3.982 \(\int \frac{1}{(c x)^{3/2} (a-b x^2)^{3/4}} \, dx\)

Optimal. Leaf size=27 \[ -\frac{2 \sqrt [4]{a-b x^2}}{a c \sqrt{c x}} \]

[Out]

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

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Rubi [A]  time = 0.0067443, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {264} \[ -\frac{2 \sqrt [4]{a-b x^2}}{a c \sqrt{c x}} \]

Antiderivative was successfully verified.

[In]

Int[1/((c*x)^(3/2)*(a - b*x^2)^(3/4)),x]

[Out]

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

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{1}{(c x)^{3/2} \left (a-b x^2\right )^{3/4}} \, dx &=-\frac{2 \sqrt [4]{a-b x^2}}{a c \sqrt{c x}}\\ \end{align*}

Mathematica [A]  time = 0.005298, size = 25, normalized size = 0.93 \[ -\frac{2 x \sqrt [4]{a-b x^2}}{a (c x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((c*x)^(3/2)*(a - b*x^2)^(3/4)),x]

[Out]

(-2*x*(a - b*x^2)^(1/4))/(a*(c*x)^(3/2))

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Maple [A]  time = 0.004, size = 22, normalized size = 0.8 \begin{align*} -2\,{\frac{x\sqrt [4]{-b{x}^{2}+a}}{a \left ( cx \right ) ^{3/2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(c*x)^(3/2)/(-b*x^2+a)^(3/4),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (-b x^{2} + a\right )}^{\frac{3}{4}} \left (c x\right )^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x)^(3/2)/(-b*x^2+a)^(3/4),x, algorithm="maxima")

[Out]

integrate(1/((-b*x^2 + a)^(3/4)*(c*x)^(3/2)), x)

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Fricas [A]  time = 2.12474, size = 58, normalized size = 2.15 \begin{align*} -\frac{2 \,{\left (-b x^{2} + a\right )}^{\frac{1}{4}} \sqrt{c x}}{a c^{2} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x)^(3/2)/(-b*x^2+a)^(3/4),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 7.03884, size = 94, normalized size = 3.48 \begin{align*} \begin{cases} \frac{\sqrt [4]{b} \sqrt [4]{\frac{a}{b x^{2}} - 1} \Gamma \left (- \frac{1}{4}\right )}{2 a c^{\frac{3}{2}} \Gamma \left (\frac{3}{4}\right )} & \text{for}\: \frac{\left |{a}\right |}{\left |{b}\right | \left |{x^{2}}\right |} > 1 \\- \frac{\sqrt [4]{b} \sqrt [4]{- \frac{a}{b x^{2}} + 1} e^{- \frac{3 i \pi }{4}} \Gamma \left (- \frac{1}{4}\right )}{2 a c^{\frac{3}{2}} \Gamma \left (\frac{3}{4}\right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x)**(3/2)/(-b*x**2+a)**(3/4),x)

[Out]

Piecewise((b**(1/4)*(a/(b*x**2) - 1)**(1/4)*gamma(-1/4)/(2*a*c**(3/2)*gamma(3/4)), Abs(a)/(Abs(b)*Abs(x**2)) >
1), (-b**(1/4)*(-a/(b*x**2) + 1)**(1/4)*exp(-3*I*pi/4)*gamma(-1/4)/(2*a*c**(3/2)*gamma(3/4)), True))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (-b x^{2} + a\right )}^{\frac{3}{4}} \left (c x\right )^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x)^(3/2)/(-b*x^2+a)^(3/4),x, algorithm="giac")

[Out]

integrate(1/((-b*x^2 + a)^(3/4)*(c*x)^(3/2)), x)