Optimal. Leaf size=504 \[ -\frac {11 d^3 (d x)^{7/2}}{16 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{11/2}}{4 b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {77 d^5 (d x)^{3/2} \left (a+b x^2\right )}{48 b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {77 a^{3/4} d^{13/2} \left (a+b x^2\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{64 \sqrt {2} b^{15/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {77 a^{3/4} d^{13/2} \left (a+b x^2\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{64 \sqrt {2} b^{15/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {77 a^{3/4} d^{13/2} \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{32 \sqrt {2} b^{15/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {77 a^{3/4} d^{13/2} \left (a+b x^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{32 \sqrt {2} b^{15/4} \sqrt {a^2+2 a b x^2+b^2 x^4}} \]
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Rubi [A] time = 0.37, antiderivative size = 504, 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, 288, 321, 329, 297, 1162, 617, 204, 1165, 628} \begin {gather*} \frac {77 d^5 (d x)^{3/2} \left (a+b x^2\right )}{48 b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {11 d^3 (d x)^{7/2}}{16 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {77 a^{3/4} d^{13/2} \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 )}{64 \sqrt {2} b^{15/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {77 a^{3/4} d^{13/2} \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 )}{64 \sqrt {2} b^{15/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {77 a^{3/4} d^{13/2} \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 )}{32 \sqrt {2} b^{15/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {77 a^{3/4} d^{13/2} \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 )}{32 \sqrt {2} b^{15/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{11/2}}{4 b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}} \end {gather*}
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)^{13/2}}{\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 {(d x)^{13/2}}{\left (a b+b^2 x^2\right )^3} \, dx}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac {d (d x)^{11/2}}{4 b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (11 d^2 \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}{8 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac {11 d^3 (d x)^{7/2}}{16 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{11/2}}{4 b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (77 d^4 \left (a b+b^2 x^2\right )\right ) \int \frac {(d x)^{5/2}}{a b+b^2 x^2} \, dx}{32 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac {11 d^3 (d x)^{7/2}}{16 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{11/2}}{4 b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {77 d^5 (d x)^{3/2} \left (a+b x^2\right )}{48 b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (77 a d^6 \left (a b+b^2 x^2\right )\right ) \int \frac {\sqrt {d x}}{a b+b^2 x^2} \, dx}{32 b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac {11 d^3 (d x)^{7/2}}{16 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{11/2}}{4 b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {77 d^5 (d x)^{3/2} \left (a+b x^2\right )}{48 b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (77 a d^5 \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 b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac {11 d^3 (d x)^{7/2}}{16 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{11/2}}{4 b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {77 d^5 (d x)^{3/2} \left (a+b x^2\right )}{48 b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (77 a d^5 \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 b^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (77 a d^5 \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 b^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac {11 d^3 (d x)^{7/2}}{16 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{11/2}}{4 b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {77 d^5 (d x)^{3/2} \left (a+b x^2\right )}{48 b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (77 a^{3/4} d^{13/2} \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 )}{64 \sqrt {2} b^{19/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (77 a^{3/4} d^{13/2} \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 )}{64 \sqrt {2} b^{19/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (77 a d^7 \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 b^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (77 a d^7 \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 b^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac {11 d^3 (d x)^{7/2}}{16 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{11/2}}{4 b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {77 d^5 (d x)^{3/2} \left (a+b x^2\right )}{48 b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {77 a^{3/4} d^{13/2} \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 )}{64 \sqrt {2} b^{15/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {77 a^{3/4} d^{13/2} \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 )}{64 \sqrt {2} b^{15/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (77 a^{3/4} d^{13/2} \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 )}{32 \sqrt {2} b^{19/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (77 a^{3/4} d^{13/2} \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 )}{32 \sqrt {2} b^{19/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac {11 d^3 (d x)^{7/2}}{16 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{11/2}}{4 b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {77 d^5 (d x)^{3/2} \left (a+b x^2\right )}{48 b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {77 a^{3/4} d^{13/2} \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 )}{32 \sqrt {2} b^{15/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {77 a^{3/4} d^{13/2} \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 )}{32 \sqrt {2} b^{15/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {77 a^{3/4} d^{13/2} \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 )}{64 \sqrt {2} b^{15/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {77 a^{3/4} d^{13/2} \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 )}{64 \sqrt {2} b^{15/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 88, normalized size = 0.17 \begin {gather*} -\frac {2 d^5 (d x)^{3/2} \left (-77 a^2+77 \left (a+b x^2\right )^2 \, _2F_1\left (\frac {3}{4},3;\frac {7}{4};-\frac {b x^2}{a}\right )-55 a b x^2-5 b^2 x^4\right )}{15 b^3 \left (a+b x^2\right ) \sqrt {\left (a+b x^2\right )^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 97.89, size = 255, normalized size = 0.51 \begin {gather*} \frac {\left (a d^2+b d^2 x^2\right ) \left (\frac {77 a^{3/4} d^{13/2} \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} b^{15/4}}+\frac {77 a^{3/4} d^{13/2} \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} b^{15/4}}+\frac {d^5 (d x)^{3/2} \left (77 a^2 d^4+121 a b d^4 x^2+32 b^2 d^4 x^4\right )}{48 b^3 \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 = 0.95, size = 341, normalized size = 0.68 \begin {gather*} \frac {924 \, \left (-\frac {a^{3} d^{26}}{b^{15}}\right )^{\frac {1}{4}} {\left (b^{5} x^{4} + 2 \, a b^{4} x^{2} + a^{2} b^{3}\right )} \arctan \left (-\frac {\left (-\frac {a^{3} d^{26}}{b^{15}}\right )^{\frac {1}{4}} \sqrt {d x} a^{2} b^{4} d^{19} - \sqrt {a^{4} d^{39} x - \sqrt {-\frac {a^{3} d^{26}}{b^{15}}} a^{3} b^{7} d^{26}} \left (-\frac {a^{3} d^{26}}{b^{15}}\right )^{\frac {1}{4}} b^{4}}{a^{3} d^{26}}\right ) - 231 \, \left (-\frac {a^{3} d^{26}}{b^{15}}\right )^{\frac {1}{4}} {\left (b^{5} x^{4} + 2 \, a b^{4} x^{2} + a^{2} b^{3}\right )} \log \left (456533 \, \sqrt {d x} a^{2} d^{19} + 456533 \, \left (-\frac {a^{3} d^{26}}{b^{15}}\right )^{\frac {3}{4}} b^{11}\right ) + 231 \, \left (-\frac {a^{3} d^{26}}{b^{15}}\right )^{\frac {1}{4}} {\left (b^{5} x^{4} + 2 \, a b^{4} x^{2} + a^{2} b^{3}\right )} \log \left (456533 \, \sqrt {d x} a^{2} d^{19} - 456533 \, \left (-\frac {a^{3} d^{26}}{b^{15}}\right )^{\frac {3}{4}} b^{11}\right ) + 4 \, {\left (32 \, b^{2} d^{6} x^{5} + 121 \, a b d^{6} x^{3} + 77 \, a^{2} d^{6} x\right )} \sqrt {d x}}{192 \, {\left (b^{5} x^{4} + 2 \, a b^{4} x^{2} + a^{2} b^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.41, size = 399, normalized size = 0.79 \begin {gather*} \frac {1}{384} \, d^{6} {\left (\frac {256 \, \sqrt {d x} x}{b^{3} \mathrm {sgn}\left (b d^{4} x^{2} + a d^{4}\right )} + \frac {24 \, {\left (19 \, \sqrt {d x} a b d^{4} x^{3} + 15 \, \sqrt {d x} a^{2} d^{4} x\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{2} b^{3} \mathrm {sgn}\left (b d^{4} x^{2} + a d^{4}\right )} - \frac {462 \, \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^{6} d \mathrm {sgn}\left (b d^{4} x^{2} + a d^{4}\right )} - \frac {462 \, \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^{6} d \mathrm {sgn}\left (b d^{4} x^{2} + a d^{4}\right )} + \frac {231 \, \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^{6} d \mathrm {sgn}\left (b d^{4} x^{2} + a d^{4}\right )} - \frac {231 \, \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^{6} d \mathrm {sgn}\left (b d^{4} x^{2} + a d^{4}\right )}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 679, normalized size = 1.35 \begin {gather*} \frac {\left (-462 \sqrt {2}\, a \,b^{2} d^{4} 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 )-462 \sqrt {2}\, a \,b^{2} d^{4} 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 )-231 \sqrt {2}\, a \,b^{2} d^{4} 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 )-924 \sqrt {2}\, a^{2} b \,d^{4} 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 )-924 \sqrt {2}\, a^{2} b \,d^{4} 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 )-462 \sqrt {2}\, a^{2} b \,d^{4} 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 )+256 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \left (d x \right )^{\frac {3}{2}} b^{3} d^{2} x^{4}-462 \sqrt {2}\, a^{3} d^{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 )-462 \sqrt {2}\, a^{3} d^{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 )-231 \sqrt {2}\, a^{3} d^{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 )+512 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \left (d x \right )^{\frac {3}{2}} a \,b^{2} d^{2} x^{2}+616 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \left (d x \right )^{\frac {3}{2}} a^{2} b \,d^{2}+456 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \left (d x \right )^{\frac {7}{2}} a \,b^{2}\right ) \left (b \,x^{2}+a \right ) d^{3}}{384 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \left (\left (b \,x^{2}+a \right )^{2}\right )^{\frac {3}{2}} b^{4}} \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 {a^{2} d^{\frac {13}{2}} x^{\frac {3}{2}}}{2 \, {\left (a b^{4} x^{2} + a^{2} b^{3} + {\left (b^{5} x^{2} + a b^{4}\right )} x^{2}\right )}} - 2 \, a d^{\frac {13}{2}} \int \frac {\sqrt {x}}{b^{4} x^{2} + a b^{3}}\,{d x} + d^{\frac {13}{2}} \int \frac {x^{\frac {5}{2}}}{b^{3} x^{2} + a b^{2}}\,{d x} + \frac {19 \, a d^{\frac {13}{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 )}}{128 \, b^{3}} + \frac {19 \, a b d^{\frac {13}{2}} x^{\frac {7}{2}} + 23 \, a^{2} d^{\frac {13}{2}} x^{\frac {3}{2}}}{16 \, {\left (b^{5} x^{4} + 2 \, a b^{4} x^{2} + a^{2} b^{3}\right )}} \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 {{\left (d\,x\right )}^{13/2}}{{\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(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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