3.2.0 ∫ −32x7−12x8+16x9+e2(144x3+576x5+576x7)+e(240x4+72x5+192x6+96x7−192x8) x6+e2(9+36x2+36x4)+e(−6x3−12x5) dx [182]

Integrand size = 100, antiderivative size = 26 \[ \int \frac {-32 x^7-12 x^8+16 x^9+e^2 \left (144 x^3+576 x^5+576 x^7\right )+e \left (240 x^4+72 x^5+192 x^6+96 x^7-192 x^8\right )}{x^6+e^2 \left (9+36 x^2+36 x^4\right )+e \left (-6 x^3-12 x^5\right )} \, dx=4 x^3 \left (x-\frac {4+x}{-e \left (6+\frac {3}{x^2}\right )+x}\right ) \]

Rubi [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 3 vs. order 1 in optimal.

Time = 116.90 (sec) , antiderivative size = 6830, normalized size of antiderivative = 262.69, number of steps used = 20, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2099, 2126, 2106, 2104, 836, 814, 648, 632, 210, 642} \[ \int \frac {-32 x^7-12 x^8+16 x^9+e^2 \left (144 x^3+576 x^5+576 x^7\right )+e \left (240 x^4+72 x^5+192 x^6+96 x^7-192 x^8\right )}{x^6+e^2 \left (9+36 x^2+36 x^4\right )+e \left (-6 x^3-12 x^5\right )} \, dx =\text {Too large to display} \]

Rubi steps \begin{align*} \text {integral}& = \int \left (-48 e (2+3 e)-16 (2+3 e) x-12 x^2+16 x^3+\frac {108 e^2 \left (4 e \left (8+15 e+144 e^2+216 e^3\right )+4 \left (1+2 e+24 e^2+36 e^3\right ) x+\left (1+80 e+144 e^2+1152 e^3+1728 e^4\right ) x^2\right )}{\left (3 e+6 e x^2-x^3\right )^2}+\frac {48 e \left (-18 e \left (1+2 e+24 e^2+36 e^3\right )-\left (1+3 e+72 e^2+108 e^3\right ) x\right )}{3 e+6 e x^2-x^3}\right ) \, dx \\ & = -48 e (2+3 e) x-8 (2+3 e) x^2-4 x^3+4 x^4+(48 e) \int \frac {-18 e \left (1+2 e+24 e^2+36 e^3\right )-\left (1+3 e+72 e^2+108 e^3\right ) x}{3 e+6 e x^2-x^3} \, dx+\left (108 e^2\right ) \int \frac {4 e \left (8+15 e+144 e^2+216 e^3\right )+4 \left (1+2 e+24 e^2+36 e^3\right ) x+\left (1+80 e+144 e^2+1152 e^3+1728 e^4\right ) x^2}{\left (3 e+6 e x^2-x^3\right )^2} \, dx \\ & = -48 e (2+3 e) x-8 (2+3 e) x^2-4 x^3+4 x^4+\frac {36 e^2 \left (1+80 e+144 e^2+1152 e^3+1728 e^4\right )}{3 e+6 e x^2-x^3}+(48 e) \text {Subst}\left (\int \frac {\frac {1}{3} \left (6 e \left (-1-3 e-72 e^2-108 e^3\right )-54 e \left (1+2 e+24 e^2+36 e^3\right )\right )+\left (-1-3 e-72 e^2-108 e^3\right ) x}{e \left (3+16 e^2\right )+12 e^2 x-x^3} \, dx,x,-2 e+x\right )-\left (36 e^2\right ) \int \frac {-12 e \left (8+15 e+144 e^2+216 e^3\right )-12 \left (1+3 e+104 e^2+180 e^3+1152 e^4+1728 e^5\right ) x}{\left (3 e+6 e x^2-x^3\right )^2} \, dx \\ & = -48 e (2+3 e) x-8 (2+3 e) x^2-4 x^3+4 x^4+\frac {36 e^2 \left (1+80 e+144 e^2+1152 e^3+1728 e^4\right )}{3 e+6 e x^2-x^3}+(48 e) \text {Subst}\left (\int \frac {\frac {1}{3} \left (6 e \left (-1-3 e-72 e^2-108 e^3\right )-54 e \left (1+2 e+24 e^2+36 e^3\right )\right )+\left (-1-3 e-72 e^2-108 e^3\right ) x}{\left (4 e^{5/3} \sqrt [3]{\frac {2}{3+16 e^2+\sqrt {9+96 e^2}}}+\sqrt [3]{\frac {1}{2} e \left (3+16 e^2+\sqrt {9+96 e^2}\right )}-x\right ) \left (-4 e^2+16 e^{10/3} \left (\frac {2}{3+16 e^2+\sqrt {9+96 e^2}}\right )^{2/3}+\left (\frac {1}{2} e \left (3+16 e^2+\sqrt {9+96 e^2}\right )\right )^{2/3}+\frac {\left (\frac {8 e^{5/3}}{\sqrt [3]{3+16 e^2+\sqrt {9+96 e^2}}}+\sqrt [3]{2 e \left (3+16 e^2+\sqrt {9+96 e^2}\right )}\right ) x}{2^{2/3}}+x^2\right )} \, dx,x,-2 e+x\right )-\left (36 e^2\right ) \text {Subst}\left (\int \frac {\frac {1}{3} \left (-36 e \left (8+15 e+144 e^2+216 e^3\right )-72 e \left (1+3 e+104 e^2+180 e^3+1152 e^4+1728 e^5\right )\right )-12 \left (1+3 e+104 e^2+180 e^3+1152 e^4+1728 e^5\right ) x}{\left (e \left (3+16 e^2\right )+12 e^2 x-x^3\right )^2} \, dx,x,-2 e+x\right ) \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [B] (veriﬁed)

Leaf count is larger than twice the leaf count of optimal. \(89\) vs. \(2(26)=52\).

Time = 0.06 (sec) , antiderivative size = 89, normalized size of antiderivative = 3.42 \[ \int \frac {-32 x^7-12 x^8+16 x^9+e^2 \left (144 x^3+576 x^5+576 x^7\right )+e \left (240 x^4+72 x^5+192 x^6+96 x^7-192 x^8\right )}{x^6+e^2 \left (9+36 x^2+36 x^4\right )+e \left (-6 x^3-12 x^5\right )} \, dx=\frac {4 \left (x^5 \left (4+x-x^2\right )+648 e^4 \left (1+2 x^2\right )+e^2 \left (9+18 x^2-144 x^3\right )-216 e^3 \left (-2-4 x^2+x^3\right )+3 e x^3 \left (-1+x+2 x^3\right )\right )}{-x^3+e \left (3+6 x^2\right )} \]

Maple [A] (veriﬁed)

Time = 0.19 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.85


method	result	size

gosper	\(\frac {4 x^{4} \left (6 x^{2} {\mathrm e}-x^{3}+x^{2}+3 \,{\mathrm e}+4 x \right )}{6 x^{2} {\mathrm e}-x^{3}+3 \,{\mathrm e}}\)	\(48\)
norman	\(\frac {\left (4+24 \,{\mathrm e}\right ) x^{6}+16 x^{5}-4 x^{7}+12 x^{4} {\mathrm e}}{6 x^{2} {\mathrm e}-x^{3}+3 \,{\mathrm e}}\)	\(49\)
parallelrisch	\(\frac {24 x^{6} {\mathrm e}-4 x^{7}+4 x^{6}+12 x^{4} {\mathrm e}+16 x^{5}}{6 x^{2} {\mathrm e}-x^{3}+3 \,{\mathrm e}}\)	\(51\)
risch	\(-\frac {4 \left (-6 x^{6} {\mathrm e}+x^{7}-x^{6}+216 x^{3} {\mathrm e}^{3}+144 x^{3} {\mathrm e}^{2}-3 x^{4} {\mathrm e}-4 x^{5}-18 x^{2} {\mathrm e}^{2}+3 x^{3} {\mathrm e}-1296 x^{2} {\mathrm e}^{4}-864 x^{2} {\mathrm e}^{3}-9 \,{\mathrm e}^{2}-648 \,{\mathrm e}^{4}-432 \,{\mathrm e}^{3}\right )}{6 x^{2} {\mathrm e}-x^{3}+3 \,{\mathrm e}}\)	\(104\)
default	\(4 x^{4}-144 \,{\mathrm e}^{2} x -24 x^{2} {\mathrm e}-4 x^{3}-96 x \,{\mathrm e}-16 x^{2}+2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (36 \textit {\_Z}^{4} {\mathrm e}^{2}-12 \textit {\_Z}^{5} {\mathrm e}+\textit {\_Z}^{6}+36 \textit {\_Z}^{2} {\mathrm e}^{2}-6 \textit {\_Z}^{3} {\mathrm e}+9 \,{\mathrm e}^{2}\right )}{\sum }\frac {\left (4 \left (108 \,{\mathrm e}^{4}+72 \,{\mathrm e}^{3}+3 \,{\mathrm e}^{2}+{\mathrm e}\right ) \textit {\_R}^{4}+24 \left (3 \,{\mathrm e}^{3}+2 \,{\mathrm e}^{2}\right ) \textit {\_R}^{3}+9 \left (48 \,{\mathrm e}^{4}+32 \,{\mathrm e}^{3}+{\mathrm e}^{2}\right ) \textit {\_R}^{2}+12 \,{\mathrm e}^{2} \left (3 \,{\mathrm e}+2\right ) \textit {\_R} +108 \,{\mathrm e}^{4}+72 \,{\mathrm e} \,{\mathrm e}^{2}\right ) \ln \left (x -\textit {\_R} \right )}{24 \textit {\_R}^{3} {\mathrm e}^{2}-10 \textit {\_R}^{4} {\mathrm e}+\textit {\_R}^{5}+12 \,{\mathrm e}^{2} \textit {\_R} -3 \textit {\_R}^{2} {\mathrm e}}\right )\)	\(196\)

Fricas [B] (veriﬁcation not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 92 vs. \(2 (26) = 52\).

Time = 0.26 (sec) , antiderivative size = 92, normalized size of antiderivative = 3.54 \[ \int \frac {-32 x^7-12 x^8+16 x^9+e^2 \left (144 x^3+576 x^5+576 x^7\right )+e \left (240 x^4+72 x^5+192 x^6+96 x^7-192 x^8\right )}{x^6+e^2 \left (9+36 x^2+36 x^4\right )+e \left (-6 x^3-12 x^5\right )} \, dx=\frac {4 \, {\left (x^{7} - x^{6} - 4 \, x^{5} - 648 \, {\left (2 \, x^{2} + 1\right )} e^{4} + 216 \, {\left (x^{3} - 4 \, x^{2} - 2\right )} e^{3} + 9 \, {\left (16 \, x^{3} - 2 \, x^{2} - 1\right )} e^{2} - 3 \, {\left (2 \, x^{6} + x^{4} - x^{3}\right )} e\right )}}{x^{3} - 3 \, {\left (2 \, x^{2} + 1\right )} e} \]

Sympy [B] (veriﬁcation not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 105 vs. \(2 (22) = 44\).

Time = 1.31 (sec) , antiderivative size = 105, normalized size of antiderivative = 4.04 \[ \int \frac {-32 x^7-12 x^8+16 x^9+e^2 \left (144 x^3+576 x^5+576 x^7\right )+e \left (240 x^4+72 x^5+192 x^6+96 x^7-192 x^8\right )}{x^6+e^2 \left (9+36 x^2+36 x^4\right )+e \left (-6 x^3-12 x^5\right )} \, dx=4 x^{4} - 4 x^{3} + x^{2} \left (- 24 e - 16\right ) + x \left (- 144 e^{2} - 96 e\right ) + \frac {x^{2} \left (- 5184 e^{4} - 3456 e^{3} - 144 e^{2} - 48 e\right ) + x \left (- 432 e^{3} - 288 e^{2}\right ) - 2592 e^{4} - 1728 e^{3} - 36 e^{2}}{x^{3} - 6 e x^{2} - 3 e} \]

Maxima [B] (veriﬁcation not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 98 vs. \(2 (26) = 52\).

Time = 0.18 (sec) , antiderivative size = 98, normalized size of antiderivative = 3.77 \[ \int \frac {-32 x^7-12 x^8+16 x^9+e^2 \left (144 x^3+576 x^5+576 x^7\right )+e \left (240 x^4+72 x^5+192 x^6+96 x^7-192 x^8\right )}{x^6+e^2 \left (9+36 x^2+36 x^4\right )+e \left (-6 x^3-12 x^5\right )} \, dx=4 \, x^{4} - 4 \, x^{3} - 8 \, x^{2} {\left (3 \, e + 2\right )} - 48 \, x {\left (3 \, e^{2} + 2 \, e\right )} - \frac {12 \, {\left (4 \, x^{2} {\left (108 \, e^{4} + 72 \, e^{3} + 3 \, e^{2} + e\right )} + 12 \, x {\left (3 \, e^{3} + 2 \, e^{2}\right )} + 216 \, e^{4} + 144 \, e^{3} + 3 \, e^{2}\right )}}{x^{3} - 6 \, x^{2} e - 3 \, e} \]

Giac [B] (veriﬁcation not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 103 vs. \(2 (26) = 52\).

Time = 0.26 (sec) , antiderivative size = 103, normalized size of antiderivative = 3.96 \[ \int \frac {-32 x^7-12 x^8+16 x^9+e^2 \left (144 x^3+576 x^5+576 x^7\right )+e \left (240 x^4+72 x^5+192 x^6+96 x^7-192 x^8\right )}{x^6+e^2 \left (9+36 x^2+36 x^4\right )+e \left (-6 x^3-12 x^5\right )} \, dx=4 \, x^{4} - 4 \, x^{3} - 24 \, x^{2} e - 16 \, x^{2} - 144 \, x e^{2} - 96 \, x e - \frac {12 \, {\left (432 \, x^{2} e^{4} + 288 \, x^{2} e^{3} + 12 \, x^{2} e^{2} + 4 \, x^{2} e + 36 \, x e^{3} + 24 \, x e^{2} + 216 \, e^{4} + 144 \, e^{3} + 3 \, e^{2}\right )}}{x^{3} - 6 \, x^{2} e - 3 \, e} \]

Mupad [B] (veriﬁcation not implemented)

Time = 0.23 (sec) , antiderivative size = 108, normalized size of antiderivative = 4.15 \[ \int \frac {-32 x^7-12 x^8+16 x^9+e^2 \left (144 x^3+576 x^5+576 x^7\right )+e \left (240 x^4+72 x^5+192 x^6+96 x^7-192 x^8\right )}{x^6+e^2 \left (9+36 x^2+36 x^4\right )+e \left (-6 x^3-12 x^5\right )} \, dx=x\,\left (288\,\mathrm {e}+432\,{\mathrm {e}}^2-12\,\mathrm {e}\,\left (48\,\mathrm {e}+32\right )\right )-x^2\,\left (24\,\mathrm {e}+16\right )+\frac {\left (48\,\mathrm {e}+144\,{\mathrm {e}}^2+3456\,{\mathrm {e}}^3+5184\,{\mathrm {e}}^4\right )\,x^2+\left (288\,{\mathrm {e}}^2+432\,{\mathrm {e}}^3\right )\,x+36\,{\mathrm {e}}^2+1728\,{\mathrm {e}}^3+2592\,{\mathrm {e}}^4}{-x^3+6\,\mathrm {e}\,x^2+3\,\mathrm {e}}-4\,x^3+4\,x^4 \]

\(\int \frac {-32 x^7-12 x^8+16 x^9+e^2 (144 x^3+576 x^5+576 x^7)+e (240 x^4+72 x^5+192 x^6+96 x^7-192 x^8)}{x^6+e^2 (9+36 x^2+36 x^4)+e (-6 x^3-12 x^5)} \, dx\) [182]

Optimal result