3.51 \(\int \frac{\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x^{19}} \, dx\)

Optimal. Leaf size=117 \[ -\frac{a^5 A}{18 x^{18}}-\frac{a^4 (a B+5 A b)}{16 x^{16}}-\frac{5 a^3 b (a B+2 A b)}{14 x^{14}}-\frac{5 a^2 b^2 (a B+A b)}{6 x^{12}}-\frac{b^4 (5 a B+A b)}{8 x^8}-\frac{a b^3 (2 a B+A b)}{2 x^{10}}-\frac{b^5 B}{6 x^6} \]

[Out]

-(a^5*A)/(18*x^18) - (a^4*(5*A*b + a*B))/(16*x^16) - (5*a^3*b*(2*A*b + a*B))/(14
*x^14) - (5*a^2*b^2*(A*b + a*B))/(6*x^12) - (a*b^3*(A*b + 2*a*B))/(2*x^10) - (b^
4*(A*b + 5*a*B))/(8*x^8) - (b^5*B)/(6*x^6)

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Rubi [A]  time = 0.255079, 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 \[ -\frac{a^5 A}{18 x^{18}}-\frac{a^4 (a B+5 A b)}{16 x^{16}}-\frac{5 a^3 b (a B+2 A b)}{14 x^{14}}-\frac{5 a^2 b^2 (a B+A b)}{6 x^{12}}-\frac{b^4 (5 a B+A b)}{8 x^8}-\frac{a b^3 (2 a B+A b)}{2 x^{10}}-\frac{b^5 B}{6 x^6} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x^2)^5*(A + B*x^2))/x^19,x]

[Out]

-(a^5*A)/(18*x^18) - (a^4*(5*A*b + a*B))/(16*x^16) - (5*a^3*b*(2*A*b + a*B))/(14
*x^14) - (5*a^2*b^2*(A*b + a*B))/(6*x^12) - (a*b^3*(A*b + 2*a*B))/(2*x^10) - (b^
4*(A*b + 5*a*B))/(8*x^8) - (b^5*B)/(6*x^6)

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Rubi in Sympy [A]  time = 29.7642, size = 114, normalized size = 0.97 \[ - \frac{A a^{5}}{18 x^{18}} - \frac{B b^{5}}{6 x^{6}} - \frac{a^{4} \left (5 A b + B a\right )}{16 x^{16}} - \frac{5 a^{3} b \left (2 A b + B a\right )}{14 x^{14}} - \frac{5 a^{2} b^{2} \left (A b + B a\right )}{6 x^{12}} - \frac{a b^{3} \left (A b + 2 B a\right )}{2 x^{10}} - \frac{b^{4} \left (A b + 5 B a\right )}{8 x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**5*(B*x**2+A)/x**19,x)

[Out]

-A*a**5/(18*x**18) - B*b**5/(6*x**6) - a**4*(5*A*b + B*a)/(16*x**16) - 5*a**3*b*
(2*A*b + B*a)/(14*x**14) - 5*a**2*b**2*(A*b + B*a)/(6*x**12) - a*b**3*(A*b + 2*B
*a)/(2*x**10) - b**4*(A*b + 5*B*a)/(8*x**8)

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Mathematica [A]  time = 0.059305, size = 121, normalized size = 1.03 \[ -\frac{7 a^5 \left (8 A+9 B x^2\right )+45 a^4 b x^2 \left (7 A+8 B x^2\right )+120 a^3 b^2 x^4 \left (6 A+7 B x^2\right )+168 a^2 b^3 x^6 \left (5 A+6 B x^2\right )+126 a b^4 x^8 \left (4 A+5 B x^2\right )+42 b^5 x^{10} \left (3 A+4 B x^2\right )}{1008 x^{18}} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x^2)^5*(A + B*x^2))/x^19,x]

[Out]

-(42*b^5*x^10*(3*A + 4*B*x^2) + 126*a*b^4*x^8*(4*A + 5*B*x^2) + 168*a^2*b^3*x^6*
(5*A + 6*B*x^2) + 120*a^3*b^2*x^4*(6*A + 7*B*x^2) + 45*a^4*b*x^2*(7*A + 8*B*x^2)
+ 7*a^5*(8*A + 9*B*x^2))/(1008*x^18)

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Maple [A]  time = 0.01, size = 104, normalized size = 0.9 \[ -{\frac{A{a}^{5}}{18\,{x}^{18}}}-{\frac{{a}^{4} \left ( 5\,Ab+Ba \right ) }{16\,{x}^{16}}}-{\frac{5\,{a}^{3}b \left ( 2\,Ab+Ba \right ) }{14\,{x}^{14}}}-{\frac{5\,{a}^{2}{b}^{2} \left ( Ab+Ba \right ) }{6\,{x}^{12}}}-{\frac{a{b}^{3} \left ( Ab+2\,Ba \right ) }{2\,{x}^{10}}}-{\frac{{b}^{4} \left ( Ab+5\,Ba \right ) }{8\,{x}^{8}}}-{\frac{B{b}^{5}}{6\,{x}^{6}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^5*(B*x^2+A)/x^19,x)

[Out]

-1/18*a^5*A/x^18-1/16*a^4*(5*A*b+B*a)/x^16-5/14*a^3*b*(2*A*b+B*a)/x^14-5/6*a^2*b
^2*(A*b+B*a)/x^12-1/2*a*b^3*(A*b+2*B*a)/x^10-1/8*b^4*(A*b+5*B*a)/x^8-1/6*b^5*B/x
^6

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Maxima [A]  time = 1.38, size = 163, normalized size = 1.39 \[ -\frac{168 \, B b^{5} x^{12} + 126 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{10} + 504 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{8} + 840 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{6} + 56 \, A a^{5} + 360 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} + 63 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2}}{1008 \, x^{18}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(b*x^2 + a)^5/x^19,x, algorithm="maxima")

[Out]

-1/1008*(168*B*b^5*x^12 + 126*(5*B*a*b^4 + A*b^5)*x^10 + 504*(2*B*a^2*b^3 + A*a*
b^4)*x^8 + 840*(B*a^3*b^2 + A*a^2*b^3)*x^6 + 56*A*a^5 + 360*(B*a^4*b + 2*A*a^3*b
^2)*x^4 + 63*(B*a^5 + 5*A*a^4*b)*x^2)/x^18

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Fricas [A]  time = 0.218908, size = 163, normalized size = 1.39 \[ -\frac{168 \, B b^{5} x^{12} + 126 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{10} + 504 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{8} + 840 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{6} + 56 \, A a^{5} + 360 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} + 63 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2}}{1008 \, x^{18}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(b*x^2 + a)^5/x^19,x, algorithm="fricas")

[Out]

-1/1008*(168*B*b^5*x^12 + 126*(5*B*a*b^4 + A*b^5)*x^10 + 504*(2*B*a^2*b^3 + A*a*
b^4)*x^8 + 840*(B*a^3*b^2 + A*a^2*b^3)*x^6 + 56*A*a^5 + 360*(B*a^4*b + 2*A*a^3*b
^2)*x^4 + 63*(B*a^5 + 5*A*a^4*b)*x^2)/x^18

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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**2+a)**5*(B*x**2+A)/x**19,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.224814, size = 171, normalized size = 1.46 \[ -\frac{168 \, B b^{5} x^{12} + 630 \, B a b^{4} x^{10} + 126 \, A b^{5} x^{10} + 1008 \, B a^{2} b^{3} x^{8} + 504 \, A a b^{4} x^{8} + 840 \, B a^{3} b^{2} x^{6} + 840 \, A a^{2} b^{3} x^{6} + 360 \, B a^{4} b x^{4} + 720 \, A a^{3} b^{2} x^{4} + 63 \, B a^{5} x^{2} + 315 \, A a^{4} b x^{2} + 56 \, A a^{5}}{1008 \, x^{18}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(b*x^2 + a)^5/x^19,x, algorithm="giac")

[Out]

-1/1008*(168*B*b^5*x^12 + 630*B*a*b^4*x^10 + 126*A*b^5*x^10 + 1008*B*a^2*b^3*x^8
+ 504*A*a*b^4*x^8 + 840*B*a^3*b^2*x^6 + 840*A*a^2*b^3*x^6 + 360*B*a^4*b*x^4 + 7
20*A*a^3*b^2*x^4 + 63*B*a^5*x^2 + 315*A*a^4*b*x^2 + 56*A*a^5)/x^18