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

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

[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

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Rubi [A]  time = 0.14, antiderivative size = 229, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.10, size = 222, normalized size = 0.97 \[ -\frac {a^{10} (17 A+18 B x)}{306 x^{18}}-\frac {5 a^9 b (16 A+17 B x)}{136 x^{17}}-\frac {3 a^8 b^2 (15 A+16 B x)}{16 x^{16}}-\frac {4 a^7 b^3 (14 A+15 B x)}{7 x^{15}}-\frac {15 a^6 b^4 (13 A+14 B x)}{13 x^{14}}-\frac {21 a^5 b^5 (12 A+13 B x)}{13 x^{13}}-\frac {35 a^4 b^6 (11 A+12 B x)}{22 x^{12}}-\frac {12 a^3 b^7 (10 A+11 B x)}{11 x^{11}}-\frac {a^2 b^8 (9 A+10 B x)}{2 x^{10}}-\frac {5 a b^9 (8 A+9 B x)}{36 x^9}-\frac {b^{10} (7 A+8 B x)}{56 x^8} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/56*(b^10*(7*A + 8*B*x))/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)

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fricas [A]  time = 0.89, size = 243, normalized size = 1.06 \[ -\frac {350064 \, B b^{10} x^{11} + 136136 \, A a^{10} + 306306 \, {\left (10 \, B a b^{9} + A b^{10}\right )} x^{10} + 1361360 \, {\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{9} + 3675672 \, {\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{8} + 6683040 \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} + 8576568 \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} + 7916832 \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} + 5250960 \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} + 2450448 \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} + 765765 \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} + 144144 \, {\left (B a^{10} + 10 \, A a^{9} b\right )} x}{2450448 \, x^{18}} \]

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

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giac [A]  time = 0.92, size = 243, normalized size = 1.06 \[ -\frac {350064 \, B b^{10} x^{11} + 3063060 \, B a b^{9} x^{10} + 306306 \, A b^{10} x^{10} + 12252240 \, B a^{2} b^{8} x^{9} + 2722720 \, A a b^{9} x^{9} + 29405376 \, B a^{3} b^{7} x^{8} + 11027016 \, A a^{2} b^{8} x^{8} + 46781280 \, B a^{4} b^{6} x^{7} + 26732160 \, A a^{3} b^{7} x^{7} + 51459408 \, B a^{5} b^{5} x^{6} + 42882840 \, A a^{4} b^{6} x^{6} + 39584160 \, B a^{6} b^{4} x^{5} + 47500992 \, A a^{5} b^{5} x^{5} + 21003840 \, B a^{7} b^{3} x^{4} + 36756720 \, A a^{6} b^{4} x^{4} + 7351344 \, B a^{8} b^{2} x^{3} + 19603584 \, A a^{7} b^{3} x^{3} + 1531530 \, B a^{9} b x^{2} + 6891885 \, A a^{8} b^{2} x^{2} + 144144 \, B a^{10} x + 1441440 \, A a^{9} b x + 136136 \, A a^{10}}{2450448 \, x^{18}} \]

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

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maple [A]  time = 0.01, size = 208, normalized size = 0.91 \[ -\frac {B \,b^{10}}{7 x^{7}}-\frac {\left (A b +10 B a \right ) b^{9}}{8 x^{8}}-\frac {5 \left (2 A b +9 B a \right ) a \,b^{8}}{9 x^{9}}-\frac {3 \left (3 A b +8 B a \right ) a^{2} b^{7}}{2 x^{10}}-\frac {30 \left (4 A b +7 B a \right ) a^{3} b^{6}}{11 x^{11}}-\frac {7 \left (5 A b +6 B a \right ) a^{4} b^{5}}{2 x^{12}}-\frac {42 \left (6 A b +5 B a \right ) a^{5} b^{4}}{13 x^{13}}-\frac {15 \left (7 A b +4 B a \right ) a^{6} b^{3}}{7 x^{14}}-\frac {\left (8 A b +3 B a \right ) a^{7} b^{2}}{x^{15}}-\frac {A \,a^{10}}{18 x^{18}}-\frac {5 \left (9 A b +2 B a \right ) a^{8} b}{16 x^{16}}-\frac {\left (10 A b +B a \right ) a^{9}}{17 x^{17}} \]

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

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maxima [A]  time = 1.12, size = 243, normalized size = 1.06 \[ -\frac {350064 \, B b^{10} x^{11} + 136136 \, A a^{10} + 306306 \, {\left (10 \, B a b^{9} + A b^{10}\right )} x^{10} + 1361360 \, {\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{9} + 3675672 \, {\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{8} + 6683040 \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} + 8576568 \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} + 7916832 \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} + 5250960 \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} + 2450448 \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} + 765765 \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} + 144144 \, {\left (B a^{10} + 10 \, A a^{9} b\right )} x}{2450448 \, x^{18}} \]

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

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mupad [B]  time = 0.39, size = 235, normalized size = 1.03 \[ -\frac {x\,\left (\frac {B\,a^{10}}{17}+\frac {10\,A\,b\,a^9}{17}\right )+\frac {A\,a^{10}}{18}+x^9\,\left (5\,B\,a^2\,b^8+\frac {10\,A\,a\,b^9}{9}\right )+x^2\,\left (\frac {5\,B\,a^9\,b}{8}+\frac {45\,A\,a^8\,b^2}{16}\right )+x^{10}\,\left (\frac {A\,b^{10}}{8}+\frac {5\,B\,a\,b^9}{4}\right )+x^3\,\left (3\,B\,a^8\,b^2+8\,A\,a^7\,b^3\right )+x^8\,\left (12\,B\,a^3\,b^7+\frac {9\,A\,a^2\,b^8}{2}\right )+x^6\,\left (21\,B\,a^5\,b^5+\frac {35\,A\,a^4\,b^6}{2}\right )+x^4\,\left (\frac {60\,B\,a^7\,b^3}{7}+15\,A\,a^6\,b^4\right )+x^7\,\left (\frac {210\,B\,a^4\,b^6}{11}+\frac {120\,A\,a^3\,b^7}{11}\right )+x^5\,\left (\frac {210\,B\,a^6\,b^4}{13}+\frac {252\,A\,a^5\,b^5}{13}\right )+\frac {B\,b^{10}\,x^{11}}{7}}{x^{18}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(a + b*x)^10)/x^19,x)

[Out]

-(x*((B*a^10)/17 + (10*A*a^9*b)/17) + (A*a^10)/18 + x^9*(5*B*a^2*b^8 + (10*A*a*b^9)/9) + x^2*((45*A*a^8*b^2)/1
6 + (5*B*a^9*b)/8) + x^10*((A*b^10)/8 + (5*B*a*b^9)/4) + x^3*(8*A*a^7*b^3 + 3*B*a^8*b^2) + x^8*((9*A*a^2*b^8)/
2 + 12*B*a^3*b^7) + x^6*((35*A*a^4*b^6)/2 + 21*B*a^5*b^5) + x^4*(15*A*a^6*b^4 + (60*B*a^7*b^3)/7) + x^7*((120*
A*a^3*b^7)/11 + (210*B*a^4*b^6)/11) + x^5*((252*A*a^5*b^5)/13 + (210*B*a^6*b^4)/13) + (B*b^10*x^11)/7)/x^18

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sympy [A]  time = 165.44, size = 260, normalized size = 1.14 \[ \frac {- 136136 A a^{10} - 350064 B b^{10} x^{11} + x^{10} \left (- 306306 A b^{10} - 3063060 B a b^{9}\right ) + x^{9} \left (- 2722720 A a b^{9} - 12252240 B a^{2} b^{8}\right ) + x^{8} \left (- 11027016 A a^{2} b^{8} - 29405376 B a^{3} b^{7}\right ) + x^{7} \left (- 26732160 A a^{3} b^{7} - 46781280 B a^{4} b^{6}\right ) + x^{6} \left (- 42882840 A a^{4} b^{6} - 51459408 B a^{5} b^{5}\right ) + x^{5} \left (- 47500992 A a^{5} b^{5} - 39584160 B a^{6} b^{4}\right ) + x^{4} \left (- 36756720 A a^{6} b^{4} - 21003840 B a^{7} b^{3}\right ) + x^{3} \left (- 19603584 A a^{7} b^{3} - 7351344 B a^{8} b^{2}\right ) + x^{2} \left (- 6891885 A a^{8} b^{2} - 1531530 B a^{9} b\right ) + x \left (- 1441440 A a^{9} b - 144144 B a^{10}\right )}{2450448 x^{18}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-136136*A*a**10 - 350064*B*b**10*x**11 + x**10*(-306306*A*b**10 - 3063060*B*a*b**9) + x**9*(-2722720*A*a*b**9
- 12252240*B*a**2*b**8) + x**8*(-11027016*A*a**2*b**8 - 29405376*B*a**3*b**7) + x**7*(-26732160*A*a**3*b**7 -
46781280*B*a**4*b**6) + x**6*(-42882840*A*a**4*b**6 - 51459408*B*a**5*b**5) + x**5*(-47500992*A*a**5*b**5 - 3
9584160*B*a**6*b**4) + x**4*(-36756720*A*a**6*b**4 - 21003840*B*a**7*b**3) + x**3*(-19603584*A*a**7*b**3 - 735
1344*B*a**8*b**2) + x**2*(-6891885*A*a**8*b**2 - 1531530*B*a**9*b) + x*(-1441440*A*a**9*b - 144144*B*a**10))/(
2450448*x**18)

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