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

Optimal. Leaf size=113 \[ -\frac{a^5 A}{4 x^4}-\frac{a^4 (a B+5 A b)}{x}+\frac{5}{2} a^3 b x^2 (a B+2 A b)+2 a^2 b^2 x^5 (a B+A b)+\frac{1}{11} b^4 x^{11} (5 a B+A b)+\frac{5}{8} a b^3 x^8 (2 a B+A b)+\frac{1}{14} b^5 B x^{14} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.209665, antiderivative size = 113, 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 \[ -\frac{a^5 A}{4 x^4}-\frac{a^4 (a B+5 A b)}{x}+\frac{5}{2} a^3 b x^2 (a B+2 A b)+2 a^2 b^2 x^5 (a B+A b)+\frac{1}{11} b^4 x^{11} (5 a B+A b)+\frac{5}{8} a b^3 x^8 (2 a B+A b)+\frac{1}{14} b^5 B x^{14} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{A a^{5}}{4 x^{4}} + \frac{B b^{5} x^{14}}{14} - \frac{a^{4} \left (5 A b + B a\right )}{x} + 5 a^{3} b \left (2 A b + B a\right ) \int x\, dx + 2 a^{2} b^{2} x^{5} \left (A b + B a\right ) + \frac{5 a b^{3} x^{8} \left (A b + 2 B a\right )}{8} + \frac{b^{4} x^{11} \left (A b + 5 B a\right )}{11} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-A*a**5/(4*x**4) + B*b**5*x**14/14 - a**4*(5*A*b + B*a)/x + 5*a**3*b*(2*A*b + B*
a)*Integral(x, x) + 2*a**2*b**2*x**5*(A*b + B*a) + 5*a*b**3*x**8*(A*b + 2*B*a)/8
+ b**4*x**11*(A*b + 5*B*a)/11

_______________________________________________________________________________________

Mathematica [A]  time = 0.0699595, size = 115, normalized size = 1.02 \[ -\frac{a^5 A}{4 x^4}+\frac{5}{2} a^3 b x^2 (a B+2 A b)+2 a^2 b^2 x^5 (a B+A b)+\frac{a^5 (-B)-5 a^4 A b}{x}+\frac{1}{11} b^4 x^{11} (5 a B+A b)+\frac{5}{8} a b^3 x^8 (2 a B+A b)+\frac{1}{14} b^5 B x^{14} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.009, size = 123, normalized size = 1.1 \[{\frac{{b}^{5}B{x}^{14}}{14}}+{\frac{A{x}^{11}{b}^{5}}{11}}+{\frac{5\,B{x}^{11}a{b}^{4}}{11}}+{\frac{5\,A{x}^{8}a{b}^{4}}{8}}+{\frac{5\,B{x}^{8}{a}^{2}{b}^{3}}{4}}+2\,A{x}^{5}{a}^{2}{b}^{3}+2\,B{x}^{5}{a}^{3}{b}^{2}+5\,A{x}^{2}{a}^{3}{b}^{2}+{\frac{5\,B{x}^{2}{a}^{4}b}{2}}-{\frac{A{a}^{5}}{4\,{x}^{4}}}-{\frac{{a}^{4} \left ( 5\,Ab+Ba \right ) }{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 1.36066, size = 163, normalized size = 1.44 \[ \frac{1}{14} \, B b^{5} x^{14} + \frac{1}{11} \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{11} + \frac{5}{8} \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{8} + 2 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{5} + \frac{5}{2} \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} - \frac{A a^{5} + 4 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{3}}{4 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.220852, size = 163, normalized size = 1.44 \[ \frac{44 \, B b^{5} x^{18} + 56 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{15} + 385 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{12} + 1232 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{9} + 1540 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{6} - 154 \, A a^{5} - 616 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{3}}{616 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/616*(44*B*b^5*x^18 + 56*(5*B*a*b^4 + A*b^5)*x^15 + 385*(2*B*a^2*b^3 + A*a*b^4)
*x^12 + 1232*(B*a^3*b^2 + A*a^2*b^3)*x^9 + 1540*(B*a^4*b + 2*A*a^3*b^2)*x^6 - 15
4*A*a^5 - 616*(B*a^5 + 5*A*a^4*b)*x^3)/x^4

_______________________________________________________________________________________

Sympy [A]  time = 2.17528, size = 131, normalized size = 1.16 \[ \frac{B b^{5} x^{14}}{14} + x^{11} \left (\frac{A b^{5}}{11} + \frac{5 B a b^{4}}{11}\right ) + x^{8} \left (\frac{5 A a b^{4}}{8} + \frac{5 B a^{2} b^{3}}{4}\right ) + x^{5} \left (2 A a^{2} b^{3} + 2 B a^{3} b^{2}\right ) + x^{2} \left (5 A a^{3} b^{2} + \frac{5 B a^{4} b}{2}\right ) - \frac{A a^{5} + x^{3} \left (20 A a^{4} b + 4 B a^{5}\right )}{4 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.215891, size = 171, normalized size = 1.51 \[ \frac{1}{14} \, B b^{5} x^{14} + \frac{5}{11} \, B a b^{4} x^{11} + \frac{1}{11} \, A b^{5} x^{11} + \frac{5}{4} \, B a^{2} b^{3} x^{8} + \frac{5}{8} \, A a b^{4} x^{8} + 2 \, B a^{3} b^{2} x^{5} + 2 \, A a^{2} b^{3} x^{5} + \frac{5}{2} \, B a^{4} b x^{2} + 5 \, A a^{3} b^{2} x^{2} - \frac{4 \, B a^{5} x^{3} + 20 \, A a^{4} b x^{3} + A a^{5}}{4 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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