) + 280*a^3*b^7*e^3*(A*e*(d^7 + 16*d^6*e

*x + 120*d^5*e^2*x^2 + 560*d^4*e^3*x^3 + 1820*d^3*e^4*x^4 + 4368*d^2*e^5*x^5 + 8008*d*e^6*x^6 + 11440*e^7*x^7)

+ B*(d^8 + 16*d^7*e*x + 120*d^6*e^2*x^2 + 560*d^5*e^3*x^3 + 1820*d^4*e^4*x^4 + 4368*d^3*e^5*x^5 + 8008*d^2*e^

6*x^6 + 11440*d*e^7*x^7 + 12870*e^8*x^8)) + 15*a^2*b^8*e^2*(7*A*e*(d^8 + 16*d^7*e*x + 120*d^6*e^2*x^2 + 560*d^

5*e^3*x^3 + 1820*d^4*e^4*x^4 + 4368*d^3*e^5*x^5 + 8008*d^2*e^6*x^6 + 11440*d*e^7*x^7 + 12870*e^8*x^8) + 9*B*(d

^9 + 16*d^8*e*x + 120*d^7*e^2*x^2 + 560*d^6*e^3*x^3 + 1820*d^5*e^4*x^4 + 4368*d^4*e^5*x^5 + 8008*d^3*e^6*x^6 +

11440*d^2*e^7*x^7 + 12870*d*e^8*x^8 + 11440*e^9*x^9)) + 10*a*b^9*e*(3*A*e*(d^9 + 16*d^8*e*x + 120*d^7*e^2*x^2

+ 560*d^6*e^3*x^3 + 1820*d^5*e^4*x^4 + 4368*d^4*e^5*x^5 + 8008*d^3*e^6*x^6 + 11440*d^2*e^7*x^7 + 12870*d*e^8*

x^8 + 11440*e^9*x^9) + 5*B*(d^10 + 16*d^9*e*x + 120*d^8*e^2*x^2 + 560*d^7*e^3*x^3 + 1820*d^6*e^4*x^4 + 4368*d^

5*e^5*x^5 + 8008*d^4*e^6*x^6 + 11440*d^3*e^7*x^7 + 12870*d^2*e^8*x^8 + 11440*d*e^9*x^9 + 8008*e^10*x^10)) + b^

10*(5*A*e*(d^10 + 16*d^9*e*x + 120*d^8*e^2*x^2 + 560*d^7*e^3*x^3 + 1820*d^6*e^4*x^4 + 4368*d^5*e^5*x^5 + 8008*

d^4*e^6*x^6 + 11440*d^3*e^7*x^7 + 12870*d^2*e^8*x^8 + 11440*d*e^9*x^9 + 8008*e^10*x^10) + 11*B*(d^11 + 16*d^10

*e*x + 120*d^9*e^2*x^2 + 560*d^8*e^3*x^3 + 1820*d^7*e^4*x^4 + 4368*d^6*e^5*x^5 + 8008*d^5*e^6*x^6 + 11440*d^4*

e^7*x^7 + 12870*d^3*e^8*x^8 + 11440*d^2*e^9*x^9 + 8008*d*e^10*x^10 + 4368*e^11*x^11)))/(e^12*(d + e*x)^16)

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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{17}} \, dx \end {gather*}

Veriﬁcation is not applicable to the result.

[In]

IntegrateAlgebraic[((a + b*x)^10*(A + B*x))/(d + e*x)^17,x]

[Out]

IntegrateAlgebraic[((a + b*x)^10*(A + B*x))/(d + e*x)^17, x]

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fricas [B] time = 1.37, size = 1984, normalized size = 6.96

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)^17,x, algorithm="fricas")

[Out]

-1/240240*(48048*B*b^10*e^11*x^11 + 11*B*b^10*d^11 + 15015*A*a^10*e^11 + 5*(10*B*a*b^9 + A*b^10)*d^10*e + 15*(

9*B*a^2*b^8 + 2*A*a*b^9)*d^9*e^2 + 35*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^8*e^3 + 70*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^7

*e^4 + 126*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^6*e^5 + 210*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^5*e^6 + 330*(4*B*a^7*b^3 +

7*A*a^6*b^4)*d^4*e^7 + 495*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^3*e^8 + 715*(2*B*a^9*b + 9*A*a^8*b^2)*d^2*e^9 + 1001*

(B*a^10 + 10*A*a^9*b)*d*e^10 + 8008*(11*B*b^10*d*e^10 + 5*(10*B*a*b^9 + A*b^10)*e^11)*x^10 + 11440*(11*B*b^10*

d^2*e^9 + 5*(10*B*a*b^9 + A*b^10)*d*e^10 + 15*(9*B*a^2*b^8 + 2*A*a*b^9)*e^11)*x^9 + 12870*(11*B*b^10*d^3*e^8 +

5*(10*B*a*b^9 + A*b^10)*d^2*e^9 + 15*(9*B*a^2*b^8 + 2*A*a*b^9)*d*e^10 + 35*(8*B*a^3*b^7 + 3*A*a^2*b^8)*e^11)*

x^8 + 11440*(11*B*b^10*d^4*e^7 + 5*(10*B*a*b^9 + A*b^10)*d^3*e^8 + 15*(9*B*a^2*b^8 + 2*A*a*b^9)*d^2*e^9 + 35*(

8*B*a^3*b^7 + 3*A*a^2*b^8)*d*e^10 + 70*(7*B*a^4*b^6 + 4*A*a^3*b^7)*e^11)*x^7 + 8008*(11*B*b^10*d^5*e^6 + 5*(10

*B*a*b^9 + A*b^10)*d^4*e^7 + 15*(9*B*a^2*b^8 + 2*A*a*b^9)*d^3*e^8 + 35*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^2*e^9 + 7

0*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d*e^10 + 126*(6*B*a^5*b^5 + 5*A*a^4*b^6)*e^11)*x^6 + 4368*(11*B*b^10*d^6*e^5 + 5

*(10*B*a*b^9 + A*b^10)*d^5*e^6 + 15*(9*B*a^2*b^8 + 2*A*a*b^9)*d^4*e^7 + 35*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^3*e^8

+ 70*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^2*e^9 + 126*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d*e^10 + 210*(5*B*a^6*b^4 + 6*A*a^

5*b^5)*e^11)*x^5 + 1820*(11*B*b^10*d^7*e^4 + 5*(10*B*a*b^9 + A*b^10)*d^6*e^5 + 15*(9*B*a^2*b^8 + 2*A*a*b^9)*d^

5*e^6 + 35*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^4*e^7 + 70*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^3*e^8 + 126*(6*B*a^5*b^5 + 5

*A*a^4*b^6)*d^2*e^9 + 210*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d*e^10 + 330*(4*B*a^7*b^3 + 7*A*a^6*b^4)*e^11)*x^4 + 560

*(11*B*b^10*d^8*e^3 + 5*(10*B*a*b^9 + A*b^10)*d^7*e^4 + 15*(9*B*a^2*b^8 + 2*A*a*b^9)*d^6*e^5 + 35*(8*B*a^3*b^7

+ 3*A*a^2*b^8)*d^5*e^6 + 70*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^4*e^7 + 126*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^3*e^8 + 2

10*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^2*e^9 + 330*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d*e^10 + 495*(3*B*a^8*b^2 + 8*A*a^7*b

^3)*e^11)*x^3 + 120*(11*B*b^10*d^9*e^2 + 5*(10*B*a*b^9 + A*b^10)*d^8*e^3 + 15*(9*B*a^2*b^8 + 2*A*a*b^9)*d^7*e^

4 + 35*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^6*e^5 + 70*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^5*e^6 + 126*(6*B*a^5*b^5 + 5*A*a

^4*b^6)*d^4*e^7 + 210*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^3*e^8 + 330*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^2*e^9 + 495*(3*B

*a^8*b^2 + 8*A*a^7*b^3)*d*e^10 + 715*(2*B*a^9*b + 9*A*a^8*b^2)*e^11)*x^2 + 16*(11*B*b^10*d^10*e + 5*(10*B*a*b^

9 + A*b^10)*d^9*e^2 + 15*(9*B*a^2*b^8 + 2*A*a*b^9)*d^8*e^3 + 35*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^7*e^4 + 70*(7*B*

a^4*b^6 + 4*A*a^3*b^7)*d^6*e^5 + 126*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^5*e^6 + 210*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^4

*e^7 + 330*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^3*e^8 + 495*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^2*e^9 + 715*(2*B*a^9*b + 9*

A*a^8*b^2)*d*e^10 + 1001*(B*a^10 + 10*A*a^9*b)*e^11)*x)/(e^28*x^16 + 16*d*e^27*x^15 + 120*d^2*e^26*x^14 + 560*

d^3*e^25*x^13 + 1820*d^4*e^24*x^12 + 4368*d^5*e^23*x^11 + 8008*d^6*e^22*x^10 + 11440*d^7*e^21*x^9 + 12870*d^8*

e^20*x^8 + 11440*d^9*e^19*x^7 + 8008*d^10*e^18*x^6 + 4368*d^11*e^17*x^5 + 1820*d^12*e^16*x^4 + 560*d^13*e^15*x

^3 + 120*d^14*e^14*x^2 + 16*d^15*e^13*x + d^16*e^12)

________________________________________________________________________________________

giac [B] time = 1.31, size = 2096, normalized size = 7.35

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)^17,x, algorithm="giac")

[Out]

-1/240240*(48048*B*b^10*x^11*e^11 + 88088*B*b^10*d*x^10*e^10 + 125840*B*b^10*d^2*x^9*e^9 + 141570*B*b^10*d^3*x

^8*e^8 + 125840*B*b^10*d^4*x^7*e^7 + 88088*B*b^10*d^5*x^6*e^6 + 48048*B*b^10*d^6*x^5*e^5 + 20020*B*b^10*d^7*x^

4*e^4 + 6160*B*b^10*d^8*x^3*e^3 + 1320*B*b^10*d^9*x^2*e^2 + 176*B*b^10*d^10*x*e + 11*B*b^10*d^11 + 400400*B*a*

b^9*x^10*e^11 + 40040*A*b^10*x^10*e^11 + 572000*B*a*b^9*d*x^9*e^10 + 57200*A*b^10*d*x^9*e^10 + 643500*B*a*b^9*

d^2*x^8*e^9 + 64350*A*b^10*d^2*x^8*e^9 + 572000*B*a*b^9*d^3*x^7*e^8 + 57200*A*b^10*d^3*x^7*e^8 + 400400*B*a*b^

9*d^4*x^6*e^7 + 40040*A*b^10*d^4*x^6*e^7 + 218400*B*a*b^9*d^5*x^5*e^6 + 21840*A*b^10*d^5*x^5*e^6 + 91000*B*a*b

^9*d^6*x^4*e^5 + 9100*A*b^10*d^6*x^4*e^5 + 28000*B*a*b^9*d^7*x^3*e^4 + 2800*A*b^10*d^7*x^3*e^4 + 6000*B*a*b^9*

d^8*x^2*e^3 + 600*A*b^10*d^8*x^2*e^3 + 800*B*a*b^9*d^9*x*e^2 + 80*A*b^10*d^9*x*e^2 + 50*B*a*b^9*d^10*e + 5*A*b

^10*d^10*e + 1544400*B*a^2*b^8*x^9*e^11 + 343200*A*a*b^9*x^9*e^11 + 1737450*B*a^2*b^8*d*x^8*e^10 + 386100*A*a*

b^9*d*x^8*e^10 + 1544400*B*a^2*b^8*d^2*x^7*e^9 + 343200*A*a*b^9*d^2*x^7*e^9 + 1081080*B*a^2*b^8*d^3*x^6*e^8 +

240240*A*a*b^9*d^3*x^6*e^8 + 589680*B*a^2*b^8*d^4*x^5*e^7 + 131040*A*a*b^9*d^4*x^5*e^7 + 245700*B*a^2*b^8*d^5*

x^4*e^6 + 54600*A*a*b^9*d^5*x^4*e^6 + 75600*B*a^2*b^8*d^6*x^3*e^5 + 16800*A*a*b^9*d^6*x^3*e^5 + 16200*B*a^2*b^

8*d^7*x^2*e^4 + 3600*A*a*b^9*d^7*x^2*e^4 + 2160*B*a^2*b^8*d^8*x*e^3 + 480*A*a*b^9*d^8*x*e^3 + 135*B*a^2*b^8*d^

9*e^2 + 30*A*a*b^9*d^9*e^2 + 3603600*B*a^3*b^7*x^8*e^11 + 1351350*A*a^2*b^8*x^8*e^11 + 3203200*B*a^3*b^7*d*x^7

*e^10 + 1201200*A*a^2*b^8*d*x^7*e^10 + 2242240*B*a^3*b^7*d^2*x^6*e^9 + 840840*A*a^2*b^8*d^2*x^6*e^9 + 1223040*

B*a^3*b^7*d^3*x^5*e^8 + 458640*A*a^2*b^8*d^3*x^5*e^8 + 509600*B*a^3*b^7*d^4*x^4*e^7 + 191100*A*a^2*b^8*d^4*x^4

*e^7 + 156800*B*a^3*b^7*d^5*x^3*e^6 + 58800*A*a^2*b^8*d^5*x^3*e^6 + 33600*B*a^3*b^7*d^6*x^2*e^5 + 12600*A*a^2*

b^8*d^6*x^2*e^5 + 4480*B*a^3*b^7*d^7*x*e^4 + 1680*A*a^2*b^8*d^7*x*e^4 + 280*B*a^3*b^7*d^8*e^3 + 105*A*a^2*b^8*

d^8*e^3 + 5605600*B*a^4*b^6*x^7*e^11 + 3203200*A*a^3*b^7*x^7*e^11 + 3923920*B*a^4*b^6*d*x^6*e^10 + 2242240*A*a

^3*b^7*d*x^6*e^10 + 2140320*B*a^4*b^6*d^2*x^5*e^9 + 1223040*A*a^3*b^7*d^2*x^5*e^9 + 891800*B*a^4*b^6*d^3*x^4*e

^8 + 509600*A*a^3*b^7*d^3*x^4*e^8 + 274400*B*a^4*b^6*d^4*x^3*e^7 + 156800*A*a^3*b^7*d^4*x^3*e^7 + 58800*B*a^4*

b^6*d^5*x^2*e^6 + 33600*A*a^3*b^7*d^5*x^2*e^6 + 7840*B*a^4*b^6*d^6*x*e^5 + 4480*A*a^3*b^7*d^6*x*e^5 + 490*B*a^

4*b^6*d^7*e^4 + 280*A*a^3*b^7*d^7*e^4 + 6054048*B*a^5*b^5*x^6*e^11 + 5045040*A*a^4*b^6*x^6*e^11 + 3302208*B*a^

5*b^5*d*x^5*e^10 + 2751840*A*a^4*b^6*d*x^5*e^10 + 1375920*B*a^5*b^5*d^2*x^4*e^9 + 1146600*A*a^4*b^6*d^2*x^4*e^

9 + 423360*B*a^5*b^5*d^3*x^3*e^8 + 352800*A*a^4*b^6*d^3*x^3*e^8 + 90720*B*a^5*b^5*d^4*x^2*e^7 + 75600*A*a^4*b^

6*d^4*x^2*e^7 + 12096*B*a^5*b^5*d^5*x*e^6 + 10080*A*a^4*b^6*d^5*x*e^6 + 756*B*a^5*b^5*d^6*e^5 + 630*A*a^4*b^6*

d^6*e^5 + 4586400*B*a^6*b^4*x^5*e^11 + 5503680*A*a^5*b^5*x^5*e^11 + 1911000*B*a^6*b^4*d*x^4*e^10 + 2293200*A*a

^5*b^5*d*x^4*e^10 + 588000*B*a^6*b^4*d^2*x^3*e^9 + 705600*A*a^5*b^5*d^2*x^3*e^9 + 126000*B*a^6*b^4*d^3*x^2*e^8

+ 151200*A*a^5*b^5*d^3*x^2*e^8 + 16800*B*a^6*b^4*d^4*x*e^7 + 20160*A*a^5*b^5*d^4*x*e^7 + 1050*B*a^6*b^4*d^5*e

^6 + 1260*A*a^5*b^5*d^5*e^6 + 2402400*B*a^7*b^3*x^4*e^11 + 4204200*A*a^6*b^4*x^4*e^11 + 739200*B*a^7*b^3*d*x^3

*e^10 + 1293600*A*a^6*b^4*d*x^3*e^10 + 158400*B*a^7*b^3*d^2*x^2*e^9 + 277200*A*a^6*b^4*d^2*x^2*e^9 + 21120*B*a

^7*b^3*d^3*x*e^8 + 36960*A*a^6*b^4*d^3*x*e^8 + 1320*B*a^7*b^3*d^4*e^7 + 2310*A*a^6*b^4*d^4*e^7 + 831600*B*a^8*

b^2*x^3*e^11 + 2217600*A*a^7*b^3*x^3*e^11 + 178200*B*a^8*b^2*d*x^2*e^10 + 475200*A*a^7*b^3*d*x^2*e^10 + 23760*

B*a^8*b^2*d^2*x*e^9 + 63360*A*a^7*b^3*d^2*x*e^9 + 1485*B*a^8*b^2*d^3*e^8 + 3960*A*a^7*b^3*d^3*e^8 + 171600*B*a

^9*b*x^2*e^11 + 772200*A*a^8*b^2*x^2*e^11 + 22880*B*a^9*b*d*x*e^10 + 102960*A*a^8*b^2*d*x*e^10 + 1430*B*a^9*b*

d^2*e^9 + 6435*A*a^8*b^2*d^2*e^9 + 16016*B*a^10*x*e^11 + 160160*A*a^9*b*x*e^11 + 1001*B*a^10*d*e^10 + 10010*A*

a^9*b*d*e^10 + 15015*A*a^10*e^11)*e^(-12)/(x*e + d)^16

________________________________________________________________________________________

maple [B] time = 0.01, size = 1942, normalized size = 6.81

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)^17,x)

[Out]

-1/5*B*b^10/e^12/(e*x+d)^5-1/15*(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^12/(e*x+d)^15-15/13*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)^13-5/14*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)^14-15/8*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)^8-1/16*(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)^16-5/2*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^1

2/(e*x+d)^12-42/11*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)^11-21/5*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)^10-10/3*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)^9

-1/6*b^9*(A*b*e+10*B*a*e-11*B*b*d)/e^12/(e*x+d)^6-5/7*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)^7

________________________________________________________________________________________

maxima [B] time = 1.38, size = 1984, normalized size = 6.96

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)^17,x, algorithm="maxima")