Optimal. Leaf size=26 \[ \frac{\left (c x^2\right )^p (a+b x)^{-2 p}}{2 a p} \]
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Rubi [A] time = 0.0061454, antiderivative size = 26, 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} \[ \frac{\left (c x^2\right )^p (a+b x)^{-2 p}}{2 a p} \]
Antiderivative was successfully verified.
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Rule 15
Rule 37
Rubi steps
\begin{align*} \int \frac{\left (c x^2\right )^p (a+b x)^{-1-2 p}}{x} \, dx &=\left (x^{-2 p} \left (c x^2\right )^p\right ) \int x^{-1+2 p} (a+b x)^{-1-2 p} \, dx\\ &=\frac{\left (c x^2\right )^p (a+b x)^{-2 p}}{2 a p}\\ \end{align*}
Mathematica [A] time = 0.0071077, size = 26, normalized size = 1. \[ \frac{\left (c x^2\right )^p (a+b x)^{-2 p}}{2 a p} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 25, normalized size = 1. \begin{align*}{\frac{ \left ( bx+a \right ) ^{-2\,p} \left ( c{x}^{2} \right ) ^{p}}{2\,ap}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.06811, size = 36, normalized size = 1.38 \begin{align*} \frac{c^{p} e^{\left (-2 \, p \log \left (b x + a\right ) + 2 \, p \log \left (x\right )\right )}}{2 \, a p} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.72138, size = 70, normalized size = 2.69 \begin{align*} \frac{{\left (b x + a\right )} \left (c x^{2}\right )^{p}{\left (b x + a\right )}^{-2 \, p - 1}}{2 \, a p} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 70.4082, size = 264, normalized size = 10.15 \begin{align*} \begin{cases} - \frac{b^{- 2 p} c^{p} x^{- 2 p} \left (x^{2}\right )^{p}}{b x} & \text{for}\: a = 0 \\\frac{0^{- 2 p - 1} c^{p} \left (x^{2}\right )^{p}}{2 p} & \text{for}\: a = - b x \\\frac{c^{p} \left (0^{\frac{1}{p}}\right )^{- 2 p - 1} \left (x^{2}\right )^{p}}{2 p} & \text{for}\: a = 0^{\frac{1}{p}} - b x \\\frac{\log{\left (x \right )}}{a} - \frac{\log{\left (\frac{a}{b} + x \right )}}{a} & \text{for}\: p = 0 \\\frac{a^{2} c^{p} \left (x^{2}\right )^{p}}{2 a^{3} p \left (a + b x\right )^{2 p} + 4 a^{2} b p x \left (a + b x\right )^{2 p} + 2 a b^{2} p x^{2} \left (a + b x\right )^{2 p}} + \frac{a b c^{p} x \left (x^{2}\right )^{p}}{2 a^{3} p \left (a + b x\right )^{2 p} + 4 a^{2} b p x \left (a + b x\right )^{2 p} + 2 a b^{2} p x^{2} \left (a + b x\right )^{2 p}} + \frac{b c^{p} x \left (x^{2}\right )^{p}}{2 a^{2} p \left (a + b x\right )^{2 p} + 2 a b p x \left (a + b x\right )^{2 p}} & \text{otherwise} \end{cases} \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 \frac{\left (c x^{2}\right )^{p}{\left (b x + a\right )}^{-2 \, p - 1}}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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