3.2753 \(\int \left (a+\frac{b}{x^2}\right )^2 (c x)^m \, dx\)

Optimal. Leaf size=61 \[ \frac{a^2 (c x)^{m+1}}{c (m+1)}-\frac{2 a b c (c x)^{m-1}}{1-m}-\frac{b^2 c^3 (c x)^{m-3}}{3-m} \]

[Out]

-((b^2*c^3*(c*x)^(-3 + m))/(3 - m)) - (2*a*b*c*(c*x)^(-1 + m))/(1 - m) + (a^2*(c
*x)^(1 + m))/(c*(1 + m))

_______________________________________________________________________________________

Rubi [A]  time = 0.07653, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ \frac{a^2 (c x)^{m+1}}{c (m+1)}-\frac{2 a b c (c x)^{m-1}}{1-m}-\frac{b^2 c^3 (c x)^{m-3}}{3-m} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x^2)^2*(c*x)^m,x]

[Out]

-((b^2*c^3*(c*x)^(-3 + m))/(3 - m)) - (2*a*b*c*(c*x)^(-1 + m))/(1 - m) + (a^2*(c
*x)^(1 + m))/(c*(1 + m))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 13.4688, size = 48, normalized size = 0.79 \[ \frac{a^{2} \left (c x\right )^{m + 1}}{c \left (m + 1\right )} - \frac{2 a b c \left (c x\right )^{m - 1}}{- m + 1} - \frac{b^{2} c^{3} \left (c x\right )^{m - 3}}{- m + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b/x**2)**2*(c*x)**m,x)

[Out]

a**2*(c*x)**(m + 1)/(c*(m + 1)) - 2*a*b*c*(c*x)**(m - 1)/(-m + 1) - b**2*c**3*(c
*x)**(m - 3)/(-m + 3)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0520785, size = 62, normalized size = 1.02 \[ \frac{x^4 \left (a+\frac{b}{x^2}\right )^2 (c x)^m \left (\frac{a^2 x}{m+1}+\frac{2 a b}{(m-1) x}+\frac{b^2}{(m-3) x^3}\right )}{\left (a x^2+b\right )^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x^2)^2*(c*x)^m,x]

[Out]

((a + b/x^2)^2*x^4*(c*x)^m*(b^2/((-3 + m)*x^3) + (2*a*b)/((-1 + m)*x) + (a^2*x)/
(1 + m)))/(b + a*x^2)^2

_______________________________________________________________________________________

Maple [A]  time = 0.007, size = 90, normalized size = 1.5 \[{\frac{ \left ( cx \right ) ^{m} \left ({a}^{2}{m}^{2}{x}^{4}-4\,{a}^{2}m{x}^{4}+3\,{x}^{4}{a}^{2}+2\,ab{m}^{2}{x}^{2}-4\,abm{x}^{2}-6\,ab{x}^{2}+{b}^{2}{m}^{2}-{b}^{2} \right ) }{{x}^{3} \left ( 1+m \right ) \left ( -1+m \right ) \left ( -3+m \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^2)^2*(c*x)^m,x)

[Out]

(c*x)^m*(a^2*m^2*x^4-4*a^2*m*x^4+3*a^2*x^4+2*a*b*m^2*x^2-4*a*b*m*x^2-6*a*b*x^2+b
^2*m^2-b^2)/x^3/(1+m)/(-1+m)/(-3+m)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x)^m*(a + b/x^2)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.230261, size = 109, normalized size = 1.79 \[ \frac{{\left ({\left (a^{2} m^{2} - 4 \, a^{2} m + 3 \, a^{2}\right )} x^{4} + b^{2} m^{2} + 2 \,{\left (a b m^{2} - 2 \, a b m - 3 \, a b\right )} x^{2} - b^{2}\right )} \left (c x\right )^{m}}{{\left (m^{3} - 3 \, m^{2} - m + 3\right )} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x)^m*(a + b/x^2)^2,x, algorithm="fricas")

[Out]

((a^2*m^2 - 4*a^2*m + 3*a^2)*x^4 + b^2*m^2 + 2*(a*b*m^2 - 2*a*b*m - 3*a*b)*x^2 -
b^2)*(c*x)^m/((m^3 - 3*m^2 - m + 3)*x^3)

_______________________________________________________________________________________

Sympy [A]  time = 3.59968, size = 401, normalized size = 6.57 \[ \begin{cases} \frac{a^{2} \log{\left (x \right )} - \frac{a b}{x^{2}} - \frac{b^{2}}{4 x^{4}}}{c} & \text{for}\: m = -1 \\c \left (\frac{a^{2} x^{2}}{2} + 2 a b \log{\left (x \right )} - \frac{b^{2}}{2 x^{2}}\right ) & \text{for}\: m = 1 \\c^{3} \left (\frac{a^{2} x^{4}}{4} + a b x^{2} + b^{2} \log{\left (x \right )}\right ) & \text{for}\: m = 3 \\\frac{a^{2} c^{m} m^{2} x^{4} x^{m}}{m^{3} x^{3} - 3 m^{2} x^{3} - m x^{3} + 3 x^{3}} - \frac{4 a^{2} c^{m} m x^{4} x^{m}}{m^{3} x^{3} - 3 m^{2} x^{3} - m x^{3} + 3 x^{3}} + \frac{3 a^{2} c^{m} x^{4} x^{m}}{m^{3} x^{3} - 3 m^{2} x^{3} - m x^{3} + 3 x^{3}} + \frac{2 a b c^{m} m^{2} x^{2} x^{m}}{m^{3} x^{3} - 3 m^{2} x^{3} - m x^{3} + 3 x^{3}} - \frac{4 a b c^{m} m x^{2} x^{m}}{m^{3} x^{3} - 3 m^{2} x^{3} - m x^{3} + 3 x^{3}} - \frac{6 a b c^{m} x^{2} x^{m}}{m^{3} x^{3} - 3 m^{2} x^{3} - m x^{3} + 3 x^{3}} + \frac{b^{2} c^{m} m^{2} x^{m}}{m^{3} x^{3} - 3 m^{2} x^{3} - m x^{3} + 3 x^{3}} - \frac{b^{2} c^{m} x^{m}}{m^{3} x^{3} - 3 m^{2} x^{3} - m x^{3} + 3 x^{3}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x**2)**2*(c*x)**m,x)

[Out]

Piecewise(((a**2*log(x) - a*b/x**2 - b**2/(4*x**4))/c, Eq(m, -1)), (c*(a**2*x**2
/2 + 2*a*b*log(x) - b**2/(2*x**2)), Eq(m, 1)), (c**3*(a**2*x**4/4 + a*b*x**2 + b
**2*log(x)), Eq(m, 3)), (a**2*c**m*m**2*x**4*x**m/(m**3*x**3 - 3*m**2*x**3 - m*x
**3 + 3*x**3) - 4*a**2*c**m*m*x**4*x**m/(m**3*x**3 - 3*m**2*x**3 - m*x**3 + 3*x*
*3) + 3*a**2*c**m*x**4*x**m/(m**3*x**3 - 3*m**2*x**3 - m*x**3 + 3*x**3) + 2*a*b*
c**m*m**2*x**2*x**m/(m**3*x**3 - 3*m**2*x**3 - m*x**3 + 3*x**3) - 4*a*b*c**m*m*x
**2*x**m/(m**3*x**3 - 3*m**2*x**3 - m*x**3 + 3*x**3) - 6*a*b*c**m*x**2*x**m/(m**
3*x**3 - 3*m**2*x**3 - m*x**3 + 3*x**3) + b**2*c**m*m**2*x**m/(m**3*x**3 - 3*m**
2*x**3 - m*x**3 + 3*x**3) - b**2*c**m*x**m/(m**3*x**3 - 3*m**2*x**3 - m*x**3 + 3
*x**3), True))

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \left (c x\right )^{m}{\left (a + \frac{b}{x^{2}}\right )}^{2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x)^m*(a + b/x^2)^2,x, algorithm="giac")

[Out]

integrate((c*x)^m*(a + b/x^2)^2, x)