3.404 \(\int \frac{1}{c^4 x^4 (\frac{a}{x^2}+b x^n)^{3/2}} \, dx\)

Optimal. Leaf size=72 \[ \frac{2}{a c^4 (n+2) x \sqrt{\frac{a}{x^2}+b x^n}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a}}{x \sqrt{\frac{a}{x^2}+b x^n}}\right )}{a^{3/2} c^4 (n+2)} \]

[Out]

2/(a*c^4*(2 + n)*x*Sqrt[a/x^2 + b*x^n]) - (2*ArcTanh[Sqrt[a]/(x*Sqrt[a/x^2 + b*x^n])])/(a^(3/2)*c^4*(2 + n))

________________________________________________________________________________________

Rubi [A]  time = 0.155999, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {12, 2030, 2029, 206} \[ \frac{2}{a c^4 (n+2) x \sqrt{\frac{a}{x^2}+b x^n}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a}}{x \sqrt{\frac{a}{x^2}+b x^n}}\right )}{a^{3/2} c^4 (n+2)} \]

Antiderivative was successfully verified.

[In]

Int[1/(c^4*x^4*(a/x^2 + b*x^n)^(3/2)),x]

[Out]

2/(a*c^4*(2 + n)*x*Sqrt[a/x^2 + b*x^n]) - (2*ArcTanh[Sqrt[a]/(x*Sqrt[a/x^2 + b*x^n])])/(a^(3/2)*c^4*(2 + n))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2030

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> -Simp[(c^(j - 1)*(c*x)^(m - j
+ 1)*(a*x^j + b*x^n)^(p + 1))/(a*(n - j)*(p + 1)), x] + Dist[(c^j*(m + n*p + n - j + 1))/(a*(n - j)*(p + 1)),
Int[(c*x)^(m - j)*(a*x^j + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, c, j, m, n}, x] && ILtQ[p + 1/2, 0] && NeQ[n
, j] && EqQ[Simplify[m + j*p + 1], 0] && (IntegerQ[j] || GtQ[c, 0])

Rule 2029

Int[(x_)^(m_.)/Sqrt[(a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.)], x_Symbol] :> Dist[-2/(n - j), Subst[Int[1/(1 - a*x^2
), x], x, x^(j/2)/Sqrt[a*x^j + b*x^n]], x] /; FreeQ[{a, b, j, n}, x] && EqQ[m, j/2 - 1] && NeQ[n, j]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{1}{c^4 x^4 \left (\frac{a}{x^2}+b x^n\right )^{3/2}} \, dx &=\frac{\int \frac{1}{x^4 \left (\frac{a}{x^2}+b x^n\right )^{3/2}} \, dx}{c^4}\\ &=\frac{2}{a c^4 (2+n) x \sqrt{\frac{a}{x^2}+b x^n}}+\frac{\int \frac{1}{x^2 \sqrt{\frac{a}{x^2}+b x^n}} \, dx}{a c^4}\\ &=\frac{2}{a c^4 (2+n) x \sqrt{\frac{a}{x^2}+b x^n}}-\frac{2 \operatorname{Subst}\left (\int \frac{1}{1-a x^2} \, dx,x,\frac{1}{x \sqrt{\frac{a}{x^2}+b x^n}}\right )}{a c^4 (2+n)}\\ &=\frac{2}{a c^4 (2+n) x \sqrt{\frac{a}{x^2}+b x^n}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a}}{x \sqrt{\frac{a}{x^2}+b x^n}}\right )}{a^{3/2} c^4 (2+n)}\\ \end{align*}

Mathematica [C]  time = 0.0408202, size = 51, normalized size = 0.71 \[ \frac{2 \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};\frac{b x^{n+2}}{a}+1\right )}{a c^4 (n+2) x \sqrt{\frac{a}{x^2}+b x^n}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(c^4*x^4*(a/x^2 + b*x^n)^(3/2)),x]

[Out]

(2*Hypergeometric2F1[-1/2, 1, 1/2, 1 + (b*x^(2 + n))/a])/(a*c^4*(2 + n)*x*Sqrt[a/x^2 + b*x^n])

________________________________________________________________________________________

Maple [F]  time = 0.326, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{c}^{4}{x}^{4}} \left ({\frac{a}{{x}^{2}}}+b{x}^{n} \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/c^4/x^4/(a/x^2+b*x^n)^(3/2),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/c^4/x^4/(a/x^2+b*x^n)^(3/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{1}{a x^{2} \sqrt{\frac{a}{x^{2}} + b x^{n}} + b x^{4} x^{n} \sqrt{\frac{a}{x^{2}} + b x^{n}}}\, dx}{c^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(a*x**2*sqrt(a/x**2 + b*x**n) + b*x**4*x**n*sqrt(a/x**2 + b*x**n)), x)/c**4

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/c^4/x^4/(a/x^2+b*x^n)^(3/2),x, algorithm="giac")

[Out]

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