Optimal. Leaf size=401 \[ -\frac{9 \left (c d^2-a e^2\right )^5 \left (a e^2+c d^2+2 c d e x\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{1024 c^5 d^5 e^2}+\frac{3 \left (c d^2-a e^2\right )^3 \left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{128 c^4 d^4 e}+\frac{3 \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{40 c^3 d^3}+\frac{3 (d+e x) \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{28 c^2 d^2}+\frac{9 \left (c d^2-a e^2\right )^7 \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2048 c^{11/2} d^{11/2} e^{5/2}}+\frac{(d+e x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{7 c d} \]
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Rubi [A] time = 0.372587, antiderivative size = 401, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.135, Rules used = {670, 640, 612, 621, 206} \[ -\frac{9 \left (c d^2-a e^2\right )^5 \left (a e^2+c d^2+2 c d e x\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{1024 c^5 d^5 e^2}+\frac{3 \left (c d^2-a e^2\right )^3 \left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{128 c^4 d^4 e}+\frac{3 \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{40 c^3 d^3}+\frac{3 (d+e x) \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{28 c^2 d^2}+\frac{9 \left (c d^2-a e^2\right )^7 \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2048 c^{11/2} d^{11/2} e^{5/2}}+\frac{(d+e x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{7 c d} \]
Antiderivative was successfully verified.
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Rule 670
Rule 640
Rule 612
Rule 621
Rule 206
Rubi steps
\begin{align*} \int (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} \, dx &=\frac{(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}}{7 c d}+\frac{\left (9 \left (d^2-\frac{a e^2}{c}\right )\right ) \int (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} \, dx}{14 d}\\ &=\frac{3 \left (c d^2-a e^2\right ) (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{28 c^2 d^2}+\frac{(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}}{7 c d}+\frac{\left (3 \left (d^2-\frac{a e^2}{c}\right )^2\right ) \int (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2} \, dx}{8 d^2}\\ &=\frac{3 \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{40 c^3 d^3}+\frac{3 \left (c d^2-a e^2\right ) (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{28 c^2 d^2}+\frac{(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}}{7 c d}+\frac{\left (3 \left (d^2-\frac{a e^2}{c}\right )^3\right ) \int \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2} \, dx}{16 d^3}\\ &=\frac{3 \left (c d^2-a e^2\right )^3 \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{128 c^4 d^4 e}+\frac{3 \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{40 c^3 d^3}+\frac{3 \left (c d^2-a e^2\right ) (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{28 c^2 d^2}+\frac{(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}}{7 c d}-\frac{\left (9 \left (c d^2-a e^2\right )^5\right ) \int \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx}{256 c^4 d^4 e}\\ &=-\frac{9 \left (c d^2-a e^2\right )^5 \left (c d^2+a e^2+2 c d e x\right ) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{1024 c^5 d^5 e^2}+\frac{3 \left (c d^2-a e^2\right )^3 \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{128 c^4 d^4 e}+\frac{3 \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{40 c^3 d^3}+\frac{3 \left (c d^2-a e^2\right ) (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{28 c^2 d^2}+\frac{(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}}{7 c d}+\frac{\left (9 \left (c d^2-a e^2\right )^7\right ) \int \frac{1}{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{2048 c^5 d^5 e^2}\\ &=-\frac{9 \left (c d^2-a e^2\right )^5 \left (c d^2+a e^2+2 c d e x\right ) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{1024 c^5 d^5 e^2}+\frac{3 \left (c d^2-a e^2\right )^3 \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{128 c^4 d^4 e}+\frac{3 \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{40 c^3 d^3}+\frac{3 \left (c d^2-a e^2\right ) (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{28 c^2 d^2}+\frac{(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}}{7 c d}+\frac{\left (9 \left (c d^2-a e^2\right )^7\right ) \operatorname{Subst}\left (\int \frac{1}{4 c d e-x^2} \, dx,x,\frac{c d^2+a e^2+2 c d e x}{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{1024 c^5 d^5 e^2}\\ &=-\frac{9 \left (c d^2-a e^2\right )^5 \left (c d^2+a e^2+2 c d e x\right ) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{1024 c^5 d^5 e^2}+\frac{3 \left (c d^2-a e^2\right )^3 \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{128 c^4 d^4 e}+\frac{3 \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{40 c^3 d^3}+\frac{3 \left (c d^2-a e^2\right ) (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{28 c^2 d^2}+\frac{(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}}{7 c d}+\frac{9 \left (c d^2-a e^2\right )^7 \tanh ^{-1}\left (\frac{c d^2+a e^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{2048 c^{11/2} d^{11/2} e^{5/2}}\\ \end{align*}
Mathematica [B] time = 6.25364, size = 1196, normalized size = 2.98 \[ \frac{2 \left (c d^2-a e^2\right )^4 (a e+c d x) ((a e+c d x) (d+e x))^{3/2} \left (\frac{c d e (a e+c d x)}{\left (c d^2-a e^2\right ) \left (\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}\right )}+1\right )^{11/2} \left (\frac{5}{14} \left (\frac{1}{\frac{c d e (a e+c d x)}{\left (c d^2-a e^2\right ) \left (\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}\right )}+1}+\frac{3}{4 \left (\frac{c d e (a e+c d x)}{\left (c d^2-a e^2\right ) \left (\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}\right )}+1\right )^2}+\frac{21}{40 \left (\frac{c d e (a e+c d x)}{\left (c d^2-a e^2\right ) \left (\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}\right )}+1\right )^3}+\frac{21}{64 \left (\frac{c d e (a e+c d x)}{\left (c d^2-a e^2\right ) \left (\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}\right )}+1\right )^4}+\frac{21}{128 \left (\frac{c d e (a e+c d x)}{\left (c d^2-a e^2\right ) \left (\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}\right )}+1\right )^5}\right )-\frac{45 \left (c d^2-a e^2\right )^3 \left (\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}\right )^3 \left (-\frac{4 c^2 d^2 e^2 (a e+c d x)^2}{3 \left (c d^2-a e^2\right )^2 \left (\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}\right )^2}+\frac{2 c d e (a e+c d x)}{\left (c d^2-a e^2\right ) \left (\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}\right )}-\frac{2 \sqrt{c} \sqrt{d} \sqrt{e} \sinh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a e+c d x}}{\sqrt{c d^2-a e^2} \sqrt{\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}}}\right ) \sqrt{a e+c d x}}{\sqrt{c d^2-a e^2} \sqrt{\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}} \sqrt{\frac{c d e (a e+c d x)}{\left (c d^2-a e^2\right ) \left (\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}\right )}+1}}\right )}{4096 c^3 d^3 e^3 (a e+c d x)^3 \left (\frac{c d e (a e+c d x)}{\left (c d^2-a e^2\right ) \left (\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}\right )}+1\right )^5}\right )}{5 c^5 d^5 \left (\frac{c d}{\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}}\right )^{9/2} (d+e x) \sqrt{\frac{c d (d+e x)}{c d^2-a e^2}}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.057, size = 1586, normalized size = 4. \begin{align*} \text{result too large to display} \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 [A] time = 2.4543, size = 2804, normalized size = 6.99 \begin{align*} \text{result too large to display} \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 [A] time = 1.31157, size = 824, normalized size = 2.05 \begin{align*} \frac{1}{35840} \, \sqrt{c d x^{2} e + c d^{2} x + a x e^{2} + a d e}{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \,{\left (10 \,{\left (4 \, c d x e^{4} + \frac{{\left (19 \, c^{7} d^{8} e^{9} + 5 \, a c^{6} d^{6} e^{11}\right )} e^{\left (-6\right )}}{c^{6} d^{6}}\right )} x + \frac{{\left (351 \, c^{7} d^{9} e^{8} + 248 \, a c^{6} d^{7} e^{10} + a^{2} c^{5} d^{5} e^{12}\right )} e^{\left (-6\right )}}{c^{6} d^{6}}\right )} x + \frac{{\left (2441 \, c^{7} d^{10} e^{7} + 3909 \, a c^{6} d^{8} e^{9} + 59 \, a^{2} c^{5} d^{6} e^{11} - 9 \, a^{3} c^{4} d^{4} e^{13}\right )} e^{\left (-6\right )}}{c^{6} d^{6}}\right )} x + \frac{{\left (1771 \, c^{7} d^{11} e^{6} + 7562 \, a c^{6} d^{9} e^{8} + 384 \, a^{2} c^{5} d^{7} e^{10} - 138 \, a^{3} c^{4} d^{5} e^{12} + 21 \, a^{4} c^{3} d^{3} e^{14}\right )} e^{\left (-6\right )}}{c^{6} d^{6}}\right )} x + \frac{{\left (105 \, c^{7} d^{12} e^{5} + 13643 \, a c^{6} d^{10} e^{7} + 2962 \, a^{2} c^{5} d^{8} e^{9} - 1938 \, a^{3} c^{4} d^{6} e^{11} + 693 \, a^{4} c^{3} d^{4} e^{13} - 105 \, a^{5} c^{2} d^{2} e^{15}\right )} e^{\left (-6\right )}}{c^{6} d^{6}}\right )} x - \frac{{\left (315 \, c^{7} d^{13} e^{4} - 2100 \, a c^{6} d^{11} e^{6} - 8393 \, a^{2} c^{5} d^{9} e^{8} + 9216 \, a^{3} c^{4} d^{7} e^{10} - 5943 \, a^{4} c^{3} d^{5} e^{12} + 2100 \, a^{5} c^{2} d^{3} e^{14} - 315 \, a^{6} c d e^{16}\right )} e^{\left (-6\right )}}{c^{6} d^{6}}\right )} - \frac{9 \,{\left (c^{7} d^{14} - 7 \, a c^{6} d^{12} e^{2} + 21 \, a^{2} c^{5} d^{10} e^{4} - 35 \, a^{3} c^{4} d^{8} e^{6} + 35 \, a^{4} c^{3} d^{6} e^{8} - 21 \, a^{5} c^{2} d^{4} e^{10} + 7 \, a^{6} c d^{2} e^{12} - a^{7} e^{14}\right )} \sqrt{c d} e^{\left (-\frac{5}{2}\right )} \log \left ({\left | -\sqrt{c d} c d^{2} e^{\frac{1}{2}} - 2 \,{\left (\sqrt{c d} x e^{\frac{1}{2}} - \sqrt{c d x^{2} e + c d^{2} x + a x e^{2} + a d e}\right )} c d e - \sqrt{c d} a e^{\frac{5}{2}} \right |}\right )}{2048 \, c^{6} d^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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