Optimal. Leaf size=231 \[ \frac{32 c^3 d^3 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{35 (d+e x) \left (c d^2-a e^2\right )^4}+\frac{16 c^2 d^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{35 (d+e x)^2 \left (c d^2-a e^2\right )^3}+\frac{12 c d \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{35 (d+e x)^3 \left (c d^2-a e^2\right )^2}+\frac{2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{7 (d+e x)^4 \left (c d^2-a e^2\right )} \]
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Rubi [A] time = 0.115357, antiderivative size = 231, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.054, Rules used = {658, 650} \[ \frac{32 c^3 d^3 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{35 (d+e x) \left (c d^2-a e^2\right )^4}+\frac{16 c^2 d^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{35 (d+e x)^2 \left (c d^2-a e^2\right )^3}+\frac{12 c d \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{35 (d+e x)^3 \left (c d^2-a e^2\right )^2}+\frac{2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{7 (d+e x)^4 \left (c d^2-a e^2\right )} \]
Antiderivative was successfully verified.
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Rule 658
Rule 650
Rubi steps
\begin{align*} \int \frac{1}{(d+e x)^4 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx &=\frac{2 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{7 \left (c d^2-a e^2\right ) (d+e x)^4}+\frac{(6 c d) \int \frac{1}{(d+e x)^3 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{7 \left (c d^2-a e^2\right )}\\ &=\frac{2 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{7 \left (c d^2-a e^2\right ) (d+e x)^4}+\frac{12 c d \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{35 \left (c d^2-a e^2\right )^2 (d+e x)^3}+\frac{\left (24 c^2 d^2\right ) \int \frac{1}{(d+e x)^2 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{35 \left (c d^2-a e^2\right )^2}\\ &=\frac{2 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{7 \left (c d^2-a e^2\right ) (d+e x)^4}+\frac{12 c d \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{35 \left (c d^2-a e^2\right )^2 (d+e x)^3}+\frac{16 c^2 d^2 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{35 \left (c d^2-a e^2\right )^3 (d+e x)^2}+\frac{\left (16 c^3 d^3\right ) \int \frac{1}{(d+e x) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{35 \left (c d^2-a e^2\right )^3}\\ &=\frac{2 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{7 \left (c d^2-a e^2\right ) (d+e x)^4}+\frac{12 c d \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{35 \left (c d^2-a e^2\right )^2 (d+e x)^3}+\frac{16 c^2 d^2 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{35 \left (c d^2-a e^2\right )^3 (d+e x)^2}+\frac{32 c^3 d^3 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{35 \left (c d^2-a e^2\right )^4 (d+e x)}\\ \end{align*}
Mathematica [A] time = 0.0690381, size = 138, normalized size = 0.6 \[ \frac{2 \sqrt{(d+e x) (a e+c d x)} \left (3 a^2 c d e^4 (7 d+2 e x)-5 a^3 e^6-a c^2 d^2 e^2 \left (35 d^2+28 d e x+8 e^2 x^2\right )+c^3 d^3 \left (70 d^2 e x+35 d^3+56 d e^2 x^2+16 e^3 x^3\right )\right )}{35 (d+e x)^4 \left (c d^2-a e^2\right )^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.049, size = 217, normalized size = 0.9 \begin{align*} -{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( -16\,{c}^{3}{d}^{3}{e}^{3}{x}^{3}+8\,a{c}^{2}{d}^{2}{e}^{4}{x}^{2}-56\,{c}^{3}{d}^{4}{e}^{2}{x}^{2}-6\,{a}^{2}cd{e}^{5}x+28\,a{c}^{2}{d}^{3}{e}^{3}x-70\,{c}^{3}{d}^{5}ex+5\,{a}^{3}{e}^{6}-21\,{a}^{2}c{d}^{2}{e}^{4}+35\,a{c}^{2}{d}^{4}{e}^{2}-35\,{c}^{3}{d}^{6} \right ) }{35\, \left ({a}^{4}{e}^{8}-4\,{a}^{3}c{d}^{2}{e}^{6}+6\,{a}^{2}{c}^{2}{d}^{4}{e}^{4}-4\,a{c}^{3}{d}^{6}{e}^{2}+{c}^{4}{d}^{8} \right ) \left ( ex+d \right ) ^{3}}{\frac{1}{\sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade}}}} \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 = 34.0082, size = 910, normalized size = 3.94 \begin{align*} \frac{2 \,{\left (16 \, c^{3} d^{3} e^{3} x^{3} + 35 \, c^{3} d^{6} - 35 \, a c^{2} d^{4} e^{2} + 21 \, a^{2} c d^{2} e^{4} - 5 \, a^{3} e^{6} + 8 \,{\left (7 \, c^{3} d^{4} e^{2} - a c^{2} d^{2} e^{4}\right )} x^{2} + 2 \,{\left (35 \, c^{3} d^{5} e - 14 \, a c^{2} d^{3} e^{3} + 3 \, a^{2} c d e^{5}\right )} x\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}}{35 \,{\left (c^{4} d^{12} - 4 \, a c^{3} d^{10} e^{2} + 6 \, a^{2} c^{2} d^{8} e^{4} - 4 \, a^{3} c d^{6} e^{6} + a^{4} d^{4} e^{8} +{\left (c^{4} d^{8} e^{4} - 4 \, a c^{3} d^{6} e^{6} + 6 \, a^{2} c^{2} d^{4} e^{8} - 4 \, a^{3} c d^{2} e^{10} + a^{4} e^{12}\right )} x^{4} + 4 \,{\left (c^{4} d^{9} e^{3} - 4 \, a c^{3} d^{7} e^{5} + 6 \, a^{2} c^{2} d^{5} e^{7} - 4 \, a^{3} c d^{3} e^{9} + a^{4} d e^{11}\right )} x^{3} + 6 \,{\left (c^{4} d^{10} e^{2} - 4 \, a c^{3} d^{8} e^{4} + 6 \, a^{2} c^{2} d^{6} e^{6} - 4 \, a^{3} c d^{4} e^{8} + a^{4} d^{2} e^{10}\right )} x^{2} + 4 \,{\left (c^{4} d^{11} e - 4 \, a c^{3} d^{9} e^{3} + 6 \, a^{2} c^{2} d^{7} e^{5} - 4 \, a^{3} c d^{5} e^{7} + a^{4} d^{3} e^{9}\right )} x\right )}} \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(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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