3.2760 \(\int (a+\frac{b}{x^2})^2 (c x)^m \, dx\)

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))

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Rubi [A]  time = 0.0287716, 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, Rules used = {270} \[ \frac{a^2 (c x)^{m+1}}{c (m+1)}-\frac{2 a b c (c x)^{m-1}}{1-m}-\frac{b^2 c^3 (c x)^{m-3}}{3-m} \]

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))

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int \left (a+\frac{b}{x^2}\right )^2 (c x)^m \, dx &=\int \left (b^2 c^4 (c x)^{-4+m}+2 a b c^2 (c x)^{-2+m}+a^2 (c x)^m\right ) \, dx\\ &=-\frac{b^2 c^3 (c x)^{-3+m}}{3-m}-\frac{2 a b c (c x)^{-1+m}}{1-m}+\frac{a^2 (c x)^{1+m}}{c (1+m)}\\ \end{align*}

Mathematica [A]  time = 0.0288922, size = 64, normalized size = 1.05 \[ \frac{(c x)^m \left (a^2 \left (m^2-4 m+3\right ) x^4+2 a b \left (m^2-2 m-3\right ) x^2+b^2 \left (m^2-1\right )\right )}{(m-3) (m-1) (m+1) x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x^2)^2*(c*x)^m,x]

[Out]

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

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Maple [A]  time = 0.003, size = 90, normalized size = 1.5 \begin{align*}{\frac{ \left ( cx \right ) ^{m} \left ({a}^{2}{m}^{2}{x}^{4}-4\,{a}^{2}m{x}^{4}+3\,{a}^{2}{x}^{4}+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 ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+1/x^2*b)^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)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.36459, size = 166, normalized size = 2.72 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^2)^2*(c*x)^m,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)

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Sympy [A]  time = 1.03755, size = 401, normalized size = 6.57 \begin{align*} \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} \end{align*}

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))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (c x\right )^{m}{\left (a + \frac{b}{x^{2}}\right )}^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((c*x)^m*(a + b/x^2)^2, x)