Optimal. Leaf size=283 \[ \frac {(d x)^{3/2}}{2 a d \left (a+b x^2\right )}-\frac {\sqrt {d} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{4 \sqrt {2} a^{5/4} b^{3/4}}+\frac {\sqrt {d} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{4 \sqrt {2} a^{5/4} b^{3/4}}+\frac {\sqrt {d} \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{8 \sqrt {2} a^{5/4} b^{3/4}}-\frac {\sqrt {d} \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{8 \sqrt {2} a^{5/4} b^{3/4}} \]
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Rubi [A]
time = 0.18, antiderivative size = 283, normalized size of antiderivative = 1.00, number of steps
used = 12, number of rules used = 9, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.321, Rules used = {28, 296, 335,
303, 1176, 631, 210, 1179, 642} \begin {gather*} -\frac {\sqrt {d} \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{4 \sqrt {2} a^{5/4} b^{3/4}}+\frac {\sqrt {d} \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{4 \sqrt {2} a^{5/4} b^{3/4}}+\frac {\sqrt {d} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{8 \sqrt {2} a^{5/4} b^{3/4}}-\frac {\sqrt {d} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{8 \sqrt {2} a^{5/4} b^{3/4}}+\frac {(d x)^{3/2}}{2 a d \left (a+b x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 28
Rule 210
Rule 296
Rule 303
Rule 335
Rule 631
Rule 642
Rule 1176
Rule 1179
Rubi steps
\begin {align*} \int \frac {\sqrt {d x}}{a^2+2 a b x^2+b^2 x^4} \, dx &=b^2 \int \frac {\sqrt {d x}}{\left (a b+b^2 x^2\right )^2} \, dx\\ &=\frac {(d x)^{3/2}}{2 a d \left (a+b x^2\right )}+\frac {b \int \frac {\sqrt {d x}}{a b+b^2 x^2} \, dx}{4 a}\\ &=\frac {(d x)^{3/2}}{2 a d \left (a+b x^2\right )}+\frac {b \text {Subst}\left (\int \frac {x^2}{a b+\frac {b^2 x^4}{d^2}} \, dx,x,\sqrt {d x}\right )}{2 a d}\\ &=\frac {(d x)^{3/2}}{2 a d \left (a+b x^2\right )}-\frac {\sqrt {b} \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 )}{4 a d}+\frac {\sqrt {b} \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 )}{4 a d}\\ &=\frac {(d x)^{3/2}}{2 a d \left (a+b x^2\right )}+\frac {\sqrt {d} \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 )}{8 \sqrt {2} a^{5/4} b^{3/4}}+\frac {\sqrt {d} \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 )}{8 \sqrt {2} a^{5/4} b^{3/4}}+\frac {d \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 )}{8 a b}+\frac {d \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 )}{8 a b}\\ &=\frac {(d x)^{3/2}}{2 a d \left (a+b x^2\right )}+\frac {\sqrt {d} \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{8 \sqrt {2} a^{5/4} b^{3/4}}-\frac {\sqrt {d} \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{8 \sqrt {2} a^{5/4} b^{3/4}}+\frac {\sqrt {d} \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 )}{4 \sqrt {2} a^{5/4} b^{3/4}}-\frac {\sqrt {d} \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 )}{4 \sqrt {2} a^{5/4} b^{3/4}}\\ &=\frac {(d x)^{3/2}}{2 a d \left (a+b x^2\right )}-\frac {\sqrt {d} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{4 \sqrt {2} a^{5/4} b^{3/4}}+\frac {\sqrt {d} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{4 \sqrt {2} a^{5/4} b^{3/4}}+\frac {\sqrt {d} \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{8 \sqrt {2} a^{5/4} b^{3/4}}-\frac {\sqrt {d} \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{8 \sqrt {2} a^{5/4} b^{3/4}}\\ \end {align*}
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Mathematica [A]
time = 0.24, size = 152, normalized size = 0.54 \begin {gather*} -\frac {\sqrt {d x} \left (-4 \sqrt [4]{a} b^{3/4} x^{3/2}+\sqrt {2} \left (a+b x^2\right ) \tan ^{-1}\left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )+\sqrt {2} \left (a+b x^2\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )\right )}{8 a^{5/4} b^{3/4} \sqrt {x} \left (a+b x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 180, normalized size = 0.64
method | result | size |
derivativedivides | \(2 d^{3} \left (\frac {\left (d x \right )^{\frac {3}{2}}}{4 a \,d^{2} \left (d^{2} x^{2} b +a \,d^{2}\right )}+\frac {\sqrt {2}\, \left (\ln \left (\frac {d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 a \,d^{2} b \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )\) | \(180\) |
default | \(2 d^{3} \left (\frac {\left (d x \right )^{\frac {3}{2}}}{4 a \,d^{2} \left (d^{2} x^{2} b +a \,d^{2}\right )}+\frac {\sqrt {2}\, \left (\ln \left (\frac {d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 a \,d^{2} b \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )\) | \(180\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 255, normalized size = 0.90 \begin {gather*} \frac {\frac {8 \, \left (d x\right )^{\frac {3}{2}} d^{2}}{a b d^{2} x^{2} + a^{2} d^{2}} + \frac {d^{2} {\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}}{16 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 232, normalized size = 0.82 \begin {gather*} -\frac {4 \, {\left (a b x^{2} + a^{2}\right )} \left (-\frac {d^{2}}{a^{5} b^{3}}\right )^{\frac {1}{4}} \arctan \left (-\frac {\sqrt {d x} a b d \left (-\frac {d^{2}}{a^{5} b^{3}}\right )^{\frac {1}{4}} - \sqrt {-a^{3} b d^{2} \sqrt {-\frac {d^{2}}{a^{5} b^{3}}} + d^{3} x} a b \left (-\frac {d^{2}}{a^{5} b^{3}}\right )^{\frac {1}{4}}}{d^{2}}\right ) - {\left (a b x^{2} + a^{2}\right )} \left (-\frac {d^{2}}{a^{5} b^{3}}\right )^{\frac {1}{4}} \log \left (a^{4} b^{2} \left (-\frac {d^{2}}{a^{5} b^{3}}\right )^{\frac {3}{4}} + \sqrt {d x} d\right ) + {\left (a b x^{2} + a^{2}\right )} \left (-\frac {d^{2}}{a^{5} b^{3}}\right )^{\frac {1}{4}} \log \left (-a^{4} b^{2} \left (-\frac {d^{2}}{a^{5} b^{3}}\right )^{\frac {3}{4}} + \sqrt {d x} d\right ) - 4 \, \sqrt {d x} x}{8 \, {\left (a b x^{2} + a^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 2.54, size = 78, normalized size = 0.28 \begin {gather*} \frac {2 d^{3} \left (d x\right )^{\frac {3}{2}}}{4 a^{2} d^{4} + 4 a b d^{4} x^{2}} + 2 d^{3} \operatorname {RootSum} {\left (65536 t^{4} a^{5} b^{3} d^{10} + 1, \left ( t \mapsto t \log {\left (4096 t^{3} a^{4} b^{2} d^{8} + \sqrt {d x} \right )} \right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.30, size = 264, normalized size = 0.93 \begin {gather*} \frac {\frac {8 \, \sqrt {d x} d^{3} x}{{\left (b d^{2} x^{2} + a d^{2}\right )} a} + \frac {2 \, \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^{2} b^{3}} + \frac {2 \, \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^{2} b^{3}} - \frac {\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^{2} b^{3}} + \frac {\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^{2} b^{3}}}{16 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.11, size = 90, normalized size = 0.32 \begin {gather*} \frac {\sqrt {d}\,\mathrm {atanh}\left (\frac {b^{1/4}\,\sqrt {d\,x}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )}{4\,{\left (-a\right )}^{5/4}\,b^{3/4}}-\frac {\sqrt {d}\,\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {d\,x}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )}{4\,{\left (-a\right )}^{5/4}\,b^{3/4}}+\frac {d\,{\left (d\,x\right )}^{3/2}}{2\,a\,\left (b\,d^2\,x^2+a\,d^2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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