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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]