3.1084 \(\int \frac{(a+b x)^{10} (A+B x)}{(d+e x)^{13}} \, dx\)

Optimal. Leaf size=86 \[ \frac{(a+b x)^{11} (-12 a B e+A b e+11 b B d)}{132 e (d+e x)^{11} (b d-a e)^2}-\frac{(a+b x)^{11} (B d-A e)}{12 e (d+e x)^{12} (b d-a e)} \]

[Out]

-((B*d - A*e)*(a + b*x)^11)/(12*e*(b*d - a*e)*(d + e*x)^12) + ((11*b*B*d + A*b*e
- 12*a*B*e)*(a + b*x)^11)/(132*e*(b*d - a*e)^2*(d + e*x)^11)

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Rubi [A]  time = 0.126572, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{(a+b x)^{11} (-12 a B e+A b e+11 b B d)}{132 e (d+e x)^{11} (b d-a e)^2}-\frac{(a+b x)^{11} (B d-A e)}{12 e (d+e x)^{12} (b d-a e)} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)^10*(A + B*x))/(d + e*x)^13,x]

[Out]

-((B*d - A*e)*(a + b*x)^11)/(12*e*(b*d - a*e)*(d + e*x)^12) + ((11*b*B*d + A*b*e
- 12*a*B*e)*(a + b*x)^11)/(132*e*(b*d - a*e)^2*(d + e*x)^11)

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Rubi in Sympy [A]  time = 12.5811, size = 73, normalized size = 0.85 \[ - \frac{\left (a + b x\right )^{11} \left (- A b e + B \left (12 a e - 11 b d\right )\right )}{132 e \left (d + e x\right )^{11} \left (a e - b d\right )^{2}} - \frac{\left (a + b x\right )^{11} \left (A e - B d\right )}{12 e \left (d + e x\right )^{12} \left (a e - b d\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**10*(B*x+A)/(e*x+d)**13,x)

[Out]

-(a + b*x)**11*(-A*b*e + B*(12*a*e - 11*b*d))/(132*e*(d + e*x)**11*(a*e - b*d)**
2) - (a + b*x)**11*(A*e - B*d)/(12*e*(d + e*x)**12*(a*e - b*d))

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Mathematica [B]  time = 2.8009, size = 1421, normalized size = 16.52 \[ -\frac{\left (A e \left (d^{10}+12 e x d^9+66 e^2 x^2 d^8+220 e^3 x^3 d^7+495 e^4 x^4 d^6+792 e^5 x^5 d^5+924 e^6 x^6 d^4+792 e^7 x^7 d^3+495 e^8 x^8 d^2+220 e^9 x^9 d+66 e^{10} x^{10}\right )+11 B \left (d^{11}+12 e x d^{10}+66 e^2 x^2 d^9+220 e^3 x^3 d^8+495 e^4 x^4 d^7+792 e^5 x^5 d^6+924 e^6 x^6 d^5+792 e^7 x^7 d^4+495 e^8 x^8 d^3+220 e^9 x^9 d^2+66 e^{10} x^{10} d+12 e^{11} x^{11}\right )\right ) b^{10}+2 a e \left (A e \left (d^9+12 e x d^8+66 e^2 x^2 d^7+220 e^3 x^3 d^6+495 e^4 x^4 d^5+792 e^5 x^5 d^4+924 e^6 x^6 d^3+792 e^7 x^7 d^2+495 e^8 x^8 d+220 e^9 x^9\right )+5 B \left (d^{10}+12 e x d^9+66 e^2 x^2 d^8+220 e^3 x^3 d^7+495 e^4 x^4 d^6+792 e^5 x^5 d^5+924 e^6 x^6 d^4+792 e^7 x^7 d^3+495 e^8 x^8 d^2+220 e^9 x^9 d+66 e^{10} x^{10}\right )\right ) b^9+3 a^2 e^2 \left (A e \left (d^8+12 e x d^7+66 e^2 x^2 d^6+220 e^3 x^3 d^5+495 e^4 x^4 d^4+792 e^5 x^5 d^3+924 e^6 x^6 d^2+792 e^7 x^7 d+495 e^8 x^8\right )+3 B \left (d^9+12 e x d^8+66 e^2 x^2 d^7+220 e^3 x^3 d^6+495 e^4 x^4 d^5+792 e^5 x^5 d^4+924 e^6 x^6 d^3+792 e^7 x^7 d^2+495 e^8 x^8 d+220 e^9 x^9\right )\right ) b^8+4 a^3 e^3 \left (A e \left (d^7+12 e x d^6+66 e^2 x^2 d^5+220 e^3 x^3 d^4+495 e^4 x^4 d^3+792 e^5 x^5 d^2+924 e^6 x^6 d+792 e^7 x^7\right )+2 B \left (d^8+12 e x d^7+66 e^2 x^2 d^6+220 e^3 x^3 d^5+495 e^4 x^4 d^4+792 e^5 x^5 d^3+924 e^6 x^6 d^2+792 e^7 x^7 d+495 e^8 x^8\right )\right ) b^7+a^4 e^4 \left (5 A e \left (d^6+12 e x d^5+66 e^2 x^2 d^4+220 e^3 x^3 d^3+495 e^4 x^4 d^2+792 e^5 x^5 d+924 e^6 x^6\right )+7 B \left (d^7+12 e x d^6+66 e^2 x^2 d^5+220 e^3 x^3 d^4+495 e^4 x^4 d^3+792 e^5 x^5 d^2+924 e^6 x^6 d+792 e^7 x^7\right )\right ) b^6+6 a^5 e^5 \left (A e \left (d^5+12 e x d^4+66 e^2 x^2 d^3+220 e^3 x^3 d^2+495 e^4 x^4 d+792 e^5 x^5\right )+B \left (d^6+12 e x d^5+66 e^2 x^2 d^4+220 e^3 x^3 d^3+495 e^4 x^4 d^2+792 e^5 x^5 d+924 e^6 x^6\right )\right ) b^5+a^6 e^6 \left (7 A e \left (d^4+12 e x d^3+66 e^2 x^2 d^2+220 e^3 x^3 d+495 e^4 x^4\right )+5 B \left (d^5+12 e x d^4+66 e^2 x^2 d^3+220 e^3 x^3 d^2+495 e^4 x^4 d+792 e^5 x^5\right )\right ) b^4+4 a^7 e^7 \left (2 A e \left (d^3+12 e x d^2+66 e^2 x^2 d+220 e^3 x^3\right )+B \left (d^4+12 e x d^3+66 e^2 x^2 d^2+220 e^3 x^3 d+495 e^4 x^4\right )\right ) b^3+3 a^8 e^8 \left (3 A e \left (d^2+12 e x d+66 e^2 x^2\right )+B \left (d^3+12 e x d^2+66 e^2 x^2 d+220 e^3 x^3\right )\right ) b^2+2 a^9 e^9 \left (5 A e (d+12 e x)+B \left (d^2+12 e x d+66 e^2 x^2\right )\right ) b+a^{10} e^{10} (11 A e+B (d+12 e x))}{132 e^{12} (d+e x)^{12}} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)^10*(A + B*x))/(d + e*x)^13,x]

[Out]

-(a^10*e^10*(11*A*e + B*(d + 12*e*x)) + 2*a^9*b*e^9*(5*A*e*(d + 12*e*x) + B*(d^2
+ 12*d*e*x + 66*e^2*x^2)) + 3*a^8*b^2*e^8*(3*A*e*(d^2 + 12*d*e*x + 66*e^2*x^2)
+ B*(d^3 + 12*d^2*e*x + 66*d*e^2*x^2 + 220*e^3*x^3)) + 4*a^7*b^3*e^7*(2*A*e*(d^3
+ 12*d^2*e*x + 66*d*e^2*x^2 + 220*e^3*x^3) + B*(d^4 + 12*d^3*e*x + 66*d^2*e^2*x
^2 + 220*d*e^3*x^3 + 495*e^4*x^4)) + a^6*b^4*e^6*(7*A*e*(d^4 + 12*d^3*e*x + 66*d
^2*e^2*x^2 + 220*d*e^3*x^3 + 495*e^4*x^4) + 5*B*(d^5 + 12*d^4*e*x + 66*d^3*e^2*x
^2 + 220*d^2*e^3*x^3 + 495*d*e^4*x^4 + 792*e^5*x^5)) + 6*a^5*b^5*e^5*(A*e*(d^5 +
12*d^4*e*x + 66*d^3*e^2*x^2 + 220*d^2*e^3*x^3 + 495*d*e^4*x^4 + 792*e^5*x^5) +
B*(d^6 + 12*d^5*e*x + 66*d^4*e^2*x^2 + 220*d^3*e^3*x^3 + 495*d^2*e^4*x^4 + 792*d
*e^5*x^5 + 924*e^6*x^6)) + a^4*b^6*e^4*(5*A*e*(d^6 + 12*d^5*e*x + 66*d^4*e^2*x^2
+ 220*d^3*e^3*x^3 + 495*d^2*e^4*x^4 + 792*d*e^5*x^5 + 924*e^6*x^6) + 7*B*(d^7 +
12*d^6*e*x + 66*d^5*e^2*x^2 + 220*d^4*e^3*x^3 + 495*d^3*e^4*x^4 + 792*d^2*e^5*x
^5 + 924*d*e^6*x^6 + 792*e^7*x^7)) + 4*a^3*b^7*e^3*(A*e*(d^7 + 12*d^6*e*x + 66*d
^5*e^2*x^2 + 220*d^4*e^3*x^3 + 495*d^3*e^4*x^4 + 792*d^2*e^5*x^5 + 924*d*e^6*x^6
+ 792*e^7*x^7) + 2*B*(d^8 + 12*d^7*e*x + 66*d^6*e^2*x^2 + 220*d^5*e^3*x^3 + 495
*d^4*e^4*x^4 + 792*d^3*e^5*x^5 + 924*d^2*e^6*x^6 + 792*d*e^7*x^7 + 495*e^8*x^8))
+ 3*a^2*b^8*e^2*(A*e*(d^8 + 12*d^7*e*x + 66*d^6*e^2*x^2 + 220*d^5*e^3*x^3 + 495
*d^4*e^4*x^4 + 792*d^3*e^5*x^5 + 924*d^2*e^6*x^6 + 792*d*e^7*x^7 + 495*e^8*x^8)
+ 3*B*(d^9 + 12*d^8*e*x + 66*d^7*e^2*x^2 + 220*d^6*e^3*x^3 + 495*d^5*e^4*x^4 + 7
92*d^4*e^5*x^5 + 924*d^3*e^6*x^6 + 792*d^2*e^7*x^7 + 495*d*e^8*x^8 + 220*e^9*x^9
)) + 2*a*b^9*e*(A*e*(d^9 + 12*d^8*e*x + 66*d^7*e^2*x^2 + 220*d^6*e^3*x^3 + 495*d
^5*e^4*x^4 + 792*d^4*e^5*x^5 + 924*d^3*e^6*x^6 + 792*d^2*e^7*x^7 + 495*d*e^8*x^8
+ 220*e^9*x^9) + 5*B*(d^10 + 12*d^9*e*x + 66*d^8*e^2*x^2 + 220*d^7*e^3*x^3 + 49
5*d^6*e^4*x^4 + 792*d^5*e^5*x^5 + 924*d^4*e^6*x^6 + 792*d^3*e^7*x^7 + 495*d^2*e^
8*x^8 + 220*d*e^9*x^9 + 66*e^10*x^10)) + b^10*(A*e*(d^10 + 12*d^9*e*x + 66*d^8*e
^2*x^2 + 220*d^7*e^3*x^3 + 495*d^6*e^4*x^4 + 792*d^5*e^5*x^5 + 924*d^4*e^6*x^6 +
792*d^3*e^7*x^7 + 495*d^2*e^8*x^8 + 220*d*e^9*x^9 + 66*e^10*x^10) + 11*B*(d^11
+ 12*d^10*e*x + 66*d^9*e^2*x^2 + 220*d^8*e^3*x^3 + 495*d^7*e^4*x^4 + 792*d^6*e^5
*x^5 + 924*d^5*e^6*x^6 + 792*d^4*e^7*x^7 + 495*d^3*e^8*x^8 + 220*d^2*e^9*x^9 + 6
6*d*e^10*x^10 + 12*e^11*x^11)))/(132*e^12*(d + e*x)^12)

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Maple [B]  time = 0.018, size = 1942, normalized size = 22.6 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^10*(B*x+A)/(e*x+d)^13,x)

[Out]

-5/3*b^2*(8*A*a^7*b*e^8-56*A*a^6*b^2*d*e^7+168*A*a^5*b^3*d^2*e^6-280*A*a^4*b^4*d
^3*e^5+280*A*a^3*b^5*d^4*e^4-168*A*a^2*b^6*d^5*e^3+56*A*a*b^7*d^6*e^2-8*A*b^8*d^
7*e+3*B*a^8*e^8-32*B*a^7*b*d*e^7+140*B*a^6*b^2*d^2*e^6-336*B*a^5*b^3*d^3*e^5+490
*B*a^4*b^4*d^4*e^4-448*B*a^3*b^5*d^5*e^3+252*B*a^2*b^6*d^6*e^2-80*B*a*b^7*d^7*e+
11*B*b^8*d^8)/e^12/(e*x+d)^9-1/2*b*(9*A*a^8*b*e^9-72*A*a^7*b^2*d*e^8+252*A*a^6*b
^3*d^2*e^7-504*A*a^5*b^4*d^3*e^6+630*A*a^4*b^5*d^4*e^5-504*A*a^3*b^6*d^5*e^4+252
*A*a^2*b^7*d^6*e^3-72*A*a*b^8*d^7*e^2+9*A*b^9*d^8*e+2*B*a^9*e^9-27*B*a^8*b*d*e^8
+144*B*a^7*b^2*d^2*e^7-420*B*a^6*b^3*d^3*e^6+756*B*a^5*b^4*d^4*e^5-882*B*a^4*b^5
*d^5*e^4+672*B*a^3*b^6*d^6*e^3-324*B*a^2*b^7*d^7*e^2+90*B*a*b^8*d^8*e-11*B*b^9*d
^9)/e^12/(e*x+d)^10-7*b^5*(5*A*a^4*b*e^5-20*A*a^3*b^2*d*e^4+30*A*a^2*b^3*d^2*e^3
-20*A*a*b^4*d^3*e^2+5*A*b^5*d^4*e+6*B*a^5*e^5-35*B*a^4*b*d*e^4+80*B*a^3*b^2*d^2*
e^3-90*B*a^2*b^3*d^3*e^2+50*B*a*b^4*d^4*e-11*B*b^5*d^5)/e^12/(e*x+d)^6-15/4*b^3*
(7*A*a^6*b*e^7-42*A*a^5*b^2*d*e^6+105*A*a^4*b^3*d^2*e^5-140*A*a^3*b^4*d^3*e^4+10
5*A*a^2*b^5*d^4*e^3-42*A*a*b^6*d^5*e^2+7*A*b^7*d^6*e+4*B*a^7*e^7-35*B*a^6*b*d*e^
6+126*B*a^5*b^2*d^2*e^5-245*B*a^4*b^3*d^3*e^4+280*B*a^3*b^4*d^4*e^3-189*B*a^2*b^
5*d^5*e^2+70*B*a*b^6*d^6*e-11*B*b^7*d^7)/e^12/(e*x+d)^8-6*b^4*(6*A*a^5*b*e^6-30*
A*a^4*b^2*d*e^5+60*A*a^3*b^3*d^2*e^4-60*A*a^2*b^4*d^3*e^3+30*A*a*b^5*d^4*e^2-6*A
*b^6*d^5*e+5*B*a^6*e^6-36*B*a^5*b*d*e^5+105*B*a^4*b^2*d^2*e^4-160*B*a^3*b^3*d^3*
e^3+135*B*a^2*b^4*d^4*e^2-60*B*a*b^5*d^5*e+11*B*b^6*d^6)/e^12/(e*x+d)^7-5/3*b^8*
(2*A*a*b*e^2-2*A*b^2*d*e+9*B*a^2*e^2-20*B*a*b*d*e+11*B*b^2*d^2)/e^12/(e*x+d)^3-1
/12*(A*a^10*e^11-10*A*a^9*b*d*e^10+45*A*a^8*b^2*d^2*e^9-120*A*a^7*b^3*d^3*e^8+21
0*A*a^6*b^4*d^4*e^7-252*A*a^5*b^5*d^5*e^6+210*A*a^4*b^6*d^6*e^5-120*A*a^3*b^7*d^
7*e^4+45*A*a^2*b^8*d^8*e^3-10*A*a*b^9*d^9*e^2+A*b^10*d^10*e-B*a^10*d*e^10+10*B*a
^9*b*d^2*e^9-45*B*a^8*b^2*d^3*e^8+120*B*a^7*b^3*d^4*e^7-210*B*a^6*b^4*d^5*e^6+25
2*B*a^5*b^5*d^6*e^5-210*B*a^4*b^6*d^7*e^4+120*B*a^3*b^7*d^8*e^3-45*B*a^2*b^8*d^9
*e^2+10*B*a*b^9*d^10*e-B*b^10*d^11)/e^12/(e*x+d)^12-1/11*(10*A*a^9*b*e^10-90*A*a
^8*b^2*d*e^9+360*A*a^7*b^3*d^2*e^8-840*A*a^6*b^4*d^3*e^7+1260*A*a^5*b^5*d^4*e^6-
1260*A*a^4*b^6*d^5*e^5+840*A*a^3*b^7*d^6*e^4-360*A*a^2*b^8*d^7*e^3+90*A*a*b^9*d^
8*e^2-10*A*b^10*d^9*e+B*a^10*e^10-20*B*a^9*b*d*e^9+135*B*a^8*b^2*d^2*e^8-480*B*a
^7*b^3*d^3*e^7+1050*B*a^6*b^4*d^4*e^6-1512*B*a^5*b^5*d^5*e^5+1470*B*a^4*b^6*d^6*
e^4-960*B*a^3*b^7*d^7*e^3+405*B*a^2*b^8*d^8*e^2-100*B*a*b^9*d^9*e+11*B*b^10*d^10
)/e^12/(e*x+d)^11-B*b^10/e^12/(e*x+d)-1/2*b^9*(A*b*e+10*B*a*e-11*B*b*d)/e^12/(e*
x+d)^2-15/4*b^7*(3*A*a^2*b*e^3-6*A*a*b^2*d*e^2+3*A*b^3*d^2*e+8*B*a^3*e^3-27*B*a^
2*b*d*e^2+30*B*a*b^2*d^2*e-11*B*b^3*d^3)/e^12/(e*x+d)^4-6*b^6*(4*A*a^3*b*e^4-12*
A*a^2*b^2*d*e^3+12*A*a*b^3*d^2*e^2-4*A*b^4*d^3*e+7*B*a^4*e^4-32*B*a^3*b*d*e^3+54
*B*a^2*b^2*d^2*e^2-40*B*a*b^3*d^3*e+11*B*b^4*d^4)/e^12/(e*x+d)^5

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Maxima [A]  time = 1.53474, size = 2531, normalized size = 29.43 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^10/(e*x + d)^13,x, algorithm="maxima")

[Out]

-1/132*(132*B*b^10*e^11*x^11 + 11*B*b^10*d^11 + 11*A*a^10*e^11 + (10*B*a*b^9 + A
*b^10)*d^10*e + (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 + (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 + (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
+ (3*B*a^8*b^2 + 8*A*a^7*b^3)*d^3*e^8 + (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 + 66*(11*B*b^10*d*e^10 + (10*B*a*b^9 + A*b^10)*e^11)*x^
10 + 220*(11*B*b^10*d^2*e^9 + (10*B*a*b^9 + A*b^10)*d*e^10 + (9*B*a^2*b^8 + 2*A*
a*b^9)*e^11)*x^9 + 495*(11*B*b^10*d^3*e^8 + (10*B*a*b^9 + A*b^10)*d^2*e^9 + (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 + 792*(11*B
*b^10*d^4*e^7 + (10*B*a*b^9 + A*b^10)*d^3*e^8 + (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 + (7*B*a^4*b^6 + 4*A*a^3*b^7)*e^11)*x^7 +
924*(11*B*b^10*d^5*e^6 + (10*B*a*b^9 + A*b^10)*d^4*e^7 + (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 + (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 + 792*(11*B*b^10*d^6*e^5 + (10*B*
a*b^9 + A*b^10)*d^5*e^6 + (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 + (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 + (5*B*a^6*b^4 + 6*A*a^5*b^5)*e^11)*x^5 + 495*(11*B*b^10*d^7*e^4 +
(10*B*a*b^9 + A*b^10)*d^6*e^5 + (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 + (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 + (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 + 220*(11*B*b^10*d^8*e^3 + (10*B*a*b^9 + A*b^10)*d^7*e^4 + (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 + (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 + (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 + (3*B*a^8*b^2 + 8*A*
a^7*b^3)*e^11)*x^3 + 66*(11*B*b^10*d^9*e^2 + (10*B*a*b^9 + A*b^10)*d^8*e^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 + (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 + (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 + (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 + 12*(11*B*b^10*d^10*e +
(10*B*a*b^9 + A*b^10)*d^9*e^2 + (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 + (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 + (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 + (3*B*a^8*b^2 + 8*A*a^7*b^3)*d^2*e^9 + (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^24*x^12 + 12*d*e^23*x^11 + 66*d^2*
e^22*x^10 + 220*d^3*e^21*x^9 + 495*d^4*e^20*x^8 + 792*d^5*e^19*x^7 + 924*d^6*e^1
8*x^6 + 792*d^7*e^17*x^5 + 495*d^8*e^16*x^4 + 220*d^9*e^15*x^3 + 66*d^10*e^14*x^
2 + 12*d^11*e^13*x + d^12*e^12)

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Fricas [A]  time = 0.218395, size = 2531, normalized size = 29.43 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^10/(e*x + d)^13,x, algorithm="fricas")

[Out]

-1/132*(132*B*b^10*e^11*x^11 + 11*B*b^10*d^11 + 11*A*a^10*e^11 + (10*B*a*b^9 + A
*b^10)*d^10*e + (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 + (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 + (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
+ (3*B*a^8*b^2 + 8*A*a^7*b^3)*d^3*e^8 + (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 + 66*(11*B*b^10*d*e^10 + (10*B*a*b^9 + A*b^10)*e^11)*x^
10 + 220*(11*B*b^10*d^2*e^9 + (10*B*a*b^9 + A*b^10)*d*e^10 + (9*B*a^2*b^8 + 2*A*
a*b^9)*e^11)*x^9 + 495*(11*B*b^10*d^3*e^8 + (10*B*a*b^9 + A*b^10)*d^2*e^9 + (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 + 792*(11*B
*b^10*d^4*e^7 + (10*B*a*b^9 + A*b^10)*d^3*e^8 + (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 + (7*B*a^4*b^6 + 4*A*a^3*b^7)*e^11)*x^7 +
924*(11*B*b^10*d^5*e^6 + (10*B*a*b^9 + A*b^10)*d^4*e^7 + (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 + (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 + 792*(11*B*b^10*d^6*e^5 + (10*B*
a*b^9 + A*b^10)*d^5*e^6 + (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 + (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 + (5*B*a^6*b^4 + 6*A*a^5*b^5)*e^11)*x^5 + 495*(11*B*b^10*d^7*e^4 +
(10*B*a*b^9 + A*b^10)*d^6*e^5 + (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 + (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 + (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 + 220*(11*B*b^10*d^8*e^3 + (10*B*a*b^9 + A*b^10)*d^7*e^4 + (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 + (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 + (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 + (3*B*a^8*b^2 + 8*A*
a^7*b^3)*e^11)*x^3 + 66*(11*B*b^10*d^9*e^2 + (10*B*a*b^9 + A*b^10)*d^8*e^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 + (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 + (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 + (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 + 12*(11*B*b^10*d^10*e +
(10*B*a*b^9 + A*b^10)*d^9*e^2 + (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 + (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 + (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 + (3*B*a^8*b^2 + 8*A*a^7*b^3)*d^2*e^9 + (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^24*x^12 + 12*d*e^23*x^11 + 66*d^2*
e^22*x^10 + 220*d^3*e^21*x^9 + 495*d^4*e^20*x^8 + 792*d^5*e^19*x^7 + 924*d^6*e^1
8*x^6 + 792*d^7*e^17*x^5 + 495*d^8*e^16*x^4 + 220*d^9*e^15*x^3 + 66*d^10*e^14*x^
2 + 12*d^11*e^13*x + d^12*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)**13,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.217602, size = 1, normalized size = 0.01 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^10/(e*x + d)^13,x, algorithm="giac")

[Out]

Done