3.19 \(\int \frac{\left (a+b x^3\right )^2 \left (A+B x^3\right )}{x^6} \, dx\)

Optimal. Leaf size=50 \[ -\frac{a^2 A}{5 x^5}-\frac{a (a B+2 A b)}{2 x^2}+b x (2 a B+A b)+\frac{1}{4} b^2 B x^4 \]

[Out]

-(a^2*A)/(5*x^5) - (a*(2*A*b + a*B))/(2*x^2) + b*(A*b + 2*a*B)*x + (b^2*B*x^4)/4

_______________________________________________________________________________________

Rubi [A]  time = 0.0937064, antiderivative size = 50, 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 \[ -\frac{a^2 A}{5 x^5}-\frac{a (a B+2 A b)}{2 x^2}+b x (2 a B+A b)+\frac{1}{4} b^2 B x^4 \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x^3)^2*(A + B*x^3))/x^6,x]

[Out]

-(a^2*A)/(5*x^5) - (a*(2*A*b + a*B))/(2*x^2) + b*(A*b + 2*a*B)*x + (b^2*B*x^4)/4

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**3+a)**2*(B*x**3+A)/x**6,x)

[Out]

-A*a**2/(5*x**5) + B*b**2*x**4/4 - a*(2*A*b + B*a)/(2*x**2) + b*(A*b + 2*B*a)*In
tegral(A, x)/A

_______________________________________________________________________________________

Mathematica [A]  time = 0.0366249, size = 50, normalized size = 1. \[ -\frac{a^2 A}{5 x^5}-\frac{a (a B+2 A b)}{2 x^2}+b x (2 a B+A b)+\frac{1}{4} b^2 B x^4 \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x^3)^2*(A + B*x^3))/x^6,x]

[Out]

-(a^2*A)/(5*x^5) - (a*(2*A*b + a*B))/(2*x^2) + b*(A*b + 2*a*B)*x + (b^2*B*x^4)/4

_______________________________________________________________________________________

Maple [A]  time = 0.007, size = 46, normalized size = 0.9 \[{\frac{{b}^{2}B{x}^{4}}{4}}+Ax{b}^{2}+2\,Bxab-{\frac{a \left ( 2\,Ab+Ba \right ) }{2\,{x}^{2}}}-{\frac{A{a}^{2}}{5\,{x}^{5}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^3+a)^2*(B*x^3+A)/x^6,x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 1.41779, size = 69, normalized size = 1.38 \[ \frac{1}{4} \, B b^{2} x^{4} +{\left (2 \, B a b + A b^{2}\right )} x - \frac{5 \,{\left (B a^{2} + 2 \, A a b\right )} x^{3} + 2 \, A a^{2}}{10 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*(b*x^3 + a)^2/x^6,x, algorithm="maxima")

[Out]

1/4*B*b^2*x^4 + (2*B*a*b + A*b^2)*x - 1/10*(5*(B*a^2 + 2*A*a*b)*x^3 + 2*A*a^2)/x
^5

_______________________________________________________________________________________

Fricas [A]  time = 0.221617, size = 72, normalized size = 1.44 \[ \frac{5 \, B b^{2} x^{9} + 20 \,{\left (2 \, B a b + A b^{2}\right )} x^{6} - 10 \,{\left (B a^{2} + 2 \, A a b\right )} x^{3} - 4 \, A a^{2}}{20 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*(b*x^3 + a)^2/x^6,x, algorithm="fricas")

[Out]

1/20*(5*B*b^2*x^9 + 20*(2*B*a*b + A*b^2)*x^6 - 10*(B*a^2 + 2*A*a*b)*x^3 - 4*A*a^
2)/x^5

_______________________________________________________________________________________

Sympy [A]  time = 1.90952, size = 51, normalized size = 1.02 \[ \frac{B b^{2} x^{4}}{4} + x \left (A b^{2} + 2 B a b\right ) - \frac{2 A a^{2} + x^{3} \left (10 A a b + 5 B a^{2}\right )}{10 x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**3+a)**2*(B*x**3+A)/x**6,x)

[Out]

B*b**2*x**4/4 + x*(A*b**2 + 2*B*a*b) - (2*A*a**2 + x**3*(10*A*a*b + 5*B*a**2))/(
10*x**5)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.212217, size = 69, normalized size = 1.38 \[ \frac{1}{4} \, B b^{2} x^{4} + 2 \, B a b x + A b^{2} x - \frac{5 \, B a^{2} x^{3} + 10 \, A a b x^{3} + 2 \, A a^{2}}{10 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*(b*x^3 + a)^2/x^6,x, algorithm="giac")

[Out]

1/4*B*b^2*x^4 + 2*B*a*b*x + A*b^2*x - 1/10*(5*B*a^2*x^3 + 10*A*a*b*x^3 + 2*A*a^2
)/x^5