3.541 \(\int \frac{1}{x^2 (a^2+2 a b x^n+b^2 x^{2 n})^{3/2}} \, dx\)

Optimal. Leaf size=65 \[ -\frac{\left (a+b x^n\right ) \, _2F_1\left (3,-\frac{1}{n};-\frac{1-n}{n};-\frac{b x^n}{a}\right )}{a^3 x \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}} \]

[Out]

-(((a + b*x^n)*Hypergeometric2F1[3, -n^(-1), -((1 - n)/n), -((b*x^n)/a)])/(a^3*x*Sqrt[a^2 + 2*a*b*x^n + b^2*x^
(2*n)]))

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Rubi [A]  time = 0.0259398, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {1355, 364} \[ -\frac{\left (a+b x^n\right ) \, _2F_1\left (3,-\frac{1}{n};-\frac{1-n}{n};-\frac{b x^n}{a}\right )}{a^3 x \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^2*(a^2 + 2*a*b*x^n + b^2*x^(2*n))^(3/2)),x]

[Out]

-(((a + b*x^n)*Hypergeometric2F1[3, -n^(-1), -((1 - n)/n), -((b*x^n)/a)])/(a^3*x*Sqrt[a^2 + 2*a*b*x^n + b^2*x^
(2*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 364

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rubi steps

\begin{align*} \int \frac{1}{x^2 \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{3/2}} \, dx &=\frac{\left (b^2 \left (a b+b^2 x^n\right )\right ) \int \frac{1}{x^2 \left (a b+b^2 x^n\right )^3} \, dx}{\sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}\\ &=-\frac{\left (a+b x^n\right ) \, _2F_1\left (3,-\frac{1}{n};-\frac{1-n}{n};-\frac{b x^n}{a}\right )}{a^3 x \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}\\ \end{align*}

Mathematica [A]  time = 0.0153982, size = 53, normalized size = 0.82 \[ -\frac{\left (a+b x^n\right )^3 \, _2F_1\left (3,-\frac{1}{n};1-\frac{1}{n};-\frac{b x^n}{a}\right )}{a^3 x \left (\left (a+b x^n\right )^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^2*(a^2 + 2*a*b*x^n + b^2*x^(2*n))^(3/2)),x]

[Out]

-(((a + b*x^n)^3*Hypergeometric2F1[3, -n^(-1), 1 - n^(-1), -((b*x^n)/a)])/(a^3*x*((a + b*x^n)^2)^(3/2)))

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Maple [F]  time = 0.059, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2}} \left ({a}^{2}+2\,ab{x}^{n}+{b}^{2}{x}^{2\,n} \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\left (2 \, n^{2} + 3 \, n + 1\right )} \int \frac{1}{2 \,{\left (a^{2} b n^{2} x^{2} x^{n} + a^{3} n^{2} x^{2}\right )}}\,{d x} + \frac{b{\left (2 \, n + 1\right )} x^{n} + a{\left (3 \, n + 1\right )}}{2 \,{\left (a^{2} b^{2} n^{2} x x^{2 \, n} + 2 \, a^{3} b n^{2} x x^{n} + a^{4} n^{2} x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}}}{b^{4} x^{2} x^{4 \, n} + 4 \, a^{2} b^{2} x^{2} x^{2 \, n} + 4 \, a^{3} b x^{2} x^{n} + a^{4} x^{2} + 2 \,{\left (2 \, a b^{3} x^{2} x^{n} + a^{2} b^{2} x^{2}\right )} x^{2 \, n}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(b^2*x^(2*n) + 2*a*b*x^n + a^2)/(b^4*x^2*x^(4*n) + 4*a^2*b^2*x^2*x^(2*n) + 4*a^3*b*x^2*x^n + a^4*
x^2 + 2*(2*a*b^3*x^2*x^n + a^2*b^2*x^2)*x^(2*n)), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{2} \left (\left (a + b x^{n}\right )^{2}\right )^{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}\right )}^{\frac{3}{2}} x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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