Optimal. Leaf size=600 \[ -\frac {17 d^3 (d x)^{13/2}}{96 b^2 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{17/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {3315 \sqrt [4]{a} d^{19/2} \left (a+b x^2\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{4096 \sqrt {2} b^{21/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {3315 \sqrt [4]{a} d^{19/2} \left (a+b x^2\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{4096 \sqrt {2} b^{21/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {3315 \sqrt [4]{a} d^{19/2} \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{2048 \sqrt {2} b^{21/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {3315 \sqrt [4]{a} d^{19/2} \left (a+b x^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{2048 \sqrt {2} b^{21/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {3315 d^9 \sqrt {d x} \left (a+b x^2\right )}{1024 b^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {663 d^7 (d x)^{5/2}}{1024 b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {221 d^5 (d x)^{9/2}}{768 b^3 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}} \]
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Rubi [A] time = 0.46, 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, 211, 1165, 628, 1162, 617, 204} \[ \frac {3315 d^9 \sqrt {d x} \left (a+b x^2\right )}{1024 b^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {663 d^7 (d x)^{5/2}}{1024 b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {221 d^5 (d x)^{9/2}}{768 b^3 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {17 d^3 (d x)^{13/2}}{96 b^2 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {3315 \sqrt [4]{a} d^{19/2} \left (a+b x^2\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{4096 \sqrt {2} b^{21/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {3315 \sqrt [4]{a} d^{19/2} \left (a+b x^2\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{4096 \sqrt {2} b^{21/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {3315 \sqrt [4]{a} d^{19/2} \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{2048 \sqrt {2} b^{21/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {3315 \sqrt [4]{a} d^{19/2} \left (a+b x^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{2048 \sqrt {2} b^{21/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{17/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}} \]
Antiderivative was successfully verified.
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Rule 204
Rule 211
Rule 288
Rule 321
Rule 329
Rule 617
Rule 628
Rule 1112
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 )^{5/2}} \, dx &=\frac {\left (b^4 \left (a b+b^2 x^2\right )\right ) \int \frac {(d x)^{19/2}}{\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)^{17/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (17 b^2 d^2 \left (a b+b^2 x^2\right )\right ) \int \frac {(d x)^{15/2}}{\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)^{17/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {17 d^3 (d x)^{13/2}}{96 b^2 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (221 d^4 \left (a b+b^2 x^2\right )\right ) \int \frac {(d x)^{11/2}}{\left (a b+b^2 x^2\right )^3} \, dx}{192 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac {d (d x)^{17/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {17 d^3 (d x)^{13/2}}{96 b^2 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {221 d^5 (d x)^{9/2}}{768 b^3 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (663 d^6 \left (a b+b^2 x^2\right )\right ) \int \frac {(d x)^{7/2}}{\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 {663 d^7 (d x)^{5/2}}{1024 b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{17/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {17 d^3 (d x)^{13/2}}{96 b^2 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {221 d^5 (d x)^{9/2}}{768 b^3 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (3315 d^8 \left (a b+b^2 x^2\right )\right ) \int \frac {(d x)^{3/2}}{a b+b^2 x^2} \, dx}{2048 b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac {663 d^7 (d x)^{5/2}}{1024 b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{17/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {17 d^3 (d x)^{13/2}}{96 b^2 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {221 d^5 (d x)^{9/2}}{768 b^3 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {3315 d^9 \sqrt {d x} \left (a+b x^2\right )}{1024 b^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (3315 a d^{10} \left (a b+b^2 x^2\right )\right ) \int \frac {1}{\sqrt {d x} \left (a b+b^2 x^2\right )} \, dx}{2048 b^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac {663 d^7 (d x)^{5/2}}{1024 b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{17/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {17 d^3 (d x)^{13/2}}{96 b^2 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {221 d^5 (d x)^{9/2}}{768 b^3 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {3315 d^9 \sqrt {d x} \left (a+b x^2\right )}{1024 b^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (3315 a d^9 \left (a b+b^2 x^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{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 {663 d^7 (d x)^{5/2}}{1024 b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{17/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {17 d^3 (d x)^{13/2}}{96 b^2 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {221 d^5 (d x)^{9/2}}{768 b^3 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {3315 d^9 \sqrt {d x} \left (a+b x^2\right )}{1024 b^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (3315 \sqrt {a} d^8 \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^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (3315 \sqrt {a} d^8 \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^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac {663 d^7 (d x)^{5/2}}{1024 b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{17/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {17 d^3 (d x)^{13/2}}{96 b^2 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {221 d^5 (d x)^{9/2}}{768 b^3 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {3315 d^9 \sqrt {d x} \left (a+b x^2\right )}{1024 b^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (3315 \sqrt [4]{a} d^{19/2} \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^{25/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (3315 \sqrt [4]{a} d^{19/2} \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^{25/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (3315 \sqrt {a} d^{10} \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^{13/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (3315 \sqrt {a} d^{10} \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^{13/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac {663 d^7 (d x)^{5/2}}{1024 b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{17/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {17 d^3 (d x)^{13/2}}{96 b^2 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {221 d^5 (d x)^{9/2}}{768 b^3 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {3315 d^9 \sqrt {d x} \left (a+b x^2\right )}{1024 b^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {3315 \sqrt [4]{a} d^{19/2} \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^{21/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {3315 \sqrt [4]{a} d^{19/2} \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^{21/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (3315 \sqrt [4]{a} d^{19/2} \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^{25/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (3315 \sqrt [4]{a} d^{19/2} \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^{25/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac {663 d^7 (d x)^{5/2}}{1024 b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{17/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {17 d^3 (d x)^{13/2}}{96 b^2 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {221 d^5 (d x)^{9/2}}{768 b^3 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {3315 d^9 \sqrt {d x} \left (a+b x^2\right )}{1024 b^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {3315 \sqrt [4]{a} d^{19/2} \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^{21/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {3315 \sqrt [4]{a} d^{19/2} \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^{21/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {3315 \sqrt [4]{a} d^{19/2} \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^{21/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {3315 \sqrt [4]{a} d^{19/2} \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^{21/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ \end {align*}
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Mathematica [A] time = 0.28, size = 384, normalized size = 0.64 \[ \frac {(d x)^{19/2} \left (a+b x^2\right ) \left (10183680 a^4 \sqrt [4]{b} \sqrt {x}+32587776 a^3 b^{5/4} x^{5/2}-848640 a^3 \sqrt [4]{b} \sqrt {x} \left (a+b x^2\right )+39829504 a^2 b^{9/4} x^{9/2}-1166880 a^2 \sqrt [4]{b} \sqrt {x} \left (a+b x^2\right )^2+21446656 a b^{13/4} x^{13/2}-2042040 a \sqrt [4]{b} \sqrt {x} \left (a+b x^2\right )^3+765765 \sqrt {2} \sqrt [4]{a} \left (a+b x^2\right )^4 \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )-765765 \sqrt {2} \sqrt [4]{a} \left (a+b x^2\right )^4 \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )+1531530 \sqrt {2} \sqrt [4]{a} \left (a+b x^2\right )^4 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )-1531530 \sqrt {2} \sqrt [4]{a} \left (a+b x^2\right )^4 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )+3784704 b^{17/4} x^{17/2}\right )}{1892352 b^{21/4} x^{19/2} \left (\left (a+b x^2\right )^2\right )^{5/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.84, size = 421, normalized size = 0.70 \[ -\frac {39780 \, \left (-\frac {a d^{38}}{b^{21}}\right )^{\frac {1}{4}} {\left (b^{9} x^{8} + 4 \, a b^{8} x^{6} + 6 \, a^{2} b^{7} x^{4} + 4 \, a^{3} b^{6} x^{2} + a^{4} b^{5}\right )} \arctan \left (-\frac {\left (-\frac {a d^{38}}{b^{21}}\right )^{\frac {3}{4}} \sqrt {d x} b^{16} d^{9} - \sqrt {d^{19} x + \sqrt {-\frac {a d^{38}}{b^{21}}} b^{10}} \left (-\frac {a d^{38}}{b^{21}}\right )^{\frac {3}{4}} b^{16}}{a d^{38}}\right ) + 9945 \, \left (-\frac {a d^{38}}{b^{21}}\right )^{\frac {1}{4}} {\left (b^{9} x^{8} + 4 \, a b^{8} x^{6} + 6 \, a^{2} b^{7} x^{4} + 4 \, a^{3} b^{6} x^{2} + a^{4} b^{5}\right )} \log \left (3315 \, \sqrt {d x} d^{9} + 3315 \, \left (-\frac {a d^{38}}{b^{21}}\right )^{\frac {1}{4}} b^{5}\right ) - 9945 \, \left (-\frac {a d^{38}}{b^{21}}\right )^{\frac {1}{4}} {\left (b^{9} x^{8} + 4 \, a b^{8} x^{6} + 6 \, a^{2} b^{7} x^{4} + 4 \, a^{3} b^{6} x^{2} + a^{4} b^{5}\right )} \log \left (3315 \, \sqrt {d x} d^{9} - 3315 \, \left (-\frac {a d^{38}}{b^{21}}\right )^{\frac {1}{4}} b^{5}\right ) - 4 \, {\left (6144 \, b^{4} d^{9} x^{8} + 31501 \, a b^{3} d^{9} x^{6} + 52819 \, a^{2} b^{2} d^{9} x^{4} + 37791 \, a^{3} b d^{9} x^{2} + 9945 \, a^{4} d^{9}\right )} \sqrt {d x}}{12288 \, {\left (b^{9} x^{8} + 4 \, a b^{8} x^{6} + 6 \, a^{2} b^{7} x^{4} + 4 \, a^{3} b^{6} x^{2} + a^{4} b^{5}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.42, size = 423, normalized size = 0.70 \[ -\frac {1}{24576} \, d^{9} {\left (\frac {19890 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{b^{6} \mathrm {sgn}\left (b d^{4} x^{2} + a d^{4}\right )} + \frac {19890 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{b^{6} \mathrm {sgn}\left (b d^{4} x^{2} + a d^{4}\right )} + \frac {9945 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \log \left (d x + \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{b^{6} \mathrm {sgn}\left (b d^{4} x^{2} + a d^{4}\right )} - \frac {9945 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \log \left (d x - \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{b^{6} \mathrm {sgn}\left (b d^{4} x^{2} + a d^{4}\right )} - \frac {49152 \, \sqrt {d x}}{b^{5} \mathrm {sgn}\left (b d^{4} x^{2} + a d^{4}\right )} - \frac {8 \, {\left (6925 \, \sqrt {d x} a b^{3} d^{8} x^{6} + 15955 \, \sqrt {d x} a^{2} b^{2} d^{8} x^{4} + 13215 \, \sqrt {d x} a^{3} b d^{8} x^{2} + 3801 \, \sqrt {d x} a^{4} d^{8}\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{4} b^{5} \mathrm {sgn}\left (b d^{4} x^{2} + a d^{4}\right )}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.03, size = 1202, normalized size = 2.00 \[ \text {result too large to display} \]
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 \[ d^{\frac {19}{2}} \int \frac {x^{\frac {3}{2}}}{b^{5} x^{2} + a b^{4}}\,{d x} - \frac {1267 \, {\left (\frac {2 \, \sqrt {2} \sqrt {a} \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}}} + \frac {2 \, \sqrt {2} \sqrt {a} \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}}} + \frac {\sqrt {2} a^{\frac {1}{4}} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{b^{\frac {1}{4}}} - \frac {\sqrt {2} a^{\frac {1}{4}} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{b^{\frac {1}{4}}}\right )} d^{\frac {19}{2}}}{8192 \, b^{5}} + \frac {1853 \, a b^{3} d^{\frac {19}{2}} x^{\frac {13}{2}} + 6515 \, a^{2} b^{2} d^{\frac {19}{2}} x^{\frac {9}{2}} + 8079 \, a^{3} b d^{\frac {19}{2}} x^{\frac {5}{2}} + 3801 \, a^{4} d^{\frac {19}{2}} \sqrt {x}}{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 (317 \, a b^{4} d^{\frac {19}{2}} x^{5} + 738 \, a^{2} b^{3} d^{\frac {19}{2}} x^{3} + 453 \, a^{3} b^{2} d^{\frac {19}{2}} x\right )} x^{\frac {11}{2}} + 2 \, {\left (243 \, a^{2} b^{3} d^{\frac {19}{2}} x^{5} + 582 \, a^{3} b^{2} d^{\frac {19}{2}} x^{3} + 371 \, a^{4} b d^{\frac {19}{2}} x\right )} x^{\frac {7}{2}} + {\left (201 \, a^{3} b^{2} d^{\frac {19}{2}} x^{5} + 490 \, a^{4} b d^{\frac {19}{2}} x^{3} + 321 \, a^{5} d^{\frac {19}{2}} x\right )} x^{\frac {3}{2}}}{192 \, {\left (a^{3} b^{7} x^{6} + 3 \, a^{4} b^{6} x^{4} + 3 \, a^{5} b^{5} x^{2} + a^{6} b^{4} + {\left (b^{10} x^{6} + 3 \, a b^{9} x^{4} + 3 \, a^{2} b^{8} x^{2} + a^{3} b^{7}\right )} x^{6} + 3 \, {\left (a b^{9} x^{6} + 3 \, a^{2} b^{8} x^{4} + 3 \, a^{3} b^{7} x^{2} + a^{4} b^{6}\right )} x^{4} + 3 \, {\left (a^{2} b^{8} x^{6} + 3 \, a^{3} b^{7} x^{4} + 3 \, a^{4} b^{6} x^{2} + a^{5} b^{5}\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 \[ \int \frac {{\left (d\,x\right )}^{19/2}}{{\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^{5/2}} \,d x \]
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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