3.168 \(\int \frac{(a+b x)^{10} (A+B x)}{x^{21}} \, dx\)

Optimal. Leaf size=229 \[ -\frac{15 a^7 b^2 (3 a B+8 A b)}{17 x^{17}}-\frac{15 a^6 b^3 (4 a B+7 A b)}{8 x^{16}}-\frac{14 a^5 b^4 (5 a B+6 A b)}{5 x^{15}}-\frac{3 a^4 b^5 (6 a B+5 A b)}{x^{14}}-\frac{30 a^3 b^6 (7 a B+4 A b)}{13 x^{13}}-\frac{5 a^2 b^7 (8 a B+3 A b)}{4 x^{12}}-\frac{a^9 (a B+10 A b)}{19 x^{19}}-\frac{5 a^8 b (2 a B+9 A b)}{18 x^{18}}-\frac{a^{10} A}{20 x^{20}}-\frac{5 a b^8 (9 a B+2 A b)}{11 x^{11}}-\frac{b^9 (10 a B+A b)}{10 x^{10}}-\frac{b^{10} B}{9 x^9} \]

[Out]

-(a^10*A)/(20*x^20) - (a^9*(10*A*b + a*B))/(19*x^19) - (5*a^8*b*(9*A*b + 2*a*B))/(18*x^18) - (15*a^7*b^2*(8*A*
b + 3*a*B))/(17*x^17) - (15*a^6*b^3*(7*A*b + 4*a*B))/(8*x^16) - (14*a^5*b^4*(6*A*b + 5*a*B))/(5*x^15) - (3*a^4
*b^5*(5*A*b + 6*a*B))/x^14 - (30*a^3*b^6*(4*A*b + 7*a*B))/(13*x^13) - (5*a^2*b^7*(3*A*b + 8*a*B))/(4*x^12) - (
5*a*b^8*(2*A*b + 9*a*B))/(11*x^11) - (b^9*(A*b + 10*a*B))/(10*x^10) - (b^10*B)/(9*x^9)

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Rubi [A]  time = 0.144066, antiderivative size = 229, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {76} \[ -\frac{15 a^7 b^2 (3 a B+8 A b)}{17 x^{17}}-\frac{15 a^6 b^3 (4 a B+7 A b)}{8 x^{16}}-\frac{14 a^5 b^4 (5 a B+6 A b)}{5 x^{15}}-\frac{3 a^4 b^5 (6 a B+5 A b)}{x^{14}}-\frac{30 a^3 b^6 (7 a B+4 A b)}{13 x^{13}}-\frac{5 a^2 b^7 (8 a B+3 A b)}{4 x^{12}}-\frac{a^9 (a B+10 A b)}{19 x^{19}}-\frac{5 a^8 b (2 a B+9 A b)}{18 x^{18}}-\frac{a^{10} A}{20 x^{20}}-\frac{5 a b^8 (9 a B+2 A b)}{11 x^{11}}-\frac{b^9 (10 a B+A b)}{10 x^{10}}-\frac{b^{10} B}{9 x^9} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^10*(A + B*x))/x^21,x]

[Out]

-(a^10*A)/(20*x^20) - (a^9*(10*A*b + a*B))/(19*x^19) - (5*a^8*b*(9*A*b + 2*a*B))/(18*x^18) - (15*a^7*b^2*(8*A*
b + 3*a*B))/(17*x^17) - (15*a^6*b^3*(7*A*b + 4*a*B))/(8*x^16) - (14*a^5*b^4*(6*A*b + 5*a*B))/(5*x^15) - (3*a^4
*b^5*(5*A*b + 6*a*B))/x^14 - (30*a^3*b^6*(4*A*b + 7*a*B))/(13*x^13) - (5*a^2*b^7*(3*A*b + 8*a*B))/(4*x^12) - (
5*a*b^8*(2*A*b + 9*a*B))/(11*x^11) - (b^9*(A*b + 10*a*B))/(10*x^10) - (b^10*B)/(9*x^9)

Rule 76

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*} \int \frac{(a+b x)^{10} (A+B x)}{x^{21}} \, dx &=\int \left (\frac{a^{10} A}{x^{21}}+\frac{a^9 (10 A b+a B)}{x^{20}}+\frac{5 a^8 b (9 A b+2 a B)}{x^{19}}+\frac{15 a^7 b^2 (8 A b+3 a B)}{x^{18}}+\frac{30 a^6 b^3 (7 A b+4 a B)}{x^{17}}+\frac{42 a^5 b^4 (6 A b+5 a B)}{x^{16}}+\frac{42 a^4 b^5 (5 A b+6 a B)}{x^{15}}+\frac{30 a^3 b^6 (4 A b+7 a B)}{x^{14}}+\frac{15 a^2 b^7 (3 A b+8 a B)}{x^{13}}+\frac{5 a b^8 (2 A b+9 a B)}{x^{12}}+\frac{b^9 (A b+10 a B)}{x^{11}}+\frac{b^{10} B}{x^{10}}\right ) \, dx\\ &=-\frac{a^{10} A}{20 x^{20}}-\frac{a^9 (10 A b+a B)}{19 x^{19}}-\frac{5 a^8 b (9 A b+2 a B)}{18 x^{18}}-\frac{15 a^7 b^2 (8 A b+3 a B)}{17 x^{17}}-\frac{15 a^6 b^3 (7 A b+4 a B)}{8 x^{16}}-\frac{14 a^5 b^4 (6 A b+5 a B)}{5 x^{15}}-\frac{3 a^4 b^5 (5 A b+6 a B)}{x^{14}}-\frac{30 a^3 b^6 (4 A b+7 a B)}{13 x^{13}}-\frac{5 a^2 b^7 (3 A b+8 a B)}{4 x^{12}}-\frac{5 a b^8 (2 A b+9 a B)}{11 x^{11}}-\frac{b^9 (A b+10 a B)}{10 x^{10}}-\frac{b^{10} B}{9 x^9}\\ \end{align*}

Mathematica [A]  time = 0.0676566, size = 222, normalized size = 0.97 \[ -\frac{5 a^8 b^2 (17 A+18 B x)}{34 x^{18}}-\frac{15 a^7 b^3 (16 A+17 B x)}{34 x^{17}}-\frac{7 a^6 b^4 (15 A+16 B x)}{8 x^{16}}-\frac{6 a^5 b^5 (14 A+15 B x)}{5 x^{15}}-\frac{15 a^4 b^6 (13 A+14 B x)}{13 x^{14}}-\frac{10 a^3 b^7 (12 A+13 B x)}{13 x^{13}}-\frac{15 a^2 b^8 (11 A+12 B x)}{44 x^{12}}-\frac{5 a^9 b (18 A+19 B x)}{171 x^{19}}-\frac{a^{10} (19 A+20 B x)}{380 x^{20}}-\frac{a b^9 (10 A+11 B x)}{11 x^{11}}-\frac{b^{10} (9 A+10 B x)}{90 x^{10}} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^10*(A + B*x))/x^21,x]

[Out]

-(b^10*(9*A + 10*B*x))/(90*x^10) - (a*b^9*(10*A + 11*B*x))/(11*x^11) - (15*a^2*b^8*(11*A + 12*B*x))/(44*x^12)
- (10*a^3*b^7*(12*A + 13*B*x))/(13*x^13) - (15*a^4*b^6*(13*A + 14*B*x))/(13*x^14) - (6*a^5*b^5*(14*A + 15*B*x)
)/(5*x^15) - (7*a^6*b^4*(15*A + 16*B*x))/(8*x^16) - (15*a^7*b^3*(16*A + 17*B*x))/(34*x^17) - (5*a^8*b^2*(17*A
+ 18*B*x))/(34*x^18) - (5*a^9*b*(18*A + 19*B*x))/(171*x^19) - (a^10*(19*A + 20*B*x))/(380*x^20)

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Maple [A]  time = 0.009, size = 208, normalized size = 0.9 \begin{align*} -{\frac{A{a}^{10}}{20\,{x}^{20}}}-{\frac{{a}^{9} \left ( 10\,Ab+Ba \right ) }{19\,{x}^{19}}}-{\frac{5\,{a}^{8}b \left ( 9\,Ab+2\,Ba \right ) }{18\,{x}^{18}}}-{\frac{15\,{a}^{7}{b}^{2} \left ( 8\,Ab+3\,Ba \right ) }{17\,{x}^{17}}}-{\frac{15\,{a}^{6}{b}^{3} \left ( 7\,Ab+4\,Ba \right ) }{8\,{x}^{16}}}-{\frac{14\,{a}^{5}{b}^{4} \left ( 6\,Ab+5\,Ba \right ) }{5\,{x}^{15}}}-3\,{\frac{{a}^{4}{b}^{5} \left ( 5\,Ab+6\,Ba \right ) }{{x}^{14}}}-{\frac{30\,{a}^{3}{b}^{6} \left ( 4\,Ab+7\,Ba \right ) }{13\,{x}^{13}}}-{\frac{5\,{a}^{2}{b}^{7} \left ( 3\,Ab+8\,Ba \right ) }{4\,{x}^{12}}}-{\frac{5\,a{b}^{8} \left ( 2\,Ab+9\,Ba \right ) }{11\,{x}^{11}}}-{\frac{{b}^{9} \left ( Ab+10\,Ba \right ) }{10\,{x}^{10}}}-{\frac{B{b}^{10}}{9\,{x}^{9}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^10*(B*x+A)/x^21,x)

[Out]

-1/20*a^10*A/x^20-1/19*a^9*(10*A*b+B*a)/x^19-5/18*a^8*b*(9*A*b+2*B*a)/x^18-15/17*a^7*b^2*(8*A*b+3*B*a)/x^17-15
/8*a^6*b^3*(7*A*b+4*B*a)/x^16-14/5*a^5*b^4*(6*A*b+5*B*a)/x^15-3*a^4*b^5*(5*A*b+6*B*a)/x^14-30/13*a^3*b^6*(4*A*
b+7*B*a)/x^13-5/4*a^2*b^7*(3*A*b+8*B*a)/x^12-5/11*a*b^8*(2*A*b+9*B*a)/x^11-1/10*b^9*(A*b+10*B*a)/x^10-1/9*b^10
*B/x^9

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Maxima [A]  time = 1.03884, size = 328, normalized size = 1.43 \begin{align*} -\frac{1847560 \, B b^{10} x^{11} + 831402 \, A a^{10} + 1662804 \,{\left (10 \, B a b^{9} + A b^{10}\right )} x^{10} + 7558200 \,{\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{9} + 20785050 \,{\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{8} + 38372400 \,{\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} + 49884120 \,{\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} + 46558512 \,{\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} + 31177575 \,{\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} + 14671800 \,{\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} + 4618900 \,{\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} + 875160 \,{\left (B a^{10} + 10 \, A a^{9} b\right )} x}{16628040 \, x^{20}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10*(B*x+A)/x^21,x, algorithm="maxima")

[Out]

-1/16628040*(1847560*B*b^10*x^11 + 831402*A*a^10 + 1662804*(10*B*a*b^9 + A*b^10)*x^10 + 7558200*(9*B*a^2*b^8 +
2*A*a*b^9)*x^9 + 20785050*(8*B*a^3*b^7 + 3*A*a^2*b^8)*x^8 + 38372400*(7*B*a^4*b^6 + 4*A*a^3*b^7)*x^7 + 498841
20*(6*B*a^5*b^5 + 5*A*a^4*b^6)*x^6 + 46558512*(5*B*a^6*b^4 + 6*A*a^5*b^5)*x^5 + 31177575*(4*B*a^7*b^3 + 7*A*a^
6*b^4)*x^4 + 14671800*(3*B*a^8*b^2 + 8*A*a^7*b^3)*x^3 + 4618900*(2*B*a^9*b + 9*A*a^8*b^2)*x^2 + 875160*(B*a^10
+ 10*A*a^9*b)*x)/x^20

________________________________________________________________________________________

Fricas [A]  time = 1.44187, size = 626, normalized size = 2.73 \begin{align*} -\frac{1847560 \, B b^{10} x^{11} + 831402 \, A a^{10} + 1662804 \,{\left (10 \, B a b^{9} + A b^{10}\right )} x^{10} + 7558200 \,{\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{9} + 20785050 \,{\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{8} + 38372400 \,{\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} + 49884120 \,{\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} + 46558512 \,{\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} + 31177575 \,{\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} + 14671800 \,{\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} + 4618900 \,{\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} + 875160 \,{\left (B a^{10} + 10 \, A a^{9} b\right )} x}{16628040 \, x^{20}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10*(B*x+A)/x^21,x, algorithm="fricas")

[Out]

-1/16628040*(1847560*B*b^10*x^11 + 831402*A*a^10 + 1662804*(10*B*a*b^9 + A*b^10)*x^10 + 7558200*(9*B*a^2*b^8 +
2*A*a*b^9)*x^9 + 20785050*(8*B*a^3*b^7 + 3*A*a^2*b^8)*x^8 + 38372400*(7*B*a^4*b^6 + 4*A*a^3*b^7)*x^7 + 498841
20*(6*B*a^5*b^5 + 5*A*a^4*b^6)*x^6 + 46558512*(5*B*a^6*b^4 + 6*A*a^5*b^5)*x^5 + 31177575*(4*B*a^7*b^3 + 7*A*a^
6*b^4)*x^4 + 14671800*(3*B*a^8*b^2 + 8*A*a^7*b^3)*x^3 + 4618900*(2*B*a^9*b + 9*A*a^8*b^2)*x^2 + 875160*(B*a^10
+ 10*A*a^9*b)*x)/x^20

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**10*(B*x+A)/x**21,x)

[Out]

Timed out

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Giac [A]  time = 1.20952, size = 328, normalized size = 1.43 \begin{align*} -\frac{1847560 \, B b^{10} x^{11} + 16628040 \, B a b^{9} x^{10} + 1662804 \, A b^{10} x^{10} + 68023800 \, B a^{2} b^{8} x^{9} + 15116400 \, A a b^{9} x^{9} + 166280400 \, B a^{3} b^{7} x^{8} + 62355150 \, A a^{2} b^{8} x^{8} + 268606800 \, B a^{4} b^{6} x^{7} + 153489600 \, A a^{3} b^{7} x^{7} + 299304720 \, B a^{5} b^{5} x^{6} + 249420600 \, A a^{4} b^{6} x^{6} + 232792560 \, B a^{6} b^{4} x^{5} + 279351072 \, A a^{5} b^{5} x^{5} + 124710300 \, B a^{7} b^{3} x^{4} + 218243025 \, A a^{6} b^{4} x^{4} + 44015400 \, B a^{8} b^{2} x^{3} + 117374400 \, A a^{7} b^{3} x^{3} + 9237800 \, B a^{9} b x^{2} + 41570100 \, A a^{8} b^{2} x^{2} + 875160 \, B a^{10} x + 8751600 \, A a^{9} b x + 831402 \, A a^{10}}{16628040 \, x^{20}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10*(B*x+A)/x^21,x, algorithm="giac")

[Out]

-1/16628040*(1847560*B*b^10*x^11 + 16628040*B*a*b^9*x^10 + 1662804*A*b^10*x^10 + 68023800*B*a^2*b^8*x^9 + 1511
6400*A*a*b^9*x^9 + 166280400*B*a^3*b^7*x^8 + 62355150*A*a^2*b^8*x^8 + 268606800*B*a^4*b^6*x^7 + 153489600*A*a^
3*b^7*x^7 + 299304720*B*a^5*b^5*x^6 + 249420600*A*a^4*b^6*x^6 + 232792560*B*a^6*b^4*x^5 + 279351072*A*a^5*b^5*
x^5 + 124710300*B*a^7*b^3*x^4 + 218243025*A*a^6*b^4*x^4 + 44015400*B*a^8*b^2*x^3 + 117374400*A*a^7*b^3*x^3 + 9
237800*B*a^9*b*x^2 + 41570100*A*a^8*b^2*x^2 + 875160*B*a^10*x + 8751600*A*a^9*b*x + 831402*A*a^10)/x^20