Optimal. Leaf size=1605 \[ -\frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt {-b d^4+4 b c e d^2-b \sqrt {d^2-4 c e} \left (d^2-2 c e\right ) d-2 a e^4-2 b c^2 e^2} x}{e \left (d+\sqrt {d^2-4 c e}\right ) \sqrt {b x^4+a}}\right ) e^2}{\sqrt {2} \sqrt {d^2-4 c e} \sqrt {-2 a e^4-b \left (d^4+\sqrt {d^2-4 c e} d^3-4 c e d^2-2 c e \sqrt {d^2-4 c e} d+2 c^2 e^2\right )}}+\frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt {-b d^4+4 b c e d^2+b \sqrt {d^2-4 c e} \left (d^2-2 c e\right ) d-2 a e^4-2 b c^2 e^2} x}{e \left (d-\sqrt {d^2-4 c e}\right ) \sqrt {b x^4+a}}\right ) e^2}{\sqrt {2} \sqrt {d^2-4 c e} \sqrt {-2 a e^4-b \left (d^4-\sqrt {d^2-4 c e} d^3-4 c e d^2+2 c e \sqrt {d^2-4 c e} d+2 c^2 e^2\right )}}-\frac {\tanh ^{-1}\left (\frac {4 a e^2+b \left (d-\sqrt {d^2-4 c e}\right )^2 x^2}{2 \sqrt {2} \sqrt {b d^4-4 b c e d^2-b \sqrt {d^2-4 c e} \left (d^2-2 c e\right ) d+2 a e^4+2 b c^2 e^2} \sqrt {b x^4+a}}\right ) e^2}{\sqrt {2} \sqrt {d^2-4 c e} \sqrt {b d^4-4 b c e d^2-b \sqrt {d^2-4 c e} \left (d^2-2 c e\right ) d+2 a e^4+2 b c^2 e^2}}+\frac {\tanh ^{-1}\left (\frac {4 a e^2+b \left (d+\sqrt {d^2-4 c e}\right )^2 x^2}{2 \sqrt {2} \sqrt {b d^4-4 b c e d^2+b \sqrt {d^2-4 c e} \left (d^2-2 c e\right ) d+2 a e^4+2 b c^2 e^2} \sqrt {b x^4+a}}\right ) e^2}{\sqrt {2} \sqrt {d^2-4 c e} \sqrt {b d^4-4 b c e d^2+b \sqrt {d^2-4 c e} \left (d^2-2 c e\right ) d+2 a e^4+2 b c^2 e^2}}+\frac {\sqrt [4]{b} \left (d-\sqrt {d^2-4 c e}\right ) \left (\sqrt {b} x^2+\sqrt {a}\right ) \sqrt {\frac {b x^4+a}{\left (\sqrt {b} x^2+\sqrt {a}\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right ) e}{2 \sqrt [4]{a} \sqrt {d^2-4 c e} \left (2 \sqrt {a} e^2+\sqrt {b} \left (d^2-\sqrt {d^2-4 c e} d-2 c e\right )\right ) \sqrt {b x^4+a}}-\frac {\sqrt [4]{b} \left (d+\sqrt {d^2-4 c e}\right ) \left (\sqrt {b} x^2+\sqrt {a}\right ) \sqrt {\frac {b x^4+a}{\left (\sqrt {b} x^2+\sqrt {a}\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right ) e}{2 \sqrt [4]{a} \sqrt {d^2-4 c e} \left (2 \sqrt {a} e^2+\sqrt {b} \left (d^2+\sqrt {d^2-4 c e} d-2 c e\right )\right ) \sqrt {b x^4+a}}+\frac {\left (2 \sqrt {a} e^2-\sqrt {b} \left (d^2-\sqrt {d^2-4 c e} d-2 c e\right )\right ) \left (\sqrt {b} x^2+\sqrt {a}\right ) \sqrt {\frac {b x^4+a}{\left (\sqrt {b} x^2+\sqrt {a}\right )^2}} \Pi \left (\frac {\left (2 \sqrt {a} e^2+\sqrt {b} \left (d^2-\sqrt {d^2-4 c e} d-2 c e\right )\right )^2}{4 \sqrt {a} \sqrt {b} e^2 \left (d-\sqrt {d^2-4 c e}\right )^2};2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right ) e}{2 \sqrt [4]{a} \sqrt [4]{b} \sqrt {d^2-4 c e} \left (d-\sqrt {d^2-4 c e}\right ) \left (2 \sqrt {a} e^2+\sqrt {b} \left (d^2-\sqrt {d^2-4 c e} d-2 c e\right )\right ) \sqrt {b x^4+a}}-\frac {\left (2 \sqrt {a} e^2-\sqrt {b} \left (d^2+\sqrt {d^2-4 c e} d-2 c e\right )\right ) \left (\sqrt {b} x^2+\sqrt {a}\right ) \sqrt {\frac {b x^4+a}{\left (\sqrt {b} x^2+\sqrt {a}\right )^2}} \Pi \left (\frac {\left (2 \sqrt {a} e^2+\sqrt {b} \left (d^2+\sqrt {d^2-4 c e} d-2 c e\right )\right )^2}{4 \sqrt {a} \sqrt {b} e^2 \left (d+\sqrt {d^2-4 c e}\right )^2};2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right ) e}{2 \sqrt [4]{a} \sqrt [4]{b} \sqrt {d^2-4 c e} \left (d+\sqrt {d^2-4 c e}\right ) \left (2 \sqrt {a} e^2+\sqrt {b} \left (d^2+\sqrt {d^2-4 c e} d-2 c e\right )\right ) \sqrt {b x^4+a}} \]
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Rubi [A] time = 9.68, antiderivative size = 1605, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 8, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6728, 1725, 1217, 220, 1707, 1248, 725, 206} \[ \text {result too large to display} \]
Antiderivative was successfully verified.
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Rule 206
Rule 220
Rule 725
Rule 1217
Rule 1248
Rule 1707
Rule 1725
Rule 6728
Rubi steps
\begin {align*} \int \frac {1}{\left (c+d x+e x^2\right ) \sqrt {a+b x^4}} \, dx &=\int \left (\frac {2 e}{\sqrt {d^2-4 c e} \left (d-\sqrt {d^2-4 c e}+2 e x\right ) \sqrt {a+b x^4}}-\frac {2 e}{\sqrt {d^2-4 c e} \left (d+\sqrt {d^2-4 c e}+2 e x\right ) \sqrt {a+b x^4}}\right ) \, dx\\ &=\frac {(2 e) \int \frac {1}{\left (d-\sqrt {d^2-4 c e}+2 e x\right ) \sqrt {a+b x^4}} \, dx}{\sqrt {d^2-4 c e}}-\frac {(2 e) \int \frac {1}{\left (d+\sqrt {d^2-4 c e}+2 e x\right ) \sqrt {a+b x^4}} \, dx}{\sqrt {d^2-4 c e}}\\ &=-\frac {\left (4 e^2\right ) \int \frac {x}{\left (\left (d-\sqrt {d^2-4 c e}\right )^2-4 e^2 x^2\right ) \sqrt {a+b x^4}} \, dx}{\sqrt {d^2-4 c e}}+\frac {\left (4 e^2\right ) \int \frac {x}{\left (\left (d+\sqrt {d^2-4 c e}\right )^2-4 e^2 x^2\right ) \sqrt {a+b x^4}} \, dx}{\sqrt {d^2-4 c e}}-\left (2 e \left (1-\frac {d}{\sqrt {d^2-4 c e}}\right )\right ) \int \frac {1}{\left (\left (d-\sqrt {d^2-4 c e}\right )^2-4 e^2 x^2\right ) \sqrt {a+b x^4}} \, dx-\left (2 e \left (1+\frac {d}{\sqrt {d^2-4 c e}}\right )\right ) \int \frac {1}{\left (\left (d+\sqrt {d^2-4 c e}\right )^2-4 e^2 x^2\right ) \sqrt {a+b x^4}} \, dx\\ &=-\frac {\left (2 e^2\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\left (d-\sqrt {d^2-4 c e}\right )^2-4 e^2 x\right ) \sqrt {a+b x^2}} \, dx,x,x^2\right )}{\sqrt {d^2-4 c e}}+\frac {\left (2 e^2\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\left (d+\sqrt {d^2-4 c e}\right )^2-4 e^2 x\right ) \sqrt {a+b x^2}} \, dx,x,x^2\right )}{\sqrt {d^2-4 c e}}-\frac {\left (\sqrt {b} e \left (1-\frac {d}{\sqrt {d^2-4 c e}}\right )\right ) \int \frac {1}{\sqrt {a+b x^4}} \, dx}{2 \sqrt {a} e^2+\sqrt {b} \left (d^2-2 c e-d \sqrt {d^2-4 c e}\right )}-\frac {\left (4 \sqrt {a} e^3 \left (1-\frac {d}{\sqrt {d^2-4 c e}}\right )\right ) \int \frac {1+\frac {\sqrt {b} x^2}{\sqrt {a}}}{\left (\left (d-\sqrt {d^2-4 c e}\right )^2-4 e^2 x^2\right ) \sqrt {a+b x^4}} \, dx}{2 \sqrt {a} e^2+\sqrt {b} \left (d^2-2 c e-d \sqrt {d^2-4 c e}\right )}-\frac {\left (\sqrt {b} e \left (1+\frac {d}{\sqrt {d^2-4 c e}}\right )\right ) \int \frac {1}{\sqrt {a+b x^4}} \, dx}{2 \sqrt {a} e^2+\sqrt {b} \left (d^2-2 c e+d \sqrt {d^2-4 c e}\right )}-\frac {\left (4 \sqrt {a} e^3 \left (1+\frac {d}{\sqrt {d^2-4 c e}}\right )\right ) \int \frac {1+\frac {\sqrt {b} x^2}{\sqrt {a}}}{\left (\left (d+\sqrt {d^2-4 c e}\right )^2-4 e^2 x^2\right ) \sqrt {a+b x^4}} \, dx}{2 \sqrt {a} e^2+\sqrt {b} \left (d^2-2 c e+d \sqrt {d^2-4 c e}\right )}\\ &=-\frac {e^2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {-b d^4+4 b c d^2 e-2 b c^2 e^2-2 a e^4-b d \sqrt {d^2-4 c e} \left (d^2-2 c e\right )} x}{e \left (d+\sqrt {d^2-4 c e}\right ) \sqrt {a+b x^4}}\right )}{\sqrt {2} \sqrt {d^2-4 c e} \sqrt {-2 a e^4-b \left (d^4-4 c d^2 e+2 c^2 e^2+d^3 \sqrt {d^2-4 c e}-2 c d e \sqrt {d^2-4 c e}\right )}}+\frac {e^2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {-b d^4+4 b c d^2 e-2 b c^2 e^2-2 a e^4+b d \sqrt {d^2-4 c e} \left (d^2-2 c e\right )} x}{e \left (d-\sqrt {d^2-4 c e}\right ) \sqrt {a+b x^4}}\right )}{\sqrt {2} \sqrt {d^2-4 c e} \sqrt {-2 a e^4-b \left (d^4-4 c d^2 e+2 c^2 e^2-d^3 \sqrt {d^2-4 c e}+2 c d e \sqrt {d^2-4 c e}\right )}}-\frac {\sqrt [4]{b} e \left (1-\frac {d}{\sqrt {d^2-4 c e}}\right ) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 \sqrt [4]{a} \left (2 \sqrt {a} e^2+\sqrt {b} \left (d^2-2 c e-d \sqrt {d^2-4 c e}\right )\right ) \sqrt {a+b x^4}}-\frac {\sqrt [4]{b} e \left (1+\frac {d}{\sqrt {d^2-4 c e}}\right ) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 \sqrt [4]{a} \left (2 \sqrt {a} e^2+\sqrt {b} \left (d^2-2 c e+d \sqrt {d^2-4 c e}\right )\right ) \sqrt {a+b x^4}}-\frac {\sqrt [4]{a} e \left (1-\frac {d}{\sqrt {d^2-4 c e}}\right ) \left (4 e^2-\frac {\sqrt {b} \left (d-\sqrt {d^2-4 c e}\right )^2}{\sqrt {a}}\right ) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \Pi \left (\frac {\left (2 \sqrt {a} e^2+\sqrt {b} \left (d^2-2 c e-d \sqrt {d^2-4 c e}\right )\right )^2}{4 \sqrt {a} \sqrt {b} e^2 \left (d-\sqrt {d^2-4 c e}\right )^2};2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{4 \sqrt [4]{b} \left (d-\sqrt {d^2-4 c e}\right )^2 \left (2 \sqrt {a} e^2+\sqrt {b} \left (d^2-2 c e-d \sqrt {d^2-4 c e}\right )\right ) \sqrt {a+b x^4}}-\frac {\sqrt [4]{a} e \left (1+\frac {d}{\sqrt {d^2-4 c e}}\right ) \left (4 e^2-\frac {\sqrt {b} \left (d+\sqrt {d^2-4 c e}\right )^2}{\sqrt {a}}\right ) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \Pi \left (\frac {\left (2 \sqrt {a} e^2+\sqrt {b} \left (d^2-2 c e+d \sqrt {d^2-4 c e}\right )\right )^2}{4 \sqrt {a} \sqrt {b} e^2 \left (d+\sqrt {d^2-4 c e}\right )^2};2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{4 \sqrt [4]{b} \left (d+\sqrt {d^2-4 c e}\right )^2 \left (2 \sqrt {a} e^2+\sqrt {b} \left (d^2-2 c e+d \sqrt {d^2-4 c e}\right )\right ) \sqrt {a+b x^4}}+\frac {\left (2 e^2\right ) \operatorname {Subst}\left (\int \frac {1}{16 a e^4+b \left (d-\sqrt {d^2-4 c e}\right )^4-x^2} \, dx,x,\frac {-4 a e^2-b \left (d-\sqrt {d^2-4 c e}\right )^2 x^2}{\sqrt {a+b x^4}}\right )}{\sqrt {d^2-4 c e}}-\frac {\left (2 e^2\right ) \operatorname {Subst}\left (\int \frac {1}{16 a e^4+b \left (d+\sqrt {d^2-4 c e}\right )^4-x^2} \, dx,x,\frac {-4 a e^2-b \left (d+\sqrt {d^2-4 c e}\right )^2 x^2}{\sqrt {a+b x^4}}\right )}{\sqrt {d^2-4 c e}}\\ &=-\frac {e^2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {-b d^4+4 b c d^2 e-2 b c^2 e^2-2 a e^4-b d \sqrt {d^2-4 c e} \left (d^2-2 c e\right )} x}{e \left (d+\sqrt {d^2-4 c e}\right ) \sqrt {a+b x^4}}\right )}{\sqrt {2} \sqrt {d^2-4 c e} \sqrt {-2 a e^4-b \left (d^4-4 c d^2 e+2 c^2 e^2+d^3 \sqrt {d^2-4 c e}-2 c d e \sqrt {d^2-4 c e}\right )}}+\frac {e^2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {-b d^4+4 b c d^2 e-2 b c^2 e^2-2 a e^4+b d \sqrt {d^2-4 c e} \left (d^2-2 c e\right )} x}{e \left (d-\sqrt {d^2-4 c e}\right ) \sqrt {a+b x^4}}\right )}{\sqrt {2} \sqrt {d^2-4 c e} \sqrt {-2 a e^4-b \left (d^4-4 c d^2 e+2 c^2 e^2-d^3 \sqrt {d^2-4 c e}+2 c d e \sqrt {d^2-4 c e}\right )}}-\frac {e^2 \tanh ^{-1}\left (\frac {4 a e^2+b \left (d-\sqrt {d^2-4 c e}\right )^2 x^2}{2 \sqrt {2} \sqrt {b d^4-4 b c d^2 e+2 b c^2 e^2+2 a e^4-b d \sqrt {d^2-4 c e} \left (d^2-2 c e\right )} \sqrt {a+b x^4}}\right )}{\sqrt {2} \sqrt {d^2-4 c e} \sqrt {b d^4-4 b c d^2 e+2 b c^2 e^2+2 a e^4-b d \sqrt {d^2-4 c e} \left (d^2-2 c e\right )}}+\frac {e^2 \tanh ^{-1}\left (\frac {4 a e^2+b \left (d+\sqrt {d^2-4 c e}\right )^2 x^2}{2 \sqrt {2} \sqrt {b d^4-4 b c d^2 e+2 b c^2 e^2+2 a e^4+b d \sqrt {d^2-4 c e} \left (d^2-2 c e\right )} \sqrt {a+b x^4}}\right )}{\sqrt {2} \sqrt {d^2-4 c e} \sqrt {b d^4-4 b c d^2 e+2 b c^2 e^2+2 a e^4+b d \sqrt {d^2-4 c e} \left (d^2-2 c e\right )}}-\frac {\sqrt [4]{b} e \left (1-\frac {d}{\sqrt {d^2-4 c e}}\right ) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 \sqrt [4]{a} \left (2 \sqrt {a} e^2+\sqrt {b} \left (d^2-2 c e-d \sqrt {d^2-4 c e}\right )\right ) \sqrt {a+b x^4}}-\frac {\sqrt [4]{b} e \left (1+\frac {d}{\sqrt {d^2-4 c e}}\right ) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 \sqrt [4]{a} \left (2 \sqrt {a} e^2+\sqrt {b} \left (d^2-2 c e+d \sqrt {d^2-4 c e}\right )\right ) \sqrt {a+b x^4}}-\frac {\sqrt [4]{a} e \left (1-\frac {d}{\sqrt {d^2-4 c e}}\right ) \left (4 e^2-\frac {\sqrt {b} \left (d-\sqrt {d^2-4 c e}\right )^2}{\sqrt {a}}\right ) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \Pi \left (\frac {\left (2 \sqrt {a} e^2+\sqrt {b} \left (d^2-2 c e-d \sqrt {d^2-4 c e}\right )\right )^2}{4 \sqrt {a} \sqrt {b} e^2 \left (d-\sqrt {d^2-4 c e}\right )^2};2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{4 \sqrt [4]{b} \left (d-\sqrt {d^2-4 c e}\right )^2 \left (2 \sqrt {a} e^2+\sqrt {b} \left (d^2-2 c e-d \sqrt {d^2-4 c e}\right )\right ) \sqrt {a+b x^4}}-\frac {\sqrt [4]{a} e \left (1+\frac {d}{\sqrt {d^2-4 c e}}\right ) \left (4 e^2-\frac {\sqrt {b} \left (d+\sqrt {d^2-4 c e}\right )^2}{\sqrt {a}}\right ) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \Pi \left (\frac {\left (2 \sqrt {a} e^2+\sqrt {b} \left (d^2-2 c e+d \sqrt {d^2-4 c e}\right )\right )^2}{4 \sqrt {a} \sqrt {b} e^2 \left (d+\sqrt {d^2-4 c e}\right )^2};2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{4 \sqrt [4]{b} \left (d+\sqrt {d^2-4 c e}\right )^2 \left (2 \sqrt {a} e^2+\sqrt {b} \left (d^2-2 c e+d \sqrt {d^2-4 c e}\right )\right ) \sqrt {a+b x^4}}\\ \end {align*}
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Mathematica [C] time = 7.37, size = 1416, normalized size = 0.88 \[ -\frac {i \sqrt {1-\frac {i \sqrt {b} x^2}{\sqrt {a}}} \sqrt {\frac {i \sqrt {b} x^2}{\sqrt {a}}+1} \Pi \left (-\frac {2 i \sqrt {a} e^2}{\sqrt {b} \left (-d^2+2 c e-\sqrt {d^4-4 c d^2 e}\right )};\left .i \sinh ^{-1}\left (\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}} x\right )\right |-1\right ) d^2}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}} e \left (-d^2+2 c e-\sqrt {d^4-4 c d^2 e}\right ) \left (\frac {d^2-2 c e+\sqrt {d^4-4 c d^2 e}}{2 e^2}-\frac {d^2-2 c e-\sqrt {d^4-4 c d^2 e}}{2 e^2}\right ) \sqrt {b x^4+a}}-\frac {i \sqrt {1-\frac {i \sqrt {b} x^2}{\sqrt {a}}} \sqrt {\frac {i \sqrt {b} x^2}{\sqrt {a}}+1} \Pi \left (-\frac {2 i \sqrt {a} e^2}{\sqrt {b} \left (-d^2+2 c e+\sqrt {d^4-4 c d^2 e}\right )};\left .i \sinh ^{-1}\left (\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}} x\right )\right |-1\right ) d^2}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}} e \left (-d^2+2 c e+\sqrt {d^4-4 c d^2 e}\right ) \left (\frac {d^2-2 c e-\sqrt {d^4-4 c d^2 e}}{2 e^2}-\frac {d^2-2 c e+\sqrt {d^4-4 c d^2 e}}{2 e^2}\right ) \sqrt {b x^4+a}}-\frac {\sqrt {2} e^2 \left (\frac {\tanh ^{-1}\left (\frac {2 a e^2+b \left (d^2-\sqrt {d^2-4 c e} d-2 c e\right ) x^2}{\sqrt {4 a e^4+b \left (2 d^4-2 \sqrt {d^2-4 c e} d^3-8 c e d^2+4 c e \sqrt {d^2-4 c e} d+4 c^2 e^2\right )} \sqrt {b x^4+a}}\right )}{2 \sqrt {2 a e^4+b \left (d^4-\sqrt {d^2-4 c e} d^3-4 c e d^2+2 c e \sqrt {d^2-4 c e} d+2 c^2 e^2\right )}}-\frac {\tanh ^{-1}\left (\frac {2 a e^2+b \left (d^2+\sqrt {d^2-4 c e} d-2 c e\right ) x^2}{\sqrt {4 a e^4+2 b \left (d^4+\sqrt {d^2-4 c e} d^3-4 c e d^2-2 c e \sqrt {d^2-4 c e} d+2 c^2 e^2\right )} \sqrt {b x^4+a}}\right )}{2 \sqrt {2 a e^4+b \left (d^4+\sqrt {d^2-4 c e} d^3-4 c e d^2-2 c e \sqrt {d^2-4 c e} d+2 c^2 e^2\right )}}\right )}{\sqrt {d^2-4 c e}}-\frac {i \sqrt {d^4-4 c d^2 e} \sqrt {1-\frac {i \sqrt {b} x^2}{\sqrt {a}}} \sqrt {\frac {i \sqrt {b} x^2}{\sqrt {a}}+1} \Pi \left (-\frac {2 i \sqrt {a} e^2}{\sqrt {b} \left (-d^2+2 c e-\sqrt {d^4-4 c d^2 e}\right )};\left .i \sinh ^{-1}\left (\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}} x\right )\right |-1\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}} e \left (-d^2+2 c e-\sqrt {d^4-4 c d^2 e}\right ) \left (\frac {d^2-2 c e+\sqrt {d^4-4 c d^2 e}}{2 e^2}-\frac {d^2-2 c e-\sqrt {d^4-4 c d^2 e}}{2 e^2}\right ) \sqrt {b x^4+a}}+\frac {i \sqrt {d^4-4 c d^2 e} \sqrt {1-\frac {i \sqrt {b} x^2}{\sqrt {a}}} \sqrt {\frac {i \sqrt {b} x^2}{\sqrt {a}}+1} \Pi \left (-\frac {2 i \sqrt {a} e^2}{\sqrt {b} \left (-d^2+2 c e+\sqrt {d^4-4 c d^2 e}\right )};\left .i \sinh ^{-1}\left (\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}} x\right )\right |-1\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}} e \left (-d^2+2 c e+\sqrt {d^4-4 c d^2 e}\right ) \left (\frac {d^2-2 c e-\sqrt {d^4-4 c d^2 e}}{2 e^2}-\frac {d^2-2 c e+\sqrt {d^4-4 c d^2 e}}{2 e^2}\right ) \sqrt {b x^4+a}} \]
Warning: Unable to verify antiderivative.
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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 [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {b x^{4} + a} {\left (e x^{2} + d x + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.10, size = 1153, normalized size = 0.72 \[ \text {result too large to display} \]
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 \[ \int \frac {1}{\sqrt {b x^{4} + a} {\left (e x^{2} + d x + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{\sqrt {b\,x^4+a}\,\left (e\,x^2+d\,x+c\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {a + b x^{4}} \left (c + d x + e x^{2}\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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