Optimal. Leaf size=38 \[ -\frac{1}{6 e (d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}} \]
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Rubi [A] time = 0.0206601, 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} \[ -\frac{1}{6 e (d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}} \]
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 )^{5/2}} \, dx &=c \int \frac{1}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{7/2}} \, dx\\ &=-\frac{1}{6 e (d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.0177792, size = 26, normalized size = 0.68 \[ -\frac{c (d+e x)}{6 e \left (c (d+e x)^2\right )^{7/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.042, size = 35, normalized size = 0.9 \begin{align*} -{\frac{1}{6\,e \left ( ex+d \right ) } \left ( c{e}^{2}{x}^{2}+2\,cdex+c{d}^{2} \right ) ^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.16159, size = 120, normalized size = 3.16 \begin{align*} -\frac{1}{6 \,{\left (c^{\frac{5}{2}} e^{7} x^{6} + 6 \, c^{\frac{5}{2}} d e^{6} x^{5} + 15 \, c^{\frac{5}{2}} d^{2} e^{5} x^{4} + 20 \, c^{\frac{5}{2}} d^{3} e^{4} x^{3} + 15 \, c^{\frac{5}{2}} d^{4} e^{3} x^{2} + 6 \, c^{\frac{5}{2}} d^{5} e^{2} x + c^{\frac{5}{2}} d^{6} e\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.26976, size = 254, normalized size = 6.68 \begin{align*} -\frac{\sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}}}{6 \,{\left (c^{3} e^{8} x^{7} + 7 \, c^{3} d e^{7} x^{6} + 21 \, c^{3} d^{2} e^{6} x^{5} + 35 \, c^{3} d^{3} e^{5} x^{4} + 35 \, c^{3} d^{4} e^{4} x^{3} + 21 \, c^{3} d^{5} e^{3} x^{2} + 7 \, c^{3} d^{6} e^{2} x + c^{3} d^{7} 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{5}{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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