3.369 \(\int \frac{\sqrt{a+b x^n}}{c x} \, dx\)

Optimal. Leaf size=51 \[ \frac{2 \sqrt{a+b x^n}}{c n}-\frac{2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b x^n}}{\sqrt{a}}\right )}{c n} \]

[Out]

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

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Rubi [A]  time = 0.0764388, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278 \[ \frac{2 \sqrt{a+b x^n}}{c n}-\frac{2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b x^n}}{\sqrt{a}}\right )}{c n} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + b*x^n]/(c*x),x]

[Out]

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

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Rubi in Sympy [A]  time = 9.40493, size = 41, normalized size = 0.8 \[ - \frac{2 \sqrt{a} \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{n}}}{\sqrt{a}} \right )}}{c n} + \frac{2 \sqrt{a + b x^{n}}}{c n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0347111, size = 45, normalized size = 0.88 \[ \frac{2 \left (\sqrt{a+b x^n}-\sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b x^n}}{\sqrt{a}}\right )\right )}{c n} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a + b*x^n]/(c*x),x]

[Out]

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

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Maple [A]  time = 0.001, size = 39, normalized size = 0.8 \[{\frac{1}{cn} \left ( 2\,\sqrt{a+b{x}^{n}}-2\,\sqrt{a}{\it Artanh} \left ({\frac{\sqrt{a+b{x}^{n}}}{\sqrt{a}}} \right ) \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*x^n)^(1/2)/c/x,x)

[Out]

1/c/n*(2*(a+b*x^n)^(1/2)-2*a^(1/2)*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(sqrt(b*x^n + a)/(c*x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.241772, size = 1, normalized size = 0.02 \[ \left [\frac{\sqrt{a} \log \left (\frac{b x^{n} - 2 \, \sqrt{b x^{n} + a} \sqrt{a} + 2 \, a}{x^{n}}\right ) + 2 \, \sqrt{b x^{n} + a}}{c n}, -\frac{2 \,{\left (\sqrt{-a} \arctan \left (\frac{\sqrt{b x^{n} + a}}{\sqrt{-a}}\right ) - \sqrt{b x^{n} + a}\right )}}{c n}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[(sqrt(a)*log((b*x^n - 2*sqrt(b*x^n + a)*sqrt(a) + 2*a)/x^n) + 2*sqrt(b*x^n + a)
)/(c*n), -2*(sqrt(-a)*arctan(sqrt(b*x^n + a)/sqrt(-a)) - sqrt(b*x^n + a))/(c*n)]

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

[Out]

Exception raised: TypeError

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{b x^{n} + a}}{c x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(sqrt(b*x^n + a)/(c*x), x)