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

Optimal. Leaf size=21 \[ -\frac{\left (b x^2\right )^p}{2 (1-p) x^2} \]

[Out]

-(b*x^2)^p/(2*(1 - p)*x^2)

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Rubi [A]  time = 0.0145627, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ -\frac{\left (b x^2\right )^p}{2 (1-p) x^2} \]

Antiderivative was successfully verified.

[In]  Int[(b*x^2)^p/x^3,x]

[Out]

-(b*x^2)^p/(2*(1 - p)*x^2)

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Rubi in Sympy [A]  time = 3.27109, size = 15, normalized size = 0.71 \[ - \frac{b \left (b x^{2}\right )^{p - 1}}{2 \left (- p + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-b*(b*x**2)**(p - 1)/(2*(-p + 1))

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Mathematica [A]  time = 0.00453064, size = 18, normalized size = 0.86 \[ \frac{\left (b x^2\right )^p}{(2 p-2) x^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(b*x^2)^p/x^3,x]

[Out]

(b*x^2)^p/((-2 + 2*p)*x^2)

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Maple [A]  time = 0.003, size = 18, normalized size = 0.9 \[{\frac{ \left ( b{x}^{2} \right ) ^{p}}{2\,{x}^{2} \left ( -1+p \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2)^p/x^3,x)

[Out]

1/2/x^2/(-1+p)*(b*x^2)^p

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Maxima [A]  time = 1.45204, size = 24, normalized size = 1.14 \[ \frac{b^{p}{\left (x^{2}\right )}^{p}}{2 \,{\left (p - 1\right )} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*b^p*(x^2)^p/((p - 1)*x^2)

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Fricas [A]  time = 0.231593, size = 23, normalized size = 1.1 \[ \frac{\left (b x^{2}\right )^{p}}{2 \,{\left (p - 1\right )} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(b*x^2)^p/((p - 1)*x^2)

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Sympy [A]  time = 1.84948, size = 24, normalized size = 1.14 \[ \begin{cases} \frac{b^{p} \left (x^{2}\right )^{p}}{2 p x^{2} - 2 x^{2}} & \text{for}\: p \neq 1 \\b \log{\left (x \right )} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2)**p/x**3,x)

[Out]

Piecewise((b**p*(x**2)**p/(2*p*x**2 - 2*x**2), Ne(p, 1)), (b*log(x), True))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (b x^{2}\right )^{p}}{x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^2)^p/x^3, x)