Optimal. Leaf size=17 \[ \left (11-\frac {x^2}{4}+\log (e+4 x)\right )^2 \]
________________________________________________________________________________________
Rubi [A] time = 0.14, antiderivative size = 21, normalized size of antiderivative = 1.24, number of steps used = 2, number of rules used = 2, integrand size = 49, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.041, Rules used = {6741, 6686} \begin {gather*} \frac {1}{16} \left (-x^2+4 \log (4 x+e)+44\right )^2 \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 6686
Rule 6741
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\left (8-e x-4 x^2\right ) \left (44-x^2+4 \log (e+4 x)\right )}{4 e+16 x} \, dx\\ &=\frac {1}{16} \left (44-x^2+4 \log (e+4 x)\right )^2\\ \end {aligned} \end {gather*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 19, normalized size = 1.12 \begin {gather*} \frac {1}{16} \left (-44+x^2-4 \log (e+4 x)\right )^2 \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.59, size = 34, normalized size = 2.00 \begin {gather*} \frac {1}{16} \, x^{4} - \frac {11}{2} \, x^{2} - \frac {1}{2} \, {\left (x^{2} - 44\right )} \log \left (4 \, x + e\right ) + \log \left (4 \, x + e\right )^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.14, size = 41, normalized size = 2.41 \begin {gather*} \frac {1}{16} \, x^{4} - \frac {1}{2} \, x^{2} \log \left (4 \, x + e\right ) - \frac {11}{2} \, x^{2} + \log \left (4 \, x + e\right )^{2} + 22 \, \log \left (4 \, x + e\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.42, size = 42, normalized size = 2.47
method | result | size |
norman | \(\ln \left ({\mathrm e}+4 x \right )^{2}+22 \ln \left ({\mathrm e}+4 x \right )-\frac {11 x^{2}}{2}+\frac {x^{4}}{16}-\frac {\ln \left ({\mathrm e}+4 x \right ) x^{2}}{2}\) | \(42\) |
risch | \(\ln \left ({\mathrm e}+4 x \right )^{2}+22 \ln \left ({\mathrm e}+4 x \right )-\frac {11 x^{2}}{2}+\frac {x^{4}}{16}-\frac {\ln \left ({\mathrm e}+4 x \right ) x^{2}}{2}\) | \(42\) |
derivativedivides | \(-\frac {\left ({\mathrm e}+4 x \right ) {\mathrm e}^{3}}{1024}+\frac {3 \,{\mathrm e}^{2} \left ({\mathrm e}+4 x \right )^{2}}{2048}-\frac {{\mathrm e} \left ({\mathrm e}+4 x \right )^{3}}{1024}+\frac {\left ({\mathrm e}+4 x \right )^{4}}{4096}+\frac {{\mathrm e} \left (\left ({\mathrm e}+4 x \right ) \ln \left ({\mathrm e}+4 x \right )-{\mathrm e}-4 x \right )}{16}-\frac {\ln \left ({\mathrm e}+4 x \right ) \left ({\mathrm e}+4 x \right )^{2}}{32}-\frac {11 \left ({\mathrm e}+4 x \right )^{2}}{32}-\frac {{\mathrm e}^{2} \ln \left ({\mathrm e}+4 x \right )}{32}+\frac {3 \,{\mathrm e} \left ({\mathrm e}+4 x \right )}{4}+\ln \left ({\mathrm e}+4 x \right )^{2}+22 \ln \left ({\mathrm e}+4 x \right )\) | \(144\) |
default | \(-\frac {\left ({\mathrm e}+4 x \right ) {\mathrm e}^{3}}{1024}+\frac {3 \,{\mathrm e}^{2} \left ({\mathrm e}+4 x \right )^{2}}{2048}-\frac {{\mathrm e} \left ({\mathrm e}+4 x \right )^{3}}{1024}+\frac {\left ({\mathrm e}+4 x \right )^{4}}{4096}+\frac {{\mathrm e} \left (\left ({\mathrm e}+4 x \right ) \ln \left ({\mathrm e}+4 x \right )-{\mathrm e}-4 x \right )}{16}-\frac {\ln \left ({\mathrm e}+4 x \right ) \left ({\mathrm e}+4 x \right )^{2}}{32}-\frac {11 \left ({\mathrm e}+4 x \right )^{2}}{32}-\frac {{\mathrm e}^{2} \ln \left ({\mathrm e}+4 x \right )}{32}+\frac {3 \,{\mathrm e} \left ({\mathrm e}+4 x \right )}{4}+\ln \left ({\mathrm e}+4 x \right )^{2}+22 \ln \left ({\mathrm e}+4 x \right )\) | \(144\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.45, size = 225, normalized size = 13.24 \begin {gather*} \frac {1}{16} \, x^{4} - \frac {1}{48} \, x^{3} e + \frac {1}{128} \, x^{2} e^{2} + \frac {1}{16} \, {\left (e \log \left (4 \, x + e\right ) - 4 \, x\right )} e \log \left (4 \, x + e\right ) + \frac {1}{32} \, e^{2} \log \left (4 \, x + e\right )^{2} - \frac {11}{2} \, x^{2} - \frac {1}{256} \, x e^{3} + \frac {1}{3072} \, {\left (64 \, x^{3} - 24 \, x^{2} e + 12 \, x e^{2} - 3 \, e^{3} \log \left (4 \, x + e\right )\right )} e - \frac {1}{32} \, {\left (e \log \left (4 \, x + e\right )^{2} + 2 \, e \log \left (4 \, x + e\right ) - 8 \, x\right )} e + \frac {11}{16} \, {\left (e \log \left (4 \, x + e\right ) - 4 \, x\right )} e + \frac {5}{2} \, x e - \frac {1}{16} \, {\left (8 \, x^{2} - 4 \, x e + e^{2} \log \left (4 \, x + e\right )\right )} \log \left (4 \, x + e\right ) + \frac {1}{1024} \, e^{4} \log \left (4 \, x + e\right ) - \frac {5}{8} \, e^{2} \log \left (4 \, x + e\right ) + \log \left (4 \, x + e\right )^{2} + 22 \, \log \left (4 \, x + e\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.25, size = 41, normalized size = 2.41 \begin {gather*} \frac {x^4}{16}-\frac {x^2\,\ln \left (4\,x+\mathrm {e}\right )}{2}-\frac {11\,x^2}{2}+{\ln \left (4\,x+\mathrm {e}\right )}^2+22\,\ln \left (4\,x+\mathrm {e}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 0.18, size = 44, normalized size = 2.59 \begin {gather*} \frac {x^{4}}{16} - \frac {x^{2} \log {\left (4 x + e \right )}}{2} - \frac {11 x^{2}}{2} + \log {\left (4 x + e \right )}^{2} + 22 \log {\left (4 x + e \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________