3.413 \(\int \frac{1}{\sqrt{\frac{a-b x^5}{x^3}}} \, dx\)

Optimal. Leaf size=33 \[ \frac{2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{\frac{a}{x^3}-b x^2}}\right )}{5 \sqrt{b}} \]

[Out]

(2*ArcTan[(Sqrt[b]*x)/Sqrt[a/x^3 - b*x^2]])/(5*Sqrt[b])

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Rubi [A]  time = 0.0161781, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {1979, 2008, 203} \[ \frac{2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{\frac{a}{x^3}-b x^2}}\right )}{5 \sqrt{b}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[(a - b*x^5)/x^3],x]

[Out]

(2*ArcTan[(Sqrt[b]*x)/Sqrt[a/x^3 - b*x^2]])/(5*Sqrt[b])

Rule 1979

Int[(u_)^(p_), x_Symbol] :> Int[ExpandToSum[u, x]^p, x] /; FreeQ[p, x] && GeneralizedBinomialQ[u, x] &&  !Gene
ralizedBinomialMatchQ[u, x]

Rule 2008

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

Rule 203

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

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{\frac{a-b x^5}{x^3}}} \, dx &=\int \frac{1}{\sqrt{\frac{a}{x^3}-b x^2}} \, dx\\ &=\frac{2}{5} \operatorname{Subst}\left (\int \frac{1}{1+b x^2} \, dx,x,\frac{x}{\sqrt{\frac{a}{x^3}-b x^2}}\right )\\ &=\frac{2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{\frac{a}{x^3}-b x^2}}\right )}{5 \sqrt{b}}\\ \end{align*}

Mathematica [A]  time = 0.0248042, size = 66, normalized size = 2. \[ \frac{2 \sqrt{a-b x^5} \tan ^{-1}\left (\frac{\sqrt{b} x^{5/2}}{\sqrt{a-b x^5}}\right )}{5 \sqrt{b} x^{3/2} \sqrt{\frac{a-b x^5}{x^3}}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[(a - b*x^5)/x^3],x]

[Out]

(2*Sqrt[a - b*x^5]*ArcTan[(Sqrt[b]*x^(5/2))/Sqrt[a - b*x^5]])/(5*Sqrt[b]*x^(3/2)*Sqrt[(a - b*x^5)/x^3])

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Maple [F]  time = 0.031, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{\sqrt{{\frac{-b{x}^{5}+a}{{x}^{3}}}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{-\frac{b x^{5} - a}{x^{3}}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/sqrt(-(b*x^5 - a)/x^3), x)

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Fricas [A]  time = 3.25213, size = 248, normalized size = 7.52 \begin{align*} \left [-\frac{\sqrt{-b} \log \left (-8 \, b^{2} x^{10} + 8 \, a b x^{5} - a^{2} + 4 \,{\left (2 \, b x^{9} - a x^{4}\right )} \sqrt{-b} \sqrt{-\frac{b x^{5} - a}{x^{3}}}\right )}{10 \, b}, -\frac{\arctan \left (\frac{2 \, \sqrt{b} x^{4} \sqrt{-\frac{b x^{5} - a}{x^{3}}}}{2 \, b x^{5} - a}\right )}{5 \, \sqrt{b}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/10*sqrt(-b)*log(-8*b^2*x^10 + 8*a*b*x^5 - a^2 + 4*(2*b*x^9 - a*x^4)*sqrt(-b)*sqrt(-(b*x^5 - a)/x^3))/b, -1
/5*arctan(2*sqrt(b)*x^4*sqrt(-(b*x^5 - a)/x^3)/(2*b*x^5 - a))/sqrt(b)]

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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/((-b*x**5+a)/x**3)**(1/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError