3.972 \(\int \frac{1}{(c x)^{3/2} \left (a+b x^2\right )^{3/4}} \, dx\)

Optimal. Leaf size=26 \[ -\frac{2 \sqrt [4]{a+b x^2}}{a c \sqrt{c x}} \]

[Out]

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

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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]

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

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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]

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

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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]

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

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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]

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

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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]

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

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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]

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

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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]

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

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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]

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