3.48 \(\int \frac{\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x^{16}} \, dx\)

Optimal. Leaf size=117 \[ -\frac{a^5 A}{15 x^{15}}-\frac{a^4 (a B+5 A b)}{13 x^{13}}-\frac{5 a^3 b (a B+2 A b)}{11 x^{11}}-\frac{10 a^2 b^2 (a B+A b)}{9 x^9}-\frac{b^4 (5 a B+A b)}{5 x^5}-\frac{5 a b^3 (2 a B+A b)}{7 x^7}-\frac{b^5 B}{3 x^3} \]

[Out]

-(a^5*A)/(15*x^15) - (a^4*(5*A*b + a*B))/(13*x^13) - (5*a^3*b*(2*A*b + a*B))/(11
*x^11) - (10*a^2*b^2*(A*b + a*B))/(9*x^9) - (5*a*b^3*(A*b + 2*a*B))/(7*x^7) - (b
^4*(A*b + 5*a*B))/(5*x^5) - (b^5*B)/(3*x^3)

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Rubi [A]  time = 0.196982, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05 \[ -\frac{a^5 A}{15 x^{15}}-\frac{a^4 (a B+5 A b)}{13 x^{13}}-\frac{5 a^3 b (a B+2 A b)}{11 x^{11}}-\frac{10 a^2 b^2 (a B+A b)}{9 x^9}-\frac{b^4 (5 a B+A b)}{5 x^5}-\frac{5 a b^3 (2 a B+A b)}{7 x^7}-\frac{b^5 B}{3 x^3} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x^2)^5*(A + B*x^2))/x^16,x]

[Out]

-(a^5*A)/(15*x^15) - (a^4*(5*A*b + a*B))/(13*x^13) - (5*a^3*b*(2*A*b + a*B))/(11
*x^11) - (10*a^2*b^2*(A*b + a*B))/(9*x^9) - (5*a*b^3*(A*b + 2*a*B))/(7*x^7) - (b
^4*(A*b + 5*a*B))/(5*x^5) - (b^5*B)/(3*x^3)

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Rubi in Sympy [A]  time = 24.2368, size = 116, normalized size = 0.99 \[ - \frac{A a^{5}}{15 x^{15}} - \frac{B b^{5}}{3 x^{3}} - \frac{a^{4} \left (5 A b + B a\right )}{13 x^{13}} - \frac{5 a^{3} b \left (2 A b + B a\right )}{11 x^{11}} - \frac{10 a^{2} b^{2} \left (A b + B a\right )}{9 x^{9}} - \frac{5 a b^{3} \left (A b + 2 B a\right )}{7 x^{7}} - \frac{b^{4} \left (A b + 5 B a\right )}{5 x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**5*(B*x**2+A)/x**16,x)

[Out]

-A*a**5/(15*x**15) - B*b**5/(3*x**3) - a**4*(5*A*b + B*a)/(13*x**13) - 5*a**3*b*
(2*A*b + B*a)/(11*x**11) - 10*a**2*b**2*(A*b + B*a)/(9*x**9) - 5*a*b**3*(A*b + 2
*B*a)/(7*x**7) - b**4*(A*b + 5*B*a)/(5*x**5)

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Mathematica [A]  time = 0.0545849, size = 121, normalized size = 1.03 \[ -\frac{231 a^5 \left (13 A+15 B x^2\right )+1575 a^4 b x^2 \left (11 A+13 B x^2\right )+4550 a^3 b^2 x^4 \left (9 A+11 B x^2\right )+7150 a^2 b^3 x^6 \left (7 A+9 B x^2\right )+6435 a b^4 x^8 \left (5 A+7 B x^2\right )+3003 b^5 x^{10} \left (3 A+5 B x^2\right )}{45045 x^{15}} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x^2)^5*(A + B*x^2))/x^16,x]

[Out]

-(3003*b^5*x^10*(3*A + 5*B*x^2) + 6435*a*b^4*x^8*(5*A + 7*B*x^2) + 7150*a^2*b^3*
x^6*(7*A + 9*B*x^2) + 4550*a^3*b^2*x^4*(9*A + 11*B*x^2) + 1575*a^4*b*x^2*(11*A +
13*B*x^2) + 231*a^5*(13*A + 15*B*x^2))/(45045*x^15)

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Maple [A]  time = 0.009, size = 104, normalized size = 0.9 \[ -{\frac{A{a}^{5}}{15\,{x}^{15}}}-{\frac{{a}^{4} \left ( 5\,Ab+Ba \right ) }{13\,{x}^{13}}}-{\frac{5\,{a}^{3}b \left ( 2\,Ab+Ba \right ) }{11\,{x}^{11}}}-{\frac{10\,{a}^{2}{b}^{2} \left ( Ab+Ba \right ) }{9\,{x}^{9}}}-{\frac{5\,a{b}^{3} \left ( Ab+2\,Ba \right ) }{7\,{x}^{7}}}-{\frac{{b}^{4} \left ( Ab+5\,Ba \right ) }{5\,{x}^{5}}}-{\frac{B{b}^{5}}{3\,{x}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^5*(B*x^2+A)/x^16,x)

[Out]

-1/15*a^5*A/x^15-1/13*a^4*(5*A*b+B*a)/x^13-5/11*a^3*b*(2*A*b+B*a)/x^11-10/9*a^2*
b^2*(A*b+B*a)/x^9-5/7*a*b^3*(A*b+2*B*a)/x^7-1/5*b^4*(A*b+5*B*a)/x^5-1/3*b^5*B/x^
3

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Maxima [A]  time = 1.32595, size = 163, normalized size = 1.39 \[ -\frac{15015 \, B b^{5} x^{12} + 9009 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{10} + 32175 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{8} + 50050 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{6} + 3003 \, A a^{5} + 20475 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} + 3465 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2}}{45045 \, x^{15}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(b*x^2 + a)^5/x^16,x, algorithm="maxima")

[Out]

-1/45045*(15015*B*b^5*x^12 + 9009*(5*B*a*b^4 + A*b^5)*x^10 + 32175*(2*B*a^2*b^3
+ A*a*b^4)*x^8 + 50050*(B*a^3*b^2 + A*a^2*b^3)*x^6 + 3003*A*a^5 + 20475*(B*a^4*b
+ 2*A*a^3*b^2)*x^4 + 3465*(B*a^5 + 5*A*a^4*b)*x^2)/x^15

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Fricas [A]  time = 0.240123, size = 163, normalized size = 1.39 \[ -\frac{15015 \, B b^{5} x^{12} + 9009 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{10} + 32175 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{8} + 50050 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{6} + 3003 \, A a^{5} + 20475 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} + 3465 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2}}{45045 \, x^{15}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(b*x^2 + a)^5/x^16,x, algorithm="fricas")

[Out]

-1/45045*(15015*B*b^5*x^12 + 9009*(5*B*a*b^4 + A*b^5)*x^10 + 32175*(2*B*a^2*b^3
+ A*a*b^4)*x^8 + 50050*(B*a^3*b^2 + A*a^2*b^3)*x^6 + 3003*A*a^5 + 20475*(B*a^4*b
+ 2*A*a^3*b^2)*x^4 + 3465*(B*a^5 + 5*A*a^4*b)*x^2)/x^15

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Sympy [A]  time = 105.46, size = 128, normalized size = 1.09 \[ - \frac{3003 A a^{5} + 15015 B b^{5} x^{12} + x^{10} \left (9009 A b^{5} + 45045 B a b^{4}\right ) + x^{8} \left (32175 A a b^{4} + 64350 B a^{2} b^{3}\right ) + x^{6} \left (50050 A a^{2} b^{3} + 50050 B a^{3} b^{2}\right ) + x^{4} \left (40950 A a^{3} b^{2} + 20475 B a^{4} b\right ) + x^{2} \left (17325 A a^{4} b + 3465 B a^{5}\right )}{45045 x^{15}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**5*(B*x**2+A)/x**16,x)

[Out]

-(3003*A*a**5 + 15015*B*b**5*x**12 + x**10*(9009*A*b**5 + 45045*B*a*b**4) + x**8
*(32175*A*a*b**4 + 64350*B*a**2*b**3) + x**6*(50050*A*a**2*b**3 + 50050*B*a**3*b
**2) + x**4*(40950*A*a**3*b**2 + 20475*B*a**4*b) + x**2*(17325*A*a**4*b + 3465*B
*a**5))/(45045*x**15)

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GIAC/XCAS [A]  time = 0.227017, size = 171, normalized size = 1.46 \[ -\frac{15015 \, B b^{5} x^{12} + 45045 \, B a b^{4} x^{10} + 9009 \, A b^{5} x^{10} + 64350 \, B a^{2} b^{3} x^{8} + 32175 \, A a b^{4} x^{8} + 50050 \, B a^{3} b^{2} x^{6} + 50050 \, A a^{2} b^{3} x^{6} + 20475 \, B a^{4} b x^{4} + 40950 \, A a^{3} b^{2} x^{4} + 3465 \, B a^{5} x^{2} + 17325 \, A a^{4} b x^{2} + 3003 \, A a^{5}}{45045 \, x^{15}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(b*x^2 + a)^5/x^16,x, algorithm="giac")

[Out]

-1/45045*(15015*B*b^5*x^12 + 45045*B*a*b^4*x^10 + 9009*A*b^5*x^10 + 64350*B*a^2*
b^3*x^8 + 32175*A*a*b^4*x^8 + 50050*B*a^3*b^2*x^6 + 50050*A*a^2*b^3*x^6 + 20475*
B*a^4*b*x^4 + 40950*A*a^3*b^2*x^4 + 3465*B*a^5*x^2 + 17325*A*a^4*b*x^2 + 3003*A*
a^5)/x^15