3.22 \(\int \frac{\left (a+b x^2\right )^2 \left (A+B x^2\right )}{x^9} \, dx\)

Optimal. Leaf size=48 \[ \frac{\left (a+b x^2\right )^3 (A b-4 a B)}{24 a^2 x^6}-\frac{A \left (a+b x^2\right )^3}{8 a x^8} \]

[Out]

-(A*(a + b*x^2)^3)/(8*a*x^8) + ((A*b - 4*a*B)*(a + b*x^2)^3)/(24*a^2*x^6)

_______________________________________________________________________________________

Rubi [A]  time = 0.107831, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ \frac{\left (a+b x^2\right )^3 (A b-4 a B)}{24 a^2 x^6}-\frac{A \left (a+b x^2\right )^3}{8 a x^8} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x^2)^2*(A + B*x^2))/x^9,x]

[Out]

-(A*(a + b*x^2)^3)/(8*a*x^8) + ((A*b - 4*a*B)*(a + b*x^2)^3)/(24*a^2*x^6)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 15.6937, size = 51, normalized size = 1.06 \[ - \frac{A a^{2}}{8 x^{8}} - \frac{B b^{2}}{2 x^{2}} - \frac{a \left (2 A b + B a\right )}{6 x^{6}} - \frac{b \left (A b + 2 B a\right )}{4 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**2*(B*x**2+A)/x**9,x)

[Out]

-A*a**2/(8*x**8) - B*b**2/(2*x**2) - a*(2*A*b + B*a)/(6*x**6) - b*(A*b + 2*B*a)/
(4*x**4)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0285649, size = 55, normalized size = 1.15 \[ -\frac{a^2 \left (3 A+4 B x^2\right )+4 a b x^2 \left (2 A+3 B x^2\right )+6 b^2 x^4 \left (A+2 B x^2\right )}{24 x^8} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x^2)^2*(A + B*x^2))/x^9,x]

[Out]

-(6*b^2*x^4*(A + 2*B*x^2) + 4*a*b*x^2*(2*A + 3*B*x^2) + a^2*(3*A + 4*B*x^2))/(24
*x^8)

_______________________________________________________________________________________

Maple [A]  time = 0.008, size = 48, normalized size = 1. \[ -{\frac{a \left ( 2\,Ab+Ba \right ) }{6\,{x}^{6}}}-{\frac{b \left ( Ab+2\,Ba \right ) }{4\,{x}^{4}}}-{\frac{A{a}^{2}}{8\,{x}^{8}}}-{\frac{B{b}^{2}}{2\,{x}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^2*(B*x^2+A)/x^9,x)

[Out]

-1/6*a*(2*A*b+B*a)/x^6-1/4*b*(A*b+2*B*a)/x^4-1/8*A*a^2/x^8-1/2*B*b^2/x^2

_______________________________________________________________________________________

Maxima [A]  time = 1.34502, size = 72, normalized size = 1.5 \[ -\frac{12 \, B b^{2} x^{6} + 6 \,{\left (2 \, B a b + A b^{2}\right )} x^{4} + 3 \, A a^{2} + 4 \,{\left (B a^{2} + 2 \, A a b\right )} x^{2}}{24 \, x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(b*x^2 + a)^2/x^9,x, algorithm="maxima")

[Out]

-1/24*(12*B*b^2*x^6 + 6*(2*B*a*b + A*b^2)*x^4 + 3*A*a^2 + 4*(B*a^2 + 2*A*a*b)*x^
2)/x^8

_______________________________________________________________________________________

Fricas [A]  time = 0.234853, size = 72, normalized size = 1.5 \[ -\frac{12 \, B b^{2} x^{6} + 6 \,{\left (2 \, B a b + A b^{2}\right )} x^{4} + 3 \, A a^{2} + 4 \,{\left (B a^{2} + 2 \, A a b\right )} x^{2}}{24 \, x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(b*x^2 + a)^2/x^9,x, algorithm="fricas")

[Out]

-1/24*(12*B*b^2*x^6 + 6*(2*B*a*b + A*b^2)*x^4 + 3*A*a^2 + 4*(B*a^2 + 2*A*a*b)*x^
2)/x^8

_______________________________________________________________________________________

Sympy [A]  time = 6.60162, size = 56, normalized size = 1.17 \[ - \frac{3 A a^{2} + 12 B b^{2} x^{6} + x^{4} \left (6 A b^{2} + 12 B a b\right ) + x^{2} \left (8 A a b + 4 B a^{2}\right )}{24 x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**2*(B*x**2+A)/x**9,x)

[Out]

-(3*A*a**2 + 12*B*b**2*x**6 + x**4*(6*A*b**2 + 12*B*a*b) + x**2*(8*A*a*b + 4*B*a
**2))/(24*x**8)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.221465, size = 74, normalized size = 1.54 \[ -\frac{12 \, B b^{2} x^{6} + 12 \, B a b x^{4} + 6 \, A b^{2} x^{4} + 4 \, B a^{2} x^{2} + 8 \, A a b x^{2} + 3 \, A a^{2}}{24 \, x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(b*x^2 + a)^2/x^9,x, algorithm="giac")

[Out]

-1/24*(12*B*b^2*x^6 + 12*B*a*b*x^4 + 6*A*b^2*x^4 + 4*B*a^2*x^2 + 8*A*a*b*x^2 + 3
*A*a^2)/x^8