Optimal. Leaf size=506 \[ \frac {9}{16 a^2 d \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1}{4 a d \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4} \left (a+b x^2\right )}-\frac {45 \sqrt [4]{b} \left (a+b x^2\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{64 \sqrt {2} a^{13/4} d^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {45 \sqrt [4]{b} \left (a+b x^2\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{64 \sqrt {2} a^{13/4} d^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {45 \sqrt [4]{b} \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{32 \sqrt {2} a^{13/4} d^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {45 \sqrt [4]{b} \left (a+b x^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{32 \sqrt {2} a^{13/4} d^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {45 \left (a+b x^2\right )}{16 a^3 d \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4}} \]
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Rubi [A] time = 0.38, antiderivative size = 506, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 10, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1112, 290, 325, 329, 297, 1162, 617, 204, 1165, 628} \begin {gather*} -\frac {45 \sqrt [4]{b} \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 )}{64 \sqrt {2} a^{13/4} d^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {45 \sqrt [4]{b} \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 )}{64 \sqrt {2} a^{13/4} d^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {45 \sqrt [4]{b} \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 )}{32 \sqrt {2} a^{13/4} d^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {45 \sqrt [4]{b} \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 )}{32 \sqrt {2} a^{13/4} d^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {45 \left (a+b x^2\right )}{16 a^3 d \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {9}{16 a^2 d \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1}{4 a d \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4} \left (a+b x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 204
Rule 290
Rule 297
Rule 325
Rule 329
Rule 617
Rule 628
Rule 1112
Rule 1162
Rule 1165
Rubi steps
\begin {align*} \int \frac {1}{(d x)^{3/2} \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \, dx &=\frac {\left (b^2 \left (a b+b^2 x^2\right )\right ) \int \frac {1}{(d x)^{3/2} \left (a b+b^2 x^2\right )^3} \, dx}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {1}{4 a d \sqrt {d x} \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (9 b \left (a b+b^2 x^2\right )\right ) \int \frac {1}{(d x)^{3/2} \left (a b+b^2 x^2\right )^2} \, dx}{8 a \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {9}{16 a^2 d \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1}{4 a d \sqrt {d x} \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (45 \left (a b+b^2 x^2\right )\right ) \int \frac {1}{(d x)^{3/2} \left (a b+b^2 x^2\right )} \, dx}{32 a^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {9}{16 a^2 d \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1}{4 a d \sqrt {d x} \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {45 \left (a+b x^2\right )}{16 a^3 d \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (45 b \left (a b+b^2 x^2\right )\right ) \int \frac {\sqrt {d x}}{a b+b^2 x^2} \, dx}{32 a^3 d^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {9}{16 a^2 d \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1}{4 a d \sqrt {d x} \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {45 \left (a+b x^2\right )}{16 a^3 d \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (45 b \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 )}{16 a^3 d^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {9}{16 a^2 d \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1}{4 a d \sqrt {d x} \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {45 \left (a+b x^2\right )}{16 a^3 d \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (45 \sqrt {b} \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 )}{32 a^3 d^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (45 \sqrt {b} \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 )}{32 a^3 d^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {9}{16 a^2 d \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1}{4 a d \sqrt {d x} \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {45 \left (a+b x^2\right )}{16 a^3 d \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (45 \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 )}{64 \sqrt {2} a^{13/4} b^{3/4} d^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (45 \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 )}{64 \sqrt {2} a^{13/4} b^{3/4} d^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (45 \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 )}{64 a^3 b d \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (45 \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 )}{64 a^3 b d \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {9}{16 a^2 d \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1}{4 a d \sqrt {d x} \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {45 \left (a+b x^2\right )}{16 a^3 d \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {45 \sqrt [4]{b} \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 )}{64 \sqrt {2} a^{13/4} d^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {45 \sqrt [4]{b} \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 )}{64 \sqrt {2} a^{13/4} d^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (45 \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 )}{32 \sqrt {2} a^{13/4} b^{3/4} d^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (45 \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 )}{32 \sqrt {2} a^{13/4} b^{3/4} d^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {9}{16 a^2 d \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1}{4 a d \sqrt {d x} \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {45 \left (a+b x^2\right )}{16 a^3 d \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {45 \sqrt [4]{b} \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 )}{32 \sqrt {2} a^{13/4} d^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {45 \sqrt [4]{b} \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 )}{32 \sqrt {2} a^{13/4} d^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {45 \sqrt [4]{b} \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 )}{64 \sqrt {2} a^{13/4} d^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {45 \sqrt [4]{b} \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 )}{64 \sqrt {2} a^{13/4} d^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 52, normalized size = 0.10 \begin {gather*} -\frac {2 x \left (a+b x^2\right )^3 \, _2F_1\left (-\frac {1}{4},3;\frac {3}{4};-\frac {b x^2}{a}\right )}{a^3 (d x)^{3/2} \left (\left (a+b x^2\right )^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 91.79, size = 255, normalized size = 0.50 \begin {gather*} \frac {\left (a d^2+b d^2 x^2\right ) \left (\frac {45 \sqrt [4]{b} \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 )}{32 \sqrt {2} a^{13/4} d^{3/2}}+\frac {45 \sqrt [4]{b} \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 )}{32 \sqrt {2} a^{13/4} d^{3/2}}+\frac {-32 a^2 d^4-81 a b d^4 x^2-45 b^2 d^4 x^4}{16 a^3 d \sqrt {d x} \left (a d^2+b d^2 x^2\right )^2}\right )}{d^2 \sqrt {\frac {\left (a d^2+b d^2 x^2\right )^2}{d^4}}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 2.18, size = 343, normalized size = 0.68 \begin {gather*} \frac {180 \, {\left (a^{3} b^{2} d^{2} x^{5} + 2 \, a^{4} b d^{2} x^{3} + a^{5} d^{2} x\right )} \left (-\frac {b}{a^{13} d^{6}}\right )^{\frac {1}{4}} \arctan \left (-\frac {91125 \, \sqrt {d x} a^{3} b d \left (-\frac {b}{a^{13} d^{6}}\right )^{\frac {1}{4}} - \sqrt {-8303765625 \, a^{7} b d^{4} \sqrt {-\frac {b}{a^{13} d^{6}}} + 8303765625 \, b^{2} d x} a^{3} d \left (-\frac {b}{a^{13} d^{6}}\right )^{\frac {1}{4}}}{91125 \, b}\right ) - 45 \, {\left (a^{3} b^{2} d^{2} x^{5} + 2 \, a^{4} b d^{2} x^{3} + a^{5} d^{2} x\right )} \left (-\frac {b}{a^{13} d^{6}}\right )^{\frac {1}{4}} \log \left (91125 \, a^{10} d^{5} \left (-\frac {b}{a^{13} d^{6}}\right )^{\frac {3}{4}} + 91125 \, \sqrt {d x} b\right ) + 45 \, {\left (a^{3} b^{2} d^{2} x^{5} + 2 \, a^{4} b d^{2} x^{3} + a^{5} d^{2} x\right )} \left (-\frac {b}{a^{13} d^{6}}\right )^{\frac {1}{4}} \log \left (-91125 \, a^{10} d^{5} \left (-\frac {b}{a^{13} d^{6}}\right )^{\frac {3}{4}} + 91125 \, \sqrt {d x} b\right ) - 4 \, {\left (45 \, b^{2} x^{4} + 81 \, a b x^{2} + 32 \, a^{2}\right )} \sqrt {d x}}{64 \, {\left (a^{3} b^{2} d^{2} x^{5} + 2 \, a^{4} b d^{2} x^{3} + a^{5} d^{2} x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.31, size = 410, normalized size = 0.81 \begin {gather*} -\frac {\frac {256}{\sqrt {d x} a^{3} \mathrm {sgn}\left (b d^{4} x^{2} + a d^{4}\right )} + \frac {8 \, {\left (13 \, \sqrt {d x} b^{2} d^{3} x^{3} + 17 \, \sqrt {d x} a b d^{3} x\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{2} a^{3} \mathrm {sgn}\left (b d^{4} x^{2} + a d^{4}\right )} + \frac {90 \, \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 )}{a^{4} b^{2} d^{2} \mathrm {sgn}\left (b d^{4} x^{2} + a d^{4}\right )} + \frac {90 \, \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 )}{a^{4} b^{2} d^{2} \mathrm {sgn}\left (b d^{4} x^{2} + a d^{4}\right )} - \frac {45 \, \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 )}{a^{4} b^{2} d^{2} \mathrm {sgn}\left (b d^{4} x^{2} + a d^{4}\right )} + \frac {45 \, \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 )}{a^{4} b^{2} d^{2} \mathrm {sgn}\left (b d^{4} x^{2} + a d^{4}\right )}}{128 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 645, normalized size = 1.27 \begin {gather*} -\frac {\left (90 \sqrt {2}\, \sqrt {d x}\, b^{2} x^{4} \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}-\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )+90 \sqrt {2}\, \sqrt {d x}\, b^{2} x^{4} \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}+\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )+45 \sqrt {2}\, \sqrt {d x}\, b^{2} x^{4} \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 )+360 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} b^{2} x^{4}+180 \sqrt {2}\, \sqrt {d x}\, a b \,x^{2} \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}-\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )+180 \sqrt {2}\, \sqrt {d x}\, a b \,x^{2} \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}+\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )+90 \sqrt {2}\, \sqrt {d x}\, a b \,x^{2} \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 )+648 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} a b \,x^{2}+90 \sqrt {2}\, \sqrt {d x}\, a^{2} \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}-\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )+90 \sqrt {2}\, \sqrt {d x}\, a^{2} \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}+\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )+45 \sqrt {2}\, \sqrt {d x}\, a^{2} \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 )+256 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} a^{2}\right ) \left (b \,x^{2}+a \right )}{128 \sqrt {d x}\, \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \left (\left (b \,x^{2}+a \right )^{2}\right )^{\frac {3}{2}} a^{3} d} \end {gather*}
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 \begin {gather*} -\frac {b x^{\frac {3}{2}}}{2 \, {\left (a^{3} b d^{\frac {3}{2}} x^{2} + a^{4} d^{\frac {3}{2}} + {\left (a^{2} b^{2} d^{\frac {3}{2}} x^{2} + a^{3} b d^{\frac {3}{2}}\right )} x^{2}\right )}} - \frac {13 \, b^{2} x^{\frac {7}{2}} + 9 \, a b x^{\frac {3}{2}}}{16 \, {\left (a^{3} b^{2} d^{\frac {3}{2}} x^{4} + 2 \, a^{4} b d^{\frac {3}{2}} x^{2} + a^{5} d^{\frac {3}{2}}\right )}} - \frac {13 \, b {\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 )}}{128 \, a^{3} d^{\frac {3}{2}}} + \int \frac {1}{{\left (a^{2} b d^{\frac {3}{2}} x^{2} + a^{3} d^{\frac {3}{2}}\right )} x^{\frac {3}{2}}}\,{d x} \end {gather*}
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 \begin {gather*} \int \frac {1}{{\left (d\,x\right )}^{3/2}\,{\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^{3/2}} \,d x \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 {3}{2}} \left (\left (a + b x^{2}\right )^{2}\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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