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3.1072 \(\int \frac{(a+b x)^{10} (A+B x)}{d+e x} \, dx\)

Optimal. Leaf size=348 \[ -\frac{(b d-a e)^{10} (B d-A e) \log (d+e x)}{e^{12}}+\frac{b x (b d-a e)^9 (B d-A e)}{e^{11}}-\frac{(a+b x)^2 (b d-a e)^8 (B d-A e)}{2 e^{10}}+\frac{(a+b x)^3 (b d-a e)^7 (B d-A e)}{3 e^9}-\frac{(a+b x)^4 (b d-a e)^6 (B d-A e)}{4 e^8}+\frac{(a+b x)^5 (b d-a e)^5 (B d-A e)}{5 e^7}-\frac{(a+b x)^6 (b d-a e)^4 (B d-A e)}{6 e^6}+\frac{(a+b x)^7 (b d-a e)^3 (B d-A e)}{7 e^5}-\frac{(a+b x)^8 (b d-a e)^2 (B d-A e)}{8 e^4}+\frac{(a+b x)^9 (b d-a e) (B d-A e)}{9 e^3}-\frac{(a+b x)^{10} (B d-A e)}{10 e^2}+\frac{B (a+b x)^{11}}{11 b e} \]

[Out]

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

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Rubi [A]  time = 0.749002, antiderivative size = 348, 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 \[ -\frac{(b d-a e)^{10} (B d-A e) \log (d+e x)}{e^{12}}+\frac{b x (b d-a e)^9 (B d-A e)}{e^{11}}-\frac{(a+b x)^2 (b d-a e)^8 (B d-A e)}{2 e^{10}}+\frac{(a+b x)^3 (b d-a e)^7 (B d-A e)}{3 e^9}-\frac{(a+b x)^4 (b d-a e)^6 (B d-A e)}{4 e^8}+\frac{(a+b x)^5 (b d-a e)^5 (B d-A e)}{5 e^7}-\frac{(a+b x)^6 (b d-a e)^4 (B d-A e)}{6 e^6}+\frac{(a+b x)^7 (b d-a e)^3 (B d-A e)}{7 e^5}-\frac{(a+b x)^8 (b d-a e)^2 (B d-A e)}{8 e^4}+\frac{(a+b x)^9 (b d-a e) (B d-A e)}{9 e^3}-\frac{(a+b x)^{10} (B d-A e)}{10 e^2}+\frac{B (a+b x)^{11}}{11 b e} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{B \left (a + b x\right )^{11}}{11 b e} + \frac{\left (a + b x\right )^{10} \left (A e - B d\right )}{10 e^{2}} + \frac{\left (a + b x\right )^{9} \left (A e - B d\right ) \left (a e - b d\right )}{9 e^{3}} + \frac{\left (a + b x\right )^{8} \left (A e - B d\right ) \left (a e - b d\right )^{2}}{8 e^{4}} + \frac{\left (a + b x\right )^{7} \left (A e - B d\right ) \left (a e - b d\right )^{3}}{7 e^{5}} + \frac{\left (a + b x\right )^{6} \left (A e - B d\right ) \left (a e - b d\right )^{4}}{6 e^{6}} + \frac{\left (a + b x\right )^{5} \left (A e - B d\right ) \left (a e - b d\right )^{5}}{5 e^{7}} + \frac{\left (a + b x\right )^{4} \left (A e - B d\right ) \left (a e - b d\right )^{6}}{4 e^{8}} + \frac{\left (a + b x\right )^{3} \left (A e - B d\right ) \left (a e - b d\right )^{7}}{3 e^{9}} + \frac{\left (a + b x\right )^{2} \left (A e - B d\right ) \left (a e - b d\right )^{8}}{2 e^{10}} + \frac{\left (A e - B d\right ) \left (a e - b d\right )^{9} \int b\, dx}{e^{11}} + \frac{\left (A e - B d\right ) \left (a e - b d\right )^{10} \log{\left (d + e x \right )}}{e^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [B]  time = 6.14148, size = 1915, normalized size = 5.5 \[ \text{result too large to display} \]

Antiderivative was successfully verified.

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

[Out]

((b^10*B*d^10 - A*b^10*d^9*e - 10*a*b^9*B*d^9*e + 10*a*A*b^9*d^8*e^2 + 45*a^2*b^
8*B*d^8*e^2 - 45*a^2*A*b^8*d^7*e^3 - 120*a^3*b^7*B*d^7*e^3 + 120*a^3*A*b^7*d^6*e
^4 + 210*a^4*b^6*B*d^6*e^4 - 210*a^4*A*b^6*d^5*e^5 - 252*a^5*b^5*B*d^5*e^5 + 252
*a^5*A*b^5*d^4*e^6 + 210*a^6*b^4*B*d^4*e^6 - 210*a^6*A*b^4*d^3*e^7 - 120*a^7*b^3
*B*d^3*e^7 + 120*a^7*A*b^3*d^2*e^8 + 45*a^8*b^2*B*d^2*e^8 - 45*a^8*A*b^2*d*e^9 -
 10*a^9*b*B*d*e^9 + 10*a^9*A*b*e^10 + a^10*B*e^10)*x)/e^11 + ((-(b^10*B*d^9) + A
*b^10*d^8*e + 10*a*b^9*B*d^8*e - 10*a*A*b^9*d^7*e^2 - 45*a^2*b^8*B*d^7*e^2 + 45*
a^2*A*b^8*d^6*e^3 + 120*a^3*b^7*B*d^6*e^3 - 120*a^3*A*b^7*d^5*e^4 - 210*a^4*b^6*
B*d^5*e^4 + 210*a^4*A*b^6*d^4*e^5 + 252*a^5*b^5*B*d^4*e^5 - 252*a^5*A*b^5*d^3*e^
6 - 210*a^6*b^4*B*d^3*e^6 + 210*a^6*A*b^4*d^2*e^7 + 120*a^7*b^3*B*d^2*e^7 - 120*
a^7*A*b^3*d*e^8 - 45*a^8*b^2*B*d*e^8 + 45*a^8*A*b^2*e^9 + 10*a^9*b*B*e^9)*x^2)/(
2*e^10) + ((b^10*B*d^8 - A*b^10*d^7*e - 10*a*b^9*B*d^7*e + 10*a*A*b^9*d^6*e^2 +
45*a^2*b^8*B*d^6*e^2 - 45*a^2*A*b^8*d^5*e^3 - 120*a^3*b^7*B*d^5*e^3 + 120*a^3*A*
b^7*d^4*e^4 + 210*a^4*b^6*B*d^4*e^4 - 210*a^4*A*b^6*d^3*e^5 - 252*a^5*b^5*B*d^3*
e^5 + 252*a^5*A*b^5*d^2*e^6 + 210*a^6*b^4*B*d^2*e^6 - 210*a^6*A*b^4*d*e^7 - 120*
a^7*b^3*B*d*e^7 + 120*a^7*A*b^3*e^8 + 45*a^8*b^2*B*e^8)*x^3)/(3*e^9) + ((-(b^10*
B*d^7) + A*b^10*d^6*e + 10*a*b^9*B*d^6*e - 10*a*A*b^9*d^5*e^2 - 45*a^2*b^8*B*d^5
*e^2 + 45*a^2*A*b^8*d^4*e^3 + 120*a^3*b^7*B*d^4*e^3 - 120*a^3*A*b^7*d^3*e^4 - 21
0*a^4*b^6*B*d^3*e^4 + 210*a^4*A*b^6*d^2*e^5 + 252*a^5*b^5*B*d^2*e^5 - 252*a^5*A*
b^5*d*e^6 - 210*a^6*b^4*B*d*e^6 + 210*a^6*A*b^4*e^7 + 120*a^7*b^3*B*e^7)*x^4)/(4
*e^8) - (b^4*(-(b^6*B*d^6) + A*b^6*d^5*e + 10*a*b^5*B*d^5*e - 10*a*A*b^5*d^4*e^2
 - 45*a^2*b^4*B*d^4*e^2 + 45*a^2*A*b^4*d^3*e^3 + 120*a^3*b^3*B*d^3*e^3 - 120*a^3
*A*b^3*d^2*e^4 - 210*a^4*b^2*B*d^2*e^4 + 210*a^4*A*b^2*d*e^5 + 252*a^5*b*B*d*e^5
 - 252*a^5*A*b*e^6 - 210*a^6*B*e^6)*x^5)/(5*e^7) + (b^5*(-(b^5*B*d^5) + A*b^5*d^
4*e + 10*a*b^4*B*d^4*e - 10*a*A*b^4*d^3*e^2 - 45*a^2*b^3*B*d^3*e^2 + 45*a^2*A*b^
3*d^2*e^3 + 120*a^3*b^2*B*d^2*e^3 - 120*a^3*A*b^2*d*e^4 - 210*a^4*b*B*d*e^4 + 21
0*a^4*A*b*e^5 + 252*a^5*B*e^5)*x^6)/(6*e^6) - (b^6*(-(b^4*B*d^4) + A*b^4*d^3*e +
 10*a*b^3*B*d^3*e - 10*a*A*b^3*d^2*e^2 - 45*a^2*b^2*B*d^2*e^2 + 45*a^2*A*b^2*d*e
^3 + 120*a^3*b*B*d*e^3 - 120*a^3*A*b*e^4 - 210*a^4*B*e^4)*x^7)/(7*e^5) + (b^7*(-
(b^3*B*d^3) + A*b^3*d^2*e + 10*a*b^2*B*d^2*e - 10*a*A*b^2*d*e^2 - 45*a^2*b*B*d*e
^2 + 45*a^2*A*b*e^3 + 120*a^3*B*e^3)*x^8)/(8*e^4) - (b^8*(-(b^2*B*d^2) + A*b^2*d
*e + 10*a*b*B*d*e - 10*a*A*b*e^2 - 45*a^2*B*e^2)*x^9)/(9*e^3) + (b^9*(-(b*B*d) +
 A*b*e + 10*a*B*e)*x^10)/(10*e^2) + (b^10*B*x^11)/(11*e) + ((-(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 - 252*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)*Log[d + e*x])/e^12

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Maple [B]  time = 0.021, size = 2357, normalized size = 6.8 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/e*ln(e*x+d)*a^10*A+1/e*B*a^10*x+1/11/e*B*b^10*x^11+1/10/e*A*x^10*b^10+126/e^5*
B*x^2*a^5*b^5*d^4-105/e^6*B*x^2*a^4*b^6*d^5+60/e^7*B*x^2*a^3*b^7*d^6+84/e^3*A*x^
3*a^5*b^5*d^2-70/e^4*A*x^3*a^4*b^6*d^3+40/e^5*A*x^3*a^3*b^7*d^4+5/e^9*B*x^2*a*b^
9*d^8-45/e^2*A*a^8*b^2*d*x+120/e^3*A*a^7*b^3*d^2*x-210/e^4*A*a^6*b^4*d^3*x-2/e^6
*B*x^5*a*b^9*d^5-63/e^2*A*x^4*a^5*b^5*d+105/2/e^3*A*x^4*a^4*b^6*d^2-30/e^4*A*x^4
*a^3*b^7*d^3-10/9/e^2*B*x^9*a*b^9*d-5/4/e^2*A*x^8*a*b^9*d+9/e^5*B*x^5*a^2*b^8*d^
4-45/2/e^8*B*x^2*a^2*b^8*d^7+15/e^7*B*x^3*a^2*b^8*d^6-10/3/e^8*B*x^3*a*b^9*d^7-6
0/e^2*A*x^2*a^7*b^3*d+30/e^5*B*x^4*a^3*b^7*d^4-45/4/e^6*B*x^4*a^2*b^8*d^5+5/2/e^
7*B*x^4*a*b^9*d^6-70/e^2*A*x^3*a^6*b^4*d+45/4/e^5*A*x^4*a^2*b^8*d^4-5/2/e^6*A*x^
4*a*b^9*d^5-105/2/e^2*B*x^4*a^6*b^4*d+63/e^3*B*x^4*a^5*b^5*d^2-105/2/e^4*B*x^4*a
^4*b^6*d^3-45/7/e^2*A*x^7*a^2*b^8*d+10/7/e^3*A*x^7*a*b^9*d^2-120/7/e^2*B*x^7*a^3
*b^7*d-5/3/e^4*A*x^6*a*b^9*d^3-35/e^2*B*x^6*a^4*b^6*d+20/e^3*B*x^6*a^3*b^7*d^2-1
5/2/e^4*B*x^6*a^2*b^8*d^3+5/3/e^5*B*x^6*a*b^9*d^4+1/5/e^7*B*x^5*b^10*d^6+1/8/e^3
*A*x^8*b^10*d^2-120/e^8*ln(e*x+d)*A*a^3*b^7*d^7+45/e^9*ln(e*x+d)*A*a^2*b^8*d^8-1
0/e^10*ln(e*x+d)*A*a*b^9*d^9+10/e^3*ln(e*x+d)*B*a^9*b*d^2-15/e^6*A*x^3*a^2*b^8*d
^5+10/3/e^7*A*x^3*a*b^9*d^6-40/e^2*B*x^3*a^7*b^3*d-5/e^8*A*x^2*a*b^9*d^7-45/2/e^
2*B*x^2*a^8*b^2*d+60/e^3*B*x^2*a^7*b^3*d^2-105/e^4*B*x^2*a^6*b^4*d^3+70/e^3*B*x^
3*a^6*b^4*d^2-84/e^4*B*x^3*a^5*b^5*d^3+70/e^5*B*x^3*a^4*b^6*d^4-40/e^6*B*x^3*a^3
*b^7*d^5+252/e^5*A*a^5*b^5*d^4*x-210/e^6*A*a^4*b^6*d^5*x+120/e^7*A*a^3*b^7*d^6*x
-45/e^8*A*a^2*b^8*d^7*x+10/e^9*A*a*b^9*d^8*x-10/e^2*B*a^9*b*d*x+45/e^3*B*a^8*b^2
*d^2*x-120/e^4*B*a^7*b^3*d^3*x+210/e^5*B*a^6*b^4*d^4*x-252/e^6*B*a^5*b^5*d^5*x+2
10/e^7*B*a^4*b^6*d^6*x-120/e^8*B*a^3*b^7*d^7*x+45/e^9*B*a^2*b^8*d^8*x-10/e^10*B*
a*b^9*d^9*x+105/e^3*A*x^2*a^6*b^4*d^2-126/e^4*A*x^2*a^5*b^5*d^3+105/e^5*A*x^2*a^
4*b^6*d^4-60/e^6*A*x^2*a^3*b^7*d^5+45/2/e^7*A*x^2*a^2*b^8*d^6+45/7/e^3*B*x^7*a^2
*b^8*d^2-10/7/e^4*B*x^7*a*b^9*d^3-20/e^2*A*x^6*a^3*b^7*d+15/2/e^3*A*x^6*a^2*b^8*
d^2+5/4/e^3*B*x^8*a*b^9*d^2-42/e^2*A*x^5*a^4*b^6*d-45/8/e^2*B*x^8*a^2*b^8*d+24/e
^3*A*x^5*a^3*b^7*d^2-9/e^4*A*x^5*a^2*b^8*d^3+2/e^5*A*x^5*a*b^9*d^4-252/5/e^2*B*x
^5*a^5*b^5*d+42/e^3*B*x^5*a^4*b^6*d^2-24/e^4*B*x^5*a^3*b^7*d^3-45/e^4*ln(e*x+d)*
B*a^8*b^2*d^3+120/e^5*ln(e*x+d)*B*a^7*b^3*d^4-210/e^6*ln(e*x+d)*B*a^6*b^4*d^5+25
2/e^7*ln(e*x+d)*B*a^5*b^5*d^6-210/e^8*ln(e*x+d)*B*a^4*b^6*d^7+120/e^9*ln(e*x+d)*
B*a^3*b^7*d^8-45/e^10*ln(e*x+d)*B*a^2*b^8*d^9+10/e^11*ln(e*x+d)*B*a*b^9*d^10-10/
e^2*ln(e*x+d)*A*a^9*b*d+45/e^3*ln(e*x+d)*A*a^8*b^2*d^2-120/e^4*ln(e*x+d)*A*a^7*b
^3*d^3+210/e^5*ln(e*x+d)*A*a^6*b^4*d^4-252/e^6*ln(e*x+d)*A*a^5*b^5*d^5+210/e^7*l
n(e*x+d)*A*a^4*b^6*d^6+1/9/e^3*B*x^9*b^10*d^2+45/8/e*A*x^8*a^2*b^8-1/9/e^2*A*x^9
*b^10*d+10/9/e*A*x^9*a*b^9-1/10/e^2*B*x^10*b^10*d+1/e^11*b^10*B*d^10*x-1/4/e^8*B
*x^4*b^10*d^7+30/e*B*x^4*a^7*b^3+5/e*B*x^9*a^2*b^8+5/e*B*x^2*a^9*b-1/2/e^10*B*x^
2*b^10*d^9+10/e*A*a^9*b*x-1/e^10*A*b^10*d^9*x+1/7/e^5*B*x^7*b^10*d^4+1/e*B*x^10*
a*b^9+1/2/e^9*A*x^2*b^10*d^8+45/2/e*A*x^2*a^8*b^2+15/e*B*x^3*a^8*b^2+40/e*A*x^3*
a^7*b^3-1/3/e^8*A*x^3*b^10*d^7+1/4/e^7*A*x^4*b^10*d^6-1/6/e^6*B*x^6*b^10*d^5+252
/5/e*A*x^5*a^5*b^5-1/5/e^6*A*x^5*b^10*d^5+42/e*B*x^5*a^6*b^4-1/7/e^4*A*x^7*b^10*
d^3+120/7/e*A*x^7*a^3*b^7-1/8/e^4*B*x^8*b^10*d^3+15/e*B*x^8*a^3*b^7+1/6/e^5*A*x^
6*b^10*d^4+42/e*B*x^6*a^5*b^5+30/e*B*x^7*a^4*b^6+105/2/e*A*x^4*a^6*b^4+1/3/e^9*B
*x^3*b^10*d^8+35/e*A*x^6*a^4*b^6+1/e^11*ln(e*x+d)*A*b^10*d^10-1/e^2*ln(e*x+d)*B*
a^10*d-1/e^12*ln(e*x+d)*b^10*B*d^11

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Maxima [A]  time = 1.39255, size = 2435, normalized size = 7. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/27720*(2520*B*b^10*e^10*x^11 - 2772*(B*b^10*d*e^9 - (10*B*a*b^9 + A*b^10)*e^10
)*x^10 + 3080*(B*b^10*d^2*e^8 - (10*B*a*b^9 + A*b^10)*d*e^9 + 5*(9*B*a^2*b^8 + 2
*A*a*b^9)*e^10)*x^9 - 3465*(B*b^10*d^3*e^7 - (10*B*a*b^9 + A*b^10)*d^2*e^8 + 5*(
9*B*a^2*b^8 + 2*A*a*b^9)*d*e^9 - 15*(8*B*a^3*b^7 + 3*A*a^2*b^8)*e^10)*x^8 + 3960
*(B*b^10*d^4*e^6 - (10*B*a*b^9 + A*b^10)*d^3*e^7 + 5*(9*B*a^2*b^8 + 2*A*a*b^9)*d
^2*e^8 - 15*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d*e^9 + 30*(7*B*a^4*b^6 + 4*A*a^3*b^7)*e
^10)*x^7 - 4620*(B*b^10*d^5*e^5 - (10*B*a*b^9 + A*b^10)*d^4*e^6 + 5*(9*B*a^2*b^8
 + 2*A*a*b^9)*d^3*e^7 - 15*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^2*e^8 + 30*(7*B*a^4*b^6
 + 4*A*a^3*b^7)*d*e^9 - 42*(6*B*a^5*b^5 + 5*A*a^4*b^6)*e^10)*x^6 + 5544*(B*b^10*
d^6*e^4 - (10*B*a*b^9 + A*b^10)*d^5*e^5 + 5*(9*B*a^2*b^8 + 2*A*a*b^9)*d^4*e^6 -
15*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^3*e^7 + 30*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^2*e^8
- 42*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d*e^9 + 42*(5*B*a^6*b^4 + 6*A*a^5*b^5)*e^10)*x^
5 - 6930*(B*b^10*d^7*e^3 - (10*B*a*b^9 + A*b^10)*d^6*e^4 + 5*(9*B*a^2*b^8 + 2*A*
a*b^9)*d^5*e^5 - 15*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^4*e^6 + 30*(7*B*a^4*b^6 + 4*A*
a^3*b^7)*d^3*e^7 - 42*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^2*e^8 + 42*(5*B*a^6*b^4 + 6*
A*a^5*b^5)*d*e^9 - 30*(4*B*a^7*b^3 + 7*A*a^6*b^4)*e^10)*x^4 + 9240*(B*b^10*d^8*e
^2 - (10*B*a*b^9 + A*b^10)*d^7*e^3 + 5*(9*B*a^2*b^8 + 2*A*a*b^9)*d^6*e^4 - 15*(8
*B*a^3*b^7 + 3*A*a^2*b^8)*d^5*e^5 + 30*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^4*e^6 - 42*
(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^3*e^7 + 42*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^2*e^8 - 3
0*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d*e^9 + 15*(3*B*a^8*b^2 + 8*A*a^7*b^3)*e^10)*x^3 -
 13860*(B*b^10*d^9*e - (10*B*a*b^9 + A*b^10)*d^8*e^2 + 5*(9*B*a^2*b^8 + 2*A*a*b^
9)*d^7*e^3 - 15*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^6*e^4 + 30*(7*B*a^4*b^6 + 4*A*a^3*
b^7)*d^5*e^5 - 42*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^4*e^6 + 42*(5*B*a^6*b^4 + 6*A*a^
5*b^5)*d^3*e^7 - 30*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^2*e^8 + 15*(3*B*a^8*b^2 + 8*A*
a^7*b^3)*d*e^9 - 5*(2*B*a^9*b + 9*A*a^8*b^2)*e^10)*x^2 + 27720*(B*b^10*d^10 - (1
0*B*a*b^9 + A*b^10)*d^9*e + 5*(9*B*a^2*b^8 + 2*A*a*b^9)*d^8*e^2 - 15*(8*B*a^3*b^
7 + 3*A*a^2*b^8)*d^7*e^3 + 30*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^6*e^4 - 42*(6*B*a^5*
b^5 + 5*A*a^4*b^6)*d^5*e^5 + 42*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^4*e^6 - 30*(4*B*a^
7*b^3 + 7*A*a^6*b^4)*d^3*e^7 + 15*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^2*e^8 - 5*(2*B*a
^9*b + 9*A*a^8*b^2)*d*e^9 + (B*a^10 + 10*A*a^9*b)*e^10)*x)/e^11 - (B*b^10*d^11 -
 A*a^10*e^11 - (10*B*a*b^9 + A*b^10)*d^10*e + 5*(9*B*a^2*b^8 + 2*A*a*b^9)*d^9*e^
2 - 15*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^8*e^3 + 30*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^7*
e^4 - 42*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^6*e^5 + 42*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^
5*e^6 - 30*(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 + (B*a^10 + 10*A*a^9*b)*d*e^10)*lo
g(e*x + d)/e^12

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Fricas [A]  time = 0.215134, size = 2437, normalized size = 7. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/27720*(2520*B*b^10*e^11*x^11 - 2772*(B*b^10*d*e^10 - (10*B*a*b^9 + A*b^10)*e^1
1)*x^10 + 3080*(B*b^10*d^2*e^9 - (10*B*a*b^9 + A*b^10)*d*e^10 + 5*(9*B*a^2*b^8 +
 2*A*a*b^9)*e^11)*x^9 - 3465*(B*b^10*d^3*e^8 - (10*B*a*b^9 + A*b^10)*d^2*e^9 + 5
*(9*B*a^2*b^8 + 2*A*a*b^9)*d*e^10 - 15*(8*B*a^3*b^7 + 3*A*a^2*b^8)*e^11)*x^8 + 3
960*(B*b^10*d^4*e^7 - (10*B*a*b^9 + A*b^10)*d^3*e^8 + 5*(9*B*a^2*b^8 + 2*A*a*b^9
)*d^2*e^9 - 15*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d*e^10 + 30*(7*B*a^4*b^6 + 4*A*a^3*b^
7)*e^11)*x^7 - 4620*(B*b^10*d^5*e^6 - (10*B*a*b^9 + A*b^10)*d^4*e^7 + 5*(9*B*a^2
*b^8 + 2*A*a*b^9)*d^3*e^8 - 15*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^2*e^9 + 30*(7*B*a^4
*b^6 + 4*A*a^3*b^7)*d*e^10 - 42*(6*B*a^5*b^5 + 5*A*a^4*b^6)*e^11)*x^6 + 5544*(B*
b^10*d^6*e^5 - (10*B*a*b^9 + A*b^10)*d^5*e^6 + 5*(9*B*a^2*b^8 + 2*A*a*b^9)*d^4*e
^7 - 15*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^3*e^8 + 30*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^2
*e^9 - 42*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d*e^10 + 42*(5*B*a^6*b^4 + 6*A*a^5*b^5)*e^
11)*x^5 - 6930*(B*b^10*d^7*e^4 - (10*B*a*b^9 + A*b^10)*d^6*e^5 + 5*(9*B*a^2*b^8
+ 2*A*a*b^9)*d^5*e^6 - 15*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^4*e^7 + 30*(7*B*a^4*b^6
+ 4*A*a^3*b^7)*d^3*e^8 - 42*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^2*e^9 + 42*(5*B*a^6*b^
4 + 6*A*a^5*b^5)*d*e^10 - 30*(4*B*a^7*b^3 + 7*A*a^6*b^4)*e^11)*x^4 + 9240*(B*b^1
0*d^8*e^3 - (10*B*a*b^9 + A*b^10)*d^7*e^4 + 5*(9*B*a^2*b^8 + 2*A*a*b^9)*d^6*e^5
- 15*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^5*e^6 + 30*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^4*e^
7 - 42*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^3*e^8 + 42*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^2*
e^9 - 30*(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^1
1)*x^3 - 13860*(B*b^10*d^9*e^2 - (10*B*a*b^9 + A*b^10)*d^8*e^3 + 5*(9*B*a^2*b^8
+ 2*A*a*b^9)*d^7*e^4 - 15*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^6*e^5 + 30*(7*B*a^4*b^6
+ 4*A*a^3*b^7)*d^5*e^6 - 42*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^4*e^7 + 42*(5*B*a^6*b^
4 + 6*A*a^5*b^5)*d^3*e^8 - 30*(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 + 27720*(B*b^1
0*d^10*e - (10*B*a*b^9 + A*b^10)*d^9*e^2 + 5*(9*B*a^2*b^8 + 2*A*a*b^9)*d^8*e^3 -
 15*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^7*e^4 + 30*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^6*e^5
 - 42*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^5*e^6 + 42*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^4*e
^7 - 30*(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 + (B*a^10 + 10*A*a^9*b)*e^11)*x - 2772
0*(B*b^10*d^11 - A*a^10*e^11 - (10*B*a*b^9 + A*b^10)*d^10*e + 5*(9*B*a^2*b^8 + 2
*A*a*b^9)*d^9*e^2 - 15*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^8*e^3 + 30*(7*B*a^4*b^6 + 4
*A*a^3*b^7)*d^7*e^4 - 42*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^6*e^5 + 42*(5*B*a^6*b^4 +
 6*A*a^5*b^5)*d^5*e^6 - 30*(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 + (B*a^10 + 10*A*a
^9*b)*d*e^10)*log(e*x + d))/e^12

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Sympy [A]  time = 18.0486, size = 1844, normalized size = 5.3 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*b**10*x**11/(11*e) + x**10*(A*b**10*e + 10*B*a*b**9*e - B*b**10*d)/(10*e**2) +
 x**9*(10*A*a*b**9*e**2 - A*b**10*d*e + 45*B*a**2*b**8*e**2 - 10*B*a*b**9*d*e +
B*b**10*d**2)/(9*e**3) + x**8*(45*A*a**2*b**8*e**3 - 10*A*a*b**9*d*e**2 + A*b**1
0*d**2*e + 120*B*a**3*b**7*e**3 - 45*B*a**2*b**8*d*e**2 + 10*B*a*b**9*d**2*e - B
*b**10*d**3)/(8*e**4) + x**7*(120*A*a**3*b**7*e**4 - 45*A*a**2*b**8*d*e**3 + 10*
A*a*b**9*d**2*e**2 - A*b**10*d**3*e + 210*B*a**4*b**6*e**4 - 120*B*a**3*b**7*d*e
**3 + 45*B*a**2*b**8*d**2*e**2 - 10*B*a*b**9*d**3*e + B*b**10*d**4)/(7*e**5) + x
**6*(210*A*a**4*b**6*e**5 - 120*A*a**3*b**7*d*e**4 + 45*A*a**2*b**8*d**2*e**3 -
10*A*a*b**9*d**3*e**2 + A*b**10*d**4*e + 252*B*a**5*b**5*e**5 - 210*B*a**4*b**6*
d*e**4 + 120*B*a**3*b**7*d**2*e**3 - 45*B*a**2*b**8*d**3*e**2 + 10*B*a*b**9*d**4
*e - B*b**10*d**5)/(6*e**6) + x**5*(252*A*a**5*b**5*e**6 - 210*A*a**4*b**6*d*e**
5 + 120*A*a**3*b**7*d**2*e**4 - 45*A*a**2*b**8*d**3*e**3 + 10*A*a*b**9*d**4*e**2
 - A*b**10*d**5*e + 210*B*a**6*b**4*e**6 - 252*B*a**5*b**5*d*e**5 + 210*B*a**4*b
**6*d**2*e**4 - 120*B*a**3*b**7*d**3*e**3 + 45*B*a**2*b**8*d**4*e**2 - 10*B*a*b*
*9*d**5*e + B*b**10*d**6)/(5*e**7) + x**4*(210*A*a**6*b**4*e**7 - 252*A*a**5*b**
5*d*e**6 + 210*A*a**4*b**6*d**2*e**5 - 120*A*a**3*b**7*d**3*e**4 + 45*A*a**2*b**
8*d**4*e**3 - 10*A*a*b**9*d**5*e**2 + A*b**10*d**6*e + 120*B*a**7*b**3*e**7 - 21
0*B*a**6*b**4*d*e**6 + 252*B*a**5*b**5*d**2*e**5 - 210*B*a**4*b**6*d**3*e**4 + 1
20*B*a**3*b**7*d**4*e**3 - 45*B*a**2*b**8*d**5*e**2 + 10*B*a*b**9*d**6*e - B*b**
10*d**7)/(4*e**8) + x**3*(120*A*a**7*b**3*e**8 - 210*A*a**6*b**4*d*e**7 + 252*A*
a**5*b**5*d**2*e**6 - 210*A*a**4*b**6*d**3*e**5 + 120*A*a**3*b**7*d**4*e**4 - 45
*A*a**2*b**8*d**5*e**3 + 10*A*a*b**9*d**6*e**2 - A*b**10*d**7*e + 45*B*a**8*b**2
*e**8 - 120*B*a**7*b**3*d*e**7 + 210*B*a**6*b**4*d**2*e**6 - 252*B*a**5*b**5*d**
3*e**5 + 210*B*a**4*b**6*d**4*e**4 - 120*B*a**3*b**7*d**5*e**3 + 45*B*a**2*b**8*
d**6*e**2 - 10*B*a*b**9*d**7*e + B*b**10*d**8)/(3*e**9) + x**2*(45*A*a**8*b**2*e
**9 - 120*A*a**7*b**3*d*e**8 + 210*A*a**6*b**4*d**2*e**7 - 252*A*a**5*b**5*d**3*
e**6 + 210*A*a**4*b**6*d**4*e**5 - 120*A*a**3*b**7*d**5*e**4 + 45*A*a**2*b**8*d*
*6*e**3 - 10*A*a*b**9*d**7*e**2 + A*b**10*d**8*e + 10*B*a**9*b*e**9 - 45*B*a**8*
b**2*d*e**8 + 120*B*a**7*b**3*d**2*e**7 - 210*B*a**6*b**4*d**3*e**6 + 252*B*a**5
*b**5*d**4*e**5 - 210*B*a**4*b**6*d**5*e**4 + 120*B*a**3*b**7*d**6*e**3 - 45*B*a
**2*b**8*d**7*e**2 + 10*B*a*b**9*d**8*e - B*b**10*d**9)/(2*e**10) + x*(10*A*a**9
*b*e**10 - 45*A*a**8*b**2*d*e**9 + 120*A*a**7*b**3*d**2*e**8 - 210*A*a**6*b**4*d
**3*e**7 + 252*A*a**5*b**5*d**4*e**6 - 210*A*a**4*b**6*d**5*e**5 + 120*A*a**3*b*
*7*d**6*e**4 - 45*A*a**2*b**8*d**7*e**3 + 10*A*a*b**9*d**8*e**2 - A*b**10*d**9*e
 + B*a**10*e**10 - 10*B*a**9*b*d*e**9 + 45*B*a**8*b**2*d**2*e**8 - 120*B*a**7*b*
*3*d**3*e**7 + 210*B*a**6*b**4*d**4*e**6 - 252*B*a**5*b**5*d**5*e**5 + 210*B*a**
4*b**6*d**6*e**4 - 120*B*a**3*b**7*d**7*e**3 + 45*B*a**2*b**8*d**8*e**2 - 10*B*a
*b**9*d**9*e + B*b**10*d**10)/e**11 - (-A*e + B*d)*(a*e - b*d)**10*log(d + e*x)/
e**12

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.212216, 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),x, algorithm="giac")

[Out]

Done

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