3.2692 \(\int \frac{x^{-1-\frac{3 n}{2}}}{\sqrt{a+b x^n}} \, dx\)

Optimal. Leaf size=58 \[ \frac{4 b x^{-n/2} \sqrt{a+b x^n}}{3 a^2 n}-\frac{2 x^{-3 n/2} \sqrt{a+b x^n}}{3 a n} \]

[Out]

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

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Rubi [A]  time = 0.0146927, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {271, 264} \[ \frac{4 b x^{-n/2} \sqrt{a+b x^n}}{3 a^2 n}-\frac{2 x^{-3 n/2} \sqrt{a+b x^n}}{3 a n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 271

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x^(m + 1)*(a + b*x^n)^(p + 1))/(a*(m + 1)), x]
- Dist[(b*(m + n*(p + 1) + 1))/(a*(m + 1)), Int[x^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, m, n, p}, x]
&& ILtQ[Simplify[(m + 1)/n + p + 1], 0] && NeQ[m, -1]

Rule 264

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a
*c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rubi steps

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

Mathematica [A]  time = 0.0132765, size = 36, normalized size = 0.62 \[ -\frac{2 x^{-3 n/2} \left (a-2 b x^n\right ) \sqrt{a+b x^n}}{3 a^2 n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.079, size = 0, normalized size = 0. \begin{align*} \int{{x}^{-1-{\frac{3\,n}{2}}}{\frac{1}{\sqrt{a+b{x}^{n}}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: UnboundLocalError

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Sympy [A]  time = 16.8277, size = 51, normalized size = 0.88 \begin{align*} - \frac{2 \sqrt{b} x^{- n} \sqrt{\frac{a x^{- n}}{b} + 1}}{3 a n} + \frac{4 b^{\frac{3}{2}} \sqrt{\frac{a x^{- n}}{b} + 1}}{3 a^{2} n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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