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))^(1/2*(1+n)/n))

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Rubi [A]  time = 0.01, antiderivative size = 43, normalized size of antiderivative = 1.00, 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 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+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*}

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Mathematica [A]  time = 0.06, 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)))

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fricas [A]  time = 0.95, size = 45, normalized size = 1.05 \[ \frac {b x x^{n} + a x}{{\left (b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}\right )}^{\frac {n + 1}{2 \, n}} a} \]

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)

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

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)

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maple [A]  time = 0.04, size = 51, normalized size = 1.19 \[ \left (\frac {b x \,{\mathrm e}^{n \ln \relax (x )}}{a}+x \right ) {\mathrm e}^{-\frac {\left (n +1\right ) \ln \left (2 a b \,{\mathrm e}^{n \ln \relax (x )}+b^{2} {\mathrm e}^{2 n \ln \relax (x )}+a^{2}\right )}{2 n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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]

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

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