3.165 \(\int \frac{(c x)^m}{\sqrt{b x^n}} \, dx\)

Optimal. Leaf size=27 \[ \frac{2 x (c x)^m}{(2 m-n+2) \sqrt{b x^n}} \]

[Out]

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

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Rubi [A]  time = 0.0183033, antiderivative size = 27, 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 \[ \frac{2 x (c x)^m}{(2 m-n+2) \sqrt{b x^n}} \]

Antiderivative was successfully verified.

[In]  Int[(c*x)^m/Sqrt[b*x^n],x]

[Out]

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

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Rubi in Sympy [A]  time = 6.05094, size = 39, normalized size = 1.44 \[ \frac{2 x^{- m} x^{- \frac{n}{2}} x^{m - \frac{n}{2} + 1} \sqrt{b x^{n}} \left (c x\right )^{m}}{b \left (2 m - n + 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x)**m/(b*x**n)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.00952749, size = 26, normalized size = 0.96 \[ \frac{x (c x)^m}{\left (m-\frac{n}{2}+1\right ) \sqrt{b x^n}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.002, size = 26, normalized size = 1. \[ 2\,{\frac{x \left ( cx \right ) ^{m}}{ \left ( 2+2\,m-n \right ) \sqrt{b{x}^{n}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x)^m/(b*x^n)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.46602, size = 36, normalized size = 1.33 \[ \frac{2 \, c^{m} x x^{m}}{\sqrt{b}{\left (2 \, m - n + 2\right )} \sqrt{x^{n}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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/sqrt(b*x^n),x, algorithm="fricas")

[Out]

Exception raised: TypeError

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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**n)**(1/2),x)

[Out]

Exception raised: TypeError

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((c*x)^m/sqrt(b*x^n), x)