3.988 \(\int x^2 \left (c x^2\right )^p (a+b x)^{-4-2 p} \, dx\)

Optimal. Leaf size=32 \[ \frac{x^3 \left (c x^2\right )^p (a+b x)^{-2 p-3}}{a (2 p+3)} \]

[Out]

(x^3*(c*x^2)^p*(a + b*x)^(-3 - 2*p))/(a*(3 + 2*p))

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Rubi [A]  time = 0.0266024, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{x^3 \left (c x^2\right )^p (a+b x)^{-2 p-3}}{a (2 p+3)} \]

Antiderivative was successfully verified.

[In]  Int[x^2*(c*x^2)^p*(a + b*x)^(-4 - 2*p),x]

[Out]

(x^3*(c*x^2)^p*(a + b*x)^(-3 - 2*p))/(a*(3 + 2*p))

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Rubi in Sympy [A]  time = 10.0906, size = 36, normalized size = 1.12 \[ \frac{x^{- 2 p} x^{2 p + 3} \left (c x^{2}\right )^{p} \left (a + b x\right )^{- 2 p - 3}}{a \left (2 p + 3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2*(c*x**2)**p*(b*x+a)**(-4-2*p),x)

[Out]

x**(-2*p)*x**(2*p + 3)*(c*x**2)**p*(a + b*x)**(-2*p - 3)/(a*(2*p + 3))

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Mathematica [A]  time = 0.0667539, size = 32, normalized size = 1. \[ \frac{x^3 \left (c x^2\right )^p (a+b x)^{-2 p-3}}{2 a p+3 a} \]

Antiderivative was successfully verified.

[In]  Integrate[x^2*(c*x^2)^p*(a + b*x)^(-4 - 2*p),x]

[Out]

(x^3*(c*x^2)^p*(a + b*x)^(-3 - 2*p))/(3*a + 2*a*p)

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Maple [A]  time = 0.005, size = 33, normalized size = 1. \[{\frac{{x}^{3} \left ( c{x}^{2} \right ) ^{p} \left ( bx+a \right ) ^{-3-2\,p}}{a \left ( 3+2\,p \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(c*x^2)^p*(b*x+a)^(-4-2*p),x)

[Out]

x^3*(c*x^2)^p*(b*x+a)^(-3-2*p)/a/(3+2*p)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \left (c x^{2}\right )^{p}{\left (b x + a\right )}^{-2 \, p - 4} x^{2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2)^p*(b*x + a)^(-2*p - 4)*x^2,x, algorithm="maxima")

[Out]

integrate((c*x^2)^p*(b*x + a)^(-2*p - 4)*x^2, x)

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Fricas [A]  time = 0.23271, size = 54, normalized size = 1.69 \[ \frac{{\left (b x^{4} + a x^{3}\right )} \left (c x^{2}\right )^{p}{\left (b x + a\right )}^{-2 \, p - 4}}{2 \, a p + 3 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2)^p*(b*x + a)^(-2*p - 4)*x^2,x, algorithm="fricas")

[Out]

(b*x^4 + a*x^3)*(c*x^2)^p*(b*x + a)^(-2*p - 4)/(2*a*p + 3*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(x**2*(c*x**2)**p*(b*x+a)**(-4-2*p),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \left (c x^{2}\right )^{p}{\left (b x + a\right )}^{-2 \, p - 4} x^{2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2)^p*(b*x + a)^(-2*p - 4)*x^2,x, algorithm="giac")

[Out]

integrate((c*x^2)^p*(b*x + a)^(-2*p - 4)*x^2, x)