\(\int \frac {(c x)^m}{(b x^n)^{3/2}} \, dx\) [166]

Optimal result

Integrand size = 15, antiderivative size = 36 \[ \int \frac {(c x)^m}{\left (b x^n\right )^{3/2}} \, dx=\frac {2 x^{1-n} (c x)^m}{b (2+2 m-3 n) \sqrt {b x^n}} \]

[Out]

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

Rubi [A] (verified)

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

[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*} \text {integral}& = \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*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.72 \[ \int \frac {(c x)^m}{\left (b x^n\right )^{3/2}} \, dx=\frac {x (c x)^m}{\left (1+m-\frac {3 n}{2}\right ) \left (b x^n\right )^{3/2}} \]

[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))

Maple [A] (verified)

Time = 0.05 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.72

method	result	size
gosper	\(\frac {2 x \left (c x \right )^{m}}{\left (2+2 m -3 n \right ) \left (b \,x^{n}\right )^{\frac {3}{2}}}\)	\(26\)

[In]

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

[Out]

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

Fricas [F(-2)]

Exception generated. \[ \int \frac {(c x)^m}{\left (b x^n\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \]

[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)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (29) = 58\).

Time = 2.13 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.94 \[ \int \frac {(c x)^m}{\left (b x^n\right )^{3/2}} \, dx=\begin {cases} \frac {2 x \left (c x\right )^{m}}{2 m \left (b x^{n}\right )^{\frac {3}{2}} - 3 n \left (b x^{n}\right )^{\frac {3}{2}} + 2 \left (b x^{n}\right )^{\frac {3}{2}}} & \text {for}\: m \neq \frac {3 n}{2} - 1 \\\frac {x \left (c x\right )^{\frac {3 n}{2} - 1} \log {\left (x \right )}}{\left (b x^{n}\right )^{\frac {3}{2}}} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.75 \[ \int \frac {(c x)^m}{\left (b x^n\right )^{3/2}} \, dx=\frac {2 \, c^{m} x x^{m}}{b^{\frac {3}{2}} {\left (2 \, m - 3 \, n + 2\right )} {\left (x^{n}\right )}^{\frac {3}{2}}} \]

[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))

Giac [F]

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

[In]

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

[Out]

integrate((c*x)^m/(b*x^n)^(3/2), x)

Mupad [B] (verification not implemented)

Time = 5.49 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.94 \[ \int \frac {(c x)^m}{\left (b x^n\right )^{3/2}} \, dx=\frac {2\,x^{1-2\,n}\,\sqrt {b\,x^n}\,{\left (c\,x\right )}^m}{b^2\,\left (2\,m-3\,n+2\right )} \]

[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))