3.2463 \(\int \frac{(a+b x^n)^2}{x^2} \, dx\)

Optimal. Leaf size=44 \[ -\frac{a^2}{x}-\frac{2 a b x^{n-1}}{1-n}-\frac{b^2 x^{2 n-1}}{1-2 n} \]

[Out]

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

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Rubi [A]  time = 0.0218228, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {270} \[ -\frac{a^2}{x}-\frac{2 a b x^{n-1}}{1-n}-\frac{b^2 x^{2 n-1}}{1-2 n} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^n)^2/x^2,x]

[Out]

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

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

Mathematica [A]  time = 0.03754, size = 38, normalized size = 0.86 \[ \frac{-a^2+\frac{2 a b x^n}{n-1}+\frac{b^2 x^{2 n}}{2 n-1}}{x} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^n)^2/x^2,x]

[Out]

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

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Maple [A]  time = 0.009, size = 43, normalized size = 1. \begin{align*}{\frac{1}{x} \left ({\frac{{b}^{2} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{-1+2\,n}}-{a}^{2}+2\,{\frac{ab{{\rm e}^{n\ln \left ( x \right ) }}}{-1+n}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b^2/(-1+2*n)*exp(n*ln(x))^2-a^2+2*a*b/(-1+n)*exp(n*ln(x)))/x

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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^n)^2/x^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.06412, size = 135, normalized size = 3.07 \begin{align*} -\frac{2 \, a^{2} n^{2} - 3 \, a^{2} n + a^{2} -{\left (b^{2} n - b^{2}\right )} x^{2 \, n} - 2 \,{\left (2 \, a b n - a b\right )} x^{n}}{{\left (2 \, n^{2} - 3 \, n + 1\right )} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.685071, size = 190, normalized size = 4.32 \begin{align*} \begin{cases} - \frac{a^{2}}{x} - \frac{4 a b}{\sqrt{x}} + b^{2} \log{\left (x \right )} & \text{for}\: n = \frac{1}{2} \\- \frac{a^{2}}{x} + 2 a b \log{\left (x \right )} + b^{2} x & \text{for}\: n = 1 \\- \frac{2 a^{2} n^{2}}{2 n^{2} x - 3 n x + x} + \frac{3 a^{2} n}{2 n^{2} x - 3 n x + x} - \frac{a^{2}}{2 n^{2} x - 3 n x + x} + \frac{4 a b n x^{n}}{2 n^{2} x - 3 n x + x} - \frac{2 a b x^{n}}{2 n^{2} x - 3 n x + x} + \frac{b^{2} n x^{2 n}}{2 n^{2} x - 3 n x + x} - \frac{b^{2} x^{2 n}}{2 n^{2} x - 3 n x + x} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x**n)**2/x**2,x)

[Out]

Piecewise((-a**2/x - 4*a*b/sqrt(x) + b**2*log(x), Eq(n, 1/2)), (-a**2/x + 2*a*b*log(x) + b**2*x, Eq(n, 1)), (-
2*a**2*n**2/(2*n**2*x - 3*n*x + x) + 3*a**2*n/(2*n**2*x - 3*n*x + x) - a**2/(2*n**2*x - 3*n*x + x) + 4*a*b*n*x
**n/(2*n**2*x - 3*n*x + x) - 2*a*b*x**n/(2*n**2*x - 3*n*x + x) + b**2*n*x**(2*n)/(2*n**2*x - 3*n*x + x) - b**2
*x**(2*n)/(2*n**2*x - 3*n*x + x), True))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*x^n + a)^2/x^2, x)