Optimal. Leaf size=16 \[ 16 e^8 x^3 (16+\log (e+x))^2 \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(144\) vs. \(2(16)=32\).
time = 0.54, antiderivative size = 144, normalized size of antiderivative = 9.00, number
of steps used = 23, number of rules used = 14, integrand size = 65, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.215, Rules
used = {6820, 12, 6874, 78, 2465, 2436, 2332, 2442, 45, 2437, 2338, 2445, 2458, 2372}
\begin {gather*} \frac {36832 e^8 x^3}{9}+16 e^8 x^3 \log ^2(x+e)+\frac {1568}{3} e^8 x^3 \log (x+e)+\frac {40 e^9 x^2}{3}-16 e^9 x^2 \log (x+e)+\frac {112 e^{10} x}{3}+\frac {32}{9} e^8 (x+e)^3-24 e^9 (x+e)^2-\frac {32}{3} e^8 (x+e)^3 \log (x+e)+48 e^9 (x+e)^2 \log (x+e)-64 e^{10} (x+e) \log (x+e)+\frac {80}{3} e^{11} \log (x+e) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 45
Rule 78
Rule 2332
Rule 2338
Rule 2372
Rule 2436
Rule 2437
Rule 2442
Rule 2445
Rule 2458
Rule 2465
Rule 6820
Rule 6874
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {16 e^8 x^2 (16+\log (e+x)) (48 e+50 x+3 (e+x) \log (e+x))}{e+x} \, dx\\ &=\left (16 e^8\right ) \int \frac {x^2 (16+\log (e+x)) (48 e+50 x+3 (e+x) \log (e+x))}{e+x} \, dx\\ &=\left (16 e^8\right ) \int \left (\frac {32 x^2 (24 e+25 x)}{e+x}+\frac {2 x^2 (48 e+49 x) \log (e+x)}{e+x}+3 x^2 \log ^2(e+x)\right ) \, dx\\ &=\left (32 e^8\right ) \int \frac {x^2 (48 e+49 x) \log (e+x)}{e+x} \, dx+\left (48 e^8\right ) \int x^2 \log ^2(e+x) \, dx+\left (512 e^8\right ) \int \frac {x^2 (24 e+25 x)}{e+x} \, dx\\ &=16 e^8 x^3 \log ^2(e+x)-\left (32 e^8\right ) \int \frac {x^3 \log (e+x)}{e+x} \, dx+\left (32 e^8\right ) \int \left (e^2 \log (e+x)-e x \log (e+x)+49 x^2 \log (e+x)-\frac {e^3 \log (e+x)}{e+x}\right ) \, dx+\left (512 e^8\right ) \int \left (e^2-e x+25 x^2-\frac {e^3}{e+x}\right ) \, dx\\ &=512 e^{10} x-256 e^9 x^2+\frac {12800 e^8 x^3}{3}-512 e^{11} \log (e+x)+16 e^8 x^3 \log ^2(e+x)-\left (32 e^8\right ) \text {Subst}\left (\int \frac {(-e+x)^3 \log (x)}{x} \, dx,x,e+x\right )+\left (1568 e^8\right ) \int x^2 \log (e+x) \, dx-\left (32 e^9\right ) \int x \log (e+x) \, dx+\left (32 e^{10}\right ) \int \log (e+x) \, dx-\left (32 e^{11}\right ) \int \frac {\log (e+x)}{e+x} \, dx\\ &=512 e^{10} x-256 e^9 x^2+\frac {12800 e^8 x^3}{3}-512 e^{11} \log (e+x)-16 e^9 x^2 \log (e+x)+\frac {1568}{3} e^8 x^3 \log (e+x)+16 e^8 x^3 \log ^2(e+x)-\frac {16}{3} e^8 \log (e+x) \left (18 e^2 (e+x)-9 e (e+x)^2+2 (e+x)^3-6 e^3 \log (e+x)\right )+\left (32 e^8\right ) \text {Subst}\left (\int \left (3 e^2-\frac {3 e x}{2}+\frac {x^2}{3}-\frac {e^3 \log (x)}{x}\right ) \, dx,x,e+x\right )-\frac {1}{3} \left (1568 e^8\right ) \int \frac {x^3}{e+x} \, dx+\left (16 e^9\right ) \int \frac {x^2}{e+x} \, dx+\left (32 e^{10}\right ) \text {Subst}(\int \log (x) \, dx,x,e+x)-\left (32 e^{11}\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,e+x\right )\\ &=576 e^{10} x-256 e^9 x^2+\frac {12800 e^8 x^3}{3}-24 e^9 (e+x)^2+\frac {32}{9} e^8 (e+x)^3-512 e^{11} \log (e+x)-16 e^9 x^2 \log (e+x)+\frac {1568}{3} e^8 x^3 \log (e+x)+32 e^{10} (e+x) \log (e+x)-16 e^{11} \log ^2(e+x)+16 e^8 x^3 \log ^2(e+x)-\frac {16}{3} e^8 \log (e+x) \left (18 e^2 (e+x)-9 e (e+x)^2+2 (e+x)^3-6 e^3 \log (e+x)\right )-\frac {1}{3} \left (1568 e^8\right ) \int \left (e^2-e x+x^2-\frac {e^3}{e+x}\right ) \, dx+\left (16 e^9\right ) \int \left (-e+x+\frac {e^2}{e+x}\right ) \, dx-\left (32 e^{11}\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,e+x\right )\\ &=\frac {112 e^{10} x}{3}+\frac {40 e^9 x^2}{3}+\frac {36832 e^8 x^3}{9}-24 e^9 (e+x)^2+\frac {32}{9} e^8 (e+x)^3+\frac {80}{3} e^{11} \log (e+x)-16 e^9 x^2 \log (e+x)+\frac {1568}{3} e^8 x^3 \log (e+x)+32 e^{10} (e+x) \log (e+x)-32 e^{11} \log ^2(e+x)+16 e^8 x^3 \log ^2(e+x)-\frac {16}{3} e^8 \log (e+x) \left (18 e^2 (e+x)-9 e (e+x)^2+2 (e+x)^3-6 e^3 \log (e+x)\right )\\ \end {aligned} \end {gather*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(34\) vs. \(2(16)=32\).
time = 0.05, size = 34, normalized size = 2.12 \begin {gather*} 16 e^8 \left (256 \left (e^3+x^3\right )+32 x^3 \log (e+x)+x^3 \log ^2(e+x)\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(288\) vs.
\(2(18)=36\).
time = 0.07, size = 289, normalized size = 18.06
method | result | size |
risch | \(16 x^{3} {\mathrm e}^{8} \ln \left (x +{\mathrm e}\right )^{2}+512 x^{3} {\mathrm e}^{8} \ln \left (x +{\mathrm e}\right )+4096 x^{3} {\mathrm e}^{8}\) | \(35\) |
norman | \(16 x^{3} {\mathrm e}^{8} \ln \left (x +{\mathrm e}\right )^{2}+512 x^{3} {\mathrm e}^{8} \ln \left (x +{\mathrm e}\right )+4096 x^{3} {\mathrm e}^{8}\) | \(41\) |
derivativedivides | \(48 \,{\mathrm e}^{8} {\mathrm e}^{2} \left (\left (x +{\mathrm e}\right ) \ln \left (x +{\mathrm e}\right )^{2}-2 \left (x +{\mathrm e}\right ) \ln \left (x +{\mathrm e}\right )+2 x +2 \,{\mathrm e}\right )-96 \,{\mathrm e}^{8} {\mathrm e} \left (\frac {\left (x +{\mathrm e}\right )^{2} \ln \left (x +{\mathrm e}\right )^{2}}{2}-\frac {\left (x +{\mathrm e}\right )^{2} \ln \left (x +{\mathrm e}\right )}{2}+\frac {\left (x +{\mathrm e}\right )^{2}}{4}\right )+48 \,{\mathrm e}^{8} \left (\frac {\ln \left (x +{\mathrm e}\right )^{2} \left (x +{\mathrm e}\right )^{3}}{3}-\frac {2 \ln \left (x +{\mathrm e}\right ) \left (x +{\mathrm e}\right )^{3}}{9}+\frac {2 \left (x +{\mathrm e}\right )^{3}}{27}\right )-16 \,{\mathrm e}^{8} {\mathrm e}^{3} \ln \left (x +{\mathrm e}\right )^{2}+1632 \,{\mathrm e}^{8} {\mathrm e}^{2} \left (\left (x +{\mathrm e}\right ) \ln \left (x +{\mathrm e}\right )-x -{\mathrm e}\right )-3168 \,{\mathrm e}^{8} {\mathrm e} \left (\frac {\left (x +{\mathrm e}\right )^{2} \ln \left (x +{\mathrm e}\right )}{2}-\frac {\left (x +{\mathrm e}\right )^{2}}{4}\right )+1568 \,{\mathrm e}^{8} \left (\frac {\ln \left (x +{\mathrm e}\right ) \left (x +{\mathrm e}\right )^{3}}{3}-\frac {\left (x +{\mathrm e}\right )^{3}}{9}\right )-512 \,{\mathrm e}^{8} {\mathrm e}^{3} \ln \left (x +{\mathrm e}\right )+13824 \,{\mathrm e}^{8} {\mathrm e}^{2} \left (x +{\mathrm e}\right )-13056 \,{\mathrm e}^{8} {\mathrm e} \left (x +{\mathrm e}\right )^{2}+\frac {12800 \,{\mathrm e}^{8} \left (x +{\mathrm e}\right )^{3}}{3}\) | \(289\) |
default | \(48 \,{\mathrm e}^{8} {\mathrm e}^{2} \left (\left (x +{\mathrm e}\right ) \ln \left (x +{\mathrm e}\right )^{2}-2 \left (x +{\mathrm e}\right ) \ln \left (x +{\mathrm e}\right )+2 x +2 \,{\mathrm e}\right )-96 \,{\mathrm e}^{8} {\mathrm e} \left (\frac {\left (x +{\mathrm e}\right )^{2} \ln \left (x +{\mathrm e}\right )^{2}}{2}-\frac {\left (x +{\mathrm e}\right )^{2} \ln \left (x +{\mathrm e}\right )}{2}+\frac {\left (x +{\mathrm e}\right )^{2}}{4}\right )+48 \,{\mathrm e}^{8} \left (\frac {\ln \left (x +{\mathrm e}\right )^{2} \left (x +{\mathrm e}\right )^{3}}{3}-\frac {2 \ln \left (x +{\mathrm e}\right ) \left (x +{\mathrm e}\right )^{3}}{9}+\frac {2 \left (x +{\mathrm e}\right )^{3}}{27}\right )-16 \,{\mathrm e}^{8} {\mathrm e}^{3} \ln \left (x +{\mathrm e}\right )^{2}+1632 \,{\mathrm e}^{8} {\mathrm e}^{2} \left (\left (x +{\mathrm e}\right ) \ln \left (x +{\mathrm e}\right )-x -{\mathrm e}\right )-3168 \,{\mathrm e}^{8} {\mathrm e} \left (\frac {\left (x +{\mathrm e}\right )^{2} \ln \left (x +{\mathrm e}\right )}{2}-\frac {\left (x +{\mathrm e}\right )^{2}}{4}\right )+1568 \,{\mathrm e}^{8} \left (\frac {\ln \left (x +{\mathrm e}\right ) \left (x +{\mathrm e}\right )^{3}}{3}-\frac {\left (x +{\mathrm e}\right )^{3}}{9}\right )-512 \,{\mathrm e}^{8} {\mathrm e}^{3} \ln \left (x +{\mathrm e}\right )+13824 \,{\mathrm e}^{8} {\mathrm e}^{2} \left (x +{\mathrm e}\right )-13056 \,{\mathrm e}^{8} {\mathrm e} \left (x +{\mathrm e}\right )^{2}+\frac {12800 \,{\mathrm e}^{8} \left (x +{\mathrm e}\right )^{3}}{3}\) | \(289\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 367 vs.
\(2 (16) = 32\).
time = 0.26, size = 367, normalized size = 22.94 \begin {gather*} 768 \, {\left (x^{2} - 2 \, x e + 2 \, e^{2} \log \left (x + e\right )\right )} e^{9} \log \left (x + e\right ) + \frac {784}{3} \, {\left (2 \, x^{3} - 3 \, x^{2} e + 6 \, x e^{2} - 6 \, e^{3} \log \left (x + e\right )\right )} e^{8} \log \left (x + e\right ) + 4 \, {\left (4 \, e^{2} \log \left (x + e\right )^{3} + 3 \, {\left (2 \, \log \left (x + e\right )^{2} - 2 \, \log \left (x + e\right ) + 1\right )} {\left (x + e\right )}^{2} - 24 \, {\left (e \log \left (x + e\right )^{2} - 2 \, e \log \left (x + e\right ) + 2 \, e\right )} {\left (x + e\right )}\right )} e^{9} - 384 \, {\left (2 \, e^{2} \log \left (x + e\right )^{2} + x^{2} - 6 \, x e + 6 \, e^{2} \log \left (x + e\right )\right )} e^{9} + 6144 \, {\left (x^{2} - 2 \, x e + 2 \, e^{2} \log \left (x + e\right )\right )} e^{9} + \frac {4}{9} \, {\left (4 \, {\left (9 \, \log \left (x + e\right )^{2} - 6 \, \log \left (x + e\right ) + 2\right )} {\left (x + e\right )}^{3} - 36 \, e^{3} \log \left (x + e\right )^{3} - 81 \, {\left (2 \, e \log \left (x + e\right )^{2} - 2 \, e \log \left (x + e\right ) + e\right )} {\left (x + e\right )}^{2} + 324 \, {\left (e^{2} \log \left (x + e\right )^{2} - 2 \, e^{2} \log \left (x + e\right ) + 2 \, e^{2}\right )} {\left (x + e\right )}\right )} e^{8} - \frac {392}{9} \, {\left (4 \, x^{3} - 15 \, x^{2} e - 18 \, e^{3} \log \left (x + e\right )^{2} + 66 \, x e^{2} - 66 \, e^{3} \log \left (x + e\right )\right )} e^{8} + \frac {6400}{3} \, {\left (2 \, x^{3} - 3 \, x^{2} e + 6 \, x e^{2} - 6 \, e^{3} \log \left (x + e\right )\right )} e^{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 34 vs.
\(2 (16) = 32\).
time = 0.38, size = 34, normalized size = 2.12 \begin {gather*} 16 \, x^{3} e^{8} \log \left (x + e\right )^{2} + 512 \, x^{3} e^{8} \log \left (x + e\right ) + 4096 \, x^{3} e^{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 39 vs.
\(2 (17) = 34\).
time = 0.08, size = 39, normalized size = 2.44 \begin {gather*} 16 x^{3} e^{8} \log {\left (x + e \right )}^{2} + 512 x^{3} e^{8} \log {\left (x + e \right )} + 4096 x^{3} e^{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 34 vs.
\(2 (16) = 32\).
time = 0.42, size = 34, normalized size = 2.12 \begin {gather*} 16 \, x^{3} e^{8} \log \left (x + e\right )^{2} + 512 \, x^{3} e^{8} \log \left (x + e\right ) + 4096 \, x^{3} e^{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.74, size = 16, normalized size = 1.00 \begin {gather*} 16\,x^3\,{\mathrm {e}}^8\,{\left (\ln \left (x+\mathrm {e}\right )+16\right )}^2 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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