Optimal. Leaf size=252 \[ \frac{256 c^3 d^3 e \left (a e^2+c d^2+2 c d e x\right )}{21 \left (c d^2-a e^2\right )^6 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{32 c^2 d^2 \left (a e^2+c d^2+2 c d e x\right )}{21 \left (c d^2-a e^2\right )^4 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}+\frac{4 c d}{7 (d+e x) \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}+\frac{2}{7 (d+e x)^2 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \]
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Rubi [A] time = 0.101749, antiderivative size = 252, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.081, Rules used = {658, 614, 613} \[ \frac{256 c^3 d^3 e \left (a e^2+c d^2+2 c d e x\right )}{21 \left (c d^2-a e^2\right )^6 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{32 c^2 d^2 \left (a e^2+c d^2+2 c d e x\right )}{21 \left (c d^2-a e^2\right )^4 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}+\frac{4 c d}{7 (d+e x) \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}+\frac{2}{7 (d+e x)^2 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 658
Rule 614
Rule 613
Rubi steps
\begin{align*} \int \frac{1}{(d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx &=\frac{2}{7 \left (c d^2-a e^2\right ) (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac{(10 c d) \int \frac{1}{(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx}{7 \left (c d^2-a e^2\right )}\\ &=\frac{2}{7 \left (c d^2-a e^2\right ) (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac{4 c d}{7 \left (c d^2-a e^2\right )^2 (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac{\left (16 c^2 d^2\right ) \int \frac{1}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx}{7 \left (c d^2-a e^2\right )^2}\\ &=\frac{2}{7 \left (c d^2-a e^2\right ) (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac{4 c d}{7 \left (c d^2-a e^2\right )^2 (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac{32 c^2 d^2 \left (c d^2+a e^2+2 c d e x\right )}{21 \left (c d^2-a e^2\right )^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac{\left (128 c^3 d^3 e\right ) \int \frac{1}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx}{21 \left (c d^2-a e^2\right )^4}\\ &=\frac{2}{7 \left (c d^2-a e^2\right ) (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac{4 c d}{7 \left (c d^2-a e^2\right )^2 (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac{32 c^2 d^2 \left (c d^2+a e^2+2 c d e x\right )}{21 \left (c d^2-a e^2\right )^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac{256 c^3 d^3 e \left (c d^2+a e^2+2 c d e x\right )}{21 \left (c d^2-a e^2\right )^6 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\\ \end{align*}
Mathematica [A] time = 0.125281, size = 259, normalized size = 1.03 \[ \frac{2 \left (-2 a^3 c^2 d^2 e^6 \left (35 d^2+28 d e x+8 e^2 x^2\right )+6 a^2 c^3 d^3 e^4 \left (70 d^2 e x+35 d^3+56 d e^2 x^2+16 e^3 x^3\right )+3 a^4 c d e^8 (7 d+2 e x)-3 a^5 e^{10}+3 a c^4 d^4 e^2 \left (560 d^2 e^2 x^2+280 d^3 e x+35 d^4+448 d e^3 x^3+128 e^4 x^4\right )+c^5 d^5 \left (560 d^3 e^2 x^2+1120 d^2 e^3 x^3+70 d^4 e x-7 d^5+896 d e^4 x^4+256 e^5 x^5\right )\right )}{21 (d+e x)^2 \left (c d^2-a e^2\right )^6 ((d+e x) (a e+c d x))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.051, size = 412, normalized size = 1.6 \begin{align*} -{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( -256\,{c}^{5}{d}^{5}{e}^{5}{x}^{5}-384\,a{c}^{4}{d}^{4}{e}^{6}{x}^{4}-896\,{c}^{5}{d}^{6}{e}^{4}{x}^{4}-96\,{a}^{2}{c}^{3}{d}^{3}{e}^{7}{x}^{3}-1344\,a{c}^{4}{d}^{5}{e}^{5}{x}^{3}-1120\,{c}^{5}{d}^{7}{e}^{3}{x}^{3}+16\,{a}^{3}{c}^{2}{d}^{2}{e}^{8}{x}^{2}-336\,{a}^{2}{c}^{3}{d}^{4}{e}^{6}{x}^{2}-1680\,a{c}^{4}{d}^{6}{e}^{4}{x}^{2}-560\,{c}^{5}{d}^{8}{e}^{2}{x}^{2}-6\,{a}^{4}cd{e}^{9}x+56\,{a}^{3}{c}^{2}{d}^{3}{e}^{7}x-420\,{a}^{2}{c}^{3}{d}^{5}{e}^{5}x-840\,a{c}^{4}{d}^{7}{e}^{3}x-70\,{c}^{5}{d}^{9}ex+3\,{a}^{5}{e}^{10}-21\,{a}^{4}c{d}^{2}{e}^{8}+70\,{a}^{3}{c}^{2}{d}^{4}{e}^{6}-210\,{a}^{2}{c}^{3}{d}^{6}{e}^{4}-105\,a{c}^{4}{d}^{8}{e}^{2}+7\,{c}^{5}{d}^{10} \right ) }{ \left ( 21\,ex+21\,d \right ) \left ({a}^{6}{e}^{12}-6\,{a}^{5}c{d}^{2}{e}^{10}+15\,{a}^{4}{c}^{2}{d}^{4}{e}^{8}-20\,{a}^{3}{c}^{3}{d}^{6}{e}^{6}+15\,{a}^{2}{c}^{4}{d}^{8}{e}^{4}-6\,a{c}^{5}{d}^{10}{e}^{2}+{c}^{6}{d}^{12} \right ) } \left ( cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade \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 [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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