3.992 \(\int \frac{\left (c x^2\right )^p (a+b x)^{-2 p}}{x^2} \, dx\)

Optimal. Leaf size=33 \[ -\frac{\left (c x^2\right )^p (a+b x)^{1-2 p}}{a (1-2 p) x} \]

[Out]

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

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Rubi [A]  time = 0.0286186, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ -\frac{\left (c x^2\right )^p (a+b x)^{1-2 p}}{a (1-2 p) x} \]

Antiderivative was successfully verified.

[In]  Int[(c*x^2)^p/(x^2*(a + b*x)^(2*p)),x]

[Out]

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

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Rubi in Sympy [A]  time = 14.373, size = 36, normalized size = 1.09 \[ - \frac{x^{- 2 p} x^{2 p - 1} \left (c x^{2}\right )^{p} \left (a + b x\right )^{- 2 p + 1}}{a \left (- 2 p + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2)**p/x**2/((b*x+a)**(2*p)),x)

[Out]

-x**(-2*p)*x**(2*p - 1)*(c*x**2)**p*(a + b*x)**(-2*p + 1)/(a*(-2*p + 1))

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Mathematica [A]  time = 0.0345089, size = 31, normalized size = 0.94 \[ -\frac{\left (c x^2\right )^p (a+b x)^{1-2 p}}{a x-2 a p x} \]

Antiderivative was successfully verified.

[In]  Integrate[(c*x^2)^p/(x^2*(a + b*x)^(2*p)),x]

[Out]

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

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Maple [A]  time = 0.004, size = 38, normalized size = 1.2 \[{\frac{ \left ( bx+a \right ) \left ( c{x}^{2} \right ) ^{p}}{ \left ( 2\,p-1 \right ) ax \left ( bx+a \right ) ^{2\,p}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2)^p/x^2/((b*x+a)^(2*p)),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (c x^{2}\right )^{p}{\left (b x + a\right )}^{-2 \, p}}{x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((c*x^2)^p*(b*x + a)^(-2*p)/x^2, x)

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Fricas [A]  time = 0.233072, size = 50, normalized size = 1.52 \[ \frac{{\left (b x + a\right )} \left (c x^{2}\right )^{p}}{{\left (2 \, a p - a\right )}{\left (b x + a\right )}^{2 \, p} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(b*x + a)*(c*x^2)^p/((2*a*p - a)*(b*x + a)^(2*p)*x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2)**p/x**2/((b*x+a)**(2*p)),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((c*x^2)^p/((b*x + a)^(2*p)*x^2), x)