Optimal. Leaf size=39 \[ \frac{x \left (c x^2\right )^p (d x)^m (a+b x)^{-m-2 p-1}}{a (m+2 p+1)} \]
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Rubi [A] time = 0.0103912, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {15, 20, 37} \[ \frac{x \left (c x^2\right )^p (d x)^m (a+b x)^{-m-2 p-1}}{a (m+2 p+1)} \]
Antiderivative was successfully verified.
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Rule 15
Rule 20
Rule 37
Rubi steps
\begin{align*} \int (d x)^m \left (c x^2\right )^p (a+b x)^{-2-m-2 p} \, dx &=\left (x^{-2 p} \left (c x^2\right )^p\right ) \int x^{2 p} (d x)^m (a+b x)^{-2-m-2 p} \, dx\\ &=\left (x^{-m-2 p} (d x)^m \left (c x^2\right )^p\right ) \int x^{m+2 p} (a+b x)^{-2-m-2 p} \, dx\\ &=\frac{x (d x)^m \left (c x^2\right )^p (a+b x)^{-1-m-2 p}}{a (1+m+2 p)}\\ \end{align*}
Mathematica [A] time = 0.015029, size = 39, normalized size = 1. \[ \frac{x \left (c x^2\right )^p (d x)^m (a+b x)^{-m-2 p-1}}{a (m+2 p+1)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 40, normalized size = 1. \begin{align*}{\frac{x \left ( dx \right ) ^{m} \left ( c{x}^{2} \right ) ^{p} \left ( bx+a \right ) ^{-1-m-2\,p}}{a \left ( 1+m+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 )}^{-m - 2 \, p - 2} \left (d x\right )^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.59401, size = 132, normalized size = 3.38 \begin{align*} \frac{{\left (b x^{2} + a x\right )}{\left (b x + a\right )}^{-m - 2 \, p - 2} \left (d x\right )^{m} e^{\left (2 \, p \log \left (d x\right ) + p \log \left (\frac{c}{d^{2}}\right )\right )}}{a m + 2 \, a p + 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 )}^{-m - 2 \, p - 2} \left (d x\right )^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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