Optimal. Leaf size=283 \[ \frac{\sqrt{d} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{8 \sqrt{2} a^{5/4} b^{3/4}}-\frac{\sqrt{d} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{8 \sqrt{2} a^{5/4} b^{3/4}}-\frac{\sqrt{d} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{4 \sqrt{2} a^{5/4} b^{3/4}}+\frac{\sqrt{d} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{4 \sqrt{2} a^{5/4} b^{3/4}}+\frac{(d x)^{3/2}}{2 a d \left (a+b x^2\right )} \]
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Rubi [A] time = 0.271493, antiderivative size = 283, normalized size of antiderivative = 1., 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, 297, 1162, 617, 204, 1165, 628} \[ \frac{\sqrt{d} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{8 \sqrt{2} a^{5/4} b^{3/4}}-\frac{\sqrt{d} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{8 \sqrt{2} a^{5/4} b^{3/4}}-\frac{\sqrt{d} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{4 \sqrt{2} a^{5/4} b^{3/4}}+\frac{\sqrt{d} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{4 \sqrt{2} a^{5/4} b^{3/4}}+\frac{(d x)^{3/2}}{2 a d \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
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Rule 28
Rule 290
Rule 329
Rule 297
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
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 \operatorname{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} \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 d}+\frac{\sqrt{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 d}\\ &=\frac{(d x)^{3/2}}{2 a d \left (a+b x^2\right )}+\frac{\sqrt{d} \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^{5/4} b^{3/4}}+\frac{\sqrt{d} \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^{5/4} b^{3/4}}+\frac{d \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 b}+\frac{d \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 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} \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^{5/4} b^{3/4}}-\frac{\sqrt{d} \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^{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*}
Mathematica [C] time = 0.0059782, size = 32, normalized size = 0.11 \[ \frac{2 x \sqrt{d x} \, _2F_1\left (\frac{3}{4},2;\frac{7}{4};-\frac{b x^2}{a}\right )}{3 a^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.056, size = 210, normalized size = 0.7 \begin{align*}{\frac{d}{2\,a \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) } \left ( dx \right ) ^{{\frac{3}{2}}}}+{\frac{d\sqrt{2}}{16\,ab}\ln \left ({ \left ( dx-\sqrt [4]{{\frac{a{d}^{2}}{b}}}\sqrt{dx}\sqrt{2}+\sqrt{{\frac{a{d}^{2}}{b}}} \right ) \left ( dx+\sqrt [4]{{\frac{a{d}^{2}}{b}}}\sqrt{dx}\sqrt{2}+\sqrt{{\frac{a{d}^{2}}{b}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}}+{\frac{d\sqrt{2}}{8\,ab}\arctan \left ({\sqrt{2}\sqrt{dx}{\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}}+{\frac{d\sqrt{2}}{8\,ab}\arctan \left ({\sqrt{2}\sqrt{dx}{\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.38718, size = 531, normalized size = 1.88 \begin{align*} -\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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 6.39636, size = 78, normalized size = 0.28 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.25684, size = 367, normalized size = 1.3 \begin{align*} \frac{\sqrt{d x} d^{2} x}{2 \,{\left (b d^{2} x^{2} + a d^{2}\right )} a} + \frac{\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 )}{8 \, a^{2} b^{3} d} + \frac{\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 )}{8 \, a^{2} b^{3} d} - \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 )}{16 \, a^{2} b^{3} d} + \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 )}{16 \, a^{2} b^{3} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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