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]

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

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Rubi [A]  time = 0.01, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, 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*}

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Mathematica [A]  time = 0.01, 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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fricas [A]  time = 1.27, size = 35, normalized size = 1.21 \[ \frac {2 \, {\left (b x^{2} - a\right )} {\left (-b x^{2} + a\right )}^{\frac {1}{4}} \sqrt {c x}}{5 \, a c^{4} x^{3}} \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (-b x^{2} + a\right )}^{\frac {1}{4}}}{\left (c x\right )^{\frac {7}{2}}}\,{d x} \]

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]

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

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maple [A]  time = 0.00, size = 22, normalized size = 0.76 \[ -\frac {2 \left (-b \,x^{2}+a \right )^{\frac {5}{4}} x}{5 \left (c x \right )^{\frac {7}{2}} a} \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (-b x^{2} + a\right )}^{\frac {1}{4}}}{\left (c x\right )^{\frac {7}{2}}}\,{d x} \]

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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mupad [B]  time = 4.85, size = 38, normalized size = 1.31 \[ -\frac {{\left (a-b\,x^2\right )}^{1/4}\,\left (\frac {2}{5\,c^3}-\frac {2\,b\,x^2}{5\,a\,c^3}\right )}{x^2\,\sqrt {c\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a - b*x^2)^(1/4)/(c*x)^(7/2),x)

[Out]

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

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sympy [B]  time = 8.76, size = 178, normalized size = 6.14 \[ \begin {cases} \frac {\sqrt [4]{b} \sqrt [4]{\frac {a}{b x^{2}} - 1} \Gamma \left (- \frac {5}{4}\right )}{2 c^{\frac {7}{2}} x^{2} \Gamma \left (- \frac {1}{4}\right )} - \frac {b^{\frac {5}{4}} \sqrt [4]{\frac {a}{b x^{2}} - 1} \Gamma \left (- \frac {5}{4}\right )}{2 a c^{\frac {7}{2}} \Gamma \left (- \frac {1}{4}\right )} & \text {for}\: \left |{\frac {a}{b x^{2}}}\right | > 1 \\\frac {\sqrt [4]{b} \sqrt [4]{- \frac {a}{b x^{2}} + 1} e^{\frac {i \pi }{4}} \Gamma \left (- \frac {5}{4}\right )}{2 c^{\frac {7}{2}} x^{2} \Gamma \left (- \frac {1}{4}\right )} - \frac {b^{\frac {5}{4}} \sqrt [4]{- \frac {a}{b x^{2}} + 1} e^{\frac {i \pi }{4}} \Gamma \left (- \frac {5}{4}\right )}{2 a c^{\frac {7}{2}} \Gamma \left (- \frac {1}{4}\right )} & \text {otherwise} \end {cases} \]

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/(b*x**2)) > 1), (b**(1/4)*(-a/(b*x**2) + 1)**(1/4)*ex
p(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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