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 = 0.210333, 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 \[ -\frac{2 \sqrt{x} \tanh ^{-1}\left (\frac{\sqrt{a}}{x^{3/2} \sqrt{\frac{a}{x^3}+b x^n}}\right )}{\sqrt{a} c^2 (n+3) \sqrt{c x}} \]

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])

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Rubi in Sympy [A]  time = 15.9454, size = 51, normalized size = 0.94 \[ - \frac{2 \sqrt{c x} \operatorname{atanh}{\left (\frac{\sqrt{a}}{x^{\frac{3}{2}} \sqrt{\frac{a}{x^{3}} + b x^{n}}} \right )}}{\sqrt{a} c^{3} \sqrt{x} \left (n + 3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(c*x)**(5/2)/(a/x**3+b*x**n)**(1/2),x)

[Out]

-2*sqrt(c*x)*atanh(sqrt(a)/(x**(3/2)*sqrt(a/x**3 + b*x**n)))/(sqrt(a)*c**3*sqrt(
x)*(n + 3))

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Mathematica [A]  time = 0.176505, size = 83, normalized size = 1.54 \[ \frac{2 x \sqrt{a+b x^{n+3}} \left (\log \left (x^{\frac{n+3}{2}}\right )-\log \left (\sqrt{a} \sqrt{a+b x^{n+3}}+a\right )\right )}{\sqrt{a} (n+3) (c x)^{5/2} \sqrt{\frac{a}{x^3}+b x^n}} \]

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

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Maple [F]  time = 0.061, size = 0, normalized size = 0. \[ \int{1 \left ( cx \right ) ^{-{\frac{5}{2}}}{\frac{1}{\sqrt{{\frac{a}{{x}^{3}}}+b{x}^{n}}}}}\, dx \]

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. \[ \int \frac{1}{\sqrt{b x^{n} + \frac{a}{x^{3}}} \left (c x\right )^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x^n + a/x^3)*(c*x)^(5/2)),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. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x^n + a/x^3)*(c*x)^(5/2)),x, algorithm="fricas")

[Out]

Exception raised: TypeError

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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