3.1085 \(\int \frac{(a+b x)^{10} (A+B x)}{(d+e x)^{14}} \, dx\)

Optimal. Leaf size=135 \[ \frac{b (a+b x)^{11} (-13 a B e+2 A b e+11 b B d)}{1716 e (d+e x)^{11} (b d-a e)^3}+\frac{(a+b x)^{11} (-13 a B e+2 A b e+11 b B d)}{156 e (d+e x)^{12} (b d-a e)^2}-\frac{(a+b x)^{11} (B d-A e)}{13 e (d+e x)^{13} (b d-a e)} \]

[Out]

-((B*d - A*e)*(a + b*x)^11)/(13*e*(b*d - a*e)*(d + e*x)^13) + ((11*b*B*d + 2*A*b
*e - 13*a*B*e)*(a + b*x)^11)/(156*e*(b*d - a*e)^2*(d + e*x)^12) + (b*(11*b*B*d +
2*A*b*e - 13*a*B*e)*(a + b*x)^11)/(1716*e*(b*d - a*e)^3*(d + e*x)^11)

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Rubi [A]  time = 0.183297, antiderivative size = 135, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ \frac{b (a+b x)^{11} (-13 a B e+2 A b e+11 b B d)}{1716 e (d+e x)^{11} (b d-a e)^3}+\frac{(a+b x)^{11} (-13 a B e+2 A b e+11 b B d)}{156 e (d+e x)^{12} (b d-a e)^2}-\frac{(a+b x)^{11} (B d-A e)}{13 e (d+e x)^{13} (b d-a e)} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)^10*(A + B*x))/(d + e*x)^14,x]

[Out]

-((B*d - A*e)*(a + b*x)^11)/(13*e*(b*d - a*e)*(d + e*x)^13) + ((11*b*B*d + 2*A*b
*e - 13*a*B*e)*(a + b*x)^11)/(156*e*(b*d - a*e)^2*(d + e*x)^12) + (b*(11*b*B*d +
2*A*b*e - 13*a*B*e)*(a + b*x)^11)/(1716*e*(b*d - a*e)^3*(d + e*x)^11)

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Rubi in Sympy [A]  time = 24.7609, size = 122, normalized size = 0.9 \[ - \frac{b \left (a + b x\right )^{11} \left (2 A b e - 13 B a e + 11 B b d\right )}{1716 e \left (d + e x\right )^{11} \left (a e - b d\right )^{3}} + \frac{\left (a + b x\right )^{11} \left (2 A b e - 13 B a e + 11 B b d\right )}{156 e \left (d + e x\right )^{12} \left (a e - b d\right )^{2}} - \frac{\left (a + b x\right )^{11} \left (A e - B d\right )}{13 e \left (d + e x\right )^{13} \left (a e - b d\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**10*(B*x+A)/(e*x+d)**14,x)

[Out]

-b*(a + b*x)**11*(2*A*b*e - 13*B*a*e + 11*B*b*d)/(1716*e*(d + e*x)**11*(a*e - b*
d)**3) + (a + b*x)**11*(2*A*b*e - 13*B*a*e + 11*B*b*d)/(156*e*(d + e*x)**12*(a*e
- b*d)**2) - (a + b*x)**11*(A*e - B*d)/(13*e*(d + e*x)**13*(a*e - b*d))

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Mathematica [B]  time = 3.03138, size = 1433, normalized size = 10.61 \[ -\frac{\left (2 A e \left (d^{10}+13 e x d^9+78 e^2 x^2 d^8+286 e^3 x^3 d^7+715 e^4 x^4 d^6+1287 e^5 x^5 d^5+1716 e^6 x^6 d^4+1716 e^7 x^7 d^3+1287 e^8 x^8 d^2+715 e^9 x^9 d+286 e^{10} x^{10}\right )+11 B \left (d^{11}+13 e x d^{10}+78 e^2 x^2 d^9+286 e^3 x^3 d^8+715 e^4 x^4 d^7+1287 e^5 x^5 d^6+1716 e^6 x^6 d^5+1716 e^7 x^7 d^4+1287 e^8 x^8 d^3+715 e^9 x^9 d^2+286 e^{10} x^{10} d+78 e^{11} x^{11}\right )\right ) b^{10}+2 a e \left (3 A e \left (d^9+13 e x d^8+78 e^2 x^2 d^7+286 e^3 x^3 d^6+715 e^4 x^4 d^5+1287 e^5 x^5 d^4+1716 e^6 x^6 d^3+1716 e^7 x^7 d^2+1287 e^8 x^8 d+715 e^9 x^9\right )+10 B \left (d^{10}+13 e x d^9+78 e^2 x^2 d^8+286 e^3 x^3 d^7+715 e^4 x^4 d^6+1287 e^5 x^5 d^5+1716 e^6 x^6 d^4+1716 e^7 x^7 d^3+1287 e^8 x^8 d^2+715 e^9 x^9 d+286 e^{10} x^{10}\right )\right ) b^9+3 a^2 e^2 \left (4 A e \left (d^8+13 e x d^7+78 e^2 x^2 d^6+286 e^3 x^3 d^5+715 e^4 x^4 d^4+1287 e^5 x^5 d^3+1716 e^6 x^6 d^2+1716 e^7 x^7 d+1287 e^8 x^8\right )+9 B \left (d^9+13 e x d^8+78 e^2 x^2 d^7+286 e^3 x^3 d^6+715 e^4 x^4 d^5+1287 e^5 x^5 d^4+1716 e^6 x^6 d^3+1716 e^7 x^7 d^2+1287 e^8 x^8 d+715 e^9 x^9\right )\right ) b^8+4 a^3 e^3 \left (5 A e \left (d^7+13 e x d^6+78 e^2 x^2 d^5+286 e^3 x^3 d^4+715 e^4 x^4 d^3+1287 e^5 x^5 d^2+1716 e^6 x^6 d+1716 e^7 x^7\right )+8 B \left (d^8+13 e x d^7+78 e^2 x^2 d^6+286 e^3 x^3 d^5+715 e^4 x^4 d^4+1287 e^5 x^5 d^3+1716 e^6 x^6 d^2+1716 e^7 x^7 d+1287 e^8 x^8\right )\right ) b^7+5 a^4 e^4 \left (6 A e \left (d^6+13 e x d^5+78 e^2 x^2 d^4+286 e^3 x^3 d^3+715 e^4 x^4 d^2+1287 e^5 x^5 d+1716 e^6 x^6\right )+7 B \left (d^7+13 e x d^6+78 e^2 x^2 d^5+286 e^3 x^3 d^4+715 e^4 x^4 d^3+1287 e^5 x^5 d^2+1716 e^6 x^6 d+1716 e^7 x^7\right )\right ) b^6+6 a^5 e^5 \left (7 A e \left (d^5+13 e x d^4+78 e^2 x^2 d^3+286 e^3 x^3 d^2+715 e^4 x^4 d+1287 e^5 x^5\right )+6 B \left (d^6+13 e x d^5+78 e^2 x^2 d^4+286 e^3 x^3 d^3+715 e^4 x^4 d^2+1287 e^5 x^5 d+1716 e^6 x^6\right )\right ) b^5+7 a^6 e^6 \left (8 A e \left (d^4+13 e x d^3+78 e^2 x^2 d^2+286 e^3 x^3 d+715 e^4 x^4\right )+5 B \left (d^5+13 e x d^4+78 e^2 x^2 d^3+286 e^3 x^3 d^2+715 e^4 x^4 d+1287 e^5 x^5\right )\right ) b^4+8 a^7 e^7 \left (9 A e \left (d^3+13 e x d^2+78 e^2 x^2 d+286 e^3 x^3\right )+4 B \left (d^4+13 e x d^3+78 e^2 x^2 d^2+286 e^3 x^3 d+715 e^4 x^4\right )\right ) b^3+9 a^8 e^8 \left (10 A e \left (d^2+13 e x d+78 e^2 x^2\right )+3 B \left (d^3+13 e x d^2+78 e^2 x^2 d+286 e^3 x^3\right )\right ) b^2+10 a^9 e^9 \left (11 A e (d+13 e x)+2 B \left (d^2+13 e x d+78 e^2 x^2\right )\right ) b+11 a^{10} e^{10} (12 A e+B (d+13 e x))}{1716 e^{12} (d+e x)^{13}} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)^10*(A + B*x))/(d + e*x)^14,x]

[Out]

-(11*a^10*e^10*(12*A*e + B*(d + 13*e*x)) + 10*a^9*b*e^9*(11*A*e*(d + 13*e*x) + 2
*B*(d^2 + 13*d*e*x + 78*e^2*x^2)) + 9*a^8*b^2*e^8*(10*A*e*(d^2 + 13*d*e*x + 78*e
^2*x^2) + 3*B*(d^3 + 13*d^2*e*x + 78*d*e^2*x^2 + 286*e^3*x^3)) + 8*a^7*b^3*e^7*(
9*A*e*(d^3 + 13*d^2*e*x + 78*d*e^2*x^2 + 286*e^3*x^3) + 4*B*(d^4 + 13*d^3*e*x +
78*d^2*e^2*x^2 + 286*d*e^3*x^3 + 715*e^4*x^4)) + 7*a^6*b^4*e^6*(8*A*e*(d^4 + 13*
d^3*e*x + 78*d^2*e^2*x^2 + 286*d*e^3*x^3 + 715*e^4*x^4) + 5*B*(d^5 + 13*d^4*e*x
+ 78*d^3*e^2*x^2 + 286*d^2*e^3*x^3 + 715*d*e^4*x^4 + 1287*e^5*x^5)) + 6*a^5*b^5*
e^5*(7*A*e*(d^5 + 13*d^4*e*x + 78*d^3*e^2*x^2 + 286*d^2*e^3*x^3 + 715*d*e^4*x^4
+ 1287*e^5*x^5) + 6*B*(d^6 + 13*d^5*e*x + 78*d^4*e^2*x^2 + 286*d^3*e^3*x^3 + 715
*d^2*e^4*x^4 + 1287*d*e^5*x^5 + 1716*e^6*x^6)) + 5*a^4*b^6*e^4*(6*A*e*(d^6 + 13*
d^5*e*x + 78*d^4*e^2*x^2 + 286*d^3*e^3*x^3 + 715*d^2*e^4*x^4 + 1287*d*e^5*x^5 +
1716*e^6*x^6) + 7*B*(d^7 + 13*d^6*e*x + 78*d^5*e^2*x^2 + 286*d^4*e^3*x^3 + 715*d
^3*e^4*x^4 + 1287*d^2*e^5*x^5 + 1716*d*e^6*x^6 + 1716*e^7*x^7)) + 4*a^3*b^7*e^3*
(5*A*e*(d^7 + 13*d^6*e*x + 78*d^5*e^2*x^2 + 286*d^4*e^3*x^3 + 715*d^3*e^4*x^4 +
1287*d^2*e^5*x^5 + 1716*d*e^6*x^6 + 1716*e^7*x^7) + 8*B*(d^8 + 13*d^7*e*x + 78*d
^6*e^2*x^2 + 286*d^5*e^3*x^3 + 715*d^4*e^4*x^4 + 1287*d^3*e^5*x^5 + 1716*d^2*e^6
*x^6 + 1716*d*e^7*x^7 + 1287*e^8*x^8)) + 3*a^2*b^8*e^2*(4*A*e*(d^8 + 13*d^7*e*x
+ 78*d^6*e^2*x^2 + 286*d^5*e^3*x^3 + 715*d^4*e^4*x^4 + 1287*d^3*e^5*x^5 + 1716*d
^2*e^6*x^6 + 1716*d*e^7*x^7 + 1287*e^8*x^8) + 9*B*(d^9 + 13*d^8*e*x + 78*d^7*e^2
*x^2 + 286*d^6*e^3*x^3 + 715*d^5*e^4*x^4 + 1287*d^4*e^5*x^5 + 1716*d^3*e^6*x^6 +
1716*d^2*e^7*x^7 + 1287*d*e^8*x^8 + 715*e^9*x^9)) + 2*a*b^9*e*(3*A*e*(d^9 + 13*
d^8*e*x + 78*d^7*e^2*x^2 + 286*d^6*e^3*x^3 + 715*d^5*e^4*x^4 + 1287*d^4*e^5*x^5
+ 1716*d^3*e^6*x^6 + 1716*d^2*e^7*x^7 + 1287*d*e^8*x^8 + 715*e^9*x^9) + 10*B*(d^
10 + 13*d^9*e*x + 78*d^8*e^2*x^2 + 286*d^7*e^3*x^3 + 715*d^6*e^4*x^4 + 1287*d^5*
e^5*x^5 + 1716*d^4*e^6*x^6 + 1716*d^3*e^7*x^7 + 1287*d^2*e^8*x^8 + 715*d*e^9*x^9
+ 286*e^10*x^10)) + b^10*(2*A*e*(d^10 + 13*d^9*e*x + 78*d^8*e^2*x^2 + 286*d^7*e
^3*x^3 + 715*d^6*e^4*x^4 + 1287*d^5*e^5*x^5 + 1716*d^4*e^6*x^6 + 1716*d^3*e^7*x^
7 + 1287*d^2*e^8*x^8 + 715*d*e^9*x^9 + 286*e^10*x^10) + 11*B*(d^11 + 13*d^10*e*x
+ 78*d^9*e^2*x^2 + 286*d^8*e^3*x^3 + 715*d^7*e^4*x^4 + 1287*d^6*e^5*x^5 + 1716*
d^5*e^6*x^6 + 1716*d^4*e^7*x^7 + 1287*d^3*e^8*x^8 + 715*d^2*e^9*x^9 + 286*d*e^10
*x^10 + 78*e^11*x^11)))/(1716*e^12*(d + e*x)^13)

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Maple [B]  time = 0.016, size = 1942, normalized size = 14.4 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^10*(B*x+A)/(e*x+d)^14,x)

[Out]

-10/3*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+105*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^3-18
9*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)^9-3/2*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)^10-5*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)^6-21/4*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)^8-6*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)^7-1/3*b^9*(A*b*e+10*B*a*e-11*B*b*d)/e^12/(e*x+d)^3-1/12*(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)^12-5/11*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+25
2*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)^11-1/13*(A*a^10*e^11-10*A*a^9*b*d*e^10+45*A*a^8*b^2*d^2*e^9-12
0*A*a^7*b^3*d^3*e^8+210*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^7-21
0*B*a^6*b^4*d^5*e^6+252*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)^13-1/2*B*
b^10/e^12/(e*x+d)^2-5/4*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)^4-3*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)^5

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Maxima [A]  time = 1.53606, size = 2634, normalized size = 19.51 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^10/(e*x + d)^14,x, algorithm="maxima")

[Out]

-1/1716*(858*B*b^10*e^11*x^11 + 11*B*b^10*d^11 + 132*A*a^10*e^11 + 2*(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 + 4*(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*(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 + 8*(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 + 10*(2*B*a^9*b + 9*A*a^
8*b^2)*d^2*e^9 + 11*(B*a^10 + 10*A*a^9*b)*d*e^10 + 286*(11*B*b^10*d*e^10 + 2*(10
*B*a*b^9 + A*b^10)*e^11)*x^10 + 715*(11*B*b^10*d^2*e^9 + 2*(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 + 1287*(11*B*b^10*d^3*e^8 + 2*(1
0*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 + 4*(8*B*a^3*b^
7 + 3*A*a^2*b^8)*e^11)*x^8 + 1716*(11*B*b^10*d^4*e^7 + 2*(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 + 4*(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 + 1716*(11*B*b^10*d^5*e^6 + 2*(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 + 4*(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*(6*B*a^5*b^5
+ 5*A*a^4*b^6)*e^11)*x^6 + 1287*(11*B*b^10*d^6*e^5 + 2*(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 + 4*(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*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d*e^1
0 + 7*(5*B*a^6*b^4 + 6*A*a^5*b^5)*e^11)*x^5 + 715*(11*B*b^10*d^7*e^4 + 2*(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 + 4*(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*(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 + 8*(4*B*a^7*b^3 + 7
*A*a^6*b^4)*e^11)*x^4 + 286*(11*B*b^10*d^8*e^3 + 2*(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 + 4*(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*(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 + 8*(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 + 78*(11*B*b^10*d^9*e^2 + 2*(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 + 4*(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*(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 + 8*(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 + 10*(2*B*a^9*b + 9*A*a^8
*b^2)*e^11)*x^2 + 13*(11*B*b^10*d^10*e + 2*(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 + 4*(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*(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 + 8*(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 + 10*(2*B*a^9*b + 9*A*a^8*b^2)*d*e^10 + 11*(B*a^1
0 + 10*A*a^9*b)*e^11)*x)/(e^25*x^13 + 13*d*e^24*x^12 + 78*d^2*e^23*x^11 + 286*d^
3*e^22*x^10 + 715*d^4*e^21*x^9 + 1287*d^5*e^20*x^8 + 1716*d^6*e^19*x^7 + 1716*d^
7*e^18*x^6 + 1287*d^8*e^17*x^5 + 715*d^9*e^16*x^4 + 286*d^10*e^15*x^3 + 78*d^11*
e^14*x^2 + 13*d^12*e^13*x + d^13*e^12)

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Fricas [A]  time = 0.217003, size = 2634, normalized size = 19.51 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^10/(e*x + d)^14,x, algorithm="fricas")

[Out]

-1/1716*(858*B*b^10*e^11*x^11 + 11*B*b^10*d^11 + 132*A*a^10*e^11 + 2*(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 + 4*(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*(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 + 8*(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 + 10*(2*B*a^9*b + 9*A*a^
8*b^2)*d^2*e^9 + 11*(B*a^10 + 10*A*a^9*b)*d*e^10 + 286*(11*B*b^10*d*e^10 + 2*(10
*B*a*b^9 + A*b^10)*e^11)*x^10 + 715*(11*B*b^10*d^2*e^9 + 2*(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 + 1287*(11*B*b^10*d^3*e^8 + 2*(1
0*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 + 4*(8*B*a^3*b^
7 + 3*A*a^2*b^8)*e^11)*x^8 + 1716*(11*B*b^10*d^4*e^7 + 2*(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 + 4*(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 + 1716*(11*B*b^10*d^5*e^6 + 2*(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 + 4*(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*(6*B*a^5*b^5
+ 5*A*a^4*b^6)*e^11)*x^6 + 1287*(11*B*b^10*d^6*e^5 + 2*(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 + 4*(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*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d*e^1
0 + 7*(5*B*a^6*b^4 + 6*A*a^5*b^5)*e^11)*x^5 + 715*(11*B*b^10*d^7*e^4 + 2*(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 + 4*(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*(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 + 8*(4*B*a^7*b^3 + 7
*A*a^6*b^4)*e^11)*x^4 + 286*(11*B*b^10*d^8*e^3 + 2*(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 + 4*(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*(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 + 8*(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 + 78*(11*B*b^10*d^9*e^2 + 2*(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 + 4*(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*(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 + 8*(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 + 10*(2*B*a^9*b + 9*A*a^8
*b^2)*e^11)*x^2 + 13*(11*B*b^10*d^10*e + 2*(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 + 4*(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*(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 + 8*(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 + 10*(2*B*a^9*b + 9*A*a^8*b^2)*d*e^10 + 11*(B*a^1
0 + 10*A*a^9*b)*e^11)*x)/(e^25*x^13 + 13*d*e^24*x^12 + 78*d^2*e^23*x^11 + 286*d^
3*e^22*x^10 + 715*d^4*e^21*x^9 + 1287*d^5*e^20*x^8 + 1716*d^6*e^19*x^7 + 1716*d^
7*e^18*x^6 + 1287*d^8*e^17*x^5 + 715*d^9*e^16*x^4 + 286*d^10*e^15*x^3 + 78*d^11*
e^14*x^2 + 13*d^12*e^13*x + d^13*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)**14,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.215198, 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)^14,x, algorithm="giac")

[Out]

Done