Optimal. Leaf size=352 \[ \frac {385 b^{3/4} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{256 \sqrt {2} a^{19/4} d^{5/2}}-\frac {385 b^{3/4} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{256 \sqrt {2} a^{19/4} d^{5/2}}+\frac {385 b^{3/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{128 \sqrt {2} a^{19/4} d^{5/2}}-\frac {385 b^{3/4} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{128 \sqrt {2} a^{19/4} d^{5/2}}-\frac {385}{192 a^4 d (d x)^{3/2}}+\frac {55}{64 a^3 d (d x)^{3/2} \left (a+b x^2\right )}+\frac {5}{16 a^2 d (d x)^{3/2} \left (a+b x^2\right )^2}+\frac {1}{6 a d (d x)^{3/2} \left (a+b x^2\right )^3} \]
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Rubi [A] time = 0.39, antiderivative size = 352, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 10, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {28, 290, 325, 329, 211, 1165, 628, 1162, 617, 204} \begin {gather*} \frac {385 b^{3/4} \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} a^{19/4} d^{5/2}}-\frac {385 b^{3/4} \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} a^{19/4} d^{5/2}}+\frac {385 b^{3/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{128 \sqrt {2} a^{19/4} d^{5/2}}-\frac {385 b^{3/4} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{128 \sqrt {2} a^{19/4} d^{5/2}}+\frac {55}{64 a^3 d (d x)^{3/2} \left (a+b x^2\right )}+\frac {5}{16 a^2 d (d x)^{3/2} \left (a+b x^2\right )^2}-\frac {385}{192 a^4 d (d x)^{3/2}}+\frac {1}{6 a d (d x)^{3/2} \left (a+b x^2\right )^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 28
Rule 204
Rule 211
Rule 290
Rule 325
Rule 329
Rule 617
Rule 628
Rule 1162
Rule 1165
Rubi steps
\begin {align*} \int \frac {1}{(d x)^{5/2} \left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx &=b^4 \int \frac {1}{(d x)^{5/2} \left (a b+b^2 x^2\right )^4} \, dx\\ &=\frac {1}{6 a d (d x)^{3/2} \left (a+b x^2\right )^3}+\frac {\left (5 b^3\right ) \int \frac {1}{(d x)^{5/2} \left (a b+b^2 x^2\right )^3} \, dx}{4 a}\\ &=\frac {1}{6 a d (d x)^{3/2} \left (a+b x^2\right )^3}+\frac {5}{16 a^2 d (d x)^{3/2} \left (a+b x^2\right )^2}+\frac {\left (55 b^2\right ) \int \frac {1}{(d x)^{5/2} \left (a b+b^2 x^2\right )^2} \, dx}{32 a^2}\\ &=\frac {1}{6 a d (d x)^{3/2} \left (a+b x^2\right )^3}+\frac {5}{16 a^2 d (d x)^{3/2} \left (a+b x^2\right )^2}+\frac {55}{64 a^3 d (d x)^{3/2} \left (a+b x^2\right )}+\frac {(385 b) \int \frac {1}{(d x)^{5/2} \left (a b+b^2 x^2\right )} \, dx}{128 a^3}\\ &=-\frac {385}{192 a^4 d (d x)^{3/2}}+\frac {1}{6 a d (d x)^{3/2} \left (a+b x^2\right )^3}+\frac {5}{16 a^2 d (d x)^{3/2} \left (a+b x^2\right )^2}+\frac {55}{64 a^3 d (d x)^{3/2} \left (a+b x^2\right )}-\frac {\left (385 b^2\right ) \int \frac {1}{\sqrt {d x} \left (a b+b^2 x^2\right )} \, dx}{128 a^4 d^2}\\ &=-\frac {385}{192 a^4 d (d x)^{3/2}}+\frac {1}{6 a d (d x)^{3/2} \left (a+b x^2\right )^3}+\frac {5}{16 a^2 d (d x)^{3/2} \left (a+b x^2\right )^2}+\frac {55}{64 a^3 d (d x)^{3/2} \left (a+b x^2\right )}-\frac {\left (385 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{a b+\frac {b^2 x^4}{d^2}} \, dx,x,\sqrt {d x}\right )}{64 a^4 d^3}\\ &=-\frac {385}{192 a^4 d (d x)^{3/2}}+\frac {1}{6 a d (d x)^{3/2} \left (a+b x^2\right )^3}+\frac {5}{16 a^2 d (d x)^{3/2} \left (a+b x^2\right )^2}+\frac {55}{64 a^3 d (d x)^{3/2} \left (a+b x^2\right )}-\frac {\left (385 b^2\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 a^{9/2} d^4}-\frac {\left (385 b^2\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 a^{9/2} d^4}\\ &=-\frac {385}{192 a^4 d (d x)^{3/2}}+\frac {1}{6 a d (d x)^{3/2} \left (a+b x^2\right )^3}+\frac {5}{16 a^2 d (d x)^{3/2} \left (a+b x^2\right )^2}+\frac {55}{64 a^3 d (d x)^{3/2} \left (a+b x^2\right )}+\frac {\left (385 b^{3/4}\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} a^{19/4} d^{5/2}}+\frac {\left (385 b^{3/4}\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} a^{19/4} d^{5/2}}-\frac {\left (385 \sqrt {b}\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 a^{9/2} d^2}-\frac {\left (385 \sqrt {b}\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 a^{9/2} d^2}\\ &=-\frac {385}{192 a^4 d (d x)^{3/2}}+\frac {1}{6 a d (d x)^{3/2} \left (a+b x^2\right )^3}+\frac {5}{16 a^2 d (d x)^{3/2} \left (a+b x^2\right )^2}+\frac {55}{64 a^3 d (d x)^{3/2} \left (a+b x^2\right )}+\frac {385 b^{3/4} \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} a^{19/4} d^{5/2}}-\frac {385 b^{3/4} \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} a^{19/4} d^{5/2}}-\frac {\left (385 b^{3/4}\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} a^{19/4} d^{5/2}}+\frac {\left (385 b^{3/4}\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} a^{19/4} d^{5/2}}\\ &=-\frac {385}{192 a^4 d (d x)^{3/2}}+\frac {1}{6 a d (d x)^{3/2} \left (a+b x^2\right )^3}+\frac {5}{16 a^2 d (d x)^{3/2} \left (a+b x^2\right )^2}+\frac {55}{64 a^3 d (d x)^{3/2} \left (a+b x^2\right )}+\frac {385 b^{3/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{128 \sqrt {2} a^{19/4} d^{5/2}}-\frac {385 b^{3/4} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{128 \sqrt {2} a^{19/4} d^{5/2}}+\frac {385 b^{3/4} \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} a^{19/4} d^{5/2}}-\frac {385 b^{3/4} \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} a^{19/4} d^{5/2}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 32, normalized size = 0.09 \begin {gather*} -\frac {2 x \, _2F_1\left (-\frac {3}{4},4;\frac {1}{4};-\frac {b x^2}{a}\right )}{3 a^4 (d x)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.82, size = 227, normalized size = 0.64 \begin {gather*} \frac {385 b^{3/4} \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} a^{19/4} d^{5/2}}-\frac {385 b^{3/4} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d} \sqrt {d x}}{\sqrt {a} d+\sqrt {b} d x}\right )}{128 \sqrt {2} a^{19/4} d^{5/2}}+\frac {-128 a^3 d^6-765 a^2 b d^6 x^2-990 a b^2 d^6 x^4-385 b^3 d^6 x^6}{192 a^4 d (d x)^{3/2} \left (a d^2+b d^2 x^2\right )^3} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.89, size = 434, normalized size = 1.23 \begin {gather*} -\frac {4620 \, {\left (a^{4} b^{3} d^{3} x^{8} + 3 \, a^{5} b^{2} d^{3} x^{6} + 3 \, a^{6} b d^{3} x^{4} + a^{7} d^{3} x^{2}\right )} \left (-\frac {b^{3}}{a^{19} d^{10}}\right )^{\frac {1}{4}} \arctan \left (-\frac {\sqrt {d x} a^{14} b d^{7} \left (-\frac {b^{3}}{a^{19} d^{10}}\right )^{\frac {3}{4}} - \sqrt {a^{10} d^{6} \sqrt {-\frac {b^{3}}{a^{19} d^{10}}} + b^{2} d x} a^{14} d^{7} \left (-\frac {b^{3}}{a^{19} d^{10}}\right )^{\frac {3}{4}}}{b^{3}}\right ) + 1155 \, {\left (a^{4} b^{3} d^{3} x^{8} + 3 \, a^{5} b^{2} d^{3} x^{6} + 3 \, a^{6} b d^{3} x^{4} + a^{7} d^{3} x^{2}\right )} \left (-\frac {b^{3}}{a^{19} d^{10}}\right )^{\frac {1}{4}} \log \left (385 \, a^{5} d^{3} \left (-\frac {b^{3}}{a^{19} d^{10}}\right )^{\frac {1}{4}} + 385 \, \sqrt {d x} b\right ) - 1155 \, {\left (a^{4} b^{3} d^{3} x^{8} + 3 \, a^{5} b^{2} d^{3} x^{6} + 3 \, a^{6} b d^{3} x^{4} + a^{7} d^{3} x^{2}\right )} \left (-\frac {b^{3}}{a^{19} d^{10}}\right )^{\frac {1}{4}} \log \left (-385 \, a^{5} d^{3} \left (-\frac {b^{3}}{a^{19} d^{10}}\right )^{\frac {1}{4}} + 385 \, \sqrt {d x} b\right ) + 4 \, {\left (385 \, b^{3} x^{6} + 990 \, a b^{2} x^{4} + 765 \, a^{2} b x^{2} + 128 \, a^{3}\right )} \sqrt {d x}}{768 \, {\left (a^{4} b^{3} d^{3} x^{8} + 3 \, a^{5} b^{2} d^{3} x^{6} + 3 \, a^{6} b d^{3} x^{4} + a^{7} d^{3} x^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 308, normalized size = 0.88 \begin {gather*} -\frac {385 \, \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 )}{256 \, a^{5} d^{3}} - \frac {385 \, \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 )}{256 \, a^{5} d^{3}} - \frac {385 \, \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 )}{512 \, a^{5} d^{3}} + \frac {385 \, \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 )}{512 \, a^{5} d^{3}} - \frac {385 \, b^{3} d^{6} x^{6} + 990 \, a b^{2} d^{6} x^{4} + 765 \, a^{2} b d^{6} x^{2} + 128 \, a^{3} d^{6}}{192 \, {\left (\sqrt {d x} b d^{2} x^{2} + \sqrt {d x} a d^{2}\right )}^{3} a^{4} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 288, normalized size = 0.82 \begin {gather*} -\frac {127 \sqrt {d x}\, b \,d^{3}}{64 \left (b \,d^{2} x^{2}+d^{2} a \right )^{3} a^{2}}-\frac {101 \left (d x \right )^{\frac {5}{2}} b^{2} d}{32 \left (b \,d^{2} x^{2}+d^{2} a \right )^{3} a^{3}}-\frac {257 \left (d x \right )^{\frac {9}{2}} b^{3}}{192 \left (b \,d^{2} x^{2}+d^{2} a \right )^{3} a^{4} d}-\frac {2}{3 \left (d x \right )^{\frac {3}{2}} a^{4} d}-\frac {385 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, b \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )}{256 a^{5} d^{3}}-\frac {385 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, b \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )}{256 a^{5} d^{3}}-\frac {385 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, b \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 a^{5} d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.07, size = 335, normalized size = 0.95 \begin {gather*} -\frac {\frac {8 \, {\left (385 \, b^{3} d^{6} x^{6} + 990 \, a b^{2} d^{6} x^{4} + 765 \, a^{2} b d^{6} x^{2} + 128 \, a^{3} d^{6}\right )}}{\left (d x\right )^{\frac {15}{2}} a^{4} b^{3} + 3 \, \left (d x\right )^{\frac {11}{2}} a^{5} b^{2} d^{2} + 3 \, \left (d x\right )^{\frac {7}{2}} a^{6} b d^{4} + \left (d x\right )^{\frac {3}{2}} a^{7} d^{6}} + \frac {1155 \, {\left (\frac {\sqrt {2} b^{\frac {3}{4}} \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}}} - \frac {\sqrt {2} b^{\frac {3}{4}} \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}}} + \frac {2 \, \sqrt {2} b \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} d} + \frac {2 \, \sqrt {2} b \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} d}\right )}}{a^{4}}}{1536 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.25, size = 166, normalized size = 0.47 \begin {gather*} \frac {385\,{\left (-b\right )}^{3/4}\,\mathrm {atan}\left (\frac {{\left (-b\right )}^{1/4}\,\sqrt {d\,x}}{a^{1/4}\,\sqrt {d}}\right )}{128\,a^{19/4}\,d^{5/2}}-\frac {\frac {2\,d^5}{3\,a}+\frac {255\,b\,d^5\,x^2}{64\,a^2}+\frac {165\,b^2\,d^5\,x^4}{32\,a^3}+\frac {385\,b^3\,d^5\,x^6}{192\,a^4}}{b^3\,{\left (d\,x\right )}^{15/2}+a^3\,d^6\,{\left (d\,x\right )}^{3/2}+3\,a^2\,b\,d^4\,{\left (d\,x\right )}^{7/2}+3\,a\,b^2\,d^2\,{\left (d\,x\right )}^{11/2}}+\frac {385\,{\left (-b\right )}^{3/4}\,\mathrm {atanh}\left (\frac {{\left (-b\right )}^{1/4}\,\sqrt {d\,x}}{a^{1/4}\,\sqrt {d}}\right )}{128\,a^{19/4}\,d^{5/2}} \end {gather*}
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 \begin {gather*} \int \frac {1}{\left (d x\right )^{\frac {5}{2}} \left (a + b x^{2}\right )^{4}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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