3.131 \(\int \frac{(a+b x)^{10} (A+B x)}{x^{15}} \, dx\)

Optimal. Leaf size=101 \[ \frac{b^2 (a+b x)^{11} (3 A b-14 a B)}{12012 a^4 x^{11}}-\frac{b (a+b x)^{11} (3 A b-14 a B)}{1092 a^3 x^{12}}+\frac{(a+b x)^{11} (3 A b-14 a B)}{182 a^2 x^{13}}-\frac{A (a+b x)^{11}}{14 a x^{14}} \]

[Out]

-(A*(a + b*x)^11)/(14*a*x^14) + ((3*A*b - 14*a*B)*(a + b*x)^11)/(182*a^2*x^13) -
(b*(3*A*b - 14*a*B)*(a + b*x)^11)/(1092*a^3*x^12) + (b^2*(3*A*b - 14*a*B)*(a +
b*x)^11)/(12012*a^4*x^11)

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Rubi [A]  time = 0.128684, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188 \[ \frac{b^2 (a+b x)^{11} (3 A b-14 a B)}{12012 a^4 x^{11}}-\frac{b (a+b x)^{11} (3 A b-14 a B)}{1092 a^3 x^{12}}+\frac{(a+b x)^{11} (3 A b-14 a B)}{182 a^2 x^{13}}-\frac{A (a+b x)^{11}}{14 a x^{14}} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)^10*(A + B*x))/x^15,x]

[Out]

-(A*(a + b*x)^11)/(14*a*x^14) + ((3*A*b - 14*a*B)*(a + b*x)^11)/(182*a^2*x^13) -
(b*(3*A*b - 14*a*B)*(a + b*x)^11)/(1092*a^3*x^12) + (b^2*(3*A*b - 14*a*B)*(a +
b*x)^11)/(12012*a^4*x^11)

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Rubi in Sympy [A]  time = 18.0422, size = 95, normalized size = 0.94 \[ - \frac{A \left (a + b x\right )^{11}}{14 a x^{14}} + \frac{\left (a + b x\right )^{11} \left (3 A b - 14 B a\right )}{182 a^{2} x^{13}} - \frac{b \left (a + b x\right )^{11} \left (3 A b - 14 B a\right )}{1092 a^{3} x^{12}} + \frac{b^{2} \left (a + b x\right )^{11} \left (3 A b - 14 B a\right )}{12012 a^{4} x^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**10*(B*x+A)/x**15,x)

[Out]

-A*(a + b*x)**11/(14*a*x**14) + (a + b*x)**11*(3*A*b - 14*B*a)/(182*a**2*x**13)
- b*(a + b*x)**11*(3*A*b - 14*B*a)/(1092*a**3*x**12) + b**2*(a + b*x)**11*(3*A*b
- 14*B*a)/(12012*a**4*x**11)

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Mathematica [A]  time = 0.10791, size = 202, normalized size = 2. \[ -\frac{66 a^{10} (13 A+14 B x)+770 a^9 b x (12 A+13 B x)+4095 a^8 b^2 x^2 (11 A+12 B x)+13104 a^7 b^3 x^3 (10 A+11 B x)+28028 a^6 b^4 x^4 (9 A+10 B x)+42042 a^5 b^5 x^5 (8 A+9 B x)+45045 a^4 b^6 x^6 (7 A+8 B x)+34320 a^3 b^7 x^7 (6 A+7 B x)+18018 a^2 b^8 x^8 (5 A+6 B x)+6006 a b^9 x^9 (4 A+5 B x)+1001 b^{10} x^{10} (3 A+4 B x)}{12012 x^{14}} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)^10*(A + B*x))/x^15,x]

[Out]

-(1001*b^10*x^10*(3*A + 4*B*x) + 6006*a*b^9*x^9*(4*A + 5*B*x) + 18018*a^2*b^8*x^
8*(5*A + 6*B*x) + 34320*a^3*b^7*x^7*(6*A + 7*B*x) + 45045*a^4*b^6*x^6*(7*A + 8*B
*x) + 42042*a^5*b^5*x^5*(8*A + 9*B*x) + 28028*a^6*b^4*x^4*(9*A + 10*B*x) + 13104
*a^7*b^3*x^3*(10*A + 11*B*x) + 4095*a^8*b^2*x^2*(11*A + 12*B*x) + 770*a^9*b*x*(1
2*A + 13*B*x) + 66*a^10*(13*A + 14*B*x))/(12012*x^14)

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Maple [B]  time = 0.01, size = 208, normalized size = 2.1 \[ -{\frac{21\,{a}^{4}{b}^{5} \left ( 5\,Ab+6\,Ba \right ) }{4\,{x}^{8}}}-{\frac{15\,{a}^{7}{b}^{2} \left ( 8\,Ab+3\,Ba \right ) }{11\,{x}^{11}}}-{\frac{30\,{a}^{3}{b}^{6} \left ( 4\,Ab+7\,Ba \right ) }{7\,{x}^{7}}}-{\frac{14\,{a}^{5}{b}^{4} \left ( 6\,Ab+5\,Ba \right ) }{3\,{x}^{9}}}-{\frac{a{b}^{8} \left ( 2\,Ab+9\,Ba \right ) }{{x}^{5}}}-{\frac{B{b}^{10}}{3\,{x}^{3}}}-{\frac{5\,{a}^{8}b \left ( 9\,Ab+2\,Ba \right ) }{12\,{x}^{12}}}-{\frac{A{a}^{10}}{14\,{x}^{14}}}-{\frac{{a}^{9} \left ( 10\,Ab+Ba \right ) }{13\,{x}^{13}}}-{\frac{{b}^{9} \left ( Ab+10\,Ba \right ) }{4\,{x}^{4}}}-3\,{\frac{{a}^{6}{b}^{3} \left ( 7\,Ab+4\,Ba \right ) }{{x}^{10}}}-{\frac{5\,{a}^{2}{b}^{7} \left ( 3\,Ab+8\,Ba \right ) }{2\,{x}^{6}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^10*(B*x+A)/x^15,x)

[Out]

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

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Maxima [A]  time = 1.35019, size = 328, normalized size = 3.25 \[ -\frac{4004 \, B b^{10} x^{11} + 858 \, A a^{10} + 3003 \,{\left (10 \, B a b^{9} + A b^{10}\right )} x^{10} + 12012 \,{\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{9} + 30030 \,{\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{8} + 51480 \,{\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} + 63063 \,{\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} + 56056 \,{\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} + 36036 \,{\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} + 16380 \,{\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} + 5005 \,{\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} + 924 \,{\left (B a^{10} + 10 \, A a^{9} b\right )} x}{12012 \, x^{14}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^10/x^15,x, algorithm="maxima")

[Out]

-1/12012*(4004*B*b^10*x^11 + 858*A*a^10 + 3003*(10*B*a*b^9 + A*b^10)*x^10 + 1201
2*(9*B*a^2*b^8 + 2*A*a*b^9)*x^9 + 30030*(8*B*a^3*b^7 + 3*A*a^2*b^8)*x^8 + 51480*
(7*B*a^4*b^6 + 4*A*a^3*b^7)*x^7 + 63063*(6*B*a^5*b^5 + 5*A*a^4*b^6)*x^6 + 56056*
(5*B*a^6*b^4 + 6*A*a^5*b^5)*x^5 + 36036*(4*B*a^7*b^3 + 7*A*a^6*b^4)*x^4 + 16380*
(3*B*a^8*b^2 + 8*A*a^7*b^3)*x^3 + 5005*(2*B*a^9*b + 9*A*a^8*b^2)*x^2 + 924*(B*a^
10 + 10*A*a^9*b)*x)/x^14

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Fricas [A]  time = 0.196113, size = 328, normalized size = 3.25 \[ -\frac{4004 \, B b^{10} x^{11} + 858 \, A a^{10} + 3003 \,{\left (10 \, B a b^{9} + A b^{10}\right )} x^{10} + 12012 \,{\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{9} + 30030 \,{\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{8} + 51480 \,{\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} + 63063 \,{\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} + 56056 \,{\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} + 36036 \,{\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} + 16380 \,{\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} + 5005 \,{\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} + 924 \,{\left (B a^{10} + 10 \, A a^{9} b\right )} x}{12012 \, x^{14}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^10/x^15,x, algorithm="fricas")

[Out]

-1/12012*(4004*B*b^10*x^11 + 858*A*a^10 + 3003*(10*B*a*b^9 + A*b^10)*x^10 + 1201
2*(9*B*a^2*b^8 + 2*A*a*b^9)*x^9 + 30030*(8*B*a^3*b^7 + 3*A*a^2*b^8)*x^8 + 51480*
(7*B*a^4*b^6 + 4*A*a^3*b^7)*x^7 + 63063*(6*B*a^5*b^5 + 5*A*a^4*b^6)*x^6 + 56056*
(5*B*a^6*b^4 + 6*A*a^5*b^5)*x^5 + 36036*(4*B*a^7*b^3 + 7*A*a^6*b^4)*x^4 + 16380*
(3*B*a^8*b^2 + 8*A*a^7*b^3)*x^3 + 5005*(2*B*a^9*b + 9*A*a^8*b^2)*x^2 + 924*(B*a^
10 + 10*A*a^9*b)*x)/x^14

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**10*(B*x+A)/x**15,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.287153, size = 328, normalized size = 3.25 \[ -\frac{4004 \, B b^{10} x^{11} + 30030 \, B a b^{9} x^{10} + 3003 \, A b^{10} x^{10} + 108108 \, B a^{2} b^{8} x^{9} + 24024 \, A a b^{9} x^{9} + 240240 \, B a^{3} b^{7} x^{8} + 90090 \, A a^{2} b^{8} x^{8} + 360360 \, B a^{4} b^{6} x^{7} + 205920 \, A a^{3} b^{7} x^{7} + 378378 \, B a^{5} b^{5} x^{6} + 315315 \, A a^{4} b^{6} x^{6} + 280280 \, B a^{6} b^{4} x^{5} + 336336 \, A a^{5} b^{5} x^{5} + 144144 \, B a^{7} b^{3} x^{4} + 252252 \, A a^{6} b^{4} x^{4} + 49140 \, B a^{8} b^{2} x^{3} + 131040 \, A a^{7} b^{3} x^{3} + 10010 \, B a^{9} b x^{2} + 45045 \, A a^{8} b^{2} x^{2} + 924 \, B a^{10} x + 9240 \, A a^{9} b x + 858 \, A a^{10}}{12012 \, x^{14}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^10/x^15,x, algorithm="giac")

[Out]

-1/12012*(4004*B*b^10*x^11 + 30030*B*a*b^9*x^10 + 3003*A*b^10*x^10 + 108108*B*a^
2*b^8*x^9 + 24024*A*a*b^9*x^9 + 240240*B*a^3*b^7*x^8 + 90090*A*a^2*b^8*x^8 + 360
360*B*a^4*b^6*x^7 + 205920*A*a^3*b^7*x^7 + 378378*B*a^5*b^5*x^6 + 315315*A*a^4*b
^6*x^6 + 280280*B*a^6*b^4*x^5 + 336336*A*a^5*b^5*x^5 + 144144*B*a^7*b^3*x^4 + 25
2252*A*a^6*b^4*x^4 + 49140*B*a^8*b^2*x^3 + 131040*A*a^7*b^3*x^3 + 10010*B*a^9*b*
x^2 + 45045*A*a^8*b^2*x^2 + 924*B*a^10*x + 9240*A*a^9*b*x + 858*A*a^10)/x^14