3.543 \(\int \left (a^2+b^2 x^{-\frac{2}{1+2 p}}+2 a b x^{-\frac{1}{1+2 p}}\right )^p \, dx\)

Optimal. Leaf size=52 \[ \frac{x \left (a+b x^{\frac{1}{-2 p-1}}\right ) \left (a^2+2 a b x^{\frac{1}{-2 p-1}}+b^2 x^{-\frac{2}{2 p+1}}\right )^p}{a} \]

[Out]

(x*(a + b*x^(-1 - 2*p)^(-1))*(a^2 + 2*a*b*x^(-1 - 2*p)^(-1) + b^2/x^(2/(1 + 2*p)
))^p)/a

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Rubi [A]  time = 0.0385906, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056 \[ \frac{x \left (a+b x^{\frac{1}{-2 p-1}}\right ) \left (a^2+2 a b x^{\frac{1}{-2 p-1}}+b^2 x^{-\frac{2}{2 p+1}}\right )^p}{a} \]

Antiderivative was successfully verified.

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

[Out]

(x*(a + b*x^(-1 - 2*p)^(-1))*(a^2 + 2*a*b*x^(-1 - 2*p)^(-1) + b^2/x^(2/(1 + 2*p)
))^p)/a

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Rubi in Sympy [A]  time = 2.5979, size = 49, normalized size = 0.94 \[ \frac{x \left (2 a + 2 b x^{- \frac{1}{2 p + 1}}\right ) \left (a^{2} + 2 a b x^{- \frac{1}{2 p + 1}} + b^{2} x^{- \frac{2}{2 p + 1}}\right )^{p}}{2 a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*(2*a + 2*b*x**(-1/(2*p + 1)))*(a**2 + 2*a*b*x**(-1/(2*p + 1)) + b**2*x**(-2/(2
*p + 1)))**p/(2*a)

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Mathematica [B]  time = 0.165466, size = 121, normalized size = 2.33 \[ \frac{x^{\frac{2 p}{2 p+1}} \left (x^{-\frac{2}{2 p+1}} \left (a x^{\frac{1}{2 p+1}}+b\right )^2\right )^p \left (\frac{a x^{\frac{1}{2 p+1}}}{b}+1\right )^{-2 p} \left (a x^{\frac{1}{2 p+1}} \left (\frac{a x^{\frac{1}{2 p+1}}}{b}+1\right )^{2 p}+b \left (\left (\frac{a x^{\frac{1}{2 p+1}}}{b}+1\right )^{2 p}-1\right )\right )}{a} \]

Antiderivative was successfully verified.

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

[Out]

(x^((2*p)/(1 + 2*p))*((b + a*x^(1 + 2*p)^(-1))^2/x^(2/(1 + 2*p)))^p*(a*x^(1 + 2*
p)^(-1)*(1 + (a*x^(1 + 2*p)^(-1))/b)^(2*p) + b*(-1 + (1 + (a*x^(1 + 2*p)^(-1))/b
)^(2*p))))/(a*(1 + (a*x^(1 + 2*p)^(-1))/b)^(2*p))

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Maple [F]  time = 0.354, size = 0, normalized size = 0. \[ \int \left ({a}^{2}+{{b}^{2} \left ({x}^{2\, \left ( 1+2\,p \right ) ^{-1}} \right ) ^{-1}}+2\,{\frac{ab}{{x}^{ \left ( 1+2\,p \right ) ^{-1}}}} \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

int((a^2+b^2/(x^(2/(1+2*p)))+2*a*b/(x^(1/(1+2*p))))^p,x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((2*a*b*x^(-1/(2*p + 1)) + b^2*x^(-2/(2*p + 1)) + a^2)^p, x)

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Fricas [A]  time = 0.29391, size = 107, normalized size = 2.06 \[ \frac{{\left (a x x^{\left (\frac{1}{2 \, p + 1}\right )} + b x\right )} \left (\frac{a^{2} x^{\frac{2}{2 \, p + 1}} + 2 \, a b x^{\left (\frac{1}{2 \, p + 1}\right )} + b^{2}}{x^{\frac{2}{2 \, p + 1}}}\right )^{p}}{a x^{\left (\frac{1}{2 \, p + 1}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(a*x*x^(1/(2*p + 1)) + b*x)*((a^2*x^(2/(2*p + 1)) + 2*a*b*x^(1/(2*p + 1)) + b^2)
/x^(2/(2*p + 1)))^p/(a*x^(1/(2*p + 1)))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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