Optimal. Leaf size=35 \[ -\frac{\left (c x^2\right )^p (a+b x)^{2-2 p}}{2 a (1-p) x^2} \]
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Rubi [A] time = 0.01086, antiderivative size = 35, 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} \[ -\frac{\left (c x^2\right )^p (a+b x)^{2-2 p}}{2 a (1-p) x^2} \]
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^3} \, dx &=\left (x^{-2 p} \left (c x^2\right )^p\right ) \int x^{-3+2 p} (a+b x)^{1-2 p} \, dx\\ &=-\frac{\left (c x^2\right )^p (a+b x)^{2-2 p}}{2 a (1-p) x^2}\\ \end{align*}
Mathematica [A] time = 0.0106003, size = 32, normalized size = 0.91 \[ \frac{\left (c x^2\right )^p (a+b x)^{2-2 p}}{a (2 p-2) x^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 32, normalized size = 0.9 \begin{align*}{\frac{ \left ( bx+a \right ) ^{2-2\,p} \left ( c{x}^{2} \right ) ^{p}}{2\,{x}^{2}a \left ( p-1 \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 \frac{\left (c x^{2}\right )^{p}{\left (b x + a\right )}^{-2 \, p + 1}}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.78616, size = 84, normalized size = 2.4 \begin{align*} \frac{{\left (b x + a\right )} \left (c x^{2}\right )^{p}{\left (b x + a\right )}^{-2 \, p + 1}}{2 \,{\left (a p - a\right )} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \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^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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