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

Optimal. Leaf size=96 \[ -\frac{b^2 x^{n-2} \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{(2-n) \left (a b+b^2 x^n\right )}-\frac{a \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{2 x^2 \left (a+b x^n\right )} \]

[Out]

-(a*Sqrt[a^2 + 2*a*b*x^n + b^2*x^(2*n)])/(2*x^2*(a + b*x^n)) - (b^2*x^(-2 + n)*Sqrt[a^2 + 2*a*b*x^n + b^2*x^(2
*n)])/((2 - n)*(a*b + b^2*x^n))

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Rubi [A]  time = 0.0293044, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {1355, 14} \[ -\frac{b^2 x^{n-2} \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{(2-n) \left (a b+b^2 x^n\right )}-\frac{a \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{2 x^2 \left (a+b x^n\right )} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a^2 + 2*a*b*x^n + b^2*x^(2*n)]/x^3,x]

[Out]

-(a*Sqrt[a^2 + 2*a*b*x^n + b^2*x^(2*n)])/(2*x^2*(a + b*x^n)) - (b^2*x^(-2 + n)*Sqrt[a^2 + 2*a*b*x^n + b^2*x^(2
*n)])/((2 - n)*(a*b + b^2*x^n))

Rule 1355

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.))^(p_), x_Symbol] :> Dist[(a + b*x^n + c*x^
(2*n))^FracPart[p]/(c^IntPart[p]*(b/2 + c*x^n)^(2*FracPart[p])), Int[(d*x)^m*(b/2 + c*x^n)^(2*p), x], x] /; Fr
eeQ[{a, b, c, d, m, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p - 1/2]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

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

Mathematica [A]  time = 0.0260375, size = 47, normalized size = 0.49 \[ \frac{\sqrt{\left (a+b x^n\right )^2} \left (2 b x^n-a (n-2)\right )}{2 (n-2) x^2 \left (a+b x^n\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a^2 + 2*a*b*x^n + b^2*x^(2*n)]/x^3,x]

[Out]

(Sqrt[(a + b*x^n)^2]*(-(a*(-2 + n)) + 2*b*x^n))/(2*(-2 + n)*x^2*(a + b*x^n))

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Maple [A]  time = 0.019, size = 61, normalized size = 0.6 \begin{align*} -{\frac{a}{ \left ( 2\,a+2\,b{x}^{n} \right ){x}^{2}}\sqrt{ \left ( a+b{x}^{n} \right ) ^{2}}}+{\frac{b{x}^{n}}{ \left ( a+b{x}^{n} \right ) \left ( -2+n \right ){x}^{2}}\sqrt{ \left ( a+b{x}^{n} \right ) ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.98262, size = 30, normalized size = 0.31 \begin{align*} -\frac{a{\left (n - 2\right )} - 2 \, b x^{n}}{2 \,{\left (n - 2\right )} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a^2+2*a*b*x^n+b^2*x^(2*n))^(1/2)/x^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.54761, size = 57, normalized size = 0.59 \begin{align*} -\frac{a n - 2 \, b x^{n} - 2 \, a}{2 \,{\left (n - 2\right )} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a^2+2*a*b*x^n+b^2*x^(2*n))^(1/2)/x^3,x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{\left (a + b x^{n}\right )^{2}}}{x^{3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a**2+2*a*b*x**n+b**2*x**(2*n))**(1/2)/x**3,x)

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}}}{x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a^2+2*a*b*x^n+b^2*x^(2*n))^(1/2)/x^3,x, algorithm="giac")

[Out]

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