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3.1080 \(\int \frac{(a+b x)^{10} (A+B x)}{(d+e x)^9} \, dx\)

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

[Out]

(5*b^8*(b*d - a*e)*(11*b*B*d - 2*A*b*e - 9*a*B*e)*x)/e^11 + ((b*d - a*e)^10*(B*d
 - A*e))/(8*e^12*(d + e*x)^8) - ((b*d - a*e)^9*(11*b*B*d - 10*A*b*e - a*B*e))/(7
*e^12*(d + e*x)^7) + (5*b*(b*d - a*e)^8*(11*b*B*d - 9*A*b*e - 2*a*B*e))/(6*e^12*
(d + e*x)^6) - (3*b^2*(b*d - a*e)^7*(11*b*B*d - 8*A*b*e - 3*a*B*e))/(e^12*(d + e
*x)^5) + (15*b^3*(b*d - a*e)^6*(11*b*B*d - 7*A*b*e - 4*a*B*e))/(2*e^12*(d + e*x)
^4) - (14*b^4*(b*d - a*e)^5*(11*b*B*d - 6*A*b*e - 5*a*B*e))/(e^12*(d + e*x)^3) +
 (21*b^5*(b*d - a*e)^4*(11*b*B*d - 5*A*b*e - 6*a*B*e))/(e^12*(d + e*x)^2) - (30*
b^6*(b*d - a*e)^3*(11*b*B*d - 4*A*b*e - 7*a*B*e))/(e^12*(d + e*x)) - (b^9*(11*b*
B*d - A*b*e - 10*a*B*e)*(d + e*x)^2)/(2*e^12) + (b^10*B*(d + e*x)^3)/(3*e^12) -
(15*b^7*(b*d - a*e)^2*(11*b*B*d - 3*A*b*e - 8*a*B*e)*Log[d + e*x])/e^12

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Rubi [A]  time = 2.61882, 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 \[ -\frac{b^9 (d+e x)^2 (-10 a B e-A b e+11 b B d)}{2 e^{12}}+\frac{5 b^8 x (b d-a e) (-9 a B e-2 A b e+11 b B d)}{e^{11}}-\frac{15 b^7 (b d-a e)^2 \log (d+e x) (-8 a B e-3 A b e+11 b B d)}{e^{12}}-\frac{30 b^6 (b d-a e)^3 (-7 a B e-4 A b e+11 b B d)}{e^{12} (d+e x)}+\frac{21 b^5 (b d-a e)^4 (-6 a B e-5 A b e+11 b B d)}{e^{12} (d+e x)^2}-\frac{14 b^4 (b d-a e)^5 (-5 a B e-6 A b e+11 b B d)}{e^{12} (d+e x)^3}+\frac{15 b^3 (b d-a e)^6 (-4 a B e-7 A b e+11 b B d)}{2 e^{12} (d+e x)^4}-\frac{3 b^2 (b d-a e)^7 (-3 a B e-8 A b e+11 b B d)}{e^{12} (d+e x)^5}+\frac{5 b (b d-a e)^8 (-2 a B e-9 A b e+11 b B d)}{6 e^{12} (d+e x)^6}-\frac{(b d-a e)^9 (-a B e-10 A b e+11 b B d)}{7 e^{12} (d+e x)^7}+\frac{(b d-a e)^{10} (B d-A e)}{8 e^{12} (d+e x)^8}+\frac{b^{10} B (d+e x)^3}{3 e^{12}} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)^10*(A + B*x))/(d + e*x)^9,x]

[Out]

(5*b^8*(b*d - a*e)*(11*b*B*d - 2*A*b*e - 9*a*B*e)*x)/e^11 + ((b*d - a*e)^10*(B*d
 - A*e))/(8*e^12*(d + e*x)^8) - ((b*d - a*e)^9*(11*b*B*d - 10*A*b*e - a*B*e))/(7
*e^12*(d + e*x)^7) + (5*b*(b*d - a*e)^8*(11*b*B*d - 9*A*b*e - 2*a*B*e))/(6*e^12*
(d + e*x)^6) - (3*b^2*(b*d - a*e)^7*(11*b*B*d - 8*A*b*e - 3*a*B*e))/(e^12*(d + e
*x)^5) + (15*b^3*(b*d - a*e)^6*(11*b*B*d - 7*A*b*e - 4*a*B*e))/(2*e^12*(d + e*x)
^4) - (14*b^4*(b*d - a*e)^5*(11*b*B*d - 6*A*b*e - 5*a*B*e))/(e^12*(d + e*x)^3) +
 (21*b^5*(b*d - a*e)^4*(11*b*B*d - 5*A*b*e - 6*a*B*e))/(e^12*(d + e*x)^2) - (30*
b^6*(b*d - a*e)^3*(11*b*B*d - 4*A*b*e - 7*a*B*e))/(e^12*(d + e*x)) - (b^9*(11*b*
B*d - A*b*e - 10*a*B*e)*(d + e*x)^2)/(2*e^12) + (b^10*B*(d + e*x)^3)/(3*e^12) -
(15*b^7*(b*d - a*e)^2*(11*b*B*d - 3*A*b*e - 8*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)**9,x)

[Out]

Timed out

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Mathematica [A]  time = 1.13854, size = 415, normalized size = 0.93 \[ \frac{-168 b^8 e x \left (-45 a^2 B e^2-10 a b e (A e-9 B d)+9 b^2 d (A e-5 B d)\right )+84 b^9 e^2 x^2 (10 a B e+A b e-9 b B d)-2520 b^7 (b d-a e)^2 \log (d+e x) (-8 a B e-3 A b e+11 b B d)-\frac{5040 b^6 (b d-a e)^3 (-7 a B e-4 A b e+11 b B d)}{d+e x}+\frac{3528 b^5 (b d-a e)^4 (-6 a B e-5 A b e+11 b B d)}{(d+e x)^2}-\frac{2352 b^4 (b d-a e)^5 (-5 a B e-6 A b e+11 b B d)}{(d+e x)^3}+\frac{1260 b^3 (b d-a e)^6 (-4 a B e-7 A b e+11 b B d)}{(d+e x)^4}-\frac{504 b^2 (b d-a e)^7 (-3 a B e-8 A b e+11 b B d)}{(d+e x)^5}+\frac{140 b (b d-a e)^8 (-2 a B e-9 A b e+11 b B d)}{(d+e x)^6}-\frac{24 (b d-a e)^9 (-a B e-10 A b e+11 b B d)}{(d+e x)^7}+\frac{21 (b d-a e)^{10} (B d-A e)}{(d+e x)^8}+56 b^{10} B e^3 x^3}{168 e^{12}} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)^10*(A + B*x))/(d + e*x)^9,x]

[Out]

(-168*b^8*e*(-45*a^2*B*e^2 - 10*a*b*e*(-9*B*d + A*e) + 9*b^2*d*(-5*B*d + A*e))*x
 + 84*b^9*e^2*(-9*b*B*d + A*b*e + 10*a*B*e)*x^2 + 56*b^10*B*e^3*x^3 + (21*(b*d -
 a*e)^10*(B*d - A*e))/(d + e*x)^8 - (24*(b*d - a*e)^9*(11*b*B*d - 10*A*b*e - a*B
*e))/(d + e*x)^7 + (140*b*(b*d - a*e)^8*(11*b*B*d - 9*A*b*e - 2*a*B*e))/(d + e*x
)^6 - (504*b^2*(b*d - a*e)^7*(11*b*B*d - 8*A*b*e - 3*a*B*e))/(d + e*x)^5 + (1260
*b^3*(b*d - a*e)^6*(11*b*B*d - 7*A*b*e - 4*a*B*e))/(d + e*x)^4 - (2352*b^4*(b*d
- a*e)^5*(11*b*B*d - 6*A*b*e - 5*a*B*e))/(d + e*x)^3 + (3528*b^5*(b*d - a*e)^4*(
11*b*B*d - 5*A*b*e - 6*a*B*e))/(d + e*x)^2 - (5040*b^6*(b*d - a*e)^3*(11*b*B*d -
 4*A*b*e - 7*a*B*e))/(d + e*x) - 2520*b^7*(b*d - a*e)^2*(11*b*B*d - 3*A*b*e - 8*
a*B*e)*Log[d + e*x])/(168*e^12)

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Maple [B]  time = 0.044, size = 2857, normalized size = 6.4 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^10*(B*x+A)/(e*x+d)^9,x)

[Out]

1/2*b^10/e^9*A*x^2-1/8/e/(e*x+d)^8*a^10*A-1/7/e^2/(e*x+d)^7*B*a^10+1/3*b^10/e^9*
B*x^3-330*b^10/e^12/(e*x+d)*B*d^4-105*b^6/e^7/(e*x+d)^2*A*a^4-105*b^10/e^11/(e*x
+d)^2*A*d^4-126*b^5/e^7/(e*x+d)^2*B*a^5+231*b^10/e^12/(e*x+d)^2*B*d^5-105/2*b^4/
e^5/(e*x+d)^4*A*a^6-105/2*b^10/e^11/(e*x+d)^4*A*d^6+120*b^10/e^11/(e*x+d)*A*d^3-
210*b^6/e^8/(e*x+d)*B*a^4-45/8/e^3/(e*x+d)^8*A*d^2*a^8*b^2+15/e^4/(e*x+d)^8*A*d^
3*a^7*b^3-105/4/e^5/(e*x+d)^8*A*d^4*a^6*b^4+63/2/e^6/(e*x+d)^8*A*d^5*a^5*b^5-105
/4/e^7/(e*x+d)^8*A*d^6*a^4*b^6+15/e^8/(e*x+d)^8*A*d^7*a^3*b^7-45/8/e^9/(e*x+d)^8
*A*d^8*a^2*b^8+5/4/e^10/(e*x+d)^8*A*a*b^9*d^9-5/4/e^3/(e*x+d)^8*B*d^2*a^9*b+45/8
/e^4/(e*x+d)^8*B*d^3*a^8*b^2-15/e^5/(e*x+d)^8*B*d^4*a^7*b^3+105/4/e^6/(e*x+d)^8*
B*d^5*a^6*b^4-63/2/e^7/(e*x+d)^8*B*d^6*a^5*b^5+105/4/e^8/(e*x+d)^8*B*d^7*a^4*b^6
-15/e^9/(e*x+d)^8*B*d^8*a^3*b^7+45/8/e^10/(e*x+d)^8*B*a^2*b^8*d^9-5/4/e^11/(e*x+
d)^8*B*a*b^9*d^10+90/7/e^3/(e*x+d)^7*A*a^8*b^2*d-360/7/e^4/(e*x+d)^7*A*a^7*b^3*d
^2+120/e^5/(e*x+d)^7*A*a^6*b^4*d^3-180/e^6/(e*x+d)^7*A*a^5*b^5*d^4+180/e^7/(e*x+
d)^7*A*a^4*b^6*d^5-120/e^8/(e*x+d)^7*A*a^3*b^7*d^6+360/7/e^9/(e*x+d)^7*A*a^2*b^8
*d^7-90/7/e^10/(e*x+d)^7*A*a*b^9*d^8+20/7/e^3/(e*x+d)^7*B*a^9*b*d-135/7/e^4/(e*x
+d)^7*B*a^8*b^2*d^2+480/7/e^5/(e*x+d)^7*B*a^7*b^3*d^3-150/e^6/(e*x+d)^7*B*a^6*b^
4*d^4+216/e^7/(e*x+d)^7*B*a^5*b^5*d^5-210/e^8/(e*x+d)^7*B*a^4*b^6*d^6+960/7/e^9/
(e*x+d)^7*B*a^3*b^7*d^7-405/7/e^10/(e*x+d)^7*B*a^2*b^8*d^8+100/7/e^11/(e*x+d)^7*
B*a*b^9*d^9+525/2*b^4/e^6/(e*x+d)^4*B*a^6*d-945*b^5/e^7/(e*x+d)^4*B*a^5*d^2+3675
/2*b^6/e^8/(e*x+d)^4*B*a^4*d^3-2100*b^7/e^9/(e*x+d)^4*B*a^3*d^4+2835/2*b^8/e^10/
(e*x+d)^4*B*a^2*d^5-525*b^9/e^11/(e*x+d)^4*B*a*d^6-90*b^9/e^10*B*a*d*x-90*b^9/e^
10*ln(e*x+d)*A*a*d-405*b^8/e^10*ln(e*x+d)*B*a^2*d+450*b^9/e^11*ln(e*x+d)*B*a*d^2
+60*b^3/e^4/(e*x+d)^6*A*a^7*d-210*b^4/e^5/(e*x+d)^6*A*a^6*d^2+420*b^5/e^6/(e*x+d
)^6*A*a^5*d^3-525*b^6/e^7/(e*x+d)^6*A*a^4*d^4+420*b^7/e^8/(e*x+d)^6*A*a^3*d^5-21
0*b^8/e^9/(e*x+d)^6*A*a^2*d^6+60*b^9/e^10/(e*x+d)^6*A*a*d^7+45/2*b^2/e^4/(e*x+d)
^6*B*a^8*d+168*b^4/e^5/(e*x+d)^5*A*a^6*d-504*b^5/e^6/(e*x+d)^5*A*a^5*d^2+840*b^6
/e^7/(e*x+d)^5*A*a^4*d^3-840*b^7/e^8/(e*x+d)^5*A*a^3*d^4+504*b^8/e^9/(e*x+d)^5*A
*a^2*d^5-168*b^9/e^10/(e*x+d)^5*A*a*d^6+96*b^3/e^5/(e*x+d)^5*B*a^7*d-420*b^4/e^6
/(e*x+d)^5*B*a^6*d^2+1008*b^5/e^7/(e*x+d)^5*B*a^5*d^3-1470*b^6/e^8/(e*x+d)^5*B*a
^4*d^4+1344*b^7/e^9/(e*x+d)^5*B*a^3*d^5-756*b^8/e^10/(e*x+d)^5*B*a^2*d^6+240*b^9
/e^11/(e*x+d)^5*B*a*d^7-120*b^3/e^5/(e*x+d)^6*B*a^7*d^2+350*b^4/e^6/(e*x+d)^6*B*
a^6*d^3-630*b^5/e^7/(e*x+d)^6*B*a^5*d^4+735*b^6/e^8/(e*x+d)^6*B*a^4*d^5-560*b^7/
e^9/(e*x+d)^6*B*a^3*d^6+270*b^8/e^10/(e*x+d)^6*B*a^2*d^7-75*b^9/e^11/(e*x+d)^6*B
*a*d^8+5/4/e^2/(e*x+d)^8*A*d*a^9*b+420*b^6/e^7/(e*x+d)^3*A*a^4*d-840*b^7/e^8/(e*
x+d)^3*A*a^3*d^2+840*b^8/e^9/(e*x+d)^3*A*a^2*d^3-420*b^9/e^10/(e*x+d)^3*A*a*d^4+
504*b^5/e^7/(e*x+d)^3*B*a^5*d-1470*b^6/e^8/(e*x+d)^3*B*a^4*d^2+2240*b^7/e^9/(e*x
+d)^3*B*a^3*d^3-1890*b^8/e^10/(e*x+d)^3*B*a^2*d^4+840*b^9/e^11/(e*x+d)^3*B*a*d^5
+360*b^8/e^9/(e*x+d)*A*a^2*d-360*b^9/e^10/(e*x+d)*A*a*d^2+960*b^7/e^9/(e*x+d)*B*
a^3*d-1620*b^8/e^10/(e*x+d)*B*a^2*d^2+1200*b^9/e^11/(e*x+d)*B*a*d^3+420*b^7/e^8/
(e*x+d)^2*A*a^3*d-630*b^8/e^9/(e*x+d)^2*A*a^2*d^2+420*b^9/e^10/(e*x+d)^2*A*a*d^3
+735*b^6/e^8/(e*x+d)^2*B*a^4*d-1680*b^7/e^9/(e*x+d)^2*B*a^3*d^2+1890*b^8/e^10/(e
*x+d)^2*B*a^2*d^3-1050*b^9/e^11/(e*x+d)^2*B*a*d^4+315*b^5/e^6/(e*x+d)^4*A*a^5*d-
1575/2*b^6/e^7/(e*x+d)^4*A*a^4*d^2+1050*b^7/e^8/(e*x+d)^4*A*a^3*d^3-1575/2*b^8/e
^9/(e*x+d)^4*A*a^2*d^4+315*b^9/e^10/(e*x+d)^4*A*a*d^5-9*b^10/e^10*A*d*x+45*b^8/e
^9*B*a^2*x+45*b^10/e^11*B*d^2*x+45*b^8/e^9*ln(e*x+d)*A*a^2+45*b^10/e^11*ln(e*x+d
)*A*d^2+120*b^7/e^9*ln(e*x+d)*B*a^3-165*b^10/e^12*ln(e*x+d)*B*d^3-15/2*b^2/e^3/(
e*x+d)^6*A*a^8-15/2*b^10/e^11/(e*x+d)^6*A*d^8-5/3*b/e^3/(e*x+d)^6*B*a^9+55/6*b^1
0/e^12/(e*x+d)^6*B*d^9-1/8/e^11/(e*x+d)^8*A*b^10*d^10-30*b^3/e^5/(e*x+d)^4*B*a^7
+165/2*b^10/e^12/(e*x+d)^4*B*d^7-24*b^3/e^4/(e*x+d)^5*A*a^7+24*b^10/e^11/(e*x+d)
^5*A*d^7-9*b^2/e^4/(e*x+d)^5*B*a^8-33*b^10/e^12/(e*x+d)^5*B*d^8+5*b^9/e^9*B*x^2*
a-9/2*b^10/e^10*B*x^2*d+10*b^9/e^9*A*a*x+1/8/e^2/(e*x+d)^8*B*d*a^10+1/8/e^12/(e*
x+d)^8*b^10*B*d^11-10/7/e^2/(e*x+d)^7*A*a^9*b+10/7/e^11/(e*x+d)^7*A*b^10*d^9-11/
7/e^12/(e*x+d)^7*b^10*B*d^10-84*b^5/e^6/(e*x+d)^3*A*a^5+84*b^10/e^11/(e*x+d)^3*A
*d^5-70*b^4/e^6/(e*x+d)^3*B*a^6-154*b^10/e^12/(e*x+d)^3*B*d^6-120*b^7/e^8/(e*x+d
)*A*a^3

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Maxima [A]  time = 2.64546, size = 2554, normalized size = 5.74 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^10/(e*x + d)^9,x, algorithm="maxima")

[Out]

-1/168*(32891*B*b^10*d^11 + 21*A*a^10*e^11 - 10803*(10*B*a*b^9 + A*b^10)*d^10*e
+ 13827*(9*B*a^2*b^8 + 2*A*a*b^9)*d^9*e^2 - 6849*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^8
*e^3 + 630*(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 + 42*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^5*e^6 + 18*(4*B*a^7*b^3 + 7*A*a^6*b^
4)*d^4*e^7 + 9*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^3*e^8 + 5*(2*B*a^9*b + 9*A*a^8*b^2)
*d^2*e^9 + 3*(B*a^10 + 10*A*a^9*b)*d*e^10 + 5040*(11*B*b^10*d^4*e^7 - 4*(10*B*a*
b^9 + A*b^10)*d^3*e^8 + 6*(9*B*a^2*b^8 + 2*A*a*b^9)*d^2*e^9 - 4*(8*B*a^3*b^7 + 3
*A*a^2*b^8)*d*e^10 + (7*B*a^4*b^6 + 4*A*a^3*b^7)*e^11)*x^7 + 3528*(99*B*b^10*d^5
*e^6 - 35*(10*B*a*b^9 + A*b^10)*d^4*e^7 + 50*(9*B*a^2*b^8 + 2*A*a*b^9)*d^3*e^8 -
 30*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^2*e^9 + 5*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d*e^10 +
 (6*B*a^5*b^5 + 5*A*a^4*b^6)*e^11)*x^6 + 2352*(407*B*b^10*d^6*e^5 - 141*(10*B*a*
b^9 + A*b^10)*d^5*e^6 + 195*(9*B*a^2*b^8 + 2*A*a*b^9)*d^4*e^7 - 110*(8*B*a^3*b^7
 + 3*A*a^2*b^8)*d^3*e^8 + 15*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^2*e^9 + 3*(6*B*a^5*b^
5 + 5*A*a^4*b^6)*d*e^10 + (5*B*a^6*b^4 + 6*A*a^5*b^5)*e^11)*x^5 + 420*(3509*B*b^
10*d^7*e^4 - 1197*(10*B*a*b^9 + A*b^10)*d^6*e^5 + 1617*(9*B*a^2*b^8 + 2*A*a*b^9)
*d^5*e^6 - 875*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^4*e^7 + 105*(7*B*a^4*b^6 + 4*A*a^3*
b^7)*d^3*e^8 + 21*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^2*e^9 + 7*(5*B*a^6*b^4 + 6*A*a^5
*b^5)*d*e^10 + 3*(4*B*a^7*b^3 + 7*A*a^6*b^4)*e^11)*x^4 + 168*(8173*B*b^10*d^8*e^
3 - 2754*(10*B*a*b^9 + A*b^10)*d^7*e^4 + 3654*(9*B*a^2*b^8 + 2*A*a*b^9)*d^6*e^5
- 1918*(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 + 42*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^3*e^8 + 14*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d
^2*e^9 + 6*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d*e^10 + 3*(3*B*a^8*b^2 + 8*A*a^7*b^3)*e^
11)*x^3 + 28*(27599*B*b^10*d^9*e^2 - 9207*(10*B*a*b^9 + A*b^10)*d^8*e^3 + 12042*
(9*B*a^2*b^8 + 2*A*a*b^9)*d^7*e^4 - 6174*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^6*e^5 + 6
30*(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
 + 42*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^3*e^8 + 18*(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 + 5*(2*B*a^9*b + 9*A*a^8*b^2)*e^11)*x^
2 + 8*(30371*B*b^10*d^10*e - 10047*(10*B*a*b^9 + A*b^10)*d^9*e^2 + 12987*(9*B*a^
2*b^8 + 2*A*a*b^9)*d^8*e^3 - 6534*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^7*e^4 + 630*(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 + 42*(
5*B*a^6*b^4 + 6*A*a^5*b^5)*d^4*e^7 + 18*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^3*e^8 + 9*
(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^2*e^9 + 5*(2*B*a^9*b + 9*A*a^8*b^2)*d*e^10 + 3*(B*
a^10 + 10*A*a^9*b)*e^11)*x)/(e^20*x^8 + 8*d*e^19*x^7 + 28*d^2*e^18*x^6 + 56*d^3*
e^17*x^5 + 70*d^4*e^16*x^4 + 56*d^5*e^15*x^3 + 28*d^6*e^14*x^2 + 8*d^7*e^13*x +
d^8*e^12) + 1/6*(2*B*b^10*e^2*x^3 - 3*(9*B*b^10*d*e - (10*B*a*b^9 + A*b^10)*e^2)
*x^2 + 6*(45*B*b^10*d^2 - 9*(10*B*a*b^9 + A*b^10)*d*e + 5*(9*B*a^2*b^8 + 2*A*a*b
^9)*e^2)*x)/e^11 - 15*(11*B*b^10*d^3 - 3*(10*B*a*b^9 + A*b^10)*d^2*e + 3*(9*B*a^
2*b^8 + 2*A*a*b^9)*d*e^2 - (8*B*a^3*b^7 + 3*A*a^2*b^8)*e^3)*log(e*x + d)/e^12

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Fricas [A]  time = 0.239727, size = 3614, normalized size = 8.12 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^10/(e*x + d)^9,x, algorithm="fricas")

[Out]

1/168*(56*B*b^10*e^11*x^11 - 32891*B*b^10*d^11 - 21*A*a^10*e^11 + 10803*(10*B*a*
b^9 + A*b^10)*d^10*e - 13827*(9*B*a^2*b^8 + 2*A*a*b^9)*d^9*e^2 + 6849*(8*B*a^3*b
^7 + 3*A*a^2*b^8)*d^8*e^3 - 630*(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 - 42*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^5*e^6 - 18*(4*B
*a^7*b^3 + 7*A*a^6*b^4)*d^4*e^7 - 9*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^3*e^8 - 5*(2*B
*a^9*b + 9*A*a^8*b^2)*d^2*e^9 - 3*(B*a^10 + 10*A*a^9*b)*d*e^10 - 28*(11*B*b^10*d
*e^10 - 3*(10*B*a*b^9 + A*b^10)*e^11)*x^10 + 280*(11*B*b^10*d^2*e^9 - 3*(10*B*a*
b^9 + A*b^10)*d*e^10 + 3*(9*B*a^2*b^8 + 2*A*a*b^9)*e^11)*x^9 + 112*(379*B*b^10*d
^3*e^8 - 87*(10*B*a*b^9 + A*b^10)*d^2*e^9 + 60*(9*B*a^2*b^8 + 2*A*a*b^9)*d*e^10)
*x^8 + 112*(1052*B*b^10*d^4*e^7 - 156*(10*B*a*b^9 + A*b^10)*d^3*e^8 - 60*(9*B*a^
2*b^8 + 2*A*a*b^9)*d^2*e^9 + 180*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d*e^10 - 45*(7*B*a^
4*b^6 + 4*A*a^3*b^7)*e^11)*x^7 + 392*(62*B*b^10*d^5*e^6 + 114*(10*B*a*b^9 + A*b^
10)*d^4*e^7 - 330*(9*B*a^2*b^8 + 2*A*a*b^9)*d^3*e^8 + 270*(8*B*a^3*b^7 + 3*A*a^2
*b^8)*d^2*e^9 - 45*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d*e^10 - 9*(6*B*a^5*b^5 + 5*A*a^4
*b^6)*e^11)*x^6 - 784*(598*B*b^10*d^6*e^5 - 294*(10*B*a*b^9 + A*b^10)*d^5*e^6 +
510*(9*B*a^2*b^8 + 2*A*a*b^9)*d^4*e^7 - 330*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^3*e^8
+ 45*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^2*e^9 + 9*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d*e^10
+ 3*(5*B*a^6*b^4 + 6*A*a^5*b^5)*e^11)*x^5 - 140*(7651*B*b^10*d^7*e^4 - 3003*(10*
B*a*b^9 + A*b^10)*d^6*e^5 + 4515*(9*B*a^2*b^8 + 2*A*a*b^9)*d^5*e^6 - 2625*(8*B*a
^3*b^7 + 3*A*a^2*b^8)*d^4*e^7 + 315*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^3*e^8 + 63*(6*
B*a^5*b^5 + 5*A*a^4*b^6)*d^2*e^9 + 21*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d*e^10 + 9*(4*
B*a^7*b^3 + 7*A*a^6*b^4)*e^11)*x^4 - 56*(20846*B*b^10*d^8*e^3 - 7518*(10*B*a*b^9
 + A*b^10)*d^7*e^4 + 10542*(9*B*a^2*b^8 + 2*A*a*b^9)*d^6*e^5 - 5754*(8*B*a^3*b^7
 + 3*A*a^2*b^8)*d^5*e^6 + 630*(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 + 42*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^2*e^9 + 18*(4*B*a
^7*b^3 + 7*A*a^6*b^4)*d*e^10 + 9*(3*B*a^8*b^2 + 8*A*a^7*b^3)*e^11)*x^3 - 28*(254
66*B*b^10*d^9*e^2 - 8778*(10*B*a*b^9 + A*b^10)*d^8*e^3 + 11802*(9*B*a^2*b^8 + 2*
A*a*b^9)*d^7*e^4 - 6174*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^6*e^5 + 630*(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 + 42*(5*B*a^6*b^
4 + 6*A*a^5*b^5)*d^3*e^8 + 18*(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 + 5*(2*B*a^9*b + 9*A*a^8*b^2)*e^11)*x^2 - 8*(29426*B*b^
10*d^10*e - 9858*(10*B*a*b^9 + A*b^10)*d^9*e^2 + 12882*(9*B*a^2*b^8 + 2*A*a*b^9)
*d^8*e^3 - 6534*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^7*e^4 + 630*(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 + 42*(5*B*a^6*b^4 + 6*A*
a^5*b^5)*d^4*e^7 + 18*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^3*e^8 + 9*(3*B*a^8*b^2 + 8*A
*a^7*b^3)*d^2*e^9 + 5*(2*B*a^9*b + 9*A*a^8*b^2)*d*e^10 + 3*(B*a^10 + 10*A*a^9*b)
*e^11)*x - 2520*(11*B*b^10*d^11 - 3*(10*B*a*b^9 + A*b^10)*d^10*e + 3*(9*B*a^2*b^
8 + 2*A*a*b^9)*d^9*e^2 - (8*B*a^3*b^7 + 3*A*a^2*b^8)*d^8*e^3 + (11*B*b^10*d^3*e^
8 - 3*(10*B*a*b^9 + A*b^10)*d^2*e^9 + 3*(9*B*a^2*b^8 + 2*A*a*b^9)*d*e^10 - (8*B*
a^3*b^7 + 3*A*a^2*b^8)*e^11)*x^8 + 8*(11*B*b^10*d^4*e^7 - 3*(10*B*a*b^9 + A*b^10
)*d^3*e^8 + 3*(9*B*a^2*b^8 + 2*A*a*b^9)*d^2*e^9 - (8*B*a^3*b^7 + 3*A*a^2*b^8)*d*
e^10)*x^7 + 28*(11*B*b^10*d^5*e^6 - 3*(10*B*a*b^9 + A*b^10)*d^4*e^7 + 3*(9*B*a^2
*b^8 + 2*A*a*b^9)*d^3*e^8 - (8*B*a^3*b^7 + 3*A*a^2*b^8)*d^2*e^9)*x^6 + 56*(11*B*
b^10*d^6*e^5 - 3*(10*B*a*b^9 + A*b^10)*d^5*e^6 + 3*(9*B*a^2*b^8 + 2*A*a*b^9)*d^4
*e^7 - (8*B*a^3*b^7 + 3*A*a^2*b^8)*d^3*e^8)*x^5 + 70*(11*B*b^10*d^7*e^4 - 3*(10*
B*a*b^9 + A*b^10)*d^6*e^5 + 3*(9*B*a^2*b^8 + 2*A*a*b^9)*d^5*e^6 - (8*B*a^3*b^7 +
 3*A*a^2*b^8)*d^4*e^7)*x^4 + 56*(11*B*b^10*d^8*e^3 - 3*(10*B*a*b^9 + A*b^10)*d^7
*e^4 + 3*(9*B*a^2*b^8 + 2*A*a*b^9)*d^6*e^5 - (8*B*a^3*b^7 + 3*A*a^2*b^8)*d^5*e^6
)*x^3 + 28*(11*B*b^10*d^9*e^2 - 3*(10*B*a*b^9 + A*b^10)*d^8*e^3 + 3*(9*B*a^2*b^8
 + 2*A*a*b^9)*d^7*e^4 - (8*B*a^3*b^7 + 3*A*a^2*b^8)*d^6*e^5)*x^2 + 8*(11*B*b^10*
d^10*e - 3*(10*B*a*b^9 + A*b^10)*d^9*e^2 + 3*(9*B*a^2*b^8 + 2*A*a*b^9)*d^8*e^3 -
 (8*B*a^3*b^7 + 3*A*a^2*b^8)*d^7*e^4)*x)*log(e*x + d))/(e^20*x^8 + 8*d*e^19*x^7
+ 28*d^2*e^18*x^6 + 56*d^3*e^17*x^5 + 70*d^4*e^16*x^4 + 56*d^5*e^15*x^3 + 28*d^6
*e^14*x^2 + 8*d^7*e^13*x + d^8*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)**9,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.213116, 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)^9,x, algorithm="giac")

[Out]

Done

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