\(\int \frac {(b x^n)^{3/2}}{x} \, dx\) [139]

Optimal result

Integrand size = 13, antiderivative size = 20 \[ \int \frac {\left (b x^n\right )^{3/2}}{x} \, dx=\frac {2 b x^n \sqrt {b x^n}}{3 n} \]

[Out]

2/3*b*x^n*(b*x^n)^(1/2)/n

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {15, 30} \[ \int \frac {\left (b x^n\right )^{3/2}}{x} \, dx=\frac {2 b x^n \sqrt {b x^n}}{3 n} \]

[In]

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

[Out]

(2*b*x^n*Sqrt[b*x^n])/(3*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 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}& = \left (b x^{-n/2} \sqrt {b x^n}\right ) \int x^{-1+\frac {3 n}{2}} \, dx \\ & = \frac {2 b x^n \sqrt {b x^n}}{3 n} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.80 \[ \int \frac {\left (b x^n\right )^{3/2}}{x} \, dx=\frac {2 \left (b x^n\right )^{3/2}}{3 n} \]

[In]

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

[Out]

(2*(b*x^n)^(3/2))/(3*n)

Maple [A] (verified)

Time = 0.08 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.65

method	result	size
gosper	\(\frac {2 \left (b \,x^{n}\right )^{\frac {3}{2}}}{3 n}\)	\(13\)
derivativedivides	\(\frac {2 \left (b \,x^{n}\right )^{\frac {3}{2}}}{3 n}\)	\(13\)
default	\(\frac {2 \left (b \,x^{n}\right )^{\frac {3}{2}}}{3 n}\)	\(13\)
risch	\(\frac {2 b^{2} x^{2 n}}{3 n \sqrt {b \,x^{n}}}\)	\(21\)

[In]

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

[Out]

2/3/n*(b*x^n)^(3/2)

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.80 \[ \int \frac {\left (b x^n\right )^{3/2}}{x} \, dx=\frac {2 \, \sqrt {b x^{n}} b x^{n}}{3 \, n} \]

[In]

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

[Out]

2/3*sqrt(b*x^n)*b*x^n/n

Sympy [A] (verification not implemented)

Time = 0.43 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\left (b x^n\right )^{3/2}}{x} \, dx=\begin {cases} \frac {2 \left (b x^{n}\right )^{\frac {3}{2}}}{3 n} & \text {for}\: n \neq 0 \\b^{\frac {3}{2}} \log {\left (x \right )} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((2*(b*x**n)**(3/2)/(3*n), Ne(n, 0)), (b**(3/2)*log(x), True))

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.60 \[ \int \frac {\left (b x^n\right )^{3/2}}{x} \, dx=\frac {2 \, \left (b x^{n}\right )^{\frac {3}{2}}}{3 \, n} \]

[In]

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

[Out]

2/3*(b*x^n)^(3/2)/n

Giac [F]

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

[In]

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

[Out]

integrate((b*x^n)^(3/2)/x, x)

Mupad [B] (verification not implemented)

Time = 5.44 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.80 \[ \int \frac {\left (b x^n\right )^{3/2}}{x} \, dx=\frac {2\,b\,x^n\,\sqrt {b\,x^n}}{3\,n} \]

[In]

int((b*x^n)^(3/2)/x,x)

[Out]

(2*b*x^n*(b*x^n)^(1/2))/(3*n)