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

Optimal. Leaf size=28 \[ \frac{\sqrt{b x^2} (c x)^{m+2}}{c^2 (m+2) x} \]

[Out]

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

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Rubi [A]  time = 0.0256201, antiderivative size = 28, 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 \[ \frac{\sqrt{b x^2} (c x)^{m+2}}{c^2 (m+2) x} \]

Antiderivative was successfully verified.

[In]  Int[(c*x)^m*Sqrt[b*x^2],x]

[Out]

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

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Rubi in Sympy [A]  time = 3.92424, size = 26, normalized size = 0.93 \[ \frac{x^{- m} x^{m + 2} \sqrt{b x^{2}} \left (c x\right )^{m}}{x \left (m + 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.00616127, size = 21, normalized size = 0.75 \[ \frac{x \sqrt{b x^2} (c x)^m}{m+2} \]

Antiderivative was successfully verified.

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

[Out]

(x*(c*x)^m*Sqrt[b*x^2])/(2 + m)

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Maple [A]  time = 0.003, size = 20, normalized size = 0.7 \[{\frac{x \left ( cx \right ) ^{m}}{2+m}\sqrt{b{x}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x)^m*(b*x^2)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.45679, size = 24, normalized size = 0.86 \[ \frac{\sqrt{b} c^{m} x^{2} x^{m}}{m + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(b)*c^m*x^2*x^m/(m + 2)

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Fricas [A]  time = 0.275655, size = 26, normalized size = 0.93 \[ \frac{\sqrt{b x^{2}} \left (c x\right )^{m} x}{m + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(b*x^2)*(c*x)^m*x/(m + 2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError