3.1.19 \(\int \frac {(b x^2)^{3/2}}{x^5} \, dx\) [19]

Optimal. Leaf size=15 \[ -\frac {b \sqrt {b x^2}}{x^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.00, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {15, 30} \begin {gather*} -\frac {b \sqrt {b x^2}}{x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(b*x^2)^(3/2)/x^5,x]

[Out]

-((b*Sqrt[b*x^2])/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 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin {align*} \int \frac {\left (b x^2\right )^{3/2}}{x^5} \, dx &=\frac {\left (b \sqrt {b x^2}\right ) \int \frac {1}{x^2} \, dx}{x}\\ &=-\frac {b \sqrt {b x^2}}{x^2}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.00, size = 14, normalized size = 0.93 \begin {gather*} -\frac {\left (b x^2\right )^{3/2}}{x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(b*x^2)^(3/2)/x^5,x]

[Out]

-((b*x^2)^(3/2)/x^4)

________________________________________________________________________________________

Maple [A]
time = 0.02, size = 13, normalized size = 0.87

method	result	size
gosper	\(-\frac {\left (b \,x^{2}\right )^{\frac {3}{2}}}{x^{4}}\)	\(13\)
default	\(-\frac {\left (b \,x^{2}\right )^{\frac {3}{2}}}{x^{4}}\)	\(13\)
risch	\(-\frac {b \sqrt {b \,x^{2}}}{x^{2}}\)	\(14\)
trager	\(\frac {b \left (x -1\right ) \sqrt {b \,x^{2}}}{x^{2}}\)	\(16\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2)^(3/2)/x^5,x,method=_RETURNVERBOSE)

[Out]

-(b*x^2)^(3/2)/x^4

________________________________________________________________________________________

Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2)^(3/2)/x^5,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: Error executing code in Maxima: expt: undefined: 0 to a negative e
xponent.

________________________________________________________________________________________

Fricas [A]
time = 0.34, size = 13, normalized size = 0.87 \begin {gather*} -\frac {\sqrt {b x^{2}} b}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2)^(3/2)/x^5,x, algorithm="fricas")

[Out]

-sqrt(b*x^2)*b/x^2

________________________________________________________________________________________

Sympy [A]
time = 0.23, size = 12, normalized size = 0.80 \begin {gather*} - \frac {\left (b x^{2}\right )^{\frac {3}{2}}}{x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2)**(3/2)/x**5,x)

[Out]

-(b*x**2)**(3/2)/x**4

________________________________________________________________________________________

Giac [A]
time = 2.17, size = 10, normalized size = 0.67 \begin {gather*} -\frac {b^{\frac {3}{2}} \mathrm {sgn}\left (x\right )}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2)^(3/2)/x^5,x, algorithm="giac")

[Out]

-b^(3/2)*sgn(x)/x

________________________________________________________________________________________

Mupad [B]
time = 0.90, size = 10, normalized size = 0.67 \begin {gather*} -\frac {b^{3/2}}{\sqrt {x^2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2)^(3/2)/x^5,x)

[Out]

-b^(3/2)/(x^2)^(1/2)

________________________________________________________________________________________