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

Optimal. Leaf size=90 \[ \frac{2}{a c^2 (n+1) \sqrt{c x} \sqrt{\frac{a}{x}+b x^n}}-\frac{2 \sqrt{x} \tanh ^{-1}\left (\frac{\sqrt{a}}{\sqrt{x} \sqrt{\frac{a}{x}+b x^n}}\right )}{a^{3/2} c^2 (n+1) \sqrt{c x}} \]

[Out]

2/(a*c^2*(1 + n)*Sqrt[c*x]*Sqrt[a/x + b*x^n]) - (2*Sqrt[x]*ArcTanh[Sqrt[a]/(Sqrt
[x]*Sqrt[a/x + b*x^n])])/(a^(3/2)*c^2*(1 + n)*Sqrt[c*x])

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Rubi [A]  time = 0.289163, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174 \[ \frac{2}{a c^2 (n+1) \sqrt{c x} \sqrt{\frac{a}{x}+b x^n}}-\frac{2 \sqrt{x} \tanh ^{-1}\left (\frac{\sqrt{a}}{\sqrt{x} \sqrt{\frac{a}{x}+b x^n}}\right )}{a^{3/2} c^2 (n+1) \sqrt{c x}} \]

Antiderivative was successfully verified.

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

[Out]

2/(a*c^2*(1 + n)*Sqrt[c*x]*Sqrt[a/x + b*x^n]) - (2*Sqrt[x]*ArcTanh[Sqrt[a]/(Sqrt
[x]*Sqrt[a/x + b*x^n])])/(a^(3/2)*c^2*(1 + n)*Sqrt[c*x])

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Rubi in Sympy [A]  time = 24.7318, size = 76, normalized size = 0.84 \[ \frac{2}{a c^{2} \sqrt{c x} \left (n + 1\right ) \sqrt{\frac{a}{x} + b x^{n}}} - \frac{2 \sqrt{c x} \operatorname{atanh}{\left (\frac{\sqrt{a}}{\sqrt{x} \sqrt{\frac{a}{x} + b x^{n}}} \right )}}{a^{\frac{3}{2}} c^{3} \sqrt{x} \left (n + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/(a*c**2*sqrt(c*x)*(n + 1)*sqrt(a/x + b*x**n)) - 2*sqrt(c*x)*atanh(sqrt(a)/(sqr
t(x)*sqrt(a/x + b*x**n)))/(a**(3/2)*c**3*sqrt(x)*(n + 1))

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Mathematica [A]  time = 0.196703, size = 104, normalized size = 1.16 \[ \frac{2 \left (-\sqrt{a+b x^{n+1}} \log \left (\sqrt{a} \sqrt{a+b x^{n+1}}+a\right )+\log \left (x^{\frac{n+1}{2}}\right ) \sqrt{a+b x^{n+1}}+\sqrt{a}\right )}{a^{3/2} c^2 (n+1) \sqrt{c x} \sqrt{\frac{a}{x}+b x^n}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(Sqrt[a] + Sqrt[a + b*x^(1 + n)]*Log[x^((1 + n)/2)] - Sqrt[a + b*x^(1 + n)]*L
og[a + Sqrt[a]*Sqrt[a + b*x^(1 + n)]]))/(a^(3/2)*c^2*(1 + n)*Sqrt[c*x]*Sqrt[a/x
+ b*x^n])

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Maple [F]  time = 0.057, size = 0, normalized size = 0. \[ \int{1 \left ( cx \right ) ^{-{\frac{5}{2}}} \left ({\frac{a}{x}}+b{x}^{n} \right ) ^{-{\frac{3}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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