Optimal. Leaf size=60 \[ \frac {(d+e x)^{-2 (1+p)} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{1+p}}{\left (c d^2-a e^2\right ) (1+p)} \]
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Rubi [A]
time = 0.01, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.026, Rules used = {664}
\begin {gather*} \frac {(d+e x)^{-2 (p+1)} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{p+1}}{(p+1) \left (c d^2-a e^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 664
Rubi steps
\begin {align*} \int (d+e x)^{-2-2 p} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p \, dx &=\frac {(d+e x)^{-2 (1+p)} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{1+p}}{\left (c d^2-a e^2\right ) (1+p)}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 49, normalized size = 0.82 \begin {gather*} \frac {(d+e x)^{-2 (1+p)} ((a e+c d x) (d+e x))^{1+p}}{\left (c d^2-a e^2\right ) (1+p)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.78, size = 75, normalized size = 1.25
method | result | size |
gosper | \(-\frac {\left (c d x +a e \right ) \left (e x +d \right )^{-1-2 p} \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{p}}{a \,e^{2} p -c \,d^{2} p +e^{2} a -c \,d^{2}}\) | \(75\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.33, size = 88, normalized size = 1.47 \begin {gather*} \frac {{\left (c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e\right )} {\left (c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e\right )}^{p} {\left (x e + d\right )}^{-2 \, p - 2}}{c d^{2} p + c d^{2} - {\left (a p + a\right )} e^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 = 0.85, size = 150, normalized size = 2.50 \begin {gather*} -\left (\frac {x\,\left (c\,d^2+a\,e^2\right )}{\left (a\,e^2-c\,d^2\right )\,\left (p+1\right )\,{\left (d+e\,x\right )}^{2\,p+2}}+\frac {a\,d\,e}{\left (a\,e^2-c\,d^2\right )\,\left (p+1\right )\,{\left (d+e\,x\right )}^{2\,p+2}}+\frac {c\,d\,e\,x^2}{\left (a\,e^2-c\,d^2\right )\,\left (p+1\right )\,{\left (d+e\,x\right )}^{2\,p+2}}\right )\,{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^p \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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