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3.10.53 \(\int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^4} \, dx\)

Optimal. Leaf size=445 \[ -\frac {b^9 (d+e x)^7 (-10 a B e-A b e+11 b B d)}{7 e^{12}}+\frac {5 b^8 (d+e x)^6 (b d-a e) (-9 a B e-2 A b e+11 b B d)}{6 e^{12}}-\frac {3 b^7 (d+e x)^5 (b d-a e)^2 (-8 a B e-3 A b e+11 b B d)}{e^{12}}+\frac {15 b^6 (d+e x)^4 (b d-a e)^3 (-7 a B e-4 A b e+11 b B d)}{2 e^{12}}-\frac {14 b^5 (d+e x)^3 (b d-a e)^4 (-6 a B e-5 A b e+11 b B d)}{e^{12}}+\frac {21 b^4 (d+e x)^2 (b d-a e)^5 (-5 a B e-6 A b e+11 b B d)}{e^{12}}-\frac {30 b^3 x (b d-a e)^6 (-4 a B e-7 A b e+11 b B d)}{e^{11}}+\frac {15 b^2 (b d-a e)^7 \log (d+e x) (-3 a B e-8 A b e+11 b B d)}{e^{12}}+\frac {5 b (b d-a e)^8 (-2 a B e-9 A b e+11 b B d)}{e^{12} (d+e x)}-\frac {(b d-a e)^9 (-a B e-10 A b e+11 b B d)}{2 e^{12} (d+e x)^2}+\frac {(b d-a e)^{10} (B d-A e)}{3 e^{12} (d+e x)^3}+\frac {b^{10} B (d+e x)^8}{8 e^{12}} \]

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Rubi [A]  time = 1.39, antiderivative size = 445, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {77} \begin {gather*} -\frac {b^9 (d+e x)^7 (-10 a B e-A b e+11 b B d)}{7 e^{12}}+\frac {5 b^8 (d+e x)^6 (b d-a e) (-9 a B e-2 A b e+11 b B d)}{6 e^{12}}-\frac {3 b^7 (d+e x)^5 (b d-a e)^2 (-8 a B e-3 A b e+11 b B d)}{e^{12}}+\frac {15 b^6 (d+e x)^4 (b d-a e)^3 (-7 a B e-4 A b e+11 b B d)}{2 e^{12}}-\frac {14 b^5 (d+e x)^3 (b d-a e)^4 (-6 a B e-5 A b e+11 b B d)}{e^{12}}+\frac {21 b^4 (d+e x)^2 (b d-a e)^5 (-5 a B e-6 A b e+11 b B d)}{e^{12}}-\frac {30 b^3 x (b d-a e)^6 (-4 a B e-7 A b e+11 b B d)}{e^{11}}+\frac {15 b^2 (b d-a e)^7 \log (d+e x) (-3 a B e-8 A b e+11 b B d)}{e^{12}}+\frac {5 b (b d-a e)^8 (-2 a B e-9 A b e+11 b B d)}{e^{12} (d+e x)}-\frac {(b d-a e)^9 (-a B e-10 A b e+11 b B d)}{2 e^{12} (d+e x)^2}+\frac {(b d-a e)^{10} (B d-A e)}{3 e^{12} (d+e x)^3}+\frac {b^{10} B (d+e x)^8}{8 e^{12}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^10*(A + B*x))/(d + e*x)^4,x]

[Out]

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

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

\begin {align*} \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^4} \, dx &=\int \left (\frac {30 b^3 (b d-a e)^6 (-11 b B d+7 A b e+4 a B e)}{e^{11}}+\frac {(-b d+a e)^{10} (-B d+A e)}{e^{11} (d+e x)^4}+\frac {(-b d+a e)^9 (-11 b B d+10 A b e+a B e)}{e^{11} (d+e x)^3}+\frac {5 b (b d-a e)^8 (-11 b B d+9 A b e+2 a B e)}{e^{11} (d+e x)^2}-\frac {15 b^2 (b d-a e)^7 (-11 b B d+8 A b e+3 a B e)}{e^{11} (d+e x)}-\frac {42 b^4 (b d-a e)^5 (-11 b B d+6 A b e+5 a B e) (d+e x)}{e^{11}}+\frac {42 b^5 (b d-a e)^4 (-11 b B d+5 A b e+6 a B e) (d+e x)^2}{e^{11}}-\frac {30 b^6 (b d-a e)^3 (-11 b B d+4 A b e+7 a B e) (d+e x)^3}{e^{11}}+\frac {15 b^7 (b d-a e)^2 (-11 b B d+3 A b e+8 a B e) (d+e x)^4}{e^{11}}-\frac {5 b^8 (b d-a e) (-11 b B d+2 A b e+9 a B e) (d+e x)^5}{e^{11}}+\frac {b^9 (-11 b B d+A b e+10 a B e) (d+e x)^6}{e^{11}}+\frac {b^{10} B (d+e x)^7}{e^{11}}\right ) \, dx\\ &=-\frac {30 b^3 (b d-a e)^6 (11 b B d-7 A b e-4 a B e) x}{e^{11}}+\frac {(b d-a e)^{10} (B d-A e)}{3 e^{12} (d+e x)^3}-\frac {(b d-a e)^9 (11 b B d-10 A b e-a B e)}{2 e^{12} (d+e x)^2}+\frac {5 b (b d-a e)^8 (11 b B d-9 A b e-2 a B e)}{e^{12} (d+e x)}+\frac {21 b^4 (b d-a e)^5 (11 b B d-6 A b e-5 a B e) (d+e x)^2}{e^{12}}-\frac {14 b^5 (b d-a e)^4 (11 b B d-5 A b e-6 a B e) (d+e x)^3}{e^{12}}+\frac {15 b^6 (b d-a e)^3 (11 b B d-4 A b e-7 a B e) (d+e x)^4}{2 e^{12}}-\frac {3 b^7 (b d-a e)^2 (11 b B d-3 A b e-8 a B e) (d+e x)^5}{e^{12}}+\frac {5 b^8 (b d-a e) (11 b B d-2 A b e-9 a B e) (d+e x)^6}{6 e^{12}}-\frac {b^9 (11 b B d-A b e-10 a B e) (d+e x)^7}{7 e^{12}}+\frac {b^{10} B (d+e x)^8}{8 e^{12}}+\frac {15 b^2 (b d-a e)^7 (11 b B d-8 A b e-3 a B e) \log (d+e x)}{e^{12}}\\ \end {align*}

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Mathematica [A]  time = 0.57, size = 814, normalized size = 1.83 \begin {gather*} \frac {21 B e^8 x^8 b^{10}+24 e^7 (-4 b B d+A b e+10 a B e) x^7 b^9-28 e^6 \left (2 d (2 A e-5 B d) b^2-10 a e (A e-4 B d) b-45 a^2 B e^2\right ) x^6 b^8+168 e^5 \left (2 d^2 (A e-2 B d) b^3+4 a d e (5 B d-2 A e) b^2+9 a^2 e^2 (A e-4 B d) b+24 a^3 B e^3\right ) x^5 b^7-210 e^4 \left (d^3 (4 A e-7 B d) b^4+20 a d^2 e (2 B d-A e) b^3+18 a^2 d e^2 (2 A e-5 B d) b^2-24 a^3 e^3 (A e-4 B d) b-42 a^4 B e^4\right ) x^4 b^6+56 e^3 \left (-7 d^4 (8 B d-5 A e) b^5+50 a d^3 e (7 B d-4 A e) b^4-450 a^2 d^2 e^2 (2 B d-A e) b^3+240 a^3 d e^3 (5 B d-2 A e) b^2+210 a^4 e^4 (A e-4 B d) b+252 a^5 B e^5\right ) x^3 b^5-84 e^2 \left (28 d^5 (2 A e-3 B d) b^6+70 a d^4 e (8 B d-5 A e) b^5-225 a^2 d^3 e^2 (7 B d-4 A e) b^4+1200 a^3 d^2 e^3 (2 B d-A e) b^3-420 a^4 d e^4 (5 B d-2 A e) b^2-252 a^5 e^5 (A e-4 B d) b-210 a^6 B e^6\right ) x^2 b^4+168 e \left (12 d^6 (7 A e-10 B d) b^7+280 a d^5 e (3 B d-2 A e) b^6-315 a^2 d^4 e^2 (8 B d-5 A e) b^5+600 a^3 d^3 e^3 (7 B d-4 A e) b^4-2100 a^4 d^2 e^4 (2 B d-A e) b^3+504 a^5 d e^5 (5 B d-2 A e) b^2+210 a^6 e^6 (A e-4 B d) b+120 a^7 B e^7\right ) x b^3+2520 (b d-a e)^7 (11 b B d-8 A b e-3 a B e) \log (d+e x) b^2+\frac {840 (b d-a e)^8 (11 b B d-9 A b e-2 a B e) b}{d+e x}-\frac {84 (b d-a e)^9 (11 b B d-10 A b e-a B e)}{(d+e x)^2}+\frac {56 (b d-a e)^{10} (B d-A e)}{(d+e x)^3}}{168 e^{12}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^10*(A + B*x))/(d + e*x)^4,x]

[Out]

(168*b^3*e*(120*a^7*B*e^7 - 315*a^2*b^5*d^4*e^2*(8*B*d - 5*A*e) + 600*a^3*b^4*d^3*e^3*(7*B*d - 4*A*e) + 280*a*
b^6*d^5*e*(3*B*d - 2*A*e) + 504*a^5*b^2*d*e^5*(5*B*d - 2*A*e) - 2100*a^4*b^3*d^2*e^4*(2*B*d - A*e) + 210*a^6*b
*e^6*(-4*B*d + A*e) + 12*b^7*d^6*(-10*B*d + 7*A*e))*x - 84*b^4*e^2*(-210*a^6*B*e^6 + 70*a*b^5*d^4*e*(8*B*d - 5
*A*e) - 225*a^2*b^4*d^3*e^2*(7*B*d - 4*A*e) - 420*a^4*b^2*d*e^4*(5*B*d - 2*A*e) + 1200*a^3*b^3*d^2*e^3*(2*B*d
- A*e) - 252*a^5*b*e^5*(-4*B*d + A*e) + 28*b^6*d^5*(-3*B*d + 2*A*e))*x^2 + 56*b^5*e^3*(252*a^5*B*e^5 - 7*b^5*d
^4*(8*B*d - 5*A*e) + 50*a*b^4*d^3*e*(7*B*d - 4*A*e) + 240*a^3*b^2*d*e^3*(5*B*d - 2*A*e) - 450*a^2*b^3*d^2*e^2*
(2*B*d - A*e) + 210*a^4*b*e^4*(-4*B*d + A*e))*x^3 - 210*b^6*e^4*(-42*a^4*B*e^4 + 20*a*b^3*d^2*e*(2*B*d - A*e)
- 24*a^3*b*e^3*(-4*B*d + A*e) + 18*a^2*b^2*d*e^2*(-5*B*d + 2*A*e) + b^4*d^3*(-7*B*d + 4*A*e))*x^4 + 168*b^7*e^
5*(24*a^3*B*e^3 + 4*a*b^2*d*e*(5*B*d - 2*A*e) + 9*a^2*b*e^2*(-4*B*d + A*e) + 2*b^3*d^2*(-2*B*d + A*e))*x^5 - 2
8*b^8*e^6*(-45*a^2*B*e^2 - 10*a*b*e*(-4*B*d + A*e) + 2*b^2*d*(-5*B*d + 2*A*e))*x^6 + 24*b^9*e^7*(-4*b*B*d + A*
b*e + 10*a*B*e)*x^7 + 21*b^10*B*e^8*x^8 + (56*(b*d - a*e)^10*(B*d - A*e))/(d + e*x)^3 - (84*(b*d - a*e)^9*(11*
b*B*d - 10*A*b*e - a*B*e))/(d + e*x)^2 + (840*b*(b*d - a*e)^8*(11*b*B*d - 9*A*b*e - 2*a*B*e))/(d + e*x) + 2520
*b^2*(b*d - a*e)^7*(11*b*B*d - 8*A*b*e - 3*a*B*e)*Log[d + e*x])/(168*e^12)

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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)^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[((a + b*x)^10*(A + B*x))/(d + e*x)^4,x]

[Out]

IntegrateAlgebraic[((a + b*x)^10*(A + B*x))/(d + e*x)^4, x]

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fricas [B]  time = 1.69, size = 2702, normalized size = 6.07

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)^4,x, algorithm="fricas")

[Out]

1/168*(21*B*b^10*e^11*x^11 + 8372*B*b^10*d^11 - 56*A*a^10*e^11 - 6776*(10*B*a*b^9 + A*b^10)*d^10*e + 26740*(9*
B*a^2*b^8 + 2*A*a*b^9)*d^9*e^2 - 61320*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^8*e^3 + 89880*(7*B*a^4*b^6 + 4*A*a^3*b^7)
*d^7*e^4 - 87024*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^6*e^5 + 55272*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^5*e^6 - 21840*(4*B*
a^7*b^3 + 7*A*a^6*b^4)*d^4*e^7 + 4620*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^3*e^8 - 280*(2*B*a^9*b + 9*A*a^8*b^2)*d^2*
e^9 - 28*(B*a^10 + 10*A*a^9*b)*d*e^10 - 3*(11*B*b^10*d*e^10 - 8*(10*B*a*b^9 + A*b^10)*e^11)*x^10 + 5*(11*B*b^1
0*d^2*e^9 - 8*(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 - 9*(11*B*b^10*d^3*e^8 - 8
*(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 - 56*(8*B*a^3*b^7 + 3*A*a^2*b^8)*e^11)*x^
8 + 18*(11*B*b^10*d^4*e^7 - 8*(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 - 56*(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 - 42*(11*B*b^10*d^5*e^6 - 8*(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 - 56*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^2*e^9 + 70*(7*B*
a^4*b^6 + 4*A*a^3*b^7)*d*e^10 - 56*(6*B*a^5*b^5 + 5*A*a^4*b^6)*e^11)*x^6 + 126*(11*B*b^10*d^6*e^5 - 8*(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 - 56*(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 - 56*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d*e^10 + 28*(5*B*a^6*b^4 + 6*A*a^5*b^5)*e^11
)*x^5 - 630*(11*B*b^10*d^7*e^4 - 8*(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 - 56*(
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 - 56*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d
^2*e^9 + 28*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d*e^10 - 8*(4*B*a^7*b^3 + 7*A*a^6*b^4)*e^11)*x^4 - 28*(1516*B*b^10*d^8
*e^3 - 1078*(10*B*a*b^9 + A*b^10)*d^7*e^4 + 3665*(9*B*a^2*b^8 + 2*A*a*b^9)*d^6*e^5 - 7050*(8*B*a^3*b^7 + 3*A*a
^2*b^8)*d^5*e^6 + 8340*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^4*e^7 - 6132*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^3*e^8 + 2646*(
5*B*a^6*b^4 + 6*A*a^5*b^5)*d^2*e^9 - 540*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d*e^10)*x^3 - 84*(526*B*b^10*d^9*e^2 - 35
8*(10*B*a*b^9 + A*b^10)*d^8*e^3 + 1145*(9*B*a^2*b^8 + 2*A*a*b^9)*d^7*e^4 - 2010*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^
6*e^5 + 2040*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^5*e^6 - 1092*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^4*e^7 + 126*(5*B*a^6*b^4
 + 6*A*a^5*b^5)*d^3*e^8 + 180*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^2*e^9 - 90*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d*e^10 + 10
*(2*B*a^9*b + 9*A*a^8*b^2)*e^11)*x^2 - 84*(31*B*b^10*d^10*e + 2*(10*B*a*b^9 + A*b^10)*d^9*e^2 - 115*(9*B*a^2*b
^8 + 2*A*a*b^9)*d^8*e^3 + 510*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^7*e^4 - 1110*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^6*e^5 +
 1428*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^5*e^6 - 1134*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^4*e^7 + 540*(4*B*a^7*b^3 + 7*A*
a^6*b^4)*d^3*e^8 - 135*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^2*e^9 + 10*(2*B*a^9*b + 9*A*a^8*b^2)*d*e^10 + (B*a^10 + 1
0*A*a^9*b)*e^11)*x + 2520*(11*B*b^10*d^11 - 8*(10*B*a*b^9 + A*b^10)*d^10*e + 28*(9*B*a^2*b^8 + 2*A*a*b^9)*d^9*
e^2 - 56*(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 - 56*(6*B*a^5*b^5 + 5*A*
a^4*b^6)*d^6*e^5 + 28*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^5*e^6 - 8*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^4*e^7 + (3*B*a^8*b
^2 + 8*A*a^7*b^3)*d^3*e^8 + (11*B*b^10*d^8*e^3 - 8*(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 - 56*(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 - 56*(6*B*a^5*b^5
+ 5*A*a^4*b^6)*d^3*e^8 + 28*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^2*e^9 - 8*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d*e^10 + (3*B*
a^8*b^2 + 8*A*a^7*b^3)*e^11)*x^3 + 3*(11*B*b^10*d^9*e^2 - 8*(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 - 56*(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 - 56*(6*B
*a^5*b^5 + 5*A*a^4*b^6)*d^4*e^7 + 28*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^3*e^8 - 8*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^2*e
^9 + (3*B*a^8*b^2 + 8*A*a^7*b^3)*d*e^10)*x^2 + 3*(11*B*b^10*d^10*e - 8*(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 - 56*(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 - 56*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^5*e^6 + 28*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^4*e^7 - 8*(4*B*a^7*b^3 + 7*A*a^6
*b^4)*d^3*e^8 + (3*B*a^8*b^2 + 8*A*a^7*b^3)*d^2*e^9)*x)*log(e*x + d))/(e^15*x^3 + 3*d*e^14*x^2 + 3*d^2*e^13*x
+ d^3*e^12)

________________________________________________________________________________________

giac [B]  time = 1.30, size = 1964, normalized size = 4.41

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)^4,x, algorithm="giac")

[Out]

15*(11*B*b^10*d^8 - 80*B*a*b^9*d^7*e - 8*A*b^10*d^7*e + 252*B*a^2*b^8*d^6*e^2 + 56*A*a*b^9*d^6*e^2 - 448*B*a^3
*b^7*d^5*e^3 - 168*A*a^2*b^8*d^5*e^3 + 490*B*a^4*b^6*d^4*e^4 + 280*A*a^3*b^7*d^4*e^4 - 336*B*a^5*b^5*d^3*e^5 -
 280*A*a^4*b^6*d^3*e^5 + 140*B*a^6*b^4*d^2*e^6 + 168*A*a^5*b^5*d^2*e^6 - 32*B*a^7*b^3*d*e^7 - 56*A*a^6*b^4*d*e
^7 + 3*B*a^8*b^2*e^8 + 8*A*a^7*b^3*e^8)*e^(-12)*log(abs(x*e + d)) + 1/168*(21*B*b^10*x^8*e^28 - 96*B*b^10*d*x^
7*e^27 + 280*B*b^10*d^2*x^6*e^26 - 672*B*b^10*d^3*x^5*e^25 + 1470*B*b^10*d^4*x^4*e^24 - 3136*B*b^10*d^5*x^3*e^
23 + 7056*B*b^10*d^6*x^2*e^22 - 20160*B*b^10*d^7*x*e^21 + 240*B*a*b^9*x^7*e^28 + 24*A*b^10*x^7*e^28 - 1120*B*a
*b^9*d*x^6*e^27 - 112*A*b^10*d*x^6*e^27 + 3360*B*a*b^9*d^2*x^5*e^26 + 336*A*b^10*d^2*x^5*e^26 - 8400*B*a*b^9*d
^3*x^4*e^25 - 840*A*b^10*d^3*x^4*e^25 + 19600*B*a*b^9*d^4*x^3*e^24 + 1960*A*b^10*d^4*x^3*e^24 - 47040*B*a*b^9*
d^5*x^2*e^23 - 4704*A*b^10*d^5*x^2*e^23 + 141120*B*a*b^9*d^6*x*e^22 + 14112*A*b^10*d^6*x*e^22 + 1260*B*a^2*b^8
*x^6*e^28 + 280*A*a*b^9*x^6*e^28 - 6048*B*a^2*b^8*d*x^5*e^27 - 1344*A*a*b^9*d*x^5*e^27 + 18900*B*a^2*b^8*d^2*x
^4*e^26 + 4200*A*a*b^9*d^2*x^4*e^26 - 50400*B*a^2*b^8*d^3*x^3*e^25 - 11200*A*a*b^9*d^3*x^3*e^25 + 132300*B*a^2
*b^8*d^4*x^2*e^24 + 29400*A*a*b^9*d^4*x^2*e^24 - 423360*B*a^2*b^8*d^5*x*e^23 - 94080*A*a*b^9*d^5*x*e^23 + 4032
*B*a^3*b^7*x^5*e^28 + 1512*A*a^2*b^8*x^5*e^28 - 20160*B*a^3*b^7*d*x^4*e^27 - 7560*A*a^2*b^8*d*x^4*e^27 + 67200
*B*a^3*b^7*d^2*x^3*e^26 + 25200*A*a^2*b^8*d^2*x^3*e^26 - 201600*B*a^3*b^7*d^3*x^2*e^25 - 75600*A*a^2*b^8*d^3*x
^2*e^25 + 705600*B*a^3*b^7*d^4*x*e^24 + 264600*A*a^2*b^8*d^4*x*e^24 + 8820*B*a^4*b^6*x^4*e^28 + 5040*A*a^3*b^7
*x^4*e^28 - 47040*B*a^4*b^6*d*x^3*e^27 - 26880*A*a^3*b^7*d*x^3*e^27 + 176400*B*a^4*b^6*d^2*x^2*e^26 + 100800*A
*a^3*b^7*d^2*x^2*e^26 - 705600*B*a^4*b^6*d^3*x*e^25 - 403200*A*a^3*b^7*d^3*x*e^25 + 14112*B*a^5*b^5*x^3*e^28 +
 11760*A*a^4*b^6*x^3*e^28 - 84672*B*a^5*b^5*d*x^2*e^27 - 70560*A*a^4*b^6*d*x^2*e^27 + 423360*B*a^5*b^5*d^2*x*e
^26 + 352800*A*a^4*b^6*d^2*x*e^26 + 17640*B*a^6*b^4*x^2*e^28 + 21168*A*a^5*b^5*x^2*e^28 - 141120*B*a^6*b^4*d*x
*e^27 - 169344*A*a^5*b^5*d*x*e^27 + 20160*B*a^7*b^3*x*e^28 + 35280*A*a^6*b^4*x*e^28)*e^(-32) + 1/6*(299*B*b^10
*d^11 - 2420*B*a*b^9*d^10*e - 242*A*b^10*d^10*e + 8595*B*a^2*b^8*d^9*e^2 + 1910*A*a*b^9*d^9*e^2 - 17520*B*a^3*
b^7*d^8*e^3 - 6570*A*a^2*b^8*d^8*e^3 + 22470*B*a^4*b^6*d^7*e^4 + 12840*A*a^3*b^7*d^7*e^4 - 18648*B*a^5*b^5*d^6
*e^5 - 15540*A*a^4*b^6*d^6*e^5 + 9870*B*a^6*b^4*d^5*e^6 + 11844*A*a^5*b^5*d^5*e^6 - 3120*B*a^7*b^3*d^4*e^7 - 5
460*A*a^6*b^4*d^4*e^7 + 495*B*a^8*b^2*d^3*e^8 + 1320*A*a^7*b^3*d^3*e^8 - 20*B*a^9*b*d^2*e^9 - 90*A*a^8*b^2*d^2
*e^9 - B*a^10*d*e^10 - 10*A*a^9*b*d*e^10 - 2*A*a^10*e^11 + 30*(11*B*b^10*d^9*e^2 - 90*B*a*b^9*d^8*e^3 - 9*A*b^
10*d^8*e^3 + 324*B*a^2*b^8*d^7*e^4 + 72*A*a*b^9*d^7*e^4 - 672*B*a^3*b^7*d^6*e^5 - 252*A*a^2*b^8*d^6*e^5 + 882*
B*a^4*b^6*d^5*e^6 + 504*A*a^3*b^7*d^5*e^6 - 756*B*a^5*b^5*d^4*e^7 - 630*A*a^4*b^6*d^4*e^7 + 420*B*a^6*b^4*d^3*
e^8 + 504*A*a^5*b^5*d^3*e^8 - 144*B*a^7*b^3*d^2*e^9 - 252*A*a^6*b^4*d^2*e^9 + 27*B*a^8*b^2*d*e^10 + 72*A*a^7*b
^3*d*e^10 - 2*B*a^9*b*e^11 - 9*A*a^8*b^2*e^11)*x^2 + 3*(209*B*b^10*d^10*e - 1700*B*a*b^9*d^9*e^2 - 170*A*b^10*
d^9*e^2 + 6075*B*a^2*b^8*d^8*e^3 + 1350*A*a*b^9*d^8*e^3 - 12480*B*a^3*b^7*d^7*e^4 - 4680*A*a^2*b^8*d^7*e^4 + 1
6170*B*a^4*b^6*d^6*e^5 + 9240*A*a^3*b^7*d^6*e^5 - 13608*B*a^5*b^5*d^5*e^6 - 11340*A*a^4*b^6*d^5*e^6 + 7350*B*a
^6*b^4*d^4*e^7 + 8820*A*a^5*b^5*d^4*e^7 - 2400*B*a^7*b^3*d^3*e^8 - 4200*A*a^6*b^4*d^3*e^8 + 405*B*a^8*b^2*d^2*
e^9 + 1080*A*a^7*b^3*d^2*e^9 - 20*B*a^9*b*d*e^10 - 90*A*a^8*b^2*d*e^10 - B*a^10*e^11 - 10*A*a^9*b*e^11)*x)*e^(
-12)/(x*e + d)^3

________________________________________________________________________________________

maple [B]  time = 0.03, size = 2607, normalized size = 5.86

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)^4,x)

[Out]

-1/2/e^2/(e*x+d)^2*B*a^10+1/8*b^10/e^4*B*x^8+1/7*b^10/e^4*A*x^7-1/3/e/(e*x+d)^3*a^10*A+105/2*b^6/e^4*B*x^4*a^4
+35/4*b^10/e^8*B*x^4*d^4+70*b^6/e^4*A*x^3*a^4+35/3*b^10/e^8*A*x^3*d^4+84*b^5/e^4*B*x^3*a^5-56/3*b^10/e^9*B*x^3
*d^5+126*b^5/e^4*A*x^2*a^5+5/e^11/(e*x+d)^2*A*b^10*d^9-11/2/e^12/(e*x+d)^2*b^10*B*d^10-28*b^10/e^9*A*x^2*d^5+1
05*b^4/e^4*B*x^2*a^6+42*b^10/e^10*B*x^2*d^6+210*b^4/e^4*A*a^6*x+84*b^10/e^10*A*d^6*x+10/7*b^9/e^4*B*x^7*a-4/7*
b^10/e^5*B*x^7*d+5/3*b^9/e^4*A*x^6*a-2/3*b^10/e^5*A*x^6*d+15/2*b^8/e^4*B*x^6*a^2+5/3*b^10/e^6*B*x^6*d^2+9*b^8/
e^4*A*x^5*a^2+2*b^10/e^6*A*x^5*d^2+24*b^7/e^4*B*x^5*a^3-4*b^10/e^7*B*x^5*d^3-45*b^2/e^3/(e*x+d)*A*a^8-45*b^10/
e^11/(e*x+d)*A*d^8-10*b/e^3/(e*x+d)*B*a^9+55*b^10/e^12/(e*x+d)*B*d^9-1/3/e^11/(e*x+d)^3*A*b^10*d^10+1/3/e^2/(e
*x+d)^3*B*d*a^10+1/3/e^12/(e*x+d)^3*b^10*B*d^11+120*b^3/e^4*ln(e*x+d)*A*a^7-120*b^10/e^11*ln(e*x+d)*A*d^7+45*b
^2/e^4*ln(e*x+d)*B*a^8+165*b^10/e^12*ln(e*x+d)*B*d^8-5/e^2/(e*x+d)^2*A*a^9*b+120*b^3/e^4*B*a^7*x-120*b^10/e^11
*B*d^7*x+30*b^7/e^4*A*x^4*a^3-5*b^10/e^7*A*x^4*d^3+840*b^9/e^10*ln(e*x+d)*A*a*d^6-480*b^3/e^5*ln(e*x+d)*B*a^7*
d+2100*b^4/e^6*ln(e*x+d)*B*a^6*d^2-5040*b^5/e^7*ln(e*x+d)*B*a^5*d^3+7350*b^6/e^8*ln(e*x+d)*B*a^4*d^4-6720*b^7/
e^9*ln(e*x+d)*B*a^3*d^5+3780*b^8/e^10*ln(e*x+d)*B*a^2*d^6-1200*b^9/e^11*ln(e*x+d)*B*a*d^7+45/e^3/(e*x+d)^2*A*a
^8*b^2*d-180/e^4/(e*x+d)^2*A*a^7*b^3*d^2+420/e^5/(e*x+d)^2*A*a^6*b^4*d^3-630/e^6/(e*x+d)^2*A*a^5*b^5*d^4+630/e
^7/(e*x+d)^2*A*a^4*b^6*d^5-420/e^8/(e*x+d)^2*A*a^3*b^7*d^6+180/e^9/(e*x+d)^2*A*a^2*b^8*d^7-45/e^10/(e*x+d)^2*A
*a*b^9*d^8+10/e^3/(e*x+d)^2*B*a^9*b*d-135/2/e^4/(e*x+d)^2*B*a^8*b^2*d^2+240/e^5/(e*x+d)^2*B*a^7*b^3*d^3-525/e^
6/(e*x+d)^2*B*a^6*b^4*d^4+756/e^7/(e*x+d)^2*B*a^5*b^5*d^5-735/e^8/(e*x+d)^2*B*a^4*b^6*d^6+480/e^9/(e*x+d)^2*B*
a^3*b^7*d^7+25*b^9/e^6*A*x^4*a*d^2-120*b^7/e^5*B*x^4*a^3*d+225/2*b^8/e^6*B*x^4*a^2*d^2-50*b^9/e^7*B*x^4*a*d^3+
360*b^3/e^4/(e*x+d)*A*a^7*d-1260*b^4/e^5/(e*x+d)*A*a^6*d^2+2520*b^5/e^6/(e*x+d)*A*a^5*d^3-3150*b^6/e^7/(e*x+d)
*A*a^4*d^4+2520*b^7/e^8/(e*x+d)*A*a^3*d^5-1260*b^8/e^9/(e*x+d)*A*a^2*d^6+360*b^9/e^10/(e*x+d)*A*a*d^7+135*b^2/
e^4/(e*x+d)*B*a^8*d-720*b^3/e^5/(e*x+d)*B*a^7*d^2+2100*b^4/e^6/(e*x+d)*B*a^6*d^3-3780*b^5/e^7/(e*x+d)*B*a^5*d^
4+4410*b^6/e^8/(e*x+d)*B*a^4*d^5-3360*b^7/e^9/(e*x+d)*B*a^3*d^6+1620*b^8/e^10/(e*x+d)*B*a^2*d^7-450*b^9/e^11/(
e*x+d)*B*a*d^8+10/3/e^2/(e*x+d)^3*A*d*a^9*b-15/e^3/(e*x+d)^3*A*d^2*a^8*b^2+40/e^4/(e*x+d)^3*A*d^3*a^7*b^3-70/e
^5/(e*x+d)^3*A*a^6*b^4*d^4+84/e^6/(e*x+d)^3*A*a^5*b^5*d^5-70/e^7/(e*x+d)^3*A*a^4*b^6*d^6+40/e^8/(e*x+d)^3*A*a^
3*b^7*d^7-15/e^9/(e*x+d)^3*A*a^2*b^8*d^8+10/3/e^10/(e*x+d)^3*A*a*b^9*d^9-10/3/e^3/(e*x+d)^3*B*d^2*a^9*b+15/e^4
/(e*x+d)^3*B*d^3*a^8*b^2-40/e^5/(e*x+d)^3*B*a^7*b^3*d^4+70/e^6/(e*x+d)^3*B*a^6*b^4*d^5-84/e^7/(e*x+d)^3*B*a^5*
b^5*d^6+70/e^8/(e*x+d)^3*B*a^4*b^6*d^7-40/e^9/(e*x+d)^3*B*a^3*b^7*d^8+15/e^10/(e*x+d)^3*B*a^2*b^8*d^9-10/3/e^1
1/(e*x+d)^3*B*a*b^9*d^10-405/2/e^10/(e*x+d)^2*B*a^2*b^8*d^8+50/e^11/(e*x+d)^2*B*a*b^9*d^9-160*b^7/e^5*A*x^3*a^
3*d+150*b^8/e^6*A*x^3*a^2*d^2-200/3*b^9/e^7*A*x^3*a*d^3-280*b^6/e^5*B*x^3*a^4*d+400*b^7/e^6*B*x^3*a^3*d^2-300*
b^8/e^7*B*x^3*a^2*d^3+350/3*b^9/e^8*B*x^3*a*d^4-420*b^6/e^5*A*x^2*a^4*d+600*b^7/e^6*A*x^2*a^3*d^2-450*b^8/e^7*
A*x^2*a^2*d^3+175*b^9/e^8*A*x^2*a*d^4-504*b^5/e^5*B*x^2*a^5*d+1050*b^6/e^6*B*x^2*a^4*d^2-1200*b^7/e^7*B*x^2*a^
3*d^3+1575/2*b^8/e^8*B*x^2*a^2*d^4-280*b^9/e^9*B*x^2*a*d^5-1008*b^5/e^5*A*a^5*d*x+2100*b^6/e^6*A*a^4*d^2*x-240
0*b^7/e^7*A*a^3*d^3*x+1575*b^8/e^8*A*a^2*d^4*x-560*b^9/e^9*A*a*d^5*x-840*b^4/e^5*B*a^6*d*x+2520*b^5/e^6*B*a^5*
d^2*x-4200*b^6/e^7*B*a^4*d^3*x+4200*b^7/e^8*B*a^3*d^4*x-2520*b^8/e^9*B*a^2*d^5*x+840*b^9/e^10*B*a*d^6*x-20/3*b
^9/e^5*B*x^6*a*d-8*b^9/e^5*A*x^5*a*d-36*b^8/e^5*B*x^5*a^2*d+20*b^9/e^6*B*x^5*a*d^2-45*b^8/e^5*A*x^4*a^2*d-840*
b^4/e^5*ln(e*x+d)*A*a^6*d+2520*b^5/e^6*ln(e*x+d)*A*a^5*d^2-4200*b^6/e^7*ln(e*x+d)*A*a^4*d^3+4200*b^7/e^8*ln(e*
x+d)*A*a^3*d^4-2520*b^8/e^9*ln(e*x+d)*A*a^2*d^5

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maxima [B]  time = 1.00, size = 1839, normalized size = 4.13

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)^4,x, algorithm="maxima")

[Out]

1/6*(299*B*b^10*d^11 - 2*A*a^10*e^11 - 242*(10*B*a*b^9 + A*b^10)*d^10*e + 955*(9*B*a^2*b^8 + 2*A*a*b^9)*d^9*e^
2 - 2190*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^8*e^3 + 3210*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^7*e^4 - 3108*(6*B*a^5*b^5 +
5*A*a^4*b^6)*d^6*e^5 + 1974*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^5*e^6 - 780*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^4*e^7 + 16
5*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^3*e^8 - 10*(2*B*a^9*b + 9*A*a^8*b^2)*d^2*e^9 - (B*a^10 + 10*A*a^9*b)*d*e^10 +
30*(11*B*b^10*d^9*e^2 - 9*(10*B*a*b^9 + A*b^10)*d^8*e^3 + 36*(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 + 126*(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
+ 84*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^3*e^8 - 36*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^2*e^9 + 9*(3*B*a^8*b^2 + 8*A*a^7*b
^3)*d*e^10 - (2*B*a^9*b + 9*A*a^8*b^2)*e^11)*x^2 + 3*(209*B*b^10*d^10*e - 170*(10*B*a*b^9 + A*b^10)*d^9*e^2 +
675*(9*B*a^2*b^8 + 2*A*a*b^9)*d^8*e^3 - 1560*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^7*e^4 + 2310*(7*B*a^4*b^6 + 4*A*a^3
*b^7)*d^6*e^5 - 2268*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^5*e^6 + 1470*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^4*e^7 - 600*(4*B
*a^7*b^3 + 7*A*a^6*b^4)*d^3*e^8 + 135*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^2*e^9 - 10*(2*B*a^9*b + 9*A*a^8*b^2)*d*e^1
0 - (B*a^10 + 10*A*a^9*b)*e^11)*x)/(e^15*x^3 + 3*d*e^14*x^2 + 3*d^2*e^13*x + d^3*e^12) + 1/168*(21*B*b^10*e^7*
x^8 - 24*(4*B*b^10*d*e^6 - (10*B*a*b^9 + A*b^10)*e^7)*x^7 + 28*(10*B*b^10*d^2*e^5 - 4*(10*B*a*b^9 + A*b^10)*d*
e^6 + 5*(9*B*a^2*b^8 + 2*A*a*b^9)*e^7)*x^6 - 168*(4*B*b^10*d^3*e^4 - 2*(10*B*a*b^9 + A*b^10)*d^2*e^5 + 4*(9*B*
a^2*b^8 + 2*A*a*b^9)*d*e^6 - 3*(8*B*a^3*b^7 + 3*A*a^2*b^8)*e^7)*x^5 + 210*(7*B*b^10*d^4*e^3 - 4*(10*B*a*b^9 +
A*b^10)*d^3*e^4 + 10*(9*B*a^2*b^8 + 2*A*a*b^9)*d^2*e^5 - 12*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d*e^6 + 6*(7*B*a^4*b^6
 + 4*A*a^3*b^7)*e^7)*x^4 - 56*(56*B*b^10*d^5*e^2 - 35*(10*B*a*b^9 + A*b^10)*d^4*e^3 + 100*(9*B*a^2*b^8 + 2*A*a
*b^9)*d^3*e^4 - 150*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^2*e^5 + 120*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d*e^6 - 42*(6*B*a^5*
b^5 + 5*A*a^4*b^6)*e^7)*x^3 + 84*(84*B*b^10*d^6*e - 56*(10*B*a*b^9 + A*b^10)*d^5*e^2 + 175*(9*B*a^2*b^8 + 2*A*
a*b^9)*d^4*e^3 - 300*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^3*e^4 + 300*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^2*e^5 - 168*(6*B*
a^5*b^5 + 5*A*a^4*b^6)*d*e^6 + 42*(5*B*a^6*b^4 + 6*A*a^5*b^5)*e^7)*x^2 - 168*(120*B*b^10*d^7 - 84*(10*B*a*b^9
+ A*b^10)*d^6*e + 280*(9*B*a^2*b^8 + 2*A*a*b^9)*d^5*e^2 - 525*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^4*e^3 + 600*(7*B*a
^4*b^6 + 4*A*a^3*b^7)*d^3*e^4 - 420*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^2*e^5 + 168*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d*e^
6 - 30*(4*B*a^7*b^3 + 7*A*a^6*b^4)*e^7)*x)/e^11 + 15*(11*B*b^10*d^8 - 8*(10*B*a*b^9 + A*b^10)*d^7*e + 28*(9*B*
a^2*b^8 + 2*A*a*b^9)*d^6*e^2 - 56*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^5*e^3 + 70*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^4*e^4
 - 56*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^3*e^5 + 28*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^2*e^6 - 8*(4*B*a^7*b^3 + 7*A*a^6*
b^4)*d*e^7 + (3*B*a^8*b^2 + 8*A*a^7*b^3)*e^8)*log(e*x + d)/e^12

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mupad [B]  time = 1.48, size = 5544, normalized size = 12.46

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(a + b*x)^10)/(d + e*x)^4,x)

[Out]

x^2*((2*d^3*((6*d^2*((A*b^10 + 10*B*a*b^9)/e^4 - (4*B*b^10*d)/e^5))/e^2 - (4*d*((4*d*((A*b^10 + 10*B*a*b^9)/e^
4 - (4*B*b^10*d)/e^5))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^4 + (6*B*b^10*d^2)/e^6))/e - (15*a^2*b^7*(3*A*b + 8*B*a
))/e^4 + (4*B*b^10*d^3)/e^7))/e^3 - (3*d^2*((4*d*((6*d^2*((A*b^10 + 10*B*a*b^9)/e^4 - (4*B*b^10*d)/e^5))/e^2 -
 (4*d*((4*d*((A*b^10 + 10*B*a*b^9)/e^4 - (4*B*b^10*d)/e^5))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^4 + (6*B*b^10*d^2)
/e^6))/e - (15*a^2*b^7*(3*A*b + 8*B*a))/e^4 + (4*B*b^10*d^3)/e^7))/e - (4*d^3*((A*b^10 + 10*B*a*b^9)/e^4 - (4*
B*b^10*d)/e^5))/e^3 + (6*d^2*((4*d*((A*b^10 + 10*B*a*b^9)/e^4 - (4*B*b^10*d)/e^5))/e - (5*a*b^8*(2*A*b + 9*B*a
))/e^4 + (6*B*b^10*d^2)/e^6))/e^2 + (30*a^3*b^6*(4*A*b + 7*B*a))/e^4 - (B*b^10*d^4)/e^8))/e^2 - (2*d*((6*d^2*(
(6*d^2*((A*b^10 + 10*B*a*b^9)/e^4 - (4*B*b^10*d)/e^5))/e^2 - (4*d*((4*d*((A*b^10 + 10*B*a*b^9)/e^4 - (4*B*b^10
*d)/e^5))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^4 + (6*B*b^10*d^2)/e^6))/e - (15*a^2*b^7*(3*A*b + 8*B*a))/e^4 + (4*B
*b^10*d^3)/e^7))/e^2 - (4*d*((4*d*((6*d^2*((A*b^10 + 10*B*a*b^9)/e^4 - (4*B*b^10*d)/e^5))/e^2 - (4*d*((4*d*((A
*b^10 + 10*B*a*b^9)/e^4 - (4*B*b^10*d)/e^5))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^4 + (6*B*b^10*d^2)/e^6))/e - (15*
a^2*b^7*(3*A*b + 8*B*a))/e^4 + (4*B*b^10*d^3)/e^7))/e - (4*d^3*((A*b^10 + 10*B*a*b^9)/e^4 - (4*B*b^10*d)/e^5))
/e^3 + (6*d^2*((4*d*((A*b^10 + 10*B*a*b^9)/e^4 - (4*B*b^10*d)/e^5))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^4 + (6*B*b
^10*d^2)/e^6))/e^2 + (30*a^3*b^6*(4*A*b + 7*B*a))/e^4 - (B*b^10*d^4)/e^8))/e - (d^4*((A*b^10 + 10*B*a*b^9)/e^4
 - (4*B*b^10*d)/e^5))/e^4 + (4*d^3*((4*d*((A*b^10 + 10*B*a*b^9)/e^4 - (4*B*b^10*d)/e^5))/e - (5*a*b^8*(2*A*b +
 9*B*a))/e^4 + (6*B*b^10*d^2)/e^6))/e^3 + (42*a^4*b^5*(5*A*b + 6*B*a))/e^4))/e + (d^4*((4*d*((A*b^10 + 10*B*a*
b^9)/e^4 - (4*B*b^10*d)/e^5))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^4 + (6*B*b^10*d^2)/e^6))/(2*e^4) + (21*a^5*b^4*(
6*A*b + 5*B*a))/e^4) - x^6*((2*d*((A*b^10 + 10*B*a*b^9)/e^4 - (4*B*b^10*d)/e^5))/(3*e) - (5*a*b^8*(2*A*b + 9*B
*a))/(6*e^4) + (B*b^10*d^2)/e^6) - ((2*A*a^10*e^11 - 299*B*b^10*d^11 + 242*A*b^10*d^10*e + B*a^10*d*e^10 - 191
0*A*a*b^9*d^9*e^2 + 20*B*a^9*b*d^2*e^9 + 6570*A*a^2*b^8*d^8*e^3 - 12840*A*a^3*b^7*d^7*e^4 + 15540*A*a^4*b^6*d^
6*e^5 - 11844*A*a^5*b^5*d^5*e^6 + 5460*A*a^6*b^4*d^4*e^7 - 1320*A*a^7*b^3*d^3*e^8 + 90*A*a^8*b^2*d^2*e^9 - 859
5*B*a^2*b^8*d^9*e^2 + 17520*B*a^3*b^7*d^8*e^3 - 22470*B*a^4*b^6*d^7*e^4 + 18648*B*a^5*b^5*d^6*e^5 - 9870*B*a^6
*b^4*d^5*e^6 + 3120*B*a^7*b^3*d^4*e^7 - 495*B*a^8*b^2*d^3*e^8 + 10*A*a^9*b*d*e^10 + 2420*B*a*b^9*d^10*e)/(6*e)
 + x*((B*a^10*e^10)/2 - (209*B*b^10*d^10)/2 + 5*A*a^9*b*e^10 + 85*A*b^10*d^9*e - 675*A*a*b^9*d^8*e^2 + 45*A*a^
8*b^2*d*e^9 + 2340*A*a^2*b^8*d^7*e^3 - 4620*A*a^3*b^7*d^6*e^4 + 5670*A*a^4*b^6*d^5*e^5 - 4410*A*a^5*b^5*d^4*e^
6 + 2100*A*a^6*b^4*d^3*e^7 - 540*A*a^7*b^3*d^2*e^8 - (6075*B*a^2*b^8*d^8*e^2)/2 + 6240*B*a^3*b^7*d^7*e^3 - 808
5*B*a^4*b^6*d^6*e^4 + 6804*B*a^5*b^5*d^5*e^5 - 3675*B*a^6*b^4*d^4*e^6 + 1200*B*a^7*b^3*d^3*e^7 - (405*B*a^8*b^
2*d^2*e^8)/2 + 850*B*a*b^9*d^9*e + 10*B*a^9*b*d*e^9) + x^2*(10*B*a^9*b*e^10 - 55*B*b^10*d^9*e + 45*A*a^8*b^2*e
^10 + 45*A*b^10*d^8*e^2 - 360*A*a*b^9*d^7*e^3 - 360*A*a^7*b^3*d*e^9 + 450*B*a*b^9*d^8*e^2 - 135*B*a^8*b^2*d*e^
9 + 1260*A*a^2*b^8*d^6*e^4 - 2520*A*a^3*b^7*d^5*e^5 + 3150*A*a^4*b^6*d^4*e^6 - 2520*A*a^5*b^5*d^3*e^7 + 1260*A
*a^6*b^4*d^2*e^8 - 1620*B*a^2*b^8*d^7*e^3 + 3360*B*a^3*b^7*d^6*e^4 - 4410*B*a^4*b^6*d^5*e^5 + 3780*B*a^5*b^5*d
^4*e^6 - 2100*B*a^6*b^4*d^3*e^7 + 720*B*a^7*b^3*d^2*e^8))/(d^3*e^11 + e^14*x^3 + 3*d^2*e^12*x + 3*d*e^13*x^2)
- x*((6*d^2*((6*d^2*((6*d^2*((A*b^10 + 10*B*a*b^9)/e^4 - (4*B*b^10*d)/e^5))/e^2 - (4*d*((4*d*((A*b^10 + 10*B*a
*b^9)/e^4 - (4*B*b^10*d)/e^5))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^4 + (6*B*b^10*d^2)/e^6))/e - (15*a^2*b^7*(3*A*b
 + 8*B*a))/e^4 + (4*B*b^10*d^3)/e^7))/e^2 - (4*d*((4*d*((6*d^2*((A*b^10 + 10*B*a*b^9)/e^4 - (4*B*b^10*d)/e^5))
/e^2 - (4*d*((4*d*((A*b^10 + 10*B*a*b^9)/e^4 - (4*B*b^10*d)/e^5))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^4 + (6*B*b^1
0*d^2)/e^6))/e - (15*a^2*b^7*(3*A*b + 8*B*a))/e^4 + (4*B*b^10*d^3)/e^7))/e - (4*d^3*((A*b^10 + 10*B*a*b^9)/e^4
 - (4*B*b^10*d)/e^5))/e^3 + (6*d^2*((4*d*((A*b^10 + 10*B*a*b^9)/e^4 - (4*B*b^10*d)/e^5))/e - (5*a*b^8*(2*A*b +
 9*B*a))/e^4 + (6*B*b^10*d^2)/e^6))/e^2 + (30*a^3*b^6*(4*A*b + 7*B*a))/e^4 - (B*b^10*d^4)/e^8))/e - (d^4*((A*b
^10 + 10*B*a*b^9)/e^4 - (4*B*b^10*d)/e^5))/e^4 + (4*d^3*((4*d*((A*b^10 + 10*B*a*b^9)/e^4 - (4*B*b^10*d)/e^5))/
e - (5*a*b^8*(2*A*b + 9*B*a))/e^4 + (6*B*b^10*d^2)/e^6))/e^3 + (42*a^4*b^5*(5*A*b + 6*B*a))/e^4))/e^2 - (d^4*(
(6*d^2*((A*b^10 + 10*B*a*b^9)/e^4 - (4*B*b^10*d)/e^5))/e^2 - (4*d*((4*d*((A*b^10 + 10*B*a*b^9)/e^4 - (4*B*b^10
*d)/e^5))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^4 + (6*B*b^10*d^2)/e^6))/e - (15*a^2*b^7*(3*A*b + 8*B*a))/e^4 + (4*B
*b^10*d^3)/e^7))/e^4 + (4*d^3*((4*d*((6*d^2*((A*b^10 + 10*B*a*b^9)/e^4 - (4*B*b^10*d)/e^5))/e^2 - (4*d*((4*d*(
(A*b^10 + 10*B*a*b^9)/e^4 - (4*B*b^10*d)/e^5))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^4 + (6*B*b^10*d^2)/e^6))/e - (1
5*a^2*b^7*(3*A*b + 8*B*a))/e^4 + (4*B*b^10*d^3)/e^7))/e - (4*d^3*((A*b^10 + 10*B*a*b^9)/e^4 - (4*B*b^10*d)/e^5
))/e^3 + (6*d^2*((4*d*((A*b^10 + 10*B*a*b^9)/e^4 - (4*B*b^10*d)/e^5))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^4 + (6*B
*b^10*d^2)/e^6))/e^2 + (30*a^3*b^6*(4*A*b + 7*B*a))/e^4 - (B*b^10*d^4)/e^8))/e^3 + (4*d*((4*d^3*((6*d^2*((A*b^
10 + 10*B*a*b^9)/e^4 - (4*B*b^10*d)/e^5))/e^2 - (4*d*((4*d*((A*b^10 + 10*B*a*b^9)/e^4 - (4*B*b^10*d)/e^5))/e -
 (5*a*b^8*(2*A*b + 9*B*a))/e^4 + (6*B*b^10*d^2)/e^6))/e - (15*a^2*b^7*(3*A*b + 8*B*a))/e^4 + (4*B*b^10*d^3)/e^
7))/e^3 - (6*d^2*((4*d*((6*d^2*((A*b^10 + 10*B*a*b^9)/e^4 - (4*B*b^10*d)/e^5))/e^2 - (4*d*((4*d*((A*b^10 + 10*
B*a*b^9)/e^4 - (4*B*b^10*d)/e^5))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^4 + (6*B*b^10*d^2)/e^6))/e - (15*a^2*b^7*(3*
A*b + 8*B*a))/e^4 + (4*B*b^10*d^3)/e^7))/e - (4*d^3*((A*b^10 + 10*B*a*b^9)/e^4 - (4*B*b^10*d)/e^5))/e^3 + (6*d
^2*((4*d*((A*b^10 + 10*B*a*b^9)/e^4 - (4*B*b^10*d)/e^5))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^4 + (6*B*b^10*d^2)/e^
6))/e^2 + (30*a^3*b^6*(4*A*b + 7*B*a))/e^4 - (B*b^10*d^4)/e^8))/e^2 - (4*d*((6*d^2*((6*d^2*((A*b^10 + 10*B*a*b
^9)/e^4 - (4*B*b^10*d)/e^5))/e^2 - (4*d*((4*d*((A*b^10 + 10*B*a*b^9)/e^4 - (4*B*b^10*d)/e^5))/e - (5*a*b^8*(2*
A*b + 9*B*a))/e^4 + (6*B*b^10*d^2)/e^6))/e - (15*a^2*b^7*(3*A*b + 8*B*a))/e^4 + (4*B*b^10*d^3)/e^7))/e^2 - (4*
d*((4*d*((6*d^2*((A*b^10 + 10*B*a*b^9)/e^4 - (4*B*b^10*d)/e^5))/e^2 - (4*d*((4*d*((A*b^10 + 10*B*a*b^9)/e^4 -
(4*B*b^10*d)/e^5))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^4 + (6*B*b^10*d^2)/e^6))/e - (15*a^2*b^7*(3*A*b + 8*B*a))/e
^4 + (4*B*b^10*d^3)/e^7))/e - (4*d^3*((A*b^10 + 10*B*a*b^9)/e^4 - (4*B*b^10*d)/e^5))/e^3 + (6*d^2*((4*d*((A*b^
10 + 10*B*a*b^9)/e^4 - (4*B*b^10*d)/e^5))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^4 + (6*B*b^10*d^2)/e^6))/e^2 + (30*a
^3*b^6*(4*A*b + 7*B*a))/e^4 - (B*b^10*d^4)/e^8))/e - (d^4*((A*b^10 + 10*B*a*b^9)/e^4 - (4*B*b^10*d)/e^5))/e^4
+ (4*d^3*((4*d*((A*b^10 + 10*B*a*b^9)/e^4 - (4*B*b^10*d)/e^5))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^4 + (6*B*b^10*d
^2)/e^6))/e^3 + (42*a^4*b^5*(5*A*b + 6*B*a))/e^4))/e + (d^4*((4*d*((A*b^10 + 10*B*a*b^9)/e^4 - (4*B*b^10*d)/e^
5))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^4 + (6*B*b^10*d^2)/e^6))/e^4 + (42*a^5*b^4*(6*A*b + 5*B*a))/e^4))/e - (30*
a^6*b^3*(7*A*b + 4*B*a))/e^4) - x^5*((6*d^2*((A*b^10 + 10*B*a*b^9)/e^4 - (4*B*b^10*d)/e^5))/(5*e^2) - (4*d*((4
*d*((A*b^10 + 10*B*a*b^9)/e^4 - (4*B*b^10*d)/e^5))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^4 + (6*B*b^10*d^2)/e^6))/(5
*e) - (3*a^2*b^7*(3*A*b + 8*B*a))/e^4 + (4*B*b^10*d^3)/(5*e^7)) + x^3*((2*d^2*((6*d^2*((A*b^10 + 10*B*a*b^9)/e
^4 - (4*B*b^10*d)/e^5))/e^2 - (4*d*((4*d*((A*b^10 + 10*B*a*b^9)/e^4 - (4*B*b^10*d)/e^5))/e - (5*a*b^8*(2*A*b +
 9*B*a))/e^4 + (6*B*b^10*d^2)/e^6))/e - (15*a^2*b^7*(3*A*b + 8*B*a))/e^4 + (4*B*b^10*d^3)/e^7))/e^2 - (4*d*((4
*d*((6*d^2*((A*b^10 + 10*B*a*b^9)/e^4 - (4*B*b^10*d)/e^5))/e^2 - (4*d*((4*d*((A*b^10 + 10*B*a*b^9)/e^4 - (4*B*
b^10*d)/e^5))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^4 + (6*B*b^10*d^2)/e^6))/e - (15*a^2*b^7*(3*A*b + 8*B*a))/e^4 +
(4*B*b^10*d^3)/e^7))/e - (4*d^3*((A*b^10 + 10*B*a*b^9)/e^4 - (4*B*b^10*d)/e^5))/e^3 + (6*d^2*((4*d*((A*b^10 +
10*B*a*b^9)/e^4 - (4*B*b^10*d)/e^5))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^4 + (6*B*b^10*d^2)/e^6))/e^2 + (30*a^3*b^
6*(4*A*b + 7*B*a))/e^4 - (B*b^10*d^4)/e^8))/(3*e) - (d^4*((A*b^10 + 10*B*a*b^9)/e^4 - (4*B*b^10*d)/e^5))/(3*e^
4) + (4*d^3*((4*d*((A*b^10 + 10*B*a*b^9)/e^4 - (4*B*b^10*d)/e^5))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^4 + (6*B*b^1
0*d^2)/e^6))/(3*e^3) + (14*a^4*b^5*(5*A*b + 6*B*a))/e^4) + x^7*((A*b^10 + 10*B*a*b^9)/(7*e^4) - (4*B*b^10*d)/(
7*e^5)) + x^4*((d*((6*d^2*((A*b^10 + 10*B*a*b^9)/e^4 - (4*B*b^10*d)/e^5))/e^2 - (4*d*((4*d*((A*b^10 + 10*B*a*b
^9)/e^4 - (4*B*b^10*d)/e^5))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^4 + (6*B*b^10*d^2)/e^6))/e - (15*a^2*b^7*(3*A*b +
 8*B*a))/e^4 + (4*B*b^10*d^3)/e^7))/e - (d^3*((A*b^10 + 10*B*a*b^9)/e^4 - (4*B*b^10*d)/e^5))/e^3 + (3*d^2*((4*
d*((A*b^10 + 10*B*a*b^9)/e^4 - (4*B*b^10*d)/e^5))/e - (5*a*b^8*(2*A*b + 9*B*a))/e^4 + (6*B*b^10*d^2)/e^6))/(2*
e^2) + (15*a^3*b^6*(4*A*b + 7*B*a))/(2*e^4) - (B*b^10*d^4)/(4*e^8)) + (log(d + e*x)*(165*B*b^10*d^8 - 120*A*b^
10*d^7*e + 120*A*a^7*b^3*e^8 + 45*B*a^8*b^2*e^8 + 840*A*a*b^9*d^6*e^2 - 840*A*a^6*b^4*d*e^7 - 480*B*a^7*b^3*d*
e^7 - 2520*A*a^2*b^8*d^5*e^3 + 4200*A*a^3*b^7*d^4*e^4 - 4200*A*a^4*b^6*d^3*e^5 + 2520*A*a^5*b^5*d^2*e^6 + 3780
*B*a^2*b^8*d^6*e^2 - 6720*B*a^3*b^7*d^5*e^3 + 7350*B*a^4*b^6*d^4*e^4 - 5040*B*a^5*b^5*d^3*e^5 + 2100*B*a^6*b^4
*d^2*e^6 - 1200*B*a*b^9*d^7*e))/e^12 + (B*b^10*x^8)/(8*e^4)

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sympy [F(-1)]  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)**4,x)

[Out]

Timed out

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