Optimal. Leaf size=23 \[ \frac {3 \left (e^{14 x/5}+\log (8+x)\right )}{(3+x) \log (x)} \]
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Rubi [F]
time = 3.24, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps
used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}
\begin {gather*} \int \frac {e^{14 x/5} \left (-360-165 x-15 x^2\right )+\left (45 x+15 x^2+e^{14 x/5} \left (888 x+447 x^2+42 x^3\right )\right ) \log (x)+\left (-360-165 x-15 x^2+\left (-120 x-15 x^2\right ) \log (x)\right ) \log (8+x)}{\left (360 x+285 x^2+70 x^3+5 x^4\right ) \log ^2(x)} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{14 x/5} \left (-360-165 x-15 x^2\right )+\left (45 x+15 x^2+e^{14 x/5} \left (888 x+447 x^2+42 x^3\right )\right ) \log (x)+\left (-360-165 x-15 x^2+\left (-120 x-15 x^2\right ) \log (x)\right ) \log (8+x)}{x \left (360+285 x+70 x^2+5 x^3\right ) \log ^2(x)} \, dx\\ &=\int \left (\frac {9}{(3+x)^2 (8+x) \log (x)}+\frac {3 x}{(3+x)^2 (8+x) \log (x)}+\frac {3 e^{14 x/5} \left (-15-5 x+37 x \log (x)+14 x^2 \log (x)\right )}{5 x (3+x)^2 \log ^2(x)}-\frac {3 (3+x+x \log (x)) \log (8+x)}{x (3+x)^2 \log ^2(x)}\right ) \, dx\\ &=\frac {3}{5} \int \frac {e^{14 x/5} \left (-15-5 x+37 x \log (x)+14 x^2 \log (x)\right )}{x (3+x)^2 \log ^2(x)} \, dx+3 \int \frac {x}{(3+x)^2 (8+x) \log (x)} \, dx-3 \int \frac {(3+x+x \log (x)) \log (8+x)}{x (3+x)^2 \log ^2(x)} \, dx+9 \int \frac {1}{(3+x)^2 (8+x) \log (x)} \, dx\\ &=\frac {3}{5} \int \left (-\frac {5 e^{14 x/5}}{x (3+x) \log ^2(x)}+\frac {e^{14 x/5} (37+14 x)}{(3+x)^2 \log (x)}\right ) \, dx+3 \int \frac {x}{(3+x)^2 (8+x) \log (x)} \, dx-3 \int \left (\frac {(3+x+x \log (x)) \log (8+x)}{9 x \log ^2(x)}-\frac {(3+x+x \log (x)) \log (8+x)}{3 (3+x)^2 \log ^2(x)}-\frac {(3+x+x \log (x)) \log (8+x)}{9 (3+x) \log ^2(x)}\right ) \, dx+9 \int \frac {1}{(3+x)^2 (8+x) \log (x)} \, dx\\ &=-\left (\frac {1}{3} \int \frac {(3+x+x \log (x)) \log (8+x)}{x \log ^2(x)} \, dx\right )+\frac {1}{3} \int \frac {(3+x+x \log (x)) \log (8+x)}{(3+x) \log ^2(x)} \, dx+\frac {3}{5} \int \frac {e^{14 x/5} (37+14 x)}{(3+x)^2 \log (x)} \, dx-3 \int \frac {e^{14 x/5}}{x (3+x) \log ^2(x)} \, dx+3 \int \frac {x}{(3+x)^2 (8+x) \log (x)} \, dx+9 \int \frac {1}{(3+x)^2 (8+x) \log (x)} \, dx+\int \frac {(3+x+x \log (x)) \log (8+x)}{(3+x)^2 \log ^2(x)} \, dx\\ &=-\left (\frac {1}{3} \int \left (\frac {\log (8+x)}{\log ^2(x)}+\frac {3 \log (8+x)}{x \log ^2(x)}+\frac {\log (8+x)}{\log (x)}\right ) \, dx\right )+\frac {1}{3} \int \left (\frac {3 \log (8+x)}{(3+x) \log ^2(x)}+\frac {x \log (8+x)}{(3+x) \log ^2(x)}+\frac {x \log (8+x)}{(3+x) \log (x)}\right ) \, dx+\frac {3}{5} \int \left (-\frac {5 e^{14 x/5}}{(3+x)^2 \log (x)}+\frac {14 e^{14 x/5}}{(3+x) \log (x)}\right ) \, dx-3 \int \left (\frac {e^{14 x/5}}{3 x \log ^2(x)}-\frac {e^{14 x/5}}{3 (3+x) \log ^2(x)}\right ) \, dx+3 \int \frac {x}{(3+x)^2 (8+x) \log (x)} \, dx+9 \int \frac {1}{(3+x)^2 (8+x) \log (x)} \, dx+\int \left (\frac {3 \log (8+x)}{(3+x)^2 \log ^2(x)}+\frac {x \log (8+x)}{(3+x)^2 \log ^2(x)}+\frac {x \log (8+x)}{(3+x)^2 \log (x)}\right ) \, dx\\ &=-\left (\frac {1}{3} \int \frac {\log (8+x)}{\log ^2(x)} \, dx\right )+\frac {1}{3} \int \frac {x \log (8+x)}{(3+x) \log ^2(x)} \, dx-\frac {1}{3} \int \frac {\log (8+x)}{\log (x)} \, dx+\frac {1}{3} \int \frac {x \log (8+x)}{(3+x) \log (x)} \, dx-3 \int \frac {e^{14 x/5}}{(3+x)^2 \log (x)} \, dx+3 \int \frac {x}{(3+x)^2 (8+x) \log (x)} \, dx+3 \int \frac {\log (8+x)}{(3+x)^2 \log ^2(x)} \, dx+\frac {42}{5} \int \frac {e^{14 x/5}}{(3+x) \log (x)} \, dx+9 \int \frac {1}{(3+x)^2 (8+x) \log (x)} \, dx-\int \frac {e^{14 x/5}}{x \log ^2(x)} \, dx+\int \frac {e^{14 x/5}}{(3+x) \log ^2(x)} \, dx-\int \frac {\log (8+x)}{x \log ^2(x)} \, dx+\int \frac {x \log (8+x)}{(3+x)^2 \log ^2(x)} \, dx+\int \frac {\log (8+x)}{(3+x) \log ^2(x)} \, dx+\int \frac {x \log (8+x)}{(3+x)^2 \log (x)} \, dx\\ &=-\left (\frac {1}{3} \int \frac {\log (8+x)}{\log ^2(x)} \, dx\right )-\frac {1}{3} \int \frac {\log (8+x)}{\log (x)} \, dx+\frac {1}{3} \int \left (\frac {\log (8+x)}{\log ^2(x)}-\frac {3 \log (8+x)}{(3+x) \log ^2(x)}\right ) \, dx+\frac {1}{3} \int \left (\frac {\log (8+x)}{\log (x)}-\frac {3 \log (8+x)}{(3+x) \log (x)}\right ) \, dx-3 \int \frac {e^{14 x/5}}{(3+x)^2 \log (x)} \, dx+3 \int \frac {x}{(3+x)^2 (8+x) \log (x)} \, dx+3 \int \frac {\log (8+x)}{(3+x)^2 \log ^2(x)} \, dx+\frac {42}{5} \int \frac {e^{14 x/5}}{(3+x) \log (x)} \, dx+9 \int \frac {1}{(3+x)^2 (8+x) \log (x)} \, dx-\int \frac {e^{14 x/5}}{x \log ^2(x)} \, dx+\int \frac {e^{14 x/5}}{(3+x) \log ^2(x)} \, dx-\int \frac {\log (8+x)}{x \log ^2(x)} \, dx+\int \frac {\log (8+x)}{(3+x) \log ^2(x)} \, dx+\int \left (-\frac {3 \log (8+x)}{(3+x)^2 \log ^2(x)}+\frac {\log (8+x)}{(3+x) \log ^2(x)}\right ) \, dx+\int \left (-\frac {3 \log (8+x)}{(3+x)^2 \log (x)}+\frac {\log (8+x)}{(3+x) \log (x)}\right ) \, dx\\ &=-\left (3 \int \frac {e^{14 x/5}}{(3+x)^2 \log (x)} \, dx\right )+3 \int \frac {x}{(3+x)^2 (8+x) \log (x)} \, dx-3 \int \frac {\log (8+x)}{(3+x)^2 \log (x)} \, dx+\frac {42}{5} \int \frac {e^{14 x/5}}{(3+x) \log (x)} \, dx+9 \int \frac {1}{(3+x)^2 (8+x) \log (x)} \, dx-\int \frac {e^{14 x/5}}{x \log ^2(x)} \, dx+\int \frac {e^{14 x/5}}{(3+x) \log ^2(x)} \, dx-\int \frac {\log (8+x)}{x \log ^2(x)} \, dx+\int \frac {\log (8+x)}{(3+x) \log ^2(x)} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.20, size = 23, normalized size = 1.00 \begin {gather*} \frac {3 \left (e^{14 x/5}+\log (8+x)\right )}{(3+x) \log (x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.84, size = 32, normalized size = 1.39
method | result | size |
risch | \(\frac {3 \ln \left (x +8\right )}{\left (3+x \right ) \ln \left (x \right )}+\frac {3 \,{\mathrm e}^{\frac {14 x}{5}}}{\left (3+x \right ) \ln \left (x \right )}\) | \(32\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 20, normalized size = 0.87 \begin {gather*} \frac {3 \, {\left (e^{\left (\frac {14}{5} \, x\right )} + \log \left (x + 8\right )\right )}}{{\left (x + 3\right )} \log \left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.31, size = 20, normalized size = 0.87 \begin {gather*} \frac {3 \, {\left (e^{\left (\frac {14}{5} \, x\right )} + \log \left (x + 8\right )\right )}}{{\left (x + 3\right )} \log \left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.43, size = 22, normalized size = 0.96 \begin {gather*} \frac {3 \, {\left (e^{\left (\frac {14}{5} \, x\right )} + \log \left (x + 8\right )\right )}}{x \log \left (x\right ) + 3 \, \log \left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.70, size = 20, normalized size = 0.87 \begin {gather*} \frac {3\,\left ({\mathrm {e}}^{\frac {14\,x}{5}}+\ln \left (x+8\right )\right )}{\ln \left (x\right )\,\left (x+3\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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