Optimal. Leaf size=34 \[ -\frac{c^3}{3 e \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \]
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Rubi [A] time = 0.0236374, antiderivative size = 34, 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 = {643, 629} \[ -\frac{c^3}{3 e \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 643
Rule 629
Rubi steps
\begin{align*} \int \frac{\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{(d+e x)^7} \, dx &=c^4 \int \frac{d+e x}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}} \, dx\\ &=-\frac{c^3}{3 e \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0089959, size = 27, normalized size = 0.79 \[ -\frac{\left (c (d+e x)^2\right )^{3/2}}{3 e (d+e x)^6} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.04, size = 35, normalized size = 1. \begin{align*} -{\frac{1}{3\, \left ( ex+d \right ) ^{6}e} \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 [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.44491, size = 143, normalized size = 4.21 \begin{align*} -\frac{\sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}} c}{3 \,{\left (e^{5} x^{4} + 4 \, d e^{4} x^{3} + 6 \, d^{2} e^{3} x^{2} + 4 \, d^{3} e^{2} x + d^{4} 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{\left (c \left (d + e x\right )^{2}\right )^{\frac{3}{2}}}{\left (d + e x\right )^{7}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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