3.1086 \(\int \frac{(a+b x)^{10} (A+B x)}{(d+e x)^{15}} \, dx\)

Optimal. Leaf size=185 \[ \frac{b^2 (a+b x)^{11} (-14 a B e+3 A b e+11 b B d)}{12012 e (d+e x)^{11} (b d-a e)^4}+\frac{b (a+b x)^{11} (-14 a B e+3 A b e+11 b B d)}{1092 e (d+e x)^{12} (b d-a e)^3}+\frac{(a+b x)^{11} (-14 a B e+3 A b e+11 b B d)}{182 e (d+e x)^{13} (b d-a e)^2}-\frac{(a+b x)^{11} (B d-A e)}{14 e (d+e x)^{14} (b d-a e)} \]

[Out]

-((B*d - A*e)*(a + b*x)^11)/(14*e*(b*d - a*e)*(d + e*x)^14) + ((11*b*B*d + 3*A*b
*e - 14*a*B*e)*(a + b*x)^11)/(182*e*(b*d - a*e)^2*(d + e*x)^13) + (b*(11*b*B*d +
3*A*b*e - 14*a*B*e)*(a + b*x)^11)/(1092*e*(b*d - a*e)^3*(d + e*x)^12) + (b^2*(1
1*b*B*d + 3*A*b*e - 14*a*B*e)*(a + b*x)^11)/(12012*e*(b*d - a*e)^4*(d + e*x)^11)

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Rubi [A]  time = 0.242714, antiderivative size = 185, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ \frac{b^2 (a+b x)^{11} (-14 a B e+3 A b e+11 b B d)}{12012 e (d+e x)^{11} (b d-a e)^4}+\frac{b (a+b x)^{11} (-14 a B e+3 A b e+11 b B d)}{1092 e (d+e x)^{12} (b d-a e)^3}+\frac{(a+b x)^{11} (-14 a B e+3 A b e+11 b B d)}{182 e (d+e x)^{13} (b d-a e)^2}-\frac{(a+b x)^{11} (B d-A e)}{14 e (d+e x)^{14} (b d-a e)} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)^10*(A + B*x))/(d + e*x)^15,x]

[Out]

-((B*d - A*e)*(a + b*x)^11)/(14*e*(b*d - a*e)*(d + e*x)^14) + ((11*b*B*d + 3*A*b
*e - 14*a*B*e)*(a + b*x)^11)/(182*e*(b*d - a*e)^2*(d + e*x)^13) + (b*(11*b*B*d +
3*A*b*e - 14*a*B*e)*(a + b*x)^11)/(1092*e*(b*d - a*e)^3*(d + e*x)^12) + (b^2*(1
1*b*B*d + 3*A*b*e - 14*a*B*e)*(a + b*x)^11)/(12012*e*(b*d - a*e)^4*(d + e*x)^11)

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Rubi in Sympy [A]  time = 36.7231, size = 172, normalized size = 0.93 \[ \frac{b^{2} \left (a + b x\right )^{11} \left (3 A b e - 14 B a e + 11 B b d\right )}{12012 e \left (d + e x\right )^{11} \left (a e - b d\right )^{4}} - \frac{b \left (a + b x\right )^{11} \left (3 A b e - 14 B a e + 11 B b d\right )}{1092 e \left (d + e x\right )^{12} \left (a e - b d\right )^{3}} + \frac{\left (a + b x\right )^{11} \left (3 A b e - 14 B a e + 11 B b d\right )}{182 e \left (d + e x\right )^{13} \left (a e - b d\right )^{2}} - \frac{\left (a + b x\right )^{11} \left (A e - B d\right )}{14 e \left (d + e x\right )^{14} \left (a e - b d\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**10*(B*x+A)/(e*x+d)**15,x)

[Out]

b**2*(a + b*x)**11*(3*A*b*e - 14*B*a*e + 11*B*b*d)/(12012*e*(d + e*x)**11*(a*e -
b*d)**4) - b*(a + b*x)**11*(3*A*b*e - 14*B*a*e + 11*B*b*d)/(1092*e*(d + e*x)**1
2*(a*e - b*d)**3) + (a + b*x)**11*(3*A*b*e - 14*B*a*e + 11*B*b*d)/(182*e*(d + e*
x)**13*(a*e - b*d)**2) - (a + b*x)**11*(A*e - B*d)/(14*e*(d + e*x)**14*(a*e - b*
d))

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Mathematica [B]  time = 3.10065, size = 1430, normalized size = 7.73 \[ -\frac{\left (3 A e \left (d^{10}+14 e x d^9+91 e^2 x^2 d^8+364 e^3 x^3 d^7+1001 e^4 x^4 d^6+2002 e^5 x^5 d^5+3003 e^6 x^6 d^4+3432 e^7 x^7 d^3+3003 e^8 x^8 d^2+2002 e^9 x^9 d+1001 e^{10} x^{10}\right )+11 B \left (d^{11}+14 e x d^{10}+91 e^2 x^2 d^9+364 e^3 x^3 d^8+1001 e^4 x^4 d^7+2002 e^5 x^5 d^6+3003 e^6 x^6 d^5+3432 e^7 x^7 d^4+3003 e^8 x^8 d^3+2002 e^9 x^9 d^2+1001 e^{10} x^{10} d+364 e^{11} x^{11}\right )\right ) b^{10}+6 a e \left (2 A e \left (d^9+14 e x d^8+91 e^2 x^2 d^7+364 e^3 x^3 d^6+1001 e^4 x^4 d^5+2002 e^5 x^5 d^4+3003 e^6 x^6 d^3+3432 e^7 x^7 d^2+3003 e^8 x^8 d+2002 e^9 x^9\right )+5 B \left (d^{10}+14 e x d^9+91 e^2 x^2 d^8+364 e^3 x^3 d^7+1001 e^4 x^4 d^6+2002 e^5 x^5 d^5+3003 e^6 x^6 d^4+3432 e^7 x^7 d^3+3003 e^8 x^8 d^2+2002 e^9 x^9 d+1001 e^{10} x^{10}\right )\right ) b^9+6 a^2 e^2 \left (5 A e \left (d^8+14 e x d^7+91 e^2 x^2 d^6+364 e^3 x^3 d^5+1001 e^4 x^4 d^4+2002 e^5 x^5 d^3+3003 e^6 x^6 d^2+3432 e^7 x^7 d+3003 e^8 x^8\right )+9 B \left (d^9+14 e x d^8+91 e^2 x^2 d^7+364 e^3 x^3 d^6+1001 e^4 x^4 d^5+2002 e^5 x^5 d^4+3003 e^6 x^6 d^3+3432 e^7 x^7 d^2+3003 e^8 x^8 d+2002 e^9 x^9\right )\right ) b^8+20 a^3 e^3 \left (3 A e \left (d^7+14 e x d^6+91 e^2 x^2 d^5+364 e^3 x^3 d^4+1001 e^4 x^4 d^3+2002 e^5 x^5 d^2+3003 e^6 x^6 d+3432 e^7 x^7\right )+4 B \left (d^8+14 e x d^7+91 e^2 x^2 d^6+364 e^3 x^3 d^5+1001 e^4 x^4 d^4+2002 e^5 x^5 d^3+3003 e^6 x^6 d^2+3432 e^7 x^7 d+3003 e^8 x^8\right )\right ) b^7+105 a^4 e^4 \left (A e \left (d^6+14 e x d^5+91 e^2 x^2 d^4+364 e^3 x^3 d^3+1001 e^4 x^4 d^2+2002 e^5 x^5 d+3003 e^6 x^6\right )+B \left (d^7+14 e x d^6+91 e^2 x^2 d^5+364 e^3 x^3 d^4+1001 e^4 x^4 d^3+2002 e^5 x^5 d^2+3003 e^6 x^6 d+3432 e^7 x^7\right )\right ) b^6+42 a^5 e^5 \left (4 A e \left (d^5+14 e x d^4+91 e^2 x^2 d^3+364 e^3 x^3 d^2+1001 e^4 x^4 d+2002 e^5 x^5\right )+3 B \left (d^6+14 e x d^5+91 e^2 x^2 d^4+364 e^3 x^3 d^3+1001 e^4 x^4 d^2+2002 e^5 x^5 d+3003 e^6 x^6\right )\right ) b^5+28 a^6 e^6 \left (9 A e \left (d^4+14 e x d^3+91 e^2 x^2 d^2+364 e^3 x^3 d+1001 e^4 x^4\right )+5 B \left (d^5+14 e x d^4+91 e^2 x^2 d^3+364 e^3 x^3 d^2+1001 e^4 x^4 d+2002 e^5 x^5\right )\right ) b^4+72 a^7 e^7 \left (5 A e \left (d^3+14 e x d^2+91 e^2 x^2 d+364 e^3 x^3\right )+2 B \left (d^4+14 e x d^3+91 e^2 x^2 d^2+364 e^3 x^3 d+1001 e^4 x^4\right )\right ) b^3+45 a^8 e^8 \left (11 A e \left (d^2+14 e x d+91 e^2 x^2\right )+3 B \left (d^3+14 e x d^2+91 e^2 x^2 d+364 e^3 x^3\right )\right ) b^2+110 a^9 e^9 \left (6 A e (d+14 e x)+B \left (d^2+14 e x d+91 e^2 x^2\right )\right ) b+66 a^{10} e^{10} (13 A e+B (d+14 e x))}{12012 e^{12} (d+e x)^{14}} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)^10*(A + B*x))/(d + e*x)^15,x]

[Out]

-(66*a^10*e^10*(13*A*e + B*(d + 14*e*x)) + 110*a^9*b*e^9*(6*A*e*(d + 14*e*x) + B
*(d^2 + 14*d*e*x + 91*e^2*x^2)) + 45*a^8*b^2*e^8*(11*A*e*(d^2 + 14*d*e*x + 91*e^
2*x^2) + 3*B*(d^3 + 14*d^2*e*x + 91*d*e^2*x^2 + 364*e^3*x^3)) + 72*a^7*b^3*e^7*(
5*A*e*(d^3 + 14*d^2*e*x + 91*d*e^2*x^2 + 364*e^3*x^3) + 2*B*(d^4 + 14*d^3*e*x +
91*d^2*e^2*x^2 + 364*d*e^3*x^3 + 1001*e^4*x^4)) + 28*a^6*b^4*e^6*(9*A*e*(d^4 + 1
4*d^3*e*x + 91*d^2*e^2*x^2 + 364*d*e^3*x^3 + 1001*e^4*x^4) + 5*B*(d^5 + 14*d^4*e
*x + 91*d^3*e^2*x^2 + 364*d^2*e^3*x^3 + 1001*d*e^4*x^4 + 2002*e^5*x^5)) + 42*a^5
*b^5*e^5*(4*A*e*(d^5 + 14*d^4*e*x + 91*d^3*e^2*x^2 + 364*d^2*e^3*x^3 + 1001*d*e^
4*x^4 + 2002*e^5*x^5) + 3*B*(d^6 + 14*d^5*e*x + 91*d^4*e^2*x^2 + 364*d^3*e^3*x^3
+ 1001*d^2*e^4*x^4 + 2002*d*e^5*x^5 + 3003*e^6*x^6)) + 105*a^4*b^6*e^4*(A*e*(d^
6 + 14*d^5*e*x + 91*d^4*e^2*x^2 + 364*d^3*e^3*x^3 + 1001*d^2*e^4*x^4 + 2002*d*e^
5*x^5 + 3003*e^6*x^6) + B*(d^7 + 14*d^6*e*x + 91*d^5*e^2*x^2 + 364*d^4*e^3*x^3 +
1001*d^3*e^4*x^4 + 2002*d^2*e^5*x^5 + 3003*d*e^6*x^6 + 3432*e^7*x^7)) + 20*a^3*
b^7*e^3*(3*A*e*(d^7 + 14*d^6*e*x + 91*d^5*e^2*x^2 + 364*d^4*e^3*x^3 + 1001*d^3*e
^4*x^4 + 2002*d^2*e^5*x^5 + 3003*d*e^6*x^6 + 3432*e^7*x^7) + 4*B*(d^8 + 14*d^7*e
*x + 91*d^6*e^2*x^2 + 364*d^5*e^3*x^3 + 1001*d^4*e^4*x^4 + 2002*d^3*e^5*x^5 + 30
03*d^2*e^6*x^6 + 3432*d*e^7*x^7 + 3003*e^8*x^8)) + 6*a^2*b^8*e^2*(5*A*e*(d^8 + 1
4*d^7*e*x + 91*d^6*e^2*x^2 + 364*d^5*e^3*x^3 + 1001*d^4*e^4*x^4 + 2002*d^3*e^5*x
^5 + 3003*d^2*e^6*x^6 + 3432*d*e^7*x^7 + 3003*e^8*x^8) + 9*B*(d^9 + 14*d^8*e*x +
91*d^7*e^2*x^2 + 364*d^6*e^3*x^3 + 1001*d^5*e^4*x^4 + 2002*d^4*e^5*x^5 + 3003*d
^3*e^6*x^6 + 3432*d^2*e^7*x^7 + 3003*d*e^8*x^8 + 2002*e^9*x^9)) + 6*a*b^9*e*(2*A
*e*(d^9 + 14*d^8*e*x + 91*d^7*e^2*x^2 + 364*d^6*e^3*x^3 + 1001*d^5*e^4*x^4 + 200
2*d^4*e^5*x^5 + 3003*d^3*e^6*x^6 + 3432*d^2*e^7*x^7 + 3003*d*e^8*x^8 + 2002*e^9*
x^9) + 5*B*(d^10 + 14*d^9*e*x + 91*d^8*e^2*x^2 + 364*d^7*e^3*x^3 + 1001*d^6*e^4*
x^4 + 2002*d^5*e^5*x^5 + 3003*d^4*e^6*x^6 + 3432*d^3*e^7*x^7 + 3003*d^2*e^8*x^8
+ 2002*d*e^9*x^9 + 1001*e^10*x^10)) + b^10*(3*A*e*(d^10 + 14*d^9*e*x + 91*d^8*e^
2*x^2 + 364*d^7*e^3*x^3 + 1001*d^6*e^4*x^4 + 2002*d^5*e^5*x^5 + 3003*d^4*e^6*x^6
+ 3432*d^3*e^7*x^7 + 3003*d^2*e^8*x^8 + 2002*d*e^9*x^9 + 1001*e^10*x^10) + 11*B
*(d^11 + 14*d^10*e*x + 91*d^9*e^2*x^2 + 364*d^8*e^3*x^3 + 1001*d^7*e^4*x^4 + 200
2*d^6*e^5*x^5 + 3003*d^5*e^6*x^6 + 3432*d^4*e^7*x^7 + 3003*d^3*e^8*x^8 + 2002*d^
2*e^9*x^9 + 1001*d*e^10*x^10 + 364*e^11*x^11)))/(12012*e^12*(d + e*x)^14)

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Maple [B]  time = 0.018, size = 1942, normalized size = 10.5 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^10*(B*x+A)/(e*x+d)^15,x)

[Out]

-1/14*(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+
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^7-210*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)^14-14/3*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)^9-3*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^3-189*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)^10-5/2*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)^6-21/4*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)^8-30
/7*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)^7-1/3*B*b^10/e^12/(e*x+d)^3-5/12*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)^12-15/11*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)^11-1/13*(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)^13-1/4*b^9*(A*b*e+10*B*a*e-11*B*b*d)/e^12/(e*x+d)^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)^5

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Maxima [A]  time = 1.56019, size = 2649, normalized size = 14.32 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^10/(e*x + d)^15,x, algorithm="maxima")

[Out]

-1/12012*(4004*B*b^10*e^11*x^11 + 11*B*b^10*d^11 + 858*A*a^10*e^11 + 3*(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 + 10*(8*B*a^3*b^7 + 3*
A*a^2*b^8)*d^8*e^3 + 15*(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 + 28*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^5*e^6 + 36*(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 + 55*(2*B*a^9*b
+ 9*A*a^8*b^2)*d^2*e^9 + 66*(B*a^10 + 10*A*a^9*b)*d*e^10 + 1001*(11*B*b^10*d*e^1
0 + 3*(10*B*a*b^9 + A*b^10)*e^11)*x^10 + 2002*(11*B*b^10*d^2*e^9 + 3*(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 + 3003*(11*B*b^10*d^3*
e^8 + 3*(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 + 10*
(8*B*a^3*b^7 + 3*A*a^2*b^8)*e^11)*x^8 + 3432*(11*B*b^10*d^4*e^7 + 3*(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 + 10*(8*B*a^3*b^7 + 3*A*
a^2*b^8)*d*e^10 + 15*(7*B*a^4*b^6 + 4*A*a^3*b^7)*e^11)*x^7 + 3003*(11*B*b^10*d^5
*e^6 + 3*(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 + 1
0*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^2*e^9 + 15*(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 + 2002*(11*B*b^10*d^6*e^5 + 3*(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 + 10*(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 + 21*(6*B*a^5*b^5 +
5*A*a^4*b^6)*d*e^10 + 28*(5*B*a^6*b^4 + 6*A*a^5*b^5)*e^11)*x^5 + 1001*(11*B*b^1
0*d^7*e^4 + 3*(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 + 10*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^4*e^7 + 15*(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 + 28*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d*
e^10 + 36*(4*B*a^7*b^3 + 7*A*a^6*b^4)*e^11)*x^4 + 364*(11*B*b^10*d^8*e^3 + 3*(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 + 10*(8*B*a^3*b
^7 + 3*A*a^2*b^8)*d^5*e^6 + 15*(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 + 28*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^2*e^9 + 36*(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 + 91*(11
*B*b^10*d^9*e^2 + 3*(10*B*a*b^9 + A*b^10)*d^8*e^3 + 6*(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 + 15*(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 + 28*(5*B*a^6*b^4 + 6*A*a^5*b
^5)*d^3*e^8 + 36*(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 + 55*(2*B*a^9*b + 9*A*a^8*b^2)*e^11)*x^2 + 14*(11*B*b^10*d^10*e + 3
*(10*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 + 10*(8*B*a
^3*b^7 + 3*A*a^2*b^8)*d^7*e^4 + 15*(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 + 28*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^4*e^7 + 36*(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 + 55*
(2*B*a^9*b + 9*A*a^8*b^2)*d*e^10 + 66*(B*a^10 + 10*A*a^9*b)*e^11)*x)/(e^26*x^14
+ 14*d*e^25*x^13 + 91*d^2*e^24*x^12 + 364*d^3*e^23*x^11 + 1001*d^4*e^22*x^10 + 2
002*d^5*e^21*x^9 + 3003*d^6*e^20*x^8 + 3432*d^7*e^19*x^7 + 3003*d^8*e^18*x^6 + 2
002*d^9*e^17*x^5 + 1001*d^10*e^16*x^4 + 364*d^11*e^15*x^3 + 91*d^12*e^14*x^2 + 1
4*d^13*e^13*x + d^14*e^12)

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Fricas [A]  time = 0.216062, size = 2649, normalized size = 14.32 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^10/(e*x + d)^15,x, algorithm="fricas")

[Out]

-1/12012*(4004*B*b^10*e^11*x^11 + 11*B*b^10*d^11 + 858*A*a^10*e^11 + 3*(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 + 10*(8*B*a^3*b^7 + 3*
A*a^2*b^8)*d^8*e^3 + 15*(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 + 28*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^5*e^6 + 36*(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 + 55*(2*B*a^9*b
+ 9*A*a^8*b^2)*d^2*e^9 + 66*(B*a^10 + 10*A*a^9*b)*d*e^10 + 1001*(11*B*b^10*d*e^1
0 + 3*(10*B*a*b^9 + A*b^10)*e^11)*x^10 + 2002*(11*B*b^10*d^2*e^9 + 3*(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 + 3003*(11*B*b^10*d^3*
e^8 + 3*(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 + 10*
(8*B*a^3*b^7 + 3*A*a^2*b^8)*e^11)*x^8 + 3432*(11*B*b^10*d^4*e^7 + 3*(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 + 10*(8*B*a^3*b^7 + 3*A*
a^2*b^8)*d*e^10 + 15*(7*B*a^4*b^6 + 4*A*a^3*b^7)*e^11)*x^7 + 3003*(11*B*b^10*d^5
*e^6 + 3*(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 + 1
0*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^2*e^9 + 15*(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 + 2002*(11*B*b^10*d^6*e^5 + 3*(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 + 10*(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 + 21*(6*B*a^5*b^5 +
5*A*a^4*b^6)*d*e^10 + 28*(5*B*a^6*b^4 + 6*A*a^5*b^5)*e^11)*x^5 + 1001*(11*B*b^1
0*d^7*e^4 + 3*(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 + 10*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^4*e^7 + 15*(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 + 28*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d*
e^10 + 36*(4*B*a^7*b^3 + 7*A*a^6*b^4)*e^11)*x^4 + 364*(11*B*b^10*d^8*e^3 + 3*(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 + 10*(8*B*a^3*b
^7 + 3*A*a^2*b^8)*d^5*e^6 + 15*(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 + 28*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^2*e^9 + 36*(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 + 91*(11
*B*b^10*d^9*e^2 + 3*(10*B*a*b^9 + A*b^10)*d^8*e^3 + 6*(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 + 15*(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 + 28*(5*B*a^6*b^4 + 6*A*a^5*b
^5)*d^3*e^8 + 36*(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 + 55*(2*B*a^9*b + 9*A*a^8*b^2)*e^11)*x^2 + 14*(11*B*b^10*d^10*e + 3
*(10*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 + 10*(8*B*a
^3*b^7 + 3*A*a^2*b^8)*d^7*e^4 + 15*(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 + 28*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^4*e^7 + 36*(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 + 55*
(2*B*a^9*b + 9*A*a^8*b^2)*d*e^10 + 66*(B*a^10 + 10*A*a^9*b)*e^11)*x)/(e^26*x^14
+ 14*d*e^25*x^13 + 91*d^2*e^24*x^12 + 364*d^3*e^23*x^11 + 1001*d^4*e^22*x^10 + 2
002*d^5*e^21*x^9 + 3003*d^6*e^20*x^8 + 3432*d^7*e^19*x^7 + 3003*d^8*e^18*x^6 + 2
002*d^9*e^17*x^5 + 1001*d^10*e^16*x^4 + 364*d^11*e^15*x^3 + 91*d^12*e^14*x^2 + 1
4*d^13*e^13*x + d^14*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)**15,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.221746, 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)^15,x, algorithm="giac")

[Out]

Done