3.130 \(\int \frac{(a+b x)^{10} (A+B x)}{x^{14}} \, dx\)

Optimal. Leaf size=72 \[ -\frac{b (a+b x)^{11} (2 A b-13 a B)}{1716 a^3 x^{11}}+\frac{(a+b x)^{11} (2 A b-13 a B)}{156 a^2 x^{12}}-\frac{A (a+b x)^{11}}{13 a x^{13}} \]

[Out]

-(A*(a + b*x)^11)/(13*a*x^13) + ((2*A*b - 13*a*B)*(a + b*x)^11)/(156*a^2*x^12) -
(b*(2*A*b - 13*a*B)*(a + b*x)^11)/(1716*a^3*x^11)

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Rubi [A]  time = 0.0989314, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188 \[ -\frac{b (a+b x)^{11} (2 A b-13 a B)}{1716 a^3 x^{11}}+\frac{(a+b x)^{11} (2 A b-13 a B)}{156 a^2 x^{12}}-\frac{A (a+b x)^{11}}{13 a x^{13}} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)^10*(A + B*x))/x^14,x]

[Out]

-(A*(a + b*x)^11)/(13*a*x^13) + ((2*A*b - 13*a*B)*(a + b*x)^11)/(156*a^2*x^12) -
(b*(2*A*b - 13*a*B)*(a + b*x)^11)/(1716*a^3*x^11)

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Rubi in Sympy [A]  time = 12.7876, size = 66, normalized size = 0.92 \[ - \frac{A \left (a + b x\right )^{11}}{13 a x^{13}} + \frac{\left (a + b x\right )^{11} \left (2 A b - 13 B a\right )}{156 a^{2} x^{12}} - \frac{b \left (a + b x\right )^{11} \left (2 A b - 13 B a\right )}{1716 a^{3} x^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**10*(B*x+A)/x**14,x)

[Out]

-A*(a + b*x)**11/(13*a*x**13) + (a + b*x)**11*(2*A*b - 13*B*a)/(156*a**2*x**12)
- b*(a + b*x)**11*(2*A*b - 13*B*a)/(1716*a**3*x**11)

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Mathematica [B]  time = 0.103351, size = 202, normalized size = 2.81 \[ -\frac{11 a^{10} (12 A+13 B x)+130 a^9 b x (11 A+12 B x)+702 a^8 b^2 x^2 (10 A+11 B x)+2288 a^7 b^3 x^3 (9 A+10 B x)+5005 a^6 b^4 x^4 (8 A+9 B x)+7722 a^5 b^5 x^5 (7 A+8 B x)+8580 a^4 b^6 x^6 (6 A+7 B x)+6864 a^3 b^7 x^7 (5 A+6 B x)+3861 a^2 b^8 x^8 (4 A+5 B x)+1430 a b^9 x^9 (3 A+4 B x)+286 b^{10} x^{10} (2 A+3 B x)}{1716 x^{13}} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)^10*(A + B*x))/x^14,x]

[Out]

-(286*b^10*x^10*(2*A + 3*B*x) + 1430*a*b^9*x^9*(3*A + 4*B*x) + 3861*a^2*b^8*x^8*
(4*A + 5*B*x) + 6864*a^3*b^7*x^7*(5*A + 6*B*x) + 8580*a^4*b^6*x^6*(6*A + 7*B*x)
+ 7722*a^5*b^5*x^5*(7*A + 8*B*x) + 5005*a^6*b^4*x^4*(8*A + 9*B*x) + 2288*a^7*b^3
*x^3*(9*A + 10*B*x) + 702*a^8*b^2*x^2*(10*A + 11*B*x) + 130*a^9*b*x*(11*A + 12*B
*x) + 11*a^10*(12*A + 13*B*x))/(1716*x^13)

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Maple [B]  time = 0.01, size = 208, normalized size = 2.9 \[ -{\frac{21\,{a}^{5}{b}^{4} \left ( 6\,Ab+5\,Ba \right ) }{4\,{x}^{8}}}-{\frac{A{a}^{10}}{13\,{x}^{13}}}-{\frac{5\,{a}^{8}b \left ( 9\,Ab+2\,Ba \right ) }{11\,{x}^{11}}}-6\,{\frac{{a}^{4}{b}^{5} \left ( 5\,Ab+6\,Ba \right ) }{{x}^{7}}}-{\frac{10\,{a}^{6}{b}^{3} \left ( 7\,Ab+4\,Ba \right ) }{3\,{x}^{9}}}-{\frac{B{b}^{10}}{2\,{x}^{2}}}-3\,{\frac{{a}^{2}{b}^{7} \left ( 3\,Ab+8\,Ba \right ) }{{x}^{5}}}-{\frac{{b}^{9} \left ( Ab+10\,Ba \right ) }{3\,{x}^{3}}}-{\frac{{a}^{9} \left ( 10\,Ab+Ba \right ) }{12\,{x}^{12}}}-{\frac{5\,a{b}^{8} \left ( 2\,Ab+9\,Ba \right ) }{4\,{x}^{4}}}-{\frac{3\,{a}^{7}{b}^{2} \left ( 8\,Ab+3\,Ba \right ) }{2\,{x}^{10}}}-5\,{\frac{{a}^{3}{b}^{6} \left ( 4\,Ab+7\,Ba \right ) }{{x}^{6}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^10*(B*x+A)/x^14,x)

[Out]

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

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Maxima [A]  time = 1.38799, size = 328, normalized size = 4.56 \[ -\frac{858 \, B b^{10} x^{11} + 132 \, A a^{10} + 572 \,{\left (10 \, B a b^{9} + A b^{10}\right )} x^{10} + 2145 \,{\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{9} + 5148 \,{\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{8} + 8580 \,{\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} + 10296 \,{\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} + 9009 \,{\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} + 5720 \,{\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} + 2574 \,{\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} + 780 \,{\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} + 143 \,{\left (B a^{10} + 10 \, A a^{9} b\right )} x}{1716 \, x^{13}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^10/x^14,x, algorithm="maxima")

[Out]

-1/1716*(858*B*b^10*x^11 + 132*A*a^10 + 572*(10*B*a*b^9 + A*b^10)*x^10 + 2145*(9
*B*a^2*b^8 + 2*A*a*b^9)*x^9 + 5148*(8*B*a^3*b^7 + 3*A*a^2*b^8)*x^8 + 8580*(7*B*a
^4*b^6 + 4*A*a^3*b^7)*x^7 + 10296*(6*B*a^5*b^5 + 5*A*a^4*b^6)*x^6 + 9009*(5*B*a^
6*b^4 + 6*A*a^5*b^5)*x^5 + 5720*(4*B*a^7*b^3 + 7*A*a^6*b^4)*x^4 + 2574*(3*B*a^8*
b^2 + 8*A*a^7*b^3)*x^3 + 780*(2*B*a^9*b + 9*A*a^8*b^2)*x^2 + 143*(B*a^10 + 10*A*
a^9*b)*x)/x^13

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Fricas [A]  time = 0.195327, size = 328, normalized size = 4.56 \[ -\frac{858 \, B b^{10} x^{11} + 132 \, A a^{10} + 572 \,{\left (10 \, B a b^{9} + A b^{10}\right )} x^{10} + 2145 \,{\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{9} + 5148 \,{\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{8} + 8580 \,{\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} + 10296 \,{\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} + 9009 \,{\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} + 5720 \,{\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} + 2574 \,{\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} + 780 \,{\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} + 143 \,{\left (B a^{10} + 10 \, A a^{9} b\right )} x}{1716 \, x^{13}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^10/x^14,x, algorithm="fricas")

[Out]

-1/1716*(858*B*b^10*x^11 + 132*A*a^10 + 572*(10*B*a*b^9 + A*b^10)*x^10 + 2145*(9
*B*a^2*b^8 + 2*A*a*b^9)*x^9 + 5148*(8*B*a^3*b^7 + 3*A*a^2*b^8)*x^8 + 8580*(7*B*a
^4*b^6 + 4*A*a^3*b^7)*x^7 + 10296*(6*B*a^5*b^5 + 5*A*a^4*b^6)*x^6 + 9009*(5*B*a^
6*b^4 + 6*A*a^5*b^5)*x^5 + 5720*(4*B*a^7*b^3 + 7*A*a^6*b^4)*x^4 + 2574*(3*B*a^8*
b^2 + 8*A*a^7*b^3)*x^3 + 780*(2*B*a^9*b + 9*A*a^8*b^2)*x^2 + 143*(B*a^10 + 10*A*
a^9*b)*x)/x^13

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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**14,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.275503, size = 328, normalized size = 4.56 \[ -\frac{858 \, B b^{10} x^{11} + 5720 \, B a b^{9} x^{10} + 572 \, A b^{10} x^{10} + 19305 \, B a^{2} b^{8} x^{9} + 4290 \, A a b^{9} x^{9} + 41184 \, B a^{3} b^{7} x^{8} + 15444 \, A a^{2} b^{8} x^{8} + 60060 \, B a^{4} b^{6} x^{7} + 34320 \, A a^{3} b^{7} x^{7} + 61776 \, B a^{5} b^{5} x^{6} + 51480 \, A a^{4} b^{6} x^{6} + 45045 \, B a^{6} b^{4} x^{5} + 54054 \, A a^{5} b^{5} x^{5} + 22880 \, B a^{7} b^{3} x^{4} + 40040 \, A a^{6} b^{4} x^{4} + 7722 \, B a^{8} b^{2} x^{3} + 20592 \, A a^{7} b^{3} x^{3} + 1560 \, B a^{9} b x^{2} + 7020 \, A a^{8} b^{2} x^{2} + 143 \, B a^{10} x + 1430 \, A a^{9} b x + 132 \, A a^{10}}{1716 \, x^{13}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^10/x^14,x, algorithm="giac")

[Out]

-1/1716*(858*B*b^10*x^11 + 5720*B*a*b^9*x^10 + 572*A*b^10*x^10 + 19305*B*a^2*b^8
*x^9 + 4290*A*a*b^9*x^9 + 41184*B*a^3*b^7*x^8 + 15444*A*a^2*b^8*x^8 + 60060*B*a^
4*b^6*x^7 + 34320*A*a^3*b^7*x^7 + 61776*B*a^5*b^5*x^6 + 51480*A*a^4*b^6*x^6 + 45
045*B*a^6*b^4*x^5 + 54054*A*a^5*b^5*x^5 + 22880*B*a^7*b^3*x^4 + 40040*A*a^6*b^4*
x^4 + 7722*B*a^8*b^2*x^3 + 20592*A*a^7*b^3*x^3 + 1560*B*a^9*b*x^2 + 7020*A*a^8*b
^2*x^2 + 143*B*a^10*x + 1430*A*a^9*b*x + 132*A*a^10)/x^13