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 = 0.03, antiderivative size = 39, normalized size of antiderivative = 1.00, 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 629
Rule 643
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*}
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Mathematica [A] time = 0.01, 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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fricas [A] time = 1.15, size = 70, normalized size = 1.79 \[ \frac {{\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{p}}{2 \, {\left (d^{2} e p - d^{2} e + {\left (e^{3} p - e^{3}\right )} x^{2} + 2 \, {\left (d e^{2} p - d e^{2}\right )} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{p}}{{\left (e x + d\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 40, normalized size = 1.03 \[ \frac {\left (c \,e^{2} x^{2}+2 c d e x +c \,d^{2}\right )^{p}}{2 \left (e x +d \right )^{2} \left (p -1\right ) e} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.51, size = 45, normalized size = 1.15 \[ \frac {{\left (e x + d\right )}^{2 \, p} c^{p}}{2 \, {\left (e^{3} {\left (p - 1\right )} x^{2} + 2 \, d e^{2} {\left (p - 1\right )} x + d^{2} e {\left (p - 1\right )}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.49, size = 52, normalized size = 1.33 \[ \frac {{\left (c\,d^2+2\,c\,d\,e\,x+c\,e^2\,x^2\right )}^p}{2\,e^3\,\left (p-1\right )\,\left (x^2+\frac {d^2}{e^2}+\frac {2\,d\,x}{e}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.25, size = 100, normalized size = 2.56 \[ \begin {cases} \frac {c x}{d} & \text {for}\: e = 0 \wedge p = 1 \\\frac {x \left (c d^{2}\right )^{p}}{d^{3}} & \text {for}\: e = 0 \\\frac {c \log {\left (\frac {d}{e} + x \right )}}{e} & \text {for}\: p = 1 \\\frac {\left (c d^{2} + 2 c d e x + c e^{2} x^{2}\right )^{p}}{2 d^{2} e p - 2 d^{2} e + 4 d e^{2} p x - 4 d e^{2} x + 2 e^{3} p x^{2} - 2 e^{3} x^{2}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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