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

Optimal. Leaf size=49 \[ \frac{2 \sqrt{a+b (c x)^n}}{n}-\frac{2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b (c x)^n}}{\sqrt{a}}\right )}{n} \]

[Out]

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

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Rubi [A]  time = 0.101153, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.353 \[ \frac{2 \sqrt{a+b (c x)^n}}{n}-\frac{2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b (c x)^n}}{\sqrt{a}}\right )}{n} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + b*(c*x)^n]/x,x]

[Out]

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

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Rubi in Sympy [A]  time = 4.67254, size = 41, normalized size = 0.84 \[ - \frac{2 \sqrt{a} \operatorname{atanh}{\left (\frac{\sqrt{a + b \left (c x\right )^{n}}}{\sqrt{a}} \right )}}{n} + \frac{2 \sqrt{a + b \left (c x\right )^{n}}}{n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0294167, size = 46, normalized size = 0.94 \[ \frac{2 \left (\sqrt{a+b (c x)^n}-\sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b (c x)^n}}{\sqrt{a}}\right )\right )}{n} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.002, size = 40, normalized size = 0.8 \[{\frac{1}{n} \left ( 2\,\sqrt{a+b \left ( cx \right ) ^{n}}-2\,\sqrt{a}{\it Artanh} \left ({\frac{\sqrt{a+b \left ( cx \right ) ^{n}}}{\sqrt{a}}} \right ) \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*(c*x)^n)^(1/2)/x,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.284205, size = 1, normalized size = 0.02 \[ \left [\frac{\sqrt{a} \log \left (\frac{\left (c x\right )^{n} b - 2 \, \sqrt{\left (c x\right )^{n} b + a} \sqrt{a} + 2 \, a}{\left (c x\right )^{n}}\right ) + 2 \, \sqrt{\left (c x\right )^{n} b + a}}{n}, -\frac{2 \,{\left (\sqrt{-a} \arctan \left (\frac{\sqrt{\left (c x\right )^{n} b + a}}{\sqrt{-a}}\right ) - \sqrt{\left (c x\right )^{n} b + a}\right )}}{n}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{a + b \left (c x\right )^{n}}}{x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(sqrt(a + b*(c*x)**n)/x, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(sqrt((c*x)^n*b + a)/x, x)