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

Optimal. Leaf size=48 \[ -\frac{a^2 A}{5 x^5}-\frac{a (a B+2 A b)}{3 x^3}-\frac{b (2 a B+A b)}{x}+b^2 B x \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0819563, antiderivative size = 48, 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 \[ -\frac{a^2 A}{5 x^5}-\frac{a (a B+2 A b)}{3 x^3}-\frac{b (2 a B+A b)}{x}+b^2 B x \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0344791, size = 48, normalized size = 1. \[ -\frac{a^2 A}{5 x^5}-\frac{a (a B+2 A b)}{3 x^3}-\frac{b (2 a B+A b)}{x}+b^2 B x \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.008, size = 45, normalized size = 0.9 \[ -{\frac{A{a}^{2}}{5\,{x}^{5}}}-{\frac{a \left ( 2\,Ab+Ba \right ) }{3\,{x}^{3}}}-{\frac{b \left ( Ab+2\,Ba \right ) }{x}}+{b}^{2}Bx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 1.34827, size = 69, normalized size = 1.44 \[ B b^{2} x - \frac{15 \,{\left (2 \, B a b + A b^{2}\right )} x^{4} + 3 \, A a^{2} + 5 \,{\left (B a^{2} + 2 \, A a b\right )} x^{2}}{15 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*b^2*x - 1/15*(15*(2*B*a*b + A*b^2)*x^4 + 3*A*a^2 + 5*(B*a^2 + 2*A*a*b)*x^2)/x^
5

_______________________________________________________________________________________

Fricas [A]  time = 0.22893, size = 72, normalized size = 1.5 \[ \frac{15 \, B b^{2} x^{6} - 15 \,{\left (2 \, B a b + A b^{2}\right )} x^{4} - 3 \, A a^{2} - 5 \,{\left (B a^{2} + 2 \, A a b\right )} x^{2}}{15 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/15*(15*B*b^2*x^6 - 15*(2*B*a*b + A*b^2)*x^4 - 3*A*a^2 - 5*(B*a^2 + 2*A*a*b)*x^
2)/x^5

_______________________________________________________________________________________

Sympy [A]  time = 3.00719, size = 51, normalized size = 1.06 \[ B b^{2} x - \frac{3 A a^{2} + x^{4} \left (15 A b^{2} + 30 B a b\right ) + x^{2} \left (10 A a b + 5 B a^{2}\right )}{15 x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*b**2*x - (3*A*a**2 + x**4*(15*A*b**2 + 30*B*a*b) + x**2*(10*A*a*b + 5*B*a**2))
/(15*x**5)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.232235, size = 72, normalized size = 1.5 \[ B b^{2} x - \frac{30 \, B a b x^{4} + 15 \, A b^{2} x^{4} + 5 \, B a^{2} x^{2} + 10 \, A a b x^{2} + 3 \, A a^{2}}{15 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*b^2*x - 1/15*(30*B*a*b*x^4 + 15*A*b^2*x^4 + 5*B*a^2*x^2 + 10*A*a*b*x^2 + 3*A*a
^2)/x^5