Optimal. Leaf size=42 \[ \frac{c^2 (d+e x) \log (d+e x)}{e \sqrt{c d^2+2 c d e x+c e^2 x^2}} \]
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Rubi [A] time = 0.0246343, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.094, Rules used = {642, 608, 31} \[ \frac{c^2 (d+e x) \log (d+e x)}{e \sqrt{c d^2+2 c d e x+c e^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 642
Rule 608
Rule 31
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)^4} \, dx &=c^2 \int \frac{1}{\sqrt{c d^2+2 c d e x+c e^2 x^2}} \, dx\\ &=\frac{\left (c^2 \left (c d e+c e^2 x\right )\right ) \int \frac{1}{c d e+c e^2 x} \, dx}{\sqrt{c d^2+2 c d e x+c e^2 x^2}}\\ &=\frac{c^2 (d+e x) \log (d+e x)}{e \sqrt{c d^2+2 c d e x+c e^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.0038717, size = 31, normalized size = 0.74 \[ \frac{c^2 (d+e x) \log (d+e x)}{e \sqrt{c (d+e x)^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 40, normalized size = 1. \begin{align*}{\frac{\ln \left ( ex+d \right ) }{ \left ( ex+d \right ) ^{3}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 [A] time = 2.2917, size = 89, normalized size = 2.12 \begin{align*} \frac{\sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}} c \log \left (e x + d\right )}{e^{2} x + d e} \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 )^{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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