Optimal. Leaf size=26 \[ -\frac{2 \sqrt [4]{a+b x^2}}{a c \sqrt{c x}} \]
[Out]
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Rubi [A] time = 0.0282875, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053 \[ -\frac{2 \sqrt [4]{a+b x^2}}{a c \sqrt{c x}} \]
Antiderivative was successfully verified.
[In] Int[1/((c*x)^(3/2)*(a + b*x^2)^(3/4)),x]
[Out]
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Rubi in Sympy [A] time = 3.58504, size = 22, normalized size = 0.85 \[ - \frac{2 \sqrt [4]{a + b x^{2}}}{a c \sqrt{c x}} \]
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 = 0.0187059, size = 24, normalized size = 0.92 \[ -\frac{2 x \sqrt [4]{a+b x^2}}{a (c x)^{3/2}} \]
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 = 0.007, size = 21, 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 = 1.40327, size = 27, 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 = 0.233551, size = 34, normalized size = 1.31 \[ -\frac{2 \,{\left (b x^{2} + a\right )}^{\frac{1}{4}} \sqrt{c x}}{a c^{2} x} \]
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 = 28.685, size = 36, normalized size = 1.38 \[ \frac{\sqrt [4]{b} \sqrt [4]{\frac{a}{b x^{2}} + 1} \Gamma \left (- \frac{1}{4}\right )}{2 a c^{\frac{3}{2}} \Gamma \left (\frac{3}{4}\right )} \]
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]