Optimal. Leaf size=827 \[ \frac {x \left (i a^{3/2} x^6-i \sqrt {a} c x^4-a \sqrt {b} x^4+i a \sqrt {a x^4-c x^2-b} x^4-i \sqrt {a} b x^2+\sqrt {b} c x^2-\frac {1}{2} i c \sqrt {a x^4-c x^2-b} x^2-\sqrt {a} \sqrt {b} \sqrt {a x^4-c x^2-b} x^2+b^{3/2}-i b \sqrt {a x^4-c x^2-b}\right )}{\left (a x^4-b\right ) \left (-2 i a x^4+i c x^2+2 \sqrt {a} \sqrt {b} x^2-2 i \sqrt {a} \sqrt {a x^4-c x^2-b} x^2+2 i b+2 \sqrt {b} \sqrt {a x^4-c x^2-b}\right )}+\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt [4]{a} \sqrt [4]{b} \tan ^{-1}\left (\frac {\sqrt {c-(2-2 i) \sqrt [4]{a} \sqrt [4]{b} \sqrt {c}-2 i \sqrt {a} \sqrt {b}} x}{\sqrt {a} x^2+i \sqrt {b}+\sqrt {a x^4-c x^2-b}}\right )}{\sqrt {c} \sqrt {c-(2-2 i) \sqrt [4]{a} \sqrt [4]{b} \sqrt {c}-2 i \sqrt {a} \sqrt {b}}}-\frac {\tan ^{-1}\left (\frac {\sqrt {c-(2-2 i) \sqrt [4]{a} \sqrt [4]{b} \sqrt {c}-2 i \sqrt {a} \sqrt {b}} x}{\sqrt {a} x^2+i \sqrt {b}+\sqrt {a x^4-c x^2-b}}\right )}{2 \sqrt {c-(2-2 i) \sqrt [4]{a} \sqrt [4]{b} \sqrt {c}-2 i \sqrt {a} \sqrt {b}}}-\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt [4]{a} \sqrt [4]{b} \tan ^{-1}\left (\frac {\sqrt {c+(2-2 i) \sqrt [4]{a} \sqrt [4]{b} \sqrt {c}-2 i \sqrt {a} \sqrt {b}} x}{\sqrt {a} x^2+i \sqrt {b}+\sqrt {a x^4-c x^2-b}}\right )}{\sqrt {c} \sqrt {c+(2-2 i) \sqrt [4]{a} \sqrt [4]{b} \sqrt {c}-2 i \sqrt {a} \sqrt {b}}}-\frac {\tan ^{-1}\left (\frac {\sqrt {c+(2-2 i) \sqrt [4]{a} \sqrt [4]{b} \sqrt {c}-2 i \sqrt {a} \sqrt {b}} x}{\sqrt {a} x^2+i \sqrt {b}+\sqrt {a x^4-c x^2-b}}\right )}{2 \sqrt {c+(2-2 i) \sqrt [4]{a} \sqrt [4]{b} \sqrt {c}-2 i \sqrt {a} \sqrt {b}}} \]
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Rubi [C] time = 5.57, antiderivative size = 908, normalized size of antiderivative = 1.10, number of steps used = 50, number of rules used = 10, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {6742, 1226, 1202, 524, 424, 419, 1220, 537, 6725, 1208} \begin {gather*} \frac {\sqrt {a x^4-c x^2-b} x}{4 \sqrt {b} \left (\sqrt {b}-\sqrt {a} x^2\right )}+\frac {\sqrt {a x^4-c x^2-b} x}{4 \sqrt {b} \left (\sqrt {a} x^2+\sqrt {b}\right )}+\frac {\left (c+2 \sqrt {a} \sqrt {b}-\sqrt {c^2+4 a b}\right ) \sqrt {c+\sqrt {c^2+4 a b}} \sqrt {1-\frac {2 a x^2}{c-\sqrt {c^2+4 a b}}} \sqrt {1-\frac {2 a x^2}{c+\sqrt {c^2+4 a b}}} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {a} x}{\sqrt {c+\sqrt {c^2+4 a b}}}\right )|\frac {c+\sqrt {c^2+4 a b}}{c-\sqrt {c^2+4 a b}}\right )}{8 \sqrt {2} a \sqrt {b} \sqrt {a x^4-c x^2-b}}+\frac {\left (-c+2 \sqrt {a} \sqrt {b}+\sqrt {c^2+4 a b}\right ) \sqrt {c+\sqrt {c^2+4 a b}} \sqrt {1-\frac {2 a x^2}{c-\sqrt {c^2+4 a b}}} \sqrt {1-\frac {2 a x^2}{c+\sqrt {c^2+4 a b}}} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {a} x}{\sqrt {c+\sqrt {c^2+4 a b}}}\right )|\frac {c+\sqrt {c^2+4 a b}}{c-\sqrt {c^2+4 a b}}\right )}{8 \sqrt {2} a \sqrt {b} \sqrt {a x^4-c x^2-b}}-\frac {\sqrt {c+\sqrt {c^2+4 a b}} \sqrt {1-\frac {2 a x^2}{c-\sqrt {c^2+4 a b}}} \sqrt {1-\frac {2 a x^2}{c+\sqrt {c^2+4 a b}}} \Pi \left (-\frac {c+\sqrt {c^2+4 a b}}{2 \sqrt {a} \sqrt {b}};\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {a} x}{\sqrt {c+\sqrt {c^2+4 a b}}}\right )|\frac {c+\sqrt {c^2+4 a b}}{c-\sqrt {c^2+4 a b}}\right )}{2 \sqrt {2} \sqrt {a} \sqrt {a x^4-c x^2-b}}-\frac {\sqrt {c+\sqrt {c^2+4 a b}} \sqrt {1-\frac {2 a x^2}{c-\sqrt {c^2+4 a b}}} \sqrt {1-\frac {2 a x^2}{c+\sqrt {c^2+4 a b}}} \Pi \left (\frac {c+\sqrt {c^2+4 a b}}{2 \sqrt {a} \sqrt {b}};\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {a} x}{\sqrt {c+\sqrt {c^2+4 a b}}}\right )|\frac {c+\sqrt {c^2+4 a b}}{c-\sqrt {c^2+4 a b}}\right )}{2 \sqrt {2} \sqrt {a} \sqrt {a x^4-c x^2-b}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 419
Rule 424
Rule 524
Rule 537
Rule 1202
Rule 1208
Rule 1220
Rule 1226
Rule 6725
Rule 6742
Rubi steps
\begin {align*} \int \frac {\left (b+a x^4\right ) \sqrt {-b-c x^2+a x^4}}{\left (-b+a x^4\right )^2} \, dx &=\int \left (\frac {2 b \sqrt {-b-c x^2+a x^4}}{\left (-b+a x^4\right )^2}+\frac {\sqrt {-b-c x^2+a x^4}}{-b+a x^4}\right ) \, dx\\ &=(2 b) \int \frac {\sqrt {-b-c x^2+a x^4}}{\left (-b+a x^4\right )^2} \, dx+\int \frac {\sqrt {-b-c x^2+a x^4}}{-b+a x^4} \, dx\\ &=(2 b) \int \left (\frac {a \sqrt {-b-c x^2+a x^4}}{4 b \left (\sqrt {a} \sqrt {b}-a x^2\right )^2}+\frac {a \sqrt {-b-c x^2+a x^4}}{4 b \left (\sqrt {a} \sqrt {b}+a x^2\right )^2}+\frac {a \sqrt {-b-c x^2+a x^4}}{2 b \left (a b-a^2 x^4\right )}\right ) \, dx+\int \left (-\frac {\sqrt {-b-c x^2+a x^4}}{2 \sqrt {b} \left (\sqrt {b}-\sqrt {a} x^2\right )}-\frac {\sqrt {-b-c x^2+a x^4}}{2 \sqrt {b} \left (\sqrt {b}+\sqrt {a} x^2\right )}\right ) \, dx\\ &=\frac {1}{2} a \int \frac {\sqrt {-b-c x^2+a x^4}}{\left (\sqrt {a} \sqrt {b}-a x^2\right )^2} \, dx+\frac {1}{2} a \int \frac {\sqrt {-b-c x^2+a x^4}}{\left (\sqrt {a} \sqrt {b}+a x^2\right )^2} \, dx+a \int \frac {\sqrt {-b-c x^2+a x^4}}{a b-a^2 x^4} \, dx-\frac {\int \frac {\sqrt {-b-c x^2+a x^4}}{\sqrt {b}-\sqrt {a} x^2} \, dx}{2 \sqrt {b}}-\frac {\int \frac {\sqrt {-b-c x^2+a x^4}}{\sqrt {b}+\sqrt {a} x^2} \, dx}{2 \sqrt {b}}\\ &=\frac {x \sqrt {-b-c x^2+a x^4}}{4 \sqrt {b} \left (\sqrt {b}-\sqrt {a} x^2\right )}+\frac {x \sqrt {-b-c x^2+a x^4}}{4 \sqrt {b} \left (\sqrt {b}+\sqrt {a} x^2\right )}+a \int \left (\frac {\sqrt {-b-c x^2+a x^4}}{2 a \sqrt {b} \left (\sqrt {b}-\sqrt {a} x^2\right )}+\frac {\sqrt {-b-c x^2+a x^4}}{2 a \sqrt {b} \left (\sqrt {b}+\sqrt {a} x^2\right )}\right ) \, dx+\frac {\int \frac {a \sqrt {b}+\sqrt {a} c-a^{3/2} x^2}{\sqrt {-b-c x^2+a x^4}} \, dx}{2 a \sqrt {b}}+\frac {\int \frac {a \sqrt {b}-\sqrt {a} c+a^{3/2} x^2}{\sqrt {-b-c x^2+a x^4}} \, dx}{2 a \sqrt {b}}+\frac {\int \frac {\sqrt {a} \sqrt {b}-a x^2}{\sqrt {-b-c x^2+a x^4}} \, dx}{4 \sqrt {a} \sqrt {b}}+\frac {\int \frac {\sqrt {a} \sqrt {b}+a x^2}{\sqrt {-b-c x^2+a x^4}} \, dx}{4 \sqrt {a} \sqrt {b}}-\frac {1}{2} \left (\sqrt {a} \sqrt {b}\right ) \int \frac {1}{\left (\sqrt {a} \sqrt {b}-a x^2\right ) \sqrt {-b-c x^2+a x^4}} \, dx-\frac {1}{2} \left (\sqrt {a} \sqrt {b}\right ) \int \frac {1}{\left (\sqrt {a} \sqrt {b}+a x^2\right ) \sqrt {-b-c x^2+a x^4}} \, dx+\frac {c \int \frac {1}{\left (\sqrt {b}-\sqrt {a} x^2\right ) \sqrt {-b-c x^2+a x^4}} \, dx}{2 \sqrt {a}}-\frac {c \int \frac {1}{\left (\sqrt {b}+\sqrt {a} x^2\right ) \sqrt {-b-c x^2+a x^4}} \, dx}{2 \sqrt {a}}\\ &=\frac {x \sqrt {-b-c x^2+a x^4}}{4 \sqrt {b} \left (\sqrt {b}-\sqrt {a} x^2\right )}+\frac {x \sqrt {-b-c x^2+a x^4}}{4 \sqrt {b} \left (\sqrt {b}+\sqrt {a} x^2\right )}+\frac {\int \frac {\sqrt {-b-c x^2+a x^4}}{\sqrt {b}-\sqrt {a} x^2} \, dx}{2 \sqrt {b}}+\frac {\int \frac {\sqrt {-b-c x^2+a x^4}}{\sqrt {b}+\sqrt {a} x^2} \, dx}{2 \sqrt {b}}+\frac {\left (\sqrt {1+\frac {2 a x^2}{-c-\sqrt {4 a b+c^2}}} \sqrt {1+\frac {2 a x^2}{-c+\sqrt {4 a b+c^2}}}\right ) \int \frac {a \sqrt {b}+\sqrt {a} c-a^{3/2} x^2}{\sqrt {1+\frac {2 a x^2}{-c-\sqrt {4 a b+c^2}}} \sqrt {1+\frac {2 a x^2}{-c+\sqrt {4 a b+c^2}}}} \, dx}{2 a \sqrt {b} \sqrt {-b-c x^2+a x^4}}+\frac {\left (\sqrt {1+\frac {2 a x^2}{-c-\sqrt {4 a b+c^2}}} \sqrt {1+\frac {2 a x^2}{-c+\sqrt {4 a b+c^2}}}\right ) \int \frac {a \sqrt {b}-\sqrt {a} c+a^{3/2} x^2}{\sqrt {1+\frac {2 a x^2}{-c-\sqrt {4 a b+c^2}}} \sqrt {1+\frac {2 a x^2}{-c+\sqrt {4 a b+c^2}}}} \, dx}{2 a \sqrt {b} \sqrt {-b-c x^2+a x^4}}+\frac {\left (\sqrt {1+\frac {2 a x^2}{-c-\sqrt {4 a b+c^2}}} \sqrt {1+\frac {2 a x^2}{-c+\sqrt {4 a b+c^2}}}\right ) \int \frac {\sqrt {a} \sqrt {b}-a x^2}{\sqrt {1+\frac {2 a x^2}{-c-\sqrt {4 a b+c^2}}} \sqrt {1+\frac {2 a x^2}{-c+\sqrt {4 a b+c^2}}}} \, dx}{4 \sqrt {a} \sqrt {b} \sqrt {-b-c x^2+a x^4}}+\frac {\left (\sqrt {1+\frac {2 a x^2}{-c-\sqrt {4 a b+c^2}}} \sqrt {1+\frac {2 a x^2}{-c+\sqrt {4 a b+c^2}}}\right ) \int \frac {\sqrt {a} \sqrt {b}+a x^2}{\sqrt {1+\frac {2 a x^2}{-c-\sqrt {4 a b+c^2}}} \sqrt {1+\frac {2 a x^2}{-c+\sqrt {4 a b+c^2}}}} \, dx}{4 \sqrt {a} \sqrt {b} \sqrt {-b-c x^2+a x^4}}-\frac {\left (\sqrt {a} \sqrt {b} \sqrt {1+\frac {2 a x^2}{-c-\sqrt {4 a b+c^2}}} \sqrt {1+\frac {2 a x^2}{-c+\sqrt {4 a b+c^2}}}\right ) \int \frac {1}{\left (\sqrt {a} \sqrt {b}-a x^2\right ) \sqrt {1+\frac {2 a x^2}{-c-\sqrt {4 a b+c^2}}} \sqrt {1+\frac {2 a x^2}{-c+\sqrt {4 a b+c^2}}}} \, dx}{2 \sqrt {-b-c x^2+a x^4}}-\frac {\left (\sqrt {a} \sqrt {b} \sqrt {1+\frac {2 a x^2}{-c-\sqrt {4 a b+c^2}}} \sqrt {1+\frac {2 a x^2}{-c+\sqrt {4 a b+c^2}}}\right ) \int \frac {1}{\left (\sqrt {a} \sqrt {b}+a x^2\right ) \sqrt {1+\frac {2 a x^2}{-c-\sqrt {4 a b+c^2}}} \sqrt {1+\frac {2 a x^2}{-c+\sqrt {4 a b+c^2}}}} \, dx}{2 \sqrt {-b-c x^2+a x^4}}+\frac {\left (c \sqrt {1+\frac {2 a x^2}{-c-\sqrt {4 a b+c^2}}} \sqrt {1+\frac {2 a x^2}{-c+\sqrt {4 a b+c^2}}}\right ) \int \frac {1}{\left (\sqrt {b}-\sqrt {a} x^2\right ) \sqrt {1+\frac {2 a x^2}{-c-\sqrt {4 a b+c^2}}} \sqrt {1+\frac {2 a x^2}{-c+\sqrt {4 a b+c^2}}}} \, dx}{2 \sqrt {a} \sqrt {-b-c x^2+a x^4}}-\frac {\left (c \sqrt {1+\frac {2 a x^2}{-c-\sqrt {4 a b+c^2}}} \sqrt {1+\frac {2 a x^2}{-c+\sqrt {4 a b+c^2}}}\right ) \int \frac {1}{\left (\sqrt {b}+\sqrt {a} x^2\right ) \sqrt {1+\frac {2 a x^2}{-c-\sqrt {4 a b+c^2}}} \sqrt {1+\frac {2 a x^2}{-c+\sqrt {4 a b+c^2}}}} \, dx}{2 \sqrt {a} \sqrt {-b-c x^2+a x^4}}\\ &=\frac {x \sqrt {-b-c x^2+a x^4}}{4 \sqrt {b} \left (\sqrt {b}-\sqrt {a} x^2\right )}+\frac {x \sqrt {-b-c x^2+a x^4}}{4 \sqrt {b} \left (\sqrt {b}+\sqrt {a} x^2\right )}-\frac {\sqrt {c+\sqrt {4 a b+c^2}} \sqrt {1-\frac {2 a x^2}{c-\sqrt {4 a b+c^2}}} \sqrt {1-\frac {2 a x^2}{c+\sqrt {4 a b+c^2}}} \Pi \left (-\frac {c+\sqrt {4 a b+c^2}}{2 \sqrt {a} \sqrt {b}};\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {a} x}{\sqrt {c+\sqrt {4 a b+c^2}}}\right )|\frac {c+\sqrt {4 a b+c^2}}{c-\sqrt {4 a b+c^2}}\right )}{2 \sqrt {2} \sqrt {a} \sqrt {-b-c x^2+a x^4}}-\frac {c \sqrt {c+\sqrt {4 a b+c^2}} \sqrt {1-\frac {2 a x^2}{c-\sqrt {4 a b+c^2}}} \sqrt {1-\frac {2 a x^2}{c+\sqrt {4 a b+c^2}}} \Pi \left (-\frac {c+\sqrt {4 a b+c^2}}{2 \sqrt {a} \sqrt {b}};\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {a} x}{\sqrt {c+\sqrt {4 a b+c^2}}}\right )|\frac {c+\sqrt {4 a b+c^2}}{c-\sqrt {4 a b+c^2}}\right )}{2 \sqrt {2} a \sqrt {b} \sqrt {-b-c x^2+a x^4}}-\frac {\sqrt {c+\sqrt {4 a b+c^2}} \sqrt {1-\frac {2 a x^2}{c-\sqrt {4 a b+c^2}}} \sqrt {1-\frac {2 a x^2}{c+\sqrt {4 a b+c^2}}} \Pi \left (\frac {c+\sqrt {4 a b+c^2}}{2 \sqrt {a} \sqrt {b}};\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {a} x}{\sqrt {c+\sqrt {4 a b+c^2}}}\right )|\frac {c+\sqrt {4 a b+c^2}}{c-\sqrt {4 a b+c^2}}\right )}{2 \sqrt {2} \sqrt {a} \sqrt {-b-c x^2+a x^4}}+\frac {c \sqrt {c+\sqrt {4 a b+c^2}} \sqrt {1-\frac {2 a x^2}{c-\sqrt {4 a b+c^2}}} \sqrt {1-\frac {2 a x^2}{c+\sqrt {4 a b+c^2}}} \Pi \left (\frac {c+\sqrt {4 a b+c^2}}{2 \sqrt {a} \sqrt {b}};\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {a} x}{\sqrt {c+\sqrt {4 a b+c^2}}}\right )|\frac {c+\sqrt {4 a b+c^2}}{c-\sqrt {4 a b+c^2}}\right )}{2 \sqrt {2} a \sqrt {b} \sqrt {-b-c x^2+a x^4}}-\frac {\int \frac {a \sqrt {b}+\sqrt {a} c-a^{3/2} x^2}{\sqrt {-b-c x^2+a x^4}} \, dx}{2 a \sqrt {b}}-\frac {\int \frac {a \sqrt {b}-\sqrt {a} c+a^{3/2} x^2}{\sqrt {-b-c x^2+a x^4}} \, dx}{2 a \sqrt {b}}-\frac {c \int \frac {1}{\left (\sqrt {b}-\sqrt {a} x^2\right ) \sqrt {-b-c x^2+a x^4}} \, dx}{2 \sqrt {a}}+\frac {c \int \frac {1}{\left (\sqrt {b}+\sqrt {a} x^2\right ) \sqrt {-b-c x^2+a x^4}} \, dx}{2 \sqrt {a}}+\frac {\left (\left (2 \sqrt {a} \sqrt {b}-c-\sqrt {4 a b+c^2}\right ) \sqrt {1+\frac {2 a x^2}{-c-\sqrt {4 a b+c^2}}} \sqrt {1+\frac {2 a x^2}{-c+\sqrt {4 a b+c^2}}}\right ) \int \frac {1}{\sqrt {1+\frac {2 a x^2}{-c-\sqrt {4 a b+c^2}}} \sqrt {1+\frac {2 a x^2}{-c+\sqrt {4 a b+c^2}}}} \, dx}{4 \sqrt {a} \sqrt {b} \sqrt {-b-c x^2+a x^4}}+\frac {\left (\left (c-\sqrt {4 a b+c^2}\right ) \sqrt {1+\frac {2 a x^2}{-c-\sqrt {4 a b+c^2}}} \sqrt {1+\frac {2 a x^2}{-c+\sqrt {4 a b+c^2}}}\right ) \int \frac {\sqrt {1+\frac {2 a x^2}{-c+\sqrt {4 a b+c^2}}}}{\sqrt {1+\frac {2 a x^2}{-c-\sqrt {4 a b+c^2}}}} \, dx}{8 \sqrt {a} \sqrt {b} \sqrt {-b-c x^2+a x^4}}+\frac {\left (\left (2 \sqrt {a} \sqrt {b}+c-\sqrt {4 a b+c^2}\right ) \sqrt {1+\frac {2 a x^2}{-c-\sqrt {4 a b+c^2}}} \sqrt {1+\frac {2 a x^2}{-c+\sqrt {4 a b+c^2}}}\right ) \int \frac {1}{\sqrt {1+\frac {2 a x^2}{-c-\sqrt {4 a b+c^2}}} \sqrt {1+\frac {2 a x^2}{-c+\sqrt {4 a b+c^2}}}} \, dx}{8 \sqrt {a} \sqrt {b} \sqrt {-b-c x^2+a x^4}}+\frac {\left (\left (-c+\sqrt {4 a b+c^2}\right ) \sqrt {1+\frac {2 a x^2}{-c-\sqrt {4 a b+c^2}}} \sqrt {1+\frac {2 a x^2}{-c+\sqrt {4 a b+c^2}}}\right ) \int \frac {\sqrt {1+\frac {2 a x^2}{-c+\sqrt {4 a b+c^2}}}}{\sqrt {1+\frac {2 a x^2}{-c-\sqrt {4 a b+c^2}}}} \, dx}{8 \sqrt {a} \sqrt {b} \sqrt {-b-c x^2+a x^4}}+\frac {\left (\left (2 \sqrt {a} \sqrt {b}-c+\sqrt {4 a b+c^2}\right ) \sqrt {1+\frac {2 a x^2}{-c-\sqrt {4 a b+c^2}}} \sqrt {1+\frac {2 a x^2}{-c+\sqrt {4 a b+c^2}}}\right ) \int \frac {1}{\sqrt {1+\frac {2 a x^2}{-c-\sqrt {4 a b+c^2}}} \sqrt {1+\frac {2 a x^2}{-c+\sqrt {4 a b+c^2}}}} \, dx}{8 \sqrt {a} \sqrt {b} \sqrt {-b-c x^2+a x^4}}+\frac {\left (\left (2 \sqrt {a} \sqrt {b}+c+\sqrt {4 a b+c^2}\right ) \sqrt {1+\frac {2 a x^2}{-c-\sqrt {4 a b+c^2}}} \sqrt {1+\frac {2 a x^2}{-c+\sqrt {4 a b+c^2}}}\right ) \int \frac {1}{\sqrt {1+\frac {2 a x^2}{-c-\sqrt {4 a b+c^2}}} \sqrt {1+\frac {2 a x^2}{-c+\sqrt {4 a b+c^2}}}} \, dx}{4 \sqrt {a} \sqrt {b} \sqrt {-b-c x^2+a x^4}}\\ &=\frac {x \sqrt {-b-c x^2+a x^4}}{4 \sqrt {b} \left (\sqrt {b}-\sqrt {a} x^2\right )}+\frac {x \sqrt {-b-c x^2+a x^4}}{4 \sqrt {b} \left (\sqrt {b}+\sqrt {a} x^2\right )}+\frac {\left (2 \sqrt {a} \sqrt {b}-c-\sqrt {4 a b+c^2}\right ) \sqrt {c+\sqrt {4 a b+c^2}} \sqrt {1-\frac {2 a x^2}{c-\sqrt {4 a b+c^2}}} \sqrt {1-\frac {2 a x^2}{c+\sqrt {4 a b+c^2}}} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {a} x}{\sqrt {c+\sqrt {4 a b+c^2}}}\right )|\frac {c+\sqrt {4 a b+c^2}}{c-\sqrt {4 a b+c^2}}\right )}{4 \sqrt {2} a \sqrt {b} \sqrt {-b-c x^2+a x^4}}+\frac {\left (2 \sqrt {a} \sqrt {b}+c-\sqrt {4 a b+c^2}\right ) \sqrt {c+\sqrt {4 a b+c^2}} \sqrt {1-\frac {2 a x^2}{c-\sqrt {4 a b+c^2}}} \sqrt {1-\frac {2 a x^2}{c+\sqrt {4 a b+c^2}}} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {a} x}{\sqrt {c+\sqrt {4 a b+c^2}}}\right )|\frac {c+\sqrt {4 a b+c^2}}{c-\sqrt {4 a b+c^2}}\right )}{8 \sqrt {2} a \sqrt {b} \sqrt {-b-c x^2+a x^4}}+\frac {\left (2 \sqrt {a} \sqrt {b}-c+\sqrt {4 a b+c^2}\right ) \sqrt {c+\sqrt {4 a b+c^2}} \sqrt {1-\frac {2 a x^2}{c-\sqrt {4 a b+c^2}}} \sqrt {1-\frac {2 a x^2}{c+\sqrt {4 a b+c^2}}} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {a} x}{\sqrt {c+\sqrt {4 a b+c^2}}}\right )|\frac {c+\sqrt {4 a b+c^2}}{c-\sqrt {4 a b+c^2}}\right )}{8 \sqrt {2} a \sqrt {b} \sqrt {-b-c x^2+a x^4}}+\frac {\sqrt {c+\sqrt {4 a b+c^2}} \left (2 \sqrt {a} \sqrt {b}+c+\sqrt {4 a b+c^2}\right ) \sqrt {1-\frac {2 a x^2}{c-\sqrt {4 a b+c^2}}} \sqrt {1-\frac {2 a x^2}{c+\sqrt {4 a b+c^2}}} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {a} x}{\sqrt {c+\sqrt {4 a b+c^2}}}\right )|\frac {c+\sqrt {4 a b+c^2}}{c-\sqrt {4 a b+c^2}}\right )}{4 \sqrt {2} a \sqrt {b} \sqrt {-b-c x^2+a x^4}}-\frac {\sqrt {c+\sqrt {4 a b+c^2}} \sqrt {1-\frac {2 a x^2}{c-\sqrt {4 a b+c^2}}} \sqrt {1-\frac {2 a x^2}{c+\sqrt {4 a b+c^2}}} \Pi \left (-\frac {c+\sqrt {4 a b+c^2}}{2 \sqrt {a} \sqrt {b}};\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {a} x}{\sqrt {c+\sqrt {4 a b+c^2}}}\right )|\frac {c+\sqrt {4 a b+c^2}}{c-\sqrt {4 a b+c^2}}\right )}{2 \sqrt {2} \sqrt {a} \sqrt {-b-c x^2+a x^4}}-\frac {c \sqrt {c+\sqrt {4 a b+c^2}} \sqrt {1-\frac {2 a x^2}{c-\sqrt {4 a b+c^2}}} \sqrt {1-\frac {2 a x^2}{c+\sqrt {4 a b+c^2}}} \Pi \left (-\frac {c+\sqrt {4 a b+c^2}}{2 \sqrt {a} \sqrt {b}};\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {a} x}{\sqrt {c+\sqrt {4 a b+c^2}}}\right )|\frac {c+\sqrt {4 a b+c^2}}{c-\sqrt {4 a b+c^2}}\right )}{2 \sqrt {2} a \sqrt {b} \sqrt {-b-c x^2+a x^4}}-\frac {\sqrt {c+\sqrt {4 a b+c^2}} \sqrt {1-\frac {2 a x^2}{c-\sqrt {4 a b+c^2}}} \sqrt {1-\frac {2 a x^2}{c+\sqrt {4 a b+c^2}}} \Pi \left (\frac {c+\sqrt {4 a b+c^2}}{2 \sqrt {a} \sqrt {b}};\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {a} x}{\sqrt {c+\sqrt {4 a b+c^2}}}\right )|\frac {c+\sqrt {4 a b+c^2}}{c-\sqrt {4 a b+c^2}}\right )}{2 \sqrt {2} \sqrt {a} \sqrt {-b-c x^2+a x^4}}+\frac {c \sqrt {c+\sqrt {4 a b+c^2}} \sqrt {1-\frac {2 a x^2}{c-\sqrt {4 a b+c^2}}} \sqrt {1-\frac {2 a x^2}{c+\sqrt {4 a b+c^2}}} \Pi \left (\frac {c+\sqrt {4 a b+c^2}}{2 \sqrt {a} \sqrt {b}};\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {a} x}{\sqrt {c+\sqrt {4 a b+c^2}}}\right )|\frac {c+\sqrt {4 a b+c^2}}{c-\sqrt {4 a b+c^2}}\right )}{2 \sqrt {2} a \sqrt {b} \sqrt {-b-c x^2+a x^4}}-\frac {\left (\sqrt {1+\frac {2 a x^2}{-c-\sqrt {4 a b+c^2}}} \sqrt {1+\frac {2 a x^2}{-c+\sqrt {4 a b+c^2}}}\right ) \int \frac {a \sqrt {b}+\sqrt {a} c-a^{3/2} x^2}{\sqrt {1+\frac {2 a x^2}{-c-\sqrt {4 a b+c^2}}} \sqrt {1+\frac {2 a x^2}{-c+\sqrt {4 a b+c^2}}}} \, dx}{2 a \sqrt {b} \sqrt {-b-c x^2+a x^4}}-\frac {\left (\sqrt {1+\frac {2 a x^2}{-c-\sqrt {4 a b+c^2}}} \sqrt {1+\frac {2 a x^2}{-c+\sqrt {4 a b+c^2}}}\right ) \int \frac {a \sqrt {b}-\sqrt {a} c+a^{3/2} x^2}{\sqrt {1+\frac {2 a x^2}{-c-\sqrt {4 a b+c^2}}} \sqrt {1+\frac {2 a x^2}{-c+\sqrt {4 a b+c^2}}}} \, dx}{2 a \sqrt {b} \sqrt {-b-c x^2+a x^4}}-\frac {\left (c \sqrt {1+\frac {2 a x^2}{-c-\sqrt {4 a b+c^2}}} \sqrt {1+\frac {2 a x^2}{-c+\sqrt {4 a b+c^2}}}\right ) \int \frac {1}{\left (\sqrt {b}-\sqrt {a} x^2\right ) \sqrt {1+\frac {2 a x^2}{-c-\sqrt {4 a b+c^2}}} \sqrt {1+\frac {2 a x^2}{-c+\sqrt {4 a b+c^2}}}} \, dx}{2 \sqrt {a} \sqrt {-b-c x^2+a x^4}}+\frac {\left (c \sqrt {1+\frac {2 a x^2}{-c-\sqrt {4 a b+c^2}}} \sqrt {1+\frac {2 a x^2}{-c+\sqrt {4 a b+c^2}}}\right ) \int \frac {1}{\left (\sqrt {b}+\sqrt {a} x^2\right ) \sqrt {1+\frac {2 a x^2}{-c-\sqrt {4 a b+c^2}}} \sqrt {1+\frac {2 a x^2}{-c+\sqrt {4 a b+c^2}}}} \, dx}{2 \sqrt {a} \sqrt {-b-c x^2+a x^4}}\\ &=\frac {x \sqrt {-b-c x^2+a x^4}}{4 \sqrt {b} \left (\sqrt {b}-\sqrt {a} x^2\right )}+\frac {x \sqrt {-b-c x^2+a x^4}}{4 \sqrt {b} \left (\sqrt {b}+\sqrt {a} x^2\right )}+\frac {\left (2 \sqrt {a} \sqrt {b}-c-\sqrt {4 a b+c^2}\right ) \sqrt {c+\sqrt {4 a b+c^2}} \sqrt {1-\frac {2 a x^2}{c-\sqrt {4 a b+c^2}}} \sqrt {1-\frac {2 a x^2}{c+\sqrt {4 a b+c^2}}} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {a} x}{\sqrt {c+\sqrt {4 a b+c^2}}}\right )|\frac {c+\sqrt {4 a b+c^2}}{c-\sqrt {4 a b+c^2}}\right )}{4 \sqrt {2} a \sqrt {b} \sqrt {-b-c x^2+a x^4}}+\frac {\left (2 \sqrt {a} \sqrt {b}+c-\sqrt {4 a b+c^2}\right ) \sqrt {c+\sqrt {4 a b+c^2}} \sqrt {1-\frac {2 a x^2}{c-\sqrt {4 a b+c^2}}} \sqrt {1-\frac {2 a x^2}{c+\sqrt {4 a b+c^2}}} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {a} x}{\sqrt {c+\sqrt {4 a b+c^2}}}\right )|\frac {c+\sqrt {4 a b+c^2}}{c-\sqrt {4 a b+c^2}}\right )}{8 \sqrt {2} a \sqrt {b} \sqrt {-b-c x^2+a x^4}}+\frac {\left (2 \sqrt {a} \sqrt {b}-c+\sqrt {4 a b+c^2}\right ) \sqrt {c+\sqrt {4 a b+c^2}} \sqrt {1-\frac {2 a x^2}{c-\sqrt {4 a b+c^2}}} \sqrt {1-\frac {2 a x^2}{c+\sqrt {4 a b+c^2}}} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {a} x}{\sqrt {c+\sqrt {4 a b+c^2}}}\right )|\frac {c+\sqrt {4 a b+c^2}}{c-\sqrt {4 a b+c^2}}\right )}{8 \sqrt {2} a \sqrt {b} \sqrt {-b-c x^2+a x^4}}+\frac {\sqrt {c+\sqrt {4 a b+c^2}} \left (2 \sqrt {a} \sqrt {b}+c+\sqrt {4 a b+c^2}\right ) \sqrt {1-\frac {2 a x^2}{c-\sqrt {4 a b+c^2}}} \sqrt {1-\frac {2 a x^2}{c+\sqrt {4 a b+c^2}}} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {a} x}{\sqrt {c+\sqrt {4 a b+c^2}}}\right )|\frac {c+\sqrt {4 a b+c^2}}{c-\sqrt {4 a b+c^2}}\right )}{4 \sqrt {2} a \sqrt {b} \sqrt {-b-c x^2+a x^4}}-\frac {\sqrt {c+\sqrt {4 a b+c^2}} \sqrt {1-\frac {2 a x^2}{c-\sqrt {4 a b+c^2}}} \sqrt {1-\frac {2 a x^2}{c+\sqrt {4 a b+c^2}}} \Pi \left (-\frac {c+\sqrt {4 a b+c^2}}{2 \sqrt {a} \sqrt {b}};\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {a} x}{\sqrt {c+\sqrt {4 a b+c^2}}}\right )|\frac {c+\sqrt {4 a b+c^2}}{c-\sqrt {4 a b+c^2}}\right )}{2 \sqrt {2} \sqrt {a} \sqrt {-b-c x^2+a x^4}}-\frac {\sqrt {c+\sqrt {4 a b+c^2}} \sqrt {1-\frac {2 a x^2}{c-\sqrt {4 a b+c^2}}} \sqrt {1-\frac {2 a x^2}{c+\sqrt {4 a b+c^2}}} \Pi \left (\frac {c+\sqrt {4 a b+c^2}}{2 \sqrt {a} \sqrt {b}};\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {a} x}{\sqrt {c+\sqrt {4 a b+c^2}}}\right )|\frac {c+\sqrt {4 a b+c^2}}{c-\sqrt {4 a b+c^2}}\right )}{2 \sqrt {2} \sqrt {a} \sqrt {-b-c x^2+a x^4}}-\frac {\left (\left (2 \sqrt {a} \sqrt {b}-c-\sqrt {4 a b+c^2}\right ) \sqrt {1+\frac {2 a x^2}{-c-\sqrt {4 a b+c^2}}} \sqrt {1+\frac {2 a x^2}{-c+\sqrt {4 a b+c^2}}}\right ) \int \frac {1}{\sqrt {1+\frac {2 a x^2}{-c-\sqrt {4 a b+c^2}}} \sqrt {1+\frac {2 a x^2}{-c+\sqrt {4 a b+c^2}}}} \, dx}{4 \sqrt {a} \sqrt {b} \sqrt {-b-c x^2+a x^4}}-\frac {\left (\left (2 \sqrt {a} \sqrt {b}+c+\sqrt {4 a b+c^2}\right ) \sqrt {1+\frac {2 a x^2}{-c-\sqrt {4 a b+c^2}}} \sqrt {1+\frac {2 a x^2}{-c+\sqrt {4 a b+c^2}}}\right ) \int \frac {1}{\sqrt {1+\frac {2 a x^2}{-c-\sqrt {4 a b+c^2}}} \sqrt {1+\frac {2 a x^2}{-c+\sqrt {4 a b+c^2}}}} \, dx}{4 \sqrt {a} \sqrt {b} \sqrt {-b-c x^2+a x^4}}\\ &=\frac {x \sqrt {-b-c x^2+a x^4}}{4 \sqrt {b} \left (\sqrt {b}-\sqrt {a} x^2\right )}+\frac {x \sqrt {-b-c x^2+a x^4}}{4 \sqrt {b} \left (\sqrt {b}+\sqrt {a} x^2\right )}+\frac {\left (2 \sqrt {a} \sqrt {b}+c-\sqrt {4 a b+c^2}\right ) \sqrt {c+\sqrt {4 a b+c^2}} \sqrt {1-\frac {2 a x^2}{c-\sqrt {4 a b+c^2}}} \sqrt {1-\frac {2 a x^2}{c+\sqrt {4 a b+c^2}}} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {a} x}{\sqrt {c+\sqrt {4 a b+c^2}}}\right )|\frac {c+\sqrt {4 a b+c^2}}{c-\sqrt {4 a b+c^2}}\right )}{8 \sqrt {2} a \sqrt {b} \sqrt {-b-c x^2+a x^4}}+\frac {\left (2 \sqrt {a} \sqrt {b}-c+\sqrt {4 a b+c^2}\right ) \sqrt {c+\sqrt {4 a b+c^2}} \sqrt {1-\frac {2 a x^2}{c-\sqrt {4 a b+c^2}}} \sqrt {1-\frac {2 a x^2}{c+\sqrt {4 a b+c^2}}} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {a} x}{\sqrt {c+\sqrt {4 a b+c^2}}}\right )|\frac {c+\sqrt {4 a b+c^2}}{c-\sqrt {4 a b+c^2}}\right )}{8 \sqrt {2} a \sqrt {b} \sqrt {-b-c x^2+a x^4}}-\frac {\sqrt {c+\sqrt {4 a b+c^2}} \sqrt {1-\frac {2 a x^2}{c-\sqrt {4 a b+c^2}}} \sqrt {1-\frac {2 a x^2}{c+\sqrt {4 a b+c^2}}} \Pi \left (-\frac {c+\sqrt {4 a b+c^2}}{2 \sqrt {a} \sqrt {b}};\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {a} x}{\sqrt {c+\sqrt {4 a b+c^2}}}\right )|\frac {c+\sqrt {4 a b+c^2}}{c-\sqrt {4 a b+c^2}}\right )}{2 \sqrt {2} \sqrt {a} \sqrt {-b-c x^2+a x^4}}-\frac {\sqrt {c+\sqrt {4 a b+c^2}} \sqrt {1-\frac {2 a x^2}{c-\sqrt {4 a b+c^2}}} \sqrt {1-\frac {2 a x^2}{c+\sqrt {4 a b+c^2}}} \Pi \left (\frac {c+\sqrt {4 a b+c^2}}{2 \sqrt {a} \sqrt {b}};\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {a} x}{\sqrt {c+\sqrt {4 a b+c^2}}}\right )|\frac {c+\sqrt {4 a b+c^2}}{c-\sqrt {4 a b+c^2}}\right )}{2 \sqrt {2} \sqrt {a} \sqrt {-b-c x^2+a x^4}}\\ \end {align*}
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Mathematica [C] time = 2.96, size = 416, normalized size = 0.50 \begin {gather*} \frac {1}{2} \sqrt {a x^4-b-c x^2} \left (\frac {x}{b-a x^4}-\frac {i \sqrt {\frac {4 a x^2}{\sqrt {4 a b+c^2}-c}+2} \sqrt {1-\frac {2 a x^2}{\sqrt {4 a b+c^2}+c}} \left (F\left (i \sinh ^{-1}\left (\sqrt {2} \sqrt {\frac {a}{\sqrt {c^2+4 a b}-c}} x\right )|\frac {c-\sqrt {c^2+4 a b}}{c+\sqrt {c^2+4 a b}}\right )-\Pi \left (\frac {c-\sqrt {c^2+4 a b}}{2 \sqrt {a} \sqrt {b}};i \sinh ^{-1}\left (\sqrt {2} \sqrt {\frac {a}{\sqrt {c^2+4 a b}-c}} x\right )|\frac {c-\sqrt {c^2+4 a b}}{c+\sqrt {c^2+4 a b}}\right )-\Pi \left (\frac {\sqrt {c^2+4 a b}-c}{2 \sqrt {a} \sqrt {b}};i \sinh ^{-1}\left (\sqrt {2} \sqrt {\frac {a}{\sqrt {c^2+4 a b}-c}} x\right )|\frac {c-\sqrt {c^2+4 a b}}{c+\sqrt {c^2+4 a b}}\right )\right )}{2 \sqrt {\frac {a}{\sqrt {4 a b+c^2}-c}} \left (a x^4-b-c x^2\right )}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.78, size = 87, normalized size = 0.11 \begin {gather*} -\frac {x \sqrt {-b-c x^2+a x^4}}{2 \left (-b+a x^4\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {c} x \sqrt {-b-c x^2+a x^4}}{b+c x^2-a x^4}\right )}{2 \sqrt {c}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.75, size = 246, normalized size = 0.30 \begin {gather*} \left [-\frac {4 \, \sqrt {a x^{4} - c x^{2} - b} c x + {\left (a x^{4} - b\right )} \sqrt {-c} \log \left (-\frac {a^{2} x^{8} - 8 \, a c x^{6} - 2 \, {\left (a b - 4 \, c^{2}\right )} x^{4} + 8 \, b c x^{2} + b^{2} - 4 \, {\left (a x^{5} - 2 \, c x^{3} - b x\right )} \sqrt {a x^{4} - c x^{2} - b} \sqrt {-c}}{a^{2} x^{8} - 2 \, a b x^{4} + b^{2}}\right )}{8 \, {\left (a c x^{4} - b c\right )}}, -\frac {2 \, \sqrt {a x^{4} - c x^{2} - b} c x + {\left (a x^{4} - b\right )} \sqrt {c} \arctan \left (\frac {2 \, \sqrt {a x^{4} - c x^{2} - b} \sqrt {c} x}{a x^{4} - 2 \, c x^{2} - b}\right )}{4 \, {\left (a c x^{4} - b c\right )}}\right ] \end {gather*}
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 \begin {gather*} \int \frac {\sqrt {a x^{4} - c x^{2} - b} {\left (a x^{4} + b\right )}}{{\left (a x^{4} - b\right )}^{2}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.14, size = 91, normalized size = 0.11
method | result | size |
elliptic | \(\frac {\left (-\frac {\sqrt {a \,x^{4}-c \,x^{2}-b}\, \sqrt {2}}{4 x \left (\frac {a \,x^{4}-c \,x^{2}-b}{2 x^{2}}+\frac {c}{2}\right )}+\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {a \,x^{4}-c \,x^{2}-b}}{x \sqrt {c}}\right )}{2 \sqrt {c}}\right ) \sqrt {2}}{2}\) | \(91\) |
default | \(2 b \left (-\frac {x \sqrt {a \,x^{4}-c \,x^{2}-b}}{4 b \left (a \,x^{4}-b \right )}-\frac {\sqrt {4+\frac {2 \left (c +\sqrt {4 a b +c^{2}}\right ) x^{2}}{b}}\, \sqrt {4-\frac {2 \left (-c +\sqrt {4 a b +c^{2}}\right ) x^{2}}{b}}\, \EllipticF \left (\frac {x \sqrt {-\frac {2 \left (c +\sqrt {4 a b +c^{2}}\right )}{b}}}{2}, \frac {\sqrt {-4+\frac {2 c \left (-c +\sqrt {4 a b +c^{2}}\right )}{b a}}}{2}\right )}{8 b \sqrt {-\frac {2 \left (c +\sqrt {4 a b +c^{2}}\right )}{b}}\, \sqrt {a \,x^{4}-c \,x^{2}-b}}-\frac {\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (a \,\textit {\_Z}^{4}-b \right )}{\sum }\frac {\left (-c \,\underline {\hspace {1.25 ex}}\alpha ^{2}-b \right ) \left (-\frac {\arctanh \left (\frac {2 a \,x^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}-c \,\underline {\hspace {1.25 ex}}\alpha ^{2}-c \,x^{2}-2 b}{2 \sqrt {-c \,\underline {\hspace {1.25 ex}}\alpha ^{2}}\, \sqrt {a \,x^{4}-c \,x^{2}-b}}\right )}{\sqrt {-c \,\underline {\hspace {1.25 ex}}\alpha ^{2}}}-\frac {a \sqrt {2}\, \underline {\hspace {1.25 ex}}\alpha ^{3} \sqrt {2+\frac {c \,x^{2}}{b}+\frac {x^{2} \sqrt {4 a b +c^{2}}}{b}}\, \sqrt {2+\frac {c \,x^{2}}{b}-\frac {x^{2} \sqrt {4 a b +c^{2}}}{b}}\, \EllipticPi \left (\sqrt {-\frac {c +\sqrt {4 a b +c^{2}}}{2 b}}\, x , \frac {\underline {\hspace {1.25 ex}}\alpha ^{2} \left (c -\sqrt {4 a b +c^{2}}\right )}{2 b}, \frac {\sqrt {2}\, \sqrt {\frac {-c +\sqrt {4 a b +c^{2}}}{b}}}{2 \sqrt {-\frac {c +\sqrt {4 a b +c^{2}}}{2 b}}}\right )}{b \sqrt {-\frac {c +\sqrt {4 a b +c^{2}}}{b}}\, \sqrt {a \,x^{4}-c \,x^{2}-b}}\right )}{\underline {\hspace {1.25 ex}}\alpha ^{3}}}{16 a b}\right )+\frac {\sqrt {4+\frac {2 \left (c +\sqrt {4 a b +c^{2}}\right ) x^{2}}{b}}\, \sqrt {4-\frac {2 \left (-c +\sqrt {4 a b +c^{2}}\right ) x^{2}}{b}}\, \EllipticF \left (\frac {x \sqrt {-\frac {2 \left (c +\sqrt {4 a b +c^{2}}\right )}{b}}}{2}, \frac {\sqrt {-4+\frac {2 c \left (-c +\sqrt {4 a b +c^{2}}\right )}{b a}}}{2}\right )}{2 \sqrt {-\frac {2 \left (c +\sqrt {4 a b +c^{2}}\right )}{b}}\, \sqrt {a \,x^{4}-c \,x^{2}-b}}-\frac {c \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (a \,\textit {\_Z}^{4}-b \right )}{\sum }\frac {-\frac {\arctanh \left (\frac {2 a \,x^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}-c \,\underline {\hspace {1.25 ex}}\alpha ^{2}-c \,x^{2}-2 b}{2 \sqrt {-c \,\underline {\hspace {1.25 ex}}\alpha ^{2}}\, \sqrt {a \,x^{4}-c \,x^{2}-b}}\right )}{\sqrt {-c \,\underline {\hspace {1.25 ex}}\alpha ^{2}}}-\frac {a \sqrt {2}\, \underline {\hspace {1.25 ex}}\alpha ^{3} \sqrt {2+\frac {c \,x^{2}}{b}+\frac {x^{2} \sqrt {4 a b +c^{2}}}{b}}\, \sqrt {2+\frac {c \,x^{2}}{b}-\frac {x^{2} \sqrt {4 a b +c^{2}}}{b}}\, \EllipticPi \left (\sqrt {-\frac {c +\sqrt {4 a b +c^{2}}}{2 b}}\, x , \frac {\underline {\hspace {1.25 ex}}\alpha ^{2} \left (c -\sqrt {4 a b +c^{2}}\right )}{2 b}, \frac {\sqrt {2}\, \sqrt {\frac {-c +\sqrt {4 a b +c^{2}}}{b}}}{2 \sqrt {-\frac {c +\sqrt {4 a b +c^{2}}}{2 b}}}\right )}{b \sqrt {-\frac {c +\sqrt {4 a b +c^{2}}}{b}}\, \sqrt {a \,x^{4}-c \,x^{2}-b}}}{\underline {\hspace {1.25 ex}}\alpha }\right )}{8 a}\) | \(899\) |
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 \begin {gather*} \int \frac {\sqrt {a x^{4} - c x^{2} - b} {\left (a x^{4} + b\right )}}{{\left (a x^{4} - b\right )}^{2}}\,{d x} \end {gather*}
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 \begin {gather*} \int \frac {\left (a\,x^4+b\right )\,\sqrt {a\,x^4-c\,x^2-b}}{{\left (b-a\,x^4\right )}^2} \,d x \end {gather*}
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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