Optimal. Leaf size=33 \[ -\frac{\left (c x^2\right )^p (a+b x)^{1-2 p}}{a (1-2 p) x} \]
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Rubi [A] time = 0.0108223, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {15, 37} \[ -\frac{\left (c x^2\right )^p (a+b x)^{1-2 p}}{a (1-2 p) x} \]
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)^{-2 p}}{x^2} \, dx &=\left (x^{-2 p} \left (c x^2\right )^p\right ) \int x^{-2+2 p} (a+b x)^{-2 p} \, dx\\ &=-\frac{\left (c x^2\right )^p (a+b x)^{1-2 p}}{a (1-2 p) x}\\ \end{align*}
Mathematica [A] time = 0.0098735, size = 32, normalized size = 0.97 \[ \frac{\left (c x^2\right )^p (a+b x)^{1-2 p}}{a (2 p-1) x} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 38, normalized size = 1.2 \begin{align*}{\frac{ \left ( bx+a \right ) \left ( c{x}^{2} \right ) ^{p}}{ \left ( 2\,p-1 \right ) ax \left ( bx+a \right ) ^{2\,p}}} \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} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.63047, size = 72, normalized size = 2.18 \begin{align*} \frac{{\left (b x + a\right )} \left (c x^{2}\right )^{p}}{{\left (2 \, a p - a\right )}{\left (b x + a\right )}^{2 \, p} x} \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} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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