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

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 = 0.00725945, 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 \[ \frac{x \log (x)}{b \sqrt{b x^2}} \]

Antiderivative was successfully verified.

[In]  Int[x^2/(b*x^2)^(3/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 2.31844, size = 15, normalized size = 0.94 \[ \frac{\sqrt{b x^{2}} \log{\left (x \right )}}{b^{2} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2/(b*x**2)**(3/2),x)

[Out]

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

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Mathematica [A]  time = 0.00254674, 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 = 0.004, size = 14, normalized size = 0.9 \[{{x}^{3}\ln \left ( x \right ) \left ( b{x}^{2} \right ) ^{-{\frac{3}{2}}}} \]

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 = 1.4396, size = 8, normalized size = 0.5 \[ \frac{\log \left (x\right )}{b^{\frac{3}{2}}} \]

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 = 0.241001, size = 22, normalized size = 1.38 \[ \frac{\sqrt{b x^{2}} \log \left (x\right )}{b^{2} x} \]

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. \[ \int \frac{x^{2}}{\left (b x^{2}\right )^{\frac{3}{2}}}\, dx \]

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/XCAS [A]  time = 0.217083, size = 28, normalized size = 1.75 \[ -\frac{{\rm ln}\left ({\left | -\sqrt{b} x + \sqrt{b x^{2}} \right |}\right )}{b^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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