3.64 \(\int \frac{(c x)^m}{\sqrt{b x^2}} \, dx\)

Optimal. Leaf size=19 \[ \frac{x (c x)^m}{m \sqrt{b x^2}} \]

[Out]

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

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Rubi [A]  time = 0.0148562, antiderivative size = 19, 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 \[ \frac{x (c x)^m}{m \sqrt{b x^2}} \]

Antiderivative was successfully verified.

[In]  Int[(c*x)^m/Sqrt[b*x^2],x]

[Out]

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

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Rubi in Sympy [A]  time = 4.2583, size = 17, normalized size = 0.89 \[ \frac{\sqrt{b x^{2}} \left (c x\right )^{m}}{b m x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.00452264, size = 19, normalized size = 1. \[ \frac{x (c x)^m}{m \sqrt{b x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.004, size = 18, normalized size = 1. \[{\frac{x \left ( cx \right ) ^{m}}{m}{\frac{1}{\sqrt{b{x}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x)^m/(b*x^2)^(1/2),x)

[Out]

x*(c*x)^m/m/(b*x^2)^(1/2)

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Maxima [A]  time = 1.46272, size = 18, normalized size = 0.95 \[ \frac{c^{m} x^{m}}{\sqrt{b} m} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

c^m*x^m/(sqrt(b)*m)

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Fricas [A]  time = 0.239581, size = 30, normalized size = 1.58 \[ \frac{\sqrt{b x^{2}} \left (c x\right )^{m}}{b m x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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)**(1/2),x)

[Out]

Exception raised: TypeError

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((c*x)^m/sqrt(b*x^2), x)