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