3.65 \(\int \frac{(c x)^m}{\left (b x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=32 \[ -\frac{c^2 x (c x)^{m-2}}{b (2-m) \sqrt{b x^2}} \]

[Out]

-((c^2*x*(c*x)^(-2 + m))/(b*(2 - m)*Sqrt[b*x^2]))

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Rubi [A]  time = 0.0317743, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{c^2 x (c x)^{m-2}}{b (2-m) \sqrt{b x^2}} \]

Antiderivative was successfully verified.

[In]  Int[(c*x)^m/(b*x^2)^(3/2),x]

[Out]

-((c^2*x*(c*x)^(-2 + m))/(b*(2 - m)*Sqrt[b*x^2]))

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Rubi in Sympy [A]  time = 4.87524, size = 31, normalized size = 0.97 \[ - \frac{x^{- m} x^{m - 2} \sqrt{b x^{2}} \left (c x\right )^{m}}{b^{2} x \left (- m + 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x)**m/(b*x**2)**(3/2),x)

[Out]

-x**(-m)*x**(m - 2)*sqrt(b*x**2)*(c*x)**m/(b**2*x*(-m + 2))

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Mathematica [A]  time = 0.00724154, size = 21, normalized size = 0.66 \[ \frac{x (c x)^m}{(m-2) \left (b x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(c*x)^m/(b*x^2)^(3/2),x]

[Out]

(x*(c*x)^m)/((-2 + m)*(b*x^2)^(3/2))

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Maple [A]  time = 0.003, size = 20, normalized size = 0.6 \[{\frac{x \left ( cx \right ) ^{m}}{-2+m} \left ( b{x}^{2} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x)^m/(b*x^2)^(3/2),x)

[Out]

x/(-2+m)*(c*x)^m/(b*x^2)^(3/2)

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Maxima [A]  time = 1.46008, size = 24, normalized size = 0.75 \[ \frac{c^{m} x^{m}}{b^{\frac{3}{2}}{\left (m - 2\right )} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x)^m/(b*x^2)^(3/2),x, algorithm="maxima")

[Out]

c^m*x^m/(b^(3/2)*(m - 2)*x^2)

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Fricas [A]  time = 0.23211, size = 39, normalized size = 1.22 \[ \frac{\sqrt{b x^{2}} \left (c x\right )^{m}}{{\left (b^{2} m - 2 \, b^{2}\right )} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x)^m/(b*x^2)^(3/2),x, algorithm="fricas")

[Out]

sqrt(b*x^2)*(c*x)^m/((b^2*m - 2*b^2)*x^3)

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x)**m/(b*x**2)**(3/2),x)

[Out]

Exception raised: TypeError

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (c x\right )^{m}}{\left (b x^{2}\right )^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x)^m/(b*x^2)^(3/2),x, algorithm="giac")

[Out]

integrate((c*x)^m/(b*x^2)^(3/2), x)