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

Optimal. Leaf size=30 \[ \frac{x \left (c x^2\right )^p (a+b x)^{-2 p-1}}{a (2 p+1)} \]

[Out]

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

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Rubi [A]  time = 0.0260095, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{x \left (c x^2\right )^p (a+b x)^{-2 p-1}}{a (2 p+1)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 8.37141, size = 36, normalized size = 1.2 \[ \frac{x^{- 2 p} x^{2 p + 1} \left (c x^{2}\right )^{p} \left (a + b x\right )^{- 2 p - 1}}{a \left (2 p + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2)**p*(b*x+a)**(-2-2*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 = 0.0572782, size = 28, normalized size = 0.93 \[ \frac{x \left (c x^2\right )^p (a+b x)^{-2 p-1}}{2 a p+a} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.004, size = 31, normalized size = 1. \[{\frac{x \left ( c{x}^{2} \right ) ^{p} \left ( bx+a \right ) ^{-1-2\,p}}{a \left ( 1+2\,p \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.227413, size = 49, normalized size = 1.63 \[ \frac{{\left (b x^{2} + a x\right )} \left (c x^{2}\right )^{p}{\left (b x + a\right )}^{-2 \, p - 2}}{2 \, a p + a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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*(b*x+a)**(-2-2*p),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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