3.2.42 \(\int \frac {(b x^n)^{3/2}}{x^4} \, dx\) [142]

Optimal. Leaf size=26 \[ -\frac {2 b x^{-3+n} \sqrt {b x^n}}{3 (2-n)} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 26, 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 {2 b x^{n-3} \sqrt {b x^n}}{3 (2-n)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(b*x^n)^(3/2)/x^4,x]

[Out]

(-2*b*x^(-3 + n)*Sqrt[b*x^n])/(3*(2 - 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*} \int \frac {\left (b x^n\right )^{3/2}}{x^4} \, dx &=\left (b x^{-n/2} \sqrt {b x^n}\right ) \int x^{-4+\frac {3 n}{2}} \, dx\\ &=-\frac {2 b x^{-3+n} \sqrt {b x^n}}{3 (2-n)}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 22, normalized size = 0.85 \begin {gather*} \frac {\left (b x^n\right )^{3/2}}{\left (-3+\frac {3 n}{2}\right ) x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(b*x^n)^(3/2)/x^4,x]

[Out]

(b*x^n)^(3/2)/((-3 + (3*n)/2)*x^3)

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Maple [A]
time = 0.02, size = 18, normalized size = 0.69

method	result	size
gosper	\(\frac {2 \left (b \,x^{n}\right )^{\frac {3}{2}}}{3 x^{3} \left (-2+n \right )}\)	\(18\)
risch	\(\frac {2 b^{2} x^{2 n}}{3 \left (-2+n \right ) x^{3} \sqrt {b \,x^{n}}}\)	\(26\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^n)^(3/2)/x^4,x,method=_RETURNVERBOSE)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^n)^(3/2)/x^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume((3*n)/2-4>0)', see `assume?` f
or more deta

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^n)^(3/2)/x^4,x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (ha
s polynomial part)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \begin {cases} \frac {2 \left (b x^{n}\right )^{\frac {3}{2}}}{3 n x^{3} - 6 x^{3}} & \text {for}\: n \neq 2 \\\int \frac {\left (b x^{2}\right )^{\frac {3}{2}}}{x^{4}}\, dx & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**n)**(3/2)/x**4,x)

[Out]

Piecewise((2*(b*x**n)**(3/2)/(3*n*x**3 - 6*x**3), Ne(n, 2)), (Integral((b*x**2)**(3/2)/x**4, x), True))

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^n)^(3/2)/x^4,x, algorithm="giac")

[Out]

integrate((b*x^n)^(3/2)/x^4, x)

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Mupad [B]
time = 0.96, size = 22, normalized size = 0.85 \begin {gather*} \frac {2\,b\,x^{n-3}\,\sqrt {b\,x^n}}{3\,n-6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^n)^(3/2)/x^4,x)

[Out]

(2*b*x^(n - 3)*(b*x^n)^(1/2))/(3*n - 6)

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