Optimal. Leaf size=556 \[ \frac {5 \sqrt {d x}}{32 a^2 d \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\sqrt {d x}}{8 a d \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {1155 \left (a+b x^2\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{4096 \sqrt {2} a^{19/4} \sqrt [4]{b} \sqrt {d} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1155 \left (a+b x^2\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{4096 \sqrt {2} a^{19/4} \sqrt [4]{b} \sqrt {d} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {1155 \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{2048 \sqrt {2} a^{19/4} \sqrt [4]{b} \sqrt {d} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1155 \left (a+b x^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{2048 \sqrt {2} a^{19/4} \sqrt [4]{b} \sqrt {d} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {385 \sqrt {d x}}{1024 a^4 d \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {55 \sqrt {d x}}{256 a^3 d \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}} \]
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Rubi [A] time = 0.43, antiderivative size = 556, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 9, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {1112, 290, 329, 211, 1165, 628, 1162, 617, 204} \[ \frac {385 \sqrt {d x}}{1024 a^4 d \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {55 \sqrt {d x}}{256 a^3 d \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {5 \sqrt {d x}}{32 a^2 d \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\sqrt {d x}}{8 a d \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {1155 \left (a+b x^2\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{4096 \sqrt {2} a^{19/4} \sqrt [4]{b} \sqrt {d} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1155 \left (a+b x^2\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{4096 \sqrt {2} a^{19/4} \sqrt [4]{b} \sqrt {d} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {1155 \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{2048 \sqrt {2} a^{19/4} \sqrt [4]{b} \sqrt {d} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1155 \left (a+b x^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{2048 \sqrt {2} a^{19/4} \sqrt [4]{b} \sqrt {d} \sqrt {a^2+2 a b x^2+b^2 x^4}} \]
Antiderivative was successfully verified.
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Rule 204
Rule 211
Rule 290
Rule 329
Rule 617
Rule 628
Rule 1112
Rule 1162
Rule 1165
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {d x} \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 {1}{\sqrt {d x} \left (a b+b^2 x^2\right )^5} \, dx}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {\sqrt {d x}}{8 a d \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (15 b^3 \left (a b+b^2 x^2\right )\right ) \int \frac {1}{\sqrt {d x} \left (a b+b^2 x^2\right )^4} \, dx}{16 a \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {\sqrt {d x}}{8 a d \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {5 \sqrt {d x}}{32 a^2 d \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (55 b^2 \left (a b+b^2 x^2\right )\right ) \int \frac {1}{\sqrt {d x} \left (a b+b^2 x^2\right )^3} \, dx}{64 a^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {\sqrt {d x}}{8 a d \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {5 \sqrt {d x}}{32 a^2 d \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {55 \sqrt {d x}}{256 a^3 d \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (385 b \left (a b+b^2 x^2\right )\right ) \int \frac {1}{\sqrt {d x} \left (a b+b^2 x^2\right )^2} \, dx}{512 a^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {385 \sqrt {d x}}{1024 a^4 d \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\sqrt {d x}}{8 a d \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {5 \sqrt {d x}}{32 a^2 d \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {55 \sqrt {d x}}{256 a^3 d \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (1155 \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 a^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {385 \sqrt {d x}}{1024 a^4 d \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\sqrt {d x}}{8 a d \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {5 \sqrt {d x}}{32 a^2 d \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {55 \sqrt {d x}}{256 a^3 d \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (1155 \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 a^4 d \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {385 \sqrt {d x}}{1024 a^4 d \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\sqrt {d x}}{8 a d \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {5 \sqrt {d x}}{32 a^2 d \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {55 \sqrt {d x}}{256 a^3 d \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (1155 \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 a^{9/2} d^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (1155 \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 a^{9/2} d^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {385 \sqrt {d x}}{1024 a^4 d \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\sqrt {d x}}{8 a d \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {5 \sqrt {d x}}{32 a^2 d \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {55 \sqrt {d x}}{256 a^3 d \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (1155 \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 a^{9/2} b^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (1155 \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 a^{9/2} b^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (1155 \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} a^{19/4} b^{5/4} \sqrt {d} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (1155 \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} a^{19/4} b^{5/4} \sqrt {d} \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {385 \sqrt {d x}}{1024 a^4 d \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\sqrt {d x}}{8 a d \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {5 \sqrt {d x}}{32 a^2 d \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {55 \sqrt {d x}}{256 a^3 d \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {1155 \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} a^{19/4} \sqrt [4]{b} \sqrt {d} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1155 \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} a^{19/4} \sqrt [4]{b} \sqrt {d} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (1155 \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} a^{19/4} b^{5/4} \sqrt {d} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (1155 \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} a^{19/4} b^{5/4} \sqrt {d} \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {385 \sqrt {d x}}{1024 a^4 d \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\sqrt {d x}}{8 a d \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {5 \sqrt {d x}}{32 a^2 d \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {55 \sqrt {d x}}{256 a^3 d \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {1155 \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} a^{19/4} \sqrt [4]{b} \sqrt {d} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1155 \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} a^{19/4} \sqrt [4]{b} \sqrt {d} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {1155 \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} a^{19/4} \sqrt [4]{b} \sqrt {d} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1155 \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} a^{19/4} \sqrt [4]{b} \sqrt {d} \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 319, normalized size = 0.57 \[ \frac {\sqrt {x} \left (a+b x^2\right ) \left (3080 a^{3/4} \sqrt {x} \left (a+b x^2\right )^3+1760 a^{7/4} \sqrt {x} \left (a+b x^2\right )^2+1280 a^{11/4} \sqrt {x} \left (a+b x^2\right )+1024 a^{15/4} \sqrt {x}-\frac {1155 \sqrt {2} \left (a+b x^2\right )^4 \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{\sqrt [4]{b}}+\frac {1155 \sqrt {2} \left (a+b x^2\right )^4 \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{\sqrt [4]{b}}-\frac {2310 \sqrt {2} \left (a+b x^2\right )^4 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt [4]{b}}+\frac {2310 \sqrt {2} \left (a+b x^2\right )^4 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt [4]{b}}\right )}{8192 a^{19/4} \sqrt {d x} \left (\left (a+b x^2\right )^2\right )^{5/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.94, size = 416, normalized size = 0.75 \[ \frac {4620 \, {\left (a^{4} b^{4} d x^{8} + 4 \, a^{5} b^{3} d x^{6} + 6 \, a^{6} b^{2} d x^{4} + 4 \, a^{7} b d x^{2} + a^{8} d\right )} \left (-\frac {1}{a^{19} b d^{2}}\right )^{\frac {1}{4}} \arctan \left (\sqrt {a^{10} d^{2} \sqrt {-\frac {1}{a^{19} b d^{2}}} + d x} a^{14} b d \left (-\frac {1}{a^{19} b d^{2}}\right )^{\frac {3}{4}} - \sqrt {d x} a^{14} b d \left (-\frac {1}{a^{19} b d^{2}}\right )^{\frac {3}{4}}\right ) + 1155 \, {\left (a^{4} b^{4} d x^{8} + 4 \, a^{5} b^{3} d x^{6} + 6 \, a^{6} b^{2} d x^{4} + 4 \, a^{7} b d x^{2} + a^{8} d\right )} \left (-\frac {1}{a^{19} b d^{2}}\right )^{\frac {1}{4}} \log \left (a^{5} d \left (-\frac {1}{a^{19} b d^{2}}\right )^{\frac {1}{4}} + \sqrt {d x}\right ) - 1155 \, {\left (a^{4} b^{4} d x^{8} + 4 \, a^{5} b^{3} d x^{6} + 6 \, a^{6} b^{2} d x^{4} + 4 \, a^{7} b d x^{2} + a^{8} d\right )} \left (-\frac {1}{a^{19} b d^{2}}\right )^{\frac {1}{4}} \log \left (-a^{5} d \left (-\frac {1}{a^{19} b d^{2}}\right )^{\frac {1}{4}} + \sqrt {d x}\right ) + 4 \, {\left (385 \, b^{3} x^{6} + 1375 \, a b^{2} x^{4} + 1755 \, a^{2} b x^{2} + 893 \, a^{3}\right )} \sqrt {d x}}{4096 \, {\left (a^{4} b^{4} d x^{8} + 4 \, a^{5} b^{3} d x^{6} + 6 \, a^{6} b^{2} d x^{4} + 4 \, a^{7} b d x^{2} + a^{8} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.33, size = 412, normalized size = 0.74 \[ \frac {1155 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{4096 \, a^{5} b d \mathrm {sgn}\left (b d^{4} x^{2} + a d^{4}\right )} + \frac {1155 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{4096 \, a^{5} b d \mathrm {sgn}\left (b d^{4} x^{2} + a d^{4}\right )} + \frac {1155 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \log \left (d x + \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{8192 \, a^{5} b d \mathrm {sgn}\left (b d^{4} x^{2} + a d^{4}\right )} - \frac {1155 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \log \left (d x - \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{8192 \, a^{5} b d \mathrm {sgn}\left (b d^{4} x^{2} + a d^{4}\right )} + \frac {385 \, \sqrt {d x} b^{3} d^{7} x^{6} + 1375 \, \sqrt {d x} a b^{2} d^{7} x^{4} + 1755 \, \sqrt {d x} a^{2} b d^{7} x^{2} + 893 \, \sqrt {d x} a^{3} d^{7}}{1024 \, {\left (b d^{2} x^{2} + a d^{2}\right )}^{4} a^{4} \mathrm {sgn}\left (b d^{4} x^{2} + a d^{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 1133, normalized size = 2.04 \[ \text {result too large to display} \]
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 \[ \frac {5267 \, b^{3} x^{\frac {13}{2}} + 11645 \, a b^{2} x^{\frac {9}{2}} + 9441 \, a^{2} b x^{\frac {5}{2}} + 2679 \, a^{3} \sqrt {x}}{3072 \, {\left (a^{4} b^{4} \sqrt {d} x^{8} + 4 \, a^{5} b^{3} \sqrt {d} x^{6} + 6 \, a^{6} b^{2} \sqrt {d} x^{4} + 4 \, a^{7} b \sqrt {d} x^{2} + a^{8} \sqrt {d}\right )}} - \frac {{\left (257 \, b^{5} \sqrt {d} x^{5} + 378 \, a b^{4} \sqrt {d} x^{3} + 153 \, a^{2} b^{3} \sqrt {d} x\right )} x^{\frac {11}{2}} + 2 \, {\left (303 \, a b^{4} \sqrt {d} x^{5} + 462 \, a^{2} b^{3} \sqrt {d} x^{3} + 191 \, a^{3} b^{2} \sqrt {d} x\right )} x^{\frac {7}{2}} + {\left (381 \, a^{2} b^{3} \sqrt {d} x^{5} + 610 \, a^{3} b^{2} \sqrt {d} x^{3} + 261 \, a^{4} b \sqrt {d} x\right )} x^{\frac {3}{2}}}{192 \, {\left (a^{7} b^{3} d x^{6} + 3 \, a^{8} b^{2} d x^{4} + 3 \, a^{9} b d x^{2} + a^{10} d + {\left (a^{4} b^{6} d x^{6} + 3 \, a^{5} b^{5} d x^{4} + 3 \, a^{6} b^{4} d x^{2} + a^{7} b^{3} d\right )} x^{6} + 3 \, {\left (a^{5} b^{5} d x^{6} + 3 \, a^{6} b^{4} d x^{4} + 3 \, a^{7} b^{3} d x^{2} + a^{8} b^{2} d\right )} x^{4} + 3 \, {\left (a^{6} b^{4} d x^{6} + 3 \, a^{7} b^{3} d x^{4} + 3 \, a^{8} b^{2} d x^{2} + a^{9} b d\right )} x^{2}\right )}} - \frac {893 \, {\left (\frac {2 \, \sqrt {2} \sqrt {d} \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 {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {2 \, \sqrt {2} \sqrt {d} \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 {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {\sqrt {2} \sqrt {d} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}} - \frac {\sqrt {2} \sqrt {d} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}}\right )}}{8192 \, a^{4} d} + \int \frac {1}{{\left (a^{4} b \sqrt {d} x^{2} + a^{5} \sqrt {d}\right )} \sqrt {x}}\,{d x} \]
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 \[ \int \frac {1}{\sqrt {d\,x}\,{\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {d x} \left (\left (a + b x^{2}\right )^{2}\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
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