Optimal. Leaf size=341 \[ -\frac{7 \left (c d^2-a e^2\right )^4 \left (a e^2+c d^2+2 c d e x\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{512 c^4 d^4 e^2}+\frac{7 \left (c d^2-a e^2\right )^2 \left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{192 c^3 d^3 e}+\frac{7 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{60 c^2 d^2}+\frac{7 \left (c d^2-a e^2\right )^6 \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{1024 c^{9/2} d^{9/2} e^{5/2}}+\frac{(d+e x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{6 c d} \]
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Rubi [A] time = 0.263849, antiderivative size = 341, normalized size of antiderivative = 1., number of steps used = 6, 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} \[ -\frac{7 \left (c d^2-a e^2\right )^4 \left (a e^2+c d^2+2 c d e x\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{512 c^4 d^4 e^2}+\frac{7 \left (c d^2-a e^2\right )^2 \left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{192 c^3 d^3 e}+\frac{7 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{60 c^2 d^2}+\frac{7 \left (c d^2-a e^2\right )^6 \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{1024 c^{9/2} d^{9/2} e^{5/2}}+\frac{(d+e x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{6 c d} \]
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)^2 \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) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{6 c d}+\frac{\left (7 \left (d^2-\frac{a e^2}{c}\right )\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}{12 d}\\ &=\frac{7 \left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{60 c^2 d^2}+\frac{(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{6 c d}+\frac{\left (7 \left (d^2-\frac{a e^2}{c}\right )^2\right ) \int \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2} \, dx}{24 d^2}\\ &=\frac{7 \left (c d^2-a e^2\right )^2 \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}}{192 c^3 d^3 e}+\frac{7 \left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{60 c^2 d^2}+\frac{(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{6 c d}-\frac{\left (7 \left (c d^2-a e^2\right )^4\right ) \int \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx}{128 c^3 d^3 e}\\ &=-\frac{7 \left (c d^2-a e^2\right )^4 \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}}{512 c^4 d^4 e^2}+\frac{7 \left (c d^2-a e^2\right )^2 \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}}{192 c^3 d^3 e}+\frac{7 \left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{60 c^2 d^2}+\frac{(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{6 c d}+\frac{\left (7 \left (c d^2-a e^2\right )^6\right ) \int \frac{1}{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{1024 c^4 d^4 e^2}\\ &=-\frac{7 \left (c d^2-a e^2\right )^4 \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}}{512 c^4 d^4 e^2}+\frac{7 \left (c d^2-a e^2\right )^2 \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}}{192 c^3 d^3 e}+\frac{7 \left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{60 c^2 d^2}+\frac{(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{6 c d}+\frac{\left (7 \left (c d^2-a e^2\right )^6\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 )}{512 c^4 d^4 e^2}\\ &=-\frac{7 \left (c d^2-a e^2\right )^4 \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}}{512 c^4 d^4 e^2}+\frac{7 \left (c d^2-a e^2\right )^2 \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}}{192 c^3 d^3 e}+\frac{7 \left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{60 c^2 d^2}+\frac{(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{6 c d}+\frac{7 \left (c d^2-a e^2\right )^6 \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 )}{1024 c^{9/2} d^{9/2} e^{5/2}}\\ \end{align*}
Mathematica [A] time = 3.22449, size = 328, normalized size = 0.96 \[ \frac{(a e+c d x)^2 \sqrt{(d+e x) (a e+c d x)} \left (896 c^5 d^5 (d+e x)^2 \left (c d^2-a e^2\right )+560 c^4 d^4 (d+e x) \left (c d^2-a e^2\right )^2+\frac{70 c^3 d^3 \left (c d^2-a e^2\right )^4}{e (a e+c d x)}-\frac{105 c^3 d^3 \left (c d^2-a e^2\right )^5}{e^2 (a e+c d x)^2}+\frac{105 c^{5/2} d^{5/2} \sqrt{c d} \left (c d^2-a e^2\right )^{11/2} \sinh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a e+c d x}}{\sqrt{c d} \sqrt{c d^2-a e^2}}\right )}{e^{5/2} (a e+c d x)^{5/2} \sqrt{\frac{c d (d+e x)}{c d^2-a e^2}}}+280 \left (c^2 d^3-a c d e^2\right )^3+1280 c^6 d^6 (d+e x)^3\right )}{7680 c^7 d^7} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.053, size = 1302, normalized size = 3.8 \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 = 2.18231, size = 2271, normalized size = 6.66 \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 = 1.26031, size = 671, normalized size = 1.97 \begin{align*} \frac{1}{7680} \, \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 \, c d x e^{3} + \frac{{\left (37 \, c^{6} d^{7} e^{7} + 13 \, a c^{5} d^{5} e^{9}\right )} e^{\left (-5\right )}}{c^{5} d^{5}}\right )} x + \frac{{\left (387 \, c^{6} d^{8} e^{6} + 410 \, a c^{5} d^{6} e^{8} + 3 \, a^{2} c^{4} d^{4} e^{10}\right )} e^{\left (-5\right )}}{c^{5} d^{5}}\right )} x + \frac{{\left (377 \, c^{6} d^{9} e^{5} + 1191 \, a c^{5} d^{7} e^{7} + 39 \, a^{2} c^{4} d^{5} e^{9} - 7 \, a^{3} c^{3} d^{3} e^{11}\right )} e^{\left (-5\right )}}{c^{5} d^{5}}\right )} x + \frac{{\left (35 \, c^{6} d^{10} e^{4} + 2876 \, a c^{5} d^{8} e^{6} + 450 \, a^{2} c^{4} d^{6} e^{8} - 196 \, a^{3} c^{3} d^{4} e^{10} + 35 \, a^{4} c^{2} d^{2} e^{12}\right )} e^{\left (-5\right )}}{c^{5} d^{5}}\right )} x - \frac{{\left (105 \, c^{6} d^{11} e^{3} - 595 \, a c^{5} d^{9} e^{5} - 1686 \, a^{2} c^{4} d^{7} e^{7} + 1386 \, a^{3} c^{3} d^{5} e^{9} - 595 \, a^{4} c^{2} d^{3} e^{11} + 105 \, a^{5} c d e^{13}\right )} e^{\left (-5\right )}}{c^{5} d^{5}}\right )} - \frac{7 \,{\left (c^{6} d^{12} - 6 \, a c^{5} d^{10} e^{2} + 15 \, a^{2} c^{4} d^{8} e^{4} - 20 \, a^{3} c^{3} d^{6} e^{6} + 15 \, a^{4} c^{2} d^{4} e^{8} - 6 \, a^{5} c d^{2} e^{10} + a^{6} e^{12}\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 )}{1024 \, c^{5} d^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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