Optimal. Leaf size=38 \[ -\frac{1}{4 e (d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \]
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Rubi [A] time = 0.0202065, antiderivative size = 38, normalized size of antiderivative = 1., 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 = {642, 607} \[ -\frac{1}{4 e (d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 642
Rule 607
Rubi steps
\begin{align*} \int \frac{1}{(d+e x)^2 \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \, dx &=c \int \frac{1}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}} \, dx\\ &=-\frac{1}{4 e (d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0177119, size = 26, normalized size = 0.68 \[ -\frac{c (d+e x)}{4 e \left (c (d+e x)^2\right )^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.041, size = 35, normalized size = 0.9 \begin{align*} -{\frac{1}{4\,e \left ( ex+d \right ) } \left ( c{e}^{2}{x}^{2}+2\,cdex+c{d}^{2} \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.20297, size = 82, normalized size = 2.16 \begin{align*} -\frac{1}{4 \,{\left (c^{\frac{3}{2}} e^{5} x^{4} + 4 \, c^{\frac{3}{2}} d e^{4} x^{3} + 6 \, c^{\frac{3}{2}} d^{2} e^{3} x^{2} + 4 \, c^{\frac{3}{2}} d^{3} e^{2} x + c^{\frac{3}{2}} d^{4} e\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.42569, size = 197, normalized size = 5.18 \begin{align*} -\frac{\sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}}}{4 \,{\left (c^{2} e^{6} x^{5} + 5 \, c^{2} d e^{5} x^{4} + 10 \, c^{2} d^{2} e^{4} x^{3} + 10 \, c^{2} d^{3} e^{3} x^{2} + 5 \, c^{2} d^{4} e^{2} x + c^{2} d^{5} e\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (c \left (d + e x\right )^{2}\right )^{\frac{3}{2}} \left (d + e x\right )^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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