3.171 \(\int \frac{x^{-1+\frac{3 n}{2}}}{(b x^n)^{3/2}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0018731, antiderivative size = 22, normalized size of antiderivative = 1., 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} \[ \frac{x^{n/2} \log (x)}{b \sqrt{b x^n}} \]

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*}

Mathematica [A]  time = 0.0031429, size = 19, normalized size = 0.86 \[ \frac{x^{3 n/2} \log (x)}{\left (b x^n\right )^{3/2}} \]

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 = 0.024, size = 23, normalized size = 1.1 \begin{align*}{\frac{\ln \left ( x \right ) }{b}{x}^{{\frac{n}{2}}}{\frac{1}{\sqrt{b \left ({x}^{{\frac{n}{2}}} \right ) ^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{\frac{3}{2} \, n - 1}}{\left (b x^{n}\right )^{\frac{3}{2}}}\,{d x} \end{align*}

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 = 1.80527, size = 49, normalized size = 2.23 \begin{align*} \frac{\sqrt{b x^{n}} \log \left (x\right )}{b^{2} x^{\frac{1}{2} \, n}} \end{align*}

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(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{\frac{3}{2} \, n - 1}}{\left (b x^{n}\right )^{\frac{3}{2}}}\,{d x} \end{align*}

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)