3.2507 \(\int \frac{1}{x \left (a+b x^n\right )^{5/2}} \, dx\)

Optimal. Leaf size=69 \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+b x^n}}{\sqrt{a}}\right )}{a^{5/2} n}+\frac{2}{a^2 n \sqrt{a+b x^n}}+\frac{2}{3 a n \left (a+b x^n\right )^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.103123, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+b x^n}}{\sqrt{a}}\right )}{a^{5/2} n}+\frac{2}{a^2 n \sqrt{a+b x^n}}+\frac{2}{3 a n \left (a+b x^n\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x*(a + b*x^n)^(5/2)),x]

[Out]

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

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Rubi in Sympy [A]  time = 11.0653, size = 58, normalized size = 0.84 \[ \frac{2}{3 a n \left (a + b x^{n}\right )^{\frac{3}{2}}} + \frac{2}{a^{2} n \sqrt{a + b x^{n}}} - \frac{2 \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{n}}}{\sqrt{a}} \right )}}{a^{\frac{5}{2}} n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/(3*a*n*(a + b*x**n)**(3/2)) + 2/(a**2*n*sqrt(a + b*x**n)) - 2*atanh(sqrt(a + b
*x**n)/sqrt(a))/(a**(5/2)*n)

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Mathematica [A]  time = 0.153165, size = 60, normalized size = 0.87 \[ \frac{2 \left (\frac{\sqrt{a} \left (4 a+3 b x^n\right )}{\left (a+b x^n\right )^{3/2}}-3 \tanh ^{-1}\left (\frac{\sqrt{a+b x^n}}{\sqrt{a}}\right )\right )}{3 a^{5/2} n} \]

Antiderivative was successfully verified.

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

[Out]

(2*((Sqrt[a]*(4*a + 3*b*x^n))/(a + b*x^n)^(3/2) - 3*ArcTanh[Sqrt[a + b*x^n]/Sqrt
[a]]))/(3*a^(5/2)*n)

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Maple [A]  time = 0.01, size = 53, normalized size = 0.8 \[{\frac{1}{n} \left ( 2\,{\frac{1}{\sqrt{a+b{x}^{n}}{a}^{2}}}+{\frac{2}{3\,a} \left ( a+b{x}^{n} \right ) ^{-{\frac{3}{2}}}}-2\,{\frac{1}{{a}^{5/2}}{\it Artanh} \left ({\frac{\sqrt{a+b{x}^{n}}}{\sqrt{a}}} \right ) } \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(a+b*x^n)^(5/2),x)

[Out]

1/n*(2/a^2/(a+b*x^n)^(1/2)+2/3/a/(a+b*x^n)^(3/2)-2/a^(5/2)*arctanh((a+b*x^n)^(1/
2)/a^(1/2)))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.230397, size = 1, normalized size = 0.01 \[ \left [\frac{6 \, \sqrt{a} b x^{n} + 3 \,{\left (b x^{n} + a\right )}^{\frac{3}{2}} \log \left (\frac{\sqrt{a} b x^{n} - 2 \, \sqrt{b x^{n} + a} a + 2 \, a^{\frac{3}{2}}}{x^{n}}\right ) + 8 \, a^{\frac{3}{2}}}{3 \,{\left (a^{\frac{5}{2}} b n x^{n} + a^{\frac{7}{2}} n\right )} \sqrt{b x^{n} + a}}, \frac{2 \,{\left (3 \, \sqrt{-a} b x^{n} + 3 \,{\left (b x^{n} + a\right )}^{\frac{3}{2}} \arctan \left (\frac{a}{\sqrt{b x^{n} + a} \sqrt{-a}}\right ) + 4 \, \sqrt{-a} a\right )}}{3 \,{\left (\sqrt{-a} a^{2} b n x^{n} + \sqrt{-a} a^{3} n\right )} \sqrt{b x^{n} + a}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/3*(6*sqrt(a)*b*x^n + 3*(b*x^n + a)^(3/2)*log((sqrt(a)*b*x^n - 2*sqrt(b*x^n +
a)*a + 2*a^(3/2))/x^n) + 8*a^(3/2))/((a^(5/2)*b*n*x^n + a^(7/2)*n)*sqrt(b*x^n +
a)), 2/3*(3*sqrt(-a)*b*x^n + 3*(b*x^n + a)^(3/2)*arctan(a/(sqrt(b*x^n + a)*sqrt(
-a))) + 4*sqrt(-a)*a)/((sqrt(-a)*a^2*b*n*x^n + sqrt(-a)*a^3*n)*sqrt(b*x^n + a))]

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Sympy [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/x/(a+b*x**n)**(5/2),x)

[Out]

Exception raised: TypeError

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/((b*x^n + a)^(5/2)*x), x)