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

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0029187, 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} \[ x^{-n/2} \log (x) \sqrt{b x^n} \]

Antiderivative was successfully verified.

[In]

Int[x^(-1 - n/2)*Sqrt[b*x^n],x]

[Out]

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

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

Mathematica [A]  time = 0.0036891, size = 19, normalized size = 1. \[ x^{-n/2} \log (x) \sqrt{b x^n} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(-1 - n/2)*Sqrt[b*x^n],x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.025, size = 22, normalized size = 1.2 \begin{align*}{\ln \left ( x \right ) \sqrt{b \left ({x}^{{\frac{n}{2}}} \right ) ^{2}} \left ({x}^{{\frac{n}{2}}} \right ) ^{-1}} \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]

(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 \sqrt{b x^{n}} x^{-\frac{1}{2} \, n - 1}\,{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(sqrt(b*x^n)*x^(-1/2*n - 1), x)

________________________________________________________________________________________

Fricas [A]  time = 1.75533, size = 68, normalized size = 3.58 \begin{align*} x x^{-\frac{1}{2} \, n - 1} \sqrt{\frac{b}{x^{2} x^{-n - 2}}} \log \left (x\right ) \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]

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

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.21357, size = 8, normalized size = 0.42 \begin{align*} \sqrt{b} \log \left (x\right ) \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]

sqrt(b)*log(x)