Optimal. Leaf size=39 \[ -\frac {c \left (c d^2+2 c d e x+c e^2 x^2\right )^{-1+p}}{2 e (1-p)} \]
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Rubi [A]
time = 0.02, 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 = {657, 643}
\begin {gather*} -\frac {c \left (c d^2+2 c d e x+c e^2 x^2\right )^{p-1}}{2 e (1-p)} \end {gather*}
Antiderivative was successfully verified.
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Rule 643
Rule 657
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 = 0.02, size = 26, normalized size = 0.67 \begin {gather*} \frac {c \left (c (d+e x)^2\right )^{-1+p}}{2 e (-1+p)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.73, size = 29, normalized size = 0.74
method | result | size |
risch | \(\frac {\left (\left (e x +d \right )^{2} c \right )^{p}}{2 \left (-1+p \right ) e \left (e x +d \right )^{2}}\) | \(29\) |
gosper | \(\frac {\left (x^{2} c \,e^{2}+2 c d e x +c \,d^{2}\right )^{p}}{2 \left (e x +d \right )^{2} \left (-1+p \right ) e}\) | \(40\) |
norman | \(\frac {{\mathrm e}^{p \ln \left (x^{2} c \,e^{2}+2 c d e x +c \,d^{2}\right )}}{2 \left (-1+p \right ) e \left (e x +d \right )^{2}}\) | \(42\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 45, normalized size = 1.15 \begin {gather*} \frac {{\left (x e + d\right )}^{2 \, p} c^{p}}{2 \, {\left ({\left (p - 1\right )} x^{2} e^{3} + 2 \, d {\left (p - 1\right )} x e^{2} + d^{2} {\left (p - 1\right )} e\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.12, size = 62, normalized size = 1.59 \begin {gather*} \frac {{\left (c x^{2} e^{2} + 2 \, c d x e + c d^{2}\right )}^{p}}{2 \, {\left ({\left (p - 1\right )} x^{2} e^{3} + 2 \, {\left (d p - d\right )} x e^{2} + {\left (d^{2} p - d^{2}\right )} e\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 100 vs.
\(2 (34) = 68\).
time = 0.47, size = 100, normalized size = 2.56 \begin {gather*} \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 {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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