Optimal. Leaf size=600 \[ -\frac {19 d^3 (d x)^{15/2}}{96 b^2 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{19/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {7315 d^9 (d x)^{3/2} \left (a+b x^2\right )}{3072 b^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {1045 d^7 (d x)^{7/2}}{1024 b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {95 d^5 (d x)^{11/2}}{256 b^3 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {7315 a^{3/4} d^{21/2} \left (a+b x^2\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{4096 \sqrt {2} b^{23/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {7315 a^{3/4} d^{21/2} \left (a+b x^2\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{4096 \sqrt {2} b^{23/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {7315 a^{3/4} d^{21/2} \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{2048 \sqrt {2} b^{23/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {7315 a^{3/4} d^{21/2} \left (a+b x^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{2048 \sqrt {2} b^{23/4} \sqrt {a^2+2 a b x^2+b^2 x^4}} \]
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Rubi [A] time = 0.47, antiderivative size = 600, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 10, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1112, 288, 321, 329, 297, 1162, 617, 204, 1165, 628} \[ \frac {7315 d^9 (d x)^{3/2} \left (a+b x^2\right )}{3072 b^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {1045 d^7 (d x)^{7/2}}{1024 b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {95 d^5 (d x)^{11/2}}{256 b^3 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {19 d^3 (d x)^{15/2}}{96 b^2 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {7315 a^{3/4} d^{21/2} \left (a+b x^2\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{4096 \sqrt {2} b^{23/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {7315 a^{3/4} d^{21/2} \left (a+b x^2\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{4096 \sqrt {2} b^{23/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {7315 a^{3/4} d^{21/2} \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{2048 \sqrt {2} b^{23/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {7315 a^{3/4} d^{21/2} \left (a+b x^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{2048 \sqrt {2} b^{23/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{19/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}} \]
Antiderivative was successfully verified.
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Rule 204
Rule 288
Rule 297
Rule 321
Rule 329
Rule 617
Rule 628
Rule 1112
Rule 1162
Rule 1165
Rubi steps
\begin {align*} \int \frac {(d x)^{21/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx &=\frac {\left (b^4 \left (a b+b^2 x^2\right )\right ) \int \frac {(d x)^{21/2}}{\left (a b+b^2 x^2\right )^5} \, dx}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac {d (d x)^{19/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (19 b^2 d^2 \left (a b+b^2 x^2\right )\right ) \int \frac {(d x)^{17/2}}{\left (a b+b^2 x^2\right )^4} \, dx}{16 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac {d (d x)^{19/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {19 d^3 (d x)^{15/2}}{96 b^2 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (95 d^4 \left (a b+b^2 x^2\right )\right ) \int \frac {(d x)^{13/2}}{\left (a b+b^2 x^2\right )^3} \, dx}{64 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac {d (d x)^{19/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {19 d^3 (d x)^{15/2}}{96 b^2 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {95 d^5 (d x)^{11/2}}{256 b^3 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (1045 d^6 \left (a b+b^2 x^2\right )\right ) \int \frac {(d x)^{9/2}}{\left (a b+b^2 x^2\right )^2} \, dx}{512 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac {1045 d^7 (d x)^{7/2}}{1024 b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{19/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {19 d^3 (d x)^{15/2}}{96 b^2 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {95 d^5 (d x)^{11/2}}{256 b^3 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (7315 d^8 \left (a b+b^2 x^2\right )\right ) \int \frac {(d x)^{5/2}}{a b+b^2 x^2} \, dx}{2048 b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac {1045 d^7 (d x)^{7/2}}{1024 b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{19/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {19 d^3 (d x)^{15/2}}{96 b^2 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {95 d^5 (d x)^{11/2}}{256 b^3 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {7315 d^9 (d x)^{3/2} \left (a+b x^2\right )}{3072 b^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (7315 a d^{10} \left (a b+b^2 x^2\right )\right ) \int \frac {\sqrt {d x}}{a b+b^2 x^2} \, dx}{2048 b^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac {1045 d^7 (d x)^{7/2}}{1024 b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{19/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {19 d^3 (d x)^{15/2}}{96 b^2 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {95 d^5 (d x)^{11/2}}{256 b^3 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {7315 d^9 (d x)^{3/2} \left (a+b x^2\right )}{3072 b^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (7315 a d^9 \left (a b+b^2 x^2\right )\right ) \operatorname {Subst}\left (\int \frac {x^2}{a b+\frac {b^2 x^4}{d^2}} \, dx,x,\sqrt {d x}\right )}{1024 b^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac {1045 d^7 (d x)^{7/2}}{1024 b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{19/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {19 d^3 (d x)^{15/2}}{96 b^2 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {95 d^5 (d x)^{11/2}}{256 b^3 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {7315 d^9 (d x)^{3/2} \left (a+b x^2\right )}{3072 b^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (7315 a d^9 \left (a b+b^2 x^2\right )\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 )}{2048 b^{11/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (7315 a d^9 \left (a b+b^2 x^2\right )\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 )}{2048 b^{11/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac {1045 d^7 (d x)^{7/2}}{1024 b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{19/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {19 d^3 (d x)^{15/2}}{96 b^2 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {95 d^5 (d x)^{11/2}}{256 b^3 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {7315 d^9 (d x)^{3/2} \left (a+b x^2\right )}{3072 b^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (7315 a^{3/4} d^{21/2} \left (a b+b^2 x^2\right )\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 )}{4096 \sqrt {2} b^{27/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (7315 a^{3/4} d^{21/2} \left (a b+b^2 x^2\right )\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 )}{4096 \sqrt {2} b^{27/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (7315 a d^{11} \left (a b+b^2 x^2\right )\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 )}{4096 b^7 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (7315 a d^{11} \left (a b+b^2 x^2\right )\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 )}{4096 b^7 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac {1045 d^7 (d x)^{7/2}}{1024 b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{19/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {19 d^3 (d x)^{15/2}}{96 b^2 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {95 d^5 (d x)^{11/2}}{256 b^3 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {7315 d^9 (d x)^{3/2} \left (a+b x^2\right )}{3072 b^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {7315 a^{3/4} d^{21/2} \left (a+b x^2\right ) \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{4096 \sqrt {2} b^{23/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {7315 a^{3/4} d^{21/2} \left (a+b x^2\right ) \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{4096 \sqrt {2} b^{23/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (7315 a^{3/4} d^{21/2} \left (a b+b^2 x^2\right )\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 )}{2048 \sqrt {2} b^{27/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (7315 a^{3/4} d^{21/2} \left (a b+b^2 x^2\right )\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 )}{2048 \sqrt {2} b^{27/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac {1045 d^7 (d x)^{7/2}}{1024 b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{19/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {19 d^3 (d x)^{15/2}}{96 b^2 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {95 d^5 (d x)^{11/2}}{256 b^3 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {7315 d^9 (d x)^{3/2} \left (a+b x^2\right )}{3072 b^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {7315 a^{3/4} d^{21/2} \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{2048 \sqrt {2} b^{23/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {7315 a^{3/4} d^{21/2} \left (a+b x^2\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{2048 \sqrt {2} b^{23/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {7315 a^{3/4} d^{21/2} \left (a+b x^2\right ) \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{4096 \sqrt {2} b^{23/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {7315 a^{3/4} d^{21/2} \left (a+b x^2\right ) \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{4096 \sqrt {2} b^{23/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ \end {align*}
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Mathematica [C] time = 0.05, size = 110, normalized size = 0.18 \[ -\frac {2 d^9 (d x)^{3/2} \left (-1463 a^4-2717 a^3 b x^2-2223 a^2 b^2 x^4-741 a b^3 x^6+1463 \left (a+b x^2\right )^4 \, _2F_1\left (\frac {3}{4},5;\frac {7}{4};-\frac {b x^2}{a}\right )-39 b^4 x^8\right )}{117 b^5 \left (a+b x^2\right )^3 \sqrt {\left (a+b x^2\right )^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.72, size = 457, normalized size = 0.76 \[ \frac {87780 \, \left (-\frac {a^{3} d^{42}}{b^{23}}\right )^{\frac {1}{4}} {\left (b^{9} x^{8} + 4 \, a b^{8} x^{6} + 6 \, a^{2} b^{7} x^{4} + 4 \, a^{3} b^{6} x^{2} + a^{4} b^{5}\right )} \arctan \left (-\frac {\left (-\frac {a^{3} d^{42}}{b^{23}}\right )^{\frac {1}{4}} \sqrt {d x} a^{2} b^{6} d^{31} - \sqrt {a^{4} d^{63} x - \sqrt {-\frac {a^{3} d^{42}}{b^{23}}} a^{3} b^{11} d^{42}} \left (-\frac {a^{3} d^{42}}{b^{23}}\right )^{\frac {1}{4}} b^{6}}{a^{3} d^{42}}\right ) - 21945 \, \left (-\frac {a^{3} d^{42}}{b^{23}}\right )^{\frac {1}{4}} {\left (b^{9} x^{8} + 4 \, a b^{8} x^{6} + 6 \, a^{2} b^{7} x^{4} + 4 \, a^{3} b^{6} x^{2} + a^{4} b^{5}\right )} \log \left (391419980875 \, \sqrt {d x} a^{2} d^{31} + 391419980875 \, \left (-\frac {a^{3} d^{42}}{b^{23}}\right )^{\frac {3}{4}} b^{17}\right ) + 21945 \, \left (-\frac {a^{3} d^{42}}{b^{23}}\right )^{\frac {1}{4}} {\left (b^{9} x^{8} + 4 \, a b^{8} x^{6} + 6 \, a^{2} b^{7} x^{4} + 4 \, a^{3} b^{6} x^{2} + a^{4} b^{5}\right )} \log \left (391419980875 \, \sqrt {d x} a^{2} d^{31} - 391419980875 \, \left (-\frac {a^{3} d^{42}}{b^{23}}\right )^{\frac {3}{4}} b^{17}\right ) + 4 \, {\left (2048 \, b^{4} d^{10} x^{9} + 16967 \, a b^{3} d^{10} x^{7} + 33345 \, a^{2} b^{2} d^{10} x^{5} + 26125 \, a^{3} b d^{10} x^{3} + 7315 \, a^{4} d^{10} x\right )} \sqrt {d x}}{12288 \, {\left (b^{9} x^{8} + 4 \, a b^{8} x^{6} + 6 \, a^{2} b^{7} x^{4} + 4 \, a^{3} b^{6} x^{2} + a^{4} b^{5}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.42, size = 437, normalized size = 0.73 \[ \frac {1}{24576} \, d^{10} {\left (\frac {16384 \, \sqrt {d x} x}{b^{5} \mathrm {sgn}\left (b d^{4} x^{2} + a d^{4}\right )} - \frac {43890 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{b^{8} d \mathrm {sgn}\left (b d^{4} x^{2} + a d^{4}\right )} - \frac {43890 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{b^{8} d \mathrm {sgn}\left (b d^{4} x^{2} + a d^{4}\right )} + \frac {21945 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \log \left (d x + \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{b^{8} d \mathrm {sgn}\left (b d^{4} x^{2} + a d^{4}\right )} - \frac {21945 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \log \left (d x - \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{b^{8} d \mathrm {sgn}\left (b d^{4} x^{2} + a d^{4}\right )} + \frac {8 \, {\left (8775 \, \sqrt {d x} a b^{3} d^{8} x^{7} + 21057 \, \sqrt {d x} a^{2} b^{2} d^{8} x^{5} + 17933 \, \sqrt {d x} a^{3} b d^{8} x^{3} + 5267 \, \sqrt {d x} a^{4} d^{8} x\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{4} b^{5} \mathrm {sgn}\left (b d^{4} x^{2} + a d^{4}\right )}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.03, size = 1171, normalized size = 1.95 \[ \text {result too large to display} \]
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 \[ -4 \, a d^{\frac {21}{2}} \int \frac {\sqrt {x}}{b^{6} x^{2} + a b^{5}}\,{d x} + d^{\frac {21}{2}} \int \frac {x^{\frac {5}{2}}}{b^{5} x^{2} + a b^{4}}\,{d x} + \frac {2925 \, a d^{\frac {21}{2}} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} - \frac {\sqrt {2} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}}\right )}}{8192 \, b^{5}} + \frac {8775 \, a b^{3} d^{\frac {21}{2}} x^{\frac {15}{2}} + 29649 \, a^{2} b^{2} d^{\frac {21}{2}} x^{\frac {11}{2}} + 34285 \, a^{3} b d^{\frac {21}{2}} x^{\frac {7}{2}} + 13795 \, a^{4} d^{\frac {21}{2}} x^{\frac {3}{2}}}{3072 \, {\left (b^{9} x^{8} + 4 \, a b^{8} x^{6} + 6 \, a^{2} b^{7} x^{4} + 4 \, a^{3} b^{6} x^{2} + a^{4} b^{5}\right )}} - \frac {{\left (537 \, a^{2} b^{4} d^{\frac {21}{2}} x^{5} + 1210 \, a^{3} b^{3} d^{\frac {21}{2}} x^{3} + 705 \, a^{4} b^{2} d^{\frac {21}{2}} x\right )} x^{\frac {9}{2}} + 2 \, {\left (443 \, a^{3} b^{3} d^{\frac {21}{2}} x^{5} + 1014 \, a^{4} b^{2} d^{\frac {21}{2}} x^{3} + 603 \, a^{5} b d^{\frac {21}{2}} x\right )} x^{\frac {5}{2}} + {\left (381 \, a^{4} b^{2} d^{\frac {21}{2}} x^{5} + 882 \, a^{5} b d^{\frac {21}{2}} x^{3} + 533 \, a^{6} d^{\frac {21}{2}} x\right )} \sqrt {x}}{192 \, {\left (a^{3} b^{8} x^{6} + 3 \, a^{4} b^{7} x^{4} + 3 \, a^{5} b^{6} x^{2} + a^{6} b^{5} + {\left (b^{11} x^{6} + 3 \, a b^{10} x^{4} + 3 \, a^{2} b^{9} x^{2} + a^{3} b^{8}\right )} x^{6} + 3 \, {\left (a b^{10} x^{6} + 3 \, a^{2} b^{9} x^{4} + 3 \, a^{3} b^{8} x^{2} + a^{4} b^{7}\right )} x^{4} + 3 \, {\left (a^{2} b^{9} x^{6} + 3 \, a^{3} b^{8} x^{4} + 3 \, a^{4} b^{7} x^{2} + a^{5} b^{6}\right )} x^{2}\right )}} \]
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 \[ \int \frac {{\left (d\,x\right )}^{21/2}}{{\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^{5/2}} \,d x \]
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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