3.55 \(\int \frac{\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x^{23}} \, dx\)

Optimal. Leaf size=117 \[ -\frac{a^5 A}{22 x^{22}}-\frac{a^4 (a B+5 A b)}{20 x^{20}}-\frac{5 a^3 b (a B+2 A b)}{18 x^{18}}-\frac{5 a^2 b^2 (a B+A b)}{8 x^{16}}-\frac{b^4 (5 a B+A b)}{12 x^{12}}-\frac{5 a b^3 (2 a B+A b)}{14 x^{14}}-\frac{b^5 B}{10 x^{10}} \]

[Out]

-(a^5*A)/(22*x^22) - (a^4*(5*A*b + a*B))/(20*x^20) - (5*a^3*b*(2*A*b + a*B))/(18
*x^18) - (5*a^2*b^2*(A*b + a*B))/(8*x^16) - (5*a*b^3*(A*b + 2*a*B))/(14*x^14) -
(b^4*(A*b + 5*a*B))/(12*x^12) - (b^5*B)/(10*x^10)

_______________________________________________________________________________________

Rubi [A]  time = 0.241734, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ -\frac{a^5 A}{22 x^{22}}-\frac{a^4 (a B+5 A b)}{20 x^{20}}-\frac{5 a^3 b (a B+2 A b)}{18 x^{18}}-\frac{5 a^2 b^2 (a B+A b)}{8 x^{16}}-\frac{b^4 (5 a B+A b)}{12 x^{12}}-\frac{5 a b^3 (2 a B+A b)}{14 x^{14}}-\frac{b^5 B}{10 x^{10}} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x^2)^5*(A + B*x^2))/x^23,x]

[Out]

-(a^5*A)/(22*x^22) - (a^4*(5*A*b + a*B))/(20*x^20) - (5*a^3*b*(2*A*b + a*B))/(18
*x^18) - (5*a^2*b^2*(A*b + a*B))/(8*x^16) - (5*a*b^3*(A*b + 2*a*B))/(14*x^14) -
(b^4*(A*b + 5*a*B))/(12*x^12) - (b^5*B)/(10*x^10)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 30.111, size = 116, normalized size = 0.99 \[ - \frac{A a^{5}}{22 x^{22}} - \frac{B b^{5}}{10 x^{10}} - \frac{a^{4} \left (5 A b + B a\right )}{20 x^{20}} - \frac{5 a^{3} b \left (2 A b + B a\right )}{18 x^{18}} - \frac{5 a^{2} b^{2} \left (A b + B a\right )}{8 x^{16}} - \frac{5 a b^{3} \left (A b + 2 B a\right )}{14 x^{14}} - \frac{b^{4} \left (A b + 5 B a\right )}{12 x^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**5*(B*x**2+A)/x**23,x)

[Out]

-A*a**5/(22*x**22) - B*b**5/(10*x**10) - a**4*(5*A*b + B*a)/(20*x**20) - 5*a**3*
b*(2*A*b + B*a)/(18*x**18) - 5*a**2*b**2*(A*b + B*a)/(8*x**16) - 5*a*b**3*(A*b +
2*B*a)/(14*x**14) - b**4*(A*b + 5*B*a)/(12*x**12)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0611542, size = 121, normalized size = 1.03 \[ -\frac{126 a^5 \left (10 A+11 B x^2\right )+770 a^4 b x^2 \left (9 A+10 B x^2\right )+1925 a^3 b^2 x^4 \left (8 A+9 B x^2\right )+2475 a^2 b^3 x^6 \left (7 A+8 B x^2\right )+1650 a b^4 x^8 \left (6 A+7 B x^2\right )+462 b^5 x^{10} \left (5 A+6 B x^2\right )}{27720 x^{22}} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x^2)^5*(A + B*x^2))/x^23,x]

[Out]

-(462*b^5*x^10*(5*A + 6*B*x^2) + 1650*a*b^4*x^8*(6*A + 7*B*x^2) + 2475*a^2*b^3*x
^6*(7*A + 8*B*x^2) + 1925*a^3*b^2*x^4*(8*A + 9*B*x^2) + 770*a^4*b*x^2*(9*A + 10*
B*x^2) + 126*a^5*(10*A + 11*B*x^2))/(27720*x^22)

_______________________________________________________________________________________

Maple [A]  time = 0.01, size = 104, normalized size = 0.9 \[ -{\frac{A{a}^{5}}{22\,{x}^{22}}}-{\frac{{a}^{4} \left ( 5\,Ab+Ba \right ) }{20\,{x}^{20}}}-{\frac{5\,{a}^{3}b \left ( 2\,Ab+Ba \right ) }{18\,{x}^{18}}}-{\frac{5\,{a}^{2}{b}^{2} \left ( Ab+Ba \right ) }{8\,{x}^{16}}}-{\frac{5\,a{b}^{3} \left ( Ab+2\,Ba \right ) }{14\,{x}^{14}}}-{\frac{{b}^{4} \left ( Ab+5\,Ba \right ) }{12\,{x}^{12}}}-{\frac{B{b}^{5}}{10\,{x}^{10}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^5*(B*x^2+A)/x^23,x)

[Out]

-1/22*a^5*A/x^22-1/20*a^4*(5*A*b+B*a)/x^20-5/18*a^3*b*(2*A*b+B*a)/x^18-5/8*a^2*b
^2*(A*b+B*a)/x^16-5/14*a*b^3*(A*b+2*B*a)/x^14-1/12*b^4*(A*b+5*B*a)/x^12-1/10*b^5
*B/x^10

_______________________________________________________________________________________

Maxima [A]  time = 1.35666, size = 163, normalized size = 1.39 \[ -\frac{2772 \, B b^{5} x^{12} + 2310 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{10} + 9900 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{8} + 17325 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{6} + 1260 \, A a^{5} + 7700 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} + 1386 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2}}{27720 \, x^{22}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(b*x^2 + a)^5/x^23,x, algorithm="maxima")

[Out]

-1/27720*(2772*B*b^5*x^12 + 2310*(5*B*a*b^4 + A*b^5)*x^10 + 9900*(2*B*a^2*b^3 +
A*a*b^4)*x^8 + 17325*(B*a^3*b^2 + A*a^2*b^3)*x^6 + 1260*A*a^5 + 7700*(B*a^4*b +
2*A*a^3*b^2)*x^4 + 1386*(B*a^5 + 5*A*a^4*b)*x^2)/x^22

_______________________________________________________________________________________

Fricas [A]  time = 0.218847, size = 163, normalized size = 1.39 \[ -\frac{2772 \, B b^{5} x^{12} + 2310 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{10} + 9900 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{8} + 17325 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{6} + 1260 \, A a^{5} + 7700 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} + 1386 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2}}{27720 \, x^{22}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(b*x^2 + a)^5/x^23,x, algorithm="fricas")

[Out]

-1/27720*(2772*B*b^5*x^12 + 2310*(5*B*a*b^4 + A*b^5)*x^10 + 9900*(2*B*a^2*b^3 +
A*a*b^4)*x^8 + 17325*(B*a^3*b^2 + A*a^2*b^3)*x^6 + 1260*A*a^5 + 7700*(B*a^4*b +
2*A*a^3*b^2)*x^4 + 1386*(B*a^5 + 5*A*a^4*b)*x^2)/x^22

_______________________________________________________________________________________

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**2+a)**5*(B*x**2+A)/x**23,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.224371, size = 171, normalized size = 1.46 \[ -\frac{2772 \, B b^{5} x^{12} + 11550 \, B a b^{4} x^{10} + 2310 \, A b^{5} x^{10} + 19800 \, B a^{2} b^{3} x^{8} + 9900 \, A a b^{4} x^{8} + 17325 \, B a^{3} b^{2} x^{6} + 17325 \, A a^{2} b^{3} x^{6} + 7700 \, B a^{4} b x^{4} + 15400 \, A a^{3} b^{2} x^{4} + 1386 \, B a^{5} x^{2} + 6930 \, A a^{4} b x^{2} + 1260 \, A a^{5}}{27720 \, x^{22}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(b*x^2 + a)^5/x^23,x, algorithm="giac")

[Out]

-1/27720*(2772*B*b^5*x^12 + 11550*B*a*b^4*x^10 + 2310*A*b^5*x^10 + 19800*B*a^2*b
^3*x^8 + 9900*A*a*b^4*x^8 + 17325*B*a^3*b^2*x^6 + 17325*A*a^2*b^3*x^6 + 7700*B*a
^4*b*x^4 + 15400*A*a^3*b^2*x^4 + 1386*B*a^5*x^2 + 6930*A*a^4*b*x^2 + 1260*A*a^5)
/x^22