Optimal. Leaf size=368 \[ -\frac {663 a^{5/4} d^{19/2} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{256 \sqrt {2} b^{21/4}}+\frac {663 a^{5/4} d^{19/2} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{256 \sqrt {2} b^{21/4}}-\frac {663 a^{5/4} d^{19/2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{128 \sqrt {2} b^{21/4}}+\frac {663 a^{5/4} d^{19/2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{128 \sqrt {2} b^{21/4}}-\frac {663 a d^9 \sqrt {d x}}{64 b^5}-\frac {221 d^5 (d x)^{9/2}}{192 b^3 \left (a+b x^2\right )}-\frac {17 d^3 (d x)^{13/2}}{48 b^2 \left (a+b x^2\right )^2}-\frac {d (d x)^{17/2}}{6 b \left (a+b x^2\right )^3}+\frac {663 d^7 (d x)^{5/2}}{320 b^4} \]
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Rubi [A] time = 0.42, antiderivative size = 368, normalized size of antiderivative = 1.00, number of steps used = 16, 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} \begin {gather*} -\frac {663 a^{5/4} 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 )}{256 \sqrt {2} b^{21/4}}+\frac {663 a^{5/4} 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 )}{256 \sqrt {2} b^{21/4}}-\frac {663 a^{5/4} d^{19/2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{128 \sqrt {2} b^{21/4}}+\frac {663 a^{5/4} d^{19/2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{128 \sqrt {2} b^{21/4}}-\frac {221 d^5 (d x)^{9/2}}{192 b^3 \left (a+b x^2\right )}-\frac {17 d^3 (d x)^{13/2}}{48 b^2 \left (a+b x^2\right )^2}-\frac {663 a d^9 \sqrt {d x}}{64 b^5}-\frac {d (d x)^{17/2}}{6 b \left (a+b x^2\right )^3}+\frac {663 d^7 (d x)^{5/2}}{320 b^4} \end {gather*}
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)^{19/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx &=b^4 \int \frac {(d x)^{19/2}}{\left (a b+b^2 x^2\right )^4} \, dx\\ &=-\frac {d (d x)^{17/2}}{6 b \left (a+b x^2\right )^3}+\frac {1}{12} \left (17 b^2 d^2\right ) \int \frac {(d x)^{15/2}}{\left (a b+b^2 x^2\right )^3} \, dx\\ &=-\frac {d (d x)^{17/2}}{6 b \left (a+b x^2\right )^3}-\frac {17 d^3 (d x)^{13/2}}{48 b^2 \left (a+b x^2\right )^2}+\frac {1}{96} \left (221 d^4\right ) \int \frac {(d x)^{11/2}}{\left (a b+b^2 x^2\right )^2} \, dx\\ &=-\frac {d (d x)^{17/2}}{6 b \left (a+b x^2\right )^3}-\frac {17 d^3 (d x)^{13/2}}{48 b^2 \left (a+b x^2\right )^2}-\frac {221 d^5 (d x)^{9/2}}{192 b^3 \left (a+b x^2\right )}+\frac {\left (663 d^6\right ) \int \frac {(d x)^{7/2}}{a b+b^2 x^2} \, dx}{128 b^2}\\ &=\frac {663 d^7 (d x)^{5/2}}{320 b^4}-\frac {d (d x)^{17/2}}{6 b \left (a+b x^2\right )^3}-\frac {17 d^3 (d x)^{13/2}}{48 b^2 \left (a+b x^2\right )^2}-\frac {221 d^5 (d x)^{9/2}}{192 b^3 \left (a+b x^2\right )}-\frac {\left (663 a d^8\right ) \int \frac {(d x)^{3/2}}{a b+b^2 x^2} \, dx}{128 b^3}\\ &=-\frac {663 a d^9 \sqrt {d x}}{64 b^5}+\frac {663 d^7 (d x)^{5/2}}{320 b^4}-\frac {d (d x)^{17/2}}{6 b \left (a+b x^2\right )^3}-\frac {17 d^3 (d x)^{13/2}}{48 b^2 \left (a+b x^2\right )^2}-\frac {221 d^5 (d x)^{9/2}}{192 b^3 \left (a+b x^2\right )}+\frac {\left (663 a^2 d^{10}\right ) \int \frac {1}{\sqrt {d x} \left (a b+b^2 x^2\right )} \, dx}{128 b^4}\\ &=-\frac {663 a d^9 \sqrt {d x}}{64 b^5}+\frac {663 d^7 (d x)^{5/2}}{320 b^4}-\frac {d (d x)^{17/2}}{6 b \left (a+b x^2\right )^3}-\frac {17 d^3 (d x)^{13/2}}{48 b^2 \left (a+b x^2\right )^2}-\frac {221 d^5 (d x)^{9/2}}{192 b^3 \left (a+b x^2\right )}+\frac {\left (663 a^2 d^9\right ) \operatorname {Subst}\left (\int \frac {1}{a b+\frac {b^2 x^4}{d^2}} \, dx,x,\sqrt {d x}\right )}{64 b^4}\\ &=-\frac {663 a d^9 \sqrt {d x}}{64 b^5}+\frac {663 d^7 (d x)^{5/2}}{320 b^4}-\frac {d (d x)^{17/2}}{6 b \left (a+b x^2\right )^3}-\frac {17 d^3 (d x)^{13/2}}{48 b^2 \left (a+b x^2\right )^2}-\frac {221 d^5 (d x)^{9/2}}{192 b^3 \left (a+b x^2\right )}+\frac {\left (663 a^{3/2} d^8\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 )}{128 b^4}+\frac {\left (663 a^{3/2} d^8\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 )}{128 b^4}\\ &=-\frac {663 a d^9 \sqrt {d x}}{64 b^5}+\frac {663 d^7 (d x)^{5/2}}{320 b^4}-\frac {d (d x)^{17/2}}{6 b \left (a+b x^2\right )^3}-\frac {17 d^3 (d x)^{13/2}}{48 b^2 \left (a+b x^2\right )^2}-\frac {221 d^5 (d x)^{9/2}}{192 b^3 \left (a+b x^2\right )}-\frac {\left (663 a^{5/4} d^{19/2}\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 )}{256 \sqrt {2} b^{21/4}}-\frac {\left (663 a^{5/4} d^{19/2}\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 )}{256 \sqrt {2} b^{21/4}}+\frac {\left (663 a^{3/2} d^{10}\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 )}{256 b^{11/2}}+\frac {\left (663 a^{3/2} d^{10}\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 )}{256 b^{11/2}}\\ &=-\frac {663 a d^9 \sqrt {d x}}{64 b^5}+\frac {663 d^7 (d x)^{5/2}}{320 b^4}-\frac {d (d x)^{17/2}}{6 b \left (a+b x^2\right )^3}-\frac {17 d^3 (d x)^{13/2}}{48 b^2 \left (a+b x^2\right )^2}-\frac {221 d^5 (d x)^{9/2}}{192 b^3 \left (a+b x^2\right )}-\frac {663 a^{5/4} 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 )}{256 \sqrt {2} b^{21/4}}+\frac {663 a^{5/4} 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 )}{256 \sqrt {2} b^{21/4}}+\frac {\left (663 a^{5/4} d^{19/2}\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 )}{128 \sqrt {2} b^{21/4}}-\frac {\left (663 a^{5/4} d^{19/2}\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 )}{128 \sqrt {2} b^{21/4}}\\ &=-\frac {663 a d^9 \sqrt {d x}}{64 b^5}+\frac {663 d^7 (d x)^{5/2}}{320 b^4}-\frac {d (d x)^{17/2}}{6 b \left (a+b x^2\right )^3}-\frac {17 d^3 (d x)^{13/2}}{48 b^2 \left (a+b x^2\right )^2}-\frac {221 d^5 (d x)^{9/2}}{192 b^3 \left (a+b x^2\right )}-\frac {663 a^{5/4} d^{19/2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{128 \sqrt {2} b^{21/4}}+\frac {663 a^{5/4} d^{19/2} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{128 \sqrt {2} b^{21/4}}-\frac {663 a^{5/4} 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 )}{256 \sqrt {2} b^{21/4}}+\frac {663 a^{5/4} 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 )}{256 \sqrt {2} b^{21/4}}\\ \end {align*}
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Mathematica [A] time = 0.23, size = 347, normalized size = 0.94 \begin {gather*} \frac {d^9 \sqrt {d x} \left (\frac {-69615 \sqrt {2} a^{5/4} \left (a+b x^2\right )^3 \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )+69615 \sqrt {2} a^{5/4} \left (a+b x^2\right )^3 \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )+139230 \sqrt {2} a^{5/4} \left (a+b x^2\right )^3 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )-848640 a^4 \sqrt [4]{b} \sqrt {x}-2036736 a^3 b^{5/4} x^{5/2}+106080 a^3 \sqrt [4]{b} \sqrt {x} \left (a+b x^2\right )-1584128 a^2 b^{9/4} x^{9/2}+185640 a^2 \sqrt [4]{b} \sqrt {x} \left (a+b x^2\right )^2-365568 a b^{13/4} x^{13/2}+21504 b^{17/4} x^{17/2}}{\left (a+b x^2\right )^3}-139230 \sqrt {2} a^{5/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )\right )}{53760 b^{21/4} \sqrt {x}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.94, size = 222, normalized size = 0.60 \begin {gather*} -\frac {663 a^{5/4} d^{19/2} \tan ^{-1}\left (\frac {\frac {\sqrt [4]{a} \sqrt {d}}{\sqrt {2} \sqrt [4]{b}}-\frac {\sqrt [4]{b} \sqrt {d} x}{\sqrt {2} \sqrt [4]{a}}}{\sqrt {d x}}\right )}{128 \sqrt {2} b^{21/4}}+\frac {663 a^{5/4} d^{19/2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}}{\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x}\right )}{128 \sqrt {2} b^{21/4}}-\frac {d^9 \sqrt {d x} \left (9945 a^4+27846 a^3 b x^2+24973 a^2 b^2 x^4+6528 a b^3 x^6-384 b^4 x^8\right )}{960 b^5 \left (a+b x^2\right )^3} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.00, size = 399, normalized size = 1.08 \begin {gather*} \frac {39780 \, \left (-\frac {a^{5} d^{38}}{b^{21}}\right )^{\frac {1}{4}} {\left (b^{8} x^{6} + 3 \, a b^{7} x^{4} + 3 \, a^{2} b^{6} x^{2} + a^{3} b^{5}\right )} \arctan \left (-\frac {\left (-\frac {a^{5} d^{38}}{b^{21}}\right )^{\frac {3}{4}} \sqrt {d x} a b^{16} d^{9} - \left (-\frac {a^{5} d^{38}}{b^{21}}\right )^{\frac {3}{4}} \sqrt {a^{2} d^{19} x + \sqrt {-\frac {a^{5} d^{38}}{b^{21}}} b^{10}} b^{16}}{a^{5} d^{38}}\right ) + 9945 \, \left (-\frac {a^{5} d^{38}}{b^{21}}\right )^{\frac {1}{4}} {\left (b^{8} x^{6} + 3 \, a b^{7} x^{4} + 3 \, a^{2} b^{6} x^{2} + a^{3} b^{5}\right )} \log \left (663 \, \sqrt {d x} a d^{9} + 663 \, \left (-\frac {a^{5} d^{38}}{b^{21}}\right )^{\frac {1}{4}} b^{5}\right ) - 9945 \, \left (-\frac {a^{5} d^{38}}{b^{21}}\right )^{\frac {1}{4}} {\left (b^{8} x^{6} + 3 \, a b^{7} x^{4} + 3 \, a^{2} b^{6} x^{2} + a^{3} b^{5}\right )} \log \left (663 \, \sqrt {d x} a d^{9} - 663 \, \left (-\frac {a^{5} d^{38}}{b^{21}}\right )^{\frac {1}{4}} b^{5}\right ) + 4 \, {\left (384 \, b^{4} d^{9} x^{8} - 6528 \, a b^{3} d^{9} x^{6} - 24973 \, a^{2} b^{2} d^{9} x^{4} - 27846 \, a^{3} b d^{9} x^{2} - 9945 \, a^{4} d^{9}\right )} \sqrt {d x}}{3840 \, {\left (b^{8} x^{6} + 3 \, a b^{7} x^{4} + 3 \, a^{2} b^{6} x^{2} + a^{3} b^{5}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 336, normalized size = 0.91 \begin {gather*} \frac {1}{7680} \, d^{9} {\left (\frac {19890 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} a \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^{6}} + \frac {19890 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} a \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^{6}} + \frac {9945 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} a \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^{6}} - \frac {9945 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} a \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^{6}} - \frac {40 \, {\left (617 \, \sqrt {d x} a^{2} b^{2} d^{6} x^{4} + 1038 \, \sqrt {d x} a^{3} b d^{6} x^{2} + 453 \, \sqrt {d x} a^{4} d^{6}\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{3} b^{5}} + \frac {3072 \, {\left (\sqrt {d x} b^{16} d^{10} x^{2} - 20 \, \sqrt {d x} a b^{15} d^{10}\right )}}{b^{20} d^{10}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 306, normalized size = 0.83 \begin {gather*} -\frac {151 \sqrt {d x}\, a^{4} d^{15}}{64 \left (b \,d^{2} x^{2}+d^{2} a \right )^{3} b^{5}}-\frac {173 \left (d x \right )^{\frac {5}{2}} a^{3} d^{13}}{32 \left (b \,d^{2} x^{2}+d^{2} a \right )^{3} b^{4}}-\frac {617 \left (d x \right )^{\frac {9}{2}} a^{2} d^{11}}{192 \left (b \,d^{2} x^{2}+d^{2} a \right )^{3} b^{3}}+\frac {663 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, a \,d^{9} \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )}{256 b^{5}}+\frac {663 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, a \,d^{9} \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )}{256 b^{5}}+\frac {663 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, a \,d^{9} \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 )}{512 b^{5}}-\frac {8 \sqrt {d x}\, a \,d^{9}}{b^{5}}+\frac {2 \left (d x \right )^{\frac {5}{2}} d^{7}}{5 b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.10, size = 361, normalized size = 0.98 \begin {gather*} -\frac {\frac {40 \, {\left (617 \, \left (d x\right )^{\frac {9}{2}} a^{2} b^{2} d^{12} + 1038 \, \left (d x\right )^{\frac {5}{2}} a^{3} b d^{14} + 453 \, \sqrt {d x} a^{4} d^{16}\right )}}{b^{8} d^{6} x^{6} + 3 \, a b^{7} d^{6} x^{4} + 3 \, a^{2} b^{6} d^{6} x^{2} + a^{3} b^{5} d^{6}} - \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 )} a^{2}}{b^{5}} - \frac {3072 \, {\left (\left (d x\right )^{\frac {5}{2}} b d^{8} - 20 \, \sqrt {d x} a d^{10}\right )}}{b^{5}}}{7680 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.13, size = 188, normalized size = 0.51 \begin {gather*} \frac {2\,d^7\,{\left (d\,x\right )}^{5/2}}{5\,b^4}-\frac {\frac {151\,a^4\,d^{15}\,\sqrt {d\,x}}{64}+\frac {617\,a^2\,b^2\,d^{11}\,{\left (d\,x\right )}^{9/2}}{192}+\frac {173\,a^3\,b\,d^{13}\,{\left (d\,x\right )}^{5/2}}{32}}{a^3\,b^5\,d^6+3\,a^2\,b^6\,d^6\,x^2+3\,a\,b^7\,d^6\,x^4+b^8\,d^6\,x^6}-\frac {663\,{\left (-a\right )}^{5/4}\,d^{19/2}\,\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {d\,x}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )}{128\,b^{21/4}}-\frac {8\,a\,d^9\,\sqrt {d\,x}}{b^5}+\frac {{\left (-a\right )}^{5/4}\,d^{19/2}\,\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {d\,x}\,1{}\mathrm {i}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )\,663{}\mathrm {i}}{128\,b^{21/4}} \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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