Optimal. Leaf size=42 \[ -\frac{c (d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{p-1}}{e (1-2 p)} \]
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Rubi [A] time = 0.0256373, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {642, 609} \[ -\frac{c (d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{p-1}}{e (1-2 p)} \]
Antiderivative was successfully verified.
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Rule 642
Rule 609
Rubi steps
\begin{align*} \int \frac{\left (c d^2+2 c d e x+c e^2 x^2\right )^p}{(d+e x)^2} \, dx &=c \int \left (c d^2+2 c d e x+c e^2 x^2\right )^{-1+p} \, dx\\ &=-\frac{c (d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{-1+p}}{e (1-2 p)}\\ \end{align*}
Mathematica [A] time = 0.0166086, size = 30, normalized size = 0.71 \[ \frac{c (d+e x) \left (c (d+e x)^2\right )^{p-1}}{e (2 p-1)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.039, size = 41, normalized size = 1. \begin{align*}{\frac{ \left ( c{e}^{2}{x}^{2}+2\,cdex+c{d}^{2} \right ) ^{p}}{ \left ( ex+d \right ) \left ( -1+2\,p \right ) e}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.31708, size = 46, normalized size = 1.1 \begin{align*} \frac{{\left (e x + d\right )}^{2 \, p} c^{p}}{e^{2}{\left (2 \, p - 1\right )} x + d e{\left (2 \, p - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.44252, size = 96, normalized size = 2.29 \begin{align*} \frac{{\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{p}}{2 \, d e p - d e +{\left (2 \, e^{2} p - e^{2}\right )} 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 e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{p}}{{\left (e x + d\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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