3.105 \(\int \frac{(a+b x)^5 (A+B x)}{x^{12}} \, dx\)

Optimal. Leaf size=117 \[ -\frac{a^5 A}{11 x^{11}}-\frac{a^4 (a B+5 A b)}{10 x^{10}}-\frac{5 a^3 b (a B+2 A b)}{9 x^9}-\frac{5 a^2 b^2 (a B+A b)}{4 x^8}-\frac{b^4 (5 a B+A b)}{6 x^6}-\frac{5 a b^3 (2 a B+A b)}{7 x^7}-\frac{b^5 B}{5 x^5} \]

[Out]

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

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Rubi [A]  time = 0.163753, 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 \[ -\frac{a^5 A}{11 x^{11}}-\frac{a^4 (a B+5 A b)}{10 x^{10}}-\frac{5 a^3 b (a B+2 A b)}{9 x^9}-\frac{5 a^2 b^2 (a B+A b)}{4 x^8}-\frac{b^4 (5 a B+A b)}{6 x^6}-\frac{5 a b^3 (2 a B+A b)}{7 x^7}-\frac{b^5 B}{5 x^5} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)^5*(A + B*x))/x^12,x]

[Out]

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

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Rubi in Sympy [A]  time = 33.7867, size = 116, normalized size = 0.99 \[ - \frac{A a^{5}}{11 x^{11}} - \frac{B b^{5}}{5 x^{5}} - \frac{a^{4} \left (5 A b + B a\right )}{10 x^{10}} - \frac{5 a^{3} b \left (2 A b + B a\right )}{9 x^{9}} - \frac{5 a^{2} b^{2} \left (A b + B a\right )}{4 x^{8}} - \frac{5 a b^{3} \left (A b + 2 B a\right )}{7 x^{7}} - \frac{b^{4} \left (A b + 5 B a\right )}{6 x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**5*(B*x+A)/x**12,x)

[Out]

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

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Mathematica [A]  time = 0.0502402, size = 107, normalized size = 0.91 \[ -\frac{126 a^5 (10 A+11 B x)+770 a^4 b x (9 A+10 B x)+1925 a^3 b^2 x^2 (8 A+9 B x)+2475 a^2 b^3 x^3 (7 A+8 B x)+1650 a b^4 x^4 (6 A+7 B x)+462 b^5 x^5 (5 A+6 B x)}{13860 x^{11}} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)^5*(A + B*x))/x^12,x]

[Out]

-(462*b^5*x^5*(5*A + 6*B*x) + 1650*a*b^4*x^4*(6*A + 7*B*x) + 2475*a^2*b^3*x^3*(7
*A + 8*B*x) + 1925*a^3*b^2*x^2*(8*A + 9*B*x) + 770*a^4*b*x*(9*A + 10*B*x) + 126*
a^5*(10*A + 11*B*x))/(13860*x^11)

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Maple [A]  time = 0.009, size = 104, normalized size = 0.9 \[ -{\frac{A{a}^{5}}{11\,{x}^{11}}}-{\frac{{a}^{4} \left ( 5\,Ab+Ba \right ) }{10\,{x}^{10}}}-{\frac{5\,{a}^{3}b \left ( 2\,Ab+Ba \right ) }{9\,{x}^{9}}}-{\frac{5\,{a}^{2}{b}^{2} \left ( Ab+Ba \right ) }{4\,{x}^{8}}}-{\frac{5\,a{b}^{3} \left ( Ab+2\,Ba \right ) }{7\,{x}^{7}}}-{\frac{{b}^{4} \left ( Ab+5\,Ba \right ) }{6\,{x}^{6}}}-{\frac{B{b}^{5}}{5\,{x}^{5}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^5*(B*x+A)/x^12,x)

[Out]

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

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Maxima [A]  time = 1.35917, size = 161, normalized size = 1.38 \[ -\frac{2772 \, B b^{5} x^{6} + 1260 \, A a^{5} + 2310 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{5} + 9900 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{4} + 17325 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} + 7700 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} + 1386 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x}{13860 \, x^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^5/x^12,x, algorithm="maxima")

[Out]

-1/13860*(2772*B*b^5*x^6 + 1260*A*a^5 + 2310*(5*B*a*b^4 + A*b^5)*x^5 + 9900*(2*B
*a^2*b^3 + A*a*b^4)*x^4 + 17325*(B*a^3*b^2 + A*a^2*b^3)*x^3 + 7700*(B*a^4*b + 2*
A*a^3*b^2)*x^2 + 1386*(B*a^5 + 5*A*a^4*b)*x)/x^11

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Fricas [A]  time = 0.195143, size = 161, normalized size = 1.38 \[ -\frac{2772 \, B b^{5} x^{6} + 1260 \, A a^{5} + 2310 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{5} + 9900 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{4} + 17325 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} + 7700 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} + 1386 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x}{13860 \, x^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^5/x^12,x, algorithm="fricas")

[Out]

-1/13860*(2772*B*b^5*x^6 + 1260*A*a^5 + 2310*(5*B*a*b^4 + A*b^5)*x^5 + 9900*(2*B
*a^2*b^3 + A*a*b^4)*x^4 + 17325*(B*a^3*b^2 + A*a^2*b^3)*x^3 + 7700*(B*a^4*b + 2*
A*a^3*b^2)*x^2 + 1386*(B*a^5 + 5*A*a^4*b)*x)/x^11

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Sympy [A]  time = 38.6468, size = 126, normalized size = 1.08 \[ - \frac{1260 A a^{5} + 2772 B b^{5} x^{6} + x^{5} \left (2310 A b^{5} + 11550 B a b^{4}\right ) + x^{4} \left (9900 A a b^{4} + 19800 B a^{2} b^{3}\right ) + x^{3} \left (17325 A a^{2} b^{3} + 17325 B a^{3} b^{2}\right ) + x^{2} \left (15400 A a^{3} b^{2} + 7700 B a^{4} b\right ) + x \left (6930 A a^{4} b + 1386 B a^{5}\right )}{13860 x^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**5*(B*x+A)/x**12,x)

[Out]

-(1260*A*a**5 + 2772*B*b**5*x**6 + x**5*(2310*A*b**5 + 11550*B*a*b**4) + x**4*(9
900*A*a*b**4 + 19800*B*a**2*b**3) + x**3*(17325*A*a**2*b**3 + 17325*B*a**3*b**2)
+ x**2*(15400*A*a**3*b**2 + 7700*B*a**4*b) + x*(6930*A*a**4*b + 1386*B*a**5))/(
13860*x**11)

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GIAC/XCAS [A]  time = 0.315792, size = 166, normalized size = 1.42 \[ -\frac{2772 \, B b^{5} x^{6} + 11550 \, B a b^{4} x^{5} + 2310 \, A b^{5} x^{5} + 19800 \, B a^{2} b^{3} x^{4} + 9900 \, A a b^{4} x^{4} + 17325 \, B a^{3} b^{2} x^{3} + 17325 \, A a^{2} b^{3} x^{3} + 7700 \, B a^{4} b x^{2} + 15400 \, A a^{3} b^{2} x^{2} + 1386 \, B a^{5} x + 6930 \, A a^{4} b x + 1260 \, A a^{5}}{13860 \, x^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^5/x^12,x, algorithm="giac")

[Out]

-1/13860*(2772*B*b^5*x^6 + 11550*B*a*b^4*x^5 + 2310*A*b^5*x^5 + 19800*B*a^2*b^3*
x^4 + 9900*A*a*b^4*x^4 + 17325*B*a^3*b^2*x^3 + 17325*A*a^2*b^3*x^3 + 7700*B*a^4*
b*x^2 + 15400*A*a^3*b^2*x^2 + 1386*B*a^5*x + 6930*A*a^4*b*x + 1260*A*a^5)/x^11