3.507 \(\int \frac{1}{x \left (-a+b (c x)^n\right )^{3/2}} \, dx\)

Optimal. Leaf size=56 \[ -\frac{2 \tan ^{-1}\left (\frac{\sqrt{b (c x)^n-a}}{\sqrt{a}}\right )}{a^{3/2} n}-\frac{2}{a n \sqrt{b (c x)^n-a}} \]

[Out]

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

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Rubi [A]  time = 0.115704, antiderivative size = 56, 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 \[ -\frac{2 \tan ^{-1}\left (\frac{\sqrt{b (c x)^n-a}}{\sqrt{a}}\right )}{a^{3/2} n}-\frac{2}{a n \sqrt{b (c x)^n-a}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x*(-a + b*(c*x)^n)^(3/2)),x]

[Out]

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

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Rubi in Sympy [A]  time = 5.53749, size = 44, normalized size = 0.79 \[ - \frac{2}{a n \sqrt{- a + b \left (c x\right )^{n}}} - \frac{2 \operatorname{atan}{\left (\frac{\sqrt{- a + b \left (c x\right )^{n}}}{\sqrt{a}} \right )}}{a^{\frac{3}{2}} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x/(-a+b*(c*x)**n)**(3/2),x)

[Out]

-2/(a*n*sqrt(-a + b*(c*x)**n)) - 2*atan(sqrt(-a + b*(c*x)**n)/sqrt(a))/(a**(3/2)
*n)

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Mathematica [A]  time = 0.0864088, size = 56, normalized size = 1. \[ -\frac{2 \tan ^{-1}\left (\frac{\sqrt{b (c x)^n-a}}{\sqrt{a}}\right )}{a^{3/2} n}-\frac{2}{a n \sqrt{b (c x)^n-a}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x*(-a + b*(c*x)^n)^(3/2)),x]

[Out]

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

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Maple [A]  time = 0.008, size = 49, normalized size = 0.9 \[ -2\,{\frac{1}{{a}^{3/2}n}\arctan \left ({\frac{\sqrt{-a+b \left ( cx \right ) ^{n}}}{\sqrt{a}}} \right ) }-2\,{\frac{1}{an\sqrt{-a+b \left ( cx \right ) ^{n}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(-a+b*(c*x)^n)^(3/2),x)

[Out]

-2*arctan((-a+b*(c*x)^n)^(1/2)/a^(1/2))/a^(3/2)/n-2/a/n/(-a+b*(c*x)^n)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (\left (c x\right )^{n} b - a\right )}^{\frac{3}{2}} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(((c*x)^n*b - a)^(3/2)*x),x, algorithm="maxima")

[Out]

integrate(1/(((c*x)^n*b - a)^(3/2)*x), x)

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Fricas [A]  time = 0.282227, size = 1, normalized size = 0.02 \[ \left [\frac{\sqrt{\left (c x\right )^{n} b - a} \log \left (\frac{\left (c x\right )^{n} \sqrt{-a} b - 2 \, \sqrt{\left (c x\right )^{n} b - a} a - 2 \, \sqrt{-a} a}{\left (c x\right )^{n}}\right ) - 2 \, \sqrt{-a}}{\sqrt{\left (c x\right )^{n} b - a} \sqrt{-a} a n}, \frac{2 \,{\left (\sqrt{\left (c x\right )^{n} b - a} \arctan \left (\frac{\sqrt{a}}{\sqrt{\left (c x\right )^{n} b - a}}\right ) - \sqrt{a}\right )}}{\sqrt{\left (c x\right )^{n} b - a} a^{\frac{3}{2}} n}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(((c*x)^n*b - a)^(3/2)*x),x, algorithm="fricas")

[Out]

[(sqrt((c*x)^n*b - a)*log(((c*x)^n*sqrt(-a)*b - 2*sqrt((c*x)^n*b - a)*a - 2*sqrt
(-a)*a)/(c*x)^n) - 2*sqrt(-a))/(sqrt((c*x)^n*b - a)*sqrt(-a)*a*n), 2*(sqrt((c*x)
^n*b - a)*arctan(sqrt(a)/sqrt((c*x)^n*b - a)) - sqrt(a))/(sqrt((c*x)^n*b - a)*a^
(3/2)*n)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x \left (- a + b \left (c x\right )^{n}\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x/(-a+b*(c*x)**n)**(3/2),x)

[Out]

Integral(1/(x*(-a + b*(c*x)**n)**(3/2)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (\left (c x\right )^{n} b - a\right )}^{\frac{3}{2}} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(((c*x)^n*b - a)^(3/2)*x),x, algorithm="giac")

[Out]

integrate(1/(((c*x)^n*b - a)^(3/2)*x), x)