3.18 \(\int \frac{\left (b x^2\right )^{3/2}}{x^4} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.00736345, 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 \[ \frac{b \sqrt{b x^2} \log (x)}{x} \]

Antiderivative was successfully verified.

[In]  Int[(b*x^2)^(3/2)/x^4,x]

[Out]

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

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Rubi in Sympy [A]  time = 2.06738, size = 14, normalized size = 0.88 \[ \frac{b \sqrt{b x^{2}} \log{\left (x \right )}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2)**(3/2)/x**4,x)

[Out]

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

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Mathematica [A]  time = 0.00313391, size = 15, normalized size = 0.94 \[ \frac{\left (b x^2\right )^{3/2} \log (x)}{x^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(b*x^2)^(3/2)/x^4,x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2)^(3/2)/x^4,x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2)^(3/2)/x^4,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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Fricas [A]  time = 0.235542, size = 19, normalized size = 1.19 \[ \frac{\sqrt{b x^{2}} b \log \left (x\right )}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2)^(3/2)/x^4,x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (b x^{2}\right )^{\frac{3}{2}}}{x^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2)**(3/2)/x**4,x)

[Out]

Integral((b*x**2)**(3/2)/x**4, x)

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GIAC/XCAS [A]  time = 0.215071, size = 12, normalized size = 0.75 \[ b^{\frac{3}{2}}{\rm ln}\left ({\left | x \right |}\right ){\rm sign}\left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2)^(3/2)/x^4,x, algorithm="giac")

[Out]

b^(3/2)*ln(abs(x))*sign(x)