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

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Rubi [A]  time = 0.02, antiderivative size = 52, normalized size of antiderivative = 1.00, 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 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]

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]

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*}

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Mathematica [A]  time = 0.02, 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

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fricas [A]  time = 0.70, size = 79, normalized size = 1.52 \[ \frac {{\left (a x x^{\left (\frac {1}{2 \, p + 1}\right )} + b x\right )} \left (\frac {a^{2} x^{\frac {2}{2 \, p + 1}} + 2 \, a b x^{\left (\frac {1}{2 \, p + 1}\right )} + b^{2}}{x^{\frac {2}{2 \, p + 1}}}\right )^{p}}{a x^{\left (\frac {1}{2 \, p + 1}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a^2+b^2/(x^(2/(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)))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (a^{2} + \frac {b^{2}}{x^{\frac {2}{2 \, p + 1}}} + \frac {2 \, a b}{x^{\left (\frac {1}{2 \, p + 1}\right )}}\right )}^{p}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a^2+b^2/(x^(2/(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)

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maple [F]  time = 0.28, size = 0, normalized size = 0.00 \[ \int \left (2 a b \,x^{-\frac {1}{2 p +1}}+b^{2} x^{-\frac {2}{2 p +1}}+a^{2}\right )^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (a^{2} + \frac {b^{2}}{x^{\frac {2}{2 \, p + 1}}} + \frac {2 \, a b}{x^{\left (\frac {1}{2 \, p + 1}\right )}}\right )}^{p}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a^2+b^2/(x^(2/(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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int {\left (a^2+\frac {b^2}{x^{\frac {2}{2\,p+1}}}+\frac {2\,a\,b}{x^{\frac {1}{2\,p+1}}}\right )}^p \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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