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

Optimal. Leaf size=159 \[ \frac{1}{a^2 n \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}+\frac{1}{2 a n \left (a+b x^n\right ) \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}+\frac{\log (x) \left (a+b x^n\right )}{a^3 \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}-\frac{\left (a+b x^n\right ) \log \left (a+b x^n\right )}{a^3 n \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}} \]

[Out]

1/(a^2*n*Sqrt[a^2 + 2*a*b*x^n + b^2*x^(2*n)]) + 1/(2*a*n*(a + b*x^n)*Sqrt[a^2 +
2*a*b*x^n + b^2*x^(2*n)]) + ((a + b*x^n)*Log[x])/(a^3*Sqrt[a^2 + 2*a*b*x^n + b^2
*x^(2*n)]) - ((a + b*x^n)*Log[a + b*x^n])/(a^3*n*Sqrt[a^2 + 2*a*b*x^n + b^2*x^(2
*n)])

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Rubi [A]  time = 0.184502, antiderivative size = 159, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.107 \[ \frac{1}{a^2 n \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}+\frac{1}{2 a n \left (a+b x^n\right ) \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}+\frac{\log (x) \left (a+b x^n\right )}{a^3 \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}-\frac{\left (a+b x^n\right ) \log \left (a+b x^n\right )}{a^3 n \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}} \]

Antiderivative was successfully verified.

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

[Out]

1/(a^2*n*Sqrt[a^2 + 2*a*b*x^n + b^2*x^(2*n)]) + 1/(2*a*n*(a + b*x^n)*Sqrt[a^2 +
2*a*b*x^n + b^2*x^(2*n)]) + ((a + b*x^n)*Log[x])/(a^3*Sqrt[a^2 + 2*a*b*x^n + b^2
*x^(2*n)]) - ((a + b*x^n)*Log[a + b*x^n])/(a^3*n*Sqrt[a^2 + 2*a*b*x^n + b^2*x^(2
*n)])

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Rubi in Sympy [A]  time = 31.2413, size = 163, normalized size = 1.03 \[ \frac{2 a + 2 b x^{n}}{4 a n \left (a^{2} + 2 a b x^{n} + b^{2} x^{2 n}\right )^{\frac{3}{2}}} + \frac{1}{a^{2} n \sqrt{a^{2} + 2 a b x^{n} + b^{2} x^{2 n}}} + \frac{b \sqrt{a^{2} + 2 a b x^{n} + b^{2} x^{2 n}} \log{\left (x^{n} \right )}}{a^{3} n \left (a b + b^{2} x^{n}\right )} - \frac{b \sqrt{a^{2} + 2 a b x^{n} + b^{2} x^{2 n}} \log{\left (a + b x^{n} \right )}}{a^{3} n \left (a b + b^{2} x^{n}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(2*a + 2*b*x**n)/(4*a*n*(a**2 + 2*a*b*x**n + b**2*x**(2*n))**(3/2)) + 1/(a**2*n*
sqrt(a**2 + 2*a*b*x**n + b**2*x**(2*n))) + b*sqrt(a**2 + 2*a*b*x**n + b**2*x**(2
*n))*log(x**n)/(a**3*n*(a*b + b**2*x**n)) - b*sqrt(a**2 + 2*a*b*x**n + b**2*x**(
2*n))*log(a + b*x**n)/(a**3*n*(a*b + b**2*x**n))

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Mathematica [A]  time = 0.104482, size = 78, normalized size = 0.49 \[ \frac{a \left (3 a+2 b x^n\right )+2 n \log (x) \left (a+b x^n\right )^2-2 \left (a+b x^n\right )^2 \log \left (a+b x^n\right )}{2 a^3 n \left (a+b x^n\right ) \sqrt{\left (a+b x^n\right )^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.031, size = 104, normalized size = 0.7 \[{\frac{\ln \left ( x \right ) }{ \left ( a+b{x}^{n} \right ){a}^{3}}\sqrt{ \left ( a+b{x}^{n} \right ) ^{2}}}+{\frac{2\,b{x}^{n}+3\,a}{2\, \left ( a+b{x}^{n} \right ) ^{3}{a}^{2}n}\sqrt{ \left ( a+b{x}^{n} \right ) ^{2}}}-{\frac{1}{ \left ( a+b{x}^{n} \right ){a}^{3}n}\sqrt{ \left ( a+b{x}^{n} \right ) ^{2}}\ln \left ({x}^{n}+{\frac{a}{b}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.756369, size = 95, normalized size = 0.6 \[ \frac{2 \, b x^{n} + 3 \, a}{2 \,{\left (a^{2} b^{2} n x^{2 \, n} + 2 \, a^{3} b n x^{n} + a^{4} n\right )}} + \frac{\log \left (x\right )}{a^{3}} - \frac{\log \left (\frac{b x^{n} + a}{b}\right )}{a^{3} n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(2*b*x^n + 3*a)/(a^2*b^2*n*x^(2*n) + 2*a^3*b*n*x^n + a^4*n) + log(x)/a^3 - l
og((b*x^n + a)/b)/(a^3*n)

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Fricas [A]  time = 0.289337, size = 143, normalized size = 0.9 \[ \frac{2 \, b^{2} n x^{2 \, n} \log \left (x\right ) + 2 \, a^{2} n \log \left (x\right ) + 3 \, a^{2} + 2 \,{\left (2 \, a b n \log \left (x\right ) + a b\right )} x^{n} - 2 \,{\left (b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}\right )} \log \left (b x^{n} + a\right )}{2 \,{\left (a^{3} b^{2} n x^{2 \, n} + 2 \, a^{4} b n x^{n} + a^{5} n\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(2*b^2*n*x^(2*n)*log(x) + 2*a^2*n*log(x) + 3*a^2 + 2*(2*a*b*n*log(x) + a*b)*
x^n - 2*(b^2*x^(2*n) + 2*a*b*x^n + a^2)*log(b*x^n + a))/(a^3*b^2*n*x^(2*n) + 2*a
^4*b*n*x^n + a^5*n)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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