Optimal. Leaf size=44 \[ \frac {(d+e x)^{p+1} \left (c d^2+2 c d e x+c e^2 x^2\right )^{-p}}{e (1-p)} \]
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Rubi [A] time = 0.02, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {644, 32} \[ \frac {(d+e x)^{p+1} \left (c d^2+2 c d e x+c e^2 x^2\right )^{-p}}{e (1-p)} \]
Antiderivative was successfully verified.
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Rule 32
Rule 644
Rubi steps
\begin {align*} \int (d+e x)^p \left (c d^2+2 c d e x+c e^2 x^2\right )^{-p} \, dx &=\left ((d+e x)^{2 p} \left (c d^2+2 c d e x+c e^2 x^2\right )^{-p}\right ) \int (d+e x)^{-p} \, dx\\ &=\frac {(d+e x)^{1+p} \left (c d^2+2 c d e x+c e^2 x^2\right )^{-p}}{e (1-p)}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 31, normalized size = 0.70 \[ \frac {(d+e x)^{p+1} \left (c (d+e x)^2\right )^{-p}}{e-e p} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.09, size = 30, normalized size = 0.68 \[ -\frac {e x + d}{{\left (e p - e\right )} {\left (e x + d\right )}^{p} c^{p}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 69, normalized size = 1.57 \[ -\frac {{\left (x e + d\right )}^{p} x e^{\left (-2 \, p \log \left (x e + d\right ) - p \log \relax (c) + 1\right )} + {\left (x e + d\right )}^{p} d e^{\left (-2 \, p \log \left (x e + d\right ) - p \log \relax (c)\right )}}{p e - e} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 44, normalized size = 1.00 \[ -\frac {\left (e x +d \right )^{p +1} \left (c \,e^{2} x^{2}+2 c d e x +c \,d^{2}\right )^{-p}}{\left (p -1\right ) e} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.47, size = 29, normalized size = 0.66 \[ -\frac {e x + d}{{\left (e x + d\right )}^{p} c^{p} e {\left (p - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.47, size = 43, normalized size = 0.98 \[ -\frac {{\left (d+e\,x\right )}^{p+1}}{e\,\left (p-1\right )\,{\left (c\,d^2+2\,c\,d\,e\,x+c\,e^2\,x^2\right )}^p} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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