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

Optimal. Leaf size=29 \[ \frac{2 \sqrt{b x^n} (c x)^{m+1}}{c (2 m+n+2)} \]

[Out]

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

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Rubi [A]  time = 0.017384, antiderivative size = 25, normalized size of antiderivative = 0.86, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{2 x \sqrt{b x^n} (c x)^m}{2 m+n+2} \]

Antiderivative was successfully verified.

[In]  Int[(c*x)^m*Sqrt[b*x^n],x]

[Out]

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

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Rubi in Sympy [A]  time = 4.94397, size = 37, normalized size = 1.28 \[ \frac{2 x^{- m} x^{- \frac{n}{2}} x^{m + \frac{n}{2} + 1} \sqrt{b x^{n}} \left (c x\right )^{m}}{2 m + n + 2} \]

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/(2*m + n + 2)

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Mathematica [A]  time = 0.00890545, size = 26, normalized size = 0.9 \[ \frac{x \sqrt{b x^n} (c x)^m}{m+\frac{n}{2}+1} \]

Antiderivative was successfully verified.

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

[Out]

(x*(c*x)^m*Sqrt[b*x^n])/(1 + m + n/2)

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Maple [A]  time = 0.002, size = 24, normalized size = 0.8 \[ 2\,{\frac{x \left ( cx \right ) ^{m}\sqrt{b{x}^{n}}}{2+2\,m+n}} \]

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

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Maxima [A]  time = 1.45972, size = 34, normalized size = 1.17 \[ \frac{2 \, \sqrt{b} c^{m} x x^{m} \sqrt{x^{n}}}{2 \, m + n + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(b)*c^m*x*x^m*sqrt(x^n)/(2*m + n + 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(sqrt(b*x^n)*(c*x)^m,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 [A]  time = 0.222928, size = 39, normalized size = 1.34 \[ \frac{2 \, \sqrt{b} x e^{\left (m{\rm ln}\left (c\right ) + m{\rm ln}\left (x\right ) + \frac{1}{2} \, n{\rm ln}\left (x\right )\right )}}{2 \, m + n + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(b)*x*e^(m*ln(c) + m*ln(x) + 1/2*n*ln(x))/(2*m + n + 2)