Optimal. Leaf size=153 \[ \frac {2 \tanh ^{-1}\left (\frac {d+4 e x}{\sqrt {3 d^2-2 \sqrt {d^4-64 a e^3}}}\right )}{\sqrt {d^4-64 a e^3} \sqrt {3 d^2-2 \sqrt {d^4-64 a e^3}}}-\frac {2 \tanh ^{-1}\left (\frac {d+4 e x}{\sqrt {2 \sqrt {d^4-64 a e^3}+3 d^2}}\right )}{\sqrt {d^4-64 a e^3} \sqrt {2 \sqrt {d^4-64 a e^3}+3 d^2}} \]
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Rubi [A] time = 0.25, antiderivative size = 153, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.094, Rules used = {1106, 1093, 208} \[ \frac {2 \tanh ^{-1}\left (\frac {d+4 e x}{\sqrt {3 d^2-2 \sqrt {d^4-64 a e^3}}}\right )}{\sqrt {d^4-64 a e^3} \sqrt {3 d^2-2 \sqrt {d^4-64 a e^3}}}-\frac {2 \tanh ^{-1}\left (\frac {d+4 e x}{\sqrt {2 \sqrt {d^4-64 a e^3}+3 d^2}}\right )}{\sqrt {d^4-64 a e^3} \sqrt {2 \sqrt {d^4-64 a e^3}+3 d^2}} \]
Antiderivative was successfully verified.
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Rule 208
Rule 1093
Rule 1106
Rubi steps
\begin {align*} \int \frac {1}{8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4} \, dx &=\operatorname {Subst}\left (\int \frac {1}{\frac {1}{32} \left (\frac {5 d^4}{e}+256 a e^2\right )-3 d^2 e x^2+8 e^3 x^4} \, dx,x,\frac {d}{4 e}+x\right )\\ &=\frac {\left (4 e^2\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {3 d^2 e}{2}-e \sqrt {d^4-64 a e^3}+8 e^3 x^2} \, dx,x,\frac {d}{4 e}+x\right )}{\sqrt {d^4-64 a e^3}}-\frac {\left (4 e^2\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {3 d^2 e}{2}+e \sqrt {d^4-64 a e^3}+8 e^3 x^2} \, dx,x,\frac {d}{4 e}+x\right )}{\sqrt {d^4-64 a e^3}}\\ &=\frac {2 \tanh ^{-1}\left (\frac {d+4 e x}{\sqrt {3 d^2-2 \sqrt {d^4-64 a e^3}}}\right )}{\sqrt {d^4-64 a e^3} \sqrt {3 d^2-2 \sqrt {d^4-64 a e^3}}}-\frac {2 \tanh ^{-1}\left (\frac {d+4 e x}{\sqrt {3 d^2+2 \sqrt {d^4-64 a e^3}}}\right )}{\sqrt {d^4-64 a e^3} \sqrt {3 d^2+2 \sqrt {d^4-64 a e^3}}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 71, normalized size = 0.46 \[ -\text {RootSum}\left [8 \text {$\#$1}^4 e^3+8 \text {$\#$1}^3 d e^2-\text {$\#$1} d^3+8 a e^2\& ,\frac {\log (x-\text {$\#$1})}{-32 \text {$\#$1}^3 e^3-24 \text {$\#$1}^2 d e^2+d^3}\& \right ] \]
Antiderivative was successfully verified.
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fricas [B] time = 0.48, size = 1115, normalized size = 7.29 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{8 \, e^{3} x^{4} + 8 \, d e^{2} x^{3} - d^{3} x + 8 \, a e^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.07, size = 67, normalized size = 0.44 \[ \frac {\ln \left (-\RootOf \left (8 e^{3} \textit {\_Z}^{4}+8 e^{2} d \,\textit {\_Z}^{3}-d^{3} \textit {\_Z} +8 a \,e^{2}\right )+x \right )}{32 \RootOf \left (8 e^{3} \textit {\_Z}^{4}+8 e^{2} d \,\textit {\_Z}^{3}-d^{3} \textit {\_Z} +8 a \,e^{2}\right )^{3} e^{3}+24 \RootOf \left (8 e^{3} \textit {\_Z}^{4}+8 e^{2} d \,\textit {\_Z}^{3}-d^{3} \textit {\_Z} +8 a \,e^{2}\right )^{2} d \,e^{2}-d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{8 \, e^{3} x^{4} + 8 \, d e^{2} x^{3} - d^{3} x + 8 \, a e^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.73, size = 1264, normalized size = 8.26 \[ -\mathrm {atan}\left (\frac {d^3\,\sqrt {-262144\,a^3\,e^9+12288\,a^2\,d^4\,e^6-192\,a\,d^8\,e^3+d^{12}}\,3{}\mathrm {i}+d^9\,2{}\mathrm {i}-a\,d^5\,e^3\,256{}\mathrm {i}+a^2\,d\,e^6\,8192{}\mathrm {i}+a^2\,e^7\,x\,32768{}\mathrm {i}+d^8\,e\,x\,8{}\mathrm {i}+d^2\,e\,x\,\sqrt {-262144\,a^3\,e^9+12288\,a^2\,d^4\,e^6-192\,a\,d^8\,e^3+d^{12}}\,12{}\mathrm {i}-a\,d^4\,e^4\,x\,1024{}\mathrm {i}}{5\,d^{12}\,\sqrt {\frac {2\,\sqrt {-262144\,a^3\,e^9+12288\,a^2\,d^4\,e^6-192\,a\,d^8\,e^3+d^{12}}+3\,d^6-192\,a\,d^2\,e^3}{1048576\,a^3\,e^9-12288\,a^2\,d^4\,e^6-384\,a\,d^8\,e^3+5\,d^{12}}}+1048576\,a^3\,e^9\,\sqrt {\frac {2\,\sqrt {-262144\,a^3\,e^9+12288\,a^2\,d^4\,e^6-192\,a\,d^8\,e^3+d^{12}}+3\,d^6-192\,a\,d^2\,e^3}{1048576\,a^3\,e^9-12288\,a^2\,d^4\,e^6-384\,a\,d^8\,e^3+5\,d^{12}}}-384\,a\,d^8\,e^3\,\sqrt {\frac {2\,\sqrt {-262144\,a^3\,e^9+12288\,a^2\,d^4\,e^6-192\,a\,d^8\,e^3+d^{12}}+3\,d^6-192\,a\,d^2\,e^3}{1048576\,a^3\,e^9-12288\,a^2\,d^4\,e^6-384\,a\,d^8\,e^3+5\,d^{12}}}-12288\,a^2\,d^4\,e^6\,\sqrt {\frac {2\,\sqrt {-262144\,a^3\,e^9+12288\,a^2\,d^4\,e^6-192\,a\,d^8\,e^3+d^{12}}+3\,d^6-192\,a\,d^2\,e^3}{1048576\,a^3\,e^9-12288\,a^2\,d^4\,e^6-384\,a\,d^8\,e^3+5\,d^{12}}}}\right )\,\sqrt {\frac {2\,\sqrt {-262144\,a^3\,e^9+12288\,a^2\,d^4\,e^6-192\,a\,d^8\,e^3+d^{12}}+3\,d^6-192\,a\,d^2\,e^3}{1048576\,a^3\,e^9-12288\,a^2\,d^4\,e^6-384\,a\,d^8\,e^3+5\,d^{12}}}\,2{}\mathrm {i}+\mathrm {atan}\left (\frac {d^3\,\sqrt {-262144\,a^3\,e^9+12288\,a^2\,d^4\,e^6-192\,a\,d^8\,e^3+d^{12}}\,3{}\mathrm {i}-d^9\,2{}\mathrm {i}+a\,d^5\,e^3\,256{}\mathrm {i}-a^2\,d\,e^6\,8192{}\mathrm {i}-a^2\,e^7\,x\,32768{}\mathrm {i}-d^8\,e\,x\,8{}\mathrm {i}+d^2\,e\,x\,\sqrt {-262144\,a^3\,e^9+12288\,a^2\,d^4\,e^6-192\,a\,d^8\,e^3+d^{12}}\,12{}\mathrm {i}+a\,d^4\,e^4\,x\,1024{}\mathrm {i}}{5\,d^{12}\,\sqrt {-\frac {2\,\sqrt {-262144\,a^3\,e^9+12288\,a^2\,d^4\,e^6-192\,a\,d^8\,e^3+d^{12}}-3\,d^6+192\,a\,d^2\,e^3}{1048576\,a^3\,e^9-12288\,a^2\,d^4\,e^6-384\,a\,d^8\,e^3+5\,d^{12}}}+1048576\,a^3\,e^9\,\sqrt {-\frac {2\,\sqrt {-262144\,a^3\,e^9+12288\,a^2\,d^4\,e^6-192\,a\,d^8\,e^3+d^{12}}-3\,d^6+192\,a\,d^2\,e^3}{1048576\,a^3\,e^9-12288\,a^2\,d^4\,e^6-384\,a\,d^8\,e^3+5\,d^{12}}}-384\,a\,d^8\,e^3\,\sqrt {-\frac {2\,\sqrt {-262144\,a^3\,e^9+12288\,a^2\,d^4\,e^6-192\,a\,d^8\,e^3+d^{12}}-3\,d^6+192\,a\,d^2\,e^3}{1048576\,a^3\,e^9-12288\,a^2\,d^4\,e^6-384\,a\,d^8\,e^3+5\,d^{12}}}-12288\,a^2\,d^4\,e^6\,\sqrt {-\frac {2\,\sqrt {-262144\,a^3\,e^9+12288\,a^2\,d^4\,e^6-192\,a\,d^8\,e^3+d^{12}}-3\,d^6+192\,a\,d^2\,e^3}{1048576\,a^3\,e^9-12288\,a^2\,d^4\,e^6-384\,a\,d^8\,e^3+5\,d^{12}}}}\right )\,\sqrt {-\frac {2\,\sqrt {-262144\,a^3\,e^9+12288\,a^2\,d^4\,e^6-192\,a\,d^8\,e^3+d^{12}}-3\,d^6+192\,a\,d^2\,e^3}{1048576\,a^3\,e^9-12288\,a^2\,d^4\,e^6-384\,a\,d^8\,e^3+5\,d^{12}}}\,2{}\mathrm {i} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.83, size = 122, normalized size = 0.80 \[ \operatorname {RootSum} {\left (t^{4} \left (1048576 a^{3} e^{9} - 12288 a^{2} d^{4} e^{6} - 384 a d^{8} e^{3} + 5 d^{12}\right ) + t^{2} \left (384 a d^{2} e^{3} - 6 d^{6}\right ) + 1, \left (t \mapsto t \log {\left (x + \frac {- 49152 t^{3} a^{2} d^{2} e^{6} - 192 t^{3} a d^{6} e^{3} + 15 t^{3} d^{10} + 256 t a e^{3} - 13 t d^{4} + 2 d}{8 e} \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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