Optimal. Leaf size=16 \[ 16 e^8 x^3 (16+\log (e+x))^2 \]
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Rubi [B] time = 0.56, antiderivative size = 169, normalized size of antiderivative = 10.56, 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 = {6688, 12, 6742, 77, 2418, 2389, 2295, 2395, 43, 2390, 2301, 2398, 2411, 2334} \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)^2-32 e^{11} \log ^2(x+e)+32 e^{10} (x+e) \log (x+e)+\frac {80}{3} e^{11} \log (x+e)-\frac {16}{3} e^8 \log (x+e) \left (2 (x+e)^3-9 e (x+e)^2+18 e^2 (x+e)-6 e^3 \log (x+e)\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 43
Rule 77
Rule 2295
Rule 2301
Rule 2334
Rule 2389
Rule 2390
Rule 2395
Rule 2398
Rule 2411
Rule 2418
Rule 6688
Rule 6742
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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {Subst}(\int \log (x) \, dx,x,e+x)-\left (32 e^{11}\right ) \operatorname {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 ) \operatorname {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 [A] time = 0.10, size = 30, normalized size = 1.88 \begin {gather*} 16 e^8 \left (256 x^3+32 x^3 \log (e+x)+x^3 \log ^2(e+x)\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.67, 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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giac [B] time = 0.17, 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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maple [A] time = 0.09, size = 35, normalized size = 2.19
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] time = 0.36, 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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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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sympy [B] time = 0.18, 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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