3.2786 \(\int \frac{(c x)^{-1-\frac{3 n}{2}}}{\sqrt{a+b x^n}} \, dx\)

Optimal. Leaf size=65 \[ \frac{4 (c x)^{-3 n/2} \left (a+b x^n\right )^{3/2}}{3 a^2 c n}-\frac{2 (c x)^{-3 n/2} \sqrt{a+b x^n}}{a c n} \]

[Out]

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

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Rubi [A]  time = 0.0174776, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {273, 264} \[ \frac{4 (c x)^{-3 n/2} \left (a+b x^n\right )^{3/2}}{3 a^2 c n}-\frac{2 (c x)^{-3 n/2} \sqrt{a+b x^n}}{a c n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 273

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(
a*c*n*(p + 1)), x] + Dist[(m + n*(p + 1) + 1)/(a*n*(p + 1)), Int[(c*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[
{a, b, c, m, n, p}, x] && ILtQ[Simplify[(m + 1)/n + p + 1], 0] && NeQ[p, -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{(c x)^{-1-\frac{3 n}{2}}}{\sqrt{a+b x^n}} \, dx &=-\frac{2 (c x)^{-3 n/2} \sqrt{a+b x^n}}{a c n}-\frac{2 \int (c x)^{-1-\frac{3 n}{2}} \sqrt{a+b x^n} \, dx}{a}\\ &=-\frac{2 (c x)^{-3 n/2} \sqrt{a+b x^n}}{a c n}+\frac{4 (c x)^{-3 n/2} \left (a+b x^n\right )^{3/2}}{3 a^2 c n}\\ \end{align*}

Mathematica [A]  time = 0.0149682, size = 41, normalized size = 0.63 \[ -\frac{2 (c x)^{-3 n/2} \left (a-2 b x^n\right ) \sqrt{a+b x^n}}{3 a^2 c n} \]

Antiderivative was successfully verified.

[In]

Integrate[(c*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*c*n*(c*x)^((3*n)/2))

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Maple [F]  time = 0.055, size = 0, normalized size = 0. \begin{align*} \int{ \left ( cx \right ) ^{-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((c*x)^(-1-3/2*n)/(a+b*x^n)^(1/2),x)

[Out]

int((c*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{\left (c x\right )^{-\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((c*x)^(-1-3/2*n)/(a+b*x^n)^(1/2),x, algorithm="maxima")

[Out]

integrate((c*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((c*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 = 13.0699, size = 68, normalized size = 1.05 \begin{align*} - \frac{2 \sqrt{b} c^{- \frac{3 n}{2}} x^{- n} \sqrt{\frac{a x^{- n}}{b} + 1}}{3 a c n} + \frac{4 b^{\frac{3}{2}} c^{- \frac{3 n}{2}} \sqrt{\frac{a x^{- n}}{b} + 1}}{3 a^{2} c n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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