Optimal. Leaf size=996 \[ \frac {a x \sqrt {c+d x^4}}{4 b (b c-a d) \left (a+b x^4\right )}-\frac {\sqrt [4]{-a} (5 b c-3 a d) \tan ^{-1}\left (\frac {\sqrt {b c-a d} x}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt {c+d x^4}}\right )}{16 b^{7/4} (b c-a d)^{3/2}}+\frac {\sqrt [4]{-a} (5 b c-3 a d) \tan ^{-1}\left (\frac {\sqrt {-b c+a d} x}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt {c+d x^4}}\right )}{16 b^{7/4} (-b c+a d)^{3/2}}+\frac {(4 b c-3 a d) \left (\sqrt {c}+\sqrt {d} x^2\right ) \sqrt {\frac {c+d x^4}{\left (\sqrt {c}+\sqrt {d} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{8 b^2 \sqrt [4]{c} \sqrt [4]{d} (b c-a d) \sqrt {c+d x^4}}-\frac {a \left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {-a}}+\sqrt {d}\right ) \sqrt [4]{d} (5 b c-3 a d) \left (\sqrt {c}+\sqrt {d} x^2\right ) \sqrt {\frac {c+d x^4}{\left (\sqrt {c}+\sqrt {d} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{16 b^2 \sqrt [4]{c} (b c-a d) (b c+a d) \sqrt {c+d x^4}}-\frac {\sqrt {-a} \left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right ) \sqrt [4]{d} (5 b c-3 a d) \left (\sqrt {c}+\sqrt {d} x^2\right ) \sqrt {\frac {c+d x^4}{\left (\sqrt {c}+\sqrt {d} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{16 b^2 \sqrt [4]{c} (b c-a d) (b c+a d) \sqrt {c+d x^4}}-\frac {\left (\sqrt {b} \sqrt {c}+\sqrt {-a} \sqrt {d}\right )^2 (5 b c-3 a d) \left (\sqrt {c}+\sqrt {d} x^2\right ) \sqrt {\frac {c+d x^4}{\left (\sqrt {c}+\sqrt {d} x^2\right )^2}} \Pi \left (-\frac {\left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right )^2}{4 \sqrt {-a} \sqrt {b} \sqrt {c} \sqrt {d}};2 \tan ^{-1}\left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{32 b^2 \sqrt [4]{c} \sqrt [4]{d} (b c-a d) (b c+a d) \sqrt {c+d x^4}}-\frac {\left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right )^2 (5 b c-3 a d) \left (\sqrt {c}+\sqrt {d} x^2\right ) \sqrt {\frac {c+d x^4}{\left (\sqrt {c}+\sqrt {d} x^2\right )^2}} \Pi \left (\frac {\left (\sqrt {b} \sqrt {c}+\sqrt {-a} \sqrt {d}\right )^2}{4 \sqrt {-a} \sqrt {b} \sqrt {c} \sqrt {d}};2 \tan ^{-1}\left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{32 b^2 \sqrt [4]{c} \sqrt [4]{d} (b c-a d) (b c+a d) \sqrt {c+d x^4}} \]
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Rubi [A]
time = 0.92, antiderivative size = 996, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {481, 537, 226,
418, 1231, 1721} \begin {gather*} -\frac {(5 b c-3 a d) \left (\sqrt {d} x^2+\sqrt {c}\right ) \sqrt {\frac {d x^4+c}{\left (\sqrt {d} x^2+\sqrt {c}\right )^2}} \Pi \left (\frac {\left (\sqrt {b} \sqrt {c}+\sqrt {-a} \sqrt {d}\right )^2}{4 \sqrt {-a} \sqrt {b} \sqrt {c} \sqrt {d}};2 \text {ArcTan}\left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac {1}{2}\right ) \left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right )^2}{32 b^2 \sqrt [4]{c} \sqrt [4]{d} (b c-a d) (b c+a d) \sqrt {d x^4+c}}-\frac {\sqrt {-a} \sqrt [4]{d} (5 b c-3 a d) \left (\sqrt {d} x^2+\sqrt {c}\right ) \sqrt {\frac {d x^4+c}{\left (\sqrt {d} x^2+\sqrt {c}\right )^2}} F\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac {1}{2}\right ) \left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right )}{16 b^2 \sqrt [4]{c} (b c-a d) (b c+a d) \sqrt {d x^4+c}}-\frac {\sqrt [4]{-a} (5 b c-3 a d) \text {ArcTan}\left (\frac {\sqrt {b c-a d} x}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt {d x^4+c}}\right )}{16 b^{7/4} (b c-a d)^{3/2}}+\frac {\sqrt [4]{-a} (5 b c-3 a d) \text {ArcTan}\left (\frac {\sqrt {a d-b c} x}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt {d x^4+c}}\right )}{16 b^{7/4} (a d-b c)^{3/2}}+\frac {(4 b c-3 a d) \left (\sqrt {d} x^2+\sqrt {c}\right ) \sqrt {\frac {d x^4+c}{\left (\sqrt {d} x^2+\sqrt {c}\right )^2}} F\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{8 b^2 \sqrt [4]{c} \sqrt [4]{d} (b c-a d) \sqrt {d x^4+c}}-\frac {a \left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {-a}}+\sqrt {d}\right ) \sqrt [4]{d} (5 b c-3 a d) \left (\sqrt {d} x^2+\sqrt {c}\right ) \sqrt {\frac {d x^4+c}{\left (\sqrt {d} x^2+\sqrt {c}\right )^2}} F\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{16 b^2 \sqrt [4]{c} (b c-a d) (b c+a d) \sqrt {d x^4+c}}-\frac {\left (\sqrt {b} \sqrt {c}+\sqrt {-a} \sqrt {d}\right )^2 (5 b c-3 a d) \left (\sqrt {d} x^2+\sqrt {c}\right ) \sqrt {\frac {d x^4+c}{\left (\sqrt {d} x^2+\sqrt {c}\right )^2}} \Pi \left (-\frac {\left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right )^2}{4 \sqrt {-a} \sqrt {b} \sqrt {c} \sqrt {d}};2 \text {ArcTan}\left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{32 b^2 \sqrt [4]{c} \sqrt [4]{d} (b c-a d) (b c+a d) \sqrt {d x^4+c}}+\frac {a x \sqrt {d x^4+c}}{4 b (b c-a d) \left (b x^4+a\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 226
Rule 418
Rule 481
Rule 537
Rule 1231
Rule 1721
Rubi steps
\begin {align*} \int \frac {x^8}{\left (a+b x^4\right )^2 \sqrt {c+d x^4}} \, dx &=\frac {a x \sqrt {c+d x^4}}{4 b (b c-a d) \left (a+b x^4\right )}-\frac {\int \frac {a c+(-4 b c+3 a d) x^4}{\left (a+b x^4\right ) \sqrt {c+d x^4}} \, dx}{4 b (b c-a d)}\\ &=\frac {a x \sqrt {c+d x^4}}{4 b (b c-a d) \left (a+b x^4\right )}+\frac {(4 b c-3 a d) \int \frac {1}{\sqrt {c+d x^4}} \, dx}{4 b^2 (b c-a d)}-\frac {(a (5 b c-3 a d)) \int \frac {1}{\left (a+b x^4\right ) \sqrt {c+d x^4}} \, dx}{4 b^2 (b c-a d)}\\ &=\frac {a x \sqrt {c+d x^4}}{4 b (b c-a d) \left (a+b x^4\right )}+\frac {(4 b c-3 a d) \left (\sqrt {c}+\sqrt {d} x^2\right ) \sqrt {\frac {c+d x^4}{\left (\sqrt {c}+\sqrt {d} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{8 b^2 \sqrt [4]{c} \sqrt [4]{d} (b c-a d) \sqrt {c+d x^4}}-\frac {(5 b c-3 a d) \int \frac {1}{\left (1-\frac {\sqrt {b} x^2}{\sqrt {-a}}\right ) \sqrt {c+d x^4}} \, dx}{8 b^2 (b c-a d)}-\frac {(5 b c-3 a d) \int \frac {1}{\left (1+\frac {\sqrt {b} x^2}{\sqrt {-a}}\right ) \sqrt {c+d x^4}} \, dx}{8 b^2 (b c-a d)}\\ &=\frac {a x \sqrt {c+d x^4}}{4 b (b c-a d) \left (a+b x^4\right )}+\frac {(4 b c-3 a d) \left (\sqrt {c}+\sqrt {d} x^2\right ) \sqrt {\frac {c+d x^4}{\left (\sqrt {c}+\sqrt {d} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{8 b^2 \sqrt [4]{c} \sqrt [4]{d} (b c-a d) \sqrt {c+d x^4}}-\frac {\left (\sqrt {c} \left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right ) (5 b c-3 a d)\right ) \int \frac {1+\frac {\sqrt {d} x^2}{\sqrt {c}}}{\left (1-\frac {\sqrt {b} x^2}{\sqrt {-a}}\right ) \sqrt {c+d x^4}} \, dx}{8 b^{3/2} (b c-a d) (b c+a d)}-\frac {\left (\sqrt {c} \left (\sqrt {b} \sqrt {c}+\sqrt {-a} \sqrt {d}\right ) (5 b c-3 a d)\right ) \int \frac {1+\frac {\sqrt {d} x^2}{\sqrt {c}}}{\left (1+\frac {\sqrt {b} x^2}{\sqrt {-a}}\right ) \sqrt {c+d x^4}} \, dx}{8 b^{3/2} (b c-a d) (b c+a d)}-\frac {\left (a \left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {-a}}+\sqrt {d}\right ) \sqrt {d} (5 b c-3 a d)\right ) \int \frac {1}{\sqrt {c+d x^4}} \, dx}{8 b^2 (b c-a d) (b c+a d)}-\frac {\left (\sqrt {-a} \left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right ) \sqrt {d} (5 b c-3 a d)\right ) \int \frac {1}{\sqrt {c+d x^4}} \, dx}{8 b^2 (b c-a d) (b c+a d)}\\ &=\frac {a x \sqrt {c+d x^4}}{4 b (b c-a d) \left (a+b x^4\right )}-\frac {\sqrt [4]{-a} (5 b c-3 a d) \tan ^{-1}\left (\frac {\sqrt {b c-a d} x}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt {c+d x^4}}\right )}{16 b^{7/4} (b c-a d)^{3/2}}+\frac {\sqrt [4]{-a} (5 b c-3 a d) \tan ^{-1}\left (\frac {\sqrt {-b c+a d} x}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt {c+d x^4}}\right )}{16 b^{7/4} (-b c+a d)^{3/2}}+\frac {(4 b c-3 a d) \left (\sqrt {c}+\sqrt {d} x^2\right ) \sqrt {\frac {c+d x^4}{\left (\sqrt {c}+\sqrt {d} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{8 b^2 \sqrt [4]{c} \sqrt [4]{d} (b c-a d) \sqrt {c+d x^4}}-\frac {a \left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {-a}}+\sqrt {d}\right ) \sqrt [4]{d} (5 b c-3 a d) \left (\sqrt {c}+\sqrt {d} x^2\right ) \sqrt {\frac {c+d x^4}{\left (\sqrt {c}+\sqrt {d} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{16 b^2 \sqrt [4]{c} (b c-a d) (b c+a d) \sqrt {c+d x^4}}-\frac {\sqrt {-a} \left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right ) \sqrt [4]{d} (5 b c-3 a d) \left (\sqrt {c}+\sqrt {d} x^2\right ) \sqrt {\frac {c+d x^4}{\left (\sqrt {c}+\sqrt {d} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{16 b^2 \sqrt [4]{c} (b c-a d) (b c+a d) \sqrt {c+d x^4}}-\frac {\left (\sqrt {b} \sqrt {c}+\sqrt {-a} \sqrt {d}\right )^2 (5 b c-3 a d) \left (\sqrt {c}+\sqrt {d} x^2\right ) \sqrt {\frac {c+d x^4}{\left (\sqrt {c}+\sqrt {d} x^2\right )^2}} \Pi \left (-\frac {\left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right )^2}{4 \sqrt {-a} \sqrt {b} \sqrt {c} \sqrt {d}};2 \tan ^{-1}\left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{32 b^2 \sqrt [4]{c} \sqrt [4]{d} (b c-a d) (b c+a d) \sqrt {c+d x^4}}-\frac {\left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right )^2 (5 b c-3 a d) \left (\sqrt {c}+\sqrt {d} x^2\right ) \sqrt {\frac {c+d x^4}{\left (\sqrt {c}+\sqrt {d} x^2\right )^2}} \Pi \left (\frac {\left (\sqrt {b} \sqrt {c}+\sqrt {-a} \sqrt {d}\right )^2}{4 \sqrt {-a} \sqrt {b} \sqrt {c} \sqrt {d}};2 \tan ^{-1}\left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{32 b^2 \sqrt [4]{c} \sqrt [4]{d} (b c-a d) (b c+a d) \sqrt {c+d x^4}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 6 vs. order 4 in
optimal.
time = 10.21, size = 253, normalized size = 0.25 \begin {gather*} \frac {x \left (\frac {(4 b c-3 a d) x^4 \sqrt {1+\frac {d x^4}{c}} F_1\left (\frac {5}{4};\frac {1}{2},1;\frac {9}{4};-\frac {d x^4}{c},-\frac {b x^4}{a}\right )}{a b}+\frac {5 a \left (c+d x^4+\frac {5 a c^2 F_1\left (\frac {1}{4};\frac {1}{2},1;\frac {5}{4};-\frac {d x^4}{c},-\frac {b x^4}{a}\right )}{-5 a c F_1\left (\frac {1}{4};\frac {1}{2},1;\frac {5}{4};-\frac {d x^4}{c},-\frac {b x^4}{a}\right )+2 x^4 \left (2 b c F_1\left (\frac {5}{4};\frac {1}{2},2;\frac {9}{4};-\frac {d x^4}{c},-\frac {b x^4}{a}\right )+a d F_1\left (\frac {5}{4};\frac {3}{2},1;\frac {9}{4};-\frac {d x^4}{c},-\frac {b x^4}{a}\right )\right )}\right )}{b \left (a+b x^4\right )}\right )}{20 (b c-a d) \sqrt {c+d x^4}} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.35, size = 604, normalized size = 0.61
method | result | size |
elliptic | \(-\frac {a x \sqrt {d \,x^{4}+c}}{4 \left (a d -b c \right ) b \left (b \,x^{4}+a \right )}+\frac {\left (\frac {1}{b^{2}}-\frac {a d}{4 b^{2} \left (a d -b c \right )}\right ) \sqrt {1-\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \sqrt {1+\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \EllipticF \left (x \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}, i\right )}{\sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}\, \sqrt {d \,x^{4}+c}}-\frac {a \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{4} b +a \right )}{\sum }\frac {\left (3 a d -5 b c \right ) \left (-\frac {\arctanh \left (\frac {2 d \,x^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}+2 c}{2 \sqrt {\frac {-a d +b c}{b}}\, \sqrt {d \,x^{4}+c}}\right )}{\sqrt {\frac {-a d +b c}{b}}}+\frac {2 \underline {\hspace {1.25 ex}}\alpha ^{3} b \sqrt {1-\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \sqrt {1+\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \EllipticPi \left (x \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}, \frac {i \sqrt {c}\, \underline {\hspace {1.25 ex}}\alpha ^{2} b}{\sqrt {d}\, a}, \frac {\sqrt {-\frac {i \sqrt {d}}{\sqrt {c}}}}{\sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}}\right )}{\sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}\, a \sqrt {d \,x^{4}+c}}\right )}{\left (a d -b c \right ) \underline {\hspace {1.25 ex}}\alpha ^{3}}\right )}{32 b^{3}}\) | \(337\) |
default | \(\frac {\sqrt {1-\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \sqrt {1+\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \EllipticF \left (x \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}, i\right )}{b^{2} \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}\, \sqrt {d \,x^{4}+c}}-\frac {a \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{4} b +a \right )}{\sum }\frac {-\frac {\arctanh \left (\frac {2 d \,x^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}+2 c}{2 \sqrt {\frac {-a d +b c}{b}}\, \sqrt {d \,x^{4}+c}}\right )}{\sqrt {\frac {-a d +b c}{b}}}+\frac {2 \underline {\hspace {1.25 ex}}\alpha ^{3} b \sqrt {1-\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \sqrt {1+\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \EllipticPi \left (x \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}, \frac {i \sqrt {c}\, \underline {\hspace {1.25 ex}}\alpha ^{2} b}{\sqrt {d}\, a}, \frac {\sqrt {-\frac {i \sqrt {d}}{\sqrt {c}}}}{\sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}}\right )}{\sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}\, a \sqrt {d \,x^{4}+c}}}{\underline {\hspace {1.25 ex}}\alpha ^{3}}\right )}{4 b^{3}}+\frac {a^{2} \left (-\frac {b x \sqrt {d \,x^{4}+c}}{4 a \left (a d -b c \right ) \left (b \,x^{4}+a \right )}-\frac {d \sqrt {1-\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \sqrt {1+\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \EllipticF \left (x \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}, i\right )}{4 a \left (a d -b c \right ) \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}\, \sqrt {d \,x^{4}+c}}-\frac {\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{4} b +a \right )}{\sum }\frac {\left (-5 a d +3 b c \right ) \left (-\frac {\arctanh \left (\frac {2 d \,x^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}+2 c}{2 \sqrt {\frac {-a d +b c}{b}}\, \sqrt {d \,x^{4}+c}}\right )}{\sqrt {\frac {-a d +b c}{b}}}+\frac {2 \underline {\hspace {1.25 ex}}\alpha ^{3} b \sqrt {1-\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \sqrt {1+\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \EllipticPi \left (x \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}, \frac {i \sqrt {c}\, \underline {\hspace {1.25 ex}}\alpha ^{2} b}{\sqrt {d}\, a}, \frac {\sqrt {-\frac {i \sqrt {d}}{\sqrt {c}}}}{\sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}}\right )}{\sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}\, a \sqrt {d \,x^{4}+c}}\right )}{\left (a d -b c \right ) \underline {\hspace {1.25 ex}}\alpha ^{3}}}{32 b a}\right )}{b^{2}}\) | \(604\) |
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] Timed out
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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{8}}{\left (a + b x^{4}\right )^{2} \sqrt {c + d x^{4}}}\, dx \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*} \text {could not integrate} \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 {x^8}{{\left (b\,x^4+a\right )}^2\,\sqrt {d\,x^4+c}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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