\(\int \frac {(a+b x)^{10} (A+B x)}{x^{16}} \, dx\) [163]

Optimal result

Integrand size = 16, antiderivative size = 130 \[ \int \frac {(a+b x)^{10} (A+B x)}{x^{16}} \, dx=-\frac {A (a+b x)^{11}}{15 a x^{15}}+\frac {(4 A b-15 a B) (a+b x)^{11}}{210 a^2 x^{14}}-\frac {b (4 A b-15 a B) (a+b x)^{11}}{910 a^3 x^{13}}+\frac {b^2 (4 A b-15 a B) (a+b x)^{11}}{5460 a^4 x^{12}}-\frac {b^3 (4 A b-15 a B) (a+b x)^{11}}{60060 a^5 x^{11}} \]

[Out]

-1/15*A*(b*x+a)^11/a/x^15+1/210*(4*A*b-15*B*a)*(b*x+a)^11/a^2/x^14-1/910*b*(4*A*b-15*B*a)*(b*x+a)^11/a^3/x^13+
1/5460*b^2*(4*A*b-15*B*a)*(b*x+a)^11/a^4/x^12-1/60060*b^3*(4*A*b-15*B*a)*(b*x+a)^11/a^5/x^11

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {79, 47, 37} \[ \int \frac {(a+b x)^{10} (A+B x)}{x^{16}} \, dx=-\frac {b^3 (a+b x)^{11} (4 A b-15 a B)}{60060 a^5 x^{11}}+\frac {b^2 (a+b x)^{11} (4 A b-15 a B)}{5460 a^4 x^{12}}-\frac {b (a+b x)^{11} (4 A b-15 a B)}{910 a^3 x^{13}}+\frac {(a+b x)^{11} (4 A b-15 a B)}{210 a^2 x^{14}}-\frac {A (a+b x)^{11}}{15 a x^{15}} \]

[In]

Int[((a + b*x)^10*(A + B*x))/x^16,x]

[Out]

-1/15*(A*(a + b*x)^11)/(a*x^15) + ((4*A*b - 15*a*B)*(a + b*x)^11)/(210*a^2*x^14) - (b*(4*A*b - 15*a*B)*(a + b*
x)^11)/(910*a^3*x^13) + (b^2*(4*A*b - 15*a*B)*(a + b*x)^11)/(5460*a^4*x^12) - (b^3*(4*A*b - 15*a*B)*(a + b*x)^
11)/(60060*a^5*x^11)

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n +
1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rule 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n + 1
)/((b*c - a*d)*(m + 1))), x] - Dist[d*(Simplify[m + n + 2]/((b*c - a*d)*(m + 1))), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

Rule 79

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(-(b*e - a*f
))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1
) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e,
f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || L
tQ[p, n]))))

Rubi steps \begin{align*} \text {integral}& = -\frac {A (a+b x)^{11}}{15 a x^{15}}+\frac {(-4 A b+15 a B) \int \frac {(a+b x)^{10}}{x^{15}} \, dx}{15 a} \\ & = -\frac {A (a+b x)^{11}}{15 a x^{15}}+\frac {(4 A b-15 a B) (a+b x)^{11}}{210 a^2 x^{14}}+\frac {(b (4 A b-15 a B)) \int \frac {(a+b x)^{10}}{x^{14}} \, dx}{70 a^2} \\ & = -\frac {A (a+b x)^{11}}{15 a x^{15}}+\frac {(4 A b-15 a B) (a+b x)^{11}}{210 a^2 x^{14}}-\frac {b (4 A b-15 a B) (a+b x)^{11}}{910 a^3 x^{13}}-\frac {\left (b^2 (4 A b-15 a B)\right ) \int \frac {(a+b x)^{10}}{x^{13}} \, dx}{455 a^3} \\ & = -\frac {A (a+b x)^{11}}{15 a x^{15}}+\frac {(4 A b-15 a B) (a+b x)^{11}}{210 a^2 x^{14}}-\frac {b (4 A b-15 a B) (a+b x)^{11}}{910 a^3 x^{13}}+\frac {b^2 (4 A b-15 a B) (a+b x)^{11}}{5460 a^4 x^{12}}+\frac {\left (b^3 (4 A b-15 a B)\right ) \int \frac {(a+b x)^{10}}{x^{12}} \, dx}{5460 a^4} \\ & = -\frac {A (a+b x)^{11}}{15 a x^{15}}+\frac {(4 A b-15 a B) (a+b x)^{11}}{210 a^2 x^{14}}-\frac {b (4 A b-15 a B) (a+b x)^{11}}{910 a^3 x^{13}}+\frac {b^2 (4 A b-15 a B) (a+b x)^{11}}{5460 a^4 x^{12}}-\frac {b^3 (4 A b-15 a B) (a+b x)^{11}}{60060 a^5 x^{11}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.55 \[ \int \frac {(a+b x)^{10} (A+B x)}{x^{16}} \, dx=-\frac {3003 b^{10} x^{10} (4 A+5 B x)+20020 a b^9 x^9 (5 A+6 B x)+64350 a^2 b^8 x^8 (6 A+7 B x)+128700 a^3 b^7 x^7 (7 A+8 B x)+175175 a^4 b^6 x^6 (8 A+9 B x)+168168 a^5 b^5 x^5 (9 A+10 B x)+114660 a^6 b^4 x^4 (10 A+11 B x)+54600 a^7 b^3 x^3 (11 A+12 B x)+17325 a^8 b^2 x^2 (12 A+13 B x)+3300 a^9 b x (13 A+14 B x)+286 a^{10} (14 A+15 B x)}{60060 x^{15}} \]

[In]

Integrate[((a + b*x)^10*(A + B*x))/x^16,x]

[Out]

-1/60060*(3003*b^10*x^10*(4*A + 5*B*x) + 20020*a*b^9*x^9*(5*A + 6*B*x) + 64350*a^2*b^8*x^8*(6*A + 7*B*x) + 128
700*a^3*b^7*x^7*(7*A + 8*B*x) + 175175*a^4*b^6*x^6*(8*A + 9*B*x) + 168168*a^5*b^5*x^5*(9*A + 10*B*x) + 114660*
a^6*b^4*x^4*(10*A + 11*B*x) + 54600*a^7*b^3*x^3*(11*A + 12*B*x) + 17325*a^8*b^2*x^2*(12*A + 13*B*x) + 3300*a^9
*b*x*(13*A + 14*B*x) + 286*a^10*(14*A + 15*B*x))/x^15

Maple [A] (verified)

Time = 0.40 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.60

method	result	size
default	\(-\frac {5 a \,b^{8} \left (2 A b +9 B a \right )}{6 x^{6}}-\frac {a^{9} \left (10 A b +B a \right )}{14 x^{14}}-\frac {15 a^{2} b^{7} \left (3 A b +8 B a \right )}{7 x^{7}}-\frac {15 a^{3} b^{6} \left (4 A b +7 B a \right )}{4 x^{8}}-\frac {21 a^{5} b^{4} \left (6 A b +5 B a \right )}{5 x^{10}}-\frac {5 a^{8} b \left (9 A b +2 B a \right )}{13 x^{13}}-\frac {5 a^{7} b^{2} \left (8 A b +3 B a \right )}{4 x^{12}}-\frac {b^{10} B}{4 x^{4}}-\frac {a^{10} A}{15 x^{15}}-\frac {b^{9} \left (A b +10 B a \right )}{5 x^{5}}-\frac {14 a^{4} b^{5} \left (5 A b +6 B a \right )}{3 x^{9}}-\frac {30 a^{6} b^{3} \left (7 A b +4 B a \right )}{11 x^{11}}\)	\(208\)
norman	\(\frac {-\frac {a^{10} A}{15}+\left (-\frac {5}{7} a^{9} b A -\frac {1}{14} a^{10} B \right ) x +\left (-\frac {45}{13} a^{8} b^{2} A -\frac {10}{13} a^{9} b B \right ) x^{2}+\left (-10 a^{7} b^{3} A -\frac {15}{4} a^{8} b^{2} B \right ) x^{3}+\left (-\frac {210}{11} a^{6} b^{4} A -\frac {120}{11} a^{7} b^{3} B \right ) x^{4}+\left (-\frac {126}{5} a^{5} b^{5} A -21 a^{6} b^{4} B \right ) x^{5}+\left (-\frac {70}{3} a^{4} b^{6} A -28 a^{5} b^{5} B \right ) x^{6}+\left (-15 a^{3} b^{7} A -\frac {105}{4} a^{4} b^{6} B \right ) x^{7}+\left (-\frac {45}{7} a^{2} b^{8} A -\frac {120}{7} a^{3} b^{7} B \right ) x^{8}+\left (-\frac {5}{3} a \,b^{9} A -\frac {15}{2} a^{2} b^{8} B \right ) x^{9}+\left (-\frac {1}{5} b^{10} A -2 a \,b^{9} B \right ) x^{10}-\frac {b^{10} B \,x^{11}}{4}}{x^{15}}\)	\(235\)
risch	\(\frac {-\frac {a^{10} A}{15}+\left (-\frac {5}{7} a^{9} b A -\frac {1}{14} a^{10} B \right ) x +\left (-\frac {45}{13} a^{8} b^{2} A -\frac {10}{13} a^{9} b B \right ) x^{2}+\left (-10 a^{7} b^{3} A -\frac {15}{4} a^{8} b^{2} B \right ) x^{3}+\left (-\frac {210}{11} a^{6} b^{4} A -\frac {120}{11} a^{7} b^{3} B \right ) x^{4}+\left (-\frac {126}{5} a^{5} b^{5} A -21 a^{6} b^{4} B \right ) x^{5}+\left (-\frac {70}{3} a^{4} b^{6} A -28 a^{5} b^{5} B \right ) x^{6}+\left (-15 a^{3} b^{7} A -\frac {105}{4} a^{4} b^{6} B \right ) x^{7}+\left (-\frac {45}{7} a^{2} b^{8} A -\frac {120}{7} a^{3} b^{7} B \right ) x^{8}+\left (-\frac {5}{3} a \,b^{9} A -\frac {15}{2} a^{2} b^{8} B \right ) x^{9}+\left (-\frac {1}{5} b^{10} A -2 a \,b^{9} B \right ) x^{10}-\frac {b^{10} B \,x^{11}}{4}}{x^{15}}\)	\(235\)
gosper	\(-\frac {15015 b^{10} B \,x^{11}+12012 A \,b^{10} x^{10}+120120 B a \,b^{9} x^{10}+100100 a A \,b^{9} x^{9}+450450 B \,a^{2} b^{8} x^{9}+386100 a^{2} A \,b^{8} x^{8}+1029600 B \,a^{3} b^{7} x^{8}+900900 a^{3} A \,b^{7} x^{7}+1576575 B \,a^{4} b^{6} x^{7}+1401400 a^{4} A \,b^{6} x^{6}+1681680 B \,a^{5} b^{5} x^{6}+1513512 a^{5} A \,b^{5} x^{5}+1261260 B \,a^{6} b^{4} x^{5}+1146600 a^{6} A \,b^{4} x^{4}+655200 B \,a^{7} b^{3} x^{4}+600600 a^{7} A \,b^{3} x^{3}+225225 B \,a^{8} b^{2} x^{3}+207900 a^{8} A \,b^{2} x^{2}+46200 B \,a^{9} b \,x^{2}+42900 a^{9} A b x +4290 a^{10} B x +4004 a^{10} A}{60060 x^{15}}\)	\(244\)
parallelrisch	\(-\frac {15015 b^{10} B \,x^{11}+12012 A \,b^{10} x^{10}+120120 B a \,b^{9} x^{10}+100100 a A \,b^{9} x^{9}+450450 B \,a^{2} b^{8} x^{9}+386100 a^{2} A \,b^{8} x^{8}+1029600 B \,a^{3} b^{7} x^{8}+900900 a^{3} A \,b^{7} x^{7}+1576575 B \,a^{4} b^{6} x^{7}+1401400 a^{4} A \,b^{6} x^{6}+1681680 B \,a^{5} b^{5} x^{6}+1513512 a^{5} A \,b^{5} x^{5}+1261260 B \,a^{6} b^{4} x^{5}+1146600 a^{6} A \,b^{4} x^{4}+655200 B \,a^{7} b^{3} x^{4}+600600 a^{7} A \,b^{3} x^{3}+225225 B \,a^{8} b^{2} x^{3}+207900 a^{8} A \,b^{2} x^{2}+46200 B \,a^{9} b \,x^{2}+42900 a^{9} A b x +4290 a^{10} B x +4004 a^{10} A}{60060 x^{15}}\)	\(244\)

[In]

int((b*x+a)^10*(B*x+A)/x^16,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 243 vs. \(2 (120) = 240\).

Time = 0.22 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.87 \[ \int \frac {(a+b x)^{10} (A+B x)}{x^{16}} \, dx=-\frac {15015 \, B b^{10} x^{11} + 4004 \, A a^{10} + 12012 \, {\left (10 \, B a b^{9} + A b^{10}\right )} x^{10} + 50050 \, {\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{9} + 128700 \, {\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{8} + 225225 \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} + 280280 \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} + 252252 \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} + 163800 \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} + 75075 \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} + 23100 \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} + 4290 \, {\left (B a^{10} + 10 \, A a^{9} b\right )} x}{60060 \, x^{15}} \]

[In]

integrate((b*x+a)^10*(B*x+A)/x^16,x, algorithm="fricas")

[Out]

-1/60060*(15015*B*b^10*x^11 + 4004*A*a^10 + 12012*(10*B*a*b^9 + A*b^10)*x^10 + 50050*(9*B*a^2*b^8 + 2*A*a*b^9)
*x^9 + 128700*(8*B*a^3*b^7 + 3*A*a^2*b^8)*x^8 + 225225*(7*B*a^4*b^6 + 4*A*a^3*b^7)*x^7 + 280280*(6*B*a^5*b^5 +
5*A*a^4*b^6)*x^6 + 252252*(5*B*a^6*b^4 + 6*A*a^5*b^5)*x^5 + 163800*(4*B*a^7*b^3 + 7*A*a^6*b^4)*x^4 + 75075*(3
*B*a^8*b^2 + 8*A*a^7*b^3)*x^3 + 23100*(2*B*a^9*b + 9*A*a^8*b^2)*x^2 + 4290*(B*a^10 + 10*A*a^9*b)*x)/x^15

Sympy [F(-1)]

Timed out. \[ \int \frac {(a+b x)^{10} (A+B x)}{x^{16}} \, dx=\text {Timed out} \]

[In]

integrate((b*x+a)**10*(B*x+A)/x**16,x)

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 243 vs. \(2 (120) = 240\).

Time = 0.20 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.87 \[ \int \frac {(a+b x)^{10} (A+B x)}{x^{16}} \, dx=-\frac {15015 \, B b^{10} x^{11} + 4004 \, A a^{10} + 12012 \, {\left (10 \, B a b^{9} + A b^{10}\right )} x^{10} + 50050 \, {\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{9} + 128700 \, {\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{8} + 225225 \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} + 280280 \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} + 252252 \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} + 163800 \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} + 75075 \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} + 23100 \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} + 4290 \, {\left (B a^{10} + 10 \, A a^{9} b\right )} x}{60060 \, x^{15}} \]

[In]

integrate((b*x+a)^10*(B*x+A)/x^16,x, algorithm="maxima")

[Out]

-1/60060*(15015*B*b^10*x^11 + 4004*A*a^10 + 12012*(10*B*a*b^9 + A*b^10)*x^10 + 50050*(9*B*a^2*b^8 + 2*A*a*b^9)
*x^9 + 128700*(8*B*a^3*b^7 + 3*A*a^2*b^8)*x^8 + 225225*(7*B*a^4*b^6 + 4*A*a^3*b^7)*x^7 + 280280*(6*B*a^5*b^5 +
5*A*a^4*b^6)*x^6 + 252252*(5*B*a^6*b^4 + 6*A*a^5*b^5)*x^5 + 163800*(4*B*a^7*b^3 + 7*A*a^6*b^4)*x^4 + 75075*(3
*B*a^8*b^2 + 8*A*a^7*b^3)*x^3 + 23100*(2*B*a^9*b + 9*A*a^8*b^2)*x^2 + 4290*(B*a^10 + 10*A*a^9*b)*x)/x^15

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 243 vs. \(2 (120) = 240\).

Time = 0.30 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.87 \[ \int \frac {(a+b x)^{10} (A+B x)}{x^{16}} \, dx=-\frac {15015 \, B b^{10} x^{11} + 120120 \, B a b^{9} x^{10} + 12012 \, A b^{10} x^{10} + 450450 \, B a^{2} b^{8} x^{9} + 100100 \, A a b^{9} x^{9} + 1029600 \, B a^{3} b^{7} x^{8} + 386100 \, A a^{2} b^{8} x^{8} + 1576575 \, B a^{4} b^{6} x^{7} + 900900 \, A a^{3} b^{7} x^{7} + 1681680 \, B a^{5} b^{5} x^{6} + 1401400 \, A a^{4} b^{6} x^{6} + 1261260 \, B a^{6} b^{4} x^{5} + 1513512 \, A a^{5} b^{5} x^{5} + 655200 \, B a^{7} b^{3} x^{4} + 1146600 \, A a^{6} b^{4} x^{4} + 225225 \, B a^{8} b^{2} x^{3} + 600600 \, A a^{7} b^{3} x^{3} + 46200 \, B a^{9} b x^{2} + 207900 \, A a^{8} b^{2} x^{2} + 4290 \, B a^{10} x + 42900 \, A a^{9} b x + 4004 \, A a^{10}}{60060 \, x^{15}} \]

[In]

integrate((b*x+a)^10*(B*x+A)/x^16,x, algorithm="giac")

[Out]

-1/60060*(15015*B*b^10*x^11 + 120120*B*a*b^9*x^10 + 12012*A*b^10*x^10 + 450450*B*a^2*b^8*x^9 + 100100*A*a*b^9*
x^9 + 1029600*B*a^3*b^7*x^8 + 386100*A*a^2*b^8*x^8 + 1576575*B*a^4*b^6*x^7 + 900900*A*a^3*b^7*x^7 + 1681680*B*
a^5*b^5*x^6 + 1401400*A*a^4*b^6*x^6 + 1261260*B*a^6*b^4*x^5 + 1513512*A*a^5*b^5*x^5 + 655200*B*a^7*b^3*x^4 + 1
146600*A*a^6*b^4*x^4 + 225225*B*a^8*b^2*x^3 + 600600*A*a^7*b^3*x^3 + 46200*B*a^9*b*x^2 + 207900*A*a^8*b^2*x^2
+ 4290*B*a^10*x + 42900*A*a^9*b*x + 4004*A*a^10)/x^15

Mupad [B] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.81 \[ \int \frac {(a+b x)^{10} (A+B x)}{x^{16}} \, dx=-\frac {x\,\left (\frac {B\,a^{10}}{14}+\frac {5\,A\,b\,a^9}{7}\right )+\frac {A\,a^{10}}{15}+x^9\,\left (\frac {15\,B\,a^2\,b^8}{2}+\frac {5\,A\,a\,b^9}{3}\right )+x^2\,\left (\frac {10\,B\,a^9\,b}{13}+\frac {45\,A\,a^8\,b^2}{13}\right )+x^{10}\,\left (\frac {A\,b^{10}}{5}+2\,B\,a\,b^9\right )+x^3\,\left (\frac {15\,B\,a^8\,b^2}{4}+10\,A\,a^7\,b^3\right )+x^6\,\left (28\,B\,a^5\,b^5+\frac {70\,A\,a^4\,b^6}{3}\right )+x^7\,\left (\frac {105\,B\,a^4\,b^6}{4}+15\,A\,a^3\,b^7\right )+x^5\,\left (21\,B\,a^6\,b^4+\frac {126\,A\,a^5\,b^5}{5}\right )+x^8\,\left (\frac {120\,B\,a^3\,b^7}{7}+\frac {45\,A\,a^2\,b^8}{7}\right )+x^4\,\left (\frac {120\,B\,a^7\,b^3}{11}+\frac {210\,A\,a^6\,b^4}{11}\right )+\frac {B\,b^{10}\,x^{11}}{4}}{x^{15}} \]

[In]

int(((A + B*x)*(a + b*x)^10)/x^16,x)

[Out]

-(x*((B*a^10)/14 + (5*A*a^9*b)/7) + (A*a^10)/15 + x^9*((15*B*a^2*b^8)/2 + (5*A*a*b^9)/3) + x^2*((45*A*a^8*b^2)
/13 + (10*B*a^9*b)/13) + x^10*((A*b^10)/5 + 2*B*a*b^9) + x^3*(10*A*a^7*b^3 + (15*B*a^8*b^2)/4) + x^6*((70*A*a^
4*b^6)/3 + 28*B*a^5*b^5) + x^7*(15*A*a^3*b^7 + (105*B*a^4*b^6)/4) + x^5*((126*A*a^5*b^5)/5 + 21*B*a^6*b^4) + x
^8*((45*A*a^2*b^8)/7 + (120*B*a^3*b^7)/7) + x^4*((210*A*a^6*b^4)/11 + (120*B*a^7*b^3)/11) + (B*b^10*x^11)/4)/x
^15