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

Optimal. Leaf size=27 \[ \frac{2 x (c x)^m}{(2 m-n+2) \sqrt{b x^n}} \]

[Out]

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

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Rubi [A]  time = 0.0059093, antiderivative size = 27, 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, Rules used = {15, 20, 30} \[ \frac{2 x (c x)^m}{(2 m-n+2) \sqrt{b x^n}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0047464, size = 26, normalized size = 0.96 \[ \frac{x (c x)^m}{\left (m-\frac{n}{2}+1\right ) \sqrt{b x^n}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.001, size = 26, normalized size = 1. \begin{align*} 2\,{\frac{x \left ( cx \right ) ^{m}}{ \left ( 2+2\,m-n \right ) \sqrt{b{x}^{n}}}} \end{align*}

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

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Maxima [A]  time = 1.02594, size = 36, normalized size = 1.33 \begin{align*} \frac{2 \, c^{m} x x^{m}}{\sqrt{b}{\left (2 \, m - n + 2\right )} \sqrt{x^{n}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x)^m/(b*x^n)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x)^m/(b*x^n)^(1/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

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 [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (c x\right )^{m}}{\sqrt{b x^{n}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x)^m/(b*x^n)^(1/2),x, algorithm="giac")

[Out]

integrate((c*x)^m/sqrt(b*x^n), x)