3.161 \(\int \frac {x^m}{(b x^n)^{3/2}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {15, 30} \[ \frac {2 x^{m-n+1}}{b (2 m-3 n+2) \sqrt {b x^n}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 15

Int[(u_.)*((a_.)*(x_)^(n_))^(m_), x_Symbol] :> Dist[(a^IntPart[m]*(a*x^n)^FracPart[m])/x^(n*FracPart[m]), Int[
u*x^(m*n), x], x] /; FreeQ[{a, m, n}, x] &&  !IntegerQ[m]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin {align*} \int \frac {x^m}{\left (b x^n\right )^{3/2}} \, dx &=\frac {x^{n/2} \int x^{m-\frac {3 n}{2}} \, dx}{b \sqrt {b x^n}}\\ &=\frac {2 x^{1+m-n}}{b (2+2 m-3 n) \sqrt {b x^n}}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 25, normalized size = 0.78 \[ \frac {x^{m+1}}{\left (m-\frac {3 n}{2}+1\right ) \left (b x^n\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m/(b*x^n)^(3/2),x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (ha
s polynomial part)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{m}}{\left (b x^{n}\right )^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^m/(b*x^n)^(3/2), x)

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maple [A]  time = 0.00, size = 25, normalized size = 0.78 \[ \frac {2 x^{m +1}}{\left (2 m -3 n +2\right ) \left (b \,x^{n}\right )^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.52, size = 24, normalized size = 0.75 \[ \frac {2 \, x x^{m}}{b^{\frac {3}{2}} {\left (2 \, m - 3 \, n + 2\right )} {\left (x^{n}\right )}^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {x^m}{{\left (b\,x^n\right )}^{3/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x^m/(b*x^n)^(3/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \begin {cases} \frac {2 x x^{m}}{2 b^{\frac {3}{2}} m \left (x^{n}\right )^{\frac {3}{2}} - 3 b^{\frac {3}{2}} n \left (x^{n}\right )^{\frac {3}{2}} + 2 b^{\frac {3}{2}} \left (x^{n}\right )^{\frac {3}{2}}} & \text {for}\: m \neq \frac {3 n}{2} - 1 \\\int \frac {x^{\frac {3 n}{2} - 1}}{\left (b x^{n}\right )^{\frac {3}{2}}}\, dx & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((2*x*x**m/(2*b**(3/2)*m*(x**n)**(3/2) - 3*b**(3/2)*n*(x**n)**(3/2) + 2*b**(3/2)*(x**n)**(3/2)), Ne(m
, 3*n/2 - 1)), (Integral(x**(3*n/2 - 1)/(b*x**n)**(3/2), x), True))

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