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3.1079 \(\int \frac{(a+b x)^{10} (A+B x)}{(d+e x)^8} \, dx\)

Optimal. Leaf size=444 \[ -\frac{b^9 (d+e x)^3 (-10 a B e-A b e+11 b B d)}{3 e^{12}}+\frac{5 b^8 (d+e x)^2 (b d-a e) (-9 a B e-2 A b e+11 b B d)}{2 e^{12}}-\frac{15 b^7 x (b d-a e)^2 (-8 a B e-3 A b e+11 b B d)}{e^{11}}+\frac{30 b^6 (b d-a e)^3 \log (d+e x) (-7 a B e-4 A b e+11 b B d)}{e^{12}}+\frac{42 b^5 (b d-a e)^4 (-6 a B e-5 A b e+11 b B d)}{e^{12} (d+e x)}-\frac{21 b^4 (b d-a e)^5 (-5 a B e-6 A b e+11 b B d)}{e^{12} (d+e x)^2}+\frac{10 b^3 (b d-a e)^6 (-4 a B e-7 A b e+11 b B d)}{e^{12} (d+e x)^3}-\frac{15 b^2 (b d-a e)^7 (-3 a B e-8 A b e+11 b B d)}{4 e^{12} (d+e x)^4}+\frac{b (b d-a e)^8 (-2 a B e-9 A b e+11 b B d)}{e^{12} (d+e x)^5}-\frac{(b d-a e)^9 (-a B e-10 A b e+11 b B d)}{6 e^{12} (d+e x)^6}+\frac{(b d-a e)^{10} (B d-A e)}{7 e^{12} (d+e x)^7}+\frac{b^{10} B (d+e x)^4}{4 e^{12}} \]

[Out]

(-15*b^7*(b*d - a*e)^2*(11*b*B*d - 3*A*b*e - 8*a*B*e)*x)/e^11 + ((b*d - a*e)^10*
(B*d - A*e))/(7*e^12*(d + e*x)^7) - ((b*d - a*e)^9*(11*b*B*d - 10*A*b*e - a*B*e)
)/(6*e^12*(d + e*x)^6) + (b*(b*d - a*e)^8*(11*b*B*d - 9*A*b*e - 2*a*B*e))/(e^12*
(d + e*x)^5) - (15*b^2*(b*d - a*e)^7*(11*b*B*d - 8*A*b*e - 3*a*B*e))/(4*e^12*(d
+ e*x)^4) + (10*b^3*(b*d - a*e)^6*(11*b*B*d - 7*A*b*e - 4*a*B*e))/(e^12*(d + e*x
)^3) - (21*b^4*(b*d - a*e)^5*(11*b*B*d - 6*A*b*e - 5*a*B*e))/(e^12*(d + e*x)^2)
+ (42*b^5*(b*d - a*e)^4*(11*b*B*d - 5*A*b*e - 6*a*B*e))/(e^12*(d + e*x)) + (5*b^
8*(b*d - a*e)*(11*b*B*d - 2*A*b*e - 9*a*B*e)*(d + e*x)^2)/(2*e^12) - (b^9*(11*b*
B*d - A*b*e - 10*a*B*e)*(d + e*x)^3)/(3*e^12) + (b^10*B*(d + e*x)^4)/(4*e^12) +
(30*b^6*(b*d - a*e)^3*(11*b*B*d - 4*A*b*e - 7*a*B*e)*Log[d + e*x])/e^12

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Rubi [A]  time = 2.83424, antiderivative size = 444, 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 \[ -\frac{b^9 (d+e x)^3 (-10 a B e-A b e+11 b B d)}{3 e^{12}}+\frac{5 b^8 (d+e x)^2 (b d-a e) (-9 a B e-2 A b e+11 b B d)}{2 e^{12}}-\frac{15 b^7 x (b d-a e)^2 (-8 a B e-3 A b e+11 b B d)}{e^{11}}+\frac{30 b^6 (b d-a e)^3 \log (d+e x) (-7 a B e-4 A b e+11 b B d)}{e^{12}}+\frac{42 b^5 (b d-a e)^4 (-6 a B e-5 A b e+11 b B d)}{e^{12} (d+e x)}-\frac{21 b^4 (b d-a e)^5 (-5 a B e-6 A b e+11 b B d)}{e^{12} (d+e x)^2}+\frac{10 b^3 (b d-a e)^6 (-4 a B e-7 A b e+11 b B d)}{e^{12} (d+e x)^3}-\frac{15 b^2 (b d-a e)^7 (-3 a B e-8 A b e+11 b B d)}{4 e^{12} (d+e x)^4}+\frac{b (b d-a e)^8 (-2 a B e-9 A b e+11 b B d)}{e^{12} (d+e x)^5}-\frac{(b d-a e)^9 (-a B e-10 A b e+11 b B d)}{6 e^{12} (d+e x)^6}+\frac{(b d-a e)^{10} (B d-A e)}{7 e^{12} (d+e x)^7}+\frac{b^{10} B (d+e x)^4}{4 e^{12}} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)^10*(A + B*x))/(d + e*x)^8,x]

[Out]

(-15*b^7*(b*d - a*e)^2*(11*b*B*d - 3*A*b*e - 8*a*B*e)*x)/e^11 + ((b*d - a*e)^10*
(B*d - A*e))/(7*e^12*(d + e*x)^7) - ((b*d - a*e)^9*(11*b*B*d - 10*A*b*e - a*B*e)
)/(6*e^12*(d + e*x)^6) + (b*(b*d - a*e)^8*(11*b*B*d - 9*A*b*e - 2*a*B*e))/(e^12*
(d + e*x)^5) - (15*b^2*(b*d - a*e)^7*(11*b*B*d - 8*A*b*e - 3*a*B*e))/(4*e^12*(d
+ e*x)^4) + (10*b^3*(b*d - a*e)^6*(11*b*B*d - 7*A*b*e - 4*a*B*e))/(e^12*(d + e*x
)^3) - (21*b^4*(b*d - a*e)^5*(11*b*B*d - 6*A*b*e - 5*a*B*e))/(e^12*(d + e*x)^2)
+ (42*b^5*(b*d - a*e)^4*(11*b*B*d - 5*A*b*e - 6*a*B*e))/(e^12*(d + e*x)) + (5*b^
8*(b*d - a*e)*(11*b*B*d - 2*A*b*e - 9*a*B*e)*(d + e*x)^2)/(2*e^12) - (b^9*(11*b*
B*d - A*b*e - 10*a*B*e)*(d + e*x)^3)/(3*e^12) + (b^10*B*(d + e*x)^4)/(4*e^12) +
(30*b^6*(b*d - a*e)^3*(11*b*B*d - 4*A*b*e - 7*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)**8,x)

[Out]

Timed out

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Mathematica [A]  time = 1.06658, size = 450, normalized size = 1.01 \[ \frac{-42 b^8 e^2 x^2 \left (-45 a^2 B e^2-10 a b e (A e-8 B d)+4 b^2 d (2 A e-9 B d)\right )+84 b^7 e x \left (120 a^3 B e^3+45 a^2 b e^2 (A e-8 B d)+40 a b^2 d e (9 B d-2 A e)+12 b^3 d^2 (3 A e-10 B d)\right )+28 b^9 e^3 x^3 (10 a B e+A b e-8 b B d)+2520 b^6 (b d-a e)^3 \log (d+e x) (-7 a B e-4 A b e+11 b B d)+\frac{3528 b^5 (b d-a e)^4 (-6 a B e-5 A b e+11 b B d)}{d+e x}-\frac{1764 b^4 (b d-a e)^5 (-5 a B e-6 A b e+11 b B d)}{(d+e x)^2}+\frac{840 b^3 (b d-a e)^6 (-4 a B e-7 A b e+11 b B d)}{(d+e x)^3}-\frac{315 b^2 (b d-a e)^7 (-3 a B e-8 A b e+11 b B d)}{(d+e x)^4}+\frac{84 b (b d-a e)^8 (-2 a B e-9 A b e+11 b B d)}{(d+e x)^5}-\frac{14 (b d-a e)^9 (-a B e-10 A b e+11 b B d)}{(d+e x)^6}+\frac{12 (b d-a e)^{10} (B d-A e)}{(d+e x)^7}+21 b^{10} B e^4 x^4}{84 e^{12}} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)^10*(A + B*x))/(d + e*x)^8,x]

[Out]

(84*b^7*e*(120*a^3*B*e^3 + 40*a*b^2*d*e*(9*B*d - 2*A*e) + 45*a^2*b*e^2*(-8*B*d +
 A*e) + 12*b^3*d^2*(-10*B*d + 3*A*e))*x - 42*b^8*e^2*(-45*a^2*B*e^2 - 10*a*b*e*(
-8*B*d + A*e) + 4*b^2*d*(-9*B*d + 2*A*e))*x^2 + 28*b^9*e^3*(-8*b*B*d + A*b*e + 1
0*a*B*e)*x^3 + 21*b^10*B*e^4*x^4 + (12*(b*d - a*e)^10*(B*d - A*e))/(d + e*x)^7 -
 (14*(b*d - a*e)^9*(11*b*B*d - 10*A*b*e - a*B*e))/(d + e*x)^6 + (84*b*(b*d - a*e
)^8*(11*b*B*d - 9*A*b*e - 2*a*B*e))/(d + e*x)^5 - (315*b^2*(b*d - a*e)^7*(11*b*B
*d - 8*A*b*e - 3*a*B*e))/(d + e*x)^4 + (840*b^3*(b*d - a*e)^6*(11*b*B*d - 7*A*b*
e - 4*a*B*e))/(d + e*x)^3 - (1764*b^4*(b*d - a*e)^5*(11*b*B*d - 6*A*b*e - 5*a*B*
e))/(d + e*x)^2 + (3528*b^5*(b*d - a*e)^4*(11*b*B*d - 5*A*b*e - 6*a*B*e))/(d + e
*x) + 2520*b^6*(b*d - a*e)^3*(11*b*B*d - 4*A*b*e - 7*a*B*e)*Log[d + e*x])/(84*e^
12)

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Maple [B]  time = 0.048, size = 2823, 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)^8,x)

[Out]

-140/e^8/(e*x+d)^6*A*a^3*b^7*d^6+60/e^9/(e*x+d)^6*A*a^2*b^8*d^7-15/e^10/(e*x+d)^
6*A*a*b^9*d^8+10/3/e^3/(e*x+d)^6*B*a^9*b*d-45/2/e^4/(e*x+d)^6*B*a^8*b^2*d^2+80/e
^5/(e*x+d)^6*B*a^7*b^3*d^3-175/e^6/(e*x+d)^6*B*a^6*b^4*d^4+252/e^7/(e*x+d)^6*B*a
^5*b^5*d^5-245/e^8/(e*x+d)^6*B*a^4*b^6*d^6+160/e^9/(e*x+d)^6*B*a^3*b^7*d^7-135/2
/e^10/(e*x+d)^6*B*a^2*b^8*d^8+50/3/e^11/(e*x+d)^6*B*a*b^9*d^9+10/7/e^2/(e*x+d)^7
*A*d*a^9*b-45/7/e^3/(e*x+d)^7*A*d^2*a^8*b^2+120/7/e^4/(e*x+d)^7*A*d^3*a^7*b^3-30
/e^5/(e*x+d)^7*A*d^4*a^6*b^4+36/e^6/(e*x+d)^7*A*d^5*a^5*b^5-30/e^7/(e*x+d)^7*A*d
^6*a^4*b^6+120/7/e^8/(e*x+d)^7*A*d^7*a^3*b^7-45/7/e^9/(e*x+d)^7*A*a^2*b^8*d^8+10
/7/e^10/(e*x+d)^7*A*a*b^9*d^9-10/7/e^3/(e*x+d)^7*B*d^2*a^9*b+45/7/e^4/(e*x+d)^7*
B*d^3*a^8*b^2-120/7/e^5/(e*x+d)^7*B*d^4*a^7*b^3+30/e^6/(e*x+d)^7*B*d^5*a^6*b^4-3
6/e^7/(e*x+d)^7*B*d^6*a^5*b^5+30/e^8/(e*x+d)^7*B*d^7*a^4*b^6-120/7/e^9/(e*x+d)^7
*B*a^3*b^7*d^8+45/7/e^10/(e*x+d)^7*B*a^2*b^8*d^9-10/7/e^11/(e*x+d)^7*B*a*b^9*d^1
0+420*b^5/e^6/(e*x+d)^3*A*a^5*d-1050*b^6/e^7/(e*x+d)^3*A*a^4*d^2+1400*b^7/e^8/(e
*x+d)^3*A*a^3*d^3-1050*b^8/e^9/(e*x+d)^3*A*a^2*d^4+420*b^9/e^10/(e*x+d)^3*A*a*d^
5+350*b^4/e^6/(e*x+d)^3*B*a^6*d-1260*b^5/e^7/(e*x+d)^3*B*a^5*d^2+2450*b^6/e^8/(e
*x+d)^3*B*a^4*d^3-2800*b^7/e^9/(e*x+d)^3*B*a^3*d^4+1890*b^8/e^10/(e*x+d)^3*B*a^2
*d^5-700*b^9/e^11/(e*x+d)^3*B*a*d^6+840*b^7/e^8/(e*x+d)*A*a^3*d-1260*b^8/e^9/(e*
x+d)*A*a^2*d^2+840*b^9/e^10/(e*x+d)*A*a*d^3+1470*b^6/e^8/(e*x+d)*B*a^4*d-3360*b^
7/e^9/(e*x+d)*B*a^3*d^2+3780*b^8/e^10/(e*x+d)*B*a^2*d^3-2100*b^9/e^11/(e*x+d)*B*
a*d^4+630*b^6/e^7/(e*x+d)^2*A*a^4*d-1260*b^7/e^8/(e*x+d)^2*A*a^3*d^2+1260*b^8/e^
9/(e*x+d)^2*A*a^2*d^3-630*b^9/e^10/(e*x+d)^2*A*a*d^4+756*b^5/e^7/(e*x+d)^2*B*a^5
*d-2205*b^6/e^8/(e*x+d)^2*B*a^4*d^2+3360*b^7/e^9/(e*x+d)^2*B*a^3*d^3-2835*b^8/e^
10/(e*x+d)^2*B*a^2*d^4+1260*b^9/e^11/(e*x+d)^2*B*a*d^5+210*b^4/e^5/(e*x+d)^4*A*a
^6*d-630*b^5/e^6/(e*x+d)^4*A*a^5*d^2+1050*b^6/e^7/(e*x+d)^4*A*a^4*d^3-1050*b^7/e
^8/(e*x+d)^4*A*a^3*d^4+630*b^8/e^9/(e*x+d)^4*A*a^2*d^5-210*b^9/e^10/(e*x+d)^4*A*
a*d^6+120*b^3/e^5/(e*x+d)^4*B*a^7*d-525*b^4/e^6/(e*x+d)^4*B*a^6*d^2+1260*b^5/e^7
/(e*x+d)^4*B*a^5*d^3-3675/2*b^6/e^8/(e*x+d)^4*B*a^4*d^4+1680*b^7/e^9/(e*x+d)^4*B
*a^3*d^5-945*b^8/e^10/(e*x+d)^4*B*a^2*d^6+300*b^9/e^11/(e*x+d)^4*B*a*d^7+72*b^3/
e^4/(e*x+d)^5*A*a^7*d-252*b^4/e^5/(e*x+d)^5*A*a^6*d^2+504*b^5/e^6/(e*x+d)^5*A*a^
5*d^3-630*b^6/e^7/(e*x+d)^5*A*a^4*d^4+504*b^7/e^8/(e*x+d)^5*A*a^3*d^5-252*b^8/e^
9/(e*x+d)^5*A*a^2*d^6+72*b^9/e^10/(e*x+d)^5*A*a*d^7+27*b^2/e^4/(e*x+d)^5*B*a^8*d
-144*b^3/e^5/(e*x+d)^5*B*a^7*d^2+420*b^4/e^6/(e*x+d)^5*B*a^6*d^3-756*b^5/e^7/(e*
x+d)^5*B*a^5*d^4+882*b^6/e^8/(e*x+d)^5*B*a^4*d^5-672*b^7/e^9/(e*x+d)^5*B*a^3*d^6
+324*b^8/e^10/(e*x+d)^5*B*a^2*d^7-90*b^9/e^11/(e*x+d)^5*B*a*d^8-2*b/e^3/(e*x+d)^
5*B*a^9+11*b^10/e^12/(e*x+d)^5*B*d^9+10/3*b^9/e^8*B*x^3*a-8/3*b^10/e^9*B*x^3*d+5
*b^9/e^8*A*x^2*a-4*b^10/e^9*A*x^2*d+45/2*b^8/e^8*B*x^2*a^2+18*b^10/e^10*B*x^2*d^
2+45*b^8/e^8*A*a^2*x+36*b^10/e^10*A*d^2*x+120*b^7/e^8*B*a^3*x-120*b^10/e^11*B*d^
3*x+120*b^7/e^8*ln(e*x+d)*A*a^3-120*b^10/e^11*ln(e*x+d)*A*d^3+210*b^6/e^8*ln(e*x
+d)*B*a^4+330*b^10/e^12*ln(e*x+d)*B*d^4-5/3/e^2/(e*x+d)^6*A*a^9*b+5/3/e^11/(e*x+
d)^6*A*b^10*d^9-11/6/e^12/(e*x+d)^6*b^10*B*d^10-1/7/e^11/(e*x+d)^7*A*b^10*d^10+1
/7/e^2/(e*x+d)^7*B*d*a^10+1/7/e^12/(e*x+d)^7*b^10*B*d^11-70*b^4/e^5/(e*x+d)^3*A*
a^6-70*b^10/e^11/(e*x+d)^3*A*d^6-40*b^3/e^5/(e*x+d)^3*B*a^7+110*b^10/e^12/(e*x+d
)^3*B*d^7-30*b^3/e^4/(e*x+d)^4*A*a^7-1/6/e^2/(e*x+d)^6*B*a^10-1/7/e/(e*x+d)^7*a^
10*A+1/4*b^10/e^8*B*x^4+1/3*b^10/e^8*A*x^3-9*b^2/e^3/(e*x+d)^5*A*a^8-9*b^10/e^11
/(e*x+d)^5*A*d^8+30*b^10/e^11/(e*x+d)^4*A*d^7-45/4*b^2/e^4/(e*x+d)^4*B*a^8-165/4
*b^10/e^12/(e*x+d)^4*B*d^8-210*b^6/e^7/(e*x+d)*A*a^4-210*b^10/e^11/(e*x+d)*A*d^4
-252*b^5/e^7/(e*x+d)*B*a^5+462*b^10/e^12/(e*x+d)*B*d^5-126*b^5/e^6/(e*x+d)^2*A*a
^5+126*b^10/e^11/(e*x+d)^2*A*d^5-105*b^4/e^6/(e*x+d)^2*B*a^6-231*b^10/e^12/(e*x+
d)^2*B*d^6-40*b^9/e^9*B*x^2*a*d-80*b^9/e^9*A*a*d*x-360*b^8/e^9*B*a^2*d*x+360*b^9
/e^10*B*a*d^2*x-360*b^8/e^9*ln(e*x+d)*A*a^2*d+360*b^9/e^10*ln(e*x+d)*A*a*d^2-960
*b^7/e^9*ln(e*x+d)*B*a^3*d+1620*b^8/e^10*ln(e*x+d)*B*a^2*d^2-1200*b^9/e^11*ln(e*
x+d)*B*a*d^3+15/e^3/(e*x+d)^6*A*a^8*b^2*d-60/e^4/(e*x+d)^6*A*a^7*b^3*d^2+140/e^5
/(e*x+d)^6*A*a^6*b^4*d^3-210/e^6/(e*x+d)^6*A*a^5*b^5*d^4+210/e^7/(e*x+d)^6*A*a^4
*b^6*d^5

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Maxima [A]  time = 1.67412, size = 2542, normalized size = 5.73 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^10/(e*x + d)^8,x, algorithm="maxima")

[Out]

1/84*(25961*B*b^10*d^11 - 12*A*a^10*e^11 - 11044*(10*B*a*b^9 + A*b^10)*d^10*e +
20094*(9*B*a^2*b^8 + 2*A*a*b^9)*d^9*e^2 - 17316*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^8*
e^3 + 6534*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^7*e^4 - 504*(6*B*a^5*b^5 + 5*A*a^4*b^6)
*d^6*e^5 - 84*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^5*e^6 - 24*(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 - 4*(2*B*a^9*b + 9*A*a^8*b^2)
*d^2*e^9 - 2*(B*a^10 + 10*A*a^9*b)*d*e^10 + 3528*(11*B*b^10*d^5*e^6 - 5*(10*B*a*
b^9 + A*b^10)*d^4*e^7 + 10*(9*B*a^2*b^8 + 2*A*a*b^9)*d^3*e^8 - 10*(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 + 1764*(121*B*b^10*d^6*e^5 - 54*(10*B*a*b^9 + A*b^10)*d^5*e
^6 + 105*(9*B*a^2*b^8 + 2*A*a*b^9)*d^4*e^7 - 100*(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 - 6*(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*(1177*B*b^10*d^7*e^4 - 518*(1
0*B*a*b^9 + A*b^10)*d^6*e^5 + 987*(9*B*a^2*b^8 + 2*A*a*b^9)*d^5*e^6 - 910*(8*B*a
^3*b^7 + 3*A*a^2*b^8)*d^4*e^7 + 385*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^3*e^8 - 42*(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 - 2*(4*B
*a^7*b^3 + 7*A*a^6*b^4)*e^11)*x^4 + 105*(5863*B*b^10*d^8*e^3 - 2552*(10*B*a*b^9
+ A*b^10)*d^7*e^4 + 4788*(9*B*a^2*b^8 + 2*A*a*b^9)*d^6*e^5 - 4312*(8*B*a^3*b^7 +
 3*A*a^2*b^8)*d^5*e^6 + 1750*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^4*e^7 - 168*(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*(3*B*a^8*b^2 + 8*A*a^7*b^3)*e^11)*x^3 + 21*(20669
*B*b^10*d^9*e^2 - 8916*(10*B*a*b^9 + A*b^10)*d^8*e^3 + 16524*(9*B*a^2*b^8 + 2*A*
a*b^9)*d^7*e^4 - 14616*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^6*e^5 + 5754*(7*B*a^4*b^6 +
 4*A*a^3*b^7)*d^5*e^6 - 504*(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 - 24*(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 - 4*(2*B*a^9*b + 9*A*a^8*b^2)*e^11)*x^2 + 7*(23441*B*b^
10*d^10*e - 10036*(10*B*a*b^9 + A*b^10)*d^9*e^2 + 18414*(9*B*a^2*b^8 + 2*A*a*b^9
)*d^8*e^3 - 16056*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^7*e^4 + 6174*(7*B*a^4*b^6 + 4*A*
a^3*b^7)*d^6*e^5 - 504*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^5*e^6 - 84*(5*B*a^6*b^4 + 6
*A*a^5*b^5)*d^4*e^7 - 24*(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 - 4*(2*B*a^9*b + 9*A*a^8*b^2)*d*e^10 - 2*(B*a^10 + 10*A*a^9
*b)*e^11)*x)/(e^19*x^7 + 7*d*e^18*x^6 + 21*d^2*e^17*x^5 + 35*d^3*e^16*x^4 + 35*d
^4*e^15*x^3 + 21*d^5*e^14*x^2 + 7*d^6*e^13*x + d^7*e^12) + 1/12*(3*B*b^10*e^3*x^
4 - 4*(8*B*b^10*d*e^2 - (10*B*a*b^9 + A*b^10)*e^3)*x^3 + 6*(36*B*b^10*d^2*e - 8*
(10*B*a*b^9 + A*b^10)*d*e^2 + 5*(9*B*a^2*b^8 + 2*A*a*b^9)*e^3)*x^2 - 12*(120*B*b
^10*d^3 - 36*(10*B*a*b^9 + A*b^10)*d^2*e + 40*(9*B*a^2*b^8 + 2*A*a*b^9)*d*e^2 -
15*(8*B*a^3*b^7 + 3*A*a^2*b^8)*e^3)*x)/e^11 + 30*(11*B*b^10*d^4 - 4*(10*B*a*b^9
+ A*b^10)*d^3*e + 6*(9*B*a^2*b^8 + 2*A*a*b^9)*d^2*e^2 - 4*(8*B*a^3*b^7 + 3*A*a^2
*b^8)*d*e^3 + (7*B*a^4*b^6 + 4*A*a^3*b^7)*e^4)*log(e*x + d)/e^12

_______________________________________________________________________________________

Fricas [A]  time = 0.232631, size = 3757, normalized size = 8.46 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^10/(e*x + d)^8,x, algorithm="fricas")

[Out]

1/84*(21*B*b^10*e^11*x^11 + 25961*B*b^10*d^11 - 12*A*a^10*e^11 - 11044*(10*B*a*b
^9 + A*b^10)*d^10*e + 20094*(9*B*a^2*b^8 + 2*A*a*b^9)*d^9*e^2 - 17316*(8*B*a^3*b
^7 + 3*A*a^2*b^8)*d^8*e^3 + 6534*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^7*e^4 - 504*(6*B*
a^5*b^5 + 5*A*a^4*b^6)*d^6*e^5 - 84*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^5*e^6 - 24*(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 - 4*(2*
B*a^9*b + 9*A*a^8*b^2)*d^2*e^9 - 2*(B*a^10 + 10*A*a^9*b)*d*e^10 - 7*(11*B*b^10*d
*e^10 - 4*(10*B*a*b^9 + A*b^10)*e^11)*x^10 + 35*(11*B*b^10*d^2*e^9 - 4*(10*B*a*b
^9 + A*b^10)*d*e^10 + 6*(9*B*a^2*b^8 + 2*A*a*b^9)*e^11)*x^9 - 315*(11*B*b^10*d^3
*e^8 - 4*(10*B*a*b^9 + A*b^10)*d^2*e^9 + 6*(9*B*a^2*b^8 + 2*A*a*b^9)*d*e^10 - 4*
(8*B*a^3*b^7 + 3*A*a^2*b^8)*e^11)*x^8 - 49*(937*B*b^10*d^4*e^7 - 308*(10*B*a*b^9
 + A*b^10)*d^3*e^8 + 390*(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)*x^7 - 49*(2599*B*b^10*d^5*e^6 - 716*(10*B*a*b^9 + A*b^10)*d
^4*e^7 + 570*(9*B*a^2*b^8 + 2*A*a*b^9)*d^3*e^8 + 180*(8*B*a^3*b^7 + 3*A*a^2*b^8)
*d^2*e^9 - 360*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d*e^10 + 72*(6*B*a^5*b^5 + 5*A*a^4*b^
6)*e^11)*x^6 - 147*(619*B*b^10*d^6*e^5 + 4*(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 + 900*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^3*e^8 - 540
*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^2*e^9 + 72*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d*e^10 + 1
2*(5*B*a^6*b^4 + 6*A*a^5*b^5)*e^11)*x^5 + 35*(4907*B*b^10*d^7*e^4 - 3388*(10*B*a
*b^9 + A*b^10)*d^6*e^5 + 8610*(9*B*a^2*b^8 + 2*A*a*b^9)*d^5*e^6 - 9660*(8*B*a^3*
b^7 + 3*A*a^2*b^8)*d^4*e^7 + 4620*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^3*e^8 - 504*(6*B
*a^5*b^5 + 5*A*a^4*b^6)*d^2*e^9 - 84*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d*e^10 - 24*(4*
B*a^7*b^3 + 7*A*a^6*b^4)*e^11)*x^4 + 35*(11837*B*b^10*d^8*e^3 - 5908*(10*B*a*b^9
 + A*b^10)*d^7*e^4 + 12390*(9*B*a^2*b^8 + 2*A*a*b^9)*d^6*e^5 - 12180*(8*B*a^3*b^
7 + 3*A*a^2*b^8)*d^5*e^6 + 5250*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^4*e^7 - 504*(6*B*a
^5*b^5 + 5*A*a^4*b^6)*d^3*e^8 - 84*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^2*e^9 - 24*(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 + 21*(1
7381*B*b^10*d^9*e^2 - 7924*(10*B*a*b^9 + A*b^10)*d^8*e^3 + 15414*(9*B*a^2*b^8 +
2*A*a*b^9)*d^7*e^4 - 14196*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^6*e^5 + 5754*(7*B*a^4*b
^6 + 4*A*a^3*b^7)*d^5*e^6 - 504*(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 - 24*(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 - 4*(2*B*a^9*b + 9*A*a^8*b^2)*e^11)*x^2 + 7*(22001*
B*b^10*d^10*e - 9604*(10*B*a*b^9 + A*b^10)*d^9*e^2 + 17934*(9*B*a^2*b^8 + 2*A*a*
b^9)*d^8*e^3 - 15876*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^7*e^4 + 6174*(7*B*a^4*b^6 + 4
*A*a^3*b^7)*d^6*e^5 - 504*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^5*e^6 - 84*(5*B*a^6*b^4
+ 6*A*a^5*b^5)*d^4*e^7 - 24*(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 - 4*(2*B*a^9*b + 9*A*a^8*b^2)*d*e^10 - 2*(B*a^10 + 10*A*
a^9*b)*e^11)*x + 2520*(11*B*b^10*d^11 - 4*(10*B*a*b^9 + A*b^10)*d^10*e + 6*(9*B*
a^2*b^8 + 2*A*a*b^9)*d^9*e^2 - 4*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^8*e^3 + (7*B*a^4*
b^6 + 4*A*a^3*b^7)*d^7*e^4 + (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 + 7*(11*B*b^10*d^5*e^6 - 4*(10*B*a*b^9 +
A*b^10)*d^4*e^7 + 6*(9*B*a^2*b^8 + 2*A*a*b^9)*d^3*e^8 - 4*(8*B*a^3*b^7 + 3*A*a^2
*b^8)*d^2*e^9 + (7*B*a^4*b^6 + 4*A*a^3*b^7)*d*e^10)*x^6 + 21*(11*B*b^10*d^6*e^5
- 4*(10*B*a*b^9 + A*b^10)*d^5*e^6 + 6*(9*B*a^2*b^8 + 2*A*a*b^9)*d^4*e^7 - 4*(8*B
*a^3*b^7 + 3*A*a^2*b^8)*d^3*e^8 + (7*B*a^4*b^6 + 4*A*a^3*b^7)*d^2*e^9)*x^5 + 35*
(11*B*b^10*d^7*e^4 - 4*(10*B*a*b^9 + A*b^10)*d^6*e^5 + 6*(9*B*a^2*b^8 + 2*A*a*b^
9)*d^5*e^6 - 4*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^4*e^7 + (7*B*a^4*b^6 + 4*A*a^3*b^7)
*d^3*e^8)*x^4 + 35*(11*B*b^10*d^8*e^3 - 4*(10*B*a*b^9 + A*b^10)*d^7*e^4 + 6*(9*B
*a^2*b^8 + 2*A*a*b^9)*d^6*e^5 - 4*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^5*e^6 + (7*B*a^4
*b^6 + 4*A*a^3*b^7)*d^4*e^7)*x^3 + 21*(11*B*b^10*d^9*e^2 - 4*(10*B*a*b^9 + A*b^1
0)*d^8*e^3 + 6*(9*B*a^2*b^8 + 2*A*a*b^9)*d^7*e^4 - 4*(8*B*a^3*b^7 + 3*A*a^2*b^8)
*d^6*e^5 + (7*B*a^4*b^6 + 4*A*a^3*b^7)*d^5*e^6)*x^2 + 7*(11*B*b^10*d^10*e - 4*(1
0*B*a*b^9 + A*b^10)*d^9*e^2 + 6*(9*B*a^2*b^8 + 2*A*a*b^9)*d^8*e^3 - 4*(8*B*a^3*b
^7 + 3*A*a^2*b^8)*d^7*e^4 + (7*B*a^4*b^6 + 4*A*a^3*b^7)*d^6*e^5)*x)*log(e*x + d)
)/(e^19*x^7 + 7*d*e^18*x^6 + 21*d^2*e^17*x^5 + 35*d^3*e^16*x^4 + 35*d^4*e^15*x^3
 + 21*d^5*e^14*x^2 + 7*d^6*e^13*x + d^7*e^12)

_______________________________________________________________________________________

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)**8,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.217106, 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)^8,x, algorithm="giac")

[Out]

Done

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