3.1.32 \(\int \frac {(b x^2)^{5/2}}{x^8} \, dx\) [32]

Optimal. Leaf size=19 \[ -\frac {b^2 \sqrt {b x^2}}{2 x^3} \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 19, 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^2 \sqrt {b x^2}}{2 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(b*x^2)^(5/2)/x^8,x]

[Out]

-1/2*(b^2*Sqrt[b*x^2])/x^3

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 )^{5/2}}{x^8} \, dx &=\frac {\left (b^2 \sqrt {b x^2}\right ) \int \frac {1}{x^3} \, dx}{x}\\ &=-\frac {b^2 \sqrt {b x^2}}{2 x^3}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 16, normalized size = 0.84 \begin {gather*} -\frac {\left (b x^2\right )^{5/2}}{2 x^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(b*x^2)^(5/2)/x^8,x]

[Out]

-1/2*(b*x^2)^(5/2)/x^7

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Maple [A]
time = 0.02, size = 13, normalized size = 0.68

method	result	size
gosper	\(-\frac {\left (b \,x^{2}\right )^{\frac {5}{2}}}{2 x^{7}}\)	\(13\)
default	\(-\frac {\left (b \,x^{2}\right )^{\frac {5}{2}}}{2 x^{7}}\)	\(13\)
risch	\(-\frac {b^{2} \sqrt {b \,x^{2}}}{2 x^{3}}\)	\(16\)
trager	\(\frac {b^{2} \left (x -1\right ) \left (x +1\right ) \sqrt {b \,x^{2}}}{2 x^{3}}\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2)^(5/2)/x^8,x,method=_RETURNVERBOSE)

[Out]

-1/2*(b*x^2)^(5/2)/x^7

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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)^(5/2)/x^8,x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.34, size = 15, normalized size = 0.79 \begin {gather*} -\frac {\sqrt {b x^{2}} b^{2}}{2 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2)^(5/2)/x^8,x, algorithm="fricas")

[Out]

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

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Sympy [A]
time = 0.43, size = 14, normalized size = 0.74 \begin {gather*} - \frac {\left (b x^{2}\right )^{\frac {5}{2}}}{2 x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2)**(5/2)/x**8,x)

[Out]

-(b*x**2)**(5/2)/(2*x**7)

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Giac [A]
time = 2.29, size = 10, normalized size = 0.53 \begin {gather*} -\frac {b^{\frac {5}{2}} \mathrm {sgn}\left (x\right )}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2)^(5/2)/x^8,x, algorithm="giac")

[Out]

-1/2*b^(5/2)*sgn(x)/x^2

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Mupad [B]
time = 0.93, size = 13, normalized size = 0.68 \begin {gather*} -\frac {b^{5/2}}{2\,x\,\sqrt {x^2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2)^(5/2)/x^8,x)

[Out]

-b^(5/2)/(2*x*(x^2)^(1/2))

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