Optimal. Leaf size=42 \[ \frac{(d+e x) \log (d+e x)}{c^2 e \sqrt{c d^2+2 c d e x+c e^2 x^2}} \]
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Rubi [A] time = 0.0258827, 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} \[ \frac{(d+e x) \log (d+e x)}{c^2 e \sqrt{c d^2+2 c d e x+c e^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 642
Rule 608
Rule 31
Rubi steps
\begin{align*} \int \frac{(d+e x)^4}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}} \, dx &=\frac{\int \frac{1}{\sqrt{c d^2+2 c d e x+c e^2 x^2}} \, dx}{c^2}\\ &=\frac{\left (c d e+c e^2 x\right ) \int \frac{1}{c d e+c e^2 x} \, dx}{c^2 \sqrt{c d^2+2 c d e x+c e^2 x^2}}\\ &=\frac{(d+e x) \log (d+e x)}{c^2 e \sqrt{c d^2+2 c d e x+c e^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.0070995, size = 31, normalized size = 0.74 \[ \frac{(d+e x) \log (d+e x)}{c^2 e \sqrt{c (d+e x)^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.042, size = 40, normalized size = 1. \begin{align*}{\frac{ \left ( ex+d \right ) ^{5}\ln \left ( ex+d \right ) }{e} \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 = 1.57926, size = 606, normalized size = 14.43 \begin{align*} \frac{1}{12} \, e^{4}{\left (\frac{48 \, \sqrt{c} d e^{3} x^{3} + 108 \, \sqrt{c} d^{2} e^{2} x^{2} + 88 \, \sqrt{c} d^{3} e x + 25 \, \sqrt{c} d^{4}}{c^{3} e^{9} x^{4} + 4 \, c^{3} d e^{8} x^{3} + 6 \, c^{3} d^{2} e^{7} x^{2} + 4 \, c^{3} d^{3} e^{6} x + c^{3} d^{4} e^{5}} + \frac{12 \, \log \left (e x + d\right )}{c^{\frac{5}{2}} e^{5}}\right )} - \frac{1}{3} \, d e^{3}{\left (\frac{3 \, c^{2} d^{3} e}{\left (c e^{2}\right )^{\frac{9}{2}}{\left (x + \frac{d}{e}\right )}^{4}} + \frac{12 \, x^{2}}{{\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{\frac{3}{2}} c e^{2}} - \frac{8 \, c d^{2}}{\left (c e^{2}\right )^{\frac{7}{2}}{\left (x + \frac{d}{e}\right )}^{3}} + \frac{8 \, d^{2}}{{\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{\frac{3}{2}} c e^{4}} + \frac{6 \, d}{\left (c e^{2}\right )^{\frac{5}{2}} e{\left (x + \frac{d}{e}\right )}^{2}} - \frac{6 \, d^{3}}{\left (c e^{2}\right )^{\frac{5}{2}} e^{3}{\left (x + \frac{d}{e}\right )}^{4}}\right )} - \frac{1}{2} \, d^{2} e^{2}{\left (\frac{3 \, c^{2} d^{2} e^{2}}{\left (c e^{2}\right )^{\frac{9}{2}}{\left (x + \frac{d}{e}\right )}^{4}} - \frac{8 \, c d e}{\left (c e^{2}\right )^{\frac{7}{2}}{\left (x + \frac{d}{e}\right )}^{3}} + \frac{6}{\left (c e^{2}\right )^{\frac{5}{2}}{\left (x + \frac{d}{e}\right )}^{2}}\right )} - \frac{1}{3} \, d^{3} e{\left (\frac{4}{{\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{\frac{3}{2}} c e^{2}} - \frac{3 \, d}{\left (c e^{2}\right )^{\frac{5}{2}} e{\left (x + \frac{d}{e}\right )}^{4}}\right )} - \frac{d^{4}}{4 \, \left (c e^{2}\right )^{\frac{5}{2}}{\left (x + \frac{d}{e}\right )}^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.32264, size = 97, normalized size = 2.31 \begin{align*} \frac{\sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}} \log \left (e x + d\right )}{c^{3} e^{2} x + c^{3} 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 (d + e x\right )^{4}}{\left (c \left (d + e x\right )^{2}\right )^{\frac{5}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.55926, size = 149, normalized size = 3.55 \begin{align*} \frac{2 \,{\left (C_{0} d^{3} e^{\left (-3\right )} +{\left (3 \, C_{0} d^{2} e^{\left (-2\right )} +{\left (3 \, C_{0} d e^{\left (-1\right )} + C_{0} x\right )} x\right )} x\right )}}{{\left (c x^{2} e^{2} + 2 \, c d x e + c d^{2}\right )}^{\frac{3}{2}}} - \frac{e^{\left (-1\right )} \log \left ({\left | -\sqrt{c} d e^{2} -{\left (\sqrt{c} x e - \sqrt{c x^{2} e^{2} + 2 \, c d x e + c d^{2}}\right )} e^{2} \right |}\right )}{c^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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