Optimal. Leaf size=552 \[ -\frac {\sqrt [4]{c} \left (\sqrt {c} d^2 \left (c d^4-3 a e^4\right )-\sqrt {a} e^2 \left (3 c d^4-a e^4\right )\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} \left (a e^4+c d^4\right )^2}+\frac {\sqrt [4]{c} \left (\sqrt {c} d^2 \left (c d^4-3 a e^4\right )-\sqrt {a} e^2 \left (3 c d^4-a e^4\right )\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} \left (a e^4+c d^4\right )^2}-\frac {\sqrt [4]{c} \left (\sqrt {a} e^2 \left (3 c d^4-a e^4\right )+\sqrt {c} d^2 \left (c d^4-3 a e^4\right )\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} \left (a e^4+c d^4\right )^2}+\frac {\sqrt [4]{c} \left (\sqrt {a} e^2 \left (3 c d^4-a e^4\right )+\sqrt {c} d^2 \left (c d^4-3 a e^4\right )\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{2 \sqrt {2} a^{3/4} \left (a e^4+c d^4\right )^2}-\frac {\sqrt {c} d e \left (c d^4-a e^4\right ) \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{\sqrt {a} \left (a e^4+c d^4\right )^2}-\frac {e^3}{(d+e x) \left (a e^4+c d^4\right )}-\frac {c d^3 e^3 \log \left (a+c x^4\right )}{\left (a e^4+c d^4\right )^2}+\frac {4 c d^3 e^3 \log (d+e x)}{\left (a e^4+c d^4\right )^2} \]
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Rubi [A] time = 0.81, antiderivative size = 552, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 12, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.706, Rules used = {6725, 1876, 1248, 635, 205, 260, 1168, 1162, 617, 204, 1165, 628} \begin {gather*} -\frac {\sqrt [4]{c} \left (\sqrt {c} d^2 \left (c d^4-3 a e^4\right )-\sqrt {a} e^2 \left (3 c d^4-a e^4\right )\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} \left (a e^4+c d^4\right )^2}+\frac {\sqrt [4]{c} \left (\sqrt {c} d^2 \left (c d^4-3 a e^4\right )-\sqrt {a} e^2 \left (3 c d^4-a e^4\right )\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} \left (a e^4+c d^4\right )^2}-\frac {\sqrt [4]{c} \left (\sqrt {c} d^2 \left (c d^4-3 a e^4\right )+\sqrt {a} e^2 \left (3 c d^4-a e^4\right )\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} \left (a e^4+c d^4\right )^2}+\frac {\sqrt [4]{c} \left (\sqrt {c} d^2 \left (c d^4-3 a e^4\right )+\sqrt {a} e^2 \left (3 c d^4-a e^4\right )\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{2 \sqrt {2} a^{3/4} \left (a e^4+c d^4\right )^2}-\frac {c d^3 e^3 \log \left (a+c x^4\right )}{\left (a e^4+c d^4\right )^2}-\frac {\sqrt {c} d e \left (c d^4-a e^4\right ) \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{\sqrt {a} \left (a e^4+c d^4\right )^2}-\frac {e^3}{(d+e x) \left (a e^4+c d^4\right )}+\frac {4 c d^3 e^3 \log (d+e x)}{\left (a e^4+c d^4\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 204
Rule 205
Rule 260
Rule 617
Rule 628
Rule 635
Rule 1162
Rule 1165
Rule 1168
Rule 1248
Rule 1876
Rule 6725
Rubi steps
\begin {align*} \int \frac {1}{(d+e x)^2 \left (a+c x^4\right )} \, dx &=\int \left (\frac {e^4}{\left (c d^4+a e^4\right ) (d+e x)^2}+\frac {4 c d^3 e^4}{\left (c d^4+a e^4\right )^2 (d+e x)}+\frac {c \left (d^2 \left (c d^4-3 a e^4\right )-2 d e \left (c d^4-a e^4\right ) x+e^2 \left (3 c d^4-a e^4\right ) x^2-4 c d^3 e^3 x^3\right )}{\left (c d^4+a e^4\right )^2 \left (a+c x^4\right )}\right ) \, dx\\ &=-\frac {e^3}{\left (c d^4+a e^4\right ) (d+e x)}+\frac {4 c d^3 e^3 \log (d+e x)}{\left (c d^4+a e^4\right )^2}+\frac {c \int \frac {d^2 \left (c d^4-3 a e^4\right )-2 d e \left (c d^4-a e^4\right ) x+e^2 \left (3 c d^4-a e^4\right ) x^2-4 c d^3 e^3 x^3}{a+c x^4} \, dx}{\left (c d^4+a e^4\right )^2}\\ &=-\frac {e^3}{\left (c d^4+a e^4\right ) (d+e x)}+\frac {4 c d^3 e^3 \log (d+e x)}{\left (c d^4+a e^4\right )^2}+\frac {c \int \left (\frac {x \left (-2 d e \left (c d^4-a e^4\right )-4 c d^3 e^3 x^2\right )}{a+c x^4}+\frac {d^2 \left (c d^4-3 a e^4\right )+e^2 \left (3 c d^4-a e^4\right ) x^2}{a+c x^4}\right ) \, dx}{\left (c d^4+a e^4\right )^2}\\ &=-\frac {e^3}{\left (c d^4+a e^4\right ) (d+e x)}+\frac {4 c d^3 e^3 \log (d+e x)}{\left (c d^4+a e^4\right )^2}+\frac {c \int \frac {x \left (-2 d e \left (c d^4-a e^4\right )-4 c d^3 e^3 x^2\right )}{a+c x^4} \, dx}{\left (c d^4+a e^4\right )^2}+\frac {c \int \frac {d^2 \left (c d^4-3 a e^4\right )+e^2 \left (3 c d^4-a e^4\right ) x^2}{a+c x^4} \, dx}{\left (c d^4+a e^4\right )^2}\\ &=-\frac {e^3}{\left (c d^4+a e^4\right ) (d+e x)}+\frac {4 c d^3 e^3 \log (d+e x)}{\left (c d^4+a e^4\right )^2}+\frac {c \operatorname {Subst}\left (\int \frac {-2 d e \left (c d^4-a e^4\right )-4 c d^3 e^3 x}{a+c x^2} \, dx,x,x^2\right )}{2 \left (c d^4+a e^4\right )^2}-\frac {\left (3 c d^4 e^2-a e^6-\frac {\sqrt {c} d^2 \left (c d^4-3 a e^4\right )}{\sqrt {a}}\right ) \int \frac {\sqrt {a} \sqrt {c}-c x^2}{a+c x^4} \, dx}{2 \left (c d^4+a e^4\right )^2}+\frac {\left (3 c d^4 e^2-a e^6+\frac {\sqrt {c} d^2 \left (c d^4-3 a e^4\right )}{\sqrt {a}}\right ) \int \frac {\sqrt {a} \sqrt {c}+c x^2}{a+c x^4} \, dx}{2 \left (c d^4+a e^4\right )^2}\\ &=-\frac {e^3}{\left (c d^4+a e^4\right ) (d+e x)}+\frac {4 c d^3 e^3 \log (d+e x)}{\left (c d^4+a e^4\right )^2}-\frac {\left (2 c^2 d^3 e^3\right ) \operatorname {Subst}\left (\int \frac {x}{a+c x^2} \, dx,x,x^2\right )}{\left (c d^4+a e^4\right )^2}-\frac {\left (c d e \left (c d^4-a e^4\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a+c x^2} \, dx,x,x^2\right )}{\left (c d^4+a e^4\right )^2}+\frac {\left (\sqrt [4]{c} \left (3 c d^4 e^2-a e^6-\frac {\sqrt {c} d^2 \left (c d^4-3 a e^4\right )}{\sqrt {a}}\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}{4 \sqrt {2} \sqrt [4]{a} \left (c d^4+a e^4\right )^2}+\frac {\left (\sqrt [4]{c} \left (3 c d^4 e^2-a e^6-\frac {\sqrt {c} d^2 \left (c d^4-3 a e^4\right )}{\sqrt {a}}\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}{4 \sqrt {2} \sqrt [4]{a} \left (c d^4+a e^4\right )^2}+\frac {\left (3 c d^4 e^2-a e^6+\frac {\sqrt {c} d^2 \left (c d^4-3 a e^4\right )}{\sqrt {a}}\right ) \int \frac {1}{\frac {\sqrt {a}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{4 \left (c d^4+a e^4\right )^2}+\frac {\left (3 c d^4 e^2-a e^6+\frac {\sqrt {c} d^2 \left (c d^4-3 a e^4\right )}{\sqrt {a}}\right ) \int \frac {1}{\frac {\sqrt {a}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{4 \left (c d^4+a e^4\right )^2}\\ &=-\frac {e^3}{\left (c d^4+a e^4\right ) (d+e x)}-\frac {\sqrt {c} d e \left (c d^4-a e^4\right ) \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{\sqrt {a} \left (c d^4+a e^4\right )^2}+\frac {4 c d^3 e^3 \log (d+e x)}{\left (c d^4+a e^4\right )^2}+\frac {\sqrt [4]{c} \left (3 c d^4 e^2-a e^6-\frac {\sqrt {c} d^2 \left (c d^4-3 a e^4\right )}{\sqrt {a}}\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} \sqrt [4]{a} \left (c d^4+a e^4\right )^2}-\frac {\sqrt [4]{c} \left (3 c d^4 e^2-a e^6-\frac {\sqrt {c} d^2 \left (c d^4-3 a e^4\right )}{\sqrt {a}}\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} \sqrt [4]{a} \left (c d^4+a e^4\right )^2}-\frac {c d^3 e^3 \log \left (a+c x^4\right )}{\left (c d^4+a e^4\right )^2}+\frac {\left (\sqrt [4]{c} \left (3 c d^4 e^2-a e^6+\frac {\sqrt {c} d^2 \left (c d^4-3 a e^4\right )}{\sqrt {a}}\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 )}{2 \sqrt {2} \sqrt [4]{a} \left (c d^4+a e^4\right )^2}-\frac {\left (\sqrt [4]{c} \left (3 c d^4 e^2-a e^6+\frac {\sqrt {c} d^2 \left (c d^4-3 a e^4\right )}{\sqrt {a}}\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 )}{2 \sqrt {2} \sqrt [4]{a} \left (c d^4+a e^4\right )^2}\\ &=-\frac {e^3}{\left (c d^4+a e^4\right ) (d+e x)}-\frac {\sqrt {c} d e \left (c d^4-a e^4\right ) \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{\sqrt {a} \left (c d^4+a e^4\right )^2}-\frac {\sqrt [4]{c} \left (3 c d^4 e^2-a e^6+\frac {\sqrt {c} d^2 \left (c d^4-3 a e^4\right )}{\sqrt {a}}\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} \sqrt [4]{a} \left (c d^4+a e^4\right )^2}+\frac {\sqrt [4]{c} \left (3 c d^4 e^2-a e^6+\frac {\sqrt {c} d^2 \left (c d^4-3 a e^4\right )}{\sqrt {a}}\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} \sqrt [4]{a} \left (c d^4+a e^4\right )^2}+\frac {4 c d^3 e^3 \log (d+e x)}{\left (c d^4+a e^4\right )^2}+\frac {\sqrt [4]{c} \left (3 c d^4 e^2-a e^6-\frac {\sqrt {c} d^2 \left (c d^4-3 a e^4\right )}{\sqrt {a}}\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} \sqrt [4]{a} \left (c d^4+a e^4\right )^2}-\frac {\sqrt [4]{c} \left (3 c d^4 e^2-a e^6-\frac {\sqrt {c} d^2 \left (c d^4-3 a e^4\right )}{\sqrt {a}}\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} \sqrt [4]{a} \left (c d^4+a e^4\right )^2}-\frac {c d^3 e^3 \log \left (a+c x^4\right )}{\left (c d^4+a e^4\right )^2}\\ \end {align*}
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Mathematica [A] time = 0.61, size = 524, normalized size = 0.95 \begin {gather*} \frac {-\frac {\sqrt {2} \sqrt [4]{c} \left (a^{3/2} e^6-3 \sqrt {a} c d^4 e^2-3 a \sqrt {c} d^2 e^4+c^{3/2} d^6\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{a^{3/4}}+\frac {\sqrt {2} \sqrt [4]{c} \left (a^{3/2} e^6-3 \sqrt {a} c d^4 e^2-3 a \sqrt {c} d^2 e^4+c^{3/2} d^6\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{a^{3/4}}+\frac {2 \sqrt [4]{c} \left (\sqrt {a} e^2-\sqrt {c} d^2\right ) \left (-4 a^{3/4} \sqrt [4]{c} d e^3-4 \sqrt [4]{a} c^{3/4} d^3 e+4 \sqrt {2} \sqrt {a} \sqrt {c} d^2 e^2+\sqrt {2} a e^4+\sqrt {2} c d^4\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{a^{3/4}}+\frac {2 \sqrt [4]{c} \left (\sqrt {c} d^2-\sqrt {a} e^2\right ) \left (4 a^{3/4} \sqrt [4]{c} d e^3+4 \sqrt [4]{a} c^{3/4} d^3 e+4 \sqrt {2} \sqrt {a} \sqrt {c} d^2 e^2+\sqrt {2} a e^4+\sqrt {2} c d^4\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{a^{3/4}}-\frac {8 e^3 \left (a e^4+c d^4\right )}{d+e x}-8 c d^3 e^3 \log \left (a+c x^4\right )+32 c d^3 e^3 \log (d+e x)}{8 \left (a e^4+c d^4\right )^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{(d+e x)^2 \left (a+c x^4\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 2.78, size = 646, normalized size = 1.17 \begin {gather*} -\frac {c d^{3} e^{3} \log \left ({\left | c x^{4} + a \right |}\right )}{c^{2} d^{8} + 2 \, a c d^{4} e^{4} + a^{2} e^{8}} + \frac {4 \, c d^{3} e^{4} \log \left ({\left | x e + d \right |}\right )}{c^{2} d^{8} e + 2 \, a c d^{4} e^{5} + a^{2} e^{9}} + \frac {{\left (\left (a c^{3}\right )^{\frac {1}{4}} c^{2} d^{2} - \left (a c^{3}\right )^{\frac {3}{4}} 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 )}{2 \, {\left (\sqrt {2} a c^{3} d^{4} - 4 \, \left (a c^{3}\right )^{\frac {1}{4}} a c^{2} d^{3} e + 4 \, \sqrt {2} \sqrt {a c} a c^{2} d^{2} e^{2} + \sqrt {2} a^{2} c^{2} e^{4} - 4 \, \left (a c^{3}\right )^{\frac {3}{4}} a d e^{3}\right )}} + \frac {{\left (\left (a c^{3}\right )^{\frac {1}{4}} c^{2} d^{2} - \left (a c^{3}\right )^{\frac {3}{4}} 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 )}{2 \, {\left (\sqrt {2} a c^{3} d^{4} + 4 \, \left (a c^{3}\right )^{\frac {1}{4}} a c^{2} d^{3} e + 4 \, \sqrt {2} \sqrt {a c} a c^{2} d^{2} e^{2} + \sqrt {2} a^{2} c^{2} e^{4} + 4 \, \left (a c^{3}\right )^{\frac {3}{4}} a d e^{3}\right )}} + \frac {{\left (\sqrt {2} \left (a c^{3}\right )^{\frac {1}{4}} c^{3} d^{6} - 3 \, \sqrt {2} \left (a c^{3}\right )^{\frac {3}{4}} c d^{4} e^{2} - 3 \, \sqrt {2} \left (a c^{3}\right )^{\frac {1}{4}} a c^{2} d^{2} e^{4} + \sqrt {2} \left (a c^{3}\right )^{\frac {3}{4}} a e^{6}\right )} \log \left (x^{2} + \sqrt {2} x \left (\frac {a}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{c}}\right )}{8 \, {\left (a c^{4} d^{8} + 2 \, a^{2} c^{3} d^{4} e^{4} + a^{3} c^{2} e^{8}\right )}} - \frac {{\left (\sqrt {2} \left (a c^{3}\right )^{\frac {1}{4}} c^{3} d^{6} - 3 \, \sqrt {2} \left (a c^{3}\right )^{\frac {3}{4}} c d^{4} e^{2} - 3 \, \sqrt {2} \left (a c^{3}\right )^{\frac {1}{4}} a c^{2} d^{2} e^{4} + \sqrt {2} \left (a c^{3}\right )^{\frac {3}{4}} a e^{6}\right )} \log \left (x^{2} - \sqrt {2} x \left (\frac {a}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{c}}\right )}{8 \, {\left (a c^{4} d^{8} + 2 \, a^{2} c^{3} d^{4} e^{4} + a^{3} c^{2} e^{8}\right )}} - \frac {c d^{4} e^{3} + a e^{7}}{{\left (c d^{4} + a e^{4}\right )}^{2} {\left (x e + d\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 866, normalized size = 1.57
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.11, size = 561, normalized size = 1.02 \begin {gather*} \frac {4 \, c d^{3} e^{3} \log \left (e x + d\right )}{c^{2} d^{8} + 2 \, a c d^{4} e^{4} + a^{2} e^{8}} - \frac {e^{3}}{c d^{5} + a d e^{4} + {\left (c d^{4} e + a e^{5}\right )} x} - \frac {c {\left (\frac {\sqrt {2} {\left (4 \, \sqrt {2} a^{\frac {3}{4}} c^{\frac {5}{4}} d^{3} e^{3} - c^{2} d^{6} + 3 \, \sqrt {a} c^{\frac {3}{2}} d^{4} e^{2} + 3 \, a c d^{2} e^{4} - a^{\frac {3}{2}} \sqrt {c} e^{6}\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 {5}{4}}} + \frac {\sqrt {2} {\left (4 \, \sqrt {2} a^{\frac {3}{4}} c^{\frac {5}{4}} d^{3} e^{3} + c^{2} d^{6} - 3 \, \sqrt {a} c^{\frac {3}{2}} d^{4} e^{2} - 3 \, a c d^{2} e^{4} + a^{\frac {3}{2}} \sqrt {c} e^{6}\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 {5}{4}}} - \frac {2 \, {\left (\sqrt {2} a^{\frac {1}{4}} c^{\frac {9}{4}} d^{6} + 3 \, \sqrt {2} a^{\frac {3}{4}} c^{\frac {7}{4}} d^{4} e^{2} - 3 \, \sqrt {2} a^{\frac {5}{4}} c^{\frac {5}{4}} d^{2} e^{4} - \sqrt {2} a^{\frac {7}{4}} c^{\frac {3}{4}} e^{6} + 4 \, \sqrt {a} c^{2} d^{5} e - 4 \, a^{\frac {3}{2}} c d e^{5}\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 {5}{4}}} - \frac {2 \, {\left (\sqrt {2} a^{\frac {1}{4}} c^{\frac {9}{4}} d^{6} + 3 \, \sqrt {2} a^{\frac {3}{4}} c^{\frac {7}{4}} d^{4} e^{2} - 3 \, \sqrt {2} a^{\frac {5}{4}} c^{\frac {5}{4}} d^{2} e^{4} - \sqrt {2} a^{\frac {7}{4}} c^{\frac {3}{4}} e^{6} - 4 \, \sqrt {a} c^{2} d^{5} e + 4 \, a^{\frac {3}{2}} c d e^{5}\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 {5}{4}}}\right )}}{8 \, {\left (c^{2} d^{8} + 2 \, a c d^{4} e^{4} + a^{2} e^{8}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.78, size = 2436, normalized size = 4.41
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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