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