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

Optimal. Leaf size=36 \[ \frac {2 x^{1-n} (c x)^m}{b (2 m-3 n+2) \sqrt {b x^n}} \]

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {15, 20, 30} \[ \frac {2 x^{1-n} (c x)^m}{b (2 m-3 n+2) \sqrt {b x^n}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x^(1 - n)*(c*x)^m)/(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 20

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

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

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Mathematica [A]  time = 0.01, size = 26, normalized size = 0.72 \[ \frac {x (c x)^m}{\left (m-\frac {3 n}{2}+1\right ) \left (b x^n\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(c*x)^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((c*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 \[ \int \frac {\left (c x\right )^{m}}{\left (b x^{n}\right )^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((c*x)^m/(b*x^n)^(3/2), x)

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maple [A]  time = 0.00, size = 26, normalized size = 0.72 \[ \frac {2 x \left (c x \right )^{m}}{\left (2 m -3 n +2\right ) \left (b \,x^{n}\right )^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.52, size = 27, normalized size = 0.75 \[ \frac {2 \, c^{m} x x^{m}}{b^{\frac {3}{2}} {\left (2 \, m - 3 \, n + 2\right )} {\left (x^{n}\right )}^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 1.03, size = 34, normalized size = 0.94 \[ \frac {2\,x^{1-2\,n}\,\sqrt {b\,x^n}\,{\left (c\,x\right )}^m}{b^2\,\left (2\,m-3\,n+2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((2*c**m*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((c*x)**(3*n/2 - 1)/(b*x**n)**(3/2), x), True))

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