3.2515 \(\int \frac {1}{x^2 (a+b x^n)^{5/2}} \, dx\)

Optimal. Leaf size=49 \[ -\frac {\, _2F_1\left (1,-\frac {3}{2}-\frac {1}{n};-\frac {1-n}{n};-\frac {b x^n}{a}\right )}{a x \left (a+b x^n\right )^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.02, antiderivative size = 61, normalized size of antiderivative = 1.24, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {365, 364} \[ -\frac {\sqrt {\frac {b x^n}{a}+1} \, _2F_1\left (\frac {5}{2},-\frac {1}{n};-\frac {1-n}{n};-\frac {b x^n}{a}\right )}{a^2 x \sqrt {a+b x^n}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^2*(a + b*x^n)^(5/2)),x]

[Out]

-((Sqrt[1 + (b*x^n)/a]*Hypergeometric2F1[5/2, -n^(-1), -((1 - n)/n), -((b*x^n)/a)])/(a^2*x*Sqrt[a + b*x^n]))

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])

Rule 365

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a + b*x^n)^FracPart[p])
/(1 + (b*x^n)/a)^FracPart[p], Int[(c*x)^m*(1 + (b*x^n)/a)^p, x], 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+b x^n\right )^{5/2}} \, dx &=\frac {\sqrt {1+\frac {b x^n}{a}} \int \frac {1}{x^2 \left (1+\frac {b x^n}{a}\right )^{5/2}} \, dx}{a^2 \sqrt {a+b x^n}}\\ &=-\frac {\sqrt {1+\frac {b x^n}{a}} \, _2F_1\left (\frac {5}{2},-\frac {1}{n};-\frac {1-n}{n};-\frac {b x^n}{a}\right )}{a^2 x \sqrt {a+b x^n}}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 58, normalized size = 1.18 \[ -\frac {\sqrt {\frac {b x^n}{a}+1} \, _2F_1\left (\frac {5}{2},-\frac {1}{n};1-\frac {1}{n};-\frac {b x^n}{a}\right )}{a^2 x \sqrt {a+b x^n}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^2*(a + b*x^n)^(5/2)),x]

[Out]

-((Sqrt[1 + (b*x^n)/a]*Hypergeometric2F1[5/2, -n^(-1), 1 - n^(-1), -((b*x^n)/a)])/(a^2*x*Sqrt[a + b*x^n]))

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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(a+b*x^n)^(5/2),x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b x^{n} + a\right )}^{\frac {5}{2}} x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(a+b*x^n)^(5/2),x, algorithm="giac")

[Out]

integrate(1/((b*x^n + a)^(5/2)*x^2), x)

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maple [F]  time = 0.22, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b \,x^{n}+a \right )^{\frac {5}{2}} x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^2/(b*x^n+a)^(5/2),x)

[Out]

int(1/x^2/(b*x^n+a)^(5/2),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b x^{n} + a\right )}^{\frac {5}{2}} x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(a+b*x^n)^(5/2),x, algorithm="maxima")

[Out]

integrate(1/((b*x^n + a)^(5/2)*x^2), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{x^2\,{\left (a+b\,x^n\right )}^{5/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^2*(a + b*x^n)^(5/2)),x)

[Out]

int(1/(x^2*(a + b*x^n)^(5/2)), x)

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sympy [C]  time = 4.00, size = 42, normalized size = 0.86 \[ \frac {\Gamma \left (- \frac {1}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {5}{2}, - \frac {1}{n} \\ 1 - \frac {1}{n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{a^{\frac {5}{2}} n x \Gamma \left (1 - \frac {1}{n}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**2/(a+b*x**n)**(5/2),x)

[Out]

gamma(-1/n)*hyper((5/2, -1/n), (1 - 1/n,), b*x**n*exp_polar(I*pi)/a)/(a**(5/2)*n*x*gamma(1 - 1/n))

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