3.10.82 \(\int \frac {1}{(c x)^{3/2} (a-b x^2)^{3/4}} \, dx\) [982]

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

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 27, 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 = {270} \begin {gather*} -\frac {2 \sqrt [4]{a-b x^2}}{a c \sqrt {c x}} \end {gather*}

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 270

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*}

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Mathematica [A]
time = 0.17, size = 25, normalized size = 0.93 \begin {gather*} -\frac {2 x \sqrt [4]{a-b x^2}}{a (c x)^{3/2}} \end {gather*}

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 = 0.05, size = 22, normalized size = 0.81

method	result	size
gosper	\(-\frac {2 x \left (-b \,x^{2}+a \right )^{\frac {1}{4}}}{a \left (c x \right )^{\frac {3}{2}}}\)	\(22\)
risch	\(-\frac {2 \left (-b \,x^{2}+a \right )^{\frac {1}{4}} \left (\left (-b \,x^{2}+a \right )^{3}\right )^{\frac {1}{4}}}{\sqrt {c x}\, \left (-\left (b \,x^{2}-a \right )^{3}\right )^{\frac {1}{4}} c a}\)	\(51\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 = 0.98, size = 26, normalized size = 0.96 \begin {gather*} -\frac {2 \, {\left (-b x^{2} + a\right )}^{\frac {1}{4}} \sqrt {c x}}{a c^{2} x} \end {gather*}

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 [C] Result contains complex when optimal does not.
time = 1.54, size = 90, normalized size = 3.33 \begin {gather*} \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}\: \left |{\frac {a}{b 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 {gather*}

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/(b*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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [B]
time = 5.08, size = 23, normalized size = 0.85 \begin {gather*} -\frac {2\,{\left (a-b\,x^2\right )}^{1/4}}{a\,c\,\sqrt {c\,x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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