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3.1083 \(\int \frac{(a+b x)^{10} (A+B x)}{(d+e x)^{12}} \, dx\)

Optimal. Leaf size=321 \[ -\frac{(a+b x)^{11} (B d-A e)}{11 e (d+e x)^{11} (b d-a e)}+\frac{10 b^9 B (b d-a e)}{e^{12} (d+e x)}-\frac{45 b^8 B (b d-a e)^2}{2 e^{12} (d+e x)^2}+\frac{40 b^7 B (b d-a e)^3}{e^{12} (d+e x)^3}-\frac{105 b^6 B (b d-a e)^4}{2 e^{12} (d+e x)^4}+\frac{252 b^5 B (b d-a e)^5}{5 e^{12} (d+e x)^5}-\frac{35 b^4 B (b d-a e)^6}{e^{12} (d+e x)^6}+\frac{120 b^3 B (b d-a e)^7}{7 e^{12} (d+e x)^7}-\frac{45 b^2 B (b d-a e)^8}{8 e^{12} (d+e x)^8}+\frac{10 b B (b d-a e)^9}{9 e^{12} (d+e x)^9}-\frac{B (b d-a e)^{10}}{10 e^{12} (d+e x)^{10}}+\frac{b^{10} B \log (d+e x)}{e^{12}} \]

[Out]

-((B*d - A*e)*(a + b*x)^11)/(11*e*(b*d - a*e)*(d + e*x)^11) - (B*(b*d - a*e)^10)
/(10*e^12*(d + e*x)^10) + (10*b*B*(b*d - a*e)^9)/(9*e^12*(d + e*x)^9) - (45*b^2*
B*(b*d - a*e)^8)/(8*e^12*(d + e*x)^8) + (120*b^3*B*(b*d - a*e)^7)/(7*e^12*(d + e
*x)^7) - (35*b^4*B*(b*d - a*e)^6)/(e^12*(d + e*x)^6) + (252*b^5*B*(b*d - a*e)^5)
/(5*e^12*(d + e*x)^5) - (105*b^6*B*(b*d - a*e)^4)/(2*e^12*(d + e*x)^4) + (40*b^7
*B*(b*d - a*e)^3)/(e^12*(d + e*x)^3) - (45*b^8*B*(b*d - a*e)^2)/(2*e^12*(d + e*x
)^2) + (10*b^9*B*(b*d - a*e))/(e^12*(d + e*x)) + (b^10*B*Log[d + e*x])/e^12

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Rubi [A]  time = 1.10631, antiderivative size = 321, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ -\frac{(a+b x)^{11} (B d-A e)}{11 e (d+e x)^{11} (b d-a e)}+\frac{10 b^9 B (b d-a e)}{e^{12} (d+e x)}-\frac{45 b^8 B (b d-a e)^2}{2 e^{12} (d+e x)^2}+\frac{40 b^7 B (b d-a e)^3}{e^{12} (d+e x)^3}-\frac{105 b^6 B (b d-a e)^4}{2 e^{12} (d+e x)^4}+\frac{252 b^5 B (b d-a e)^5}{5 e^{12} (d+e x)^5}-\frac{35 b^4 B (b d-a e)^6}{e^{12} (d+e x)^6}+\frac{120 b^3 B (b d-a e)^7}{7 e^{12} (d+e x)^7}-\frac{45 b^2 B (b d-a e)^8}{8 e^{12} (d+e x)^8}+\frac{10 b B (b d-a e)^9}{9 e^{12} (d+e x)^9}-\frac{B (b d-a e)^{10}}{10 e^{12} (d+e x)^{10}}+\frac{b^{10} B \log (d+e x)}{e^{12}} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)^10*(A + B*x))/(d + e*x)^12,x]

[Out]

-((B*d - A*e)*(a + b*x)^11)/(11*e*(b*d - a*e)*(d + e*x)^11) - (B*(b*d - a*e)^10)
/(10*e^12*(d + e*x)^10) + (10*b*B*(b*d - a*e)^9)/(9*e^12*(d + e*x)^9) - (45*b^2*
B*(b*d - a*e)^8)/(8*e^12*(d + e*x)^8) + (120*b^3*B*(b*d - a*e)^7)/(7*e^12*(d + e
*x)^7) - (35*b^4*B*(b*d - a*e)^6)/(e^12*(d + e*x)^6) + (252*b^5*B*(b*d - a*e)^5)
/(5*e^12*(d + e*x)^5) - (105*b^6*B*(b*d - a*e)^4)/(2*e^12*(d + e*x)^4) + (40*b^7
*B*(b*d - a*e)^3)/(e^12*(d + e*x)^3) - (45*b^8*B*(b*d - a*e)^2)/(2*e^12*(d + e*x
)^2) + (10*b^9*B*(b*d - a*e))/(e^12*(d + e*x)) + (b^10*B*Log[d + e*x])/e^12

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Rubi in Sympy [A]  time = 146.666, size = 301, normalized size = 0.94 \[ \frac{B b^{10} \log{\left (d + e x \right )}}{e^{12}} - \frac{10 B b^{9} \left (a e - b d\right )}{e^{12} \left (d + e x\right )} - \frac{45 B b^{8} \left (a e - b d\right )^{2}}{2 e^{12} \left (d + e x\right )^{2}} - \frac{40 B b^{7} \left (a e - b d\right )^{3}}{e^{12} \left (d + e x\right )^{3}} - \frac{105 B b^{6} \left (a e - b d\right )^{4}}{2 e^{12} \left (d + e x\right )^{4}} - \frac{252 B b^{5} \left (a e - b d\right )^{5}}{5 e^{12} \left (d + e x\right )^{5}} - \frac{35 B b^{4} \left (a e - b d\right )^{6}}{e^{12} \left (d + e x\right )^{6}} - \frac{120 B b^{3} \left (a e - b d\right )^{7}}{7 e^{12} \left (d + e x\right )^{7}} - \frac{45 B b^{2} \left (a e - b d\right )^{8}}{8 e^{12} \left (d + e x\right )^{8}} - \frac{10 B b \left (a e - b d\right )^{9}}{9 e^{12} \left (d + e x\right )^{9}} - \frac{B \left (a e - b d\right )^{10}}{10 e^{12} \left (d + e x\right )^{10}} - \frac{\left (a + b x\right )^{11} \left (A e - B d\right )}{11 e \left (d + e x\right )^{11} \left (a e - b d\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**10*(B*x+A)/(e*x+d)**12,x)

[Out]

B*b**10*log(d + e*x)/e**12 - 10*B*b**9*(a*e - b*d)/(e**12*(d + e*x)) - 45*B*b**8
*(a*e - b*d)**2/(2*e**12*(d + e*x)**2) - 40*B*b**7*(a*e - b*d)**3/(e**12*(d + e*
x)**3) - 105*B*b**6*(a*e - b*d)**4/(2*e**12*(d + e*x)**4) - 252*B*b**5*(a*e - b*
d)**5/(5*e**12*(d + e*x)**5) - 35*B*b**4*(a*e - b*d)**6/(e**12*(d + e*x)**6) - 1
20*B*b**3*(a*e - b*d)**7/(7*e**12*(d + e*x)**7) - 45*B*b**2*(a*e - b*d)**8/(8*e*
*12*(d + e*x)**8) - 10*B*b*(a*e - b*d)**9/(9*e**12*(d + e*x)**9) - B*(a*e - b*d)
**10/(10*e**12*(d + e*x)**10) - (a + b*x)**11*(A*e - B*d)/(11*e*(d + e*x)**11*(a
*e - b*d))

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Mathematica [B]  time = 6.4491, size = 1992, normalized size = 6.21 \[ \text{result too large to display} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)^10*(A + B*x))/(d + e*x)^12,x]

[Out]

(b^10*B*d^11 - A*b^10*d^10*e - 10*a*b^9*B*d^10*e + 10*a*A*b^9*d^9*e^2 + 45*a^2*b
^8*B*d^9*e^2 - 45*a^2*A*b^8*d^8*e^3 - 120*a^3*b^7*B*d^8*e^3 + 120*a^3*A*b^7*d^7*
e^4 + 210*a^4*b^6*B*d^7*e^4 - 210*a^4*A*b^6*d^6*e^5 - 252*a^5*b^5*B*d^6*e^5 + 25
2*a^5*A*b^5*d^5*e^6 + 210*a^6*b^4*B*d^5*e^6 - 210*a^6*A*b^4*d^4*e^7 - 120*a^7*b^
3*B*d^4*e^7 + 120*a^7*A*b^3*d^3*e^8 + 45*a^8*b^2*B*d^3*e^8 - 45*a^8*A*b^2*d^2*e^
9 - 10*a^9*b*B*d^2*e^9 + 10*a^9*A*b*d*e^10 + a^10*B*d*e^10 - a^10*A*e^11)/(11*e^
12*(d + e*x)^11) + (-11*b^10*B*d^10 + 10*A*b^10*d^9*e + 100*a*b^9*B*d^9*e - 90*a
*A*b^9*d^8*e^2 - 405*a^2*b^8*B*d^8*e^2 + 360*a^2*A*b^8*d^7*e^3 + 960*a^3*b^7*B*d
^7*e^3 - 840*a^3*A*b^7*d^6*e^4 - 1470*a^4*b^6*B*d^6*e^4 + 1260*a^4*A*b^6*d^5*e^5
 + 1512*a^5*b^5*B*d^5*e^5 - 1260*a^5*A*b^5*d^4*e^6 - 1050*a^6*b^4*B*d^4*e^6 + 84
0*a^6*A*b^4*d^3*e^7 + 480*a^7*b^3*B*d^3*e^7 - 360*a^7*A*b^3*d^2*e^8 - 135*a^8*b^
2*B*d^2*e^8 + 90*a^8*A*b^2*d*e^9 + 20*a^9*b*B*d*e^9 - 10*a^9*A*b*e^10 - a^10*B*e
^10)/(10*e^12*(d + e*x)^10) - (5*(-11*b^10*B*d^9 + 9*A*b^10*d^8*e + 90*a*b^9*B*d
^8*e - 72*a*A*b^9*d^7*e^2 - 324*a^2*b^8*B*d^7*e^2 + 252*a^2*A*b^8*d^6*e^3 + 672*
a^3*b^7*B*d^6*e^3 - 504*a^3*A*b^7*d^5*e^4 - 882*a^4*b^6*B*d^5*e^4 + 630*a^4*A*b^
6*d^4*e^5 + 756*a^5*b^5*B*d^4*e^5 - 504*a^5*A*b^5*d^3*e^6 - 420*a^6*b^4*B*d^3*e^
6 + 252*a^6*A*b^4*d^2*e^7 + 144*a^7*b^3*B*d^2*e^7 - 72*a^7*A*b^3*d*e^8 - 27*a^8*
b^2*B*d*e^8 + 9*a^8*A*b^2*e^9 + 2*a^9*b*B*e^9))/(9*e^12*(d + e*x)^9) - (15*(11*b
^10*B*d^8 - 8*A*b^10*d^7*e - 80*a*b^9*B*d^7*e + 56*a*A*b^9*d^6*e^2 + 252*a^2*b^8
*B*d^6*e^2 - 168*a^2*A*b^8*d^5*e^3 - 448*a^3*b^7*B*d^5*e^3 + 280*a^3*A*b^7*d^4*e
^4 + 490*a^4*b^6*B*d^4*e^4 - 280*a^4*A*b^6*d^3*e^5 - 336*a^5*b^5*B*d^3*e^5 + 168
*a^5*A*b^5*d^2*e^6 + 140*a^6*b^4*B*d^2*e^6 - 56*a^6*A*b^4*d*e^7 - 32*a^7*b^3*B*d
*e^7 + 8*a^7*A*b^3*e^8 + 3*a^8*b^2*B*e^8))/(8*e^12*(d + e*x)^8) - (30*(-11*b^10*
B*d^7 + 7*A*b^10*d^6*e + 70*a*b^9*B*d^6*e - 42*a*A*b^9*d^5*e^2 - 189*a^2*b^8*B*d
^5*e^2 + 105*a^2*A*b^8*d^4*e^3 + 280*a^3*b^7*B*d^4*e^3 - 140*a^3*A*b^7*d^3*e^4 -
 245*a^4*b^6*B*d^3*e^4 + 105*a^4*A*b^6*d^2*e^5 + 126*a^5*b^5*B*d^2*e^5 - 42*a^5*
A*b^5*d*e^6 - 35*a^6*b^4*B*d*e^6 + 7*a^6*A*b^4*e^7 + 4*a^7*b^3*B*e^7))/(7*e^12*(
d + e*x)^7) - (7*(11*b^10*B*d^6 - 6*A*b^10*d^5*e - 60*a*b^9*B*d^5*e + 30*a*A*b^9
*d^4*e^2 + 135*a^2*b^8*B*d^4*e^2 - 60*a^2*A*b^8*d^3*e^3 - 160*a^3*b^7*B*d^3*e^3
+ 60*a^3*A*b^7*d^2*e^4 + 105*a^4*b^6*B*d^2*e^4 - 30*a^4*A*b^6*d*e^5 - 36*a^5*b^5
*B*d*e^5 + 6*a^5*A*b^5*e^6 + 5*a^6*b^4*B*e^6))/(e^12*(d + e*x)^6) - (42*(-11*b^1
0*B*d^5 + 5*A*b^10*d^4*e + 50*a*b^9*B*d^4*e - 20*a*A*b^9*d^3*e^2 - 90*a^2*b^8*B*
d^3*e^2 + 30*a^2*A*b^8*d^2*e^3 + 80*a^3*b^7*B*d^2*e^3 - 20*a^3*A*b^7*d*e^4 - 35*
a^4*b^6*B*d*e^4 + 5*a^4*A*b^6*e^5 + 6*a^5*b^5*B*e^5))/(5*e^12*(d + e*x)^5) - (15
*(11*b^10*B*d^4 - 4*A*b^10*d^3*e - 40*a*b^9*B*d^3*e + 12*a*A*b^9*d^2*e^2 + 54*a^
2*b^8*B*d^2*e^2 - 12*a^2*A*b^8*d*e^3 - 32*a^3*b^7*B*d*e^3 + 4*a^3*A*b^7*e^4 + 7*
a^4*b^6*B*e^4))/(2*e^12*(d + e*x)^4) - (5*(-11*b^10*B*d^3 + 3*A*b^10*d^2*e + 30*
a*b^9*B*d^2*e - 6*a*A*b^9*d*e^2 - 27*a^2*b^8*B*d*e^2 + 3*a^2*A*b^8*e^3 + 8*a^3*b
^7*B*e^3))/(e^12*(d + e*x)^3) - (5*(11*b^10*B*d^2 - 2*A*b^10*d*e - 20*a*b^9*B*d*
e + 2*a*A*b^9*e^2 + 9*a^2*b^8*B*e^2))/(2*e^12*(d + e*x)^2) + (11*b^10*B*d - A*b^
10*e - 10*a*b^9*B*e)/(e^12*(d + e*x)) + (b^10*B*Log[d + e*x])/e^12

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Maple [B]  time = 0.024, size = 2907, normalized size = 9.1 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^10*(B*x+A)/(e*x+d)^12,x)

[Out]

-140*b^4/e^5/(e*x+d)^9*A*a^6*d^2+280*b^5/e^6/(e*x+d)^9*A*a^5*d^3+30*b^9/e^10/(e*
x+d)^3*A*a*d+135*b^8/e^10/(e*x+d)^3*B*a^2*d-150*b^9/e^11/(e*x+d)^3*B*a*d^2+5*b^1
0/e^11/(e*x+d)^2*A*d-45/2*b^8/e^10/(e*x+d)^2*B*a^2-55/2*b^10/e^12/(e*x+d)^2*B*d^
2-30*b^7/e^8/(e*x+d)^4*A*a^3+30*b^10/e^11/(e*x+d)^4*A*d^3-105/2*b^6/e^8/(e*x+d)^
4*B*a^4-42*b^5/e^6/(e*x+d)^6*A*a^5+42*b^10/e^11/(e*x+d)^6*A*d^5-35*b^4/e^6/(e*x+
d)^6*B*a^6-77*b^10/e^12/(e*x+d)^6*B*d^6-15*b^3/e^4/(e*x+d)^8*A*a^7+15*b^10/e^11/
(e*x+d)^8*A*d^7-45/8*b^2/e^4/(e*x+d)^8*B*a^8-165/8*b^10/e^12/(e*x+d)^8*B*d^8-165
/2*b^10/e^12/(e*x+d)^4*B*d^4-42*b^6/e^7/(e*x+d)^5*A*a^4-42*b^10/e^11/(e*x+d)^5*A
*d^4-252/5*b^5/e^7/(e*x+d)^5*B*a^5+462/5*b^10/e^12/(e*x+d)^5*B*d^5-120/7*b^3/e^5
/(e*x+d)^7*B*a^7+330/7*b^10/e^12/(e*x+d)^7*B*d^7-15*b^8/e^9/(e*x+d)^3*A*a^2-15*b
^10/e^11/(e*x+d)^3*A*d^2-40*b^7/e^9/(e*x+d)^3*B*a^3+55*b^10/e^12/(e*x+d)^3*B*d^3
-1/11/e^11/(e*x+d)^11*A*d^10*b^10+1/11/e^2/(e*x+d)^11*a^10*B*d+b^10*B*ln(e*x+d)/
e^12-5*b^2/e^3/(e*x+d)^9*A*a^8-5*b^10/e^11/(e*x+d)^9*A*d^8-1/10/e^2/(e*x+d)^10*a
^10*B-1/11/e/(e*x+d)^11*a^10*A-b^10/e^11/(e*x+d)*A-420*b^7/e^8/(e*x+d)^6*A*a^3*d
^2+420*b^8/e^9/(e*x+d)^6*A*a^2*d^3-210*b^9/e^10/(e*x+d)^6*A*a*d^4+252*b^5/e^7/(e
*x+d)^6*B*a^5*d-735*b^6/e^8/(e*x+d)^6*B*a^4*d^2+1120*b^7/e^9/(e*x+d)^6*B*a^3*d^3
-945*b^8/e^10/(e*x+d)^6*B*a^2*d^4+420*b^9/e^11/(e*x+d)^6*B*a*d^5+105*b^4/e^5/(e*
x+d)^8*A*a^6*d-315*b^5/e^6/(e*x+d)^8*A*a^5*d^2+525*b^6/e^7/(e*x+d)^8*A*a^4*d^3-5
25*b^7/e^8/(e*x+d)^8*A*a^3*d^4+315*b^8/e^9/(e*x+d)^8*A*a^2*d^5-105*b^9/e^10/(e*x
+d)^8*A*a*d^6+60*b^3/e^5/(e*x+d)^8*B*a^7*d-525/2*b^4/e^6/(e*x+d)^8*B*a^6*d^2+630
*b^5/e^7/(e*x+d)^8*B*a^5*d^3-3675/4*b^6/e^8/(e*x+d)^8*B*a^4*d^4+840*b^7/e^9/(e*x
+d)^8*B*a^3*d^5-945/2*b^8/e^10/(e*x+d)^8*B*a^2*d^6+150*b^9/e^11/(e*x+d)^8*B*a*d^
7+180*b^5/e^6/(e*x+d)^7*A*a^5*d-450*b^6/e^7/(e*x+d)^7*A*a^4*d^2+600*b^7/e^8/(e*x
+d)^7*A*a^3*d^3-450*b^8/e^9/(e*x+d)^7*A*a^2*d^4+180*b^9/e^10/(e*x+d)^7*A*a*d^5+1
50*b^4/e^6/(e*x+d)^7*B*a^6*d-540*b^5/e^7/(e*x+d)^7*B*a^5*d^2+1050*b^6/e^8/(e*x+d
)^7*B*a^4*d^3-1200*b^7/e^9/(e*x+d)^7*B*a^3*d^4+810*b^8/e^10/(e*x+d)^7*B*a^2*d^5-
300*b^9/e^11/(e*x+d)^7*B*a*d^6-350*b^6/e^7/(e*x+d)^9*A*a^4*d^4+280*b^7/e^8/(e*x+
d)^9*A*a^3*d^5-140*b^8/e^9/(e*x+d)^9*A*a^2*d^6+40*b^9/e^10/(e*x+d)^9*A*a*d^7+15*
b^2/e^4/(e*x+d)^9*B*a^8*d-80*b^3/e^5/(e*x+d)^9*B*a^7*d^2+700/3*b^4/e^6/(e*x+d)^9
*B*a^6*d^3-420*b^5/e^7/(e*x+d)^9*B*a^5*d^4+490*b^6/e^8/(e*x+d)^9*B*a^4*d^5-1120/
3*b^7/e^9/(e*x+d)^9*B*a^3*d^6+180*b^8/e^10/(e*x+d)^9*B*a^2*d^7-50*b^9/e^11/(e*x+
d)^9*B*a*d^8+9/e^3/(e*x+d)^10*A*d*a^8*b^2-36/e^4/(e*x+d)^10*A*d^2*a^7*b^3+84/e^5
/(e*x+d)^10*A*d^3*a^6*b^4-126/e^6/(e*x+d)^10*A*d^4*a^5*b^5+126/e^7/(e*x+d)^10*A*
d^5*a^4*b^6-84/e^8/(e*x+d)^10*A*d^6*a^3*b^7+36/e^9/(e*x+d)^10*A*d^7*a^2*b^8-9/e^
10/(e*x+d)^10*A*d^8*a*b^9+2/e^3/(e*x+d)^10*B*d*a^9*b-27/2/e^4/(e*x+d)^10*B*d^2*a
^8*b^2+48/e^5/(e*x+d)^10*B*d^3*a^7*b^3+10/11/e^2/(e*x+d)^11*a^9*b*A*d-45/11/e^3/
(e*x+d)^11*A*d^2*a^8*b^2+120/11/e^4/(e*x+d)^11*A*d^3*a^7*b^3-210/11/e^5/(e*x+d)^
11*A*d^4*a^6*b^4+252/11/e^6/(e*x+d)^11*A*d^5*a^5*b^5-210/11/e^7/(e*x+d)^11*A*d^6
*a^4*b^6+120/11/e^8/(e*x+d)^11*A*d^7*a^3*b^7-45/11/e^9/(e*x+d)^11*A*d^8*a^2*b^8+
10/11/e^10/(e*x+d)^11*A*d^9*a*b^9-10/11/e^3/(e*x+d)^11*B*d^2*a^9*b+45/11/e^4/(e*
x+d)^11*B*d^3*a^8*b^2-120/11/e^5/(e*x+d)^11*B*d^4*a^7*b^3+210/11/e^6/(e*x+d)^11*
B*d^5*a^6*b^4-252/11/e^7/(e*x+d)^11*B*d^6*a^5*b^5+210/11/e^8/(e*x+d)^11*B*d^7*a^
4*b^6-120/11/e^9/(e*x+d)^11*B*d^8*a^3*b^7+45/11/e^10/(e*x+d)^11*B*d^9*a^2*b^8-10
/11/e^11/(e*x+d)^11*B*d^10*a*b^9+50*b^9/e^11/(e*x+d)^2*B*a*d+90*b^8/e^9/(e*x+d)^
4*A*a^2*d-90*b^9/e^10/(e*x+d)^4*A*a*d^2+240*b^7/e^9/(e*x+d)^4*B*a^3*d-405*b^8/e^
10/(e*x+d)^4*B*a^2*d^2+300*b^9/e^11/(e*x+d)^4*B*a*d^3+168*b^7/e^8/(e*x+d)^5*A*a^
3*d-252*b^8/e^9/(e*x+d)^5*A*a^2*d^2+168*b^9/e^10/(e*x+d)^5*A*a*d^3+294*b^6/e^8/(
e*x+d)^5*B*a^4*d-672*b^7/e^9/(e*x+d)^5*B*a^3*d^2+756*b^8/e^10/(e*x+d)^5*B*a^2*d^
3-420*b^9/e^11/(e*x+d)^5*B*a*d^4+40*b^3/e^4/(e*x+d)^9*A*a^7*d-105/e^6/(e*x+d)^10
*B*d^4*a^6*b^4+756/5/e^7/(e*x+d)^10*B*d^5*a^5*b^5-147/e^8/(e*x+d)^10*B*d^6*a^4*b
^6+96/e^9/(e*x+d)^10*B*d^7*a^3*b^7-81/2/e^10/(e*x+d)^10*B*d^8*a^2*b^8+10/e^11/(e
*x+d)^10*B*d^9*a*b^9+210*b^6/e^7/(e*x+d)^6*A*a^4*d-10/9*b/e^3/(e*x+d)^9*B*a^9+55
/9*b^10/e^12/(e*x+d)^9*B*d^9-1/e^2/(e*x+d)^10*a^9*b*A+1/e^11/(e*x+d)^10*A*d^9*b^
10-11/10/e^12/(e*x+d)^10*b^10*B*d^10-30*b^4/e^5/(e*x+d)^7*A*a^6-30*b^10/e^11/(e*
x+d)^7*A*d^6+1/11/e^12/(e*x+d)^11*b^10*B*d^11-10*b^9/e^11/(e*x+d)*B*a+11*b^10/e^
12/(e*x+d)*B*d-5*b^9/e^10/(e*x+d)^2*A*a

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Maxima [A]  time = 1.53003, size = 2608, normalized size = 8.12 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^10/(e*x + d)^12,x, algorithm="maxima")

[Out]

1/27720*(83711*B*b^10*d^11 - 2520*A*a^10*e^11 - 2520*(10*B*a*b^9 + A*b^10)*d^10*
e - 1260*(9*B*a^2*b^8 + 2*A*a*b^9)*d^9*e^2 - 840*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^8
*e^3 - 630*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^7*e^4 - 504*(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 - 360*(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 - 280*(2*B*a^9*b + 9*A*a^
8*b^2)*d^2*e^9 - 252*(B*a^10 + 10*A*a^9*b)*d*e^10 + 27720*(11*B*b^10*d*e^10 - (1
0*B*a*b^9 + A*b^10)*e^11)*x^10 + 69300*(33*B*b^10*d^2*e^9 - 2*(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 + 69300*(121*B*b^10*d^3*e^8 - 6
*(10*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 - 2*(8*B*a^3
*b^7 + 3*A*a^2*b^8)*e^11)*x^8 + 69300*(275*B*b^10*d^4*e^7 - 12*(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 - 4*(8*B*a^3*b^7 + 3*A*a^2*b^
8)*d*e^10 - 3*(7*B*a^4*b^6 + 4*A*a^3*b^7)*e^11)*x^7 + 19404*(1507*B*b^10*d^5*e^6
 - 60*(10*B*a*b^9 + A*b^10)*d^4*e^7 - 30*(9*B*a^2*b^8 + 2*A*a*b^9)*d^3*e^8 - 20*
(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 - 12
*(6*B*a^5*b^5 + 5*A*a^4*b^6)*e^11)*x^6 + 19404*(1617*B*b^10*d^6*e^5 - 60*(10*B*a
*b^9 + A*b^10)*d^5*e^6 - 30*(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 - 12*(6*B*a^5*b^
5 + 5*A*a^4*b^6)*d*e^10 - 10*(5*B*a^6*b^4 + 6*A*a^5*b^5)*e^11)*x^5 + 1980*(11979
*B*b^10*d^7*e^4 - 420*(10*B*a*b^9 + A*b^10)*d^6*e^5 - 210*(9*B*a^2*b^8 + 2*A*a*b
^9)*d^5*e^6 - 140*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^4*e^7 - 105*(7*B*a^4*b^6 + 4*A*a
^3*b^7)*d^3*e^8 - 84*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^2*e^9 - 70*(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 + 495*(25113*B*b^10*
d^8*e^3 - 840*(10*B*a*b^9 + A*b^10)*d^7*e^4 - 420*(9*B*a^2*b^8 + 2*A*a*b^9)*d^6*
e^5 - 280*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^5*e^6 - 210*(7*B*a^4*b^6 + 4*A*a^3*b^7)*
d^4*e^7 - 168*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^3*e^8 - 140*(5*B*a^6*b^4 + 6*A*a^5*b
^5)*d^2*e^9 - 120*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d*e^10 - 105*(3*B*a^8*b^2 + 8*A*a^
7*b^3)*e^11)*x^3 + 55*(78419*B*b^10*d^9*e^2 - 2520*(10*B*a*b^9 + A*b^10)*d^8*e^3
 - 1260*(9*B*a^2*b^8 + 2*A*a*b^9)*d^7*e^4 - 840*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^6*
e^5 - 630*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^5*e^6 - 504*(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 - 360*(4*B*a^7*b^3 + 7*A*a^6*b
^4)*d^2*e^9 - 315*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d*e^10 - 280*(2*B*a^9*b + 9*A*a^8*
b^2)*e^11)*x^2 + 11*(81191*B*b^10*d^10*e - 2520*(10*B*a*b^9 + A*b^10)*d^9*e^2 -
1260*(9*B*a^2*b^8 + 2*A*a*b^9)*d^8*e^3 - 840*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^7*e^4
 - 630*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^6*e^5 - 504*(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 - 360*(4*B*a^7*b^3 + 7*A*a^6*b^4)
*d^3*e^8 - 315*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^2*e^9 - 280*(2*B*a^9*b + 9*A*a^8*b^
2)*d*e^10 - 252*(B*a^10 + 10*A*a^9*b)*e^11)*x)/(e^23*x^11 + 11*d*e^22*x^10 + 55*
d^2*e^21*x^9 + 165*d^3*e^20*x^8 + 330*d^4*e^19*x^7 + 462*d^5*e^18*x^6 + 462*d^6*
e^17*x^5 + 330*d^7*e^16*x^4 + 165*d^8*e^15*x^3 + 55*d^9*e^14*x^2 + 11*d^10*e^13*
x + d^11*e^12) + B*b^10*log(e*x + d)/e^12

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Fricas [A]  time = 0.221473, size = 2820, normalized size = 8.79 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^10/(e*x + d)^12,x, algorithm="fricas")

[Out]

1/27720*(83711*B*b^10*d^11 - 2520*A*a^10*e^11 - 2520*(10*B*a*b^9 + A*b^10)*d^10*
e - 1260*(9*B*a^2*b^8 + 2*A*a*b^9)*d^9*e^2 - 840*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^8
*e^3 - 630*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^7*e^4 - 504*(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 - 360*(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 - 280*(2*B*a^9*b + 9*A*a^
8*b^2)*d^2*e^9 - 252*(B*a^10 + 10*A*a^9*b)*d*e^10 + 27720*(11*B*b^10*d*e^10 - (1
0*B*a*b^9 + A*b^10)*e^11)*x^10 + 69300*(33*B*b^10*d^2*e^9 - 2*(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 + 69300*(121*B*b^10*d^3*e^8 - 6
*(10*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 - 2*(8*B*a^3
*b^7 + 3*A*a^2*b^8)*e^11)*x^8 + 69300*(275*B*b^10*d^4*e^7 - 12*(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 - 4*(8*B*a^3*b^7 + 3*A*a^2*b^
8)*d*e^10 - 3*(7*B*a^4*b^6 + 4*A*a^3*b^7)*e^11)*x^7 + 19404*(1507*B*b^10*d^5*e^6
 - 60*(10*B*a*b^9 + A*b^10)*d^4*e^7 - 30*(9*B*a^2*b^8 + 2*A*a*b^9)*d^3*e^8 - 20*
(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 - 12
*(6*B*a^5*b^5 + 5*A*a^4*b^6)*e^11)*x^6 + 19404*(1617*B*b^10*d^6*e^5 - 60*(10*B*a
*b^9 + A*b^10)*d^5*e^6 - 30*(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 - 12*(6*B*a^5*b^
5 + 5*A*a^4*b^6)*d*e^10 - 10*(5*B*a^6*b^4 + 6*A*a^5*b^5)*e^11)*x^5 + 1980*(11979
*B*b^10*d^7*e^4 - 420*(10*B*a*b^9 + A*b^10)*d^6*e^5 - 210*(9*B*a^2*b^8 + 2*A*a*b
^9)*d^5*e^6 - 140*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^4*e^7 - 105*(7*B*a^4*b^6 + 4*A*a
^3*b^7)*d^3*e^8 - 84*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^2*e^9 - 70*(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 + 495*(25113*B*b^10*
d^8*e^3 - 840*(10*B*a*b^9 + A*b^10)*d^7*e^4 - 420*(9*B*a^2*b^8 + 2*A*a*b^9)*d^6*
e^5 - 280*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^5*e^6 - 210*(7*B*a^4*b^6 + 4*A*a^3*b^7)*
d^4*e^7 - 168*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^3*e^8 - 140*(5*B*a^6*b^4 + 6*A*a^5*b
^5)*d^2*e^9 - 120*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d*e^10 - 105*(3*B*a^8*b^2 + 8*A*a^
7*b^3)*e^11)*x^3 + 55*(78419*B*b^10*d^9*e^2 - 2520*(10*B*a*b^9 + A*b^10)*d^8*e^3
 - 1260*(9*B*a^2*b^8 + 2*A*a*b^9)*d^7*e^4 - 840*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^6*
e^5 - 630*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^5*e^6 - 504*(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 - 360*(4*B*a^7*b^3 + 7*A*a^6*b
^4)*d^2*e^9 - 315*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d*e^10 - 280*(2*B*a^9*b + 9*A*a^8*
b^2)*e^11)*x^2 + 11*(81191*B*b^10*d^10*e - 2520*(10*B*a*b^9 + A*b^10)*d^9*e^2 -
1260*(9*B*a^2*b^8 + 2*A*a*b^9)*d^8*e^3 - 840*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^7*e^4
 - 630*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^6*e^5 - 504*(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 - 360*(4*B*a^7*b^3 + 7*A*a^6*b^4)
*d^3*e^8 - 315*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^2*e^9 - 280*(2*B*a^9*b + 9*A*a^8*b^
2)*d*e^10 - 252*(B*a^10 + 10*A*a^9*b)*e^11)*x + 27720*(B*b^10*e^11*x^11 + 11*B*b
^10*d*e^10*x^10 + 55*B*b^10*d^2*e^9*x^9 + 165*B*b^10*d^3*e^8*x^8 + 330*B*b^10*d^
4*e^7*x^7 + 462*B*b^10*d^5*e^6*x^6 + 462*B*b^10*d^6*e^5*x^5 + 330*B*b^10*d^7*e^4
*x^4 + 165*B*b^10*d^8*e^3*x^3 + 55*B*b^10*d^9*e^2*x^2 + 11*B*b^10*d^10*e*x + B*b
^10*d^11)*log(e*x + d))/(e^23*x^11 + 11*d*e^22*x^10 + 55*d^2*e^21*x^9 + 165*d^3*
e^20*x^8 + 330*d^4*e^19*x^7 + 462*d^5*e^18*x^6 + 462*d^6*e^17*x^5 + 330*d^7*e^16
*x^4 + 165*d^8*e^15*x^3 + 55*d^9*e^14*x^2 + 11*d^10*e^13*x + d^11*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)**12,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.214489, 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)^12,x, algorithm="giac")

[Out]

Done

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