3.58 \(\int \frac{x^4}{\left (b x^2\right )^{5/2}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.00763255, 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 \[ \frac{x \log (x)}{b^2 \sqrt{b x^2}} \]

Antiderivative was successfully verified.

[In]  Int[x^4/(b*x^2)^(5/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 2.35479, size = 15, normalized size = 0.94 \[ \frac{\sqrt{b x^{2}} \log{\left (x \right )}}{b^{3} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4/(b*x**2)**(5/2),x)

[Out]

sqrt(b*x**2)*log(x)/(b**3*x)

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Mathematica [A]  time = 0.00234548, size = 15, normalized size = 0.94 \[ \frac{x^5 \log (x)}{\left (b x^2\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^4/(b*x^2)^(5/2),x]

[Out]

(x^5*Log[x])/(b*x^2)^(5/2)

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Maple [A]  time = 0.006, size = 14, normalized size = 0.9 \[{{x}^{5}\ln \left ( x \right ) \left ( b{x}^{2} \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4/(b*x^2)^(5/2),x)

[Out]

1/(b*x^2)^(5/2)*x^5*ln(x)

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Maxima [A]  time = 1.43954, size = 8, normalized size = 0.5 \[ \frac{\log \left (x\right )}{b^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/(b*x^2)^(5/2),x, algorithm="maxima")

[Out]

log(x)/b^(5/2)

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Fricas [A]  time = 0.242377, size = 22, normalized size = 1.38 \[ \frac{\sqrt{b x^{2}} \log \left (x\right )}{b^{3} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/(b*x^2)^(5/2),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{4}}{\left (b x^{2}\right )^{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4/(b*x**2)**(5/2),x)

[Out]

Integral(x**4/(b*x**2)**(5/2), x)

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GIAC/XCAS [A]  time = 0.234759, size = 28, normalized size = 1.75 \[ -\frac{{\rm ln}\left ({\left | -\sqrt{b} x + \sqrt{b x^{2}} \right |}\right )}{b^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/(b*x^2)^(5/2),x, algorithm="giac")

[Out]

-ln(abs(-sqrt(b)*x + sqrt(b*x^2)))/b^(5/2)