3.2785 \(\int \frac{(c x)^{-1-\frac{n}{2}}}{\sqrt{a+b x^n}} \, dx\)

Optimal. Leaf size=31 \[ -\frac{2 (c x)^{-n/2} \sqrt{a+b x^n}}{a c n} \]

[Out]

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

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Rubi [A]  time = 0.0086017, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043, Rules used = {264} \[ -\frac{2 (c x)^{-n/2} \sqrt{a+b x^n}}{a c n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.0100875, size = 31, normalized size = 1. \[ -\frac{2 x (c x)^{-\frac{n}{2}-1} \sqrt{a+b x^n}}{a n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.055, size = 0, normalized size = 0. \begin{align*} \int{ \left ( cx \right ) ^{-1-{\frac{n}{2}}}{\frac{1}{\sqrt{a+b{x}^{n}}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((c*x)^(-1-1/2*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{\left (c x\right )^{-\frac{1}{2} \, n - 1}}{\sqrt{b x^{n} + a}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: UnboundLocalError

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Sympy [A]  time = 13.0092, size = 29, normalized size = 0.94 \begin{align*} - \frac{2 \sqrt{b} c^{- \frac{n}{2}} \sqrt{\frac{a x^{- n}}{b} + 1}}{a c n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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