3.166 \(\int \frac{(a+b x)^{10} (A+B x)}{x^{19}} \, dx\)

Optimal. Leaf size=229 \[ -\frac{a^7 b^2 (3 a B+8 A b)}{x^{15}}-\frac{15 a^6 b^3 (4 a B+7 A b)}{7 x^{14}}-\frac{42 a^5 b^4 (5 a B+6 A b)}{13 x^{13}}-\frac{7 a^4 b^5 (6 a B+5 A b)}{2 x^{12}}-\frac{30 a^3 b^6 (7 a B+4 A b)}{11 x^{11}}-\frac{3 a^2 b^7 (8 a B+3 A b)}{2 x^{10}}-\frac{a^9 (a B+10 A b)}{17 x^{17}}-\frac{5 a^8 b (2 a B+9 A b)}{16 x^{16}}-\frac{a^{10} A}{18 x^{18}}-\frac{5 a b^8 (9 a B+2 A b)}{9 x^9}-\frac{b^9 (10 a B+A b)}{8 x^8}-\frac{b^{10} B}{7 x^7} \]

[Out]

-(a^10*A)/(18*x^18) - (a^9*(10*A*b + a*B))/(17*x^17) - (5*a^8*b*(9*A*b + 2*a*B))/(16*x^16) - (a^7*b^2*(8*A*b +
3*a*B))/x^15 - (15*a^6*b^3*(7*A*b + 4*a*B))/(7*x^14) - (42*a^5*b^4*(6*A*b + 5*a*B))/(13*x^13) - (7*a^4*b^5*(5
*A*b + 6*a*B))/(2*x^12) - (30*a^3*b^6*(4*A*b + 7*a*B))/(11*x^11) - (3*a^2*b^7*(3*A*b + 8*a*B))/(2*x^10) - (5*a
*b^8*(2*A*b + 9*a*B))/(9*x^9) - (b^9*(A*b + 10*a*B))/(8*x^8) - (b^10*B)/(7*x^7)

________________________________________________________________________________________

Rubi [A]  time = 0.140205, 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} \[ -\frac{a^7 b^2 (3 a B+8 A b)}{x^{15}}-\frac{15 a^6 b^3 (4 a B+7 A b)}{7 x^{14}}-\frac{42 a^5 b^4 (5 a B+6 A b)}{13 x^{13}}-\frac{7 a^4 b^5 (6 a B+5 A b)}{2 x^{12}}-\frac{30 a^3 b^6 (7 a B+4 A b)}{11 x^{11}}-\frac{3 a^2 b^7 (8 a B+3 A b)}{2 x^{10}}-\frac{a^9 (a B+10 A b)}{17 x^{17}}-\frac{5 a^8 b (2 a B+9 A b)}{16 x^{16}}-\frac{a^{10} A}{18 x^{18}}-\frac{5 a b^8 (9 a B+2 A b)}{9 x^9}-\frac{b^9 (10 a B+A b)}{8 x^8}-\frac{b^{10} B}{7 x^7} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^10*(A + B*x))/x^19,x]

[Out]

-(a^10*A)/(18*x^18) - (a^9*(10*A*b + a*B))/(17*x^17) - (5*a^8*b*(9*A*b + 2*a*B))/(16*x^16) - (a^7*b^2*(8*A*b +
3*a*B))/x^15 - (15*a^6*b^3*(7*A*b + 4*a*B))/(7*x^14) - (42*a^5*b^4*(6*A*b + 5*a*B))/(13*x^13) - (7*a^4*b^5*(5
*A*b + 6*a*B))/(2*x^12) - (30*a^3*b^6*(4*A*b + 7*a*B))/(11*x^11) - (3*a^2*b^7*(3*A*b + 8*a*B))/(2*x^10) - (5*a
*b^8*(2*A*b + 9*a*B))/(9*x^9) - (b^9*(A*b + 10*a*B))/(8*x^8) - (b^10*B)/(7*x^7)

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

Mathematica [A]  time = 0.0677452, size = 222, normalized size = 0.97 \[ -\frac{3 a^8 b^2 (15 A+16 B x)}{16 x^{16}}-\frac{4 a^7 b^3 (14 A+15 B x)}{7 x^{15}}-\frac{15 a^6 b^4 (13 A+14 B x)}{13 x^{14}}-\frac{21 a^5 b^5 (12 A+13 B x)}{13 x^{13}}-\frac{35 a^4 b^6 (11 A+12 B x)}{22 x^{12}}-\frac{12 a^3 b^7 (10 A+11 B x)}{11 x^{11}}-\frac{a^2 b^8 (9 A+10 B x)}{2 x^{10}}-\frac{5 a^9 b (16 A+17 B x)}{136 x^{17}}-\frac{a^{10} (17 A+18 B x)}{306 x^{18}}-\frac{5 a b^9 (8 A+9 B x)}{36 x^9}-\frac{b^{10} (7 A+8 B x)}{56 x^8} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^10*(A + B*x))/x^19,x]

[Out]

-(b^10*(7*A + 8*B*x))/(56*x^8) - (5*a*b^9*(8*A + 9*B*x))/(36*x^9) - (a^2*b^8*(9*A + 10*B*x))/(2*x^10) - (12*a^
3*b^7*(10*A + 11*B*x))/(11*x^11) - (35*a^4*b^6*(11*A + 12*B*x))/(22*x^12) - (21*a^5*b^5*(12*A + 13*B*x))/(13*x
^13) - (15*a^6*b^4*(13*A + 14*B*x))/(13*x^14) - (4*a^7*b^3*(14*A + 15*B*x))/(7*x^15) - (3*a^8*b^2*(15*A + 16*B
*x))/(16*x^16) - (5*a^9*b*(16*A + 17*B*x))/(136*x^17) - (a^10*(17*A + 18*B*x))/(306*x^18)

________________________________________________________________________________________

Maple [A]  time = 0.008, size = 208, normalized size = 0.9 \begin{align*} -{\frac{A{a}^{10}}{18\,{x}^{18}}}-{\frac{{a}^{9} \left ( 10\,Ab+Ba \right ) }{17\,{x}^{17}}}-{\frac{5\,{a}^{8}b \left ( 9\,Ab+2\,Ba \right ) }{16\,{x}^{16}}}-{\frac{{a}^{7}{b}^{2} \left ( 8\,Ab+3\,Ba \right ) }{{x}^{15}}}-{\frac{15\,{a}^{6}{b}^{3} \left ( 7\,Ab+4\,Ba \right ) }{7\,{x}^{14}}}-{\frac{42\,{a}^{5}{b}^{4} \left ( 6\,Ab+5\,Ba \right ) }{13\,{x}^{13}}}-{\frac{7\,{a}^{4}{b}^{5} \left ( 5\,Ab+6\,Ba \right ) }{2\,{x}^{12}}}-{\frac{30\,{a}^{3}{b}^{6} \left ( 4\,Ab+7\,Ba \right ) }{11\,{x}^{11}}}-{\frac{3\,{a}^{2}{b}^{7} \left ( 3\,Ab+8\,Ba \right ) }{2\,{x}^{10}}}-{\frac{5\,a{b}^{8} \left ( 2\,Ab+9\,Ba \right ) }{9\,{x}^{9}}}-{\frac{{b}^{9} \left ( Ab+10\,Ba \right ) }{8\,{x}^{8}}}-{\frac{B{b}^{10}}{7\,{x}^{7}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^10*(B*x+A)/x^19,x)

[Out]

-1/18*a^10*A/x^18-1/17*a^9*(10*A*b+B*a)/x^17-5/16*a^8*b*(9*A*b+2*B*a)/x^16-a^7*b^2*(8*A*b+3*B*a)/x^15-15/7*a^6
*b^3*(7*A*b+4*B*a)/x^14-42/13*a^5*b^4*(6*A*b+5*B*a)/x^13-7/2*a^4*b^5*(5*A*b+6*B*a)/x^12-30/11*a^3*b^6*(4*A*b+7
*B*a)/x^11-3/2*a^2*b^7*(3*A*b+8*B*a)/x^10-5/9*a*b^8*(2*A*b+9*B*a)/x^9-1/8*b^9*(A*b+10*B*a)/x^8-1/7*b^10*B/x^7

________________________________________________________________________________________

Maxima [A]  time = 1.03544, size = 328, normalized size = 1.43 \begin{align*} -\frac{350064 \, B b^{10} x^{11} + 136136 \, A a^{10} + 306306 \,{\left (10 \, B a b^{9} + A b^{10}\right )} x^{10} + 1361360 \,{\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{9} + 3675672 \,{\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{8} + 6683040 \,{\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} + 8576568 \,{\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} + 7916832 \,{\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} + 5250960 \,{\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} + 2450448 \,{\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} + 765765 \,{\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} + 144144 \,{\left (B a^{10} + 10 \, A a^{9} b\right )} x}{2450448 \, x^{18}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10*(B*x+A)/x^19,x, algorithm="maxima")

[Out]

-1/2450448*(350064*B*b^10*x^11 + 136136*A*a^10 + 306306*(10*B*a*b^9 + A*b^10)*x^10 + 1361360*(9*B*a^2*b^8 + 2*
A*a*b^9)*x^9 + 3675672*(8*B*a^3*b^7 + 3*A*a^2*b^8)*x^8 + 6683040*(7*B*a^4*b^6 + 4*A*a^3*b^7)*x^7 + 8576568*(6*
B*a^5*b^5 + 5*A*a^4*b^6)*x^6 + 7916832*(5*B*a^6*b^4 + 6*A*a^5*b^5)*x^5 + 5250960*(4*B*a^7*b^3 + 7*A*a^6*b^4)*x
^4 + 2450448*(3*B*a^8*b^2 + 8*A*a^7*b^3)*x^3 + 765765*(2*B*a^9*b + 9*A*a^8*b^2)*x^2 + 144144*(B*a^10 + 10*A*a^
9*b)*x)/x^18

________________________________________________________________________________________

Fricas [A]  time = 1.42859, size = 613, normalized size = 2.68 \begin{align*} -\frac{350064 \, B b^{10} x^{11} + 136136 \, A a^{10} + 306306 \,{\left (10 \, B a b^{9} + A b^{10}\right )} x^{10} + 1361360 \,{\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{9} + 3675672 \,{\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{8} + 6683040 \,{\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} + 8576568 \,{\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} + 7916832 \,{\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} + 5250960 \,{\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} + 2450448 \,{\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} + 765765 \,{\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} + 144144 \,{\left (B a^{10} + 10 \, A a^{9} b\right )} x}{2450448 \, x^{18}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10*(B*x+A)/x^19,x, algorithm="fricas")

[Out]

-1/2450448*(350064*B*b^10*x^11 + 136136*A*a^10 + 306306*(10*B*a*b^9 + A*b^10)*x^10 + 1361360*(9*B*a^2*b^8 + 2*
A*a*b^9)*x^9 + 3675672*(8*B*a^3*b^7 + 3*A*a^2*b^8)*x^8 + 6683040*(7*B*a^4*b^6 + 4*A*a^3*b^7)*x^7 + 8576568*(6*
B*a^5*b^5 + 5*A*a^4*b^6)*x^6 + 7916832*(5*B*a^6*b^4 + 6*A*a^5*b^5)*x^5 + 5250960*(4*B*a^7*b^3 + 7*A*a^6*b^4)*x
^4 + 2450448*(3*B*a^8*b^2 + 8*A*a^7*b^3)*x^3 + 765765*(2*B*a^9*b + 9*A*a^8*b^2)*x^2 + 144144*(B*a^10 + 10*A*a^
9*b)*x)/x^18

________________________________________________________________________________________

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**19,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.23544, size = 328, normalized size = 1.43 \begin{align*} -\frac{350064 \, B b^{10} x^{11} + 3063060 \, B a b^{9} x^{10} + 306306 \, A b^{10} x^{10} + 12252240 \, B a^{2} b^{8} x^{9} + 2722720 \, A a b^{9} x^{9} + 29405376 \, B a^{3} b^{7} x^{8} + 11027016 \, A a^{2} b^{8} x^{8} + 46781280 \, B a^{4} b^{6} x^{7} + 26732160 \, A a^{3} b^{7} x^{7} + 51459408 \, B a^{5} b^{5} x^{6} + 42882840 \, A a^{4} b^{6} x^{6} + 39584160 \, B a^{6} b^{4} x^{5} + 47500992 \, A a^{5} b^{5} x^{5} + 21003840 \, B a^{7} b^{3} x^{4} + 36756720 \, A a^{6} b^{4} x^{4} + 7351344 \, B a^{8} b^{2} x^{3} + 19603584 \, A a^{7} b^{3} x^{3} + 1531530 \, B a^{9} b x^{2} + 6891885 \, A a^{8} b^{2} x^{2} + 144144 \, B a^{10} x + 1441440 \, A a^{9} b x + 136136 \, A a^{10}}{2450448 \, x^{18}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10*(B*x+A)/x^19,x, algorithm="giac")

[Out]

-1/2450448*(350064*B*b^10*x^11 + 3063060*B*a*b^9*x^10 + 306306*A*b^10*x^10 + 12252240*B*a^2*b^8*x^9 + 2722720*
A*a*b^9*x^9 + 29405376*B*a^3*b^7*x^8 + 11027016*A*a^2*b^8*x^8 + 46781280*B*a^4*b^6*x^7 + 26732160*A*a^3*b^7*x^
7 + 51459408*B*a^5*b^5*x^6 + 42882840*A*a^4*b^6*x^6 + 39584160*B*a^6*b^4*x^5 + 47500992*A*a^5*b^5*x^5 + 210038
40*B*a^7*b^3*x^4 + 36756720*A*a^6*b^4*x^4 + 7351344*B*a^8*b^2*x^3 + 19603584*A*a^7*b^3*x^3 + 1531530*B*a^9*b*x
^2 + 6891885*A*a^8*b^2*x^2 + 144144*B*a^10*x + 1441440*A*a^9*b*x + 136136*A*a^10)/x^18