Optimal. Leaf size=402 \[ \frac {13923 \sqrt [4]{a} d^{23/2} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{16384 \sqrt {2} b^{25/4}}-\frac {13923 \sqrt [4]{a} d^{23/2} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{16384 \sqrt {2} b^{25/4}}+\frac {13923 \sqrt [4]{a} d^{23/2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} b^{25/4}}-\frac {13923 \sqrt [4]{a} d^{23/2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{8192 \sqrt {2} b^{25/4}}-\frac {13923 d^9 (d x)^{5/2}}{20480 b^5 \left (a+b x^2\right )}-\frac {1547 d^7 (d x)^{9/2}}{5120 b^4 \left (a+b x^2\right )^2}-\frac {119 d^5 (d x)^{13/2}}{640 b^3 \left (a+b x^2\right )^3}-\frac {21 d^3 (d x)^{17/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {d (d x)^{21/2}}{10 b \left (a+b x^2\right )^5}+\frac {13923 d^{11} \sqrt {d x}}{4096 b^6} \]
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Rubi [A] time = 0.49, antiderivative size = 402, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 10, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {28, 288, 321, 329, 211, 1165, 628, 1162, 617, 204} \[ -\frac {13923 d^9 (d x)^{5/2}}{20480 b^5 \left (a+b x^2\right )}-\frac {1547 d^7 (d x)^{9/2}}{5120 b^4 \left (a+b x^2\right )^2}-\frac {119 d^5 (d x)^{13/2}}{640 b^3 \left (a+b x^2\right )^3}-\frac {21 d^3 (d x)^{17/2}}{160 b^2 \left (a+b x^2\right )^4}+\frac {13923 \sqrt [4]{a} d^{23/2} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{16384 \sqrt {2} b^{25/4}}-\frac {13923 \sqrt [4]{a} d^{23/2} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{16384 \sqrt {2} b^{25/4}}+\frac {13923 \sqrt [4]{a} d^{23/2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} b^{25/4}}-\frac {13923 \sqrt [4]{a} d^{23/2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{8192 \sqrt {2} b^{25/4}}-\frac {d (d x)^{21/2}}{10 b \left (a+b x^2\right )^5}+\frac {13923 d^{11} \sqrt {d x}}{4096 b^6} \]
Antiderivative was successfully verified.
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Rule 28
Rule 204
Rule 211
Rule 288
Rule 321
Rule 329
Rule 617
Rule 628
Rule 1162
Rule 1165
Rubi steps
\begin {align*} \int \frac {(d x)^{23/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx &=b^6 \int \frac {(d x)^{23/2}}{\left (a b+b^2 x^2\right )^6} \, dx\\ &=-\frac {d (d x)^{21/2}}{10 b \left (a+b x^2\right )^5}+\frac {1}{20} \left (21 b^4 d^2\right ) \int \frac {(d x)^{19/2}}{\left (a b+b^2 x^2\right )^5} \, dx\\ &=-\frac {d (d x)^{21/2}}{10 b \left (a+b x^2\right )^5}-\frac {21 d^3 (d x)^{17/2}}{160 b^2 \left (a+b x^2\right )^4}+\frac {1}{320} \left (357 b^2 d^4\right ) \int \frac {(d x)^{15/2}}{\left (a b+b^2 x^2\right )^4} \, dx\\ &=-\frac {d (d x)^{21/2}}{10 b \left (a+b x^2\right )^5}-\frac {21 d^3 (d x)^{17/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {119 d^5 (d x)^{13/2}}{640 b^3 \left (a+b x^2\right )^3}+\frac {\left (1547 d^6\right ) \int \frac {(d x)^{11/2}}{\left (a b+b^2 x^2\right )^3} \, dx}{1280}\\ &=-\frac {d (d x)^{21/2}}{10 b \left (a+b x^2\right )^5}-\frac {21 d^3 (d x)^{17/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {119 d^5 (d x)^{13/2}}{640 b^3 \left (a+b x^2\right )^3}-\frac {1547 d^7 (d x)^{9/2}}{5120 b^4 \left (a+b x^2\right )^2}+\frac {\left (13923 d^8\right ) \int \frac {(d x)^{7/2}}{\left (a b+b^2 x^2\right )^2} \, dx}{10240 b^2}\\ &=-\frac {d (d x)^{21/2}}{10 b \left (a+b x^2\right )^5}-\frac {21 d^3 (d x)^{17/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {119 d^5 (d x)^{13/2}}{640 b^3 \left (a+b x^2\right )^3}-\frac {1547 d^7 (d x)^{9/2}}{5120 b^4 \left (a+b x^2\right )^2}-\frac {13923 d^9 (d x)^{5/2}}{20480 b^5 \left (a+b x^2\right )}+\frac {\left (13923 d^{10}\right ) \int \frac {(d x)^{3/2}}{a b+b^2 x^2} \, dx}{8192 b^4}\\ &=\frac {13923 d^{11} \sqrt {d x}}{4096 b^6}-\frac {d (d x)^{21/2}}{10 b \left (a+b x^2\right )^5}-\frac {21 d^3 (d x)^{17/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {119 d^5 (d x)^{13/2}}{640 b^3 \left (a+b x^2\right )^3}-\frac {1547 d^7 (d x)^{9/2}}{5120 b^4 \left (a+b x^2\right )^2}-\frac {13923 d^9 (d x)^{5/2}}{20480 b^5 \left (a+b x^2\right )}-\frac {\left (13923 a d^{12}\right ) \int \frac {1}{\sqrt {d x} \left (a b+b^2 x^2\right )} \, dx}{8192 b^5}\\ &=\frac {13923 d^{11} \sqrt {d x}}{4096 b^6}-\frac {d (d x)^{21/2}}{10 b \left (a+b x^2\right )^5}-\frac {21 d^3 (d x)^{17/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {119 d^5 (d x)^{13/2}}{640 b^3 \left (a+b x^2\right )^3}-\frac {1547 d^7 (d x)^{9/2}}{5120 b^4 \left (a+b x^2\right )^2}-\frac {13923 d^9 (d x)^{5/2}}{20480 b^5 \left (a+b x^2\right )}-\frac {\left (13923 a d^{11}\right ) \operatorname {Subst}\left (\int \frac {1}{a b+\frac {b^2 x^4}{d^2}} \, dx,x,\sqrt {d x}\right )}{4096 b^5}\\ &=\frac {13923 d^{11} \sqrt {d x}}{4096 b^6}-\frac {d (d x)^{21/2}}{10 b \left (a+b x^2\right )^5}-\frac {21 d^3 (d x)^{17/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {119 d^5 (d x)^{13/2}}{640 b^3 \left (a+b x^2\right )^3}-\frac {1547 d^7 (d x)^{9/2}}{5120 b^4 \left (a+b x^2\right )^2}-\frac {13923 d^9 (d x)^{5/2}}{20480 b^5 \left (a+b x^2\right )}-\frac {\left (13923 \sqrt {a} d^{10}\right ) \operatorname {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 b^5}-\frac {\left (13923 \sqrt {a} d^{10}\right ) \operatorname {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 b^5}\\ &=\frac {13923 d^{11} \sqrt {d x}}{4096 b^6}-\frac {d (d x)^{21/2}}{10 b \left (a+b x^2\right )^5}-\frac {21 d^3 (d x)^{17/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {119 d^5 (d x)^{13/2}}{640 b^3 \left (a+b x^2\right )^3}-\frac {1547 d^7 (d x)^{9/2}}{5120 b^4 \left (a+b x^2\right )^2}-\frac {13923 d^9 (d x)^{5/2}}{20480 b^5 \left (a+b x^2\right )}+\frac {\left (13923 \sqrt [4]{a} d^{23/2}\right ) \operatorname {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} b^{25/4}}+\frac {\left (13923 \sqrt [4]{a} d^{23/2}\right ) \operatorname {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} b^{25/4}}-\frac {\left (13923 \sqrt {a} d^{12}\right ) \operatorname {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 b^{13/2}}-\frac {\left (13923 \sqrt {a} d^{12}\right ) \operatorname {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 b^{13/2}}\\ &=\frac {13923 d^{11} \sqrt {d x}}{4096 b^6}-\frac {d (d x)^{21/2}}{10 b \left (a+b x^2\right )^5}-\frac {21 d^3 (d x)^{17/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {119 d^5 (d x)^{13/2}}{640 b^3 \left (a+b x^2\right )^3}-\frac {1547 d^7 (d x)^{9/2}}{5120 b^4 \left (a+b x^2\right )^2}-\frac {13923 d^9 (d x)^{5/2}}{20480 b^5 \left (a+b x^2\right )}+\frac {13923 \sqrt [4]{a} d^{23/2} \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} b^{25/4}}-\frac {13923 \sqrt [4]{a} d^{23/2} \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} b^{25/4}}-\frac {\left (13923 \sqrt [4]{a} d^{23/2}\right ) \operatorname {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} b^{25/4}}+\frac {\left (13923 \sqrt [4]{a} d^{23/2}\right ) \operatorname {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} b^{25/4}}\\ &=\frac {13923 d^{11} \sqrt {d x}}{4096 b^6}-\frac {d (d x)^{21/2}}{10 b \left (a+b x^2\right )^5}-\frac {21 d^3 (d x)^{17/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {119 d^5 (d x)^{13/2}}{640 b^3 \left (a+b x^2\right )^3}-\frac {1547 d^7 (d x)^{9/2}}{5120 b^4 \left (a+b x^2\right )^2}-\frac {13923 d^9 (d x)^{5/2}}{20480 b^5 \left (a+b x^2\right )}+\frac {13923 \sqrt [4]{a} d^{23/2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} b^{25/4}}-\frac {13923 \sqrt [4]{a} d^{23/2} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} b^{25/4}}+\frac {13923 \sqrt [4]{a} d^{23/2} \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} b^{25/4}}-\frac {13923 \sqrt [4]{a} d^{23/2} \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} b^{25/4}}\\ \end {align*}
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Mathematica [A] time = 0.30, size = 408, normalized size = 1.01 \[ \frac {d^{11} \sqrt {d x} \left (\frac {10862592 a^5 \sqrt [4]{b} \sqrt {x}+43450368 a^4 b^{5/4} x^{5/2}-678912 a^4 \sqrt [4]{b} \sqrt {x} \left (a+b x^2\right )+72417280 a^3 b^{9/4} x^{9/2}-848640 a^3 \sqrt [4]{b} \sqrt {x} \left (a+b x^2\right )^2+61276160 a^2 b^{13/4} x^{13/2}-1166880 a^2 \sqrt [4]{b} \sqrt {x} \left (a+b x^2\right )^3+25231360 a b^{17/4} x^{17/2}-2042040 a \sqrt [4]{b} \sqrt {x} \left (a+b x^2\right )^4+765765 \sqrt {2} \sqrt [4]{a} \left (a+b x^2\right )^5 \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )-765765 \sqrt {2} \sqrt [4]{a} \left (a+b x^2\right )^5 \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )-1531530 \sqrt {2} \sqrt [4]{a} \left (a+b x^2\right )^5 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )+3604480 b^{21/4} x^{21/2}}{\left (a+b x^2\right )^5}+1531530 \sqrt {2} \sqrt [4]{a} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )\right )}{1802240 b^{25/4} \sqrt {x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.80, size = 479, normalized size = 1.19 \[ -\frac {278460 \, \left (-\frac {a d^{46}}{b^{25}}\right )^{\frac {1}{4}} {\left (b^{11} x^{10} + 5 \, a b^{10} x^{8} + 10 \, a^{2} b^{9} x^{6} + 10 \, a^{3} b^{8} x^{4} + 5 \, a^{4} b^{7} x^{2} + a^{5} b^{6}\right )} \arctan \left (-\frac {\left (-\frac {a d^{46}}{b^{25}}\right )^{\frac {3}{4}} \sqrt {d x} b^{19} d^{11} - \sqrt {d^{23} x + \sqrt {-\frac {a d^{46}}{b^{25}}} b^{12}} \left (-\frac {a d^{46}}{b^{25}}\right )^{\frac {3}{4}} b^{19}}{a d^{46}}\right ) + 69615 \, \left (-\frac {a d^{46}}{b^{25}}\right )^{\frac {1}{4}} {\left (b^{11} x^{10} + 5 \, a b^{10} x^{8} + 10 \, a^{2} b^{9} x^{6} + 10 \, a^{3} b^{8} x^{4} + 5 \, a^{4} b^{7} x^{2} + a^{5} b^{6}\right )} \log \left (13923 \, \sqrt {d x} d^{11} + 13923 \, \left (-\frac {a d^{46}}{b^{25}}\right )^{\frac {1}{4}} b^{6}\right ) - 69615 \, \left (-\frac {a d^{46}}{b^{25}}\right )^{\frac {1}{4}} {\left (b^{11} x^{10} + 5 \, a b^{10} x^{8} + 10 \, a^{2} b^{9} x^{6} + 10 \, a^{3} b^{8} x^{4} + 5 \, a^{4} b^{7} x^{2} + a^{5} b^{6}\right )} \log \left (13923 \, \sqrt {d x} d^{11} - 13923 \, \left (-\frac {a d^{46}}{b^{25}}\right )^{\frac {1}{4}} b^{6}\right ) - 4 \, {\left (40960 \, b^{5} d^{11} x^{10} + 263515 \, a b^{4} d^{11} x^{8} + 590240 \, a^{2} b^{3} d^{11} x^{6} + 634270 \, a^{3} b^{2} d^{11} x^{4} + 334152 \, a^{4} b d^{11} x^{2} + 69615 \, a^{5} d^{11}\right )} \sqrt {d x}}{81920 \, {\left (b^{11} x^{10} + 5 \, a b^{10} x^{8} + 10 \, a^{2} b^{9} x^{6} + 10 \, a^{3} b^{8} x^{4} + 5 \, a^{4} b^{7} x^{2} + a^{5} b^{6}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.25, size = 340, normalized size = 0.85 \[ -\frac {1}{163840} \, d^{11} {\left (\frac {139230 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{b^{7}} + \frac {139230 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{b^{7}} + \frac {69615 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \log \left (d x + \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{b^{7}} - \frac {69615 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \log \left (d x - \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{b^{7}} - \frac {327680 \, \sqrt {d x}}{b^{6}} - \frac {8 \, {\left (58715 \, \sqrt {d x} a b^{4} d^{10} x^{8} + 180640 \, \sqrt {d x} a^{2} b^{3} d^{10} x^{6} + 224670 \, \sqrt {d x} a^{3} b^{2} d^{10} x^{4} + 129352 \, \sqrt {d x} a^{4} b d^{10} x^{2} + 28655 \, \sqrt {d x} a^{5} d^{10}\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{5} b^{6}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 351, normalized size = 0.87 \[ \frac {5731 \sqrt {d x}\, a^{5} d^{21}}{4096 \left (b \,d^{2} x^{2}+d^{2} a \right )^{5} b^{6}}+\frac {16169 \left (d x \right )^{\frac {5}{2}} a^{4} d^{19}}{2560 \left (b \,d^{2} x^{2}+d^{2} a \right )^{5} b^{5}}+\frac {22467 \left (d x \right )^{\frac {9}{2}} a^{3} d^{17}}{2048 \left (b \,d^{2} x^{2}+d^{2} a \right )^{5} b^{4}}+\frac {1129 \left (d x \right )^{\frac {13}{2}} a^{2} d^{15}}{128 \left (b \,d^{2} x^{2}+d^{2} a \right )^{5} b^{3}}+\frac {11743 \left (d x \right )^{\frac {17}{2}} a \,d^{13}}{4096 \left (b \,d^{2} x^{2}+d^{2} a \right )^{5} b^{2}}-\frac {13923 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, d^{11} \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )}{16384 b^{6}}-\frac {13923 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, d^{11} \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )}{16384 b^{6}}-\frac {13923 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, d^{11} \ln \left (\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )}{32768 b^{6}}+\frac {2 \sqrt {d x}\, d^{11}}{b^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.23, size = 403, normalized size = 1.00 \[ \frac {\frac {327680 \, \sqrt {d x} d^{12}}{b^{6}} + \frac {8 \, {\left (58715 \, \left (d x\right )^{\frac {17}{2}} a b^{4} d^{14} + 180640 \, \left (d x\right )^{\frac {13}{2}} a^{2} b^{3} d^{16} + 224670 \, \left (d x\right )^{\frac {9}{2}} a^{3} b^{2} d^{18} + 129352 \, \left (d x\right )^{\frac {5}{2}} a^{4} b d^{20} + 28655 \, \sqrt {d x} a^{5} d^{22}\right )}}{b^{11} d^{10} x^{10} + 5 \, a b^{10} d^{10} x^{8} + 10 \, a^{2} b^{9} d^{10} x^{6} + 10 \, a^{3} b^{8} d^{10} x^{4} + 5 \, a^{4} b^{7} d^{10} x^{2} + a^{5} b^{6} d^{10}} - \frac {69615 \, {\left (\frac {\sqrt {2} d^{14} \log \left (\sqrt {b} d x + \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {3}{4}} b^{\frac {1}{4}}} - \frac {\sqrt {2} d^{14} \log \left (\sqrt {b} d x - \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {3}{4}} b^{\frac {1}{4}}} + \frac {2 \, \sqrt {2} d^{13} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {a}} + \frac {2 \, \sqrt {2} d^{13} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {a}}\right )} a}{b^{6}}}{163840 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.36, size = 231, normalized size = 0.57 \[ \frac {\frac {5731\,a^5\,d^{21}\,\sqrt {d\,x}}{4096}+\frac {22467\,a^3\,b^2\,d^{17}\,{\left (d\,x\right )}^{9/2}}{2048}+\frac {1129\,a^2\,b^3\,d^{15}\,{\left (d\,x\right )}^{13/2}}{128}+\frac {16169\,a^4\,b\,d^{19}\,{\left (d\,x\right )}^{5/2}}{2560}+\frac {11743\,a\,b^4\,d^{13}\,{\left (d\,x\right )}^{17/2}}{4096}}{a^5\,b^6\,d^{10}+5\,a^4\,b^7\,d^{10}\,x^2+10\,a^3\,b^8\,d^{10}\,x^4+10\,a^2\,b^9\,d^{10}\,x^6+5\,a\,b^{10}\,d^{10}\,x^8+b^{11}\,d^{10}\,x^{10}}+\frac {2\,d^{11}\,\sqrt {d\,x}}{b^6}-\frac {13923\,{\left (-a\right )}^{1/4}\,d^{23/2}\,\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {d\,x}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )}{8192\,b^{25/4}}+\frac {{\left (-a\right )}^{1/4}\,d^{23/2}\,\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {d\,x}\,1{}\mathrm {i}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )\,13923{}\mathrm {i}}{8192\,b^{25/4}} \]
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 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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