Optimal. Leaf size=352 \[ -\frac {195}{64 a^4 d \sqrt {d x}}+\frac {1}{6 a d \sqrt {d x} \left (a+b x^2\right )^3}+\frac {13}{48 a^2 d \sqrt {d x} \left (a+b x^2\right )^2}+\frac {39}{64 a^3 d \sqrt {d x} \left (a+b x^2\right )}+\frac {195 \sqrt [4]{b} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{128 \sqrt {2} a^{17/4} d^{3/2}}-\frac {195 \sqrt [4]{b} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{128 \sqrt {2} a^{17/4} d^{3/2}}-\frac {195 \sqrt [4]{b} \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{256 \sqrt {2} a^{17/4} d^{3/2}}+\frac {195 \sqrt [4]{b} \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{256 \sqrt {2} a^{17/4} d^{3/2}} \]
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Rubi [A]
time = 0.26, 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, 296, 331,
335, 303, 1176, 631, 210, 1179, 642} \begin {gather*} \frac {195 \sqrt [4]{b} \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{128 \sqrt {2} a^{17/4} d^{3/2}}-\frac {195 \sqrt [4]{b} \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{128 \sqrt {2} a^{17/4} d^{3/2}}-\frac {195 \sqrt [4]{b} \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^{17/4} d^{3/2}}+\frac {195 \sqrt [4]{b} \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^{17/4} d^{3/2}}-\frac {195}{64 a^4 d \sqrt {d x}}+\frac {39}{64 a^3 d \sqrt {d x} \left (a+b x^2\right )}+\frac {13}{48 a^2 d \sqrt {d x} \left (a+b x^2\right )^2}+\frac {1}{6 a d \sqrt {d x} \left (a+b x^2\right )^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 28
Rule 210
Rule 296
Rule 303
Rule 331
Rule 335
Rule 631
Rule 642
Rule 1176
Rule 1179
Rubi steps
\begin {align*} \int \frac {1}{(d x)^{3/2} \left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx &=b^4 \int \frac {1}{(d x)^{3/2} \left (a b+b^2 x^2\right )^4} \, dx\\ &=\frac {1}{6 a d \sqrt {d x} \left (a+b x^2\right )^3}+\frac {\left (13 b^3\right ) \int \frac {1}{(d x)^{3/2} \left (a b+b^2 x^2\right )^3} \, dx}{12 a}\\ &=\frac {1}{6 a d \sqrt {d x} \left (a+b x^2\right )^3}+\frac {13}{48 a^2 d \sqrt {d x} \left (a+b x^2\right )^2}+\frac {\left (39 b^2\right ) \int \frac {1}{(d x)^{3/2} \left (a b+b^2 x^2\right )^2} \, dx}{32 a^2}\\ &=\frac {1}{6 a d \sqrt {d x} \left (a+b x^2\right )^3}+\frac {13}{48 a^2 d \sqrt {d x} \left (a+b x^2\right )^2}+\frac {39}{64 a^3 d \sqrt {d x} \left (a+b x^2\right )}+\frac {(195 b) \int \frac {1}{(d x)^{3/2} \left (a b+b^2 x^2\right )} \, dx}{128 a^3}\\ &=-\frac {195}{64 a^4 d \sqrt {d x}}+\frac {1}{6 a d \sqrt {d x} \left (a+b x^2\right )^3}+\frac {13}{48 a^2 d \sqrt {d x} \left (a+b x^2\right )^2}+\frac {39}{64 a^3 d \sqrt {d x} \left (a+b x^2\right )}-\frac {\left (195 b^2\right ) \int \frac {\sqrt {d x}}{a b+b^2 x^2} \, dx}{128 a^4 d^2}\\ &=-\frac {195}{64 a^4 d \sqrt {d x}}+\frac {1}{6 a d \sqrt {d x} \left (a+b x^2\right )^3}+\frac {13}{48 a^2 d \sqrt {d x} \left (a+b x^2\right )^2}+\frac {39}{64 a^3 d \sqrt {d x} \left (a+b x^2\right )}-\frac {\left (195 b^2\right ) \text {Subst}\left (\int \frac {x^2}{a b+\frac {b^2 x^4}{d^2}} \, dx,x,\sqrt {d x}\right )}{64 a^4 d^3}\\ &=-\frac {195}{64 a^4 d \sqrt {d x}}+\frac {1}{6 a d \sqrt {d x} \left (a+b x^2\right )^3}+\frac {13}{48 a^2 d \sqrt {d x} \left (a+b x^2\right )^2}+\frac {39}{64 a^3 d \sqrt {d x} \left (a+b x^2\right )}+\frac {\left (195 b^{3/2}\right ) \text {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^4 d^3}-\frac {\left (195 b^{3/2}\right ) \text {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^4 d^3}\\ &=-\frac {195}{64 a^4 d \sqrt {d x}}+\frac {1}{6 a d \sqrt {d x} \left (a+b x^2\right )^3}+\frac {13}{48 a^2 d \sqrt {d x} \left (a+b x^2\right )^2}+\frac {39}{64 a^3 d \sqrt {d x} \left (a+b x^2\right )}-\frac {\left (195 \sqrt [4]{b}\right ) \text {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^{17/4} d^{3/2}}-\frac {\left (195 \sqrt [4]{b}\right ) \text {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^{17/4} d^{3/2}}-\frac {195 \text {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^4 d}-\frac {195 \text {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^4 d}\\ &=-\frac {195}{64 a^4 d \sqrt {d x}}+\frac {1}{6 a d \sqrt {d x} \left (a+b x^2\right )^3}+\frac {13}{48 a^2 d \sqrt {d x} \left (a+b x^2\right )^2}+\frac {39}{64 a^3 d \sqrt {d x} \left (a+b x^2\right )}-\frac {195 \sqrt [4]{b} \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^{17/4} d^{3/2}}+\frac {195 \sqrt [4]{b} \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^{17/4} d^{3/2}}-\frac {\left (195 \sqrt [4]{b}\right ) \text {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^{17/4} d^{3/2}}+\frac {\left (195 \sqrt [4]{b}\right ) \text {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^{17/4} d^{3/2}}\\ &=-\frac {195}{64 a^4 d \sqrt {d x}}+\frac {1}{6 a d \sqrt {d x} \left (a+b x^2\right )^3}+\frac {13}{48 a^2 d \sqrt {d x} \left (a+b x^2\right )^2}+\frac {39}{64 a^3 d \sqrt {d x} \left (a+b x^2\right )}+\frac {195 \sqrt [4]{b} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{128 \sqrt {2} a^{17/4} d^{3/2}}-\frac {195 \sqrt [4]{b} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{128 \sqrt {2} a^{17/4} d^{3/2}}-\frac {195 \sqrt [4]{b} \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^{17/4} d^{3/2}}+\frac {195 \sqrt [4]{b} \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^{17/4} d^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.49, size = 173, normalized size = 0.49 \begin {gather*} \frac {x \left (-\frac {4 \sqrt [4]{a} \left (384 a^3+1469 a^2 b x^2+1638 a b^2 x^4+585 b^3 x^6\right )}{\left (a+b x^2\right )^3}+585 \sqrt {2} \sqrt [4]{b} \sqrt {x} \tan ^{-1}\left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )+585 \sqrt {2} \sqrt [4]{b} \sqrt {x} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )\right )}{768 a^{17/4} (d x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 221, normalized size = 0.63
method | result | size |
derivativedivides | \(2 d^{7} \left (-\frac {1}{a^{4} d^{8} \sqrt {d x}}-\frac {b \left (\frac {\frac {67 b^{2} \left (d x \right )^{\frac {11}{2}}}{128}+\frac {81 a b \,d^{2} \left (d x \right )^{\frac {7}{2}}}{64}+\frac {317 a^{2} d^{4} \left (d x \right )^{\frac {3}{2}}}{384}}{\left (d^{2} x^{2} b +a \,d^{2}\right )^{3}}+\frac {195 \sqrt {2}\, \left (\ln \left (\frac {d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{1024 b \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a^{4} d^{8}}\right )\) | \(221\) |
default | \(2 d^{7} \left (-\frac {1}{a^{4} d^{8} \sqrt {d x}}-\frac {b \left (\frac {\frac {67 b^{2} \left (d x \right )^{\frac {11}{2}}}{128}+\frac {81 a b \,d^{2} \left (d x \right )^{\frac {7}{2}}}{64}+\frac {317 a^{2} d^{4} \left (d x \right )^{\frac {3}{2}}}{384}}{\left (d^{2} x^{2} b +a \,d^{2}\right )^{3}}+\frac {195 \sqrt {2}\, \left (\ln \left (\frac {d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{1024 b \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a^{4} d^{8}}\right )\) | \(221\) |
risch | \(-\frac {2}{a^{4} d \sqrt {d x}}+\frac {-\frac {67 b^{3} \left (d x \right )^{\frac {11}{2}}}{64 a^{4} \left (d^{2} x^{2} b +a \,d^{2}\right )^{3}}-\frac {81 b^{2} \left (d x \right )^{\frac {7}{2}} d^{2}}{32 a^{3} \left (d^{2} x^{2} b +a \,d^{2}\right )^{3}}-\frac {317 b \left (d x \right )^{\frac {3}{2}} d^{4}}{192 a^{2} \left (d^{2} x^{2} b +a \,d^{2}\right )^{3}}-\frac {195 \sqrt {2}\, \ln \left (\frac {d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )}{512 a^{4} \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-\frac {195 \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )}{256 a^{4} \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-\frac {195 \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )}{256 a^{4} \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}}{d}\) | \(280\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 328, normalized size = 0.93 \begin {gather*} -\frac {\frac {8 \, {\left (585 \, b^{3} d^{6} x^{6} + 1638 \, a b^{2} d^{6} x^{4} + 1469 \, a^{2} b d^{6} x^{2} + 384 \, a^{3} d^{6}\right )}}{\left (d x\right )^{\frac {13}{2}} a^{4} b^{3} + 3 \, \left (d x\right )^{\frac {9}{2}} a^{5} b^{2} d^{2} + 3 \, \left (d x\right )^{\frac {5}{2}} a^{6} b d^{4} + \sqrt {d x} a^{7} d^{6}} + \frac {585 \, b {\left (\frac {2 \, \sqrt {2} \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 {b}} + \frac {2 \, \sqrt {2} \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 {b}} - \frac {\sqrt {2} \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 {1}{4}} b^{\frac {3}{4}}} + \frac {\sqrt {2} \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 {1}{4}} b^{\frac {3}{4}}}\right )}}{a^{4}}}{1536 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 410, normalized size = 1.16 \begin {gather*} \frac {2340 \, {\left (a^{4} b^{3} d^{2} x^{7} + 3 \, a^{5} b^{2} d^{2} x^{5} + 3 \, a^{6} b d^{2} x^{3} + a^{7} d^{2} x\right )} \left (-\frac {b}{a^{17} d^{6}}\right )^{\frac {1}{4}} \arctan \left (-\frac {7414875 \, \sqrt {d x} a^{4} b d \left (-\frac {b}{a^{17} d^{6}}\right )^{\frac {1}{4}} - \sqrt {-54980371265625 \, a^{9} b d^{4} \sqrt {-\frac {b}{a^{17} d^{6}}} + 54980371265625 \, b^{2} d x} a^{4} d \left (-\frac {b}{a^{17} d^{6}}\right )^{\frac {1}{4}}}{7414875 \, b}\right ) - 585 \, {\left (a^{4} b^{3} d^{2} x^{7} + 3 \, a^{5} b^{2} d^{2} x^{5} + 3 \, a^{6} b d^{2} x^{3} + a^{7} d^{2} x\right )} \left (-\frac {b}{a^{17} d^{6}}\right )^{\frac {1}{4}} \log \left (7414875 \, a^{13} d^{5} \left (-\frac {b}{a^{17} d^{6}}\right )^{\frac {3}{4}} + 7414875 \, \sqrt {d x} b\right ) + 585 \, {\left (a^{4} b^{3} d^{2} x^{7} + 3 \, a^{5} b^{2} d^{2} x^{5} + 3 \, a^{6} b d^{2} x^{3} + a^{7} d^{2} x\right )} \left (-\frac {b}{a^{17} d^{6}}\right )^{\frac {1}{4}} \log \left (-7414875 \, a^{13} d^{5} \left (-\frac {b}{a^{17} d^{6}}\right )^{\frac {3}{4}} + 7414875 \, \sqrt {d x} b\right ) - 4 \, {\left (585 \, b^{3} x^{6} + 1638 \, a b^{2} x^{4} + 1469 \, a^{2} b x^{2} + 384 \, a^{3}\right )} \sqrt {d x}}{768 \, {\left (a^{4} b^{3} d^{2} x^{7} + 3 \, a^{5} b^{2} d^{2} x^{5} + 3 \, a^{6} b d^{2} x^{3} + a^{7} d^{2} x\right )}} \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 (a + b x^{2}\right )^{4}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.81, size = 327, normalized size = 0.93 \begin {gather*} -\frac {\frac {3072}{\sqrt {d x} a^{4}} + \frac {8 \, {\left (201 \, \sqrt {d x} b^{3} d^{5} x^{5} + 486 \, \sqrt {d x} a b^{2} d^{5} x^{3} + 317 \, \sqrt {d x} a^{2} b d^{5} x\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{3} a^{4}} + \frac {1170 \, \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^{5} b^{2} d^{2}} + \frac {1170 \, \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^{5} b^{2} d^{2}} - \frac {585 \, \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^{5} b^{2} d^{2}} + \frac {585 \, \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^{5} b^{2} d^{2}}}{1536 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.14, size = 166, normalized size = 0.47 \begin {gather*} \frac {195\,{\left (-b\right )}^{1/4}\,\mathrm {atanh}\left (\frac {{\left (-b\right )}^{1/4}\,\sqrt {d\,x}}{a^{1/4}\,\sqrt {d}}\right )}{128\,a^{17/4}\,d^{3/2}}-\frac {195\,{\left (-b\right )}^{1/4}\,\mathrm {atan}\left (\frac {{\left (-b\right )}^{1/4}\,\sqrt {d\,x}}{a^{1/4}\,\sqrt {d}}\right )}{128\,a^{17/4}\,d^{3/2}}-\frac {\frac {2\,d^5}{a}+\frac {1469\,b\,d^5\,x^2}{192\,a^2}+\frac {273\,b^2\,d^5\,x^4}{32\,a^3}+\frac {195\,b^3\,d^5\,x^6}{64\,a^4}}{b^3\,{\left (d\,x\right )}^{13/2}+a^3\,d^6\,\sqrt {d\,x}+3\,a^2\,b\,d^4\,{\left (d\,x\right )}^{5/2}+3\,a\,b^2\,d^2\,{\left (d\,x\right )}^{9/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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