\(\int \frac {x^{\frac {3}{2} (-1+n)}}{(a x^{-1+n}+b x^n+c x^{1+n})^{3/2}} \, dx\) [121]

Optimal result

Integrand size = 34, antiderivative size = 51 \[ \int \frac {x^{\frac {3}{2} (-1+n)}}{\left (a x^{-1+n}+b x^n+c x^{1+n}\right )^{3/2}} \, dx=-\frac {2 x^{\frac {1}{2} (-1+n)} (b+2 c x)}{\left (b^2-4 a c\right ) \sqrt {a x^{-1+n}+b x^n+c x^{1+n}}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.029, Rules used = {1929} \[ \int \frac {x^{\frac {3}{2} (-1+n)}}{\left (a x^{-1+n}+b x^n+c x^{1+n}\right )^{3/2}} \, dx=-\frac {2 x^{\frac {n-1}{2}} (b+2 c x)}{\left (b^2-4 a c\right ) \sqrt {a x^{n-1}+b x^n+c x^{n+1}}} \]

[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 1929

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*} \text {integral}& = -\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] (verified)

Time = 0.05 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.90 \[ \int \frac {x^{\frac {3}{2} (-1+n)}}{\left (a x^{-1+n}+b x^n+c x^{1+n}\right )^{3/2}} \, dx=-\frac {2 x^{\frac {1}{2} (-1+n)} (b+2 c x)}{\left (b^2-4 a c\right ) \sqrt {x^{-1+n} (a+x (b+c x))}} \]

[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]

\[\int \frac {x^{-\frac {3}{2}+\frac {3 n}{2}}}{\left (a \,x^{-1+n}+b \,x^{n}+c \,x^{1+n}\right )^{\frac {3}{2}}}d x\]

[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)

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.63 \[ \int \frac {x^{\frac {3}{2} (-1+n)}}{\left (a x^{-1+n}+b x^n+c x^{1+n}\right )^{3/2}} \, dx=-\frac {2 \, {\left (2 \, c x^{2} + b x\right )} \sqrt {\frac {{\left (c x^{2} + b x + a\right )} x^{n + 1}}{x^{2}}}}{{\left (a b^{2} - 4 \, a^{2} c + {\left (b^{2} c - 4 \, a c^{2}\right )} x^{2} + {\left (b^{3} - 4 \, a b c\right )} x\right )} x^{\frac {1}{2} \, n + \frac {1}{2}}} \]

[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]

\[ \int \frac {x^{\frac {3}{2} (-1+n)}}{\left (a x^{-1+n}+b x^n+c x^{1+n}\right )^{3/2}} \, dx=\int \frac {x^{\frac {3 n}{2} - \frac {3}{2}}}{\left (a x^{n - 1} + b x^{n} + c x^{n + 1}\right )^{\frac {3}{2}}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {x^{\frac {3}{2} (-1+n)}}{\left (a x^{-1+n}+b x^n+c x^{1+n}\right )^{3/2}} \, dx=\int { \frac {x^{\frac {3}{2} \, n - \frac {3}{2}}}{{\left (c x^{n + 1} + a x^{n - 1} + b x^{n}\right )}^{\frac {3}{2}}} \,d x } \]

[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)

Giac [F]

\[ \int \frac {x^{\frac {3}{2} (-1+n)}}{\left (a x^{-1+n}+b x^n+c x^{1+n}\right )^{3/2}} \, dx=\int { \frac {x^{\frac {3}{2} \, n - \frac {3}{2}}}{{\left (c x^{n + 1} + a x^{n - 1} + b x^{n}\right )}^{\frac {3}{2}}} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^{\frac {3}{2} (-1+n)}}{\left (a x^{-1+n}+b x^n+c x^{1+n}\right )^{3/2}} \, dx=\int \frac {x^{\frac {3\,n}{2}-\frac {3}{2}}}{{\left (b\,x^n+a\,x^{n-1}+c\,x^{n+1}\right )}^{3/2}} \,d x \]

[In]

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

[Out]

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