3.940 \(\int \frac{\sqrt [4]{a-b x^2}}{(c x)^{7/2}} \, dx\)

Optimal. Leaf size=29 \[ -\frac{2 \left (a-b x^2\right )^{5/4}}{5 a c (c x)^{5/2}} \]

[Out]

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

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Rubi [A]  time = 0.0061529, 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} \[ -\frac{2 \left (a-b x^2\right )^{5/4}}{5 a c (c x)^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[(a - b*x^2)^(1/4)/(c*x)^(7/2),x]

[Out]

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

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{\sqrt [4]{a-b x^2}}{(c x)^{7/2}} \, dx &=-\frac{2 \left (a-b x^2\right )^{5/4}}{5 a c (c x)^{5/2}}\\ \end{align*}

Mathematica [A]  time = 0.0081107, size = 27, normalized size = 0.93 \[ -\frac{2 x \left (a-b x^2\right )^{5/4}}{5 a (c x)^{7/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a - b*x^2)^(1/4)/(c*x)^(7/2),x]

[Out]

(-2*x*(a - b*x^2)^(5/4))/(5*a*(c*x)^(7/2))

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Maple [A]  time = 0.004, size = 22, normalized size = 0.8 \begin{align*} -{\frac{2\,x}{5\,a} \left ( -b{x}^{2}+a \right ) ^{{\frac{5}{4}}} \left ( cx \right ) ^{-{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-b*x^2+a)^(1/4)/(c*x)^(7/2),x)

[Out]

-2/5*x*(-b*x^2+a)^(5/4)/a/(c*x)^(7/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (-b x^{2} + a\right )}^{\frac{1}{4}}}{\left (c x\right )^{\frac{7}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-b*x^2+a)^(1/4)/(c*x)^(7/2),x, algorithm="maxima")

[Out]

integrate((-b*x^2 + a)^(1/4)/(c*x)^(7/2), x)

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Fricas [A]  time = 2.22486, size = 78, normalized size = 2.69 \begin{align*} \frac{2 \,{\left (b x^{2} - a\right )}{\left (-b x^{2} + a\right )}^{\frac{1}{4}} \sqrt{c x}}{5 \, a c^{4} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-b*x^2+a)^(1/4)/(c*x)^(7/2),x, algorithm="fricas")

[Out]

2/5*(b*x^2 - a)*(-b*x^2 + a)^(1/4)*sqrt(c*x)/(a*c^4*x^3)

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Sympy [B]  time = 52.9355, size = 182, normalized size = 6.28 \begin{align*} \begin{cases} \frac{\sqrt [4]{b} \sqrt [4]{\frac{a}{b x^{2}} - 1} \Gamma \left (- \frac{5}{4}\right )}{2 c^{\frac{7}{2}} x^{2} \Gamma \left (- \frac{1}{4}\right )} - \frac{b^{\frac{5}{4}} \sqrt [4]{\frac{a}{b x^{2}} - 1} \Gamma \left (- \frac{5}{4}\right )}{2 a c^{\frac{7}{2}} \Gamma \left (- \frac{1}{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{i \pi }{4}} \Gamma \left (- \frac{5}{4}\right )}{2 c^{\frac{7}{2}} x^{2} \Gamma \left (- \frac{1}{4}\right )} - \frac{b^{\frac{5}{4}} \sqrt [4]{- \frac{a}{b x^{2}} + 1} e^{\frac{i \pi }{4}} \Gamma \left (- \frac{5}{4}\right )}{2 a c^{\frac{7}{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((-b*x**2+a)**(1/4)/(c*x)**(7/2),x)

[Out]

Piecewise((b**(1/4)*(a/(b*x**2) - 1)**(1/4)*gamma(-5/4)/(2*c**(7/2)*x**2*gamma(-1/4)) - b**(5/4)*(a/(b*x**2) -
1)**(1/4)*gamma(-5/4)/(2*a*c**(7/2)*gamma(-1/4)), Abs(a)/(Abs(b)*Abs(x**2)) > 1), (b**(1/4)*(-a/(b*x**2) + 1)
**(1/4)*exp(I*pi/4)*gamma(-5/4)/(2*c**(7/2)*x**2*gamma(-1/4)) - b**(5/4)*(-a/(b*x**2) + 1)**(1/4)*exp(I*pi/4)*
gamma(-5/4)/(2*a*c**(7/2)*gamma(-1/4)), True))

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Giac [A]  time = 2.43532, size = 61, normalized size = 2.1 \begin{align*} \frac{2 \,{\left (-b c^{4} x^{2} + a c^{4}\right )}^{\frac{1}{4}}{\left (b c^{2} - \frac{a c^{2}}{x^{2}}\right )}}{5 \, \sqrt{c x} a c^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-b*x^2+a)^(1/4)/(c*x)^(7/2),x, algorithm="giac")

[Out]

2/5*(-b*c^4*x^2 + a*c^4)^(1/4)*(b*c^2 - a*c^2/x^2)/(sqrt(c*x)*a*c^6)