Optimal. Leaf size=39 \[ -\frac{c \left (c d^2+2 c d e x+c e^2 x^2\right )^{p-1}}{2 e (1-p)} \]
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Rubi [A] time = 0.0256884, antiderivative size = 39, 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 = {643, 629} \[ -\frac{c \left (c d^2+2 c d e x+c e^2 x^2\right )^{p-1}}{2 e (1-p)} \]
Antiderivative was successfully verified.
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Rule 643
Rule 629
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)^3} \, dx &=c^2 \int (d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{-2+p} \, dx\\ &=-\frac{c \left (c d^2+2 c d e x+c e^2 x^2\right )^{-1+p}}{2 e (1-p)}\\ \end{align*}
Mathematica [A] time = 0.0124948, size = 26, normalized size = 0.67 \[ \frac{c \left (c (d+e x)^2\right )^{p-1}}{2 e (p-1)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.039, size = 40, normalized size = 1. \begin{align*}{\frac{ \left ( c{e}^{2}{x}^{2}+2\,cdex+c{d}^{2} \right ) ^{p}}{2\, \left ( ex+d \right ) ^{2} \left ( p-1 \right ) e}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.27352, size = 61, normalized size = 1.56 \begin{align*} \frac{{\left (e x + d\right )}^{2 \, p} c^{p}}{2 \,{\left (e^{3}{\left (p - 1\right )} x^{2} + 2 \, d e^{2}{\left (p - 1\right )} x + d^{2} e{\left (p - 1\right )}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.48723, size = 136, normalized size = 3.49 \begin{align*} \frac{{\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{p}}{2 \,{\left (d^{2} e p - d^{2} e +{\left (e^{3} p - e^{3}\right )} x^{2} + 2 \,{\left (d e^{2} p - d e^{2}\right )} x\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.16808, size = 100, normalized size = 2.56 \begin{align*} \begin{cases} \frac{c x}{d} & \text{for}\: e = 0 \wedge p = 1 \\\frac{x \left (c d^{2}\right )^{p}}{d^{3}} & \text{for}\: e = 0 \\\frac{c \log{\left (\frac{d}{e} + x \right )}}{e} & \text{for}\: p = 1 \\\frac{\left (c d^{2} + 2 c d e x + c e^{2} x^{2}\right )^{p}}{2 d^{2} e p - 2 d^{2} e + 4 d e^{2} p x - 4 d e^{2} x + 2 e^{3} p x^{2} - 2 e^{3} x^{2}} & \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 e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{p}}{{\left (e x + d\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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