Optimal. Leaf size=281 \[ -\frac{d^{3/2} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{8 \sqrt{2} a^{3/4} b^{5/4}}+\frac{d^{3/2} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{8 \sqrt{2} a^{3/4} b^{5/4}}-\frac{d^{3/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{4 \sqrt{2} a^{3/4} b^{5/4}}+\frac{d^{3/2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{4 \sqrt{2} a^{3/4} b^{5/4}}-\frac{d \sqrt{d x}}{2 b \left (a+b x^2\right )} \]
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Rubi [A] time = 0.260389, antiderivative size = 281, 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, 288, 329, 211, 1165, 628, 1162, 617, 204} \[ -\frac{d^{3/2} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{8 \sqrt{2} a^{3/4} b^{5/4}}+\frac{d^{3/2} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{8 \sqrt{2} a^{3/4} b^{5/4}}-\frac{d^{3/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{4 \sqrt{2} a^{3/4} b^{5/4}}+\frac{d^{3/2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{4 \sqrt{2} a^{3/4} b^{5/4}}-\frac{d \sqrt{d x}}{2 b \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
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Rule 28
Rule 288
Rule 329
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
Rubi steps
\begin{align*} \int \frac{(d x)^{3/2}}{a^2+2 a b x^2+b^2 x^4} \, dx &=b^2 \int \frac{(d x)^{3/2}}{\left (a b+b^2 x^2\right )^2} \, dx\\ &=-\frac{d \sqrt{d x}}{2 b \left (a+b x^2\right )}+\frac{1}{4} d^2 \int \frac{1}{\sqrt{d x} \left (a b+b^2 x^2\right )} \, dx\\ &=-\frac{d \sqrt{d x}}{2 b \left (a+b x^2\right )}+\frac{1}{2} d \operatorname{Subst}\left (\int \frac{1}{a b+\frac{b^2 x^4}{d^2}} \, dx,x,\sqrt{d x}\right )\\ &=-\frac{d \sqrt{d x}}{2 b \left (a+b x^2\right )}+\frac{\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 \sqrt{a}}+\frac{\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 \sqrt{a}}\\ &=-\frac{d \sqrt{d x}}{2 b \left (a+b x^2\right )}-\frac{d^{3/2} \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^{3/4} b^{5/4}}-\frac{d^{3/2} \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^{3/4} b^{5/4}}+\frac{d^2 \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 \sqrt{a} b^{3/2}}+\frac{d^2 \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 \sqrt{a} b^{3/2}}\\ &=-\frac{d \sqrt{d x}}{2 b \left (a+b x^2\right )}-\frac{d^{3/2} \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^{3/4} b^{5/4}}+\frac{d^{3/2} \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^{3/4} b^{5/4}}+\frac{d^{3/2} \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^{3/4} b^{5/4}}-\frac{d^{3/2} \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^{3/4} b^{5/4}}\\ &=-\frac{d \sqrt{d x}}{2 b \left (a+b x^2\right )}-\frac{d^{3/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{4 \sqrt{2} a^{3/4} b^{5/4}}+\frac{d^{3/2} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{4 \sqrt{2} a^{3/4} b^{5/4}}-\frac{d^{3/2} \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^{3/4} b^{5/4}}+\frac{d^{3/2} \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^{3/4} b^{5/4}}\\ \end{align*}
Mathematica [A] time = 0.166154, size = 210, normalized size = 0.75 \[ \frac{(d x)^{3/2} \left (-\frac{\sqrt{2} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{a^{3/4}}+\frac{\sqrt{2} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{a^{3/4}}-\frac{2 \sqrt{2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{a^{3/4}}+\frac{2 \sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{a^{3/4}}-\frac{8 \sqrt [4]{b} \sqrt{x}}{a+b x^2}\right )}{16 b^{5/4} x^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.061, size = 212, normalized size = 0.8 \begin{align*} -{\frac{{d}^{3}}{2\,b \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) }\sqrt{dx}}+{\frac{d\sqrt{2}}{16\,ab}\sqrt [4]{{\frac{a{d}^{2}}{b}}}\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{d\sqrt{2}}{8\,ab}\sqrt [4]{{\frac{a{d}^{2}}{b}}}\arctan \left ({\sqrt{2}\sqrt{dx}{\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}}+1 \right ) }+{\frac{d\sqrt{2}}{8\,ab}\sqrt [4]{{\frac{a{d}^{2}}{b}}}\arctan \left ({\sqrt{2}\sqrt{dx}{\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}}-1 \right ) } \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 = 1.37143, size = 525, normalized size = 1.87 \begin{align*} \frac{4 \,{\left (b^{2} x^{2} + a b\right )} \left (-\frac{d^{6}}{a^{3} b^{5}}\right )^{\frac{1}{4}} \arctan \left (-\frac{\sqrt{d x} a^{2} b^{4} d \left (-\frac{d^{6}}{a^{3} b^{5}}\right )^{\frac{3}{4}} - \sqrt{a^{2} b^{2} \sqrt{-\frac{d^{6}}{a^{3} b^{5}}} + d^{3} x} a^{2} b^{4} \left (-\frac{d^{6}}{a^{3} b^{5}}\right )^{\frac{3}{4}}}{d^{6}}\right ) +{\left (b^{2} x^{2} + a b\right )} \left (-\frac{d^{6}}{a^{3} b^{5}}\right )^{\frac{1}{4}} \log \left (a b \left (-\frac{d^{6}}{a^{3} b^{5}}\right )^{\frac{1}{4}} + \sqrt{d x} d\right ) -{\left (b^{2} x^{2} + a b\right )} \left (-\frac{d^{6}}{a^{3} b^{5}}\right )^{\frac{1}{4}} \log \left (-a b \left (-\frac{d^{6}}{a^{3} b^{5}}\right )^{\frac{1}{4}} + \sqrt{d x} d\right ) - 4 \, \sqrt{d x} d}{8 \,{\left (b^{2} x^{2} + a b\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (d x\right )^{\frac{3}{2}}}{\left (a + b x^{2}\right )^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20026, size = 355, normalized size = 1.26 \begin{align*} -\frac{\sqrt{d x} d^{3}}{2 \,{\left (b d^{2} x^{2} + a d^{2}\right )} b} + \frac{\sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{4}} d \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 b^{2}} + \frac{\sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{4}} d \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 b^{2}} + \frac{\sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{4}} d \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 b^{2}} - \frac{\sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{4}} d \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 b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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