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

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

________________________________________________________________________________________

Rubi [A]  time = 0.0195689, 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, Rules used = {1343, 191} \[ \frac{x \left (a+b x^{\frac{1}{-2 p-1}}\right ) \left (a^2+2 a b x^{\frac{1}{-2 p-1}}+b^2 x^{-\frac{2}{2 p+1}}\right )^p}{a} \]

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

Rule 1343

Int[((a_) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.))^(p_), x_Symbol] :> Dist[(a + b*x^n + c*x^(2*n))^p/(b + 2*c*x
^n)^(2*p), Int[(b + 2*c*x^n)^(2*p), x], x] /; FreeQ[{a, b, c, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0]

Rule 191

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^(p + 1))/a, x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rubi steps

\begin{align*} \int \left (a^2+b^2 x^{-\frac{2}{1+2 p}}+2 a b x^{-\frac{1}{1+2 p}}\right )^p \, dx &=\left (\left (a^2+b^2 x^{-\frac{2}{1+2 p}}+2 a b x^{-\frac{1}{1+2 p}}\right )^p \left (2 a b+2 b^2 x^{-\frac{1}{1+2 p}}\right )^{-2 p}\right ) \int \left (2 a b+2 b^2 x^{-\frac{1}{1+2 p}}\right )^{2 p} \, dx\\ &=\frac{x \left (a+b x^{\frac{1}{-1-2 p}}\right ) \left (a^2+2 a b x^{\frac{1}{-1-2 p}}+b^2 x^{-\frac{2}{1+2 p}}\right )^p}{a}\\ \end{align*}

Mathematica [A]  time = 0.0240551, size = 58, normalized size = 1.12 \[ \frac{x^{\frac{2 p}{2 p+1}} \left (a x^{\frac{1}{2 p+1}}+b\right ) \left (x^{-\frac{2}{2 p+1}} \left (a x^{\frac{1}{2 p+1}}+b\right )^2\right )^p}{a} \]

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

________________________________________________________________________________________

Maple [F]  time = 0.243, size = 0, normalized size = 0. \begin{align*} \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 \end{align*}

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)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

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, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.65135, size = 163, normalized size = 3.13 \begin{align*} \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 )}} \end{align*}

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, 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)))

________________________________________________________________________________________

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

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

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

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, algorithm="giac")

[Out]

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