3.1075 \(\int \frac{(a+b x)^{10} (A+B x)}{(d+e x)^4} \, dx\)

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

[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^1
2*(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 - (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 +
(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)*Log[d + e*x])/e^12

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Rubi [A]  time = 4.41532, antiderivative size = 445, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05 \[ -\frac{b^9 (d+e x)^7 (-10 a B e-A b e+11 b B d)}{7 e^{12}}+\frac{5 b^8 (d+e x)^6 (b d-a e) (-9 a B e-2 A b e+11 b B d)}{6 e^{12}}-\frac{3 b^7 (d+e x)^5 (b d-a e)^2 (-8 a B e-3 A b e+11 b B d)}{e^{12}}+\frac{15 b^6 (d+e x)^4 (b d-a e)^3 (-7 a B e-4 A b e+11 b B d)}{2 e^{12}}-\frac{14 b^5 (d+e x)^3 (b d-a e)^4 (-6 a B e-5 A b e+11 b B d)}{e^{12}}+\frac{21 b^4 (d+e x)^2 (b d-a e)^5 (-5 a B e-6 A b e+11 b B d)}{e^{12}}-\frac{30 b^3 x (b d-a e)^6 (-4 a B e-7 A b e+11 b B d)}{e^{11}}+\frac{15 b^2 (b d-a e)^7 \log (d+e x) (-3 a B e-8 A b e+11 b B d)}{e^{12}}+\frac{5 b (b d-a e)^8 (-2 a B e-9 A b e+11 b B d)}{e^{12} (d+e x)}-\frac{(b d-a e)^9 (-a B e-10 A b e+11 b B d)}{2 e^{12} (d+e x)^2}+\frac{(b d-a e)^{10} (B d-A e)}{3 e^{12} (d+e x)^3}+\frac{b^{10} B (d+e x)^8}{8 e^{12}} \]

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^1
2*(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 - (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 +
(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)*Log[d + e*x])/e^12

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Mathematica [A]  time = 1.00022, size = 814, normalized size = 1.83 \[ \frac{21 B e^8 x^8 b^{10}+24 e^7 (-4 b B d+A b e+10 a B e) x^7 b^9-28 e^6 \left (2 d (2 A e-5 B d) b^2-10 a e (A e-4 B d) b-45 a^2 B e^2\right ) x^6 b^8+168 e^5 \left (2 d^2 (A e-2 B d) b^3+4 a d e (5 B d-2 A e) b^2+9 a^2 e^2 (A e-4 B d) b+24 a^3 B e^3\right ) x^5 b^7-210 e^4 \left (d^3 (4 A e-7 B d) b^4+20 a d^2 e (2 B d-A e) b^3+18 a^2 d e^2 (2 A e-5 B d) b^2-24 a^3 e^3 (A e-4 B d) b-42 a^4 B e^4\right ) x^4 b^6+56 e^3 \left (-7 d^4 (8 B d-5 A e) b^5+50 a d^3 e (7 B d-4 A e) b^4-450 a^2 d^2 e^2 (2 B d-A e) b^3+240 a^3 d e^3 (5 B d-2 A e) b^2+210 a^4 e^4 (A e-4 B d) b+252 a^5 B e^5\right ) x^3 b^5-84 e^2 \left (28 d^5 (2 A e-3 B d) b^6+70 a d^4 e (8 B d-5 A e) b^5-225 a^2 d^3 e^2 (7 B d-4 A e) b^4+1200 a^3 d^2 e^3 (2 B d-A e) b^3-420 a^4 d e^4 (5 B d-2 A e) b^2-252 a^5 e^5 (A e-4 B d) b-210 a^6 B e^6\right ) x^2 b^4+168 e \left (12 d^6 (7 A e-10 B d) b^7+280 a d^5 e (3 B d-2 A e) b^6-315 a^2 d^4 e^2 (8 B d-5 A e) b^5+600 a^3 d^3 e^3 (7 B d-4 A e) b^4-2100 a^4 d^2 e^4 (2 B d-A e) b^3+504 a^5 d e^5 (5 B d-2 A e) b^2+210 a^6 e^6 (A e-4 B d) b+120 a^7 B e^7\right ) x b^3+2520 (b d-a e)^7 (11 b B d-8 A b e-3 a B e) \log (d+e x) b^2+\frac{840 (b d-a e)^8 (11 b B d-9 A b e-2 a B e) b}{d+e x}-\frac{84 (b d-a e)^9 (11 b B d-10 A b e-a B e)}{(d+e x)^2}+\frac{56 (b d-a e)^{10} (B d-A e)}{(d+e x)^3}}{168 e^{12}} \]

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 - 28*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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Maple [B]  time = 0.042, size = 2607, normalized size = 5.9 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

105*b^4/e^4*B*x^2*a^6+42*b^10/e^10*B*x^2*d^6+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-1/2/e^2/(e*x+d)^2*B*a^10-1/3/e^11/(e*x+d)^3*A*b^10*d^1
0+1/3/e^2/(e*x+d)^3*B*d*a^10+1/3/e^12/(e*x+d)^3*b^10*B*d^11-5/e^2/(e*x+d)^2*A*a^
9*b+5/e^11/(e*x+d)^2*A*b^10*d^9-11/2/e^12/(e*x+d)^2*b^10*B*d^10-45*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+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^1
0/e^6*A*x^5*d^2+120*b^3/e^4*ln(e*x+d)*A*a^7-120*b^10/e^11*ln(e*x+d)*A*d^7+84*b^1
0/e^10*A*d^6*x+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^11/(e*x+d)^3*B*a*b^9*d^10+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+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^4-20/3*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+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^3-160*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^4-4200*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-420*b^6/e^5*A*x^2*a^4*d+600*b^7/e^6*A*x^2*a^3*d^2-45
0*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+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^7-405/2/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^9-1008*b^5/e^5*A*a^5*d*x+2100*b^6/e^6*A*a^4*d^2*x-2400*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-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+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+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+120*b^3/
e^4*B*a^7*x-120*b^10/e^11*B*d^7*x+24*b^7/e^4*B*x^5*a^3-4*b^10/e^7*B*x^5*d^3+30*b
^7/e^4*A*x^4*a^3-5*b^10/e^7*A*x^4*d^3+105/2*b^6/e^4*B*x^4*a^4+35/4*b^10/e^8*B*x^
4*d^4+210*b^4/e^4*A*a^6*x+45*b^2/e^4*ln(e*x+d)*B*a^8+165*b^10/e^12*ln(e*x+d)*B*d
^8+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^5-28*b^10/e^9*A*x^2*d^5

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Maxima [A]  time = 1.47122, size = 2483, normalized size = 5.58 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^10/(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 + 321
0*(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 + 165*(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^10 - (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^1
0)*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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Fricas [A]  time = 0.251136, size = 3648, normalized size = 8.2 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^10/(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 - 2
1840*(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^10*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*(1
1*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 - 63
0*(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 - 358*(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 + 10*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)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

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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GIAC/XCAS [A]  time = 0.21584, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^10/(e*x + d)^4,x, algorithm="giac")

[Out]

Done