Optimal. Leaf size=27 \[ -\frac{2 \sqrt [4]{a-b x^2}}{a c \sqrt{c x}} \]
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Rubi [A] time = 0.0296647, 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 \[ -\frac{2 \sqrt [4]{a-b x^2}}{a c \sqrt{c x}} \]
Antiderivative was successfully verified.
[In] Int[1/((c*x)^(3/2)*(a - b*x^2)^(3/4)),x]
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Rubi in Sympy [A] time = 3.93105, size = 22, normalized size = 0.81 \[ - \frac{2 \sqrt [4]{a - b x^{2}}}{a c \sqrt{c x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(c*x)**(3/2)/(-b*x**2+a)**(3/4),x)
[Out]
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Mathematica [A] time = 0.0194153, 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]
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Maple [A] time = 0.006, size = 22, normalized size = 0.8 \[ -2\,{\frac{x\sqrt [4]{-b{x}^{2}+a}}{a \left ( cx \right ) ^{3/2}}} \]
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]
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Maxima [A] time = 1.39217, size = 28, normalized size = 1.04 \[ -\frac{2 \,{\left (-b x^{2} + a\right )}^{\frac{1}{4}}}{a c^{\frac{3}{2}} \sqrt{x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((-b*x^2 + a)^(3/4)*(c*x)^(3/2)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.229225, size = 35, normalized size = 1.3 \[ -\frac{2 \,{\left (-b x^{2} + a\right )}^{\frac{1}{4}} \sqrt{c x}}{a c^{2} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((-b*x^2 + a)^(3/4)*(c*x)^(3/2)),x, algorithm="fricas")
[Out]
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Sympy [A] time = 28.4942, size = 90, normalized size = 3.33 \[ \begin{cases} \frac{\sqrt [4]{b} \sqrt [4]{\frac{a}{b x^{2}} - 1} \Gamma \left (- \frac{1}{4}\right )}{2 a c^{\frac{3}{2}} \Gamma \left (\frac{3}{4}\right )} & \text{for}\: \left |{\frac{a}{b x^{2}}}\right | > 1 \\- \frac{\sqrt [4]{b} \sqrt [4]{- \frac{a}{b x^{2}} + 1} e^{\frac{5 i \pi }{4}} \Gamma \left (- \frac{1}{4}\right )}{2 a c^{\frac{3}{2}} \Gamma \left (\frac{3}{4}\right )} & \text{otherwise} \end{cases} \]
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]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (-b x^{2} + a\right )}^{\frac{3}{4}} \left (c x\right )^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((-b*x^2 + a)^(3/4)*(c*x)^(3/2)),x, algorithm="giac")
[Out]