3.2.71 \(\int \frac {x^{-1+\frac {3 n}{2}}}{(b x^n)^{3/2}} \, dx\) [171]

Optimal. Leaf size=22 \[ \frac {x^{n/2} \log (x)}{b \sqrt {b x^n}} \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {15, 29} \begin {gather*} \frac {x^{n/2} \log (x)}{b \sqrt {b x^n}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^(-1 + (3*n)/2)/(b*x^n)^(3/2),x]

[Out]

(x^(n/2)*Log[x])/(b*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 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rubi steps

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

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Mathematica [A]
time = 0.00, size = 19, normalized size = 0.86 \begin {gather*} \frac {x^{3 n/2} \log (x)}{\left (b x^n\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^(-1 + (3*n)/2)/(b*x^n)^(3/2),x]

[Out]

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

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Maple [A]
time = 0.03, size = 23, normalized size = 1.05

method	result	size
risch	\(\frac {x^{\frac {n}{2}} \ln \left (x \right )}{b \sqrt {b \,x^{n}}}\)	\(23\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.37, size = 20, normalized size = 0.91 \begin {gather*} \frac {\sqrt {b x^{n}} \log \left (x\right )}{b^{2} x^{\frac {1}{2} \, n}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(b*x^n)*log(x)/(b^2*x^(1/2*n))

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{\frac {3 n}{2} - 1}}{\left (b x^{n}\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**(3*n/2 - 1)/(b*x**n)**(3/2), x)

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.05 \begin {gather*} \int \frac {x^{\frac {3\,n}{2}-1}}{{\left (b\,x^n\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^((3*n)/2 - 1)/(b*x^n)^(3/2),x)

[Out]

int(x^((3*n)/2 - 1)/(b*x^n)^(3/2), x)

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