3.167 \(\int \frac{(a+b x)^{10} (A+B x)}{x^{20}} \, dx\)

Optimal. Leaf size=227 \[ -\frac{15 a^7 b^2 (3 a B+8 A b)}{16 x^{16}}-\frac{2 a^6 b^3 (4 a B+7 A b)}{x^{15}}-\frac{3 a^5 b^4 (5 a B+6 A b)}{x^{14}}-\frac{42 a^4 b^5 (6 a B+5 A b)}{13 x^{13}}-\frac{5 a^3 b^6 (7 a B+4 A b)}{2 x^{12}}-\frac{15 a^2 b^7 (8 a B+3 A b)}{11 x^{11}}-\frac{a^9 (a B+10 A b)}{18 x^{18}}-\frac{5 a^8 b (2 a B+9 A b)}{17 x^{17}}-\frac{a^{10} A}{19 x^{19}}-\frac{a b^8 (9 a B+2 A b)}{2 x^{10}}-\frac{b^9 (10 a B+A b)}{9 x^9}-\frac{b^{10} B}{8 x^8} \]

[Out]

-(a^10*A)/(19*x^19) - (a^9*(10*A*b + a*B))/(18*x^18) - (5*a^8*b*(9*A*b + 2*a*B))/(17*x^17) - (15*a^7*b^2*(8*A*
b + 3*a*B))/(16*x^16) - (2*a^6*b^3*(7*A*b + 4*a*B))/x^15 - (3*a^5*b^4*(6*A*b + 5*a*B))/x^14 - (42*a^4*b^5*(5*A
*b + 6*a*B))/(13*x^13) - (5*a^3*b^6*(4*A*b + 7*a*B))/(2*x^12) - (15*a^2*b^7*(3*A*b + 8*a*B))/(11*x^11) - (a*b^
8*(2*A*b + 9*a*B))/(2*x^10) - (b^9*(A*b + 10*a*B))/(9*x^9) - (b^10*B)/(8*x^8)

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Rubi [A]  time = 0.128833, antiderivative size = 227, 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} \[ -\frac{15 a^7 b^2 (3 a B+8 A b)}{16 x^{16}}-\frac{2 a^6 b^3 (4 a B+7 A b)}{x^{15}}-\frac{3 a^5 b^4 (5 a B+6 A b)}{x^{14}}-\frac{42 a^4 b^5 (6 a B+5 A b)}{13 x^{13}}-\frac{5 a^3 b^6 (7 a B+4 A b)}{2 x^{12}}-\frac{15 a^2 b^7 (8 a B+3 A b)}{11 x^{11}}-\frac{a^9 (a B+10 A b)}{18 x^{18}}-\frac{5 a^8 b (2 a B+9 A b)}{17 x^{17}}-\frac{a^{10} A}{19 x^{19}}-\frac{a b^8 (9 a B+2 A b)}{2 x^{10}}-\frac{b^9 (10 a B+A b)}{9 x^9}-\frac{b^{10} B}{8 x^8} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^10*(A + B*x))/x^20,x]

[Out]

-(a^10*A)/(19*x^19) - (a^9*(10*A*b + a*B))/(18*x^18) - (5*a^8*b*(9*A*b + 2*a*B))/(17*x^17) - (15*a^7*b^2*(8*A*
b + 3*a*B))/(16*x^16) - (2*a^6*b^3*(7*A*b + 4*a*B))/x^15 - (3*a^5*b^4*(6*A*b + 5*a*B))/x^14 - (42*a^4*b^5*(5*A
*b + 6*a*B))/(13*x^13) - (5*a^3*b^6*(4*A*b + 7*a*B))/(2*x^12) - (15*a^2*b^7*(3*A*b + 8*a*B))/(11*x^11) - (a*b^
8*(2*A*b + 9*a*B))/(2*x^10) - (b^9*(A*b + 10*a*B))/(9*x^9) - (b^10*B)/(8*x^8)

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

Mathematica [A]  time = 0.0643007, size = 220, normalized size = 0.97 \[ -\frac{45 a^8 b^2 (16 A+17 B x)}{272 x^{17}}-\frac{a^7 b^3 (15 A+16 B x)}{2 x^{16}}-\frac{a^6 b^4 (14 A+15 B x)}{x^{15}}-\frac{18 a^5 b^5 (13 A+14 B x)}{13 x^{14}}-\frac{35 a^4 b^6 (12 A+13 B x)}{26 x^{13}}-\frac{10 a^3 b^7 (11 A+12 B x)}{11 x^{12}}-\frac{9 a^2 b^8 (10 A+11 B x)}{22 x^{11}}-\frac{5 a^9 b (17 A+18 B x)}{153 x^{18}}-\frac{a^{10} (18 A+19 B x)}{342 x^{19}}-\frac{a b^9 (9 A+10 B x)}{9 x^{10}}-\frac{b^{10} (8 A+9 B x)}{72 x^9} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^10*(A + B*x))/x^20,x]

[Out]

-(b^10*(8*A + 9*B*x))/(72*x^9) - (a*b^9*(9*A + 10*B*x))/(9*x^10) - (9*a^2*b^8*(10*A + 11*B*x))/(22*x^11) - (10
*a^3*b^7*(11*A + 12*B*x))/(11*x^12) - (35*a^4*b^6*(12*A + 13*B*x))/(26*x^13) - (18*a^5*b^5*(13*A + 14*B*x))/(1
3*x^14) - (a^6*b^4*(14*A + 15*B*x))/x^15 - (a^7*b^3*(15*A + 16*B*x))/(2*x^16) - (45*a^8*b^2*(16*A + 17*B*x))/(
272*x^17) - (5*a^9*b*(17*A + 18*B*x))/(153*x^18) - (a^10*(18*A + 19*B*x))/(342*x^19)

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Maple [A]  time = 0.008, size = 208, normalized size = 0.9 \begin{align*} -{\frac{A{a}^{10}}{19\,{x}^{19}}}-{\frac{{a}^{9} \left ( 10\,Ab+Ba \right ) }{18\,{x}^{18}}}-{\frac{5\,{a}^{8}b \left ( 9\,Ab+2\,Ba \right ) }{17\,{x}^{17}}}-{\frac{15\,{a}^{7}{b}^{2} \left ( 8\,Ab+3\,Ba \right ) }{16\,{x}^{16}}}-2\,{\frac{{a}^{6}{b}^{3} \left ( 7\,Ab+4\,Ba \right ) }{{x}^{15}}}-3\,{\frac{{a}^{5}{b}^{4} \left ( 6\,Ab+5\,Ba \right ) }{{x}^{14}}}-{\frac{42\,{a}^{4}{b}^{5} \left ( 5\,Ab+6\,Ba \right ) }{13\,{x}^{13}}}-{\frac{5\,{a}^{3}{b}^{6} \left ( 4\,Ab+7\,Ba \right ) }{2\,{x}^{12}}}-{\frac{15\,{a}^{2}{b}^{7} \left ( 3\,Ab+8\,Ba \right ) }{11\,{x}^{11}}}-{\frac{a{b}^{8} \left ( 2\,Ab+9\,Ba \right ) }{2\,{x}^{10}}}-{\frac{{b}^{9} \left ( Ab+10\,Ba \right ) }{9\,{x}^{9}}}-{\frac{B{b}^{10}}{8\,{x}^{8}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^10*(B*x+A)/x^20,x)

[Out]

-1/19*a^10*A/x^19-1/18*a^9*(10*A*b+B*a)/x^18-5/17*a^8*b*(9*A*b+2*B*a)/x^17-15/16*a^7*b^2*(8*A*b+3*B*a)/x^16-2*
a^6*b^3*(7*A*b+4*B*a)/x^15-3*a^5*b^4*(6*A*b+5*B*a)/x^14-42/13*a^4*b^5*(5*A*b+6*B*a)/x^13-5/2*a^3*b^6*(4*A*b+7*
B*a)/x^12-15/11*a^2*b^7*(3*A*b+8*B*a)/x^11-1/2*a*b^8*(2*A*b+9*B*a)/x^10-1/9*b^9*(A*b+10*B*a)/x^9-1/8*b^10*B/x^
8

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Maxima [A]  time = 1.03018, size = 328, normalized size = 1.44 \begin{align*} -\frac{831402 \, B b^{10} x^{11} + 350064 \, A a^{10} + 739024 \,{\left (10 \, B a b^{9} + A b^{10}\right )} x^{10} + 3325608 \,{\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{9} + 9069840 \,{\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{8} + 16628040 \,{\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} + 21488544 \,{\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} + 19953648 \,{\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} + 13302432 \,{\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} + 6235515 \,{\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} + 1956240 \,{\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} + 369512 \,{\left (B a^{10} + 10 \, A a^{9} b\right )} x}{6651216 \, x^{19}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10*(B*x+A)/x^20,x, algorithm="maxima")

[Out]

-1/6651216*(831402*B*b^10*x^11 + 350064*A*a^10 + 739024*(10*B*a*b^9 + A*b^10)*x^10 + 3325608*(9*B*a^2*b^8 + 2*
A*a*b^9)*x^9 + 9069840*(8*B*a^3*b^7 + 3*A*a^2*b^8)*x^8 + 16628040*(7*B*a^4*b^6 + 4*A*a^3*b^7)*x^7 + 21488544*(
6*B*a^5*b^5 + 5*A*a^4*b^6)*x^6 + 19953648*(5*B*a^6*b^4 + 6*A*a^5*b^5)*x^5 + 13302432*(4*B*a^7*b^3 + 7*A*a^6*b^
4)*x^4 + 6235515*(3*B*a^8*b^2 + 8*A*a^7*b^3)*x^3 + 1956240*(2*B*a^9*b + 9*A*a^8*b^2)*x^2 + 369512*(B*a^10 + 10
*A*a^9*b)*x)/x^19

________________________________________________________________________________________

Fricas [A]  time = 1.40274, size = 620, normalized size = 2.73 \begin{align*} -\frac{831402 \, B b^{10} x^{11} + 350064 \, A a^{10} + 739024 \,{\left (10 \, B a b^{9} + A b^{10}\right )} x^{10} + 3325608 \,{\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{9} + 9069840 \,{\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{8} + 16628040 \,{\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} + 21488544 \,{\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} + 19953648 \,{\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} + 13302432 \,{\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} + 6235515 \,{\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} + 1956240 \,{\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} + 369512 \,{\left (B a^{10} + 10 \, A a^{9} b\right )} x}{6651216 \, x^{19}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10*(B*x+A)/x^20,x, algorithm="fricas")

[Out]

-1/6651216*(831402*B*b^10*x^11 + 350064*A*a^10 + 739024*(10*B*a*b^9 + A*b^10)*x^10 + 3325608*(9*B*a^2*b^8 + 2*
A*a*b^9)*x^9 + 9069840*(8*B*a^3*b^7 + 3*A*a^2*b^8)*x^8 + 16628040*(7*B*a^4*b^6 + 4*A*a^3*b^7)*x^7 + 21488544*(
6*B*a^5*b^5 + 5*A*a^4*b^6)*x^6 + 19953648*(5*B*a^6*b^4 + 6*A*a^5*b^5)*x^5 + 13302432*(4*B*a^7*b^3 + 7*A*a^6*b^
4)*x^4 + 6235515*(3*B*a^8*b^2 + 8*A*a^7*b^3)*x^3 + 1956240*(2*B*a^9*b + 9*A*a^8*b^2)*x^2 + 369512*(B*a^10 + 10
*A*a^9*b)*x)/x^19

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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**20,x)

[Out]

Timed out

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Giac [A]  time = 1.19935, size = 328, normalized size = 1.44 \begin{align*} -\frac{831402 \, B b^{10} x^{11} + 7390240 \, B a b^{9} x^{10} + 739024 \, A b^{10} x^{10} + 29930472 \, B a^{2} b^{8} x^{9} + 6651216 \, A a b^{9} x^{9} + 72558720 \, B a^{3} b^{7} x^{8} + 27209520 \, A a^{2} b^{8} x^{8} + 116396280 \, B a^{4} b^{6} x^{7} + 66512160 \, A a^{3} b^{7} x^{7} + 128931264 \, B a^{5} b^{5} x^{6} + 107442720 \, A a^{4} b^{6} x^{6} + 99768240 \, B a^{6} b^{4} x^{5} + 119721888 \, A a^{5} b^{5} x^{5} + 53209728 \, B a^{7} b^{3} x^{4} + 93117024 \, A a^{6} b^{4} x^{4} + 18706545 \, B a^{8} b^{2} x^{3} + 49884120 \, A a^{7} b^{3} x^{3} + 3912480 \, B a^{9} b x^{2} + 17606160 \, A a^{8} b^{2} x^{2} + 369512 \, B a^{10} x + 3695120 \, A a^{9} b x + 350064 \, A a^{10}}{6651216 \, x^{19}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10*(B*x+A)/x^20,x, algorithm="giac")

[Out]

-1/6651216*(831402*B*b^10*x^11 + 7390240*B*a*b^9*x^10 + 739024*A*b^10*x^10 + 29930472*B*a^2*b^8*x^9 + 6651216*
A*a*b^9*x^9 + 72558720*B*a^3*b^7*x^8 + 27209520*A*a^2*b^8*x^8 + 116396280*B*a^4*b^6*x^7 + 66512160*A*a^3*b^7*x
^7 + 128931264*B*a^5*b^5*x^6 + 107442720*A*a^4*b^6*x^6 + 99768240*B*a^6*b^4*x^5 + 119721888*A*a^5*b^5*x^5 + 53
209728*B*a^7*b^3*x^4 + 93117024*A*a^6*b^4*x^4 + 18706545*B*a^8*b^2*x^3 + 49884120*A*a^7*b^3*x^3 + 3912480*B*a^
9*b*x^2 + 17606160*A*a^8*b^2*x^2 + 369512*B*a^10*x + 3695120*A*a^9*b*x + 350064*A*a^10)/x^19