3.121 \(\int \frac{x^{\frac{3}{2} (-1+n)}}{(a x^{-1+n}+b x^n+c x^{1+n})^{3/2}} \, dx\)

Optimal. Leaf size=51 \[ -\frac{2 x^{\frac{n-1}{2}} (b+2 c x)}{\left (b^2-4 a c\right ) \sqrt{a x^{n-1}+b x^n+c x^{n+1}}} \]

[Out]

(-2*x^((-1 + n)/2)*(b + 2*c*x))/((b^2 - 4*a*c)*Sqrt[a*x^(-1 + n) + b*x^n + c*x^(1 + n)])

________________________________________________________________________________________

Rubi [A]  time = 0.0499673, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.029, Rules used = {1915} \[ -\frac{2 x^{\frac{n-1}{2}} (b+2 c x)}{\left (b^2-4 a c\right ) \sqrt{a x^{n-1}+b x^n+c x^{n+1}}} \]

Antiderivative was successfully verified.

[In]

Int[x^((3*(-1 + n))/2)/(a*x^(-1 + n) + b*x^n + c*x^(1 + n))^(3/2),x]

[Out]

(-2*x^((-1 + n)/2)*(b + 2*c*x))/((b^2 - 4*a*c)*Sqrt[a*x^(-1 + n) + b*x^n + c*x^(1 + n)])

Rule 1915

Int[(x_)^(m_.)/((b_.)*(x_)^(n_.) + (a_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.))^(3/2), x_Symbol] :> Simp[(-2*x^((n - 1
)/2)*(b + 2*c*x))/((b^2 - 4*a*c)*Sqrt[a*x^(n - 1) + b*x^n + c*x^(n + 1)]), x] /; FreeQ[{a, b, c, n}, x] && EqQ
[m, (3*(n - 1))/2] && EqQ[q, n - 1] && EqQ[r, n + 1] && NeQ[b^2 - 4*a*c, 0]

Rubi steps

\begin{align*} \int \frac{x^{\frac{3}{2} (-1+n)}}{\left (a x^{-1+n}+b x^n+c x^{1+n}\right )^{3/2}} \, dx &=-\frac{2 x^{\frac{1}{2} (-1+n)} (b+2 c x)}{\left (b^2-4 a c\right ) \sqrt{a x^{-1+n}+b x^n+c x^{1+n}}}\\ \end{align*}

Mathematica [A]  time = 0.0883845, size = 46, normalized size = 0.9 \[ -\frac{2 x^{\frac{n-1}{2}} (b+2 c x)}{\left (b^2-4 a c\right ) \sqrt{x^{n-1} (a+x (b+c x))}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^((3*(-1 + n))/2)/(a*x^(-1 + n) + b*x^n + c*x^(1 + n))^(3/2),x]

[Out]

(-2*x^((-1 + n)/2)*(b + 2*c*x))/((b^2 - 4*a*c)*Sqrt[x^(-1 + n)*(a + x*(b + c*x))])

________________________________________________________________________________________

Maple [F]  time = 0.074, size = 0, normalized size = 0. \begin{align*} \int{{x}^{-{\frac{3}{2}}+{\frac{3\,n}{2}}} \left ( a{x}^{-1+n}+b{x}^{n}+c{x}^{1+n} \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(-3/2+3/2*n)/(a*x^(-1+n)+b*x^n+c*x^(1+n))^(3/2),x)

[Out]

int(x^(-3/2+3/2*n)/(a*x^(-1+n)+b*x^n+c*x^(1+n))^(3/2),x)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{\frac{3}{2} \, n - \frac{3}{2}}}{{\left (c x^{n + 1} + a x^{n - 1} + b x^{n}\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-3/2+3/2*n)/(a*x^(-1+n)+b*x^n+c*x^(1+n))^(3/2),x, algorithm="maxima")

[Out]

integrate(x^(3/2*n - 3/2)/(c*x^(n + 1) + a*x^(n - 1) + b*x^n)^(3/2), x)

________________________________________________________________________________________

Fricas [A]  time = 1.65691, size = 186, normalized size = 3.65 \begin{align*} -\frac{2 \,{\left (2 \, c x^{2} + b x\right )} \sqrt{\frac{{\left (c x^{2} + b x + a\right )} x^{n + 1}}{x^{2}}}}{{\left (a b^{2} - 4 \, a^{2} c +{\left (b^{2} c - 4 \, a c^{2}\right )} x^{2} +{\left (b^{3} - 4 \, a b c\right )} x\right )} x^{\frac{1}{2} \, n + \frac{1}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-3/2+3/2*n)/(a*x^(-1+n)+b*x^n+c*x^(1+n))^(3/2),x, algorithm="fricas")

[Out]

-2*(2*c*x^2 + b*x)*sqrt((c*x^2 + b*x + a)*x^(n + 1)/x^2)/((a*b^2 - 4*a^2*c + (b^2*c - 4*a*c^2)*x^2 + (b^3 - 4*
a*b*c)*x)*x^(1/2*n + 1/2))

________________________________________________________________________________________

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(x**(-3/2+3/2*n)/(a*x**(-1+n)+b*x**n+c*x**(1+n))**(3/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{\frac{3}{2} \, n - \frac{3}{2}}}{{\left (c x^{n + 1} + a x^{n - 1} + b x^{n}\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-3/2+3/2*n)/(a*x^(-1+n)+b*x^n+c*x^(1+n))^(3/2),x, algorithm="giac")

[Out]

integrate(x^(3/2*n - 3/2)/(c*x^(n + 1) + a*x^(n - 1) + b*x^n)^(3/2), x)