\(\int \frac {(a+b x)^{10} (A+B x)}{x^{11}} \, dx\) [158]

Optimal result

Integrand size = 16, antiderivative size = 216 \[ \int \frac {(a+b x)^{10} (A+B x)}{x^{11}} \, dx=-\frac {a^{10} A}{10 x^{10}}-\frac {a^9 (10 A b+a B)}{9 x^9}-\frac {5 a^8 b (9 A b+2 a B)}{8 x^8}-\frac {15 a^7 b^2 (8 A b+3 a B)}{7 x^7}-\frac {5 a^6 b^3 (7 A b+4 a B)}{x^6}-\frac {42 a^5 b^4 (6 A b+5 a B)}{5 x^5}-\frac {21 a^4 b^5 (5 A b+6 a B)}{2 x^4}-\frac {10 a^3 b^6 (4 A b+7 a B)}{x^3}-\frac {15 a^2 b^7 (3 A b+8 a B)}{2 x^2}-\frac {5 a b^8 (2 A b+9 a B)}{x}+b^{10} B x+b^9 (A b+10 a B) \log (x) \]

[Out]

-1/10*a^10*A/x^10-1/9*a^9*(10*A*b+B*a)/x^9-5/8*a^8*b*(9*A*b+2*B*a)/x^8-15/7*a^7*b^2*(8*A*b+3*B*a)/x^7-5*a^6*b^
3*(7*A*b+4*B*a)/x^6-42/5*a^5*b^4*(6*A*b+5*B*a)/x^5-21/2*a^4*b^5*(5*A*b+6*B*a)/x^4-10*a^3*b^6*(4*A*b+7*B*a)/x^3
-15/2*a^2*b^7*(3*A*b+8*B*a)/x^2-5*a*b^8*(2*A*b+9*B*a)/x+b^10*B*x+b^9*(A*b+10*B*a)*ln(x)

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {77} \[ \int \frac {(a+b x)^{10} (A+B x)}{x^{11}} \, dx=-\frac {a^{10} A}{10 x^{10}}-\frac {a^9 (a B+10 A b)}{9 x^9}-\frac {5 a^8 b (2 a B+9 A b)}{8 x^8}-\frac {15 a^7 b^2 (3 a B+8 A b)}{7 x^7}-\frac {5 a^6 b^3 (4 a B+7 A b)}{x^6}-\frac {42 a^5 b^4 (5 a B+6 A b)}{5 x^5}-\frac {21 a^4 b^5 (6 a B+5 A b)}{2 x^4}-\frac {10 a^3 b^6 (7 a B+4 A b)}{x^3}-\frac {15 a^2 b^7 (8 a B+3 A b)}{2 x^2}+b^9 \log (x) (10 a B+A b)-\frac {5 a b^8 (9 a B+2 A b)}{x}+b^{10} B x \]

[In]

Int[((a + b*x)^10*(A + B*x))/x^11,x]

[Out]

-1/10*(a^10*A)/x^10 - (a^9*(10*A*b + a*B))/(9*x^9) - (5*a^8*b*(9*A*b + 2*a*B))/(8*x^8) - (15*a^7*b^2*(8*A*b +
3*a*B))/(7*x^7) - (5*a^6*b^3*(7*A*b + 4*a*B))/x^6 - (42*a^5*b^4*(6*A*b + 5*a*B))/(5*x^5) - (21*a^4*b^5*(5*A*b
+ 6*a*B))/(2*x^4) - (10*a^3*b^6*(4*A*b + 7*a*B))/x^3 - (15*a^2*b^7*(3*A*b + 8*a*B))/(2*x^2) - (5*a*b^8*(2*A*b
+ 9*a*B))/x + b^10*B*x + b^9*(A*b + 10*a*B)*Log[x]

Rule 77

Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*
x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && (NeQ[n, -1] || EqQ[p, 1]) && N
eQ[b*e + a*f, 0] && ( !IntegerQ[n] || LtQ[9*p + 5*n, 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && Rational
Q[a, b, d, e, f])) && (NeQ[n + p + 3, 0] || EqQ[p, 1])

Rubi steps \begin{align*} \text {integral}& = \int \left (b^{10} B+\frac {a^{10} A}{x^{11}}+\frac {a^9 (10 A b+a B)}{x^{10}}+\frac {5 a^8 b (9 A b+2 a B)}{x^9}+\frac {15 a^7 b^2 (8 A b+3 a B)}{x^8}+\frac {30 a^6 b^3 (7 A b+4 a B)}{x^7}+\frac {42 a^5 b^4 (6 A b+5 a B)}{x^6}+\frac {42 a^4 b^5 (5 A b+6 a B)}{x^5}+\frac {30 a^3 b^6 (4 A b+7 a B)}{x^4}+\frac {15 a^2 b^7 (3 A b+8 a B)}{x^3}+\frac {5 a b^8 (2 A b+9 a B)}{x^2}+\frac {b^9 (A b+10 a B)}{x}\right ) \, dx \\ & = -\frac {a^{10} A}{10 x^{10}}-\frac {a^9 (10 A b+a B)}{9 x^9}-\frac {5 a^8 b (9 A b+2 a B)}{8 x^8}-\frac {15 a^7 b^2 (8 A b+3 a B)}{7 x^7}-\frac {5 a^6 b^3 (7 A b+4 a B)}{x^6}-\frac {42 a^5 b^4 (6 A b+5 a B)}{5 x^5}-\frac {21 a^4 b^5 (5 A b+6 a B)}{2 x^4}-\frac {10 a^3 b^6 (4 A b+7 a B)}{x^3}-\frac {15 a^2 b^7 (3 A b+8 a B)}{2 x^2}-\frac {5 a b^8 (2 A b+9 a B)}{x}+b^{10} B x+b^9 (A b+10 a B) \log (x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 209, normalized size of antiderivative = 0.97 \[ \int \frac {(a+b x)^{10} (A+B x)}{x^{11}} \, dx=-\frac {10 a A b^9}{x}+b^{10} B x-\frac {45 a^2 b^8 (A+2 B x)}{2 x^2}-\frac {20 a^3 b^7 (2 A+3 B x)}{x^3}-\frac {35 a^4 b^6 (3 A+4 B x)}{2 x^4}-\frac {63 a^5 b^5 (4 A+5 B x)}{5 x^5}-\frac {7 a^6 b^4 (5 A+6 B x)}{x^6}-\frac {20 a^7 b^3 (6 A+7 B x)}{7 x^7}-\frac {45 a^8 b^2 (7 A+8 B x)}{56 x^8}-\frac {5 a^9 b (8 A+9 B x)}{36 x^9}-\frac {a^{10} (9 A+10 B x)}{90 x^{10}}+b^9 (A b+10 a B) \log (x) \]

[In]

Integrate[((a + b*x)^10*(A + B*x))/x^11,x]

[Out]

(-10*a*A*b^9)/x + b^10*B*x - (45*a^2*b^8*(A + 2*B*x))/(2*x^2) - (20*a^3*b^7*(2*A + 3*B*x))/x^3 - (35*a^4*b^6*(
3*A + 4*B*x))/(2*x^4) - (63*a^5*b^5*(4*A + 5*B*x))/(5*x^5) - (7*a^6*b^4*(5*A + 6*B*x))/x^6 - (20*a^7*b^3*(6*A
+ 7*B*x))/(7*x^7) - (45*a^8*b^2*(7*A + 8*B*x))/(56*x^8) - (5*a^9*b*(8*A + 9*B*x))/(36*x^9) - (a^10*(9*A + 10*B
*x))/(90*x^10) + b^9*(A*b + 10*a*B)*Log[x]

Maple [A] (verified)

Time = 0.41 (sec) , antiderivative size = 203, normalized size of antiderivative = 0.94

method	result	size
default	\(-\frac {a^{10} A}{10 x^{10}}-\frac {a^{9} \left (10 A b +B a \right )}{9 x^{9}}-\frac {5 a^{8} b \left (9 A b +2 B a \right )}{8 x^{8}}-\frac {15 a^{7} b^{2} \left (8 A b +3 B a \right )}{7 x^{7}}-\frac {5 a^{6} b^{3} \left (7 A b +4 B a \right )}{x^{6}}-\frac {42 a^{5} b^{4} \left (6 A b +5 B a \right )}{5 x^{5}}-\frac {21 a^{4} b^{5} \left (5 A b +6 B a \right )}{2 x^{4}}-\frac {10 a^{3} b^{6} \left (4 A b +7 B a \right )}{x^{3}}-\frac {15 a^{2} b^{7} \left (3 A b +8 B a \right )}{2 x^{2}}-\frac {5 a \,b^{8} \left (2 A b +9 B a \right )}{x}+b^{10} B x +b^{9} \left (A b +10 B a \right ) \ln \left (x \right )\)	\(203\)
risch	\(b^{10} B x +\frac {\left (-10 a \,b^{9} A -45 a^{2} b^{8} B \right ) x^{9}+\left (-\frac {45}{2} a^{2} b^{8} A -60 a^{3} b^{7} B \right ) x^{8}+\left (-40 a^{3} b^{7} A -70 a^{4} b^{6} B \right ) x^{7}+\left (-\frac {105}{2} a^{4} b^{6} A -63 a^{5} b^{5} B \right ) x^{6}+\left (-\frac {252}{5} a^{5} b^{5} A -42 a^{6} b^{4} B \right ) x^{5}+\left (-35 a^{6} b^{4} A -20 a^{7} b^{3} B \right ) x^{4}+\left (-\frac {120}{7} a^{7} b^{3} A -\frac {45}{7} a^{8} b^{2} B \right ) x^{3}+\left (-\frac {45}{8} a^{8} b^{2} A -\frac {5}{4} a^{9} b B \right ) x^{2}+\left (-\frac {10}{9} a^{9} b A -\frac {1}{9} a^{10} B \right ) x -\frac {a^{10} A}{10}}{x^{10}}+A \ln \left (x \right ) b^{10}+10 B \ln \left (x \right ) a \,b^{9}\)	\(231\)
norman	\(\frac {\left (-\frac {45}{2} a^{2} b^{8} A -60 a^{3} b^{7} B \right ) x^{8}+\left (-\frac {105}{2} a^{4} b^{6} A -63 a^{5} b^{5} B \right ) x^{6}+\left (-\frac {252}{5} a^{5} b^{5} A -42 a^{6} b^{4} B \right ) x^{5}+\left (-\frac {120}{7} a^{7} b^{3} A -\frac {45}{7} a^{8} b^{2} B \right ) x^{3}+\left (-\frac {45}{8} a^{8} b^{2} A -\frac {5}{4} a^{9} b B \right ) x^{2}+\left (-\frac {10}{9} a^{9} b A -\frac {1}{9} a^{10} B \right ) x +\left (-10 a \,b^{9} A -45 a^{2} b^{8} B \right ) x^{9}+\left (-40 a^{3} b^{7} A -70 a^{4} b^{6} B \right ) x^{7}+\left (-35 a^{6} b^{4} A -20 a^{7} b^{3} B \right ) x^{4}+b^{10} B \,x^{11}-\frac {a^{10} A}{10}}{x^{10}}+\left (b^{10} A +10 a \,b^{9} B \right ) \ln \left (x \right )\)	\(233\)
parallelrisch	\(\frac {2520 A \ln \left (x \right ) x^{10} b^{10}+25200 B \ln \left (x \right ) x^{10} a \,b^{9}+2520 b^{10} B \,x^{11}-25200 a A \,b^{9} x^{9}-113400 B \,a^{2} b^{8} x^{9}-56700 a^{2} A \,b^{8} x^{8}-151200 B \,a^{3} b^{7} x^{8}-100800 a^{3} A \,b^{7} x^{7}-176400 B \,a^{4} b^{6} x^{7}-132300 a^{4} A \,b^{6} x^{6}-158760 B \,a^{5} b^{5} x^{6}-127008 a^{5} A \,b^{5} x^{5}-105840 B \,a^{6} b^{4} x^{5}-88200 a^{6} A \,b^{4} x^{4}-50400 B \,a^{7} b^{3} x^{4}-43200 a^{7} A \,b^{3} x^{3}-16200 B \,a^{8} b^{2} x^{3}-14175 a^{8} A \,b^{2} x^{2}-3150 B \,a^{9} b \,x^{2}-2800 a^{9} A b x -280 a^{10} B x -252 a^{10} A}{2520 x^{10}}\)	\(248\)

[In]

int((b*x+a)^10*(B*x+A)/x^11,x,method=_RETURNVERBOSE)

[Out]

-1/10*a^10*A/x^10-1/9*a^9*(10*A*b+B*a)/x^9-5/8*a^8*b*(9*A*b+2*B*a)/x^8-15/7*a^7*b^2*(8*A*b+3*B*a)/x^7-5*a^6*b^
3*(7*A*b+4*B*a)/x^6-42/5*a^5*b^4*(6*A*b+5*B*a)/x^5-21/2*a^4*b^5*(5*A*b+6*B*a)/x^4-10*a^3*b^6*(4*A*b+7*B*a)/x^3
-15/2*a^2*b^7*(3*A*b+8*B*a)/x^2-5*a*b^8*(2*A*b+9*B*a)/x+b^10*B*x+b^9*(A*b+10*B*a)*ln(x)

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.13 \[ \int \frac {(a+b x)^{10} (A+B x)}{x^{11}} \, dx=\frac {2520 \, B b^{10} x^{11} + 2520 \, {\left (10 \, B a b^{9} + A b^{10}\right )} x^{10} \log \left (x\right ) - 252 \, A a^{10} - 12600 \, {\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{9} - 18900 \, {\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{8} - 25200 \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} - 26460 \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} - 21168 \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} - 12600 \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} - 5400 \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} - 1575 \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} - 280 \, {\left (B a^{10} + 10 \, A a^{9} b\right )} x}{2520 \, x^{10}} \]

[In]

integrate((b*x+a)^10*(B*x+A)/x^11,x, algorithm="fricas")

[Out]

1/2520*(2520*B*b^10*x^11 + 2520*(10*B*a*b^9 + A*b^10)*x^10*log(x) - 252*A*a^10 - 12600*(9*B*a^2*b^8 + 2*A*a*b^
9)*x^9 - 18900*(8*B*a^3*b^7 + 3*A*a^2*b^8)*x^8 - 25200*(7*B*a^4*b^6 + 4*A*a^3*b^7)*x^7 - 26460*(6*B*a^5*b^5 +
5*A*a^4*b^6)*x^6 - 21168*(5*B*a^6*b^4 + 6*A*a^5*b^5)*x^5 - 12600*(4*B*a^7*b^3 + 7*A*a^6*b^4)*x^4 - 5400*(3*B*a
^8*b^2 + 8*A*a^7*b^3)*x^3 - 1575*(2*B*a^9*b + 9*A*a^8*b^2)*x^2 - 280*(B*a^10 + 10*A*a^9*b)*x)/x^10

Sympy [A] (verification not implemented)

Time = 11.66 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.17 \[ \int \frac {(a+b x)^{10} (A+B x)}{x^{11}} \, dx=B b^{10} x + b^{9} \left (A b + 10 B a\right ) \log {\left (x \right )} + \frac {- 252 A a^{10} + x^{9} \left (- 25200 A a b^{9} - 113400 B a^{2} b^{8}\right ) + x^{8} \left (- 56700 A a^{2} b^{8} - 151200 B a^{3} b^{7}\right ) + x^{7} \left (- 100800 A a^{3} b^{7} - 176400 B a^{4} b^{6}\right ) + x^{6} \left (- 132300 A a^{4} b^{6} - 158760 B a^{5} b^{5}\right ) + x^{5} \left (- 127008 A a^{5} b^{5} - 105840 B a^{6} b^{4}\right ) + x^{4} \left (- 88200 A a^{6} b^{4} - 50400 B a^{7} b^{3}\right ) + x^{3} \left (- 43200 A a^{7} b^{3} - 16200 B a^{8} b^{2}\right ) + x^{2} \left (- 14175 A a^{8} b^{2} - 3150 B a^{9} b\right ) + x \left (- 2800 A a^{9} b - 280 B a^{10}\right )}{2520 x^{10}} \]

[In]

integrate((b*x+a)**10*(B*x+A)/x**11,x)

[Out]

B*b**10*x + b**9*(A*b + 10*B*a)*log(x) + (-252*A*a**10 + x**9*(-25200*A*a*b**9 - 113400*B*a**2*b**8) + x**8*(-
56700*A*a**2*b**8 - 151200*B*a**3*b**7) + x**7*(-100800*A*a**3*b**7 - 176400*B*a**4*b**6) + x**6*(-132300*A*a*
*4*b**6 - 158760*B*a**5*b**5) + x**5*(-127008*A*a**5*b**5 - 105840*B*a**6*b**4) + x**4*(-88200*A*a**6*b**4 - 5
0400*B*a**7*b**3) + x**3*(-43200*A*a**7*b**3 - 16200*B*a**8*b**2) + x**2*(-14175*A*a**8*b**2 - 3150*B*a**9*b)
+ x*(-2800*A*a**9*b - 280*B*a**10))/(2520*x**10)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.11 \[ \int \frac {(a+b x)^{10} (A+B x)}{x^{11}} \, dx=B b^{10} x + {\left (10 \, B a b^{9} + A b^{10}\right )} \log \left (x\right ) - \frac {252 \, A a^{10} + 12600 \, {\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{9} + 18900 \, {\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{8} + 25200 \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} + 26460 \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} + 21168 \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} + 12600 \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} + 5400 \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} + 1575 \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} + 280 \, {\left (B a^{10} + 10 \, A a^{9} b\right )} x}{2520 \, x^{10}} \]

[In]

integrate((b*x+a)^10*(B*x+A)/x^11,x, algorithm="maxima")

[Out]

B*b^10*x + (10*B*a*b^9 + A*b^10)*log(x) - 1/2520*(252*A*a^10 + 12600*(9*B*a^2*b^8 + 2*A*a*b^9)*x^9 + 18900*(8*
B*a^3*b^7 + 3*A*a^2*b^8)*x^8 + 25200*(7*B*a^4*b^6 + 4*A*a^3*b^7)*x^7 + 26460*(6*B*a^5*b^5 + 5*A*a^4*b^6)*x^6 +
21168*(5*B*a^6*b^4 + 6*A*a^5*b^5)*x^5 + 12600*(4*B*a^7*b^3 + 7*A*a^6*b^4)*x^4 + 5400*(3*B*a^8*b^2 + 8*A*a^7*b
^3)*x^3 + 1575*(2*B*a^9*b + 9*A*a^8*b^2)*x^2 + 280*(B*a^10 + 10*A*a^9*b)*x)/x^10

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.11 \[ \int \frac {(a+b x)^{10} (A+B x)}{x^{11}} \, dx=B b^{10} x + {\left (10 \, B a b^{9} + A b^{10}\right )} \log \left ({\left | x \right |}\right ) - \frac {252 \, A a^{10} + 12600 \, {\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{9} + 18900 \, {\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{8} + 25200 \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} + 26460 \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} + 21168 \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} + 12600 \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} + 5400 \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} + 1575 \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} + 280 \, {\left (B a^{10} + 10 \, A a^{9} b\right )} x}{2520 \, x^{10}} \]

[In]

integrate((b*x+a)^10*(B*x+A)/x^11,x, algorithm="giac")

[Out]

B*b^10*x + (10*B*a*b^9 + A*b^10)*log(abs(x)) - 1/2520*(252*A*a^10 + 12600*(9*B*a^2*b^8 + 2*A*a*b^9)*x^9 + 1890
0*(8*B*a^3*b^7 + 3*A*a^2*b^8)*x^8 + 25200*(7*B*a^4*b^6 + 4*A*a^3*b^7)*x^7 + 26460*(6*B*a^5*b^5 + 5*A*a^4*b^6)*
x^6 + 21168*(5*B*a^6*b^4 + 6*A*a^5*b^5)*x^5 + 12600*(4*B*a^7*b^3 + 7*A*a^6*b^4)*x^4 + 5400*(3*B*a^8*b^2 + 8*A*
a^7*b^3)*x^3 + 1575*(2*B*a^9*b + 9*A*a^8*b^2)*x^2 + 280*(B*a^10 + 10*A*a^9*b)*x)/x^10

Mupad [B] (verification not implemented)

Time = 0.39 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.07 \[ \int \frac {(a+b x)^{10} (A+B x)}{x^{11}} \, dx=\ln \left (x\right )\,\left (A\,b^{10}+10\,B\,a\,b^9\right )-\frac {x\,\left (\frac {B\,a^{10}}{9}+\frac {10\,A\,b\,a^9}{9}\right )+\frac {A\,a^{10}}{10}+x^2\,\left (\frac {5\,B\,a^9\,b}{4}+\frac {45\,A\,a^8\,b^2}{8}\right )+x^9\,\left (45\,B\,a^2\,b^8+10\,A\,a\,b^9\right )+x^4\,\left (20\,B\,a^7\,b^3+35\,A\,a^6\,b^4\right )+x^8\,\left (60\,B\,a^3\,b^7+\frac {45\,A\,a^2\,b^8}{2}\right )+x^7\,\left (70\,B\,a^4\,b^6+40\,A\,a^3\,b^7\right )+x^6\,\left (63\,B\,a^5\,b^5+\frac {105\,A\,a^4\,b^6}{2}\right )+x^3\,\left (\frac {45\,B\,a^8\,b^2}{7}+\frac {120\,A\,a^7\,b^3}{7}\right )+x^5\,\left (42\,B\,a^6\,b^4+\frac {252\,A\,a^5\,b^5}{5}\right )}{x^{10}}+B\,b^{10}\,x \]

[In]

int(((A + B*x)*(a + b*x)^10)/x^11,x)

[Out]

log(x)*(A*b^10 + 10*B*a*b^9) - (x*((B*a^10)/9 + (10*A*a^9*b)/9) + (A*a^10)/10 + x^2*((45*A*a^8*b^2)/8 + (5*B*a
^9*b)/4) + x^9*(45*B*a^2*b^8 + 10*A*a*b^9) + x^4*(35*A*a^6*b^4 + 20*B*a^7*b^3) + x^8*((45*A*a^2*b^8)/2 + 60*B*
a^3*b^7) + x^7*(40*A*a^3*b^7 + 70*B*a^4*b^6) + x^6*((105*A*a^4*b^6)/2 + 63*B*a^5*b^5) + x^3*((120*A*a^7*b^3)/7
+ (45*B*a^8*b^2)/7) + x^5*((252*A*a^5*b^5)/5 + 42*B*a^6*b^4))/x^10 + B*b^10*x