3.49 \(\int \frac{x^2}{(b x^2)^{3/2}} \, dx\)

Optimal. Leaf size=16 \[ \frac{x \log (x)}{b \sqrt{b x^2}} \]

[Out]

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

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Rubi [A]  time = 0.0015001, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {15, 29} \[ \frac{x \log (x)}{b \sqrt{b x^2}} \]

Antiderivative was successfully verified.

[In]

Int[x^2/(b*x^2)^(3/2),x]

[Out]

(x*Log[x])/(b*Sqrt[b*x^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 \frac{x^2}{\left (b x^2\right )^{3/2}} \, dx &=\frac{x \int \frac{1}{x} \, dx}{b \sqrt{b x^2}}\\ &=\frac{x \log (x)}{b \sqrt{b x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0013106, size = 15, normalized size = 0.94 \[ \frac{x^3 \log (x)}{\left (b x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/(b*x^2)^(3/2),x]

[Out]

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

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Maple [A]  time = 0.001, size = 14, normalized size = 0.9 \begin{align*}{{x}^{3}\ln \left ( x \right ) \left ( b{x}^{2} \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(b*x^2)^(3/2),x)

[Out]

1/(b*x^2)^(3/2)*x^3*ln(x)

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Maxima [A]  time = 1.01693, size = 8, normalized size = 0.5 \begin{align*} \frac{\log \left (x\right )}{b^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(b*x^2)^(3/2),x, algorithm="maxima")

[Out]

log(x)/b^(3/2)

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Fricas [A]  time = 1.47664, size = 38, normalized size = 2.38 \begin{align*} \frac{\sqrt{b x^{2}} \log \left (x\right )}{b^{2} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(b*x^2)^(3/2),x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2/(b*x**2)**(3/2),x)

[Out]

Integral(x**2/(b*x**2)**(3/2), x)

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Giac [A]  time = 1.16158, size = 28, normalized size = 1.75 \begin{align*} -\frac{\log \left ({\left | -\sqrt{b} x + \sqrt{b x^{2}} \right |}\right )}{b^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(b*x^2)^(3/2),x, algorithm="giac")

[Out]

-log(abs(-sqrt(b)*x + sqrt(b*x^2)))/b^(3/2)