Optimal. Leaf size=39 \[ \frac{(d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{7/2}}{8 c e} \]
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Rubi [A] time = 0.0207566, 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} \[ \frac{(d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{7/2}}{8 c e} \]
Antiderivative was successfully verified.
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Rule 642
Rule 609
Rubi steps
\begin{align*} \int (d+e x)^2 \left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2} \, dx &=\frac{\int \left (c d^2+2 c d e x+c e^2 x^2\right )^{7/2} \, dx}{c}\\ &=\frac{(d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{7/2}}{8 c e}\\ \end{align*}
Mathematica [A] time = 0.0200629, size = 28, normalized size = 0.72 \[ \frac{(d+e x) \left (c (d+e x)^2\right )^{7/2}}{8 c e} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.04, size = 106, normalized size = 2.7 \begin{align*}{\frac{x \left ({e}^{7}{x}^{7}+8\,d{e}^{6}{x}^{6}+28\,{d}^{2}{e}^{5}{x}^{5}+56\,{d}^{3}{e}^{4}{x}^{4}+70\,{d}^{4}{e}^{3}{x}^{3}+56\,{d}^{5}{e}^{2}{x}^{2}+28\,{d}^{6}ex+8\,{d}^{7} \right ) }{8\, \left ( ex+d \right ) ^{5}} \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 [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.50481, size = 270, normalized size = 6.92 \begin{align*} \frac{{\left (c^{2} e^{7} x^{8} + 8 \, c^{2} d e^{6} x^{7} + 28 \, c^{2} d^{2} e^{5} x^{6} + 56 \, c^{2} d^{3} e^{4} x^{5} + 70 \, c^{2} d^{4} e^{3} x^{4} + 56 \, c^{2} d^{5} e^{2} x^{3} + 28 \, c^{2} d^{6} e x^{2} + 8 \, c^{2} d^{7} x\right )} \sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}}}{8 \,{\left (e x + 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 \left (c \left (d + e x\right )^{2}\right )^{\frac{5}{2}} \left (d + e x\right )^{2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.20787, size = 155, normalized size = 3.97 \begin{align*} \frac{1}{8} \,{\left (c^{2} d^{7} e^{\left (-1\right )} +{\left (7 \, c^{2} d^{6} +{\left (21 \, c^{2} d^{5} e +{\left (35 \, c^{2} d^{4} e^{2} +{\left (35 \, c^{2} d^{3} e^{3} +{\left (21 \, c^{2} d^{2} e^{4} +{\left (c^{2} x e^{6} + 7 \, c^{2} d e^{5}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \sqrt{c x^{2} e^{2} + 2 \, c d x e + c d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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