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

Optimal. Leaf size=36 \[ \frac{2 x^{1-n} (c x)^m}{b (2 m-3 n+2) \sqrt{b x^n}} \]

[Out]

(2*x^(1 - n)*(c*x)^m)/(b*(2 + 2*m - 3*n)*Sqrt[b*x^n])

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Rubi [A]  time = 0.0212993, antiderivative size = 36, 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 \[ \frac{2 x^{1-n} (c x)^m}{b (2 m-3 n+2) \sqrt{b x^n}} \]

Antiderivative was successfully verified.

[In]  Int[(c*x)^m/(b*x^n)^(3/2),x]

[Out]

(2*x^(1 - n)*(c*x)^m)/(b*(2 + 2*m - 3*n)*Sqrt[b*x^n])

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Rubi in Sympy [A]  time = 6.05171, size = 44, normalized size = 1.22 \[ \frac{2 x^{- m} x^{- \frac{n}{2}} x^{m - \frac{3 n}{2} + 1} \sqrt{b x^{n}} \left (c x\right )^{m}}{b^{2} \left (2 m - 3 n + 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*x**(-m)*x**(-n/2)*x**(m - 3*n/2 + 1)*sqrt(b*x**n)*(c*x)**m/(b**2*(2*m - 3*n +
2))

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Mathematica [A]  time = 0.0119936, size = 26, normalized size = 0.72 \[ \frac{x (c x)^m}{\left (m-\frac{3 n}{2}+1\right ) \left (b x^n\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(x*(c*x)^m)/((1 + m - (3*n)/2)*(b*x^n)^(3/2))

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Maple [A]  time = 0.001, size = 26, normalized size = 0.7 \[ 2\,{\frac{x \left ( cx \right ) ^{m}}{ \left ( 2+2\,m-3\,n \right ) \left ( b{x}^{n} \right ) ^{3/2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x)^m/(b*x^n)^(3/2),x)

[Out]

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

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Maxima [A]  time = 1.45976, size = 36, normalized size = 1. \[ \frac{2 \, c^{m} x x^{m}}{b^{\frac{3}{2}}{\left (2 \, m - 3 \, n + 2\right )}{\left (x^{n}\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [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^n)^(3/2),x, algorithm="fricas")

[Out]

Exception raised: TypeError

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((c*x)^m/(b*x^n)^(3/2), x)