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

Optimal. Leaf size=28 \[ \frac{2 b \sqrt{b x^n} x^{m+n+1}}{2 m+3 n+2} \]

[Out]

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

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Rubi [A]  time = 0.0196735, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{2 b \sqrt{b x^n} x^{m+n+1}}{2 m+3 n+2} \]

Antiderivative was successfully verified.

[In]  Int[x^m*(b*x^n)^(3/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 3.54806, size = 34, normalized size = 1.21 \[ \frac{2 b x^{- \frac{n}{2}} x^{m + \frac{3 n}{2} + 1} \sqrt{b x^{n}}}{2 m + 3 n + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0124342, size = 25, normalized size = 0.89 \[ \frac{x^{m+1} \left (b x^n\right )^{3/2}}{m+\frac{3 n}{2}+1} \]

Antiderivative was successfully verified.

[In]  Integrate[x^m*(b*x^n)^(3/2),x]

[Out]

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

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Maple [A]  time = 0.001, size = 25, normalized size = 0.9 \[ 2\,{\frac{{x}^{1+m} \left ( b{x}^{n} \right ) ^{3/2}}{2+2\,m+3\,n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^m*(b*x^n)^(3/2),x)

[Out]

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

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Maxima [A]  time = 1.45279, size = 32, normalized size = 1.14 \[ \frac{2 \, b^{\frac{3}{2}} x x^{m}{\left (x^{n}\right )}^{\frac{3}{2}}}{2 \, m + 3 \, n + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*b^(3/2)*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)*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(x**m*(b*x**n)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.227339, size = 36, normalized size = 1.29 \[ \frac{2 \, b^{\frac{3}{2}} x e^{\left (m{\rm ln}\left (x\right ) + \frac{3}{2} \, n{\rm ln}\left (x\right )\right )}}{2 \, m + 3 \, n + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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