Optimal. Leaf size=335 \[ \frac {\sqrt {d x}}{6 a d \left (a+b x^2\right )^3}+\frac {11 \sqrt {d x}}{48 a^2 d \left (a+b x^2\right )^2}+\frac {77 \sqrt {d x}}{192 a^3 d \left (a+b x^2\right )}-\frac {77 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{128 \sqrt {2} a^{15/4} \sqrt [4]{b} \sqrt {d}}+\frac {77 \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{128 \sqrt {2} a^{15/4} \sqrt [4]{b} \sqrt {d}}-\frac {77 \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{256 \sqrt {2} a^{15/4} \sqrt [4]{b} \sqrt {d}}+\frac {77 \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{256 \sqrt {2} a^{15/4} \sqrt [4]{b} \sqrt {d}} \]
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Rubi [A]
time = 0.22, antiderivative size = 335, normalized size of antiderivative = 1.00, number of steps
used = 14, number of rules used = 9, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.321, Rules used = {28, 296, 335,
217, 1179, 642, 1176, 631, 210} \begin {gather*} -\frac {77 \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{128 \sqrt {2} a^{15/4} \sqrt [4]{b} \sqrt {d}}+\frac {77 \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{128 \sqrt {2} a^{15/4} \sqrt [4]{b} \sqrt {d}}-\frac {77 \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^{15/4} \sqrt [4]{b} \sqrt {d}}+\frac {77 \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^{15/4} \sqrt [4]{b} \sqrt {d}}+\frac {77 \sqrt {d x}}{192 a^3 d \left (a+b x^2\right )}+\frac {11 \sqrt {d x}}{48 a^2 d \left (a+b x^2\right )^2}+\frac {\sqrt {d x}}{6 a d \left (a+b x^2\right )^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 28
Rule 210
Rule 217
Rule 296
Rule 335
Rule 631
Rule 642
Rule 1176
Rule 1179
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {d x} \left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx &=b^4 \int \frac {1}{\sqrt {d x} \left (a b+b^2 x^2\right )^4} \, dx\\ &=\frac {\sqrt {d x}}{6 a d \left (a+b x^2\right )^3}+\frac {\left (11 b^3\right ) \int \frac {1}{\sqrt {d x} \left (a b+b^2 x^2\right )^3} \, dx}{12 a}\\ &=\frac {\sqrt {d x}}{6 a d \left (a+b x^2\right )^3}+\frac {11 \sqrt {d x}}{48 a^2 d \left (a+b x^2\right )^2}+\frac {\left (77 b^2\right ) \int \frac {1}{\sqrt {d x} \left (a b+b^2 x^2\right )^2} \, dx}{96 a^2}\\ &=\frac {\sqrt {d x}}{6 a d \left (a+b x^2\right )^3}+\frac {11 \sqrt {d x}}{48 a^2 d \left (a+b x^2\right )^2}+\frac {77 \sqrt {d x}}{192 a^3 d \left (a+b x^2\right )}+\frac {(77 b) \int \frac {1}{\sqrt {d x} \left (a b+b^2 x^2\right )} \, dx}{128 a^3}\\ &=\frac {\sqrt {d x}}{6 a d \left (a+b x^2\right )^3}+\frac {11 \sqrt {d x}}{48 a^2 d \left (a+b x^2\right )^2}+\frac {77 \sqrt {d x}}{192 a^3 d \left (a+b x^2\right )}+\frac {(77 b) \text {Subst}\left (\int \frac {1}{a b+\frac {b^2 x^4}{d^2}} \, dx,x,\sqrt {d x}\right )}{64 a^3 d}\\ &=\frac {\sqrt {d x}}{6 a d \left (a+b x^2\right )^3}+\frac {11 \sqrt {d x}}{48 a^2 d \left (a+b x^2\right )^2}+\frac {77 \sqrt {d x}}{192 a^3 d \left (a+b x^2\right )}+\frac {(77 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 )}{128 a^{7/2} d^2}+\frac {(77 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 )}{128 a^{7/2} d^2}\\ &=\frac {\sqrt {d x}}{6 a d \left (a+b x^2\right )^3}+\frac {11 \sqrt {d x}}{48 a^2 d \left (a+b x^2\right )^2}+\frac {77 \sqrt {d x}}{192 a^3 d \left (a+b x^2\right )}+\frac {77 \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^{7/2} \sqrt {b}}+\frac {77 \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^{7/2} \sqrt {b}}-\frac {77 \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^{15/4} \sqrt [4]{b} \sqrt {d}}-\frac {77 \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^{15/4} \sqrt [4]{b} \sqrt {d}}\\ &=\frac {\sqrt {d x}}{6 a d \left (a+b x^2\right )^3}+\frac {11 \sqrt {d x}}{48 a^2 d \left (a+b x^2\right )^2}+\frac {77 \sqrt {d x}}{192 a^3 d \left (a+b x^2\right )}-\frac {77 \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^{15/4} \sqrt [4]{b} \sqrt {d}}+\frac {77 \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^{15/4} \sqrt [4]{b} \sqrt {d}}+\frac {77 \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^{15/4} \sqrt [4]{b} \sqrt {d}}-\frac {77 \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^{15/4} \sqrt [4]{b} \sqrt {d}}\\ &=\frac {\sqrt {d x}}{6 a d \left (a+b x^2\right )^3}+\frac {11 \sqrt {d x}}{48 a^2 d \left (a+b x^2\right )^2}+\frac {77 \sqrt {d x}}{192 a^3 d \left (a+b x^2\right )}-\frac {77 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{128 \sqrt {2} a^{15/4} \sqrt [4]{b} \sqrt {d}}+\frac {77 \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{128 \sqrt {2} a^{15/4} \sqrt [4]{b} \sqrt {d}}-\frac {77 \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^{15/4} \sqrt [4]{b} \sqrt {d}}+\frac {77 \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^{15/4} \sqrt [4]{b} \sqrt {d}}\\ \end {align*}
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Mathematica [A]
time = 0.26, size = 161, normalized size = 0.48 \begin {gather*} \frac {\sqrt {x} \left (\frac {4 a^{3/4} \sqrt {x} \left (153 a^2+198 a b x^2+77 b^2 x^4\right )}{\left (a+b x^2\right )^3}-\frac {231 \sqrt {2} \tan ^{-1}\left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )}{\sqrt [4]{b}}+\frac {231 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{\sqrt [4]{b}}\right )}{768 a^{15/4} \sqrt {d x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 209, normalized size = 0.62
method | result | size |
derivativedivides | \(2 d^{7} \left (\frac {\frac {77 b^{2} \left (d x \right )^{\frac {9}{2}}}{384 a^{3} d^{6}}+\frac {33 b \left (d x \right )^{\frac {5}{2}}}{64 a^{2} d^{4}}+\frac {51 \sqrt {d x}}{128 a \,d^{2}}}{\left (d^{2} x^{2} b +a \,d^{2}\right )^{3}}+\frac {77 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{1024 a^{4} d^{8}}\right )\) | \(209\) |
default | \(2 d^{7} \left (\frac {\frac {77 b^{2} \left (d x \right )^{\frac {9}{2}}}{384 a^{3} d^{6}}+\frac {33 b \left (d x \right )^{\frac {5}{2}}}{64 a^{2} d^{4}}+\frac {51 \sqrt {d x}}{128 a \,d^{2}}}{\left (d^{2} x^{2} b +a \,d^{2}\right )^{3}}+\frac {77 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{1024 a^{4} d^{8}}\right )\) | \(209\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 322, normalized size = 0.96 \begin {gather*} \frac {\frac {8 \, {\left (77 \, \left (d x\right )^{\frac {9}{2}} b^{2} d^{2} + 198 \, \left (d x\right )^{\frac {5}{2}} a b d^{4} + 153 \, \sqrt {d x} a^{2} d^{6}\right )}}{a^{3} b^{3} d^{6} x^{6} + 3 \, a^{4} b^{2} d^{6} x^{4} + 3 \, a^{5} b d^{6} x^{2} + a^{6} d^{6}} + \frac {231 \, {\left (\frac {\sqrt {2} d^{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 {3}{4}} b^{\frac {1}{4}}} - \frac {\sqrt {2} d^{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 {3}{4}} b^{\frac {1}{4}}} + \frac {2 \, \sqrt {2} d \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 {a}} + \frac {2 \, \sqrt {2} d \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 {a}}\right )}}{a^{3}}}{1536 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 357, normalized size = 1.07 \begin {gather*} \frac {924 \, {\left (a^{3} b^{3} d x^{6} + 3 \, a^{4} b^{2} d x^{4} + 3 \, a^{5} b d x^{2} + a^{6} d\right )} \left (-\frac {1}{a^{15} b d^{2}}\right )^{\frac {1}{4}} \arctan \left (\sqrt {a^{8} d^{2} \sqrt {-\frac {1}{a^{15} b d^{2}}} + d x} a^{11} b d \left (-\frac {1}{a^{15} b d^{2}}\right )^{\frac {3}{4}} - \sqrt {d x} a^{11} b d \left (-\frac {1}{a^{15} b d^{2}}\right )^{\frac {3}{4}}\right ) + 231 \, {\left (a^{3} b^{3} d x^{6} + 3 \, a^{4} b^{2} d x^{4} + 3 \, a^{5} b d x^{2} + a^{6} d\right )} \left (-\frac {1}{a^{15} b d^{2}}\right )^{\frac {1}{4}} \log \left (a^{4} d \left (-\frac {1}{a^{15} b d^{2}}\right )^{\frac {1}{4}} + \sqrt {d x}\right ) - 231 \, {\left (a^{3} b^{3} d x^{6} + 3 \, a^{4} b^{2} d x^{4} + 3 \, a^{5} b d x^{2} + a^{6} d\right )} \left (-\frac {1}{a^{15} b d^{2}}\right )^{\frac {1}{4}} \log \left (-a^{4} d \left (-\frac {1}{a^{15} b d^{2}}\right )^{\frac {1}{4}} + \sqrt {d x}\right ) + 4 \, {\left (77 \, b^{2} x^{4} + 198 \, a b x^{2} + 153 \, a^{2}\right )} \sqrt {d x}}{768 \, {\left (a^{3} b^{3} d x^{6} + 3 \, a^{4} b^{2} d x^{4} + 3 \, a^{5} b d x^{2} + a^{6} d\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}{\sqrt {d x} \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 = 3.95, size = 308, normalized size = 0.92 \begin {gather*} \frac {77 \, \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 )}{256 \, a^{4} b d} + \frac {77 \, \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 )}{256 \, a^{4} b d} + \frac {77 \, \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 )}{512 \, a^{4} b d} - \frac {77 \, \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 )}{512 \, a^{4} b d} + \frac {77 \, \sqrt {d x} b^{2} d^{5} x^{4} + 198 \, \sqrt {d x} a b d^{5} x^{2} + 153 \, \sqrt {d x} a^{2} d^{5}}{192 \, {\left (b d^{2} x^{2} + a d^{2}\right )}^{3} a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.28, size = 150, normalized size = 0.45 \begin {gather*} \frac {\frac {51\,d^5\,\sqrt {d\,x}}{64\,a}+\frac {33\,b\,d^3\,{\left (d\,x\right )}^{5/2}}{32\,a^2}+\frac {77\,b^2\,d\,{\left (d\,x\right )}^{9/2}}{192\,a^3}}{a^3\,d^6+3\,a^2\,b\,d^6\,x^2+3\,a\,b^2\,d^6\,x^4+b^3\,d^6\,x^6}+\frac {77\,\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {d\,x}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )}{128\,{\left (-a\right )}^{15/4}\,b^{1/4}\,\sqrt {d}}+\frac {77\,\mathrm {atanh}\left (\frac {b^{1/4}\,\sqrt {d\,x}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )}{128\,{\left (-a\right )}^{15/4}\,b^{1/4}\,\sqrt {d}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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