3.508 \(\int \frac{1}{x \left (-a+b (c x)^n\right )^{5/2}} \, dx\)

Optimal. Leaf size=81 \[ \frac{2 \tan ^{-1}\left (\frac{\sqrt{b (c x)^n-a}}{\sqrt{a}}\right )}{a^{5/2} n}+\frac{2}{a^2 n \sqrt{b (c x)^n-a}}-\frac{2}{3 a n \left (b (c x)^n-a\right )^{3/2}} \]

[Out]

-2/(3*a*n*(-a + b*(c*x)^n)^(3/2)) + 2/(a^2*n*Sqrt[-a + b*(c*x)^n]) + (2*ArcTan[S
qrt[-a + b*(c*x)^n]/Sqrt[a]])/(a^(5/2)*n)

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Rubi [A]  time = 0.1529, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316 \[ \frac{2 \tan ^{-1}\left (\frac{\sqrt{b (c x)^n-a}}{\sqrt{a}}\right )}{a^{5/2} n}+\frac{2}{a^2 n \sqrt{b (c x)^n-a}}-\frac{2}{3 a n \left (b (c x)^n-a\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x*(-a + b*(c*x)^n)^(5/2)),x]

[Out]

-2/(3*a*n*(-a + b*(c*x)^n)^(3/2)) + 2/(a^2*n*Sqrt[-a + b*(c*x)^n]) + (2*ArcTan[S
qrt[-a + b*(c*x)^n]/Sqrt[a]])/(a^(5/2)*n)

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Rubi in Sympy [A]  time = 7.26356, size = 63, normalized size = 0.78 \[ - \frac{2}{3 a n \left (- a + b \left (c x\right )^{n}\right )^{\frac{3}{2}}} + \frac{2}{a^{2} n \sqrt{- a + b \left (c x\right )^{n}}} + \frac{2 \operatorname{atan}{\left (\frac{\sqrt{- a + b \left (c x\right )^{n}}}{\sqrt{a}} \right )}}{a^{\frac{5}{2}} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x/(-a+b*(c*x)**n)**(5/2),x)

[Out]

-2/(3*a*n*(-a + b*(c*x)**n)**(3/2)) + 2/(a**2*n*sqrt(-a + b*(c*x)**n)) + 2*atan(
sqrt(-a + b*(c*x)**n)/sqrt(a))/(a**(5/2)*n)

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Mathematica [A]  time = 0.191232, size = 70, normalized size = 0.86 \[ \frac{2 \left (\frac{\sqrt{a} \left (3 b (c x)^n-4 a\right )}{\left (b (c x)^n-a\right )^{3/2}}+3 \tan ^{-1}\left (\frac{\sqrt{b (c x)^n-a}}{\sqrt{a}}\right )\right )}{3 a^{5/2} n} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x*(-a + b*(c*x)^n)^(5/2)),x]

[Out]

(2*((Sqrt[a]*(-4*a + 3*b*(c*x)^n))/(-a + b*(c*x)^n)^(3/2) + 3*ArcTan[Sqrt[-a + b
*(c*x)^n]/Sqrt[a]]))/(3*a^(5/2)*n)

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Maple [A]  time = 0.01, size = 70, normalized size = 0.9 \[ -{\frac{2}{3\,an} \left ( -a+b \left ( cx \right ) ^{n} \right ) ^{-{\frac{3}{2}}}}+2\,{\frac{1}{{a}^{5/2}n}\arctan \left ({\frac{\sqrt{-a+b \left ( cx \right ) ^{n}}}{\sqrt{a}}} \right ) }+2\,{\frac{1}{{a}^{2}n\sqrt{-a+b \left ( cx \right ) ^{n}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(-a+b*(c*x)^n)^(5/2),x)

[Out]

-2/3/a/n/(-a+b*(c*x)^n)^(3/2)+2*arctan((-a+b*(c*x)^n)^(1/2)/a^(1/2))/a^(5/2)/n+2
/a^2/n/(-a+b*(c*x)^n)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (\left (c x\right )^{n} b - a\right )}^{\frac{5}{2}} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(((c*x)^n*b - a)^(5/2)*x),x, algorithm="maxima")

[Out]

integrate(1/(((c*x)^n*b - a)^(5/2)*x), x)

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Fricas [A]  time = 0.284092, size = 1, normalized size = 0.01 \[ \left [\frac{6 \, \left (c x\right )^{n} \sqrt{-a} b + 3 \,{\left (\left (c x\right )^{n} b - a\right )}^{\frac{3}{2}} \log \left (\frac{\left (c x\right )^{n} \sqrt{-a} b + 2 \, \sqrt{\left (c x\right )^{n} b - a} a - 2 \, \sqrt{-a} a}{\left (c x\right )^{n}}\right ) - 8 \, \sqrt{-a} a}{3 \,{\left (\left (c x\right )^{n} \sqrt{-a} a^{2} b n - \sqrt{-a} a^{3} n\right )} \sqrt{\left (c x\right )^{n} b - a}}, \frac{2 \,{\left (3 \, \left (c x\right )^{n} \sqrt{a} b - 3 \,{\left (\left (c x\right )^{n} b - a\right )}^{\frac{3}{2}} \arctan \left (\frac{\sqrt{a}}{\sqrt{\left (c x\right )^{n} b - a}}\right ) - 4 \, a^{\frac{3}{2}}\right )}}{3 \,{\left (\left (c x\right )^{n} a^{\frac{5}{2}} b n - a^{\frac{7}{2}} n\right )} \sqrt{\left (c x\right )^{n} b - a}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(((c*x)^n*b - a)^(5/2)*x),x, algorithm="fricas")

[Out]

[1/3*(6*(c*x)^n*sqrt(-a)*b + 3*((c*x)^n*b - a)^(3/2)*log(((c*x)^n*sqrt(-a)*b + 2
*sqrt((c*x)^n*b - a)*a - 2*sqrt(-a)*a)/(c*x)^n) - 8*sqrt(-a)*a)/(((c*x)^n*sqrt(-
a)*a^2*b*n - sqrt(-a)*a^3*n)*sqrt((c*x)^n*b - a)), 2/3*(3*(c*x)^n*sqrt(a)*b - 3*
((c*x)^n*b - a)^(3/2)*arctan(sqrt(a)/sqrt((c*x)^n*b - a)) - 4*a^(3/2))/(((c*x)^n
*a^(5/2)*b*n - a^(7/2)*n)*sqrt((c*x)^n*b - a))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x \left (- a + b \left (c x\right )^{n}\right )^{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x/(-a+b*(c*x)**n)**(5/2),x)

[Out]

Integral(1/(x*(-a + b*(c*x)**n)**(5/2)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (\left (c x\right )^{n} b - a\right )}^{\frac{5}{2}} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(((c*x)^n*b - a)^(5/2)*x),x, algorithm="giac")

[Out]

integrate(1/(((c*x)^n*b - a)^(5/2)*x), x)