Optimal. Leaf size=368 \[ \frac {\sqrt {d} \left (a+b x^2\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{2 \sqrt {2} \sqrt [4]{a} b^{3/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\sqrt {d} \left (a+b x^2\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{2 \sqrt {2} \sqrt [4]{a} b^{3/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\sqrt {d} \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} b^{3/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\sqrt {d} \left (a+b x^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} b^{3/4} \sqrt {a^2+2 a b x^2+b^2 x^4}} \]
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Rubi [A] time = 0.25, antiderivative size = 368, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {1112, 329, 297, 1162, 617, 204, 1165, 628} \[ \frac {\sqrt {d} \left (a+b x^2\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{2 \sqrt {2} \sqrt [4]{a} b^{3/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\sqrt {d} \left (a+b x^2\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{2 \sqrt {2} \sqrt [4]{a} b^{3/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\sqrt {d} \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} b^{3/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\sqrt {d} \left (a+b x^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} b^{3/4} \sqrt {a^2+2 a b x^2+b^2 x^4}} \]
Antiderivative was successfully verified.
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Rule 204
Rule 297
Rule 329
Rule 617
Rule 628
Rule 1112
Rule 1162
Rule 1165
Rubi steps
\begin {align*} \int \frac {\sqrt {d x}}{\sqrt {a^2+2 a b x^2+b^2 x^4}} \, dx &=\frac {\left (a b+b^2 x^2\right ) \int \frac {\sqrt {d x}}{a b+b^2 x^2} \, dx}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {\left (2 \left (a b+b^2 x^2\right )\right ) \operatorname {Subst}\left (\int \frac {x^2}{a b+\frac {b^2 x^4}{d^2}} \, dx,x,\sqrt {d x}\right )}{d \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac {\left (a b+b^2 x^2\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 )}{\sqrt {b} d \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (a b+b^2 x^2\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 )}{\sqrt {b} d \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {\left (\sqrt {d} \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 )}{2 \sqrt {2} \sqrt [4]{a} b^{7/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (\sqrt {d} \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 )}{2 \sqrt {2} \sqrt [4]{a} b^{7/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (d \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 )}{2 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (d \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 )}{2 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {\sqrt {d} \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 )}{2 \sqrt {2} \sqrt [4]{a} b^{3/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\sqrt {d} \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 )}{2 \sqrt {2} \sqrt [4]{a} b^{3/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (\sqrt {d} \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 )}{\sqrt {2} \sqrt [4]{a} b^{7/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (\sqrt {d} \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 )}{\sqrt {2} \sqrt [4]{a} b^{7/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac {\sqrt {d} \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 )}{\sqrt {2} \sqrt [4]{a} b^{3/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\sqrt {d} \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 )}{\sqrt {2} \sqrt [4]{a} b^{3/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\sqrt {d} \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 )}{2 \sqrt {2} \sqrt [4]{a} b^{3/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\sqrt {d} \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 )}{2 \sqrt {2} \sqrt [4]{a} b^{3/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 85, normalized size = 0.23 \[ \frac {\sqrt {d x} \left (a+b x^2\right ) \left (\tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{-a}}\right )+\tanh ^{-1}\left (\frac {a \sqrt [4]{b} \sqrt {x}}{(-a)^{5/4}}\right )\right )}{\sqrt [4]{-a} b^{3/4} \sqrt {x} \sqrt {\left (a+b x^2\right )^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.83, size = 173, normalized size = 0.47 \[ -2 \, \left (-\frac {d^{2}}{a b^{3}}\right )^{\frac {1}{4}} \arctan \left (-\frac {\sqrt {d x} b d \left (-\frac {d^{2}}{a b^{3}}\right )^{\frac {1}{4}} - \sqrt {-a b d^{2} \sqrt {-\frac {d^{2}}{a b^{3}}} + d^{3} x} b \left (-\frac {d^{2}}{a b^{3}}\right )^{\frac {1}{4}}}{d^{2}}\right ) + \frac {1}{2} \, \left (-\frac {d^{2}}{a b^{3}}\right )^{\frac {1}{4}} \log \left (a b^{2} \left (-\frac {d^{2}}{a b^{3}}\right )^{\frac {3}{4}} + \sqrt {d x} d\right ) - \frac {1}{2} \, \left (-\frac {d^{2}}{a b^{3}}\right )^{\frac {1}{4}} \log \left (-a b^{2} \left (-\frac {d^{2}}{a b^{3}}\right )^{\frac {3}{4}} + \sqrt {d x} d\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 242, normalized size = 0.66 \[ \frac {{\left (\frac {2 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a b^{3}} + \frac {2 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a b^{3}} - \frac {\sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \log \left (d x + \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{a b^{3}} + \frac {\sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \log \left (d x - \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{a b^{3}}\right )} \mathrm {sgn}\left (b x^{2} + a\right )}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 183, normalized size = 0.50 \[ \frac {\left (b \,x^{2}+a \right ) \sqrt {2}\, \left (2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}-\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}+\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )+\ln \left (-\frac {-d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}-\sqrt {\frac {a \,d^{2}}{b}}}{d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )\right ) d}{4 \sqrt {\left (b \,x^{2}+a \right )^{2}}\, \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.00, size = 216, normalized size = 0.59 \[ \frac {1}{4} \, d {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {b}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {b}} - \frac {\sqrt {2} \log \left (\sqrt {b} d x + \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (\sqrt {b} d x - \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {3}{4}}}\right )} \]
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 \[ \int \frac {\sqrt {d\,x}}{\sqrt {{\left (b\,x^2+a\right )}^2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 57.26, size = 41, normalized size = 0.11 \[ 2 d \operatorname {RootSum} {\left (256 t^{4} a b^{3} d^{2} + 1, \left (t \mapsto t \log {\left (64 t^{3} a b^{2} d^{2} + \sqrt {d x} \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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