3.170 \(\int \frac{x^{-1+\frac{n}{2}}}{\sqrt{b x^n}} \, dx\)

Optimal. Leaf size=19 \[ \frac{x^{n/2} \log (x)}{\sqrt{b x^n}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0025012, antiderivative size = 19, 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} \[ \frac{x^{n/2} \log (x)}{\sqrt{b x^n}} \]

Antiderivative was successfully verified.

[In]

Int[x^(-1 + n/2)/Sqrt[b*x^n],x]

[Out]

(x^(n/2)*Log[x])/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{n}{2}}}{\sqrt{b x^n}} \, dx &=\frac{x^{n/2} \int \frac{1}{x} \, dx}{\sqrt{b x^n}}\\ &=\frac{x^{n/2} \log (x)}{\sqrt{b x^n}}\\ \end{align*}

Mathematica [A]  time = 0.0027406, size = 19, normalized size = 1. \[ \frac{x^{n/2} \log (x)}{\sqrt{b x^n}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(-1 + n/2)/Sqrt[b*x^n],x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.026, size = 20, normalized size = 1.1 \begin{align*}{\ln \left ( x \right ){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+1/2*n)/(b*x^n)^(1/2),x)

[Out]

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

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{\frac{1}{2} \, n - 1}}{\sqrt{b x^{n}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.86517, size = 68, normalized size = 3.58 \begin{align*} \frac{\sqrt{b x^{2} x^{n - 2}} \log \left (x\right )}{b x x^{\frac{1}{2} \, n - 1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{\frac{n}{2} - 1}}{\sqrt{b x^{n}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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