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

Optimal. Leaf size=98 \[ \frac{2 \left (-2 a c+b^2+b c x^n\right )}{a n \left (b^2-4 a c\right ) \sqrt{a+b x^n+c x^{2 n}}}-\frac{\tanh ^{-1}\left (\frac{2 a+b x^n}{2 \sqrt{a} \sqrt{a+b x^n+c x^{2 n}}}\right )}{a^{3/2} n} \]

[Out]

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

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Rubi [A]  time = 0.166185, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ \frac{2 \left (-2 a c+b^2+b c x^n\right )}{a n \left (b^2-4 a c\right ) \sqrt{a+b x^n+c x^{2 n}}}-\frac{\tanh ^{-1}\left (\frac{2 a+b x^n}{2 \sqrt{a} \sqrt{a+b x^n+c x^{2 n}}}\right )}{a^{3/2} n} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 23.0723, size = 87, normalized size = 0.89 \[ \frac{2 \left (- 2 a c + b^{2} + b c x^{n}\right )}{a n \left (- 4 a c + b^{2}\right ) \sqrt{a + b x^{n} + c x^{2 n}}} - \frac{\operatorname{atanh}{\left (\frac{2 a + b x^{n}}{2 \sqrt{a} \sqrt{a + b x^{n} + c x^{2 n}}} \right )}}{a^{\frac{3}{2}} n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(-2*a*c + b**2 + b*c*x**n)/(a*n*(-4*a*c + b**2)*sqrt(a + b*x**n + c*x**(2*n)))
- atanh((2*a + b*x**n)/(2*sqrt(a)*sqrt(a + b*x**n + c*x**(2*n))))/(a**(3/2)*n)

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Mathematica [A]  time = 0.884211, size = 100, normalized size = 1.02 \[ \frac{\log (x)-\frac{\frac{2 \sqrt{a} \left (2 a c-b^2-b c x^n\right )}{\left (b^2-4 a c\right ) \sqrt{a+x^n \left (b+c x^n\right )}}+\log \left (2 \sqrt{a} \sqrt{a+x^n \left (b+c x^n\right )}+2 a+b x^n\right )}{n}}{a^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.02, size = 0, normalized size = 0. \[ \int{\frac{1}{x} \left ( a+b{x}^{n}+c{x}^{2\,n} \right ) ^{-{\frac{3}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

int(1/x/(a+b*x^n+c*x^(2*n))^(3/2),x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.330033, size = 1, normalized size = 0.01 \[ \left [\frac{{\left (a b^{2} - 4 \, a^{2} c +{\left (b^{2} c - 4 \, a c^{2}\right )} x^{2 \, n} +{\left (b^{3} - 4 \, a b c\right )} x^{n}\right )} \log \left (-\frac{8 \, a^{\frac{3}{2}} b x^{n} + 8 \, a^{\frac{5}{2}} +{\left (b^{2} + 4 \, a c\right )} \sqrt{a} x^{2 \, n} - 4 \,{\left (a b x^{n} + 2 \, a^{2}\right )} \sqrt{c x^{2 \, n} + b x^{n} + a}}{x^{2 \, n}}\right ) + 4 \,{\left (\sqrt{a} b c x^{n} +{\left (b^{2} - 2 \, a c\right )} \sqrt{a}\right )} \sqrt{c x^{2 \, n} + b x^{n} + a}}{2 \,{\left ({\left (a b^{2} c - 4 \, a^{2} c^{2}\right )} \sqrt{a} n x^{2 \, n} +{\left (a b^{3} - 4 \, a^{2} b c\right )} \sqrt{a} n x^{n} +{\left (a^{2} b^{2} - 4 \, a^{3} c\right )} \sqrt{a} n\right )}}, -\frac{{\left (a b^{2} - 4 \, a^{2} c +{\left (b^{2} c - 4 \, a c^{2}\right )} x^{2 \, n} +{\left (b^{3} - 4 \, a b c\right )} x^{n}\right )} \arctan \left (\frac{\sqrt{-a} b x^{n} + 2 \, \sqrt{-a} a}{2 \, \sqrt{c x^{2 \, n} + b x^{n} + a} a}\right ) - 2 \,{\left (\sqrt{-a} b c x^{n} +{\left (b^{2} - 2 \, a c\right )} \sqrt{-a}\right )} \sqrt{c x^{2 \, n} + b x^{n} + a}}{{\left (a b^{2} c - 4 \, a^{2} c^{2}\right )} \sqrt{-a} n x^{2 \, n} +{\left (a b^{3} - 4 \, a^{2} b c\right )} \sqrt{-a} n x^{n} +{\left (a^{2} b^{2} - 4 \, a^{3} c\right )} \sqrt{-a} n}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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