3.2464 \(\int \frac{(a+b x^n)^2}{x^3} \, dx\)

Optimal. Leaf size=50 \[ -\frac{a^2}{2 x^2}-\frac{2 a b x^{n-2}}{2-n}-\frac{b^2 x^{-2 (1-n)}}{2 (1-n)} \]

[Out]

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

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Rubi [A]  time = 0.0209923, antiderivative size = 50, 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} \[ -\frac{a^2}{2 x^2}-\frac{2 a b x^{n-2}}{2-n}-\frac{b^2 x^{-2 (1-n)}}{2 (1-n)} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^n)^2/x^3,x]

[Out]

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

Mathematica [A]  time = 0.0375724, size = 39, normalized size = 0.78 \[ \frac{-a^2+\frac{4 a b x^n}{n-2}+\frac{b^2 x^{2 n}}{n-1}}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^n)^2/x^3,x]

[Out]

(-a^2 + (4*a*b*x^n)/(-2 + n) + (b^2*x^(2*n))/(-1 + n))/(2*x^2)

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Maple [A]  time = 0.011, size = 42, normalized size = 0.8 \begin{align*}{\frac{1}{{x}^{2}} \left ( -{\frac{{a}^{2}}{2}}+{\frac{{b}^{2} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{-2+2\,n}}+2\,{\frac{ab{{\rm e}^{n\ln \left ( x \right ) }}}{-2+n}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*x^n)^2/x^3,x)

[Out]

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

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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^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.04656, size = 140, normalized size = 2.8 \begin{align*} -\frac{a^{2} n^{2} - 3 \, a^{2} n + 2 \, a^{2} -{\left (b^{2} n - 2 \, b^{2}\right )} x^{2 \, n} - 4 \,{\left (a b n - a b\right )} x^{n}}{2 \,{\left (n^{2} - 3 \, n + 2\right )} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^n)^2/x^3,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.663754, size = 245, normalized size = 4.9 \begin{align*} \begin{cases} - \frac{a^{2}}{2 x^{2}} - \frac{2 a b}{x} + b^{2} \log{\left (x \right )} & \text{for}\: n = 1 \\- \frac{a^{2}}{2 x^{2}} + 2 a b \log{\left (x \right )} + \frac{b^{2} x^{2}}{2} & \text{for}\: n = 2 \\- \frac{a^{2} n^{2}}{2 n^{2} x^{2} - 6 n x^{2} + 4 x^{2}} + \frac{3 a^{2} n}{2 n^{2} x^{2} - 6 n x^{2} + 4 x^{2}} - \frac{2 a^{2}}{2 n^{2} x^{2} - 6 n x^{2} + 4 x^{2}} + \frac{4 a b n x^{n}}{2 n^{2} x^{2} - 6 n x^{2} + 4 x^{2}} - \frac{4 a b x^{n}}{2 n^{2} x^{2} - 6 n x^{2} + 4 x^{2}} + \frac{b^{2} n x^{2 n}}{2 n^{2} x^{2} - 6 n x^{2} + 4 x^{2}} - \frac{2 b^{2} x^{2 n}}{2 n^{2} x^{2} - 6 n x^{2} + 4 x^{2}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x**n)**2/x**3,x)

[Out]

Piecewise((-a**2/(2*x**2) - 2*a*b/x + b**2*log(x), Eq(n, 1)), (-a**2/(2*x**2) + 2*a*b*log(x) + b**2*x**2/2, Eq
(n, 2)), (-a**2*n**2/(2*n**2*x**2 - 6*n*x**2 + 4*x**2) + 3*a**2*n/(2*n**2*x**2 - 6*n*x**2 + 4*x**2) - 2*a**2/(
2*n**2*x**2 - 6*n*x**2 + 4*x**2) + 4*a*b*n*x**n/(2*n**2*x**2 - 6*n*x**2 + 4*x**2) - 4*a*b*x**n/(2*n**2*x**2 -
6*n*x**2 + 4*x**2) + b**2*n*x**(2*n)/(2*n**2*x**2 - 6*n*x**2 + 4*x**2) - 2*b**2*x**(2*n)/(2*n**2*x**2 - 6*n*x*
*2 + 4*x**2), 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^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^n)^2/x^3,x, algorithm="giac")

[Out]

integrate((b*x^n + a)^2/x^3, x)