Optimal. Leaf size=283 \[ -\frac {3 \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{8 \sqrt {2} a^{7/4} \sqrt [4]{b} \sqrt {d}}+\frac {3 \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{8 \sqrt {2} a^{7/4} \sqrt [4]{b} \sqrt {d}}-\frac {3 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{4 \sqrt {2} a^{7/4} \sqrt [4]{b} \sqrt {d}}+\frac {3 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{4 \sqrt {2} a^{7/4} \sqrt [4]{b} \sqrt {d}}+\frac {\sqrt {d x}}{2 a d \left (a+b x^2\right )} \]
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Rubi [A] time = 0.26, 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, 290, 329, 211, 1165, 628, 1162, 617, 204} \[ -\frac {3 \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{8 \sqrt {2} a^{7/4} \sqrt [4]{b} \sqrt {d}}+\frac {3 \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{8 \sqrt {2} a^{7/4} \sqrt [4]{b} \sqrt {d}}-\frac {3 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{4 \sqrt {2} a^{7/4} \sqrt [4]{b} \sqrt {d}}+\frac {3 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{4 \sqrt {2} a^{7/4} \sqrt [4]{b} \sqrt {d}}+\frac {\sqrt {d x}}{2 a d \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
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Rule 28
Rule 204
Rule 211
Rule 290
Rule 329
Rule 617
Rule 628
Rule 1162
Rule 1165
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {d x} \left (a^2+2 a b x^2+b^2 x^4\right )} \, dx &=b^2 \int \frac {1}{\sqrt {d x} \left (a b+b^2 x^2\right )^2} \, dx\\ &=\frac {\sqrt {d x}}{2 a d \left (a+b x^2\right )}+\frac {(3 b) \int \frac {1}{\sqrt {d x} \left (a b+b^2 x^2\right )} \, dx}{4 a}\\ &=\frac {\sqrt {d x}}{2 a d \left (a+b x^2\right )}+\frac {(3 b) \operatorname {Subst}\left (\int \frac {1}{a b+\frac {b^2 x^4}{d^2}} \, dx,x,\sqrt {d x}\right )}{2 a d}\\ &=\frac {\sqrt {d x}}{2 a d \left (a+b x^2\right )}+\frac {(3 b) \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 )}{4 a^{3/2} d^2}+\frac {(3 b) \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 )}{4 a^{3/2} d^2}\\ &=\frac {\sqrt {d x}}{2 a d \left (a+b x^2\right )}+\frac {3 \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 )}{8 a^{3/2} \sqrt {b}}+\frac {3 \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 )}{8 a^{3/2} \sqrt {b}}-\frac {3 \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 )}{8 \sqrt {2} a^{7/4} \sqrt [4]{b} \sqrt {d}}-\frac {3 \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 )}{8 \sqrt {2} a^{7/4} \sqrt [4]{b} \sqrt {d}}\\ &=\frac {\sqrt {d x}}{2 a d \left (a+b x^2\right )}-\frac {3 \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^{7/4} \sqrt [4]{b} \sqrt {d}}+\frac {3 \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^{7/4} \sqrt [4]{b} \sqrt {d}}+\frac {3 \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 )}{4 \sqrt {2} a^{7/4} \sqrt [4]{b} \sqrt {d}}-\frac {3 \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 )}{4 \sqrt {2} a^{7/4} \sqrt [4]{b} \sqrt {d}}\\ &=\frac {\sqrt {d x}}{2 a d \left (a+b x^2\right )}-\frac {3 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{4 \sqrt {2} a^{7/4} \sqrt [4]{b} \sqrt {d}}+\frac {3 \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{4 \sqrt {2} a^{7/4} \sqrt [4]{b} \sqrt {d}}-\frac {3 \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^{7/4} \sqrt [4]{b} \sqrt {d}}+\frac {3 \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^{7/4} \sqrt [4]{b} \sqrt {d}}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 211, normalized size = 0.75 \[ \frac {\sqrt {x} \left (\frac {8 a^{3/4} \sqrt {x}}{a+b x^2}-\frac {3 \sqrt {2} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{\sqrt [4]{b}}+\frac {3 \sqrt {2} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{\sqrt [4]{b}}-\frac {6 \sqrt {2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt [4]{b}}+\frac {6 \sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt [4]{b}}\right )}{16 a^{7/4} \sqrt {d x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.00, size = 232, normalized size = 0.82 \[ \frac {12 \, {\left (a b d x^{2} + a^{2} d\right )} \left (-\frac {1}{a^{7} b d^{2}}\right )^{\frac {1}{4}} \arctan \left (\sqrt {a^{4} d^{2} \sqrt {-\frac {1}{a^{7} b d^{2}}} + d x} a^{5} b d \left (-\frac {1}{a^{7} b d^{2}}\right )^{\frac {3}{4}} - \sqrt {d x} a^{5} b d \left (-\frac {1}{a^{7} b d^{2}}\right )^{\frac {3}{4}}\right ) + 3 \, {\left (a b d x^{2} + a^{2} d\right )} \left (-\frac {1}{a^{7} b d^{2}}\right )^{\frac {1}{4}} \log \left (a^{2} d \left (-\frac {1}{a^{7} b d^{2}}\right )^{\frac {1}{4}} + \sqrt {d x}\right ) - 3 \, {\left (a b d x^{2} + a^{2} d\right )} \left (-\frac {1}{a^{7} b d^{2}}\right )^{\frac {1}{4}} \log \left (-a^{2} d \left (-\frac {1}{a^{7} b d^{2}}\right )^{\frac {1}{4}} + \sqrt {d x}\right ) + 4 \, \sqrt {d x}}{8 \, {\left (a b d x^{2} + a^{2} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 269, normalized size = 0.95 \[ \frac {\sqrt {d x} d}{2 \, {\left (b d^{2} x^{2} + a d^{2}\right )} a} + \frac {3 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{8 \, a^{2} b d} + \frac {3 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{8 \, a^{2} b d} + \frac {3 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \log \left (d x + \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{16 \, a^{2} b d} - \frac {3 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \log \left (d x - \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{16 \, a^{2} b d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 207, normalized size = 0.73 \[ \frac {\sqrt {d x}\, d}{2 \left (b \,d^{2} x^{2}+d^{2} a \right ) a}+\frac {3 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )}{8 a^{2} d}+\frac {3 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )}{8 a^{2} d}+\frac {3 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \ln \left (\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )}{16 a^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.00, size = 261, normalized size = 0.92 \[ \frac {\frac {8 \, \sqrt {d x} d^{2}}{a b d^{2} x^{2} + a^{2} d^{2}} + \frac {3 \, {\left (\frac {\sqrt {2} d^{2} \log \left (\sqrt {b} d x + \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {3}{4}} b^{\frac {1}{4}}} - \frac {\sqrt {2} d^{2} \log \left (\sqrt {b} d x - \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {3}{4}} b^{\frac {1}{4}}} + \frac {2 \, \sqrt {2} d \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {a}} + \frac {2 \, \sqrt {2} d \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {a}}\right )}}{a}}{16 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.10, size = 90, normalized size = 0.32 \[ \frac {3\,\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {d\,x}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )}{4\,{\left (-a\right )}^{7/4}\,b^{1/4}\,\sqrt {d}}+\frac {3\,\mathrm {atanh}\left (\frac {b^{1/4}\,\sqrt {d\,x}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )}{4\,{\left (-a\right )}^{7/4}\,b^{1/4}\,\sqrt {d}}+\frac {d\,\sqrt {d\,x}}{2\,a\,\left (b\,d^2\,x^2+a\,d^2\right )} \]
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 \[ \int \frac {1}{\sqrt {d x} \left (a + b x^{2}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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