3.104 \(\int \frac{(a+b x)^5 (A+B x)}{x^{11}} \, dx\)

Optimal. Leaf size=117 \[ -\frac{a^5 A}{10 x^{10}}-\frac{a^4 (a B+5 A b)}{9 x^9}-\frac{5 a^3 b (a B+2 A b)}{8 x^8}-\frac{10 a^2 b^2 (a B+A b)}{7 x^7}-\frac{b^4 (5 a B+A b)}{5 x^5}-\frac{5 a b^3 (2 a B+A b)}{6 x^6}-\frac{b^5 B}{4 x^4} \]

[Out]

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

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Rubi [A]  time = 0.175462, antiderivative size = 117, 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 \[ -\frac{a^5 A}{10 x^{10}}-\frac{a^4 (a B+5 A b)}{9 x^9}-\frac{5 a^3 b (a B+2 A b)}{8 x^8}-\frac{10 a^2 b^2 (a B+A b)}{7 x^7}-\frac{b^4 (5 a B+A b)}{5 x^5}-\frac{5 a b^3 (2 a B+A b)}{6 x^6}-\frac{b^5 B}{4 x^4} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)^5*(A + B*x))/x^11,x]

[Out]

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

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Rubi in Sympy [A]  time = 33.4383, size = 116, normalized size = 0.99 \[ - \frac{A a^{5}}{10 x^{10}} - \frac{B b^{5}}{4 x^{4}} - \frac{a^{4} \left (5 A b + B a\right )}{9 x^{9}} - \frac{5 a^{3} b \left (2 A b + B a\right )}{8 x^{8}} - \frac{10 a^{2} b^{2} \left (A b + B a\right )}{7 x^{7}} - \frac{5 a b^{3} \left (A b + 2 B a\right )}{6 x^{6}} - \frac{b^{4} \left (A b + 5 B a\right )}{5 x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**5*(B*x+A)/x**11,x)

[Out]

-A*a**5/(10*x**10) - B*b**5/(4*x**4) - a**4*(5*A*b + B*a)/(9*x**9) - 5*a**3*b*(2
*A*b + B*a)/(8*x**8) - 10*a**2*b**2*(A*b + B*a)/(7*x**7) - 5*a*b**3*(A*b + 2*B*a
)/(6*x**6) - b**4*(A*b + 5*B*a)/(5*x**5)

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Mathematica [A]  time = 0.0500053, size = 107, normalized size = 0.91 \[ -\frac{28 a^5 (9 A+10 B x)+175 a^4 b x (8 A+9 B x)+450 a^3 b^2 x^2 (7 A+8 B x)+600 a^2 b^3 x^3 (6 A+7 B x)+420 a b^4 x^4 (5 A+6 B x)+126 b^5 x^5 (4 A+5 B x)}{2520 x^{10}} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)^5*(A + B*x))/x^11,x]

[Out]

-(126*b^5*x^5*(4*A + 5*B*x) + 420*a*b^4*x^4*(5*A + 6*B*x) + 600*a^2*b^3*x^3*(6*A
+ 7*B*x) + 450*a^3*b^2*x^2*(7*A + 8*B*x) + 175*a^4*b*x*(8*A + 9*B*x) + 28*a^5*(
9*A + 10*B*x))/(2520*x^10)

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Maple [A]  time = 0.009, size = 104, normalized size = 0.9 \[ -{\frac{A{a}^{5}}{10\,{x}^{10}}}-{\frac{{a}^{4} \left ( 5\,Ab+Ba \right ) }{9\,{x}^{9}}}-{\frac{5\,{a}^{3}b \left ( 2\,Ab+Ba \right ) }{8\,{x}^{8}}}-{\frac{10\,{a}^{2}{b}^{2} \left ( Ab+Ba \right ) }{7\,{x}^{7}}}-{\frac{5\,a{b}^{3} \left ( Ab+2\,Ba \right ) }{6\,{x}^{6}}}-{\frac{{b}^{4} \left ( Ab+5\,Ba \right ) }{5\,{x}^{5}}}-{\frac{B{b}^{5}}{4\,{x}^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^5*(B*x+A)/x^11,x)

[Out]

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

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Maxima [A]  time = 1.34251, size = 161, normalized size = 1.38 \[ -\frac{630 \, B b^{5} x^{6} + 252 \, A a^{5} + 504 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{5} + 2100 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{4} + 3600 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} + 1575 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} + 280 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x}{2520 \, x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^5/x^11,x, algorithm="maxima")

[Out]

-1/2520*(630*B*b^5*x^6 + 252*A*a^5 + 504*(5*B*a*b^4 + A*b^5)*x^5 + 2100*(2*B*a^2
*b^3 + A*a*b^4)*x^4 + 3600*(B*a^3*b^2 + A*a^2*b^3)*x^3 + 1575*(B*a^4*b + 2*A*a^3
*b^2)*x^2 + 280*(B*a^5 + 5*A*a^4*b)*x)/x^10

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Fricas [A]  time = 0.196308, size = 161, normalized size = 1.38 \[ -\frac{630 \, B b^{5} x^{6} + 252 \, A a^{5} + 504 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{5} + 2100 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{4} + 3600 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} + 1575 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} + 280 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x}{2520 \, x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^5/x^11,x, algorithm="fricas")

[Out]

-1/2520*(630*B*b^5*x^6 + 252*A*a^5 + 504*(5*B*a*b^4 + A*b^5)*x^5 + 2100*(2*B*a^2
*b^3 + A*a*b^4)*x^4 + 3600*(B*a^3*b^2 + A*a^2*b^3)*x^3 + 1575*(B*a^4*b + 2*A*a^3
*b^2)*x^2 + 280*(B*a^5 + 5*A*a^4*b)*x)/x^10

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Sympy [A]  time = 33.8304, size = 126, normalized size = 1.08 \[ - \frac{252 A a^{5} + 630 B b^{5} x^{6} + x^{5} \left (504 A b^{5} + 2520 B a b^{4}\right ) + x^{4} \left (2100 A a b^{4} + 4200 B a^{2} b^{3}\right ) + x^{3} \left (3600 A a^{2} b^{3} + 3600 B a^{3} b^{2}\right ) + x^{2} \left (3150 A a^{3} b^{2} + 1575 B a^{4} b\right ) + x \left (1400 A a^{4} b + 280 B a^{5}\right )}{2520 x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**5*(B*x+A)/x**11,x)

[Out]

-(252*A*a**5 + 630*B*b**5*x**6 + x**5*(504*A*b**5 + 2520*B*a*b**4) + x**4*(2100*
A*a*b**4 + 4200*B*a**2*b**3) + x**3*(3600*A*a**2*b**3 + 3600*B*a**3*b**2) + x**2
*(3150*A*a**3*b**2 + 1575*B*a**4*b) + x*(1400*A*a**4*b + 280*B*a**5))/(2520*x**1
0)

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GIAC/XCAS [A]  time = 0.278103, size = 166, normalized size = 1.42 \[ -\frac{630 \, B b^{5} x^{6} + 2520 \, B a b^{4} x^{5} + 504 \, A b^{5} x^{5} + 4200 \, B a^{2} b^{3} x^{4} + 2100 \, A a b^{4} x^{4} + 3600 \, B a^{3} b^{2} x^{3} + 3600 \, A a^{2} b^{3} x^{3} + 1575 \, B a^{4} b x^{2} + 3150 \, A a^{3} b^{2} x^{2} + 280 \, B a^{5} x + 1400 \, A a^{4} b x + 252 \, A a^{5}}{2520 \, x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^5/x^11,x, algorithm="giac")

[Out]

-1/2520*(630*B*b^5*x^6 + 2520*B*a*b^4*x^5 + 504*A*b^5*x^5 + 4200*B*a^2*b^3*x^4 +
2100*A*a*b^4*x^4 + 3600*B*a^3*b^2*x^3 + 3600*A*a^2*b^3*x^3 + 1575*B*a^4*b*x^2 +
3150*A*a^3*b^2*x^2 + 280*B*a^5*x + 1400*A*a^4*b*x + 252*A*a^5)/x^10