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

Optimal. Leaf size=130 \[ \frac{2 (p+1) x \left (a+b x^{-\frac{1}{2 (p+1)}}\right ) \left (a^2+2 a b x^{-\frac{1}{2 (p+1)}}+b^2 x^{-\frac{1}{p+1}}\right )^p}{a (2 p+1)}-\frac{x \left (a+b x^{-\frac{1}{2 (p+1)}}\right )^2 \left (a^2+2 a b x^{-\frac{1}{2 (p+1)}}+b^2 x^{-\frac{1}{p+1}}\right )^p}{a^2 (2 p+1)} \]

[Out]

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

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Rubi [A]  time = 0.109703, antiderivative size = 130, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.088 \[ \frac{2 (p+1) x \left (a+b x^{-\frac{1}{2 (p+1)}}\right ) \left (a^2+2 a b x^{-\frac{1}{2 (p+1)}}+b^2 x^{-\frac{1}{p+1}}\right )^p}{a (2 p+1)}-\frac{x \left (a+b x^{-\frac{1}{2 (p+1)}}\right )^2 \left (a^2+2 a b x^{-\frac{1}{2 (p+1)}}+b^2 x^{-\frac{1}{p+1}}\right )^p}{a^2 (2 p+1)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 4.68432, size = 95, normalized size = 0.73 \[ \frac{x \left (2 a + 2 b x^{- \frac{1}{2 \left (p + 1\right )}}\right ) \left (p + 1\right ) \left (a^{2} + 2 a b x^{- \frac{1}{2 \left (p + 1\right )}} + b^{2} x^{- \frac{1}{p + 1}}\right )^{p}}{a \left (2 p + 1\right )} - \frac{x \left (a^{2} + 2 a b x^{- \frac{1}{2 \left (p + 1\right )}} + b^{2} x^{- \frac{1}{p + 1}}\right )^{p + 1}}{a^{2} \left (2 p + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.308144, size = 165, normalized size = 1.27 \[ \frac{x^{\frac{p}{p+1}} \left (x^{-\frac{1}{p+1}} \left (a x^{\frac{1}{2 p+2}}+b\right )^2\right )^p \left (\frac{a x^{\frac{1}{2 p+2}}}{b}+1\right )^{-2 p} \left (a^2 (2 p+1) x^{\frac{1}{p+1}} \left (\frac{a x^{\frac{1}{2 p+2}}}{b}+1\right )^{2 p}-b^2 \left (\left (\frac{a x^{\frac{1}{2 p+2}}}{b}+1\right )^{2 p}-1\right )+2 a b p x^{\frac{1}{2 p+2}} \left (\frac{a x^{\frac{1}{2 p+2}}}{b}+1\right )^{2 p}\right )}{a^2 (2 p+1)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.391, size = 0, normalized size = 0. \[ \int \left ({a}^{2}+{\frac{{b}^{2}}{{x}^{ \left ( 1+p \right ) ^{-1}}}}+2\,{ab \left ({x}^{1/2\, \left ( 1+p \right ) ^{-1}} \right ) ^{-1}} \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.287701, size = 139, normalized size = 1.07 \[ \frac{{\left (2 \, a b p x x^{\frac{1}{2 \,{\left (p + 1\right )}}} - b^{2} x +{\left (2 \, a^{2} p + a^{2}\right )} x x^{\left (\frac{1}{p + 1}\right )}\right )} \left (\frac{2 \, a b x^{\frac{1}{2 \,{\left (p + 1\right )}}} + a^{2} x^{\left (\frac{1}{p + 1}\right )} + b^{2}}{x^{\left (\frac{1}{p + 1}\right )}}\right )^{p}}{{\left (2 \, a^{2} p + a^{2}\right )} x^{\left (\frac{1}{p + 1}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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