\(\int \frac {1}{x^3 \sqrt {b x^n}} \, dx\) [148]

Optimal result

Integrand size = 13, antiderivative size = 19 \[ \int \frac {1}{x^3 \sqrt {b x^n}} \, dx=-\frac {2}{(4+n) x^2 \sqrt {b x^n}} \]

[Out]

-2/(4+n)/x^2/(b*x^n)^(1/2)

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 19, 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} \[ \int \frac {1}{x^3 \sqrt {b x^n}} \, dx=-\frac {2}{(n+4) x^2 \sqrt {b x^n}} \]

[In]

Int[1/(x^3*Sqrt[b*x^n]),x]

[Out]

-2/((4 + n)*x^2*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 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-\frac {n}{2}} \, dx}{\sqrt {b x^n}} \\ & = -\frac {2}{(4+n) x^2 \sqrt {b x^n}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^3 \sqrt {b x^n}} \, dx=-\frac {2}{(4+n) x^2 \sqrt {b x^n}} \]

[In]

Integrate[1/(x^3*Sqrt[b*x^n]),x]

[Out]

-2/((4 + n)*x^2*Sqrt[b*x^n])

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.95

method	result	size
gosper	\(-\frac {2}{\left (4+n \right ) x^{2} \sqrt {b \,x^{n}}}\)	\(18\)
risch	\(-\frac {2}{\left (4+n \right ) x^{2} \sqrt {b \,x^{n}}}\)	\(18\)

[In]

int(1/x^3/(b*x^n)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-2/(4+n)/x^2/(b*x^n)^(1/2)

Fricas [F(-2)]

Exception generated. \[ \int \frac {1}{x^3 \sqrt {b x^n}} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(1/x^3/(b*x^n)^(1/2),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. 46 vs. \(2 (17) = 34\).

Time = 0.77 (sec) , antiderivative size = 46, normalized size of antiderivative = 2.42 \[ \int \frac {1}{x^3 \sqrt {b x^n}} \, dx=\begin {cases} - \frac {2}{n x^{2} \sqrt {b x^{n}} + 4 x^{2} \sqrt {b x^{n}}} & \text {for}\: n \neq -4 \\\frac {\log {\left (x \right )}}{x^{2} \sqrt {\frac {b}{x^{4}}}} & \text {otherwise} \end {cases} \]

[In]

integrate(1/x**3/(b*x**n)**(1/2),x)

[Out]

Piecewise((-2/(n*x**2*sqrt(b*x**n) + 4*x**2*sqrt(b*x**n)), Ne(n, -4)), (log(x)/(x**2*sqrt(b/x**4)), True))

Maxima [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.89 \[ \int \frac {1}{x^3 \sqrt {b x^n}} \, dx=-\frac {2}{\sqrt {b x^{n}} {\left (n + 4\right )} x^{2}} \]

[In]

integrate(1/x^3/(b*x^n)^(1/2),x, algorithm="maxima")

[Out]

-2/(sqrt(b*x^n)*(n + 4)*x^2)

Giac [F]

\[ \int \frac {1}{x^3 \sqrt {b x^n}} \, dx=\int { \frac {1}{\sqrt {b x^{n}} x^{3}} \,d x } \]

[In]

integrate(1/x^3/(b*x^n)^(1/2),x, algorithm="giac")

[Out]

integrate(1/(sqrt(b*x^n)*x^3), x)

Mupad [B] (verification not implemented)

Time = 5.58 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.26 \[ \int \frac {1}{x^3 \sqrt {b x^n}} \, dx=-\frac {2\,\sqrt {b\,x^n}}{b\,x^{n+2}\,\left (n+4\right )} \]

[In]

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

[Out]

-(2*(b*x^n)^(1/2))/(b*x^(n + 2)*(n + 4))