3.2.69 \(\int x^{-1-\frac {n}{2}} \sqrt {b x^n} \, dx\) [169]

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

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 19, 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*} x^{-n/2} \log (x) \sqrt {b x^n} \end {gather*}

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*}

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Mathematica [A]
time = 0.00, size = 19, normalized size = 1.00 \begin {gather*} x^{-n/2} \sqrt {b x^n} \log (x) \end {gather*}

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)

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Maple [A]
time = 0.03, size = 22, normalized size = 1.16

method	result	size
risch	\(\sqrt {b \,x^{n}}\, x^{-\frac {n}{2}} \ln \left (x \right )\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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Fricas [A]
time = 0.37, size = 27, normalized size = 1.42 \begin {gather*} x x^{-\frac {1}{2} \, n - 1} \sqrt {\frac {b}{x^{2} x^{-n - 2}}} \log \left (x\right ) \end {gather*}

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)

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

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)

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.05 \begin {gather*} \int \frac {\sqrt {b\,x^n}}{x^{\frac {n}{2}+1}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^n)^(1/2)/x^(n/2 + 1),x)

[Out]

int((b*x^n)^(1/2)/x^(n/2 + 1), x)

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