Optimal. Leaf size=385 \[ -\frac {d (d x)^{17/2}}{10 b \left (a+b x^2\right )^5}-\frac {17 d^3 (d x)^{13/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {221 d^5 (d x)^{9/2}}{1920 b^3 \left (a+b x^2\right )^3}-\frac {663 d^7 (d x)^{5/2}}{5120 b^4 \left (a+b x^2\right )^2}-\frac {663 d^9 \sqrt {d x}}{4096 b^5 \left (a+b x^2\right )}-\frac {663 d^{19/2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{3/4} b^{21/4}}+\frac {663 d^{19/2} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{3/4} b^{21/4}}-\frac {663 d^{19/2} \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{16384 \sqrt {2} a^{3/4} b^{21/4}}+\frac {663 d^{19/2} \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{16384 \sqrt {2} a^{3/4} b^{21/4}} \]
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Rubi [A]
time = 0.28, antiderivative size = 385, normalized size of antiderivative = 1.00, number of steps
used = 16, number of rules used = 9, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.321, Rules used = {28, 294, 335,
217, 1179, 642, 1176, 631, 210} \begin {gather*} -\frac {663 d^{19/2} \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{3/4} b^{21/4}}+\frac {663 d^{19/2} \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{8192 \sqrt {2} a^{3/4} b^{21/4}}-\frac {663 d^{19/2} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{16384 \sqrt {2} a^{3/4} b^{21/4}}+\frac {663 d^{19/2} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{16384 \sqrt {2} a^{3/4} b^{21/4}}-\frac {663 d^9 \sqrt {d x}}{4096 b^5 \left (a+b x^2\right )}-\frac {663 d^7 (d x)^{5/2}}{5120 b^4 \left (a+b x^2\right )^2}-\frac {221 d^5 (d x)^{9/2}}{1920 b^3 \left (a+b x^2\right )^3}-\frac {17 d^3 (d x)^{13/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {d (d x)^{17/2}}{10 b \left (a+b x^2\right )^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 28
Rule 210
Rule 217
Rule 294
Rule 335
Rule 631
Rule 642
Rule 1176
Rule 1179
Rubi steps
\begin {align*} \int \frac {(d x)^{19/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx &=b^6 \int \frac {(d x)^{19/2}}{\left (a b+b^2 x^2\right )^6} \, dx\\ &=-\frac {d (d x)^{17/2}}{10 b \left (a+b x^2\right )^5}+\frac {1}{20} \left (17 b^4 d^2\right ) \int \frac {(d x)^{15/2}}{\left (a b+b^2 x^2\right )^5} \, dx\\ &=-\frac {d (d x)^{17/2}}{10 b \left (a+b x^2\right )^5}-\frac {17 d^3 (d x)^{13/2}}{160 b^2 \left (a+b x^2\right )^4}+\frac {1}{320} \left (221 b^2 d^4\right ) \int \frac {(d x)^{11/2}}{\left (a b+b^2 x^2\right )^4} \, dx\\ &=-\frac {d (d x)^{17/2}}{10 b \left (a+b x^2\right )^5}-\frac {17 d^3 (d x)^{13/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {221 d^5 (d x)^{9/2}}{1920 b^3 \left (a+b x^2\right )^3}+\frac {\left (663 d^6\right ) \int \frac {(d x)^{7/2}}{\left (a b+b^2 x^2\right )^3} \, dx}{1280}\\ &=-\frac {d (d x)^{17/2}}{10 b \left (a+b x^2\right )^5}-\frac {17 d^3 (d x)^{13/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {221 d^5 (d x)^{9/2}}{1920 b^3 \left (a+b x^2\right )^3}-\frac {663 d^7 (d x)^{5/2}}{5120 b^4 \left (a+b x^2\right )^2}+\frac {\left (663 d^8\right ) \int \frac {(d x)^{3/2}}{\left (a b+b^2 x^2\right )^2} \, dx}{2048 b^2}\\ &=-\frac {d (d x)^{17/2}}{10 b \left (a+b x^2\right )^5}-\frac {17 d^3 (d x)^{13/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {221 d^5 (d x)^{9/2}}{1920 b^3 \left (a+b x^2\right )^3}-\frac {663 d^7 (d x)^{5/2}}{5120 b^4 \left (a+b x^2\right )^2}-\frac {663 d^9 \sqrt {d x}}{4096 b^5 \left (a+b x^2\right )}+\frac {\left (663 d^{10}\right ) \int \frac {1}{\sqrt {d x} \left (a b+b^2 x^2\right )} \, dx}{8192 b^4}\\ &=-\frac {d (d x)^{17/2}}{10 b \left (a+b x^2\right )^5}-\frac {17 d^3 (d x)^{13/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {221 d^5 (d x)^{9/2}}{1920 b^3 \left (a+b x^2\right )^3}-\frac {663 d^7 (d x)^{5/2}}{5120 b^4 \left (a+b x^2\right )^2}-\frac {663 d^9 \sqrt {d x}}{4096 b^5 \left (a+b x^2\right )}+\frac {\left (663 d^9\right ) \text {Subst}\left (\int \frac {1}{a b+\frac {b^2 x^4}{d^2}} \, dx,x,\sqrt {d x}\right )}{4096 b^4}\\ &=-\frac {d (d x)^{17/2}}{10 b \left (a+b x^2\right )^5}-\frac {17 d^3 (d x)^{13/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {221 d^5 (d x)^{9/2}}{1920 b^3 \left (a+b x^2\right )^3}-\frac {663 d^7 (d x)^{5/2}}{5120 b^4 \left (a+b x^2\right )^2}-\frac {663 d^9 \sqrt {d x}}{4096 b^5 \left (a+b x^2\right )}+\frac {\left (663 d^8\right ) \text {Subst}\left (\int \frac {\sqrt {a} d-\sqrt {b} x^2}{a b+\frac {b^2 x^4}{d^2}} \, dx,x,\sqrt {d x}\right )}{8192 \sqrt {a} b^4}+\frac {\left (663 d^8\right ) \text {Subst}\left (\int \frac {\sqrt {a} d+\sqrt {b} x^2}{a b+\frac {b^2 x^4}{d^2}} \, dx,x,\sqrt {d x}\right )}{8192 \sqrt {a} b^4}\\ &=-\frac {d (d x)^{17/2}}{10 b \left (a+b x^2\right )^5}-\frac {17 d^3 (d x)^{13/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {221 d^5 (d x)^{9/2}}{1920 b^3 \left (a+b x^2\right )^3}-\frac {663 d^7 (d x)^{5/2}}{5120 b^4 \left (a+b x^2\right )^2}-\frac {663 d^9 \sqrt {d x}}{4096 b^5 \left (a+b x^2\right )}-\frac {\left (663 d^{19/2}\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d}}{\sqrt [4]{b}}+2 x}{-\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d} x}{\sqrt [4]{b}}-x^2} \, dx,x,\sqrt {d x}\right )}{16384 \sqrt {2} a^{3/4} b^{21/4}}-\frac {\left (663 d^{19/2}\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d}}{\sqrt [4]{b}}-2 x}{-\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d} x}{\sqrt [4]{b}}-x^2} \, dx,x,\sqrt {d x}\right )}{16384 \sqrt {2} a^{3/4} b^{21/4}}+\frac {\left (663 d^{10}\right ) \text {Subst}\left (\int \frac {1}{\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt {d x}\right )}{16384 \sqrt {a} b^{11/2}}+\frac {\left (663 d^{10}\right ) \text {Subst}\left (\int \frac {1}{\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt {d x}\right )}{16384 \sqrt {a} b^{11/2}}\\ &=-\frac {d (d x)^{17/2}}{10 b \left (a+b x^2\right )^5}-\frac {17 d^3 (d x)^{13/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {221 d^5 (d x)^{9/2}}{1920 b^3 \left (a+b x^2\right )^3}-\frac {663 d^7 (d x)^{5/2}}{5120 b^4 \left (a+b x^2\right )^2}-\frac {663 d^9 \sqrt {d x}}{4096 b^5 \left (a+b x^2\right )}-\frac {663 d^{19/2} \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{16384 \sqrt {2} a^{3/4} b^{21/4}}+\frac {663 d^{19/2} \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{16384 \sqrt {2} a^{3/4} b^{21/4}}+\frac {\left (663 d^{19/2}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{3/4} b^{21/4}}-\frac {\left (663 d^{19/2}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{3/4} b^{21/4}}\\ &=-\frac {d (d x)^{17/2}}{10 b \left (a+b x^2\right )^5}-\frac {17 d^3 (d x)^{13/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {221 d^5 (d x)^{9/2}}{1920 b^3 \left (a+b x^2\right )^3}-\frac {663 d^7 (d x)^{5/2}}{5120 b^4 \left (a+b x^2\right )^2}-\frac {663 d^9 \sqrt {d x}}{4096 b^5 \left (a+b x^2\right )}-\frac {663 d^{19/2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{3/4} b^{21/4}}+\frac {663 d^{19/2} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{3/4} b^{21/4}}-\frac {663 d^{19/2} \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{16384 \sqrt {2} a^{3/4} b^{21/4}}+\frac {663 d^{19/2} \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{16384 \sqrt {2} a^{3/4} b^{21/4}}\\ \end {align*}
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Mathematica [A]
time = 0.66, size = 205, normalized size = 0.53 \begin {gather*} \frac {d^9 \sqrt {d x} \left (-4 a^{3/4} \sqrt [4]{b} \sqrt {x} \left (9945 a^4+47736 a^3 b x^2+90610 a^2 b^2 x^4+84320 a b^3 x^6+37645 b^4 x^8\right )+9945 \sqrt {2} \left (a+b x^2\right )^5 \tan ^{-1}\left (\frac {-\sqrt {a}+\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )+9945 \sqrt {2} \left (a+b x^2\right )^5 \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )\right )}{245760 a^{3/4} b^{21/4} \sqrt {x} \left (a+b x^2\right )^5} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 241, normalized size = 0.63
method | result | size |
derivativedivides | \(2 d^{11} \left (\frac {-\frac {663 a^{4} d^{8} \sqrt {d x}}{8192 b^{5}}-\frac {1989 a^{3} d^{6} \left (d x \right )^{\frac {5}{2}}}{5120 b^{4}}-\frac {9061 a^{2} d^{4} \left (d x \right )^{\frac {9}{2}}}{12288 b^{3}}-\frac {527 a \,d^{2} \left (d x \right )^{\frac {13}{2}}}{768 b^{2}}-\frac {7529 \left (d x \right )^{\frac {17}{2}}}{24576 b}}{\left (d^{2} x^{2} b +a \,d^{2}\right )^{5}}+\frac {663 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{65536 b^{5} a \,d^{2}}\right )\) | \(241\) |
default | \(2 d^{11} \left (\frac {-\frac {663 a^{4} d^{8} \sqrt {d x}}{8192 b^{5}}-\frac {1989 a^{3} d^{6} \left (d x \right )^{\frac {5}{2}}}{5120 b^{4}}-\frac {9061 a^{2} d^{4} \left (d x \right )^{\frac {9}{2}}}{12288 b^{3}}-\frac {527 a \,d^{2} \left (d x \right )^{\frac {13}{2}}}{768 b^{2}}-\frac {7529 \left (d x \right )^{\frac {17}{2}}}{24576 b}}{\left (d^{2} x^{2} b +a \,d^{2}\right )^{5}}+\frac {663 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{65536 b^{5} a \,d^{2}}\right )\) | \(241\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 386, normalized size = 1.00 \begin {gather*} -\frac {\frac {8 \, {\left (37645 \, \left (d x\right )^{\frac {17}{2}} b^{4} d^{12} + 84320 \, \left (d x\right )^{\frac {13}{2}} a b^{3} d^{14} + 90610 \, \left (d x\right )^{\frac {9}{2}} a^{2} b^{2} d^{16} + 47736 \, \left (d x\right )^{\frac {5}{2}} a^{3} b d^{18} + 9945 \, \sqrt {d x} a^{4} d^{20}\right )}}{b^{10} d^{10} x^{10} + 5 \, a b^{9} d^{10} x^{8} + 10 \, a^{2} b^{8} d^{10} x^{6} + 10 \, a^{3} b^{7} d^{10} x^{4} + 5 \, a^{4} b^{6} d^{10} x^{2} + a^{5} b^{5} d^{10}} - \frac {9945 \, {\left (\frac {\sqrt {2} d^{12} \log \left (\sqrt {b} d x + \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {3}{4}} b^{\frac {1}{4}}} - \frac {\sqrt {2} d^{12} \log \left (\sqrt {b} d x - \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {3}{4}} b^{\frac {1}{4}}} + \frac {2 \, \sqrt {2} d^{11} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {a}} + \frac {2 \, \sqrt {2} d^{11} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {a}}\right )}}{b^{5}}}{491520 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 489, normalized size = 1.27 \begin {gather*} \frac {39780 \, {\left (b^{10} x^{10} + 5 \, a b^{9} x^{8} + 10 \, a^{2} b^{8} x^{6} + 10 \, a^{3} b^{7} x^{4} + 5 \, a^{4} b^{6} x^{2} + a^{5} b^{5}\right )} \left (-\frac {d^{38}}{a^{3} b^{21}}\right )^{\frac {1}{4}} \arctan \left (-\frac {\left (-\frac {d^{38}}{a^{3} b^{21}}\right )^{\frac {3}{4}} \sqrt {d x} a^{2} b^{16} d^{9} - \sqrt {d^{19} x + \sqrt {-\frac {d^{38}}{a^{3} b^{21}}} a^{2} b^{10}} \left (-\frac {d^{38}}{a^{3} b^{21}}\right )^{\frac {3}{4}} a^{2} b^{16}}{d^{38}}\right ) + 9945 \, {\left (b^{10} x^{10} + 5 \, a b^{9} x^{8} + 10 \, a^{2} b^{8} x^{6} + 10 \, a^{3} b^{7} x^{4} + 5 \, a^{4} b^{6} x^{2} + a^{5} b^{5}\right )} \left (-\frac {d^{38}}{a^{3} b^{21}}\right )^{\frac {1}{4}} \log \left (663 \, \sqrt {d x} d^{9} + 663 \, \left (-\frac {d^{38}}{a^{3} b^{21}}\right )^{\frac {1}{4}} a b^{5}\right ) - 9945 \, {\left (b^{10} x^{10} + 5 \, a b^{9} x^{8} + 10 \, a^{2} b^{8} x^{6} + 10 \, a^{3} b^{7} x^{4} + 5 \, a^{4} b^{6} x^{2} + a^{5} b^{5}\right )} \left (-\frac {d^{38}}{a^{3} b^{21}}\right )^{\frac {1}{4}} \log \left (663 \, \sqrt {d x} d^{9} - 663 \, \left (-\frac {d^{38}}{a^{3} b^{21}}\right )^{\frac {1}{4}} a b^{5}\right ) - 4 \, {\left (37645 \, b^{4} d^{9} x^{8} + 84320 \, a b^{3} d^{9} x^{6} + 90610 \, a^{2} b^{2} d^{9} x^{4} + 47736 \, a^{3} b d^{9} x^{2} + 9945 \, a^{4} d^{9}\right )} \sqrt {d x}}{245760 \, {\left (b^{10} x^{10} + 5 \, a b^{9} x^{8} + 10 \, a^{2} b^{8} x^{6} + 10 \, a^{3} b^{7} x^{4} + 5 \, a^{4} b^{6} x^{2} + a^{5} b^{5}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [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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Giac [A]
time = 6.36, size = 339, normalized size = 0.88 \begin {gather*} \frac {1}{491520} \, d^{9} {\left (\frac {19890 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a b^{6}} + \frac {19890 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a b^{6}} + \frac {9945 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \log \left (d x + \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{a b^{6}} - \frac {9945 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \log \left (d x - \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{a b^{6}} - \frac {8 \, {\left (37645 \, \sqrt {d x} b^{4} d^{10} x^{8} + 84320 \, \sqrt {d x} a b^{3} d^{10} x^{6} + 90610 \, \sqrt {d x} a^{2} b^{2} d^{10} x^{4} + 47736 \, \sqrt {d x} a^{3} b d^{10} x^{2} + 9945 \, \sqrt {d x} a^{4} d^{10}\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{5} b^{5}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.27, size = 213, normalized size = 0.55 \begin {gather*} -\frac {\frac {7529\,d^{11}\,{\left (d\,x\right )}^{17/2}}{12288\,b}+\frac {9061\,a^2\,d^{15}\,{\left (d\,x\right )}^{9/2}}{6144\,b^3}+\frac {1989\,a^3\,d^{17}\,{\left (d\,x\right )}^{5/2}}{2560\,b^4}+\frac {663\,a^4\,d^{19}\,\sqrt {d\,x}}{4096\,b^5}+\frac {527\,a\,d^{13}\,{\left (d\,x\right )}^{13/2}}{384\,b^2}}{a^5\,d^{10}+5\,a^4\,b\,d^{10}\,x^2+10\,a^3\,b^2\,d^{10}\,x^4+10\,a^2\,b^3\,d^{10}\,x^6+5\,a\,b^4\,d^{10}\,x^8+b^5\,d^{10}\,x^{10}}-\frac {663\,d^{19/2}\,\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {d\,x}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )}{8192\,{\left (-a\right )}^{3/4}\,b^{21/4}}-\frac {663\,d^{19/2}\,\mathrm {atanh}\left (\frac {b^{1/4}\,\sqrt {d\,x}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )}{8192\,{\left (-a\right )}^{3/4}\,b^{21/4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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