Optimal. Leaf size=39 \[ \frac{(d+e x) \sqrt{c d^2+2 c d e x+c e^2 x^2}}{2 c^3 e} \]
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Rubi [A] time = 0.0225454, 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, 609} \[ \frac{(d+e x) \sqrt{c d^2+2 c d e x+c e^2 x^2}}{2 c^3 e} \]
Antiderivative was successfully verified.
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Rule 642
Rule 609
Rubi steps
\begin{align*} \int \frac{(d+e x)^6}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}} \, dx &=\frac{\int \sqrt{c d^2+2 c d e x+c e^2 x^2} \, dx}{c^3}\\ &=\frac{(d+e x) \sqrt{c d^2+2 c d e x+c e^2 x^2}}{2 c^3 e}\\ \end{align*}
Mathematica [A] time = 0.0032829, size = 33, normalized size = 0.85 \[ \frac{x (d+e x) (2 d+e x)}{2 c^2 \sqrt{c (d+e x)^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.04, size = 40, normalized size = 1. \begin{align*}{\frac{x \left ( ex+2\,d \right ) \left ( ex+d \right ) ^{5}}{2} \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 = 1.27596, size = 313, normalized size = 8.03 \begin{align*} \frac{e^{4} x^{5}}{2 \,{\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{\frac{3}{2}} c} + \frac{5 \, d e^{3} x^{4}}{2 \,{\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{\frac{3}{2}} c} - \frac{25 \, c^{2} d^{6} e^{4}}{4 \, \left (c e^{2}\right )^{\frac{9}{2}}{\left (x + \frac{d}{e}\right )}^{4}} - \frac{10 \, d^{3} e x^{2}}{{\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{\frac{3}{2}} c} + \frac{50 \, c d^{5} e^{3}}{3 \, \left (c e^{2}\right )^{\frac{7}{2}}{\left (x + \frac{d}{e}\right )}^{3}} - \frac{26 \, d^{5}}{3 \,{\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{\frac{3}{2}} c e} - \frac{25 \, d^{4} e^{2}}{2 \, \left (c e^{2}\right )^{\frac{5}{2}}{\left (x + \frac{d}{e}\right )}^{2}} + \frac{25 \, d^{6}}{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 = 2.30854, size = 101, normalized size = 2.59 \begin{align*} \frac{\sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}}{\left (e x^{2} + 2 \, d x\right )}}{2 \,{\left (c^{3} e x + c^{3} d\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 (d + e x\right )^{6}}{\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 = 1.48774, size = 146, normalized size = 3.74 \begin{align*} -\frac{\frac{9 \, d^{5} e^{\left (-1\right )}}{c} - 4 \, C_{0} d^{3} e^{\left (-3\right )} -{\left (12 \, C_{0} d^{2} e^{\left (-2\right )} - \frac{25 \, d^{4}}{c} -{\left (\frac{20 \, d^{3} e}{c} - 12 \, C_{0} d e^{\left (-1\right )} -{\left (x{\left (\frac{x e^{4}}{c} + \frac{5 \, d e^{3}}{c}\right )} + 4 \, C_{0}\right )} x\right )} x\right )} x}{2 \,{\left (c x^{2} e^{2} + 2 \, c d x e + c d^{2}\right )}^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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