Optimal. Leaf size=349 \[ -\frac {3 d \left (\sqrt {c} d^2-\sqrt {a} e^2\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{16 \sqrt {2} a^{7/4} c^{3/4}}+\frac {3 d \left (\sqrt {c} d^2-\sqrt {a} e^2\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{16 \sqrt {2} a^{7/4} c^{3/4}}-\frac {3 d \left (\sqrt {a} e^2+\sqrt {c} d^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} c^{3/4}}+\frac {3 d \left (\sqrt {a} e^2+\sqrt {c} d^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{8 \sqrt {2} a^{7/4} c^{3/4}}+\frac {3 d^2 e \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{4 a^{3/2} \sqrt {c}}-\frac {a e^3-c x \left (d^3+3 d^2 e x+3 d e^2 x^2\right )}{4 a c \left (a+c x^4\right )} \]
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Rubi [A] time = 0.30, antiderivative size = 349, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 12, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.706, Rules used = {1854, 27, 12, 1876, 275, 205, 1168, 1162, 617, 204, 1165, 628} \[ -\frac {3 d \left (\sqrt {c} d^2-\sqrt {a} e^2\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{16 \sqrt {2} a^{7/4} c^{3/4}}+\frac {3 d \left (\sqrt {c} d^2-\sqrt {a} e^2\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{16 \sqrt {2} a^{7/4} c^{3/4}}-\frac {3 d \left (\sqrt {a} e^2+\sqrt {c} d^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} c^{3/4}}+\frac {3 d \left (\sqrt {a} e^2+\sqrt {c} d^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{8 \sqrt {2} a^{7/4} c^{3/4}}+\frac {3 d^2 e \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{4 a^{3/2} \sqrt {c}}-\frac {a e^3-c x \left (3 d^2 e x+d^3+3 d e^2 x^2\right )}{4 a c \left (a+c x^4\right )} \]
Antiderivative was successfully verified.
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Rule 12
Rule 27
Rule 204
Rule 205
Rule 275
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 1168
Rule 1854
Rule 1876
Rubi steps
\begin {align*} \int \frac {(d+e x)^3}{\left (a+c x^4\right )^2} \, dx &=-\frac {a e^3-c x \left (d^3+3 d^2 e x+3 d e^2 x^2\right )}{4 a c \left (a+c x^4\right )}-\frac {\int \frac {-3 d^3-6 d^2 e x-3 d e^2 x^2}{a+c x^4} \, dx}{4 a}\\ &=-\frac {a e^3-c x \left (d^3+3 d^2 e x+3 d e^2 x^2\right )}{4 a c \left (a+c x^4\right )}-\frac {\int -\frac {3 d (d+e x)^2}{a+c x^4} \, dx}{4 a}\\ &=-\frac {a e^3-c x \left (d^3+3 d^2 e x+3 d e^2 x^2\right )}{4 a c \left (a+c x^4\right )}+\frac {(3 d) \int \frac {(d+e x)^2}{a+c x^4} \, dx}{4 a}\\ &=-\frac {a e^3-c x \left (d^3+3 d^2 e x+3 d e^2 x^2\right )}{4 a c \left (a+c x^4\right )}+\frac {(3 d) \int \left (\frac {2 d e x}{a+c x^4}+\frac {d^2+e^2 x^2}{a+c x^4}\right ) \, dx}{4 a}\\ &=-\frac {a e^3-c x \left (d^3+3 d^2 e x+3 d e^2 x^2\right )}{4 a c \left (a+c x^4\right )}+\frac {(3 d) \int \frac {d^2+e^2 x^2}{a+c x^4} \, dx}{4 a}+\frac {\left (3 d^2 e\right ) \int \frac {x}{a+c x^4} \, dx}{2 a}\\ &=-\frac {a e^3-c x \left (d^3+3 d^2 e x+3 d e^2 x^2\right )}{4 a c \left (a+c x^4\right )}+\frac {\left (3 d^2 e\right ) \operatorname {Subst}\left (\int \frac {1}{a+c x^2} \, dx,x,x^2\right )}{4 a}+\frac {\left (3 d \left (\frac {\sqrt {c} d^2}{\sqrt {a}}-e^2\right )\right ) \int \frac {\sqrt {a} \sqrt {c}-c x^2}{a+c x^4} \, dx}{8 a c}+\frac {\left (3 d \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right )\right ) \int \frac {\sqrt {a} \sqrt {c}+c x^2}{a+c x^4} \, dx}{8 a c}\\ &=-\frac {a e^3-c x \left (d^3+3 d^2 e x+3 d e^2 x^2\right )}{4 a c \left (a+c x^4\right )}+\frac {3 d^2 e \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{4 a^{3/2} \sqrt {c}}+\frac {\left (3 d \left (\frac {\sqrt {c} d^2}{\sqrt {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 c}+\frac {\left (3 d \left (\frac {\sqrt {c} d^2}{\sqrt {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 c}-\frac {\left (3 d \left (\sqrt {c} d^2-\sqrt {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^{3/4}}-\frac {\left (3 d \left (\sqrt {c} d^2-\sqrt {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^{3/4}}\\ &=-\frac {a e^3-c x \left (d^3+3 d^2 e x+3 d e^2 x^2\right )}{4 a c \left (a+c x^4\right )}+\frac {3 d^2 e \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{4 a^{3/2} \sqrt {c}}-\frac {3 d \left (\sqrt {c} d^2-\sqrt {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^{3/4}}+\frac {3 d \left (\sqrt {c} d^2-\sqrt {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^{3/4}}+\frac {\left (3 d \left (\sqrt {c} d^2+\sqrt {a} e^2\right )\right ) \operatorname {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^{3/4}}-\frac {\left (3 d \left (\sqrt {c} d^2+\sqrt {a} e^2\right )\right ) \operatorname {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^{3/4}}\\ &=-\frac {a e^3-c x \left (d^3+3 d^2 e x+3 d e^2 x^2\right )}{4 a c \left (a+c x^4\right )}+\frac {3 d^2 e \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{4 a^{3/2} \sqrt {c}}-\frac {3 d \left (\sqrt {c} d^2+\sqrt {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^{3/4}}+\frac {3 d \left (\sqrt {c} d^2+\sqrt {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^{3/4}}-\frac {3 d \left (\sqrt {c} d^2-\sqrt {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^{3/4}}+\frac {3 d \left (\sqrt {c} d^2-\sqrt {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^{3/4}}\\ \end {align*}
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Mathematica [A] time = 0.37, size = 347, normalized size = 0.99 \[ \frac {3 \sqrt {2} \sqrt [4]{c} \left (a^{3/4} d e^2-\sqrt [4]{a} \sqrt {c} d^3\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )+3 \sqrt {2} \sqrt [4]{c} \left (\sqrt [4]{a} \sqrt {c} d^3-a^{3/4} d e^2\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )-6 \sqrt [4]{a} \sqrt [4]{c} d \left (4 \sqrt [4]{a} \sqrt [4]{c} d e+\sqrt {2} \sqrt {a} e^2+\sqrt {2} \sqrt {c} d^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )+6 \sqrt [4]{a} \sqrt [4]{c} d \left (-4 \sqrt [4]{a} \sqrt [4]{c} d e+\sqrt {2} \sqrt {a} e^2+\sqrt {2} \sqrt {c} d^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )-\frac {8 a \left (a e^3-c d x \left (d^2+3 d e x+3 e^2 x^2\right )\right )}{a+c x^4}}{32 a^2 c} \]
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.34, size = 342, normalized size = 0.98 \[ \frac {3 \, c d x^{3} e^{2} + 3 \, c d^{2} x^{2} e + c d^{3} x - a e^{3}}{4 \, {\left (c x^{4} + a\right )} a c} + \frac {3 \, \sqrt {2} {\left (2 \, \sqrt {2} \sqrt {a c} c^{2} d^{2} e + \left (a c^{3}\right )^{\frac {1}{4}} c^{2} d^{3} + \left (a c^{3}\right )^{\frac {3}{4}} d e^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{16 \, a^{2} c^{3}} + \frac {3 \, \sqrt {2} {\left (2 \, \sqrt {2} \sqrt {a c} c^{2} d^{2} e + \left (a c^{3}\right )^{\frac {1}{4}} c^{2} d^{3} + \left (a c^{3}\right )^{\frac {3}{4}} d e^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{16 \, a^{2} c^{3}} + \frac {3 \, \sqrt {2} {\left (\left (a c^{3}\right )^{\frac {1}{4}} c^{2} d^{3} - \left (a c^{3}\right )^{\frac {3}{4}} d e^{2}\right )} \log \left (x^{2} + \sqrt {2} x \left (\frac {a}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{c}}\right )}{32 \, a^{2} c^{3}} - \frac {3 \, \sqrt {2} {\left (\left (a c^{3}\right )^{\frac {1}{4}} c^{2} d^{3} - \left (a c^{3}\right )^{\frac {3}{4}} d e^{2}\right )} \log \left (x^{2} - \sqrt {2} x \left (\frac {a}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{c}}\right )}{32 \, a^{2} c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 390, normalized size = 1.12 \[ \frac {e^{3} x^{4}}{4 \left (c \,x^{4}+a \right ) a}+\frac {3 d \,e^{2} x^{3}}{4 \left (c \,x^{4}+a \right ) a}+\frac {3 d^{2} e \,x^{2}}{4 \left (c \,x^{4}+a \right ) a}+\frac {d^{3} x}{4 \left (c \,x^{4}+a \right ) a}+\frac {3 d^{2} e \arctan \left (\sqrt {\frac {c}{a}}\, x^{2}\right )}{4 \sqrt {a c}\, a}+\frac {3 \sqrt {2}\, d \,e^{2} \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )}{16 \left (\frac {a}{c}\right )^{\frac {1}{4}} a c}+\frac {3 \sqrt {2}\, d \,e^{2} \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )}{16 \left (\frac {a}{c}\right )^{\frac {1}{4}} a c}+\frac {3 \sqrt {2}\, d \,e^{2} \ln \left (\frac {x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a}{c}}}{x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a}{c}}}\right )}{32 \left (\frac {a}{c}\right )^{\frac {1}{4}} a c}+\frac {3 \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, d^{3} \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )}{16 a^{2}}+\frac {3 \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, d^{3} \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )}{16 a^{2}}+\frac {3 \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, d^{3} \ln \left (\frac {x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a}{c}}}{x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a}{c}}}\right )}{32 a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.20, size = 332, normalized size = 0.95 \[ \frac {3 \, d {\left (\frac {\sqrt {2} {\left (\sqrt {c} d^{2} - \sqrt {a} e^{2}\right )} \log \left (\sqrt {c} x^{2} + \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} c^{\frac {3}{4}}} - \frac {\sqrt {2} {\left (\sqrt {c} d^{2} - \sqrt {a} e^{2}\right )} \log \left (\sqrt {c} x^{2} - \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} c^{\frac {3}{4}}} + \frac {2 \, {\left (\sqrt {2} a^{\frac {1}{4}} c^{\frac {3}{4}} d^{2} + \sqrt {2} a^{\frac {3}{4}} c^{\frac {1}{4}} e^{2} - 4 \, \sqrt {a} \sqrt {c} d e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {c} x + \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{a^{\frac {3}{4}} \sqrt {\sqrt {a} \sqrt {c}} c^{\frac {3}{4}}} + \frac {2 \, {\left (\sqrt {2} a^{\frac {1}{4}} c^{\frac {3}{4}} d^{2} + \sqrt {2} a^{\frac {3}{4}} c^{\frac {1}{4}} e^{2} + 4 \, \sqrt {a} \sqrt {c} d e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {c} x - \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{a^{\frac {3}{4}} \sqrt {\sqrt {a} \sqrt {c}} c^{\frac {3}{4}}}\right )}}{32 \, a} + \frac {3 \, c d e^{2} x^{3} + 3 \, c d^{2} e x^{2} + c d^{3} x - a e^{3}}{4 \, {\left (a c^{2} x^{4} + a^{2} c\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.43, size = 670, normalized size = 1.92 \[ \left (\sum _{k=1}^4\ln \left (\frac {c\,d^2\,\left (27\,c\,d^5\,e^2-9\,a\,d\,e^6+36\,c\,d^4\,e^3\,x-{\mathrm {root}\left (65536\,a^7\,c^3\,z^4+27648\,a^4\,c^2\,d^4\,e^2\,z^2+3456\,a^3\,c\,d^4\,e^5\,z-3456\,a^2\,c^2\,d^8\,e\,z+162\,a\,c\,d^8\,e^4+81\,a^2\,d^4\,e^8+81\,c^2\,d^{12},z,k\right )}^2\,a^3\,c^2\,d\,256-\mathrm {root}\left (65536\,a^7\,c^3\,z^4+27648\,a^4\,c^2\,d^4\,e^2\,z^2+3456\,a^3\,c\,d^4\,e^5\,z-3456\,a^2\,c^2\,d^8\,e\,z+162\,a\,c\,d^8\,e^4+81\,a^2\,d^4\,e^8+81\,c^2\,d^{12},z,k\right )\,a\,c^2\,d^4\,x\,48+\mathrm {root}\left (65536\,a^7\,c^3\,z^4+27648\,a^4\,c^2\,d^4\,e^2\,z^2+3456\,a^3\,c\,d^4\,e^5\,z-3456\,a^2\,c^2\,d^8\,e\,z+162\,a\,c\,d^8\,e^4+81\,a^2\,d^4\,e^8+81\,c^2\,d^{12},z,k\right )\,a^2\,c\,e^4\,x\,48+{\mathrm {root}\left (65536\,a^7\,c^3\,z^4+27648\,a^4\,c^2\,d^4\,e^2\,z^2+3456\,a^3\,c\,d^4\,e^5\,z-3456\,a^2\,c^2\,d^8\,e\,z+162\,a\,c\,d^8\,e^4+81\,a^2\,d^4\,e^8+81\,c^2\,d^{12},z,k\right )}^2\,a^3\,c^2\,e\,x\,512-\mathrm {root}\left (65536\,a^7\,c^3\,z^4+27648\,a^4\,c^2\,d^4\,e^2\,z^2+3456\,a^3\,c\,d^4\,e^5\,z-3456\,a^2\,c^2\,d^8\,e\,z+162\,a\,c\,d^8\,e^4+81\,a^2\,d^4\,e^8+81\,c^2\,d^{12},z,k\right )\,a^2\,c\,d\,e^3\,192\right )\,3}{a^3\,64}\right )\,\mathrm {root}\left (65536\,a^7\,c^3\,z^4+27648\,a^4\,c^2\,d^4\,e^2\,z^2+3456\,a^3\,c\,d^4\,e^5\,z-3456\,a^2\,c^2\,d^8\,e\,z+162\,a\,c\,d^8\,e^4+81\,a^2\,d^4\,e^8+81\,c^2\,d^{12},z,k\right )\right )+\frac {\frac {d^3\,x}{4\,a}-\frac {e^3}{4\,c}+\frac {3\,d^2\,e\,x^2}{4\,a}+\frac {3\,d\,e^2\,x^3}{4\,a}}{c\,x^4+a} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 8.33, size = 350, normalized size = 1.00 \[ \operatorname {RootSum} {\left (65536 t^{4} a^{7} c^{3} + 27648 t^{2} a^{4} c^{2} d^{4} e^{2} + t \left (3456 a^{3} c d^{4} e^{5} - 3456 a^{2} c^{2} d^{8} e\right ) + 81 a^{2} d^{4} e^{8} + 162 a c d^{8} e^{4} + 81 c^{2} d^{12}, \left (t \mapsto t \log {\left (x + \frac {4096 t^{3} a^{7} c^{2} e^{6} + 28672 t^{3} a^{6} c^{3} d^{4} e^{2} - 7680 t^{2} a^{5} c^{2} d^{4} e^{5} + 1536 t^{2} a^{4} c^{3} d^{8} e + 2160 t a^{4} c d^{4} e^{8} + 9216 t a^{3} c^{2} d^{8} e^{4} + 144 t a^{2} c^{3} d^{12} + 162 a^{3} d^{4} e^{11} - 648 a^{2} c d^{8} e^{7} - 810 a c^{2} d^{12} e^{3}}{27 a^{3} d^{3} e^{12} - 891 a^{2} c d^{7} e^{8} - 891 a c^{2} d^{11} e^{4} + 27 c^{3} d^{15}} \right )} \right )\right )} + \frac {- a e^{3} + c d^{3} x + 3 c d^{2} e x^{2} + 3 c d e^{2} x^{3}}{4 a^{2} c + 4 a c^{2} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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