Optimal. Leaf size=39 \[ -\frac{c}{4 e (d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \]
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Rubi [A] time = 0.0193221, antiderivative size = 39, 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} \[ -\frac{c}{4 e (d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 642
Rule 607
Rubi steps
\begin{align*} \int \frac{1}{(d+e x)^4 \sqrt{c d^2+2 c d e x+c e^2 x^2}} \, dx &=c^2 \int \frac{1}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}} \, dx\\ &=-\frac{c}{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 = 0.0107383, size = 27, normalized size = 0.69 \[ -\frac{1}{4 e (d+e x)^3 \sqrt{c (d+e x)^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 35, normalized size = 0.9 \begin{align*} -{\frac{1}{4\, \left ( ex+d \right ) ^{3}e}{\frac{1}{\sqrt{c{e}^{2}{x}^{2}+2\,cdex+c{d}^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.15207, size = 82, normalized size = 2.1 \begin{align*} -\frac{1}{4 \,{\left (\sqrt{c} e^{5} x^{4} + 4 \, \sqrt{c} d e^{4} x^{3} + 6 \, \sqrt{c} d^{2} e^{3} x^{2} + 4 \, \sqrt{c} d^{3} e^{2} x + \sqrt{c} d^{4} e\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.27123, size = 181, normalized size = 4.64 \begin{align*} -\frac{\sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}}}{4 \,{\left (c e^{6} x^{5} + 5 \, c d e^{5} x^{4} + 10 \, c d^{2} e^{4} x^{3} + 10 \, c d^{3} e^{3} x^{2} + 5 \, c d^{4} e^{2} x + c 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}{\sqrt{c \left (d + e x\right )^{2}} \left (d + e x\right )^{4}}\, 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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