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