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

Optimal. Leaf size=53 \[ \frac{2 \sqrt{b (c x)^n-a}}{n}-\frac{2 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{b (c x)^n-a}}{\sqrt{a}}\right )}{n} \]

[Out]

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

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Rubi [A]  time = 0.106826, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316 \[ \frac{2 \sqrt{b (c x)^n-a}}{n}-\frac{2 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{b (c x)^n-a}}{\sqrt{a}}\right )}{n} \]

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

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Rubi in Sympy [A]  time = 5.22331, size = 41, normalized size = 0.77 \[ - \frac{2 \sqrt{a} \operatorname{atan}{\left (\frac{\sqrt{- a + b \left (c x\right )^{n}}}{\sqrt{a}} \right )}}{n} + \frac{2 \sqrt{- a + b \left (c x\right )^{n}}}{n} \]

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

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Mathematica [A]  time = 0.0321183, size = 50, normalized size = 0.94 \[ \frac{2 \left (\sqrt{b (c x)^n-a}-\sqrt{a} \tan ^{-1}\left (\frac{\sqrt{b (c x)^n-a}}{\sqrt{a}}\right )\right )}{n} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.005, size = 46, normalized size = 0.9 \[ -2\,{\frac{\sqrt{a}}{n}\arctan \left ({\frac{\sqrt{-a+b \left ( cx \right ) ^{n}}}{\sqrt{a}}} \right ) }+2\,{\frac{\sqrt{-a+b \left ( cx \right ) ^{n}}}{n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-a+b*(c*x)^n)^(1/2)/x,x)

[Out]

-2*arctan((-a+b*(c*x)^n)^(1/2)/a^(1/2))*a^(1/2)/n+2*(-a+b*(c*x)^n)^(1/2)/n

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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 = 0.284697, 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*s
qrt((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)