3.2759 \(\int (a+\frac {b}{x})^2 (c x)^m \, dx\)

Optimal. Leaf size=52 \[ \frac {a^2 (c x)^{m+1}}{c (m+1)}+\frac {2 a b (c x)^m}{m}-\frac {b^2 c (c x)^{m-1}}{1-m} \]

[Out]

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

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Rubi [A]  time = 0.02, antiderivative size = 52, normalized size of antiderivative = 1.00, 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} \[ \frac {a^2 (c x)^{m+1}}{c (m+1)}+\frac {2 a b (c x)^m}{m}-\frac {b^2 c (c x)^{m-1}}{1-m} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x)^2*(c*x)^m,x]

[Out]

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

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Mathematica [A]  time = 0.05, size = 36, normalized size = 0.69 \[ (c x)^m \left (\frac {a^2 x}{m+1}+\frac {2 a b}{m}+\frac {b^2}{(m-1) x}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x)^2*(c*x)^m,x]

[Out]

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

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fricas [A]  time = 0.74, size = 63, normalized size = 1.21 \[ \frac {{\left (b^{2} m^{2} + b^{2} m + {\left (a^{2} m^{2} - a^{2} m\right )} x^{2} + 2 \, {\left (a b m^{2} - a b\right )} x\right )} \left (c x\right )^{m}}{{\left (m^{3} - m\right )} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (c x\right )^{m} {\left (a + \frac {b}{x}\right )}^{2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((c*x)^m*(a + b/x)^2, x)

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maple [A]  time = 0.01, size = 68, normalized size = 1.31 \[ \frac {\left (a^{2} x^{2} m^{2}-a^{2} x^{2} m +2 a b \,m^{2} x +b^{2} m^{2}-2 a b x +b^{2} m \right ) \left (c x \right )^{m}}{\left (m +1\right ) \left (m -1\right ) m x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.58, size = 51, normalized size = 0.98 \[ \frac {2 \, a b c^{m} x^{m}}{m} + \frac {b^{2} c^{m} x^{m}}{{\left (m - 1\right )} x} + \frac {\left (c x\right )^{m + 1} a^{2}}{c {\left (m + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 1.21, size = 45, normalized size = 0.87 \[ \frac {b^2\,{\left (c\,x\right )}^m}{x\,\left (m-1\right )}+\frac {a\,{\left (c\,x\right )}^m\,\left (2\,b+2\,b\,m+a\,m\,x\right )}{m\,\left (m+1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x)^m*(a + b/x)^2,x)

[Out]

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

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sympy [A]  time = 0.57, size = 202, normalized size = 3.88 \[ \begin {cases} \frac {a^{2} \log {\relax (x )} - \frac {2 a b}{x} - \frac {b^{2}}{2 x^{2}}}{c} & \text {for}\: m = -1 \\a^{2} x + 2 a b \log {\relax (x )} - \frac {b^{2}}{x} & \text {for}\: m = 0 \\c \left (\frac {a^{2} x^{2}}{2} + 2 a b x + b^{2} \log {\relax (x )}\right ) & \text {for}\: m = 1 \\\frac {a^{2} c^{m} m^{2} x^{2} x^{m}}{m^{3} x - m x} - \frac {a^{2} c^{m} m x^{2} x^{m}}{m^{3} x - m x} + \frac {2 a b c^{m} m^{2} x x^{m}}{m^{3} x - m x} - \frac {2 a b c^{m} x x^{m}}{m^{3} x - m x} + \frac {b^{2} c^{m} m^{2} x^{m}}{m^{3} x - m x} + \frac {b^{2} c^{m} m x^{m}}{m^{3} x - m x} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise(((a**2*log(x) - 2*a*b/x - b**2/(2*x**2))/c, Eq(m, -1)), (a**2*x + 2*a*b*log(x) - b**2/x, Eq(m, 0)),
(c*(a**2*x**2/2 + 2*a*b*x + b**2*log(x)), Eq(m, 1)), (a**2*c**m*m**2*x**2*x**m/(m**3*x - m*x) - a**2*c**m*m*x*
*2*x**m/(m**3*x - m*x) + 2*a*b*c**m*m**2*x*x**m/(m**3*x - m*x) - 2*a*b*c**m*x*x**m/(m**3*x - m*x) + b**2*c**m*
m**2*x**m/(m**3*x - m*x) + b**2*c**m*m*x**m/(m**3*x - m*x), True))

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