3.957 \(\int \frac{1}{(c x)^{5/2} \sqrt [4]{a-b x^2}} \, dx\)

Optimal. Leaf size=29 \[ -\frac{2 \left (a-b x^2\right )^{3/4}}{3 a c (c x)^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0060532, antiderivative size = 29, 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} \[ -\frac{2 \left (a-b x^2\right )^{3/4}}{3 a c (c x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[1/((c*x)^(5/2)*(a - b*x^2)^(1/4)),x]

[Out]

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

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{1}{(c x)^{5/2} \sqrt [4]{a-b x^2}} \, dx &=-\frac{2 \left (a-b x^2\right )^{3/4}}{3 a c (c x)^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.0063869, size = 27, normalized size = 0.93 \[ -\frac{2 x \left (a-b x^2\right )^{3/4}}{3 a (c x)^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((c*x)^(5/2)*(a - b*x^2)^(1/4)),x]

[Out]

(-2*x*(a - b*x^2)^(3/4))/(3*a*(c*x)^(5/2))

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Maple [A]  time = 0.004, size = 22, normalized size = 0.8 \begin{align*} -{\frac{2\,x}{3\,a} \left ( -b{x}^{2}+a \right ) ^{{\frac{3}{4}}} \left ( cx \right ) ^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(c*x)^(5/2)/(-b*x^2+a)^(1/4),x)

[Out]

-2/3*x*(-b*x^2+a)^(3/4)/a/(c*x)^(5/2)

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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{1}{4}} \left (c x\right )^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x)^(5/2)/(-b*x^2+a)^(1/4),x, algorithm="maxima")

[Out]

integrate(1/((-b*x^2 + a)^(1/4)*(c*x)^(5/2)), x)

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Fricas [A]  time = 1.55234, size = 63, normalized size = 2.17 \begin{align*} -\frac{2 \,{\left (-b x^{2} + a\right )}^{\frac{3}{4}} \sqrt{c x}}{3 \, a c^{3} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x)^(5/2)/(-b*x^2+a)^(1/4),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 11.2316, size = 92, normalized size = 3.17 \begin{align*} \begin{cases} \frac{b^{\frac{3}{4}} \left (\frac{a}{b x^{2}} - 1\right )^{\frac{3}{4}} \Gamma \left (- \frac{3}{4}\right )}{2 a c^{\frac{5}{2}} \Gamma \left (\frac{1}{4}\right )} & \text{for}\: \frac{\left |{a}\right |}{\left |{b}\right | \left |{x^{2}}\right |} > 1 \\- \frac{b^{\frac{3}{4}} \left (- \frac{a}{b x^{2}} + 1\right )^{\frac{3}{4}} e^{- \frac{i \pi }{4}} \Gamma \left (- \frac{3}{4}\right )}{2 a c^{\frac{5}{2}} \Gamma \left (\frac{1}{4}\right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x)**(5/2)/(-b*x**2+a)**(1/4),x)

[Out]

Piecewise((b**(3/4)*(a/(b*x**2) - 1)**(3/4)*gamma(-3/4)/(2*a*c**(5/2)*gamma(1/4)), Abs(a)/(Abs(b)*Abs(x**2)) >
1), (-b**(3/4)*(-a/(b*x**2) + 1)**(3/4)*exp(-I*pi/4)*gamma(-3/4)/(2*a*c**(5/2)*gamma(1/4)), 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{1}{4}} \left (c x\right )^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x)^(5/2)/(-b*x^2+a)^(1/4),x, algorithm="giac")

[Out]

integrate(1/((-b*x^2 + a)^(1/4)*(c*x)^(5/2)), x)