3.397 \(\int \frac{1}{(c x)^{5/2} \sqrt{\frac{a}{x^3}+b x^n}} \, dx\)

Optimal. Leaf size=54 \[ -\frac{2 \sqrt{x} \tanh ^{-1}\left (\frac{\sqrt{a}}{x^{3/2} \sqrt{\frac{a}{x^3}+b x^n}}\right )}{\sqrt{a} c^2 (n+3) \sqrt{c x}} \]

[Out]

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

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Rubi [A]  time = 0.135793, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {2031, 2029, 206} \[ -\frac{2 \sqrt{x} \tanh ^{-1}\left (\frac{\sqrt{a}}{x^{3/2} \sqrt{\frac{a}{x^3}+b x^n}}\right )}{\sqrt{a} c^2 (n+3) \sqrt{c x}} \]

Antiderivative was successfully verified.

[In]

Int[1/((c*x)^(5/2)*Sqrt[a/x^3 + b*x^n]),x]

[Out]

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

Rule 2031

Int[((c_)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[(c^IntPart[m]*(c*x)^FracPar
t[m])/x^FracPart[m], Int[x^m*(a*x^j + b*x^n)^p, x], x] /; FreeQ[{a, b, c, j, m, n, p}, x] && IntegerQ[p + 1/2]
&& NeQ[n, j] && EqQ[Simplify[m + j*p + 1], 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 x)^{5/2} \sqrt{\frac{a}{x^3}+b x^n}} \, dx &=\frac{\sqrt{x} \int \frac{1}{x^{5/2} \sqrt{\frac{a}{x^3}+b x^n}} \, dx}{c^2 \sqrt{c x}}\\ &=-\frac{\left (2 \sqrt{x}\right ) \operatorname{Subst}\left (\int \frac{1}{1-a x^2} \, dx,x,\frac{1}{x^{3/2} \sqrt{\frac{a}{x^3}+b x^n}}\right )}{c^2 (3+n) \sqrt{c x}}\\ &=-\frac{2 \sqrt{x} \tanh ^{-1}\left (\frac{\sqrt{a}}{x^{3/2} \sqrt{\frac{a}{x^3}+b x^n}}\right )}{\sqrt{a} c^2 (3+n) \sqrt{c x}}\\ \end{align*}

Mathematica [A]  time = 0.0500456, size = 68, normalized size = 1.26 \[ -\frac{2 x \sqrt{a+b x^{n+3}} \tanh ^{-1}\left (\frac{\sqrt{a+b x^{n+3}}}{\sqrt{a}}\right )}{\sqrt{a} (n+3) (c x)^{5/2} \sqrt{\frac{a}{x^3}+b x^n}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((c*x)^(5/2)*Sqrt[a/x^3 + b*x^n]),x]

[Out]

(-2*x*Sqrt[a + b*x^(3 + n)]*ArcTanh[Sqrt[a + b*x^(3 + n)]/Sqrt[a]])/(Sqrt[a]*(3 + n)*(c*x)^(5/2)*Sqrt[a/x^3 +
b*x^n])

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Maple [F]  time = 0.329, size = 0, normalized size = 0. \begin{align*} \int{ \left ( cx \right ) ^{-{\frac{5}{2}}}{\frac{1}{\sqrt{{\frac{a}{{x}^{3}}}+b{x}^{n}}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(c*x)^(5/2)/(a/x^3+b*x^n)^(1/2),x)

[Out]

int(1/(c*x)^(5/2)/(a/x^3+b*x^n)^(1/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{b x^{n} + \frac{a}{x^{3}}} \left (c x\right )^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x)^(5/2)/(a/x^3+b*x^n)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(b*x^n + a/x^3)*(c*x)^(5/2)), x)

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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*x)^(5/2)/(a/x^3+b*x^n)^(1/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x)**(5/2)/(a/x**3+b*x**n)**(1/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{b x^{n} + \frac{a}{x^{3}}} \left (c x\right )^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x)^(5/2)/(a/x^3+b*x^n)^(1/2),x, algorithm="giac")

[Out]

integrate(1/(sqrt(b*x^n + a/x^3)*(c*x)^(5/2)), x)