3.64 \(\int \frac{(c x)^m}{\sqrt{b x^2}} \, dx\)

Optimal. Leaf size=19 \[ \frac{x (c x)^m}{m \sqrt{b x^2}} \]

[Out]

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

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Rubi [A]  time = 0.0049172, antiderivative size = 19, 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, 16, 32} \[ \frac{x (c x)^m}{m \sqrt{b x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 16

Int[(u_.)*(v_)^(m_.)*((b_)*(v_))^(n_), x_Symbol] :> Dist[1/b^m, Int[u*(b*v)^(m + n), x], x] /; FreeQ[{b, n}, x
] && IntegerQ[m]

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps

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

Mathematica [A]  time = 0.0024654, size = 19, normalized size = 1. \[ \frac{x (c x)^m}{m \sqrt{b x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.002, size = 18, normalized size = 1. \begin{align*}{\frac{x \left ( cx \right ) ^{m}}{m}{\frac{1}{\sqrt{b{x}^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x)^m/(b*x^2)^(1/2),x)

[Out]

x*(c*x)^m/m/(b*x^2)^(1/2)

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Maxima [A]  time = 0.997392, size = 18, normalized size = 0.95 \begin{align*} \frac{c^{m} x^{m}}{\sqrt{b} m} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c^m*x^m/(sqrt(b)*m)

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Fricas [A]  time = 1.81149, size = 39, normalized size = 2.05 \begin{align*} \frac{\sqrt{b x^{2}} \left (c x\right )^{m}}{b m x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(b*x^2)*(c*x)^m/(b*m*x)

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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**2)**(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^{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((c*x)^m/sqrt(b*x^2), x)