Optimal. Leaf size=363 \[ -\frac {e^3 x^3}{c \left (a+c x^4\right )}+\frac {x \left (d \left (c d^2-3 a e^2\right )+3 e \left (c d^2+a e^2\right ) x^2\right )}{4 a c \left (a+c x^4\right )}-\frac {3 \left (\sqrt {c} d+\sqrt {a} e\right ) \left (c d^2+a e^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} c^{7/4}}+\frac {3 \left (\sqrt {c} d+\sqrt {a} e\right ) \left (c d^2+a e^2\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} c^{7/4}}-\frac {3 \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2+a e^2\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{16 \sqrt {2} a^{7/4} c^{7/4}}+\frac {3 \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2+a e^2\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{16 \sqrt {2} a^{7/4} c^{7/4}} \]
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Rubi [A]
time = 0.26, antiderivative size = 363, 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, 1872,
1182, 1176, 631, 210, 1179, 642} \begin {gather*} -\frac {3 \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right ) \left (\sqrt {a} e+\sqrt {c} d\right ) \left (a e^2+c d^2\right )}{8 \sqrt {2} a^{7/4} c^{7/4}}+\frac {3 \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right ) \left (\sqrt {a} e+\sqrt {c} d\right ) \left (a e^2+c d^2\right )}{8 \sqrt {2} a^{7/4} c^{7/4}}-\frac {3 \left (\sqrt {c} d-\sqrt {a} e\right ) \left (a e^2+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^{7/4}}+\frac {3 \left (\sqrt {c} d-\sqrt {a} e\right ) \left (a e^2+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^{7/4}}+\frac {x \left (3 e x^2 \left (a e^2+c d^2\right )+d \left (c d^2-3 a e^2\right )\right )}{4 a c \left (a+c x^4\right )}-\frac {e^3 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 1221
Rule 1872
Rubi steps
\begin {align*} \int \frac {\left (d+e x^2\right )^3}{\left (a+c x^4\right )^2} \, dx &=-\frac {e^3 x^3}{c \left (a+c x^4\right )}-\frac {\int \frac {-c d^3-3 e \left (c d^2+a e^2\right ) x^2-3 c d e^2 x^4}{\left (a+c x^4\right )^2} \, dx}{c}\\ &=-\frac {e^3 x^3}{c \left (a+c x^4\right )}+\frac {x \left (d \left (c d^2-3 a e^2\right )+3 e \left (c d^2+a e^2\right ) x^2\right )}{4 a c \left (a+c x^4\right )}+\frac {\int \frac {3 c d \left (c d^2+a e^2\right )+3 c e \left (c d^2+a e^2\right ) x^2}{a+c x^4} \, dx}{4 a c^2}\\ &=-\frac {e^3 x^3}{c \left (a+c x^4\right )}+\frac {x \left (d \left (c d^2-3 a e^2\right )+3 e \left (c d^2+a e^2\right ) x^2\right )}{4 a c \left (a+c x^4\right )}+\frac {\left (3 \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2+a e^2\right )\right ) \int \frac {\sqrt {a} \sqrt {c}-c x^2}{a+c x^4} \, dx}{8 a^{3/2} c^2}+\frac {\left (3 \left (\sqrt {c} d+\sqrt {a} e\right ) \left (c d^2+a e^2\right )\right ) \int \frac {\sqrt {a} \sqrt {c}+c x^2}{a+c x^4} \, dx}{8 a^{3/2} c^2}\\ &=-\frac {e^3 x^3}{c \left (a+c x^4\right )}+\frac {x \left (d \left (c d^2-3 a e^2\right )+3 e \left (c d^2+a e^2\right ) x^2\right )}{4 a c \left (a+c x^4\right )}-\frac {\left (3 \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2+a e^2\right )\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^{7/4}}-\frac {\left (3 \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2+a e^2\right )\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^{7/4}}+\frac {\left (3 \left (\sqrt {c} d+\sqrt {a} e\right ) \left (c d^2+a e^2\right )\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^2}+\frac {\left (3 \left (\sqrt {c} d+\sqrt {a} e\right ) \left (c d^2+a e^2\right )\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^2}\\ &=-\frac {e^3 x^3}{c \left (a+c x^4\right )}+\frac {x \left (d \left (c d^2-3 a e^2\right )+3 e \left (c d^2+a e^2\right ) x^2\right )}{4 a c \left (a+c x^4\right )}-\frac {3 \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2+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^{7/4}}+\frac {3 \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2+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^{7/4}}+\frac {\left (3 \left (\sqrt {c} d+\sqrt {a} e\right ) \left (c d^2+a e^2\right )\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^{7/4}}-\frac {\left (3 \left (\sqrt {c} d+\sqrt {a} e\right ) \left (c d^2+a e^2\right )\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^{7/4}}\\ &=-\frac {e^3 x^3}{c \left (a+c x^4\right )}+\frac {x \left (d \left (c d^2-3 a e^2\right )+3 e \left (c d^2+a e^2\right ) x^2\right )}{4 a c \left (a+c x^4\right )}-\frac {3 \left (\sqrt {c} d+\sqrt {a} e\right ) \left (c d^2+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^{7/4}}+\frac {3 \left (\sqrt {c} d+\sqrt {a} e\right ) \left (c d^2+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^{7/4}}-\frac {3 \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2+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^{7/4}}+\frac {3 \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2+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^{7/4}}\\ \end {align*}
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Mathematica [A]
time = 0.16, size = 371, normalized size = 1.02 \begin {gather*} \frac {-\frac {8 a^{3/4} c^{3/4} \left (a e^2 x \left (3 d+e x^2\right )-c d^2 x \left (d+3 e x^2\right )\right )}{a+c x^4}-6 \sqrt {2} \left (c^{3/2} d^3+\sqrt {a} c d^2 e+a \sqrt {c} d e^2+a^{3/2} e^3\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )+6 \sqrt {2} \left (c^{3/2} d^3+\sqrt {a} c d^2 e+a \sqrt {c} d e^2+a^{3/2} e^3\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )+3 \sqrt {2} \left (-c^{3/2} d^3+\sqrt {a} c d^2 e-a \sqrt {c} d e^2+a^{3/2} e^3\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )+3 \sqrt {2} \left (c^{3/2} d^3-\sqrt {a} c d^2 e+a \sqrt {c} d e^2-a^{3/2} e^3\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{32 a^{7/4} c^{7/4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 284, normalized size = 0.78
method | result | size |
risch | \(\frac {-\frac {e \left (a \,e^{2}-3 c \,d^{2}\right ) x^{3}}{4 a c}-\frac {d \left (3 a \,e^{2}-c \,d^{2}\right ) x}{4 a c}}{c \,x^{4}+a}+\frac {3 \left (\munderset {\textit {\_R} =\RootOf \left (c \,\textit {\_Z}^{4}+a \right )}{\sum }\frac {\left (e \left (a \,e^{2}+c \,d^{2}\right ) \textit {\_R}^{2}+d \left (a \,e^{2}+c \,d^{2}\right )\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}}\right )}{16 a \,c^{2}}\) | \(119\) |
default | \(\frac {-\frac {e \left (a \,e^{2}-3 c \,d^{2}\right ) x^{3}}{4 a c}-\frac {d \left (3 a \,e^{2}-c \,d^{2}\right ) x}{4 a c}}{c \,x^{4}+a}+\frac {3 \left (a \,e^{2}+c \,d^{2}\right ) \left (\frac {d \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}{x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{8 a}+\frac {e \sqrt {2}\, \left (\ln \left (\frac {x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}{x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{8 c \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{4 a c}\) | \(284\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.52, size = 294, normalized size = 0.81 \begin {gather*} \frac {{\left (3 \, c d^{2} e - a e^{3}\right )} x^{3} + {\left (c d^{3} - 3 \, a d e^{2}\right )} x}{4 \, {\left (a c^{2} x^{4} + a^{2} c\right )}} + \frac {3 \, {\left (c d^{2} + a e^{2}\right )} {\left (\frac {2 \, \sqrt {2} {\left (\sqrt {c} d + \sqrt {a} e\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 (\sqrt {c} d + \sqrt {a} e\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 (\sqrt {c} d - \sqrt {a} e\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 (\sqrt {c} d - \sqrt {a} e\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}}}\right )}}{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. 2018 vs.
\(2 (278) = 556\).
time = 0.38, size = 2018, normalized size = 5.56 \begin {gather*} \frac {12 \, c d^{2} x^{3} e + 4 \, c d^{3} x - 4 \, a x^{3} e^{3} - 12 \, a d x e^{2} - 3 \, {\left (a c^{2} x^{4} + a^{2} c\right )} \sqrt {-\frac {2 \, c^{2} d^{5} e + a^{3} c^{3} \sqrt {-\frac {c^{6} d^{12} + 2 \, a c^{5} d^{10} e^{2} - a^{2} c^{4} d^{8} e^{4} - 4 \, a^{3} c^{3} d^{6} e^{6} - a^{4} c^{2} d^{4} e^{8} + 2 \, a^{5} c d^{2} e^{10} + a^{6} e^{12}}{a^{7} c^{7}}} + 4 \, a c d^{3} e^{3} + 2 \, a^{2} d e^{5}}{a^{3} c^{3}}} \log \left (-27 \, c^{5} d^{10} x - 81 \, a c^{4} d^{8} x e^{2} - 54 \, a^{2} c^{3} d^{6} x e^{4} + 54 \, a^{3} c^{2} d^{4} x e^{6} + 81 \, a^{4} c d^{2} x e^{8} + 27 \, a^{5} x e^{10} + 27 \, {\left (a^{2} c^{5} d^{7} + a^{3} c^{4} d^{5} e^{2} + a^{6} c^{5} \sqrt {-\frac {c^{6} d^{12} + 2 \, a c^{5} d^{10} e^{2} - a^{2} c^{4} d^{8} e^{4} - 4 \, a^{3} c^{3} d^{6} e^{6} - a^{4} c^{2} d^{4} e^{8} + 2 \, a^{5} c d^{2} e^{10} + a^{6} e^{12}}{a^{7} c^{7}}} e - a^{4} c^{3} d^{3} e^{4} - a^{5} c^{2} d e^{6}\right )} \sqrt {-\frac {2 \, c^{2} d^{5} e + a^{3} c^{3} \sqrt {-\frac {c^{6} d^{12} + 2 \, a c^{5} d^{10} e^{2} - a^{2} c^{4} d^{8} e^{4} - 4 \, a^{3} c^{3} d^{6} e^{6} - a^{4} c^{2} d^{4} e^{8} + 2 \, a^{5} c d^{2} e^{10} + a^{6} e^{12}}{a^{7} c^{7}}} + 4 \, a c d^{3} e^{3} + 2 \, a^{2} d e^{5}}{a^{3} c^{3}}}\right ) + 3 \, {\left (a c^{2} x^{4} + a^{2} c\right )} \sqrt {-\frac {2 \, c^{2} d^{5} e + a^{3} c^{3} \sqrt {-\frac {c^{6} d^{12} + 2 \, a c^{5} d^{10} e^{2} - a^{2} c^{4} d^{8} e^{4} - 4 \, a^{3} c^{3} d^{6} e^{6} - a^{4} c^{2} d^{4} e^{8} + 2 \, a^{5} c d^{2} e^{10} + a^{6} e^{12}}{a^{7} c^{7}}} + 4 \, a c d^{3} e^{3} + 2 \, a^{2} d e^{5}}{a^{3} c^{3}}} \log \left (-27 \, c^{5} d^{10} x - 81 \, a c^{4} d^{8} x e^{2} - 54 \, a^{2} c^{3} d^{6} x e^{4} + 54 \, a^{3} c^{2} d^{4} x e^{6} + 81 \, a^{4} c d^{2} x e^{8} + 27 \, a^{5} x e^{10} - 27 \, {\left (a^{2} c^{5} d^{7} + a^{3} c^{4} d^{5} e^{2} + a^{6} c^{5} \sqrt {-\frac {c^{6} d^{12} + 2 \, a c^{5} d^{10} e^{2} - a^{2} c^{4} d^{8} e^{4} - 4 \, a^{3} c^{3} d^{6} e^{6} - a^{4} c^{2} d^{4} e^{8} + 2 \, a^{5} c d^{2} e^{10} + a^{6} e^{12}}{a^{7} c^{7}}} e - a^{4} c^{3} d^{3} e^{4} - a^{5} c^{2} d e^{6}\right )} \sqrt {-\frac {2 \, c^{2} d^{5} e + a^{3} c^{3} \sqrt {-\frac {c^{6} d^{12} + 2 \, a c^{5} d^{10} e^{2} - a^{2} c^{4} d^{8} e^{4} - 4 \, a^{3} c^{3} d^{6} e^{6} - a^{4} c^{2} d^{4} e^{8} + 2 \, a^{5} c d^{2} e^{10} + a^{6} e^{12}}{a^{7} c^{7}}} + 4 \, a c d^{3} e^{3} + 2 \, a^{2} d e^{5}}{a^{3} c^{3}}}\right ) - 3 \, {\left (a c^{2} x^{4} + a^{2} c\right )} \sqrt {-\frac {2 \, c^{2} d^{5} e - a^{3} c^{3} \sqrt {-\frac {c^{6} d^{12} + 2 \, a c^{5} d^{10} e^{2} - a^{2} c^{4} d^{8} e^{4} - 4 \, a^{3} c^{3} d^{6} e^{6} - a^{4} c^{2} d^{4} e^{8} + 2 \, a^{5} c d^{2} e^{10} + a^{6} e^{12}}{a^{7} c^{7}}} + 4 \, a c d^{3} e^{3} + 2 \, a^{2} d e^{5}}{a^{3} c^{3}}} \log \left (-27 \, c^{5} d^{10} x - 81 \, a c^{4} d^{8} x e^{2} - 54 \, a^{2} c^{3} d^{6} x e^{4} + 54 \, a^{3} c^{2} d^{4} x e^{6} + 81 \, a^{4} c d^{2} x e^{8} + 27 \, a^{5} x e^{10} + 27 \, {\left (a^{2} c^{5} d^{7} + a^{3} c^{4} d^{5} e^{2} - a^{6} c^{5} \sqrt {-\frac {c^{6} d^{12} + 2 \, a c^{5} d^{10} e^{2} - a^{2} c^{4} d^{8} e^{4} - 4 \, a^{3} c^{3} d^{6} e^{6} - a^{4} c^{2} d^{4} e^{8} + 2 \, a^{5} c d^{2} e^{10} + a^{6} e^{12}}{a^{7} c^{7}}} e - a^{4} c^{3} d^{3} e^{4} - a^{5} c^{2} d e^{6}\right )} \sqrt {-\frac {2 \, c^{2} d^{5} e - a^{3} c^{3} \sqrt {-\frac {c^{6} d^{12} + 2 \, a c^{5} d^{10} e^{2} - a^{2} c^{4} d^{8} e^{4} - 4 \, a^{3} c^{3} d^{6} e^{6} - a^{4} c^{2} d^{4} e^{8} + 2 \, a^{5} c d^{2} e^{10} + a^{6} e^{12}}{a^{7} c^{7}}} + 4 \, a c d^{3} e^{3} + 2 \, a^{2} d e^{5}}{a^{3} c^{3}}}\right ) + 3 \, {\left (a c^{2} x^{4} + a^{2} c\right )} \sqrt {-\frac {2 \, c^{2} d^{5} e - a^{3} c^{3} \sqrt {-\frac {c^{6} d^{12} + 2 \, a c^{5} d^{10} e^{2} - a^{2} c^{4} d^{8} e^{4} - 4 \, a^{3} c^{3} d^{6} e^{6} - a^{4} c^{2} d^{4} e^{8} + 2 \, a^{5} c d^{2} e^{10} + a^{6} e^{12}}{a^{7} c^{7}}} + 4 \, a c d^{3} e^{3} + 2 \, a^{2} d e^{5}}{a^{3} c^{3}}} \log \left (-27 \, c^{5} d^{10} x - 81 \, a c^{4} d^{8} x e^{2} - 54 \, a^{2} c^{3} d^{6} x e^{4} + 54 \, a^{3} c^{2} d^{4} x e^{6} + 81 \, a^{4} c d^{2} x e^{8} + 27 \, a^{5} x e^{10} - 27 \, {\left (a^{2} c^{5} d^{7} + a^{3} c^{4} d^{5} e^{2} - a^{6} c^{5} \sqrt {-\frac {c^{6} d^{12} + 2 \, a c^{5} d^{10} e^{2} - a^{2} c^{4} d^{8} e^{4} - 4 \, a^{3} c^{3} d^{6} e^{6} - a^{4} c^{2} d^{4} e^{8} + 2 \, a^{5} c d^{2} e^{10} + a^{6} e^{12}}{a^{7} c^{7}}} e - a^{4} c^{3} d^{3} e^{4} - a^{5} c^{2} d e^{6}\right )} \sqrt {-\frac {2 \, c^{2} d^{5} e - a^{3} c^{3} \sqrt {-\frac {c^{6} d^{12} + 2 \, a c^{5} d^{10} e^{2} - a^{2} c^{4} d^{8} e^{4} - 4 \, a^{3} c^{3} d^{6} e^{6} - a^{4} c^{2} d^{4} e^{8} + 2 \, a^{5} c d^{2} e^{10} + a^{6} e^{12}}{a^{7} c^{7}}} + 4 \, a c d^{3} e^{3} + 2 \, a^{2} d e^{5}}{a^{3} c^{3}}}\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 = 1.81, size = 352, normalized size = 0.97 \begin {gather*} \operatorname {RootSum} {\left (65536 t^{4} a^{7} c^{7} + t^{2} \cdot \left (9216 a^{6} c^{4} d e^{5} + 18432 a^{5} c^{5} d^{3} e^{3} + 9216 a^{4} c^{6} d^{5} e\right ) + 81 a^{6} e^{12} + 486 a^{5} c d^{2} e^{10} + 1215 a^{4} c^{2} d^{4} e^{8} + 1620 a^{3} c^{3} d^{6} e^{6} + 1215 a^{2} c^{4} d^{8} e^{4} + 486 a c^{5} d^{10} e^{2} + 81 c^{6} d^{12}, \left ( t \mapsto t \log {\left (x + \frac {4096 t^{3} a^{6} c^{5} e + 432 t a^{5} c^{2} d e^{6} + 720 t a^{4} c^{3} d^{3} e^{4} + 144 t a^{3} c^{4} d^{5} e^{2} - 144 t a^{2} c^{5} d^{7}}{27 a^{5} e^{10} + 81 a^{4} c d^{2} e^{8} + 54 a^{3} c^{2} d^{4} e^{6} - 54 a^{2} c^{3} d^{6} e^{4} - 81 a c^{4} d^{8} e^{2} - 27 c^{5} d^{10}} \right )} \right )\right )} + \frac {x^{3} \left (- a e^{3} + 3 c d^{2} e\right ) + x \left (- 3 a d e^{2} + c d^{3}\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 = 9.01, size = 425, normalized size = 1.17 \begin {gather*} \frac {3 \, c d^{2} x^{3} e + c d^{3} x - a x^{3} e^{3} - 3 \, a d x e^{2}}{4 \, {\left (c x^{4} + a\right )} a c} + \frac {3 \, \sqrt {2} {\left (\left (a c^{3}\right )^{\frac {1}{4}} c^{3} d^{3} + \left (a c^{3}\right )^{\frac {1}{4}} a c^{2} d e^{2} + \left (a c^{3}\right )^{\frac {3}{4}} c d^{2} e + \left (a c^{3}\right )^{\frac {3}{4}} a e^{3}\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^{4}} + \frac {3 \, \sqrt {2} {\left (\left (a c^{3}\right )^{\frac {1}{4}} c^{3} d^{3} + \left (a c^{3}\right )^{\frac {1}{4}} a c^{2} d e^{2} + \left (a c^{3}\right )^{\frac {3}{4}} c d^{2} e + \left (a c^{3}\right )^{\frac {3}{4}} a e^{3}\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^{4}} + \frac {3 \, \sqrt {2} {\left (\left (a c^{3}\right )^{\frac {1}{4}} c^{3} d^{3} + \left (a c^{3}\right )^{\frac {1}{4}} a c^{2} d e^{2} - \left (a c^{3}\right )^{\frac {3}{4}} c d^{2} e - \left (a c^{3}\right )^{\frac {3}{4}} a e^{3}\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^{4}} - \frac {3 \, \sqrt {2} {\left (\left (a c^{3}\right )^{\frac {1}{4}} c^{3} d^{3} + \left (a c^{3}\right )^{\frac {1}{4}} a c^{2} d e^{2} - \left (a c^{3}\right )^{\frac {3}{4}} c d^{2} e - \left (a c^{3}\right )^{\frac {3}{4}} a e^{3}\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^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.94, size = 2560, normalized size = 7.05 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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