3.506 \(\int \frac{1}{x \sqrt{-a+b (c x)^n}} \, dx\)

Optimal. Leaf size=32 \[ \frac{2 \tan ^{-1}\left (\frac{\sqrt{b (c x)^n-a}}{\sqrt{a}}\right )}{\sqrt{a} n} \]

[Out]

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

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Rubi [A]  time = 0.0811627, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263 \[ \frac{2 \tan ^{-1}\left (\frac{\sqrt{b (c x)^n-a}}{\sqrt{a}}\right )}{\sqrt{a} n} \]

Antiderivative was successfully verified.

[In]  Int[1/(x*Sqrt[-a + b*(c*x)^n]),x]

[Out]

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

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Rubi in Sympy [A]  time = 4.07476, size = 26, normalized size = 0.81 \[ \frac{2 \operatorname{atan}{\left (\frac{\sqrt{- a + b \left (c x\right )^{n}}}{\sqrt{a}} \right )}}{\sqrt{a} n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0256412, size = 32, normalized size = 1. \[ \frac{2 \tan ^{-1}\left (\frac{\sqrt{b (c x)^n-a}}{\sqrt{a}}\right )}{\sqrt{a} n} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x*Sqrt[-a + b*(c*x)^n]),x]

[Out]

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

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Maple [A]  time = 0.007, size = 27, normalized size = 0.8 \[ 2\,{\frac{1}{n\sqrt{a}}\arctan \left ({\frac{\sqrt{-a+b \left ( cx \right ) ^{n}}}{\sqrt{a}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(-a+b*(c*x)^n)^(1/2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.289236, size = 1, normalized size = 0.03 \[ \left [\frac{\log \left (\frac{\left (c x\right )^{n} \sqrt{-a} b + 2 \, \sqrt{\left (c x\right )^{n} b - a} a - 2 \, \sqrt{-a} a}{\left (c x\right )^{n}}\right )}{\sqrt{-a} n}, -\frac{2 \, \arctan \left (\frac{\sqrt{a}}{\sqrt{\left (c x\right )^{n} b - a}}\right )}{\sqrt{a} n}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt((c*x)^n*b - a)*x),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x \sqrt{- a + b \left (c x\right )^{n}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(x*sqrt(-a + b*(c*x)**n)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt((c*x)^n*b - a)*x),x, algorithm="giac")

[Out]

integrate(1/(sqrt((c*x)^n*b - a)*x), x)