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3.1078 \(\int \frac{(a+b x)^{10} (A+B x)}{(d+e x)^7} \, dx\)

Optimal. Leaf size=447 \[ -\frac{b^9 (d+e x)^4 (-10 a B e-A b e+11 b B d)}{4 e^{12}}+\frac{5 b^8 (d+e x)^3 (b d-a e) (-9 a B e-2 A b e+11 b B d)}{3 e^{12}}-\frac{15 b^7 (d+e x)^2 (b d-a e)^2 (-8 a B e-3 A b e+11 b B d)}{2 e^{12}}+\frac{30 b^6 x (b d-a e)^3 (-7 a B e-4 A b e+11 b B d)}{e^{11}}-\frac{42 b^5 (b d-a e)^4 \log (d+e x) (-6 a B e-5 A b e+11 b B d)}{e^{12}}-\frac{42 b^4 (b d-a e)^5 (-5 a B e-6 A b e+11 b B d)}{e^{12} (d+e x)}+\frac{15 b^3 (b d-a e)^6 (-4 a B e-7 A b e+11 b B d)}{e^{12} (d+e x)^2}-\frac{5 b^2 (b d-a e)^7 (-3 a B e-8 A b e+11 b B d)}{e^{12} (d+e x)^3}+\frac{5 b (b d-a e)^8 (-2 a B e-9 A b e+11 b B d)}{4 e^{12} (d+e x)^4}-\frac{(b d-a e)^9 (-a B e-10 A b e+11 b B d)}{5 e^{12} (d+e x)^5}+\frac{(b d-a e)^{10} (B d-A e)}{6 e^{12} (d+e x)^6}+\frac{b^{10} B (d+e x)^5}{5 e^{12}} \]

[Out]

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

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Rubi [A]  time = 2.94573, antiderivative size = 447, 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 \[ -\frac{b^9 (d+e x)^4 (-10 a B e-A b e+11 b B d)}{4 e^{12}}+\frac{5 b^8 (d+e x)^3 (b d-a e) (-9 a B e-2 A b e+11 b B d)}{3 e^{12}}-\frac{15 b^7 (d+e x)^2 (b d-a e)^2 (-8 a B e-3 A b e+11 b B d)}{2 e^{12}}+\frac{30 b^6 x (b d-a e)^3 (-7 a B e-4 A b e+11 b B d)}{e^{11}}-\frac{42 b^5 (b d-a e)^4 \log (d+e x) (-6 a B e-5 A b e+11 b B d)}{e^{12}}-\frac{42 b^4 (b d-a e)^5 (-5 a B e-6 A b e+11 b B d)}{e^{12} (d+e x)}+\frac{15 b^3 (b d-a e)^6 (-4 a B e-7 A b e+11 b B d)}{e^{12} (d+e x)^2}-\frac{5 b^2 (b d-a e)^7 (-3 a B e-8 A b e+11 b B d)}{e^{12} (d+e x)^3}+\frac{5 b (b d-a e)^8 (-2 a B e-9 A b e+11 b B d)}{4 e^{12} (d+e x)^4}-\frac{(b d-a e)^9 (-a B e-10 A b e+11 b B d)}{5 e^{12} (d+e x)^5}+\frac{(b d-a e)^{10} (B d-A e)}{6 e^{12} (d+e x)^6}+\frac{b^{10} B (d+e x)^5}{5 e^{12}} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)^10*(A + B*x))/(d + e*x)^7,x]

[Out]

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

[Out]

Timed out

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Mathematica [A]  time = 1.01454, size = 505, normalized size = 1.13 \[ \frac{-20 b^8 e^3 x^3 \left (-45 a^2 B e^2-10 a b e (A e-7 B d)+7 b^2 d (A e-4 B d)\right )+30 b^7 e^2 x^2 \left (120 a^3 B e^3+45 a^2 b e^2 (A e-7 B d)+70 a b^2 d e (4 B d-A e)+28 b^3 d^2 (A e-3 B d)\right )-60 b^6 e x \left (-210 a^4 B e^4-120 a^3 b e^3 (A e-7 B d)-315 a^2 b^2 d e^2 (4 B d-A e)+280 a b^3 d^2 e (3 B d-A e)-42 b^4 d^3 (5 B d-2 A e)\right )+15 b^9 e^4 x^4 (10 a B e+A b e-7 b B d)-2520 b^5 (b d-a e)^4 \log (d+e x) (-6 a B e-5 A b e+11 b B d)-\frac{2520 b^4 (b d-a e)^5 (-5 a B e-6 A b e+11 b B d)}{d+e x}+\frac{900 b^3 (b d-a e)^6 (-4 a B e-7 A b e+11 b B d)}{(d+e x)^2}-\frac{300 b^2 (b d-a e)^7 (-3 a B e-8 A b e+11 b B d)}{(d+e x)^3}+\frac{75 b (b d-a e)^8 (-2 a B e-9 A b e+11 b B d)}{(d+e x)^4}-\frac{12 (b d-a e)^9 (-a B e-10 A b e+11 b B d)}{(d+e x)^5}+\frac{10 (b d-a e)^{10} (B d-A e)}{(d+e x)^6}+12 b^{10} B e^5 x^5}{60 e^{12}} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)^10*(A + B*x))/(d + e*x)^7,x]

[Out]

(-60*b^6*e*(-210*a^4*B*e^4 - 42*b^4*d^3*(5*B*d - 2*A*e) + 280*a*b^3*d^2*e*(3*B*d
 - A*e) - 315*a^2*b^2*d*e^2*(4*B*d - A*e) - 120*a^3*b*e^3*(-7*B*d + A*e))*x + 30
*b^7*e^2*(120*a^3*B*e^3 + 70*a*b^2*d*e*(4*B*d - A*e) + 45*a^2*b*e^2*(-7*B*d + A*
e) + 28*b^3*d^2*(-3*B*d + A*e))*x^2 - 20*b^8*e^3*(-45*a^2*B*e^2 - 10*a*b*e*(-7*B
*d + A*e) + 7*b^2*d*(-4*B*d + A*e))*x^3 + 15*b^9*e^4*(-7*b*B*d + A*b*e + 10*a*B*
e)*x^4 + 12*b^10*B*e^5*x^5 + (10*(b*d - a*e)^10*(B*d - A*e))/(d + e*x)^6 - (12*(
b*d - a*e)^9*(11*b*B*d - 10*A*b*e - a*B*e))/(d + e*x)^5 + (75*b*(b*d - a*e)^8*(1
1*b*B*d - 9*A*b*e - 2*a*B*e))/(d + e*x)^4 - (300*b^2*(b*d - a*e)^7*(11*b*B*d - 8
*A*b*e - 3*a*B*e))/(d + e*x)^3 + (900*b^3*(b*d - a*e)^6*(11*b*B*d - 7*A*b*e - 4*
a*B*e))/(d + e*x)^2 - (2520*b^4*(b*d - a*e)^5*(11*b*B*d - 6*A*b*e - 5*a*B*e))/(d
 + e*x) - 2520*b^5*(b*d - a*e)^4*(11*b*B*d - 5*A*b*e - 6*a*B*e)*Log[d + e*x])/(6
0*e^12)

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Maple [B]  time = 0.05, size = 2781, normalized size = 6.2 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^10*(B*x+A)/(e*x+d)^7,x)

[Out]

1400*b^6/e^7/(e*x+d)^3*A*a^4*d^3-1400*b^7/e^8/(e*x+d)^3*A*a^3*d^4+840*b^8/e^9/(e
*x+d)^3*A*a^2*d^5-280*b^9/e^10/(e*x+d)^3*A*a*d^6+160*b^3/e^5/(e*x+d)^3*B*a^7*d-7
00*b^4/e^6/(e*x+d)^3*B*a^6*d^2+1680*b^5/e^7/(e*x+d)^3*B*a^5*d^3-2450*b^6/e^8/(e*
x+d)^3*B*a^4*d^4+2240*b^7/e^9/(e*x+d)^3*B*a^3*d^5-1260*b^8/e^10/(e*x+d)^3*B*a^2*
d^6+400*b^9/e^11/(e*x+d)^3*B*a*d^7+1260*b^6/e^7/(e*x+d)*A*a^4*d-2520*b^7/e^8/(e*
x+d)*A*a^3*d^2+2520*b^8/e^9/(e*x+d)*A*a^2*d^3-1260*b^9/e^10/(e*x+d)*A*a*d^4+1/4*
b^10/e^7*A*x^4+1/5*b^10/e^7*B*x^5-1/6/e/(e*x+d)^6*a^10*A-1/5/e^2/(e*x+d)^5*B*a^1
0+280*b^4/e^5/(e*x+d)^3*A*a^6*d-840*b^5/e^6/(e*x+d)^3*A*a^5*d^2-35*b^9/e^8*A*x^2
*a*d-315/2*b^8/e^8*B*x^2*a^2*d+140*b^9/e^9*B*x^2*a*d^2-315*b^8/e^8*A*a^2*d*x+60*
b^7/e^7*B*x^2*a^3+120*b^7/e^7*A*a^3*x+1512*b^5/e^7/(e*x+d)*B*a^5*d-4410*b^6/e^8/
(e*x+d)*B*a^4*d^2+6720*b^7/e^9/(e*x+d)*B*a^3*d^3-5670*b^8/e^10/(e*x+d)*B*a^2*d^4
+2520*b^9/e^11/(e*x+d)*B*a*d^5+630*b^5/e^6/(e*x+d)^2*A*a^5*d-1575*b^6/e^7/(e*x+d
)^2*A*a^4*d^2+2100*b^7/e^8/(e*x+d)^2*A*a^3*d^3-1575*b^8/e^9/(e*x+d)^2*A*a^2*d^4+
630*b^9/e^10/(e*x+d)^2*A*a*d^5+525*b^4/e^6/(e*x+d)^2*B*a^6*d-1890*b^5/e^7/(e*x+d
)^2*B*a^5*d^2+3675*b^6/e^8/(e*x+d)^2*B*a^4*d^3-4200*b^7/e^9/(e*x+d)^2*B*a^3*d^4+
2835*b^8/e^10/(e*x+d)^2*B*a^2*d^5-1050*b^9/e^11/(e*x+d)^2*B*a*d^6+90*b^3/e^4/(e*
x+d)^4*A*a^7*d-315*b^4/e^5/(e*x+d)^4*A*a^6*d^2+630*b^5/e^6/(e*x+d)^4*A*a^5*d^3-1
575/2*b^6/e^7/(e*x+d)^4*A*a^4*d^4+630*b^7/e^8/(e*x+d)^4*A*a^3*d^5-315*b^8/e^9/(e
*x+d)^4*A*a^2*d^6+90*b^9/e^10/(e*x+d)^4*A*a*d^7+135/4*b^2/e^4/(e*x+d)^4*B*a^8*d-
180*b^3/e^5/(e*x+d)^4*B*a^7*d^2+525*b^4/e^6/(e*x+d)^4*B*a^6*d^3-945*b^5/e^7/(e*x
+d)^4*B*a^5*d^4+2205/2*b^6/e^8/(e*x+d)^4*B*a^4*d^5-840*b^7/e^9/(e*x+d)^4*B*a^3*d
^6+405*b^8/e^10/(e*x+d)^4*B*a^2*d^7-225/2*b^9/e^11/(e*x+d)^4*B*a*d^8+18/e^3/(e*x
+d)^5*A*a^8*b^2*d-72/e^4/(e*x+d)^5*A*a^7*b^3*d^2+168/e^5/(e*x+d)^5*A*a^6*b^4*d^3
-252/e^6/(e*x+d)^5*A*a^5*b^5*d^4+252/e^7/(e*x+d)^5*A*a^4*b^6*d^5-168/e^8/(e*x+d)
^5*A*a^3*b^7*d^6+72/e^9/(e*x+d)^5*A*a^2*b^8*d^7-18/e^10/(e*x+d)^5*A*a*b^9*d^8-7/
4*b^10/e^8*B*x^4*d+10/3*b^9/e^7*A*x^3*a-42*b^10/e^10*B*x^2*d^3-7/3*b^10/e^8*A*x^
3*d+15*b^8/e^7*B*x^3*a^2+28/3*b^10/e^9*B*x^3*d^2+45/2*b^8/e^7*A*x^2*a^2+14*b^10/
e^9*A*x^2*d^2+5/2*b^9/e^7*B*x^4*a+210*b^6/e^7*ln(e*x+d)*A*a^4+210*b^10/e^11*ln(e
*x+d)*A*d^4+252*b^5/e^7*ln(e*x+d)*B*a^5-462*b^10/e^12*ln(e*x+d)*B*d^5-1/6/e^11/(
e*x+d)^6*A*b^10*d^10+1/6/e^2/(e*x+d)^6*B*d*a^10+1/6/e^12/(e*x+d)^6*b^10*B*d^11-4
0*b^3/e^4/(e*x+d)^3*A*a^7+40*b^10/e^11/(e*x+d)^3*A*d^7-15*b^2/e^4/(e*x+d)^3*B*a^
8-55*b^10/e^12/(e*x+d)^3*B*d^8-252*b^5/e^6/(e*x+d)*A*a^5+252*b^10/e^11/(e*x+d)*A
*d^5-210*b^4/e^6/(e*x+d)*B*a^6-462*b^10/e^12/(e*x+d)*B*d^6-105*b^4/e^5/(e*x+d)^2
*A*a^6-105*b^10/e^11/(e*x+d)^2*A*d^6-60*b^3/e^5/(e*x+d)^2*B*a^7+165*b^10/e^12/(e
*x+d)^2*B*d^7-45/4*b^2/e^3/(e*x+d)^4*A*a^8-45/4*b^10/e^11/(e*x+d)^4*A*d^8-5/2*b/
e^3/(e*x+d)^4*B*a^9+55/4*b^10/e^12/(e*x+d)^4*B*d^9-2/e^2/(e*x+d)^5*A*a^9*b+2/e^1
1/(e*x+d)^5*A*b^10*d^9-11/5/e^12/(e*x+d)^5*b^10*B*d^10-84*b^10/e^10*A*d^3*x+210*
b^6/e^7*B*a^4*x+210*b^10/e^11*B*d^4*x+4/e^3/(e*x+d)^5*B*a^9*b*d-27/e^4/(e*x+d)^5
*B*a^8*b^2*d^2+96/e^5/(e*x+d)^5*B*a^7*b^3*d^3-210/e^6/(e*x+d)^5*B*a^6*b^4*d^4+15
12/5/e^7/(e*x+d)^5*B*a^5*b^5*d^5-294/e^8/(e*x+d)^5*B*a^4*b^6*d^6+192/e^9/(e*x+d)
^5*B*a^3*b^7*d^7-81/e^10/(e*x+d)^5*B*a^2*b^8*d^8+20/e^11/(e*x+d)^5*B*a*b^9*d^9+2
80*b^9/e^9*A*a*d^2*x-840*b^7/e^8*B*a^3*d*x+1260*b^8/e^9*B*a^2*d^2*x-840*b^9/e^10
*B*a*d^3*x-70/3*b^9/e^8*B*x^3*a*d-840*b^7/e^8*ln(e*x+d)*A*a^3*d+1260*b^8/e^9*ln(
e*x+d)*A*a^2*d^2-840*b^9/e^10*ln(e*x+d)*A*a*d^3-1470*b^6/e^8*ln(e*x+d)*B*a^4*d+3
360*b^7/e^9*ln(e*x+d)*B*a^3*d^2-3780*b^8/e^10*ln(e*x+d)*B*a^2*d^3+2100*b^9/e^11*
ln(e*x+d)*B*a*d^4+5/3/e^2/(e*x+d)^6*A*d*a^9*b-15/2/e^3/(e*x+d)^6*A*d^2*a^8*b^2+2
0/e^4/(e*x+d)^6*A*d^3*a^7*b^3-35/e^5/(e*x+d)^6*A*d^4*a^6*b^4+42/e^6/(e*x+d)^6*A*
d^5*a^5*b^5-35/e^7/(e*x+d)^6*A*d^6*a^4*b^6+20/e^8/(e*x+d)^6*A*a^3*b^7*d^7-15/2/e
^9/(e*x+d)^6*A*a^2*b^8*d^8+5/3/e^10/(e*x+d)^6*A*a*b^9*d^9-5/3/e^3/(e*x+d)^6*B*d^
2*a^9*b+15/2/e^4/(e*x+d)^6*B*d^3*a^8*b^2-20/e^5/(e*x+d)^6*B*d^4*a^7*b^3+35/e^6/(
e*x+d)^6*B*d^5*a^6*b^4-42/e^7/(e*x+d)^6*B*d^6*a^5*b^5+35/e^8/(e*x+d)^6*B*a^4*b^6
*d^7-20/e^9/(e*x+d)^6*B*a^3*b^7*d^8+15/2/e^10/(e*x+d)^6*B*a^2*b^8*d^9-5/3/e^11/(
e*x+d)^6*B*a*b^9*d^10

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Maxima [A]  time = 1.63018, size = 2523, normalized size = 5.64 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^10/(e*x + d)^7,x, algorithm="maxima")

[Out]

-1/60*(20417*B*b^10*d^11 + 10*A*a^10*e^11 - 10655*(10*B*a*b^9 + A*b^10)*d^10*e +
 25090*(9*B*a^2*b^8 + 2*A*a*b^9)*d^9*e^2 - 30690*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^8
*e^3 + 20070*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^7*e^4 - 6174*(6*B*a^5*b^5 + 5*A*a^4*b
^6)*d^6*e^5 + 420*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^5*e^6 + 60*(4*B*a^7*b^3 + 7*A*a^
6*b^4)*d^4*e^7 + 15*(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 + 2*(B*a^10 + 10*A*a^9*b)*d*e^10 + 2520*(11*B*b^10*d^6*e^5 - 6*(10
*B*a*b^9 + A*b^10)*d^5*e^6 + 15*(9*B*a^2*b^8 + 2*A*a*b^9)*d^4*e^7 - 20*(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 - 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 + 900*(143*B*
b^10*d^7*e^4 - 77*(10*B*a*b^9 + A*b^10)*d^6*e^5 + 189*(9*B*a^2*b^8 + 2*A*a*b^9)*
d^5*e^6 - 245*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^4*e^7 + 175*(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 + 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 + 300*(803*B*b^10*d^8*e^3 -
428*(10*B*a*b^9 + A*b^10)*d^7*e^4 + 1036*(9*B*a^2*b^8 + 2*A*a*b^9)*d^6*e^5 - 131
6*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^5*e^6 + 910*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^4*e^7
- 308*(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 + 4*(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 + 75*(3025*B*b^10*d^9*e^2 - 1599*(10*B*a*b^9 + A*b^10)*d^8*e^3 + 3828*(9*B*a^2
*b^8 + 2*A*a*b^9)*d^7*e^4 - 4788*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^6*e^5 + 3234*(7*B
*a^4*b^6 + 4*A*a^3*b^7)*d^5*e^6 - 1050*(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 + 12*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^2*e^9 + 3
*(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 + 6*(1
7897*B*b^10*d^10*e - 9395*(10*B*a*b^9 + A*b^10)*d^9*e^2 + 22290*(9*B*a^2*b^8 + 2
*A*a*b^9)*d^8*e^3 - 27540*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^7*e^4 + 18270*(7*B*a^4*b
^6 + 4*A*a^3*b^7)*d^6*e^5 - 5754*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^5*e^6 + 420*(5*B*
a^6*b^4 + 6*A*a^5*b^5)*d^4*e^7 + 60*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^3*e^8 + 15*(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 + 2*(B*a^1
0 + 10*A*a^9*b)*e^11)*x)/(e^18*x^6 + 6*d*e^17*x^5 + 15*d^2*e^16*x^4 + 20*d^3*e^1
5*x^3 + 15*d^4*e^14*x^2 + 6*d^5*e^13*x + d^6*e^12) + 1/60*(12*B*b^10*e^4*x^5 - 1
5*(7*B*b^10*d*e^3 - (10*B*a*b^9 + A*b^10)*e^4)*x^4 + 20*(28*B*b^10*d^2*e^2 - 7*(
10*B*a*b^9 + A*b^10)*d*e^3 + 5*(9*B*a^2*b^8 + 2*A*a*b^9)*e^4)*x^3 - 30*(84*B*b^1
0*d^3*e - 28*(10*B*a*b^9 + A*b^10)*d^2*e^2 + 35*(9*B*a^2*b^8 + 2*A*a*b^9)*d*e^3
- 15*(8*B*a^3*b^7 + 3*A*a^2*b^8)*e^4)*x^2 + 60*(210*B*b^10*d^4 - 84*(10*B*a*b^9
+ A*b^10)*d^3*e + 140*(9*B*a^2*b^8 + 2*A*a*b^9)*d^2*e^2 - 105*(8*B*a^3*b^7 + 3*A
*a^2*b^8)*d*e^3 + 30*(7*B*a^4*b^6 + 4*A*a^3*b^7)*e^4)*x)/e^11 - 42*(11*B*b^10*d^
5 - 5*(10*B*a*b^9 + A*b^10)*d^4*e + 10*(9*B*a^2*b^8 + 2*A*a*b^9)*d^3*e^2 - 10*(8
*B*a^3*b^7 + 3*A*a^2*b^8)*d^2*e^3 + 5*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d*e^4 - (6*B*a
^5*b^5 + 5*A*a^4*b^6)*e^5)*log(e*x + d)/e^12

_______________________________________________________________________________________

Fricas [A]  time = 0.228962, size = 3848, normalized size = 8.61 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^10/(e*x + d)^7,x, algorithm="fricas")

[Out]

1/60*(12*B*b^10*e^11*x^11 - 20417*B*b^10*d^11 - 10*A*a^10*e^11 + 10655*(10*B*a*b
^9 + A*b^10)*d^10*e - 25090*(9*B*a^2*b^8 + 2*A*a*b^9)*d^9*e^2 + 30690*(8*B*a^3*b
^7 + 3*A*a^2*b^8)*d^8*e^3 - 20070*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^7*e^4 + 6174*(6*
B*a^5*b^5 + 5*A*a^4*b^6)*d^6*e^5 - 420*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^5*e^6 - 60*
(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^4*e^7 - 15*(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 - 2*(B*a^10 + 10*A*a^9*b)*d*e^10 - 3*(11*B*b^
10*d*e^10 - 5*(10*B*a*b^9 + A*b^10)*e^11)*x^10 + 10*(11*B*b^10*d^2*e^9 - 5*(10*B
*a*b^9 + A*b^10)*d*e^10 + 10*(9*B*a^2*b^8 + 2*A*a*b^9)*e^11)*x^9 - 45*(11*B*b^10
*d^3*e^8 - 5*(10*B*a*b^9 + A*b^10)*d^2*e^9 + 10*(9*B*a^2*b^8 + 2*A*a*b^9)*d*e^10
 - 10*(8*B*a^3*b^7 + 3*A*a^2*b^8)*e^11)*x^8 + 360*(11*B*b^10*d^4*e^7 - 5*(10*B*a
*b^9 + A*b^10)*d^3*e^8 + 10*(9*B*a^2*b^8 + 2*A*a*b^9)*d^2*e^9 - 10*(8*B*a^3*b^7
+ 3*A*a^2*b^8)*d*e^10 + 5*(7*B*a^4*b^6 + 4*A*a^3*b^7)*e^11)*x^7 + (47497*B*b^10*
d^5*e^6 - 20215*(10*B*a*b^9 + A*b^10)*d^4*e^7 + 36650*(9*B*a^2*b^8 + 2*A*a*b^9)*
d^3*e^8 - 31050*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^2*e^9 + 10800*(7*B*a^4*b^6 + 4*A*a
^3*b^7)*d*e^10)*x^6 + 6*(19777*B*b^10*d^6*e^5 - 7615*(10*B*a*b^9 + A*b^10)*d^5*e
^6 + 11450*(9*B*a^2*b^8 + 2*A*a*b^9)*d^4*e^7 - 5850*(8*B*a^3*b^7 + 3*A*a^2*b^8)*
d^3*e^8 - 1800*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^2*e^9 + 2520*(6*B*a^5*b^5 + 5*A*a^4
*b^6)*d*e^10 - 420*(5*B*a^6*b^4 + 6*A*a^5*b^5)*e^11)*x^5 + 15*(5917*B*b^10*d^7*e
^4 - 1315*(10*B*a*b^9 + A*b^10)*d^6*e^5 - 1150*(9*B*a^2*b^8 + 2*A*a*b^9)*d^5*e^6
 + 6750*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^4*e^7 - 8100*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d
^3*e^8 + 3780*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^2*e^9 - 420*(5*B*a^6*b^4 + 6*A*a^5*b
^5)*d*e^10 - 60*(4*B*a^7*b^3 + 7*A*a^6*b^4)*e^11)*x^4 - 20*(3323*B*b^10*d^8*e^3
- 2885*(10*B*a*b^9 + A*b^10)*d^7*e^4 + 9550*(9*B*a^2*b^8 + 2*A*a*b^9)*d^6*e^5 -
15150*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^5*e^6 + 12300*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^
4*e^7 - 4620*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^3*e^8 + 420*(5*B*a^6*b^4 + 6*A*a^5*b^
5)*d^2*e^9 + 60*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d*e^10 + 15*(3*B*a^8*b^2 + 8*A*a^7*b
^3)*e^11)*x^3 - 15*(10253*B*b^10*d^9*e^2 - 6035*(10*B*a*b^9 + A*b^10)*d^8*e^3 +
15850*(9*B*a^2*b^8 + 2*A*a*b^9)*d^7*e^4 - 21450*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^6*
e^5 + 15450*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^5*e^6 - 5250*(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 + 60*(4*B*a^7*b^3 + 7*A*a^6
*b^4)*d^2*e^9 + 15*(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 - 6*(15797*B*b^10*d^10*e - 8555*(10*B*a*b^9 + A*b^10)*d^9*e^2 + 20
890*(9*B*a^2*b^8 + 2*A*a*b^9)*d^8*e^3 - 26490*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^7*e^
4 + 17970*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^6*e^5 - 5754*(6*B*a^5*b^5 + 5*A*a^4*b^6)
*d^5*e^6 + 420*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^4*e^7 + 60*(4*B*a^7*b^3 + 7*A*a^6*b
^4)*d^3*e^8 + 15*(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 + 2*(B*a^10 + 10*A*a^9*b)*e^11)*x - 2520*(11*B*b^10*d^11 - 5*(10*B*a*b
^9 + A*b^10)*d^10*e + 10*(9*B*a^2*b^8 + 2*A*a*b^9)*d^9*e^2 - 10*(8*B*a^3*b^7 + 3
*A*a^2*b^8)*d^8*e^3 + 5*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^7*e^4 - (6*B*a^5*b^5 + 5*A
*a^4*b^6)*d^6*e^5 + (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 + 6*(11*
B*b^10*d^6*e^5 - 5*(10*B*a*b^9 + A*b^10)*d^5*e^6 + 10*(9*B*a^2*b^8 + 2*A*a*b^9)*
d^4*e^7 - 10*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^3*e^8 + 5*(7*B*a^4*b^6 + 4*A*a^3*b^7)
*d^2*e^9 - (6*B*a^5*b^5 + 5*A*a^4*b^6)*d*e^10)*x^5 + 15*(11*B*b^10*d^7*e^4 - 5*(
10*B*a*b^9 + A*b^10)*d^6*e^5 + 10*(9*B*a^2*b^8 + 2*A*a*b^9)*d^5*e^6 - 10*(8*B*a^
3*b^7 + 3*A*a^2*b^8)*d^4*e^7 + 5*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^3*e^8 - (6*B*a^5*
b^5 + 5*A*a^4*b^6)*d^2*e^9)*x^4 + 20*(11*B*b^10*d^8*e^3 - 5*(10*B*a*b^9 + A*b^10
)*d^7*e^4 + 10*(9*B*a^2*b^8 + 2*A*a*b^9)*d^6*e^5 - 10*(8*B*a^3*b^7 + 3*A*a^2*b^8
)*d^5*e^6 + 5*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^4*e^7 - (6*B*a^5*b^5 + 5*A*a^4*b^6)*
d^3*e^8)*x^3 + 15*(11*B*b^10*d^9*e^2 - 5*(10*B*a*b^9 + A*b^10)*d^8*e^3 + 10*(9*B
*a^2*b^8 + 2*A*a*b^9)*d^7*e^4 - 10*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^6*e^5 + 5*(7*B*
a^4*b^6 + 4*A*a^3*b^7)*d^5*e^6 - (6*B*a^5*b^5 + 5*A*a^4*b^6)*d^4*e^7)*x^2 + 6*(1
1*B*b^10*d^10*e - 5*(10*B*a*b^9 + A*b^10)*d^9*e^2 + 10*(9*B*a^2*b^8 + 2*A*a*b^9)
*d^8*e^3 - 10*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^7*e^4 + 5*(7*B*a^4*b^6 + 4*A*a^3*b^7
)*d^6*e^5 - (6*B*a^5*b^5 + 5*A*a^4*b^6)*d^5*e^6)*x)*log(e*x + d))/(e^18*x^6 + 6*
d*e^17*x^5 + 15*d^2*e^16*x^4 + 20*d^3*e^15*x^3 + 15*d^4*e^14*x^2 + 6*d^5*e^13*x
+ d^6*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)**7,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.216229, 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)^7,x, algorithm="giac")

[Out]

Done

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