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

Optimal. Leaf size=75 \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+b (c x)^n}}{\sqrt{a}}\right )}{a^{5/2} n}+\frac{2}{a^2 n \sqrt{a+b (c x)^n}}+\frac{2}{3 a n \left (a+b (c x)^n\right )^{3/2}} \]

[Out]

2/(3*a*n*(a + b*(c*x)^n)^(3/2)) + 2/(a^2*n*Sqrt[a + b*(c*x)^n]) - (2*ArcTanh[Sqr
t[a + b*(c*x)^n]/Sqrt[a]])/(a^(5/2)*n)

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Rubi [A]  time = 0.146465, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.353 \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+b (c x)^n}}{\sqrt{a}}\right )}{a^{5/2} n}+\frac{2}{a^2 n \sqrt{a+b (c x)^n}}+\frac{2}{3 a n \left (a+b (c x)^n\right )^{3/2}} \]

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*ArcTanh[Sqr
t[a + b*(c*x)^n]/Sqrt[a]])/(a^(5/2)*n)

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Rubi in Sympy [A]  time = 6.5052, size = 63, normalized size = 0.84 \[ \frac{2}{3 a n \left (a + b \left (c x\right )^{n}\right )^{\frac{3}{2}}} + \frac{2}{a^{2} n \sqrt{a + b \left (c x\right )^{n}}} - \frac{2 \operatorname{atanh}{\left (\frac{\sqrt{a + b \left (c x\right )^{n}}}{\sqrt{a}} \right )}}{a^{\frac{5}{2}} n} \]

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*atanh(sq
rt(a + b*(c*x)**n)/sqrt(a))/(a**(5/2)*n)

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Mathematica [A]  time = 0.153711, size = 66, normalized size = 0.88 \[ \frac{2 \left (\frac{\sqrt{a} \left (4 a+3 b (c x)^n\right )}{\left (a+b (c x)^n\right )^{3/2}}-3 \tanh ^{-1}\left (\frac{\sqrt{a+b (c x)^n}}{\sqrt{a}}\right )\right )}{3 a^{5/2} n} \]

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

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Maple [A]  time = 0.011, size = 59, normalized size = 0.8 \[{\frac{1}{n} \left ( -2\,{\frac{1}{{a}^{5/2}}{\it Artanh} \left ({\frac{\sqrt{a+b \left ( cx \right ) ^{n}}}{\sqrt{a}}} \right ) }+2\,{\frac{1}{\sqrt{a+b \left ( cx \right ) ^{n}}{a}^{2}}}+{\frac{2}{3\,a} \left ( a+b \left ( cx \right ) ^{n} \right ) ^{-{\frac{3}{2}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(a+b*(c*x)^n)^(5/2),x)

[Out]

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

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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 = 0.288377, size = 1, normalized size = 0.01 \[ \left [\frac{6 \, \left (c x\right )^{n} \sqrt{a} b + 3 \,{\left (\left (c x\right )^{n} b + a\right )}^{\frac{3}{2}} \log \left (\frac{\left (c x\right )^{n} \sqrt{a} b - 2 \, \sqrt{\left (c x\right )^{n} b + a} a + 2 \, a^{\frac{3}{2}}}{\left (c x\right )^{n}}\right ) + 8 \, a^{\frac{3}{2}}}{3 \,{\left (\left (c x\right )^{n} a^{\frac{5}{2}} b n + a^{\frac{7}{2}} n\right )} \sqrt{\left (c x\right )^{n} b + a}}, \frac{2 \,{\left (3 \, \left (c x\right )^{n} \sqrt{-a} b + 3 \,{\left (\left (c x\right )^{n} b + a\right )}^{\frac{3}{2}} \arctan \left (\frac{a}{\sqrt{\left (c x\right )^{n} b + a} \sqrt{-a}}\right ) + 4 \, \sqrt{-a} a\right )}}{3 \,{\left (\left (c x\right )^{n} \sqrt{-a} a^{2} b n + \sqrt{-a} a^{3} n\right )} \sqrt{\left (c x\right )^{n} b + a}}\right ] \]

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*s
qrt((c*x)^n*b + a)*a + 2*a^(3/2))/(c*x)^n) + 8*a^(3/2))/(((c*x)^n*a^(5/2)*b*n +
a^(7/2)*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(a/(sqrt((c*x)^n*b + a)*sqrt(-a))) + 4*sqrt(-a)*a)/(((c*x)^n*sqrt(-a)
*a^2*b*n + sqrt(-a)*a^3*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)