3.53 \(\int \frac{\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x^{21}} \, dx\)

Optimal. Leaf size=117 \[ -\frac{a^5 A}{20 x^{20}}-\frac{a^4 (a B+5 A b)}{17 x^{17}}-\frac{5 a^3 b (a B+2 A b)}{14 x^{14}}-\frac{10 a^2 b^2 (a B+A b)}{11 x^{11}}-\frac{b^4 (5 a B+A b)}{5 x^5}-\frac{5 a b^3 (2 a B+A b)}{8 x^8}-\frac{b^5 B}{2 x^2} \]

[Out]

-(a^5*A)/(20*x^20) - (a^4*(5*A*b + a*B))/(17*x^17) - (5*a^3*b*(2*A*b + a*B))/(14
*x^14) - (10*a^2*b^2*(A*b + a*B))/(11*x^11) - (5*a*b^3*(A*b + 2*a*B))/(8*x^8) -
(b^4*(A*b + 5*a*B))/(5*x^5) - (b^5*B)/(2*x^2)

_______________________________________________________________________________________

Rubi [A]  time = 0.20038, 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 \[ -\frac{a^5 A}{20 x^{20}}-\frac{a^4 (a B+5 A b)}{17 x^{17}}-\frac{5 a^3 b (a B+2 A b)}{14 x^{14}}-\frac{10 a^2 b^2 (a B+A b)}{11 x^{11}}-\frac{b^4 (5 a B+A b)}{5 x^5}-\frac{5 a b^3 (2 a B+A b)}{8 x^8}-\frac{b^5 B}{2 x^2} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x^3)^5*(A + B*x^3))/x^21,x]

[Out]

-(a^5*A)/(20*x^20) - (a^4*(5*A*b + a*B))/(17*x^17) - (5*a^3*b*(2*A*b + a*B))/(14
*x^14) - (10*a^2*b^2*(A*b + a*B))/(11*x^11) - (5*a*b^3*(A*b + 2*a*B))/(8*x^8) -
(b^4*(A*b + 5*a*B))/(5*x^5) - (b^5*B)/(2*x^2)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 22.7601, size = 116, normalized size = 0.99 \[ - \frac{A a^{5}}{20 x^{20}} - \frac{B b^{5}}{2 x^{2}} - \frac{a^{4} \left (5 A b + B a\right )}{17 x^{17}} - \frac{5 a^{3} b \left (2 A b + B a\right )}{14 x^{14}} - \frac{10 a^{2} b^{2} \left (A b + B a\right )}{11 x^{11}} - \frac{5 a b^{3} \left (A b + 2 B a\right )}{8 x^{8}} - \frac{b^{4} \left (A b + 5 B a\right )}{5 x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**3+a)**5*(B*x**3+A)/x**21,x)

[Out]

-A*a**5/(20*x**20) - B*b**5/(2*x**2) - a**4*(5*A*b + B*a)/(17*x**17) - 5*a**3*b*
(2*A*b + B*a)/(14*x**14) - 10*a**2*b**2*(A*b + B*a)/(11*x**11) - 5*a*b**3*(A*b +
2*B*a)/(8*x**8) - b**4*(A*b + 5*B*a)/(5*x**5)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0677279, size = 121, normalized size = 1.03 \[ -\frac{154 a^5 \left (17 A+20 B x^3\right )+1100 a^4 b x^3 \left (14 A+17 B x^3\right )+3400 a^3 b^2 x^6 \left (11 A+14 B x^3\right )+5950 a^2 b^3 x^9 \left (8 A+11 B x^3\right )+6545 a b^4 x^{12} \left (5 A+8 B x^3\right )+5236 b^5 x^{15} \left (2 A+5 B x^3\right )}{52360 x^{20}} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x^3)^5*(A + B*x^3))/x^21,x]

[Out]

-(5236*b^5*x^15*(2*A + 5*B*x^3) + 6545*a*b^4*x^12*(5*A + 8*B*x^3) + 5950*a^2*b^3
*x^9*(8*A + 11*B*x^3) + 3400*a^3*b^2*x^6*(11*A + 14*B*x^3) + 1100*a^4*b*x^3*(14*
A + 17*B*x^3) + 154*a^5*(17*A + 20*B*x^3))/(52360*x^20)

_______________________________________________________________________________________

Maple [A]  time = 0.009, size = 104, normalized size = 0.9 \[ -{\frac{A{a}^{5}}{20\,{x}^{20}}}-{\frac{{a}^{4} \left ( 5\,Ab+Ba \right ) }{17\,{x}^{17}}}-{\frac{5\,{a}^{3}b \left ( 2\,Ab+Ba \right ) }{14\,{x}^{14}}}-{\frac{10\,{a}^{2}{b}^{2} \left ( Ab+Ba \right ) }{11\,{x}^{11}}}-{\frac{5\,a{b}^{3} \left ( Ab+2\,Ba \right ) }{8\,{x}^{8}}}-{\frac{{b}^{4} \left ( Ab+5\,Ba \right ) }{5\,{x}^{5}}}-{\frac{B{b}^{5}}{2\,{x}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^3+a)^5*(B*x^3+A)/x^21,x)

[Out]

-1/20*a^5*A/x^20-1/17*a^4*(5*A*b+B*a)/x^17-5/14*a^3*b*(2*A*b+B*a)/x^14-10/11*a^2
*b^2*(A*b+B*a)/x^11-5/8*a*b^3*(A*b+2*B*a)/x^8-1/5*b^4*(A*b+5*B*a)/x^5-1/2*b^5*B/
x^2

_______________________________________________________________________________________

Maxima [A]  time = 1.36565, size = 163, normalized size = 1.39 \[ -\frac{26180 \, B b^{5} x^{18} + 10472 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{15} + 32725 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{12} + 47600 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{9} + 18700 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{6} + 2618 \, A a^{5} + 3080 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{3}}{52360 \, x^{20}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*(b*x^3 + a)^5/x^21,x, algorithm="maxima")

[Out]

-1/52360*(26180*B*b^5*x^18 + 10472*(5*B*a*b^4 + A*b^5)*x^15 + 32725*(2*B*a^2*b^3
+ A*a*b^4)*x^12 + 47600*(B*a^3*b^2 + A*a^2*b^3)*x^9 + 18700*(B*a^4*b + 2*A*a^3*
b^2)*x^6 + 2618*A*a^5 + 3080*(B*a^5 + 5*A*a^4*b)*x^3)/x^20

_______________________________________________________________________________________

Fricas [A]  time = 0.210278, size = 163, normalized size = 1.39 \[ -\frac{26180 \, B b^{5} x^{18} + 10472 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{15} + 32725 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{12} + 47600 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{9} + 18700 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{6} + 2618 \, A a^{5} + 3080 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{3}}{52360 \, x^{20}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*(b*x^3 + a)^5/x^21,x, algorithm="fricas")

[Out]

-1/52360*(26180*B*b^5*x^18 + 10472*(5*B*a*b^4 + A*b^5)*x^15 + 32725*(2*B*a^2*b^3
+ A*a*b^4)*x^12 + 47600*(B*a^3*b^2 + A*a^2*b^3)*x^9 + 18700*(B*a^4*b + 2*A*a^3*
b^2)*x^6 + 2618*A*a^5 + 3080*(B*a^5 + 5*A*a^4*b)*x^3)/x^20

_______________________________________________________________________________________

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**3+a)**5*(B*x**3+A)/x**21,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.215366, size = 171, normalized size = 1.46 \[ -\frac{26180 \, B b^{5} x^{18} + 52360 \, B a b^{4} x^{15} + 10472 \, A b^{5} x^{15} + 65450 \, B a^{2} b^{3} x^{12} + 32725 \, A a b^{4} x^{12} + 47600 \, B a^{3} b^{2} x^{9} + 47600 \, A a^{2} b^{3} x^{9} + 18700 \, B a^{4} b x^{6} + 37400 \, A a^{3} b^{2} x^{6} + 3080 \, B a^{5} x^{3} + 15400 \, A a^{4} b x^{3} + 2618 \, A a^{5}}{52360 \, x^{20}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*(b*x^3 + a)^5/x^21,x, algorithm="giac")

[Out]

-1/52360*(26180*B*b^5*x^18 + 52360*B*a*b^4*x^15 + 10472*A*b^5*x^15 + 65450*B*a^2
*b^3*x^12 + 32725*A*a*b^4*x^12 + 47600*B*a^3*b^2*x^9 + 47600*A*a^2*b^3*x^9 + 187
00*B*a^4*b*x^6 + 37400*A*a^3*b^2*x^6 + 3080*B*a^5*x^3 + 15400*A*a^4*b*x^3 + 2618
*A*a^5)/x^20