Optimal. Leaf size=312 \[ \frac{1024 c^4 d^4 e \left (a e^2+c d^2+2 c d e x\right )}{63 \left (c d^2-a e^2\right )^7 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{128 c^3 d^3 \left (a e^2+c d^2+2 c d e x\right )}{63 \left (c d^2-a e^2\right )^5 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}+\frac{16 c^2 d^2}{21 (d+e x) \left (c d^2-a e^2\right )^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}+\frac{8 c d}{21 (d+e x)^2 \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}+\frac{2}{9 (d+e x)^3 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \]
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Rubi [A] time = 0.151707, antiderivative size = 312, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.081, Rules used = {658, 614, 613} \[ \frac{1024 c^4 d^4 e \left (a e^2+c d^2+2 c d e x\right )}{63 \left (c d^2-a e^2\right )^7 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{128 c^3 d^3 \left (a e^2+c d^2+2 c d e x\right )}{63 \left (c d^2-a e^2\right )^5 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}+\frac{16 c^2 d^2}{21 (d+e x) \left (c d^2-a e^2\right )^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}+\frac{8 c d}{21 (d+e x)^2 \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}+\frac{2}{9 (d+e x)^3 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 658
Rule 614
Rule 613
Rubi steps
\begin{align*} \int \frac{1}{(d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx &=\frac{2}{9 \left (c d^2-a e^2\right ) (d+e x)^3 \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) \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}{3 \left (c d^2-a e^2\right )}\\ &=\frac{2}{9 \left (c d^2-a e^2\right ) (d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac{8 c d}{21 \left (c d^2-a e^2\right )^2 (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{\left (40 c^2 d^2\right ) \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}{21 \left (c d^2-a e^2\right )^2}\\ &=\frac{2}{9 \left (c d^2-a e^2\right ) (d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac{8 c d}{21 \left (c d^2-a e^2\right )^2 (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{16 c^2 d^2}{21 \left (c d^2-a e^2\right )^3 (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 (64 c^3 d^3\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}{21 \left (c d^2-a e^2\right )^3}\\ &=\frac{2}{9 \left (c d^2-a e^2\right ) (d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac{8 c d}{21 \left (c d^2-a e^2\right )^2 (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{16 c^2 d^2}{21 \left (c d^2-a e^2\right )^3 (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{128 c^3 d^3 \left (c d^2+a e^2+2 c d e x\right )}{63 \left (c d^2-a e^2\right )^5 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac{\left (512 c^4 d^4 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}{63 \left (c d^2-a e^2\right )^5}\\ &=\frac{2}{9 \left (c d^2-a e^2\right ) (d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac{8 c d}{21 \left (c d^2-a e^2\right )^2 (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{16 c^2 d^2}{21 \left (c d^2-a e^2\right )^3 (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{128 c^3 d^3 \left (c d^2+a e^2+2 c d e x\right )}{63 \left (c d^2-a e^2\right )^5 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac{1024 c^4 d^4 e \left (c d^2+a e^2+2 c d e x\right )}{63 \left (c d^2-a e^2\right )^7 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\\ \end{align*}
Mathematica [A] time = 0.153942, size = 336, normalized size = 1.08 \[ \frac{2 \left (3 a^4 c^2 d^2 e^8 \left (63 d^2+36 d e x+8 e^2 x^2\right )-4 a^3 c^3 d^3 e^6 \left (126 d^2 e x+105 d^3+72 d e^2 x^2+16 e^3 x^3\right )+3 a^2 c^4 d^4 e^4 \left (1008 d^2 e^2 x^2+840 d^3 e x+315 d^4+576 d e^3 x^3+128 e^4 x^4\right )-6 a^5 c d e^{10} (9 d+2 e x)+7 a^6 e^{12}+6 a c^5 d^5 e^2 \left (1680 d^3 e^2 x^2+2016 d^2 e^3 x^3+630 d^4 e x+63 d^5+1152 d e^4 x^4+256 e^5 x^5\right )+c^6 d^6 \left (2520 d^4 e^2 x^2+6720 d^3 e^3 x^3+8064 d^2 e^4 x^4+252 d^5 e x-21 d^6+4608 d e^5 x^5+1024 e^6 x^6\right )\right )}{63 (d+e x)^3 \left (c d^2-a e^2\right )^7 ((d+e x) (a e+c d x))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.055, size = 536, normalized size = 1.7 \begin{align*} -{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( 1024\,{c}^{6}{d}^{6}{e}^{6}{x}^{6}+1536\,a{c}^{5}{d}^{5}{e}^{7}{x}^{5}+4608\,{c}^{6}{d}^{7}{e}^{5}{x}^{5}+384\,{a}^{2}{c}^{4}{d}^{4}{e}^{8}{x}^{4}+6912\,a{c}^{5}{d}^{6}{e}^{6}{x}^{4}+8064\,{c}^{6}{d}^{8}{e}^{4}{x}^{4}-64\,{a}^{3}{c}^{3}{d}^{3}{e}^{9}{x}^{3}+1728\,{a}^{2}{c}^{4}{d}^{5}{e}^{7}{x}^{3}+12096\,a{c}^{5}{d}^{7}{e}^{5}{x}^{3}+6720\,{c}^{6}{d}^{9}{e}^{3}{x}^{3}+24\,{a}^{4}{c}^{2}{d}^{2}{e}^{10}{x}^{2}-288\,{a}^{3}{c}^{3}{d}^{4}{e}^{8}{x}^{2}+3024\,{a}^{2}{c}^{4}{d}^{6}{e}^{6}{x}^{2}+10080\,a{c}^{5}{d}^{8}{e}^{4}{x}^{2}+2520\,{c}^{6}{d}^{10}{e}^{2}{x}^{2}-12\,{a}^{5}cd{e}^{11}x+108\,{a}^{4}{c}^{2}{d}^{3}{e}^{9}x-504\,{a}^{3}{c}^{3}{d}^{5}{e}^{7}x+2520\,{a}^{2}{c}^{4}{d}^{7}{e}^{5}x+3780\,a{c}^{5}{d}^{9}{e}^{3}x+252\,{c}^{6}{d}^{11}ex+7\,{a}^{6}{e}^{12}-54\,{a}^{5}c{d}^{2}{e}^{10}+189\,{a}^{4}{c}^{2}{d}^{4}{e}^{8}-420\,{a}^{3}{c}^{3}{d}^{6}{e}^{6}+945\,{a}^{2}{c}^{4}{d}^{8}{e}^{4}+378\,a{c}^{5}{d}^{10}{e}^{2}-21\,{c}^{6}{d}^{12} \right ) }{63\, \left ({a}^{7}{e}^{14}-7\,{a}^{6}c{d}^{2}{e}^{12}+21\,{a}^{5}{c}^{2}{d}^{4}{e}^{10}-35\,{a}^{4}{c}^{3}{d}^{6}{e}^{8}+35\,{a}^{3}{c}^{4}{d}^{8}{e}^{6}-21\,{a}^{2}{c}^{5}{d}^{10}{e}^{4}+7\,a{c}^{6}{d}^{12}{e}^{2}-{c}^{7}{d}^{14} \right ) \left ( ex+d \right ) ^{2}} \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] time = 0., size = 0, normalized size = 0. \begin{align*} \left [\mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, 1\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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