Optimal. Leaf size=504 \[ -\frac {9 d^3 (d x)^{5/2}}{16 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{9/2}}{4 b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {45 \sqrt [4]{a} d^{11/2} \left (a+b x^2\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{64 \sqrt {2} b^{13/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {45 \sqrt [4]{a} d^{11/2} \left (a+b x^2\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{64 \sqrt {2} b^{13/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {45 \sqrt [4]{a} d^{11/2} \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{32 \sqrt {2} b^{13/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {45 \sqrt [4]{a} d^{11/2} \left (a+b x^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{32 \sqrt {2} b^{13/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {45 d^5 \sqrt {d x} \left (a+b x^2\right )}{16 b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}} \]
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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, 211, 1165, 628, 1162, 617, 204} \[ \frac {45 d^5 \sqrt {d x} \left (a+b x^2\right )}{16 b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {9 d^3 (d x)^{5/2}}{16 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {45 \sqrt [4]{a} d^{11/2} \left (a+b x^2\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{64 \sqrt {2} b^{13/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {45 \sqrt [4]{a} d^{11/2} \left (a+b x^2\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{64 \sqrt {2} b^{13/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {45 \sqrt [4]{a} d^{11/2} \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{32 \sqrt {2} b^{13/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {45 \sqrt [4]{a} d^{11/2} \left (a+b x^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{32 \sqrt {2} b^{13/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{9/2}}{4 b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}} \]
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)^{11/2}}{\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)^{11/2}}{\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)^{9/2}}{4 b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (9 d^2 \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}{8 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac {9 d^3 (d x)^{5/2}}{16 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{9/2}}{4 b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (45 d^4 \left (a b+b^2 x^2\right )\right ) \int \frac {(d x)^{3/2}}{a b+b^2 x^2} \, dx}{32 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac {9 d^3 (d x)^{5/2}}{16 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{9/2}}{4 b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {45 d^5 \sqrt {d x} \left (a+b x^2\right )}{16 b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (45 a d^6 \left (a b+b^2 x^2\right )\right ) \int \frac {1}{\sqrt {d x} \left (a b+b^2 x^2\right )} \, dx}{32 b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac {9 d^3 (d x)^{5/2}}{16 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{9/2}}{4 b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {45 d^5 \sqrt {d x} \left (a+b x^2\right )}{16 b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (45 a d^5 \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 )}{16 b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac {9 d^3 (d x)^{5/2}}{16 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{9/2}}{4 b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {45 d^5 \sqrt {d x} \left (a+b x^2\right )}{16 b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (45 \sqrt {a} d^4 \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^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (45 \sqrt {a} d^4 \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^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac {9 d^3 (d x)^{5/2}}{16 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{9/2}}{4 b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {45 d^5 \sqrt {d x} \left (a+b x^2\right )}{16 b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (45 \sqrt [4]{a} d^{11/2} \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^{17/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (45 \sqrt [4]{a} d^{11/2} \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^{17/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (45 \sqrt {a} d^6 \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^{9/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (45 \sqrt {a} d^6 \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^{9/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac {9 d^3 (d x)^{5/2}}{16 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{9/2}}{4 b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {45 d^5 \sqrt {d x} \left (a+b x^2\right )}{16 b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {45 \sqrt [4]{a} d^{11/2} \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^{13/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {45 \sqrt [4]{a} d^{11/2} \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^{13/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (45 \sqrt [4]{a} d^{11/2} \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^{17/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (45 \sqrt [4]{a} d^{11/2} \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^{17/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac {9 d^3 (d x)^{5/2}}{16 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{9/2}}{4 b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {45 d^5 \sqrt {d x} \left (a+b x^2\right )}{16 b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {45 \sqrt [4]{a} d^{11/2} \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^{13/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {45 \sqrt [4]{a} d^{11/2} \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^{13/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {45 \sqrt [4]{a} d^{11/2} \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^{13/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {45 \sqrt [4]{a} d^{11/2} \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^{13/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 484, normalized size = 0.96 \[ \frac {15 a^2 (d x)^{11/2} \left (a+b x^2\right )}{4 b^3 x^5 \left (\left (a+b x^2\right )^2\right )^{3/2}}+\frac {45 \sqrt [4]{a} (d x)^{11/2} \left (a+b x^2\right )^3 \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{64 \sqrt {2} b^{13/4} x^{11/2} \left (\left (a+b x^2\right )^2\right )^{3/2}}-\frac {45 \sqrt [4]{a} (d x)^{11/2} \left (a+b x^2\right )^3 \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{64 \sqrt {2} b^{13/4} x^{11/2} \left (\left (a+b x^2\right )^2\right )^{3/2}}+\frac {45 \sqrt [4]{a} (d x)^{11/2} \left (a+b x^2\right )^3 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} b^{13/4} x^{11/2} \left (\left (a+b x^2\right )^2\right )^{3/2}}-\frac {45 \sqrt [4]{a} (d x)^{11/2} \left (a+b x^2\right )^3 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{32 \sqrt {2} b^{13/4} x^{11/2} \left (\left (a+b x^2\right )^2\right )^{3/2}}-\frac {15 a (d x)^{11/2} \left (a+b x^2\right )^2}{16 b^3 x^5 \left (\left (a+b x^2\right )^2\right )^{3/2}}+\frac {6 a (d x)^{11/2} \left (a+b x^2\right )}{b^2 x^3 \left (\left (a+b x^2\right )^2\right )^{3/2}}+\frac {2 (d x)^{11/2} \left (a+b x^2\right )}{b x \left (\left (a+b x^2\right )^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.01, size = 305, normalized size = 0.61 \[ -\frac {180 \, \left (-\frac {a d^{22}}{b^{13}}\right )^{\frac {1}{4}} {\left (b^{5} x^{4} + 2 \, a b^{4} x^{2} + a^{2} b^{3}\right )} \arctan \left (-\frac {\left (-\frac {a d^{22}}{b^{13}}\right )^{\frac {3}{4}} \sqrt {d x} b^{10} d^{5} - \sqrt {d^{11} x + \sqrt {-\frac {a d^{22}}{b^{13}}} b^{6}} \left (-\frac {a d^{22}}{b^{13}}\right )^{\frac {3}{4}} b^{10}}{a d^{22}}\right ) + 45 \, \left (-\frac {a d^{22}}{b^{13}}\right )^{\frac {1}{4}} {\left (b^{5} x^{4} + 2 \, a b^{4} x^{2} + a^{2} b^{3}\right )} \log \left (45 \, \sqrt {d x} d^{5} + 45 \, \left (-\frac {a d^{22}}{b^{13}}\right )^{\frac {1}{4}} b^{3}\right ) - 45 \, \left (-\frac {a d^{22}}{b^{13}}\right )^{\frac {1}{4}} {\left (b^{5} x^{4} + 2 \, a b^{4} x^{2} + a^{2} b^{3}\right )} \log \left (45 \, \sqrt {d x} d^{5} - 45 \, \left (-\frac {a d^{22}}{b^{13}}\right )^{\frac {1}{4}} b^{3}\right ) - 4 \, {\left (32 \, b^{2} d^{5} x^{4} + 81 \, a b d^{5} x^{2} + 45 \, a^{2} d^{5}\right )} \sqrt {d x}}{64 \, {\left (b^{5} x^{4} + 2 \, a b^{4} x^{2} + a^{2} b^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.31, size = 385, normalized size = 0.76 \[ -\frac {1}{128} \, d^{5} {\left (\frac {90 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{b^{4} \mathrm {sgn}\left (b d^{4} x^{2} + a d^{4}\right )} + \frac {90 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{b^{4} \mathrm {sgn}\left (b d^{4} x^{2} + a d^{4}\right )} + \frac {45 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \log \left (d x + \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{b^{4} \mathrm {sgn}\left (b d^{4} x^{2} + a d^{4}\right )} - \frac {45 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \log \left (d x - \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{b^{4} \mathrm {sgn}\left (b d^{4} x^{2} + a d^{4}\right )} - \frac {256 \, \sqrt {d x}}{b^{3} \mathrm {sgn}\left (b d^{4} x^{2} + a d^{4}\right )} - \frac {8 \, {\left (17 \, \sqrt {d x} a b d^{4} x^{2} + 13 \, \sqrt {d x} a^{2} d^{4}\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{2} b^{3} \mathrm {sgn}\left (b d^{4} x^{2} + a d^{4}\right )}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 696, normalized size = 1.38 \[ -\frac {\left (90 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, b^{2} d^{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 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, b^{2} d^{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 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, b^{2} d^{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 )-256 \sqrt {d x}\, b^{2} d^{2} x^{4}+180 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, a b \,d^{2} 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 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, a b \,d^{2} 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 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, a b \,d^{2} 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 )-512 \sqrt {d x}\, a b \,d^{2} x^{2}+90 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, a^{2} d^{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 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, a^{2} d^{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 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, a^{2} d^{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 )-360 \sqrt {d x}\, a^{2} d^{2}-136 \left (d x \right )^{\frac {5}{2}} a b \right ) \left (b \,x^{2}+a \right ) d^{3}}{128 \left (\left (b \,x^{2}+a \right )^{2}\right )^{\frac {3}{2}} b^{3}} \]
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 \[ \frac {a d^{\frac {11}{2}} x^{\frac {5}{2}}}{2 \, {\left (a b^{3} x^{2} + a^{2} b^{2} + {\left (b^{4} x^{2} + a b^{3}\right )} x^{2}\right )}} + d^{\frac {11}{2}} \int \frac {x^{\frac {3}{2}}}{b^{3} x^{2} + a b^{2}}\,{d x} - \frac {13 \, {\left (\frac {2 \, \sqrt {2} \sqrt {a} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}}} + \frac {2 \, \sqrt {2} \sqrt {a} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}}} + \frac {\sqrt {2} a^{\frac {1}{4}} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{b^{\frac {1}{4}}} - \frac {\sqrt {2} a^{\frac {1}{4}} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{b^{\frac {1}{4}}}\right )} d^{\frac {11}{2}}}{128 \, b^{3}} + \frac {9 \, a b d^{\frac {11}{2}} x^{\frac {5}{2}} + 13 \, a^{2} d^{\frac {11}{2}} \sqrt {x}}{16 \, {\left (b^{5} x^{4} + 2 \, a b^{4} x^{2} + a^{2} b^{3}\right )}} \]
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 \[ \int \frac {{\left (d\,x\right )}^{11/2}}{{\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^{3/2}} \,d x \]
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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