Optimal. Leaf size=32 \[ \frac{x^3 \left (c x^2\right )^p (a+b x)^{-2 p-3}}{a (2 p+3)} \]
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Rubi [A] time = 0.0086787, 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, Rules used = {15, 37} \[ \frac{x^3 \left (c x^2\right )^p (a+b x)^{-2 p-3}}{a (2 p+3)} \]
Antiderivative was successfully verified.
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Rule 15
Rule 37
Rubi steps
\begin{align*} \int x^2 \left (c x^2\right )^p (a+b x)^{-4-2 p} \, dx &=\left (x^{-2 p} \left (c x^2\right )^p\right ) \int x^{2+2 p} (a+b x)^{-4-2 p} \, dx\\ &=\frac{x^3 \left (c x^2\right )^p (a+b x)^{-3-2 p}}{a (3+2 p)}\\ \end{align*}
Mathematica [A] time = 0.0198317, size = 34, normalized size = 1.06 \[ \frac{x^3 \left (c x^2\right )^p (a+b x)^{1-2 (p+2)}}{a (2 p+3)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 33, normalized size = 1. \begin{align*}{\frac{{x}^{3} \left ( c{x}^{2} \right ) ^{p} \left ( bx+a \right ) ^{-3-2\,p}}{a \left ( 3+2\,p \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (c x^{2}\right )^{p}{\left (b x + a\right )}^{-2 \, p - 4} x^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.58956, size = 84, normalized size = 2.62 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (c x^{2}\right )^{p}{\left (b x + a\right )}^{-2 \, p - 4} x^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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