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

Optimal. Leaf size=32 \[ \frac{2 b x^{n+1} \sqrt{b x^n} (c x)^m}{2 m+3 n+2} \]

[Out]

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

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Rubi [A]  time = 0.0205336, antiderivative size = 32, 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 \[ \frac{2 b x^{n+1} \sqrt{b x^n} (c x)^m}{2 m+3 n+2} \]

Antiderivative was successfully verified.

[In]  Int[(c*x)^m*(b*x^n)^(3/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 5.27682, size = 42, normalized size = 1.31 \[ \frac{2 b x^{- m} x^{- \frac{n}{2}} x^{m + \frac{3 n}{2} + 1} \sqrt{b x^{n}} \left (c x\right )^{m}}{2 m + 3 n + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0113741, size = 26, normalized size = 0.81 \[ \frac{x \left (b x^n\right )^{3/2} (c x)^m}{m+\frac{3 n}{2}+1} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.002, size = 26, normalized size = 0.8 \[ 2\,{\frac{x \left ( cx \right ) ^{m} \left ( b{x}^{n} \right ) ^{3/2}}{2+2\,m+3\,n}} \]

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 = 1.45865, size = 36, normalized size = 1.12 \[ \frac{2 \, b^{\frac{3}{2}} c^{m} x x^{m}{\left (x^{n}\right )}^{\frac{3}{2}}}{2 \, m + 3 \, n + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*b^(3/2)*c^m*x*x^m*(x^n)^(3/2)/(2*m + 3*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((b*x^n)^(3/2)*(c*x)^m,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 [A]  time = 0.23334, size = 42, normalized size = 1.31 \[ \frac{2 \, b^{\frac{3}{2}} x e^{\left (m{\rm ln}\left (c\right ) + m{\rm ln}\left (x\right ) + \frac{3}{2} \, n{\rm ln}\left (x\right )\right )}}{2 \, m + 3 \, n + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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