Optimal. Leaf size=349 \[ -\frac {e^2 x}{3 c \left (a+c x^4\right )}+\frac {x \left (3 c d^2+a e^2+6 c d e x^2\right )}{12 a c \left (a+c x^4\right )}-\frac {\left (3 c d^2+2 \sqrt {a} \sqrt {c} d e+a e^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} c^{5/4}}+\frac {\left (3 c d^2+2 \sqrt {a} \sqrt {c} d e+a e^2\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} c^{5/4}}-\frac {\left (3 c d^2-2 \sqrt {a} \sqrt {c} d e+a e^2\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{16 \sqrt {2} a^{7/4} c^{5/4}}+\frac {\left (3 c d^2-2 \sqrt {a} \sqrt {c} d e+a e^2\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{16 \sqrt {2} a^{7/4} c^{5/4}} \]
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Rubi [A]
time = 0.20, antiderivative size = 349, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 8, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {1221, 1193,
1182, 1176, 631, 210, 1179, 642} \begin {gather*} -\frac {\text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right ) \left (2 \sqrt {a} \sqrt {c} d e+a e^2+3 c d^2\right )}{8 \sqrt {2} a^{7/4} c^{5/4}}+\frac {\text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right ) \left (2 \sqrt {a} \sqrt {c} d e+a e^2+3 c d^2\right )}{8 \sqrt {2} a^{7/4} c^{5/4}}-\frac {\left (-2 \sqrt {a} \sqrt {c} d e+a e^2+3 c d^2\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{16 \sqrt {2} a^{7/4} c^{5/4}}+\frac {\left (-2 \sqrt {a} \sqrt {c} d e+a e^2+3 c d^2\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{16 \sqrt {2} a^{7/4} c^{5/4}}+\frac {x \left (a e^2+3 c d^2+6 c d e x^2\right )}{12 a c \left (a+c x^4\right )}-\frac {e^2 x}{3 c \left (a+c x^4\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 1182
Rule 1193
Rule 1221
Rubi steps
\begin {align*} \int \frac {\left (d+e x^2\right )^2}{\left (a+c x^4\right )^2} \, dx &=-\frac {e^2 x}{3 c \left (a+c x^4\right )}-\frac {\int \frac {-3 c d^2-a e^2-6 c d e x^2}{\left (a+c x^4\right )^2} \, dx}{3 c}\\ &=-\frac {e^2 x}{3 c \left (a+c x^4\right )}+\frac {x \left (3 c d^2+a e^2+6 c d e x^2\right )}{12 a c \left (a+c x^4\right )}+\frac {\int \frac {3 \left (3 c d^2+a e^2\right )+6 c d e x^2}{a+c x^4} \, dx}{12 a c}\\ &=-\frac {e^2 x}{3 c \left (a+c x^4\right )}+\frac {x \left (3 c d^2+a e^2+6 c d e x^2\right )}{12 a c \left (a+c x^4\right )}+\frac {\left (3 c d^2-2 \sqrt {a} \sqrt {c} d e+a e^2\right ) \int \frac {\sqrt {a} \sqrt {c}-c x^2}{a+c x^4} \, dx}{8 a^{3/2} c^{3/2}}+\frac {\left (3 c d^2+2 \sqrt {a} \sqrt {c} d e+a e^2\right ) \int \frac {\sqrt {a} \sqrt {c}+c x^2}{a+c x^4} \, dx}{8 a^{3/2} c^{3/2}}\\ &=-\frac {e^2 x}{3 c \left (a+c x^4\right )}+\frac {x \left (3 c d^2+a e^2+6 c d e x^2\right )}{12 a c \left (a+c x^4\right )}-\frac {\left (3 c d^2-2 \sqrt {a} \sqrt {c} d e+a e^2\right ) \int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{c}}+2 x}{-\frac {\sqrt {a}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}-x^2} \, dx}{16 \sqrt {2} a^{7/4} c^{5/4}}-\frac {\left (3 c d^2-2 \sqrt {a} \sqrt {c} d e+a e^2\right ) \int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{c}}-2 x}{-\frac {\sqrt {a}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}-x^2} \, dx}{16 \sqrt {2} a^{7/4} c^{5/4}}+\frac {\left (3 c d^2+2 \sqrt {a} \sqrt {c} d e+a e^2\right ) \int \frac {1}{\frac {\sqrt {a}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{16 a^{3/2} c^{3/2}}+\frac {\left (3 c d^2+2 \sqrt {a} \sqrt {c} d e+a e^2\right ) \int \frac {1}{\frac {\sqrt {a}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{16 a^{3/2} c^{3/2}}\\ &=-\frac {e^2 x}{3 c \left (a+c x^4\right )}+\frac {x \left (3 c d^2+a e^2+6 c d e x^2\right )}{12 a c \left (a+c x^4\right )}-\frac {\left (3 c d^2-2 \sqrt {a} \sqrt {c} d e+a e^2\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{16 \sqrt {2} a^{7/4} c^{5/4}}+\frac {\left (3 c d^2-2 \sqrt {a} \sqrt {c} d e+a e^2\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{16 \sqrt {2} a^{7/4} c^{5/4}}+\frac {\left (3 c d^2+2 \sqrt {a} \sqrt {c} d e+a e^2\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} c^{5/4}}-\frac {\left (3 c d^2+2 \sqrt {a} \sqrt {c} d e+a e^2\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} c^{5/4}}\\ &=-\frac {e^2 x}{3 c \left (a+c x^4\right )}+\frac {x \left (3 c d^2+a e^2+6 c d e x^2\right )}{12 a c \left (a+c x^4\right )}-\frac {\left (3 c d^2+2 \sqrt {a} \sqrt {c} d e+a e^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} c^{5/4}}+\frac {\left (3 c d^2+2 \sqrt {a} \sqrt {c} d e+a e^2\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} c^{5/4}}-\frac {\left (3 c d^2-2 \sqrt {a} \sqrt {c} d e+a e^2\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{16 \sqrt {2} a^{7/4} c^{5/4}}+\frac {\left (3 c d^2-2 \sqrt {a} \sqrt {c} d e+a e^2\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{16 \sqrt {2} a^{7/4} c^{5/4}}\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 295, normalized size = 0.85 \begin {gather*} \frac {-\frac {8 a^{3/4} \sqrt [4]{c} \left (a e^2 x-c d x \left (d+2 e x^2\right )\right )}{a+c x^4}-2 \sqrt {2} \left (3 c d^2+2 \sqrt {a} \sqrt {c} d e+a e^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )+2 \sqrt {2} \left (3 c d^2+2 \sqrt {a} \sqrt {c} d e+a e^2\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )-\sqrt {2} \left (3 c d^2-2 \sqrt {a} \sqrt {c} d e+a e^2\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )+\sqrt {2} \left (3 c d^2-2 \sqrt {a} \sqrt {c} d e+a e^2\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{32 a^{7/4} c^{5/4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.16, size = 266, normalized size = 0.76
method | result | size |
risch | \(\frac {\frac {d e \,x^{3}}{2 a}-\frac {\left (a \,e^{2}-c \,d^{2}\right ) x}{4 a c}}{c \,x^{4}+a}+\frac {\munderset {\textit {\_R} =\RootOf \left (c \,\textit {\_Z}^{4}+a \right )}{\sum }\frac {\left (2 d e \,\textit {\_R}^{2}+\frac {a \,e^{2}+3 c \,d^{2}}{c}\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}}}{16 a c}\) | \(97\) |
default | \(\frac {\frac {d e \,x^{3}}{2 a}-\frac {\left (a \,e^{2}-c \,d^{2}\right ) x}{4 a c}}{c \,x^{4}+a}+\frac {\frac {\left (a \,e^{2}+3 c \,d^{2}\right ) \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}{x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{8 a}+\frac {d e \sqrt {2}\, \left (\ln \left (\frac {x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}{x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{4 \left (\frac {a}{c}\right )^{\frac {1}{4}}}}{4 a c}\) | \(266\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.52, size = 324, normalized size = 0.93 \begin {gather*} \frac {2 \, c d x^{3} e + {\left (c d^{2} - a e^{2}\right )} x}{4 \, {\left (a c^{2} x^{4} + a^{2} c\right )}} + \frac {\frac {2 \, \sqrt {2} {\left (3 \, c^{\frac {3}{2}} d^{2} + 2 \, \sqrt {a} c d e + a \sqrt {c} e^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {c} x + \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {c}} \sqrt {c}} + \frac {2 \, \sqrt {2} {\left (3 \, c^{\frac {3}{2}} d^{2} + 2 \, \sqrt {a} c d e + a \sqrt {c} e^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {c} x - \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {c}} \sqrt {c}} + \frac {\sqrt {2} {\left (3 \, c^{\frac {3}{2}} d^{2} - 2 \, \sqrt {a} c d e + a \sqrt {c} e^{2}\right )} \log \left (\sqrt {c} x^{2} + \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} c^{\frac {3}{4}}} - \frac {\sqrt {2} {\left (3 \, c^{\frac {3}{2}} d^{2} - 2 \, \sqrt {a} c d e + a \sqrt {c} e^{2}\right )} \log \left (\sqrt {c} x^{2} - \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} c^{\frac {3}{4}}}}{32 \, a c} \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. 1531 vs.
\(2 (263) = 526\).
time = 0.39, size = 1531, normalized size = 4.39 \begin {gather*} \frac {8 \, c d x^{3} e + 4 \, c d^{2} x - 4 \, a x e^{2} + {\left (a c^{2} x^{4} + a^{2} c\right )} \sqrt {-\frac {a^{3} c^{2} \sqrt {-\frac {81 \, c^{4} d^{8} + 36 \, a c^{3} d^{6} e^{2} + 22 \, a^{2} c^{2} d^{4} e^{4} + 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}}{a^{7} c^{5}}} + 12 \, c d^{3} e + 4 \, a d e^{3}}{a^{3} c^{2}}} \log \left (81 \, c^{4} d^{8} x + 108 \, a c^{3} d^{6} x e^{2} + 38 \, a^{2} c^{2} d^{4} x e^{4} + 12 \, a^{3} c d^{2} x e^{6} + a^{4} x e^{8} + {\left (2 \, a^{6} c^{4} d \sqrt {-\frac {81 \, c^{4} d^{8} + 36 \, a c^{3} d^{6} e^{2} + 22 \, a^{2} c^{2} d^{4} e^{4} + 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}}{a^{7} c^{5}}} e + 27 \, a^{2} c^{4} d^{6} + 15 \, a^{3} c^{3} d^{4} e^{2} + 5 \, a^{4} c^{2} d^{2} e^{4} + a^{5} c e^{6}\right )} \sqrt {-\frac {a^{3} c^{2} \sqrt {-\frac {81 \, c^{4} d^{8} + 36 \, a c^{3} d^{6} e^{2} + 22 \, a^{2} c^{2} d^{4} e^{4} + 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}}{a^{7} c^{5}}} + 12 \, c d^{3} e + 4 \, a d e^{3}}{a^{3} c^{2}}}\right ) - {\left (a c^{2} x^{4} + a^{2} c\right )} \sqrt {-\frac {a^{3} c^{2} \sqrt {-\frac {81 \, c^{4} d^{8} + 36 \, a c^{3} d^{6} e^{2} + 22 \, a^{2} c^{2} d^{4} e^{4} + 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}}{a^{7} c^{5}}} + 12 \, c d^{3} e + 4 \, a d e^{3}}{a^{3} c^{2}}} \log \left (81 \, c^{4} d^{8} x + 108 \, a c^{3} d^{6} x e^{2} + 38 \, a^{2} c^{2} d^{4} x e^{4} + 12 \, a^{3} c d^{2} x e^{6} + a^{4} x e^{8} - {\left (2 \, a^{6} c^{4} d \sqrt {-\frac {81 \, c^{4} d^{8} + 36 \, a c^{3} d^{6} e^{2} + 22 \, a^{2} c^{2} d^{4} e^{4} + 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}}{a^{7} c^{5}}} e + 27 \, a^{2} c^{4} d^{6} + 15 \, a^{3} c^{3} d^{4} e^{2} + 5 \, a^{4} c^{2} d^{2} e^{4} + a^{5} c e^{6}\right )} \sqrt {-\frac {a^{3} c^{2} \sqrt {-\frac {81 \, c^{4} d^{8} + 36 \, a c^{3} d^{6} e^{2} + 22 \, a^{2} c^{2} d^{4} e^{4} + 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}}{a^{7} c^{5}}} + 12 \, c d^{3} e + 4 \, a d e^{3}}{a^{3} c^{2}}}\right ) - {\left (a c^{2} x^{4} + a^{2} c\right )} \sqrt {\frac {a^{3} c^{2} \sqrt {-\frac {81 \, c^{4} d^{8} + 36 \, a c^{3} d^{6} e^{2} + 22 \, a^{2} c^{2} d^{4} e^{4} + 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}}{a^{7} c^{5}}} - 12 \, c d^{3} e - 4 \, a d e^{3}}{a^{3} c^{2}}} \log \left (81 \, c^{4} d^{8} x + 108 \, a c^{3} d^{6} x e^{2} + 38 \, a^{2} c^{2} d^{4} x e^{4} + 12 \, a^{3} c d^{2} x e^{6} + a^{4} x e^{8} + {\left (2 \, a^{6} c^{4} d \sqrt {-\frac {81 \, c^{4} d^{8} + 36 \, a c^{3} d^{6} e^{2} + 22 \, a^{2} c^{2} d^{4} e^{4} + 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}}{a^{7} c^{5}}} e - 27 \, a^{2} c^{4} d^{6} - 15 \, a^{3} c^{3} d^{4} e^{2} - 5 \, a^{4} c^{2} d^{2} e^{4} - a^{5} c e^{6}\right )} \sqrt {\frac {a^{3} c^{2} \sqrt {-\frac {81 \, c^{4} d^{8} + 36 \, a c^{3} d^{6} e^{2} + 22 \, a^{2} c^{2} d^{4} e^{4} + 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}}{a^{7} c^{5}}} - 12 \, c d^{3} e - 4 \, a d e^{3}}{a^{3} c^{2}}}\right ) + {\left (a c^{2} x^{4} + a^{2} c\right )} \sqrt {\frac {a^{3} c^{2} \sqrt {-\frac {81 \, c^{4} d^{8} + 36 \, a c^{3} d^{6} e^{2} + 22 \, a^{2} c^{2} d^{4} e^{4} + 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}}{a^{7} c^{5}}} - 12 \, c d^{3} e - 4 \, a d e^{3}}{a^{3} c^{2}}} \log \left (81 \, c^{4} d^{8} x + 108 \, a c^{3} d^{6} x e^{2} + 38 \, a^{2} c^{2} d^{4} x e^{4} + 12 \, a^{3} c d^{2} x e^{6} + a^{4} x e^{8} - {\left (2 \, a^{6} c^{4} d \sqrt {-\frac {81 \, c^{4} d^{8} + 36 \, a c^{3} d^{6} e^{2} + 22 \, a^{2} c^{2} d^{4} e^{4} + 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}}{a^{7} c^{5}}} e - 27 \, a^{2} c^{4} d^{6} - 15 \, a^{3} c^{3} d^{4} e^{2} - 5 \, a^{4} c^{2} d^{2} e^{4} - a^{5} c e^{6}\right )} \sqrt {\frac {a^{3} c^{2} \sqrt {-\frac {81 \, c^{4} d^{8} + 36 \, a c^{3} d^{6} e^{2} + 22 \, a^{2} c^{2} d^{4} e^{4} + 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}}{a^{7} c^{5}}} - 12 \, c d^{3} e - 4 \, a d e^{3}}{a^{3} c^{2}}}\right )}{16 \, {\left (a c^{2} x^{4} + a^{2} c\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.07, size = 275, normalized size = 0.79 \begin {gather*} \operatorname {RootSum} {\left (65536 t^{4} a^{7} c^{5} + t^{2} \cdot \left (2048 a^{5} c^{3} d e^{3} + 6144 a^{4} c^{4} d^{3} e\right ) + a^{4} e^{8} + 20 a^{3} c d^{2} e^{6} + 118 a^{2} c^{2} d^{4} e^{4} + 180 a c^{3} d^{6} e^{2} + 81 c^{4} d^{8}, \left ( t \mapsto t \log {\left (x + \frac {- 8192 t^{3} a^{6} c^{4} d e + 16 t a^{5} c e^{6} - 48 t a^{4} c^{2} d^{2} e^{4} - 144 t a^{3} c^{3} d^{4} e^{2} + 432 t a^{2} c^{4} d^{6}}{a^{4} e^{8} + 12 a^{3} c d^{2} e^{6} + 38 a^{2} c^{2} d^{4} e^{4} + 108 a c^{3} d^{6} e^{2} + 81 c^{4} d^{8}} \right )} \right )\right )} + \frac {2 c d e x^{3} + x \left (- a e^{2} + c d^{2}\right )}{4 a^{2} c + 4 a c^{2} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.30, size = 350, normalized size = 1.00 \begin {gather*} \frac {2 \, c d x^{3} e + c d^{2} x - a x e^{2}}{4 \, {\left (c x^{4} + a\right )} a c} + \frac {\sqrt {2} {\left (3 \, \left (a c^{3}\right )^{\frac {1}{4}} c^{2} d^{2} + \left (a c^{3}\right )^{\frac {1}{4}} a c e^{2} + 2 \, \left (a c^{3}\right )^{\frac {3}{4}} d e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{16 \, a^{2} c^{3}} + \frac {\sqrt {2} {\left (3 \, \left (a c^{3}\right )^{\frac {1}{4}} c^{2} d^{2} + \left (a c^{3}\right )^{\frac {1}{4}} a c e^{2} + 2 \, \left (a c^{3}\right )^{\frac {3}{4}} d e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{16 \, a^{2} c^{3}} + \frac {\sqrt {2} {\left (3 \, \left (a c^{3}\right )^{\frac {1}{4}} c^{2} d^{2} + \left (a c^{3}\right )^{\frac {1}{4}} a c e^{2} - 2 \, \left (a c^{3}\right )^{\frac {3}{4}} d e\right )} \log \left (x^{2} + \sqrt {2} x \left (\frac {a}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{c}}\right )}{32 \, a^{2} c^{3}} - \frac {\sqrt {2} {\left (3 \, \left (a c^{3}\right )^{\frac {1}{4}} c^{2} d^{2} + \left (a c^{3}\right )^{\frac {1}{4}} a c e^{2} - 2 \, \left (a c^{3}\right )^{\frac {3}{4}} d e\right )} \log \left (x^{2} - \sqrt {2} x \left (\frac {a}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{c}}\right )}{32 \, a^{2} c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.79, size = 1565, normalized size = 4.48 \begin {gather*} 2\,\mathrm {atanh}\left (\frac {9\,c^3\,d^4\,x\,\sqrt {\frac {9\,d^4\,\sqrt {-a^7\,c^5}}{256\,a^7\,c^3}-\frac {3\,d^3\,e}{64\,a^3\,c}-\frac {d\,e^3}{64\,a^2\,c^2}+\frac {e^4\,\sqrt {-a^7\,c^5}}{256\,a^5\,c^5}+\frac {d^2\,e^2\,\sqrt {-a^7\,c^5}}{128\,a^6\,c^4}}}{2\,\left (\frac {27\,d^6\,\sqrt {-a^7\,c^5}}{32\,a^5}-\frac {c\,d^3\,e^3}{8}-\frac {a\,d\,e^5}{16}-\frac {9\,c^2\,d^5\,e}{16\,a}+\frac {e^6\,\sqrt {-a^7\,c^5}}{32\,a^2\,c^3}+\frac {5\,d^2\,e^4\,\sqrt {-a^7\,c^5}}{32\,a^3\,c^2}+\frac {15\,d^4\,e^2\,\sqrt {-a^7\,c^5}}{32\,a^4\,c}\right )}+\frac {c\,e^4\,x\,\sqrt {\frac {9\,d^4\,\sqrt {-a^7\,c^5}}{256\,a^7\,c^3}-\frac {3\,d^3\,e}{64\,a^3\,c}-\frac {d\,e^3}{64\,a^2\,c^2}+\frac {e^4\,\sqrt {-a^7\,c^5}}{256\,a^5\,c^5}+\frac {d^2\,e^2\,\sqrt {-a^7\,c^5}}{128\,a^6\,c^4}}}{2\,\left (\frac {27\,d^6\,\sqrt {-a^7\,c^5}}{32\,a^7}-\frac {d\,e^5}{16\,a}-\frac {c\,d^3\,e^3}{8\,a^2}-\frac {9\,c^2\,d^5\,e}{16\,a^3}+\frac {e^6\,\sqrt {-a^7\,c^5}}{32\,a^4\,c^3}+\frac {5\,d^2\,e^4\,\sqrt {-a^7\,c^5}}{32\,a^5\,c^2}+\frac {15\,d^4\,e^2\,\sqrt {-a^7\,c^5}}{32\,a^6\,c}\right )}+\frac {c^2\,d^2\,e^2\,x\,\sqrt {\frac {9\,d^4\,\sqrt {-a^7\,c^5}}{256\,a^7\,c^3}-\frac {3\,d^3\,e}{64\,a^3\,c}-\frac {d\,e^3}{64\,a^2\,c^2}+\frac {e^4\,\sqrt {-a^7\,c^5}}{256\,a^5\,c^5}+\frac {d^2\,e^2\,\sqrt {-a^7\,c^5}}{128\,a^6\,c^4}}}{\frac {27\,d^6\,\sqrt {-a^7\,c^5}}{32\,a^6}-\frac {d\,e^5}{16}-\frac {c\,d^3\,e^3}{8\,a}-\frac {9\,c^2\,d^5\,e}{16\,a^2}+\frac {e^6\,\sqrt {-a^7\,c^5}}{32\,a^3\,c^3}+\frac {5\,d^2\,e^4\,\sqrt {-a^7\,c^5}}{32\,a^4\,c^2}+\frac {15\,d^4\,e^2\,\sqrt {-a^7\,c^5}}{32\,a^5\,c}}\right )\,\sqrt {\frac {a^2\,e^4\,\sqrt {-a^7\,c^5}+9\,c^2\,d^4\,\sqrt {-a^7\,c^5}-12\,a^4\,c^4\,d^3\,e-4\,a^5\,c^3\,d\,e^3+2\,a\,c\,d^2\,e^2\,\sqrt {-a^7\,c^5}}{256\,a^7\,c^5}}-2\,\mathrm {atanh}\left (\frac {9\,c^3\,d^4\,x\,\sqrt {-\frac {d\,e^3}{64\,a^2\,c^2}-\frac {3\,d^3\,e}{64\,a^3\,c}-\frac {9\,d^4\,\sqrt {-a^7\,c^5}}{256\,a^7\,c^3}-\frac {e^4\,\sqrt {-a^7\,c^5}}{256\,a^5\,c^5}-\frac {d^2\,e^2\,\sqrt {-a^7\,c^5}}{128\,a^6\,c^4}}}{2\,\left (\frac {27\,d^6\,\sqrt {-a^7\,c^5}}{32\,a^5}+\frac {c\,d^3\,e^3}{8}+\frac {a\,d\,e^5}{16}+\frac {9\,c^2\,d^5\,e}{16\,a}+\frac {e^6\,\sqrt {-a^7\,c^5}}{32\,a^2\,c^3}+\frac {5\,d^2\,e^4\,\sqrt {-a^7\,c^5}}{32\,a^3\,c^2}+\frac {15\,d^4\,e^2\,\sqrt {-a^7\,c^5}}{32\,a^4\,c}\right )}+\frac {c\,e^4\,x\,\sqrt {-\frac {d\,e^3}{64\,a^2\,c^2}-\frac {3\,d^3\,e}{64\,a^3\,c}-\frac {9\,d^4\,\sqrt {-a^7\,c^5}}{256\,a^7\,c^3}-\frac {e^4\,\sqrt {-a^7\,c^5}}{256\,a^5\,c^5}-\frac {d^2\,e^2\,\sqrt {-a^7\,c^5}}{128\,a^6\,c^4}}}{2\,\left (\frac {27\,d^6\,\sqrt {-a^7\,c^5}}{32\,a^7}+\frac {d\,e^5}{16\,a}+\frac {c\,d^3\,e^3}{8\,a^2}+\frac {9\,c^2\,d^5\,e}{16\,a^3}+\frac {e^6\,\sqrt {-a^7\,c^5}}{32\,a^4\,c^3}+\frac {5\,d^2\,e^4\,\sqrt {-a^7\,c^5}}{32\,a^5\,c^2}+\frac {15\,d^4\,e^2\,\sqrt {-a^7\,c^5}}{32\,a^6\,c}\right )}+\frac {c^2\,d^2\,e^2\,x\,\sqrt {-\frac {d\,e^3}{64\,a^2\,c^2}-\frac {3\,d^3\,e}{64\,a^3\,c}-\frac {9\,d^4\,\sqrt {-a^7\,c^5}}{256\,a^7\,c^3}-\frac {e^4\,\sqrt {-a^7\,c^5}}{256\,a^5\,c^5}-\frac {d^2\,e^2\,\sqrt {-a^7\,c^5}}{128\,a^6\,c^4}}}{\frac {d\,e^5}{16}+\frac {27\,d^6\,\sqrt {-a^7\,c^5}}{32\,a^6}+\frac {c\,d^3\,e^3}{8\,a}+\frac {9\,c^2\,d^5\,e}{16\,a^2}+\frac {e^6\,\sqrt {-a^7\,c^5}}{32\,a^3\,c^3}+\frac {5\,d^2\,e^4\,\sqrt {-a^7\,c^5}}{32\,a^4\,c^2}+\frac {15\,d^4\,e^2\,\sqrt {-a^7\,c^5}}{32\,a^5\,c}}\right )\,\sqrt {-\frac {a^2\,e^4\,\sqrt {-a^7\,c^5}+9\,c^2\,d^4\,\sqrt {-a^7\,c^5}+12\,a^4\,c^4\,d^3\,e+4\,a^5\,c^3\,d\,e^3+2\,a\,c\,d^2\,e^2\,\sqrt {-a^7\,c^5}}{256\,a^7\,c^5}}+\frac {\frac {d\,e\,x^3}{2\,a}-\frac {x\,\left (a\,e^2-c\,d^2\right )}{4\,a\,c}}{c\,x^4+a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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