3.987 \(\int x^3 \left (c x^2\right )^p (a+b x)^{-5-2 p} \, dx\)

Optimal. Leaf size=33 \[ \frac{x^4 \left (c x^2\right )^p (a+b x)^{-2 (p+2)}}{2 a (p+2)} \]

[Out]

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

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Rubi [A]  time = 0.0293616, antiderivative size = 33, 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 \[ \frac{x^4 \left (c x^2\right )^p (a+b x)^{-2 (p+2)}}{2 a (p+2)} \]

Antiderivative was successfully verified.

[In]  Int[x^3*(c*x^2)^p*(a + b*x)^(-5 - 2*p),x]

[Out]

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

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Rubi in Sympy [A]  time = 49.7463, size = 39, normalized size = 1.18 \[ \frac{x^{3} x^{- 2 p} x^{2 p + 1} \left (c x^{2}\right )^{p} \left (a + b x\right )^{- 2 p - 4}}{2 a \left (p + 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3*(c*x**2)**p*(b*x+a)**(-5-2*p),x)

[Out]

x**3*x**(-2*p)*x**(2*p + 1)*(c*x**2)**p*(a + b*x)**(-2*p - 4)/(2*a*(p + 2))

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Mathematica [A]  time = 0.0907897, size = 32, normalized size = 0.97 \[ \frac{x^4 \left (c x^2\right )^p (a+b x)^{-2 p-4}}{2 a p+4 a} \]

Antiderivative was successfully verified.

[In]  Integrate[x^3*(c*x^2)^p*(a + b*x)^(-5 - 2*p),x]

[Out]

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

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Maple [A]  time = 0.004, size = 32, normalized size = 1. \[{\frac{{x}^{4} \left ( bx+a \right ) ^{-4-2\,p} \left ( c{x}^{2} \right ) ^{p}}{2\,a \left ( 2+p \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(c*x^2)^p*(b*x+a)^(-5-2*p),x)

[Out]

1/2*x^4*(b*x+a)^(-4-2*p)/a/(2+p)*(c*x^2)^p

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \left (c x^{2}\right )^{p}{\left (b x + a\right )}^{-2 \, p - 5} x^{3}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2)^p*(b*x + a)^(-2*p - 5)*x^3,x, algorithm="maxima")

[Out]

integrate((c*x^2)^p*(b*x + a)^(-2*p - 5)*x^3, x)

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Fricas [A]  time = 0.231991, size = 54, normalized size = 1.64 \[ \frac{{\left (b x^{5} + a x^{4}\right )} \left (c x^{2}\right )^{p}{\left (b x + a\right )}^{-2 \, p - 5}}{2 \,{\left (a p + 2 \, a\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2)^p*(b*x + a)^(-2*p - 5)*x^3,x, algorithm="fricas")

[Out]

1/2*(b*x^5 + a*x^4)*(c*x^2)^p*(b*x + a)^(-2*p - 5)/(a*p + 2*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**3*(c*x**2)**p*(b*x+a)**(-5-2*p),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.224061, size = 103, normalized size = 3.12 \[ \frac{b x^{5} e^{\left (p{\rm ln}\left (c x^{2}\right ) - 2 \, p{\rm ln}\left (b x + a\right ) - 5 \,{\rm ln}\left (b x + a\right )\right )} + a x^{4} e^{\left (p{\rm ln}\left (c x^{2}\right ) - 2 \, p{\rm ln}\left (b x + a\right ) - 5 \,{\rm ln}\left (b x + a\right )\right )}}{2 \,{\left (a p + 2 \, a\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2)^p*(b*x + a)^(-2*p - 5)*x^3,x, algorithm="giac")

[Out]

1/2*(b*x^5*e^(p*ln(c*x^2) - 2*p*ln(b*x + a) - 5*ln(b*x + a)) + a*x^4*e^(p*ln(c*x
^2) - 2*p*ln(b*x + a) - 5*ln(b*x + a)))/(a*p + 2*a)