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

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

[Out]

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

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Rubi [A]  time = 0.0119664, antiderivative size = 32, 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} \[ -\frac{c^2 x (c x)^{m-2}}{b (2-m) \sqrt{b x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.004051, size = 21, normalized size = 0.66 \[ \frac{x (c x)^m}{(m-2) \left (b x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(c*x)^m)/((-2 + m)*(b*x^2)^(3/2))

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Maple [A]  time = 0., size = 20, normalized size = 0.6 \begin{align*}{\frac{x \left ( cx \right ) ^{m}}{-2+m} \left ( b{x}^{2} \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.05758, size = 24, normalized size = 0.75 \begin{align*} \frac{c^{m} x^{m}}{b^{\frac{3}{2}}{\left (m - 2\right )} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c^m*x^m/(b^(3/2)*(m - 2)*x^2)

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Fricas [A]  time = 1.79251, size = 58, normalized size = 1.81 \begin{align*} \frac{\sqrt{b x^{2}} \left (c x\right )^{m}}{{\left (b^{2} m - 2 \, b^{2}\right )} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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)**(3/2),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}}{\left (b x^{2}\right )^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((c*x)^m/(b*x^2)^(3/2), x)