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

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 32, 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, 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 = 0.01, 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))

________________________________________________________________________________________

fricas [A]  time = 0.62, size = 29, normalized size = 0.91 \[ \frac {\sqrt {b x^{2}} \left (c x\right )^{m}}{{\left (b^{2} m - 2 \, b^{2}\right )} x^{3}} \]

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)

________________________________________________________________________________________

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

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)

________________________________________________________________________________________

maple [A]  time = 0.00, size = 20, normalized size = 0.62 \[ \frac {x \left (c x \right )^{m}}{\left (m -2\right ) \left (b \,x^{2}\right )^{\frac {3}{2}}} \]

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)

________________________________________________________________________________________

maxima [A]  time = 1.42, size = 18, normalized size = 0.56 \[ \frac {c^{m} x^{m}}{b^{\frac {3}{2}} {\left (m - 2\right )} x^{2}} \]

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)

________________________________________________________________________________________

mupad [B]  time = 1.00, size = 22, normalized size = 0.69 \[ \frac {{\left (c\,x\right )}^m}{b^{3/2}\,x\,\left (m-2\right )\,\sqrt {x^2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((c**m*x*x**m/(b**(3/2)*m*(x**2)**(3/2) - 2*b**(3/2)*(x**2)**(3/2)), Ne(m, 2)), (c**2*Integral(x**2/(
b*x**2)**(3/2), x), True))

________________________________________________________________________________________