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

Optimal. Leaf size=43 \[ \frac{x \left (a+b x^n\right ) \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{-\frac{n+1}{2 n}}}{a} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0134285, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065, Rules used = {1343, 191} \[ \frac{x \left (a+b x^n\right ) \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{-\frac{n+1}{2 n}}}{a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.0609589, size = 32, normalized size = 0.74 \[ \frac{x \left (a+b x^n\right ) \left (\left (a+b x^n\right )^2\right )^{-\frac{n+1}{2 n}}}{a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.035, size = 51, normalized size = 1.2 \begin{align*}{ \left ( x+{\frac{bx{{\rm e}^{n\ln \left ( x \right ) }}}{a}} \right ) \left ({{\rm e}^{{\frac{ \left ( 1+n \right ) \ln \left ({a}^{2}+2\,ab{{\rm e}^{n\ln \left ( x \right ) }}+{b}^{2} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2} \right ) }{2\,n}}}} \right ) ^{-1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x+b/a*x*exp(n*ln(x)))/exp(1/2*(1+n)/n*ln(a^2+2*a*b*exp(n*ln(x))+b^2*exp(n*ln(x))^2))

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((a^2+2*a*b*x^n+b^2*x^(2*n))^(1/2*(1+n)/n)),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.64416, size = 93, normalized size = 2.16 \begin{align*} \frac{b x x^{n} + a x}{{\left (b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}\right )}^{\frac{n + 1}{2 \, n}} a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((a^2+2*a*b*x^n+b^2*x^(2*n))^(1/2*(1+n)/n)),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

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(1/((a**2+2*a*b*x**n+b**2*x**(2*n))**(1/2*(1+n)/n)),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}\right )}^{\frac{n + 1}{2 \, n}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((a^2+2*a*b*x^n+b^2*x^(2*n))^(1/2*(1+n)/n)),x, algorithm="giac")

[Out]

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