Optimal. Leaf size=33 \[ -\frac{\left (c x^2\right )^p (a+b x)^{3-2 p}}{a (3-2 p) x^3} \]
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Rubi [A] time = 0.0104211, antiderivative size = 33, 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} \[ -\frac{\left (c x^2\right )^p (a+b x)^{3-2 p}}{a (3-2 p) x^3} \]
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)^{2-2 p}}{x^4} \, dx &=\left (x^{-2 p} \left (c x^2\right )^p\right ) \int x^{-4+2 p} (a+b x)^{2-2 p} \, dx\\ &=-\frac{\left (c x^2\right )^p (a+b x)^{3-2 p}}{a (3-2 p) x^3}\\ \end{align*}
Mathematica [A] time = 0.0098643, size = 32, normalized size = 0.97 \[ \frac{\left (c x^2\right )^p (a+b x)^{3-2 p}}{a (2 p-3) x^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 33, normalized size = 1. \begin{align*}{\frac{ \left ( bx+a \right ) ^{3-2\,p} \left ( c{x}^{2} \right ) ^{p}}{{x}^{3}a \left ( 2\,p-3 \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 + 2}}{x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.6147, size = 84, normalized size = 2.55 \begin{align*} \frac{{\left (b x + a\right )} \left (c x^{2}\right )^{p}{\left (b x + a\right )}^{-2 \, p + 2}}{{\left (2 \, a p - 3 \, a\right )} x^{3}} \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 \frac{\left (c x^{2}\right )^{p}{\left (b x + a\right )}^{-2 \, p + 2}}{x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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