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3.1076 \(\int \frac{(a+b x)^{10} (A+B x)}{(d+e x)^5} \, dx\)

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

[Out]

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

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Rubi [A]  time = 3.83138, 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 \[ -\frac{b^9 (d+e x)^6 (-10 a B e-A b e+11 b B d)}{6 e^{12}}+\frac{b^8 (d+e x)^5 (b d-a e) (-9 a B e-2 A b e+11 b B d)}{e^{12}}-\frac{15 b^7 (d+e x)^4 (b d-a e)^2 (-8 a B e-3 A b e+11 b B d)}{4 e^{12}}+\frac{10 b^6 (d+e x)^3 (b d-a e)^3 (-7 a B e-4 A b e+11 b B d)}{e^{12}}-\frac{21 b^5 (d+e x)^2 (b d-a e)^4 (-6 a B e-5 A b e+11 b B d)}{e^{12}}+\frac{42 b^4 x (b d-a e)^5 (-5 a B e-6 A b e+11 b B d)}{e^{11}}-\frac{30 b^3 (b d-a e)^6 \log (d+e x) (-4 a B e-7 A b e+11 b B d)}{e^{12}}-\frac{15 b^2 (b d-a e)^7 (-3 a B e-8 A b e+11 b B d)}{e^{12} (d+e x)}+\frac{5 b (b d-a e)^8 (-2 a B e-9 A b e+11 b B d)}{2 e^{12} (d+e x)^2}-\frac{(b d-a e)^9 (-a B e-10 A b e+11 b B d)}{3 e^{12} (d+e x)^3}+\frac{(b d-a e)^{10} (B d-A e)}{4 e^{12} (d+e x)^4}+\frac{b^{10} B (d+e x)^7}{7 e^{12}} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)^10*(A + B*x))/(d + e*x)^5,x]

[Out]

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

[Out]

Timed out

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Mathematica [A]  time = 0.948639, size = 686, normalized size = 1.55 \[ \frac{-84 b^8 e^5 x^5 \left (-9 a^2 B e^2-2 a b e (A e-5 B d)+b^2 d (A e-3 B d)\right )+105 b^7 e^4 x^4 \left (24 a^3 B e^3+9 a^2 b e^2 (A e-5 B d)+10 a b^2 d e (3 B d-A e)+b^3 d^2 (3 A e-7 B d)\right )-140 b^6 e^3 x^3 \left (-42 a^4 B e^4-24 a^3 b e^3 (A e-5 B d)-45 a^2 b^2 d e^2 (3 B d-A e)+10 a b^3 d^2 e (7 B d-3 A e)+7 b^4 d^3 (A e-2 B d)\right )+42 b^5 e^2 x^2 \left (252 a^5 B e^5+210 a^4 b e^4 (A e-5 B d)+600 a^3 b^2 d e^3 (3 B d-A e)-225 a^2 b^3 d^2 e^2 (7 B d-3 A e)+350 a b^4 d^3 e (2 B d-A e)-14 b^5 d^4 (9 B d-5 A e)\right )-84 b^4 e x \left (-210 a^6 B e^6-252 a^5 b e^5 (A e-5 B d)-1050 a^4 b^2 d e^4 (3 B d-A e)+600 a^3 b^3 d^2 e^3 (7 B d-3 A e)-1575 a^2 b^4 d^3 e^2 (2 B d-A e)+140 a b^5 d^4 e (9 B d-5 A e)-42 b^6 d^5 (5 B d-3 A e)\right )+14 b^9 e^6 x^6 (10 a B e+A b e-5 b B d)-2520 b^3 (b d-a e)^6 \log (d+e x) (-4 a B e-7 A b e+11 b B d)-\frac{1260 b^2 (b d-a e)^7 (-3 a B e-8 A b e+11 b B d)}{d+e x}+\frac{210 b (b d-a e)^8 (-2 a B e-9 A b e+11 b B d)}{(d+e x)^2}-\frac{28 (b d-a e)^9 (-a B e-10 A b e+11 b B d)}{(d+e x)^3}+\frac{21 (b d-a e)^{10} (B d-A e)}{(d+e x)^4}+12 b^{10} B e^7 x^7}{84 e^{12}} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)^10*(A + B*x))/(d + e*x)^5,x]

[Out]

(-84*b^4*e*(-210*a^6*B*e^6 + 140*a*b^5*d^4*e*(9*B*d - 5*A*e) - 42*b^6*d^5*(5*B*d
 - 3*A*e) + 600*a^3*b^3*d^2*e^3*(7*B*d - 3*A*e) - 1575*a^2*b^4*d^3*e^2*(2*B*d -
A*e) - 1050*a^4*b^2*d*e^4*(3*B*d - A*e) - 252*a^5*b*e^5*(-5*B*d + A*e))*x + 42*b
^5*e^2*(252*a^5*B*e^5 - 14*b^5*d^4*(9*B*d - 5*A*e) - 225*a^2*b^3*d^2*e^2*(7*B*d
- 3*A*e) + 350*a*b^4*d^3*e*(2*B*d - A*e) + 600*a^3*b^2*d*e^3*(3*B*d - A*e) + 210
*a^4*b*e^4*(-5*B*d + A*e))*x^2 - 140*b^6*e^3*(-42*a^4*B*e^4 + 10*a*b^3*d^2*e*(7*
B*d - 3*A*e) - 45*a^2*b^2*d*e^2*(3*B*d - A*e) - 24*a^3*b*e^3*(-5*B*d + A*e) + 7*
b^4*d^3*(-2*B*d + A*e))*x^3 + 105*b^7*e^4*(24*a^3*B*e^3 + 10*a*b^2*d*e*(3*B*d -
A*e) + 9*a^2*b*e^2*(-5*B*d + A*e) + b^3*d^2*(-7*B*d + 3*A*e))*x^4 - 84*b^8*e^5*(
-9*a^2*B*e^2 - 2*a*b*e*(-5*B*d + A*e) + b^2*d*(-3*B*d + A*e))*x^5 + 14*b^9*e^6*(
-5*b*B*d + A*b*e + 10*a*B*e)*x^6 + 12*b^10*B*e^7*x^7 + (21*(b*d - a*e)^10*(B*d -
 A*e))/(d + e*x)^4 - (28*(b*d - a*e)^9*(11*b*B*d - 10*A*b*e - a*B*e))/(d + e*x)^
3 + (210*b*(b*d - a*e)^8*(11*b*B*d - 9*A*b*e - 2*a*B*e))/(d + e*x)^2 - (1260*b^2
*(b*d - a*e)^7*(11*b*B*d - 8*A*b*e - 3*a*B*e))/(d + e*x) - 2520*b^3*(b*d - a*e)^
6*(11*b*B*d - 7*A*b*e - 4*a*B*e)*Log[d + e*x])/(84*e^12)

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Maple [B]  time = 0.045, size = 2673, normalized size = 6. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^10*(B*x+A)/(e*x+d)^5,x)

[Out]

1/7*b^10/e^5*B*x^7+1/6*b^10/e^5*A*x^6-1/3/e^2/(e*x+d)^3*B*a^10-1/4/e/(e*x+d)^4*a
^10*A-25/2*b^9/e^6*A*x^4*a*d-10*b^9/e^6*B*x^5*a*d-75*b^8/e^6*A*x^3*a^2*d+50*b^9/
e^7*A*x^3*a*d^2-225/4*b^8/e^6*B*x^4*a^2*d+75/2*b^9/e^7*B*x^4*a*d^2-200*b^7/e^6*B
*x^3*a^3*d+225*b^8/e^7*B*x^3*a^2*d^2-350/3*b^9/e^8*B*x^3*a*d^3+900*b^7/e^7*B*x^2
*a^3*d^2-1575/2*b^8/e^8*B*x^2*a^2*d^3+350*b^9/e^9*B*x^2*a*d^4-1050*b^6/e^6*A*a^4
*d*x+1800*b^7/e^7*A*a^3*d^2*x-1575*b^8/e^8*A*a^2*d^3*x+700*b^9/e^9*A*a*d^4*x-126
0*b^5/e^6*B*a^5*d*x+3150*b^6/e^7*B*a^4*d^2*x-4200*b^7/e^8*B*a^3*d^3*x+3150*b^8/e
^9*B*a^2*d^4*x-1260*b^9/e^10*B*a*d^5*x+675/2*b^8/e^7*A*x^2*a^2*d^2-175*b^9/e^8*A
*x^2*a*d^3-525*b^6/e^6*B*x^2*a^4*d-300*b^7/e^6*A*x^2*a^3*d-1260*b^5/e^6*ln(e*x+d
)*A*a^5*d+3150*b^6/e^7*ln(e*x+d)*A*a^4*d^2-4200*b^7/e^8*ln(e*x+d)*A*a^3*d^3+3150
*b^8/e^9*ln(e*x+d)*A*a^2*d^4-1260*b^9/e^10*ln(e*x+d)*A*a*d^5-1050*b^4/e^6*ln(e*x
+d)*B*a^6*d+3780*b^5/e^7*ln(e*x+d)*B*a^5*d^2-7350*b^6/e^8*ln(e*x+d)*B*a^4*d^3+84
00*b^7/e^9*ln(e*x+d)*B*a^3*d^4-5670*b^8/e^10*ln(e*x+d)*B*a^2*d^5+2100*b^9/e^11*l
n(e*x+d)*B*a*d^6+30/e^3/(e*x+d)^3*A*a^8*b^2*d-120/e^4/(e*x+d)^3*A*a^7*b^3*d^2+28
0/e^5/(e*x+d)^3*A*a^6*b^4*d^3-420/e^6/(e*x+d)^3*A*a^5*b^5*d^4+420/e^7/(e*x+d)^3*
A*a^4*b^6*d^5-280/e^8/(e*x+d)^3*A*a^3*b^7*d^6+120/e^9/(e*x+d)^3*A*a^2*b^8*d^7-30
/e^10/(e*x+d)^3*A*a*b^9*d^8+20/3/e^3/(e*x+d)^3*B*a^9*b*d-45/e^4/(e*x+d)^3*B*a^8*
b^2*d^2+160/e^5/(e*x+d)^3*B*a^7*b^3*d^3-350/e^6/(e*x+d)^3*B*a^6*b^4*d^4+504/e^7/
(e*x+d)^3*B*a^5*b^5*d^5-490/e^8/(e*x+d)^3*B*a^4*b^6*d^6+320/e^9/(e*x+d)^3*B*a^3*
b^7*d^7-135/e^10/(e*x+d)^3*B*a^2*b^8*d^8+100/3/e^11/(e*x+d)^3*B*a*b^9*d^9+840*b^
4/e^5/(e*x+d)*A*a^6*d-2520*b^5/e^6/(e*x+d)*A*a^5*d^2+4200*b^6/e^7/(e*x+d)*A*a^4*
d^3-4200*b^7/e^8/(e*x+d)*A*a^3*d^4+2520*b^8/e^9/(e*x+d)*A*a^2*d^5-840*b^9/e^10/(
e*x+d)*A*a*d^6+480*b^3/e^5/(e*x+d)*B*a^7*d-2100*b^4/e^6/(e*x+d)*B*a^6*d^2+5040*b
^5/e^7/(e*x+d)*B*a^5*d^3-7350*b^6/e^8/(e*x+d)*B*a^4*d^4-126*b^10/e^10*A*d^5*x+25
2*b^5/e^5*A*a^5*x-63*b^10/e^10*B*x^2*d^5+126*b^5/e^5*B*x^2*a^5+210*b^4/e^5*ln(e*
x+d)*A*a^6+210*b^10/e^11*ln(e*x+d)*A*d^6+120*b^3/e^5*ln(e*x+d)*B*a^7-330*b^10/e^
12*ln(e*x+d)*B*d^7+5/3*b^9/e^5*B*x^6*a+35*b^10/e^9*A*x^2*d^4-10/3/e^2/(e*x+d)^3*
A*a^9*b+10/3/e^11/(e*x+d)^3*A*b^10*d^9-11/3/e^12/(e*x+d)^3*b^10*B*d^10-120*b^3/e
^4/(e*x+d)*A*a^7+120*b^10/e^11/(e*x+d)*A*d^7-45*b^2/e^4/(e*x+d)*B*a^8-165*b^10/e
^12/(e*x+d)*B*d^8-45/2*b^2/e^3/(e*x+d)^2*A*a^8-45/2*b^10/e^11/(e*x+d)^2*A*d^8-5*
b/e^3/(e*x+d)^2*B*a^9+55/2*b^10/e^12/(e*x+d)^2*B*d^9-1/4/e^11/(e*x+d)^4*A*b^10*d
^10+1/4/e^2/(e*x+d)^4*B*d*a^10+1/4/e^12/(e*x+d)^4*b^10*B*d^11+210*b^10/e^11*B*d^
6*x+210*b^4/e^5*B*a^6*x+6720*b^7/e^9/(e*x+d)*B*a^3*d^5-3780*b^8/e^10/(e*x+d)*B*a
^2*d^6+1200*b^9/e^11/(e*x+d)*B*a*d^7+180*b^3/e^4/(e*x+d)^2*A*a^7*d-630*b^4/e^5/(
e*x+d)^2*A*a^6*d^2+1260*b^5/e^6/(e*x+d)^2*A*a^5*d^3-1575*b^6/e^7/(e*x+d)^2*A*a^4
*d^4+1260*b^7/e^8/(e*x+d)^2*A*a^3*d^5-630*b^8/e^9/(e*x+d)^2*A*a^2*d^6+180*b^9/e^
10/(e*x+d)^2*A*a*d^7+135/2*b^2/e^4/(e*x+d)^2*B*a^8*d-360*b^3/e^5/(e*x+d)^2*B*a^7
*d^2+1050*b^4/e^6/(e*x+d)^2*B*a^6*d^3-1890*b^5/e^7/(e*x+d)^2*B*a^5*d^4+2205*b^6/
e^8/(e*x+d)^2*B*a^4*d^5-1680*b^7/e^9/(e*x+d)^2*B*a^3*d^6+810*b^8/e^10/(e*x+d)^2*
B*a^2*d^7-225*b^9/e^11/(e*x+d)^2*B*a*d^8+5/2/e^2/(e*x+d)^4*A*d*a^9*b-45/4/e^3/(e
*x+d)^4*A*d^2*a^8*b^2+30/e^4/(e*x+d)^4*A*d^3*a^7*b^3-105/2/e^5/(e*x+d)^4*A*d^4*a
^6*b^4+63/e^6/(e*x+d)^4*A*a^5*b^5*d^5-105/2/e^7/(e*x+d)^4*A*a^4*b^6*d^6+30/e^8/(
e*x+d)^4*A*a^3*b^7*d^7-45/4/e^9/(e*x+d)^4*A*a^2*b^8*d^8+5/2/e^10/(e*x+d)^4*A*a*b
^9*d^9-5/2/e^3/(e*x+d)^4*B*d^2*a^9*b+45/4/e^4/(e*x+d)^4*B*d^3*a^8*b^2-30/e^5/(e*
x+d)^4*B*d^4*a^7*b^3+105/2/e^6/(e*x+d)^4*B*a^6*b^4*d^5-63/e^7/(e*x+d)^4*B*a^5*b^
5*d^6+105/2/e^8/(e*x+d)^4*B*a^4*b^6*d^7-30/e^9/(e*x+d)^4*B*a^3*b^7*d^8+45/4/e^10
/(e*x+d)^4*B*a^2*b^8*d^9-5/2/e^11/(e*x+d)^4*B*a*b^9*d^10+105*b^6/e^5*A*x^2*a^4+7
0/3*b^10/e^9*B*x^3*d^4+70*b^6/e^5*B*x^3*a^4-35/3*b^10/e^8*A*x^3*d^3+40*b^7/e^5*A
*x^3*a^3-35/4*b^10/e^8*B*x^4*d^3+30*b^7/e^5*B*x^4*a^3+15/4*b^10/e^7*A*x^4*d^2+3*
b^10/e^7*B*x^5*d^2+45/4*b^8/e^5*A*x^4*a^2+9*b^8/e^5*B*x^5*a^2-b^10/e^6*A*x^5*d+2
*b^9/e^5*A*x^5*a-5/6*b^10/e^6*B*x^6*d

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Maxima [A]  time = 1.5774, size = 2492, normalized size = 5.61 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^10/(e*x + d)^5,x, algorithm="maxima")

[Out]

-1/12*(1691*B*b^10*d^11 + 3*A*a^10*e^11 - 1207*(10*B*a*b^9 + A*b^10)*d^10*e + 41
25*(9*B*a^2*b^8 + 2*A*a*b^9)*d^9*e^2 - 7995*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^8*e^3
+ 9570*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^7*e^4 - 7182*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^
6*e^5 + 3234*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^5*e^6 - 750*(4*B*a^7*b^3 + 7*A*a^6*b^
4)*d^4*e^7 + 45*(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 + (B*a^10 + 10*A*a^9*b)*d*e^10 + 180*(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 + 30*(187*B*b^10*d
^9*e^2 - 135*(10*B*a*b^9 + A*b^10)*d^8*e^3 + 468*(9*B*a^2*b^8 + 2*A*a*b^9)*d^7*e
^4 - 924*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^6*e^5 + 1134*(7*B*a^4*b^6 + 4*A*a^3*b^7)*
d^5*e^6 - 882*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^4*e^7 + 420*(5*B*a^6*b^4 + 6*A*a^5*b
^5)*d^3*e^8 - 108*(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 + 4*(1331*B*b^10*d^10*e - 955
*(10*B*a*b^9 + A*b^10)*d^9*e^2 + 3285*(9*B*a^2*b^8 + 2*A*a*b^9)*d^8*e^3 - 6420*(
8*B*a^3*b^7 + 3*A*a^2*b^8)*d^7*e^4 + 7770*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^6*e^5 -
5922*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^5*e^6 + 2730*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^4*
e^7 - 660*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^3*e^8 + 45*(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 + (B*a^10 + 10*A*a^9*b)*e^11)*x)/(e^
16*x^4 + 4*d*e^15*x^3 + 6*d^2*e^14*x^2 + 4*d^3*e^13*x + d^4*e^12) + 1/84*(12*B*b
^10*e^6*x^7 - 14*(5*B*b^10*d*e^5 - (10*B*a*b^9 + A*b^10)*e^6)*x^6 + 84*(3*B*b^10
*d^2*e^4 - (10*B*a*b^9 + A*b^10)*d*e^5 + (9*B*a^2*b^8 + 2*A*a*b^9)*e^6)*x^5 - 10
5*(7*B*b^10*d^3*e^3 - 3*(10*B*a*b^9 + A*b^10)*d^2*e^4 + 5*(9*B*a^2*b^8 + 2*A*a*b
^9)*d*e^5 - 3*(8*B*a^3*b^7 + 3*A*a^2*b^8)*e^6)*x^4 + 140*(14*B*b^10*d^4*e^2 - 7*
(10*B*a*b^9 + A*b^10)*d^3*e^3 + 15*(9*B*a^2*b^8 + 2*A*a*b^9)*d^2*e^4 - 15*(8*B*a
^3*b^7 + 3*A*a^2*b^8)*d*e^5 + 6*(7*B*a^4*b^6 + 4*A*a^3*b^7)*e^6)*x^3 - 42*(126*B
*b^10*d^5*e - 70*(10*B*a*b^9 + A*b^10)*d^4*e^2 + 175*(9*B*a^2*b^8 + 2*A*a*b^9)*d
^3*e^3 - 225*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^2*e^4 + 150*(7*B*a^4*b^6 + 4*A*a^3*b^
7)*d*e^5 - 42*(6*B*a^5*b^5 + 5*A*a^4*b^6)*e^6)*x^2 + 84*(210*B*b^10*d^6 - 126*(1
0*B*a*b^9 + A*b^10)*d^5*e + 350*(9*B*a^2*b^8 + 2*A*a*b^9)*d^4*e^2 - 525*(8*B*a^3
*b^7 + 3*A*a^2*b^8)*d^3*e^3 + 450*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^2*e^4 - 210*(6*B
*a^5*b^5 + 5*A*a^4*b^6)*d*e^5 + 42*(5*B*a^6*b^4 + 6*A*a^5*b^5)*e^6)*x)/e^11 - 30
*(11*B*b^10*d^7 - 7*(10*B*a*b^9 + A*b^10)*d^6*e + 21*(9*B*a^2*b^8 + 2*A*a*b^9)*d
^5*e^2 - 35*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^4*e^3 + 35*(7*B*a^4*b^6 + 4*A*a^3*b^7)
*d^3*e^4 - 21*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^2*e^5 + 7*(5*B*a^6*b^4 + 6*A*a^5*b^5
)*d*e^6 - (4*B*a^7*b^3 + 7*A*a^6*b^4)*e^7)*log(e*x + d)/e^12

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Fricas [A]  time = 0.25695, size = 3791, normalized size = 8.54 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^10/(e*x + d)^5,x, algorithm="fricas")

[Out]

1/84*(12*B*b^10*e^11*x^11 - 11837*B*b^10*d^11 - 21*A*a^10*e^11 + 8449*(10*B*a*b^
9 + A*b^10)*d^10*e - 28875*(9*B*a^2*b^8 + 2*A*a*b^9)*d^9*e^2 + 55965*(8*B*a^3*b^
7 + 3*A*a^2*b^8)*d^8*e^3 - 66990*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^7*e^4 + 50274*(6*
B*a^5*b^5 + 5*A*a^4*b^6)*d^6*e^5 - 22638*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^5*e^6 + 5
250*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^4*e^7 - 315*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^3*e^
8 - 35*(2*B*a^9*b + 9*A*a^8*b^2)*d^2*e^9 - 7*(B*a^10 + 10*A*a^9*b)*d*e^10 - 2*(1
1*B*b^10*d*e^10 - 7*(10*B*a*b^9 + A*b^10)*e^11)*x^10 + 4*(11*B*b^10*d^2*e^9 - 7*
(10*B*a*b^9 + A*b^10)*d*e^10 + 21*(9*B*a^2*b^8 + 2*A*a*b^9)*e^11)*x^9 - 9*(11*B*
b^10*d^3*e^8 - 7*(10*B*a*b^9 + A*b^10)*d^2*e^9 + 21*(9*B*a^2*b^8 + 2*A*a*b^9)*d*
e^10 - 35*(8*B*a^3*b^7 + 3*A*a^2*b^8)*e^11)*x^8 + 24*(11*B*b^10*d^4*e^7 - 7*(10*
B*a*b^9 + A*b^10)*d^3*e^8 + 21*(9*B*a^2*b^8 + 2*A*a*b^9)*d^2*e^9 - 35*(8*B*a^3*b
^7 + 3*A*a^2*b^8)*d*e^10 + 35*(7*B*a^4*b^6 + 4*A*a^3*b^7)*e^11)*x^7 - 84*(11*B*b
^10*d^5*e^6 - 7*(10*B*a*b^9 + A*b^10)*d^4*e^7 + 21*(9*B*a^2*b^8 + 2*A*a*b^9)*d^3
*e^8 - 35*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^2*e^9 + 35*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d
*e^10 - 21*(6*B*a^5*b^5 + 5*A*a^4*b^6)*e^11)*x^6 + 504*(11*B*b^10*d^6*e^5 - 7*(1
0*B*a*b^9 + A*b^10)*d^5*e^6 + 21*(9*B*a^2*b^8 + 2*A*a*b^9)*d^4*e^7 - 35*(8*B*a^3
*b^7 + 3*A*a^2*b^8)*d^3*e^8 + 35*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^2*e^9 - 21*(6*B*a
^5*b^5 + 5*A*a^4*b^6)*d*e^10 + 7*(5*B*a^6*b^4 + 6*A*a^5*b^5)*e^11)*x^5 + 7*(6559
*B*b^10*d^7*e^4 - 4043*(10*B*a*b^9 + A*b^10)*d^6*e^5 + 11625*(9*B*a^2*b^8 + 2*A*
a*b^9)*d^5*e^6 - 18255*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^4*e^7 + 16680*(7*B*a^4*b^6
+ 4*A*a^3*b^7)*d^3*e^8 - 8568*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^2*e^9 + 2016*(5*B*a^
6*b^4 + 6*A*a^5*b^5)*d*e^10)*x^4 + 28*(2599*B*b^10*d^8*e^3 - 1523*(10*B*a*b^9 +
A*b^10)*d^7*e^4 + 4065*(9*B*a^2*b^8 + 2*A*a*b^9)*d^6*e^5 - 5655*(8*B*a^3*b^7 + 3
*A*a^2*b^8)*d^5*e^6 + 4080*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^4*e^7 - 1008*(6*B*a^5*b
^5 + 5*A*a^4*b^6)*d^3*e^8 - 504*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^2*e^9 + 360*(4*B*a
^7*b^3 + 7*A*a^6*b^4)*d*e^10 - 45*(3*B*a^8*b^2 + 8*A*a^7*b^3)*e^11)*x^3 + 42*(61
9*B*b^10*d^9*e^2 - 263*(10*B*a*b^9 + A*b^10)*d^8*e^3 + 285*(9*B*a^2*b^8 + 2*A*a*
b^9)*d^7*e^4 + 645*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^6*e^5 - 2220*(7*B*a^4*b^6 + 4*A
*a^3*b^7)*d^5*e^6 + 2772*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^4*e^7 - 1764*(5*B*a^6*b^4
 + 6*A*a^5*b^5)*d^3*e^8 + 540*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^2*e^9 - 45*(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 - 28*(701*B*b^
10*d^10*e - 577*(10*B*a*b^9 + A*b^10)*d^9*e^2 + 2235*(9*B*a^2*b^8 + 2*A*a*b^9)*d
^8*e^3 - 4845*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^7*e^4 + 6420*(7*B*a^4*b^6 + 4*A*a^3*
b^7)*d^6*e^5 - 5292*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^5*e^6 + 2604*(5*B*a^6*b^4 + 6*
A*a^5*b^5)*d^4*e^7 - 660*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^3*e^8 + 45*(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 + (B*a^10 + 10*A*a^9*
b)*e^11)*x - 2520*(11*B*b^10*d^11 - 7*(10*B*a*b^9 + A*b^10)*d^10*e + 21*(9*B*a^2
*b^8 + 2*A*a*b^9)*d^9*e^2 - 35*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^8*e^3 + 35*(7*B*a^4
*b^6 + 4*A*a^3*b^7)*d^7*e^4 - 21*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^6*e^5 + 7*(5*B*a^
6*b^4 + 6*A*a^5*b^5)*d^5*e^6 - (4*B*a^7*b^3 + 7*A*a^6*b^4)*d^4*e^7 + (11*B*b^10*
d^7*e^4 - 7*(10*B*a*b^9 + A*b^10)*d^6*e^5 + 21*(9*B*a^2*b^8 + 2*A*a*b^9)*d^5*e^6
 - 35*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^4*e^7 + 35*(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 - (4*B*a^7*b^3 + 7*A*a^6*b^4)*e^11)*x^4 + 4*(11*B*b^10*d^8*e^3 - 7*(10*B*a*b^
9 + A*b^10)*d^7*e^4 + 21*(9*B*a^2*b^8 + 2*A*a*b^9)*d^6*e^5 - 35*(8*B*a^3*b^7 + 3
*A*a^2*b^8)*d^5*e^6 + 35*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^4*e^7 - 21*(6*B*a^5*b^5 +
 5*A*a^4*b^6)*d^3*e^8 + 7*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^2*e^9 - (4*B*a^7*b^3 + 7
*A*a^6*b^4)*d*e^10)*x^3 + 6*(11*B*b^10*d^9*e^2 - 7*(10*B*a*b^9 + A*b^10)*d^8*e^3
 + 21*(9*B*a^2*b^8 + 2*A*a*b^9)*d^7*e^4 - 35*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^6*e^5
 + 35*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^5*e^6 - 21*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^4*e
^7 + 7*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^3*e^8 - (4*B*a^7*b^3 + 7*A*a^6*b^4)*d^2*e^9
)*x^2 + 4*(11*B*b^10*d^10*e - 7*(10*B*a*b^9 + A*b^10)*d^9*e^2 + 21*(9*B*a^2*b^8
+ 2*A*a*b^9)*d^8*e^3 - 35*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^7*e^4 + 35*(7*B*a^4*b^6
+ 4*A*a^3*b^7)*d^6*e^5 - 21*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^5*e^6 + 7*(5*B*a^6*b^4
 + 6*A*a^5*b^5)*d^4*e^7 - (4*B*a^7*b^3 + 7*A*a^6*b^4)*d^3*e^8)*x)*log(e*x + d))/
(e^16*x^4 + 4*d*e^15*x^3 + 6*d^2*e^14*x^2 + 4*d^3*e^13*x + d^4*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)**5,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.227785, 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)^5,x, algorithm="giac")

[Out]

Done

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