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

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

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Rubi [A]  time = 0.05, antiderivative size = 51, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\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}}} \end {gather*}

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*}

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Mathematica [A]  time = 0.09, size = 46, normalized size = 0.90 \begin {gather*} -\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))}} \end {gather*}

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))])

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IntegrateAlgebraic [A]  time = 0.13, size = 73, normalized size = 1.43 \begin {gather*} -\frac {2 x^{\frac {3 (n-1)}{2}} (b+2 c x) (a+x (b+c x))^{3/2}}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} \left (x^{n-1} (a+x (b+c x))\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 1.08, size = 83, normalized size = 1.63 \begin {gather*} -\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 {gather*}

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))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 {gather*}

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)

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maple [F]  time = 0.08, size = 0, normalized size = 0.00 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 {gather*}

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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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

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

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