Optimal. Leaf size=387 \[ \frac{663 \sqrt{d} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{16384 \sqrt{2} a^{21/4} b^{3/4}}-\frac{663 \sqrt{d} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{16384 \sqrt{2} a^{21/4} b^{3/4}}-\frac{663 \sqrt{d} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{8192 \sqrt{2} a^{21/4} b^{3/4}}+\frac{663 \sqrt{d} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{8192 \sqrt{2} a^{21/4} b^{3/4}}+\frac{663 (d x)^{3/2}}{4096 a^5 d \left (a+b x^2\right )}+\frac{663 (d x)^{3/2}}{5120 a^4 d \left (a+b x^2\right )^2}+\frac{221 (d x)^{3/2}}{1920 a^3 d \left (a+b x^2\right )^3}+\frac{17 (d x)^{3/2}}{160 a^2 d \left (a+b x^2\right )^4}+\frac{(d x)^{3/2}}{10 a d \left (a+b x^2\right )^5} \]
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Rubi [A] time = 0.490223, antiderivative size = 387, normalized size of antiderivative = 1., number of steps used = 16, 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} \[ \frac{663 \sqrt{d} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{16384 \sqrt{2} a^{21/4} b^{3/4}}-\frac{663 \sqrt{d} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{16384 \sqrt{2} a^{21/4} b^{3/4}}-\frac{663 \sqrt{d} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{8192 \sqrt{2} a^{21/4} b^{3/4}}+\frac{663 \sqrt{d} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{8192 \sqrt{2} a^{21/4} b^{3/4}}+\frac{663 (d x)^{3/2}}{4096 a^5 d \left (a+b x^2\right )}+\frac{663 (d x)^{3/2}}{5120 a^4 d \left (a+b x^2\right )^2}+\frac{221 (d x)^{3/2}}{1920 a^3 d \left (a+b x^2\right )^3}+\frac{17 (d x)^{3/2}}{160 a^2 d \left (a+b x^2\right )^4}+\frac{(d x)^{3/2}}{10 a d \left (a+b x^2\right )^5} \]
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}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx &=b^6 \int \frac{\sqrt{d x}}{\left (a b+b^2 x^2\right )^6} \, dx\\ &=\frac{(d x)^{3/2}}{10 a d \left (a+b x^2\right )^5}+\frac{\left (17 b^5\right ) \int \frac{\sqrt{d x}}{\left (a b+b^2 x^2\right )^5} \, dx}{20 a}\\ &=\frac{(d x)^{3/2}}{10 a d \left (a+b x^2\right )^5}+\frac{17 (d x)^{3/2}}{160 a^2 d \left (a+b x^2\right )^4}+\frac{\left (221 b^4\right ) \int \frac{\sqrt{d x}}{\left (a b+b^2 x^2\right )^4} \, dx}{320 a^2}\\ &=\frac{(d x)^{3/2}}{10 a d \left (a+b x^2\right )^5}+\frac{17 (d x)^{3/2}}{160 a^2 d \left (a+b x^2\right )^4}+\frac{221 (d x)^{3/2}}{1920 a^3 d \left (a+b x^2\right )^3}+\frac{\left (663 b^3\right ) \int \frac{\sqrt{d x}}{\left (a b+b^2 x^2\right )^3} \, dx}{1280 a^3}\\ &=\frac{(d x)^{3/2}}{10 a d \left (a+b x^2\right )^5}+\frac{17 (d x)^{3/2}}{160 a^2 d \left (a+b x^2\right )^4}+\frac{221 (d x)^{3/2}}{1920 a^3 d \left (a+b x^2\right )^3}+\frac{663 (d x)^{3/2}}{5120 a^4 d \left (a+b x^2\right )^2}+\frac{\left (663 b^2\right ) \int \frac{\sqrt{d x}}{\left (a b+b^2 x^2\right )^2} \, dx}{2048 a^4}\\ &=\frac{(d x)^{3/2}}{10 a d \left (a+b x^2\right )^5}+\frac{17 (d x)^{3/2}}{160 a^2 d \left (a+b x^2\right )^4}+\frac{221 (d x)^{3/2}}{1920 a^3 d \left (a+b x^2\right )^3}+\frac{663 (d x)^{3/2}}{5120 a^4 d \left (a+b x^2\right )^2}+\frac{663 (d x)^{3/2}}{4096 a^5 d \left (a+b x^2\right )}+\frac{(663 b) \int \frac{\sqrt{d x}}{a b+b^2 x^2} \, dx}{8192 a^5}\\ &=\frac{(d x)^{3/2}}{10 a d \left (a+b x^2\right )^5}+\frac{17 (d x)^{3/2}}{160 a^2 d \left (a+b x^2\right )^4}+\frac{221 (d x)^{3/2}}{1920 a^3 d \left (a+b x^2\right )^3}+\frac{663 (d x)^{3/2}}{5120 a^4 d \left (a+b x^2\right )^2}+\frac{663 (d x)^{3/2}}{4096 a^5 d \left (a+b x^2\right )}+\frac{(663 b) \operatorname{Subst}\left (\int \frac{x^2}{a b+\frac{b^2 x^4}{d^2}} \, dx,x,\sqrt{d x}\right )}{4096 a^5 d}\\ &=\frac{(d x)^{3/2}}{10 a d \left (a+b x^2\right )^5}+\frac{17 (d x)^{3/2}}{160 a^2 d \left (a+b x^2\right )^4}+\frac{221 (d x)^{3/2}}{1920 a^3 d \left (a+b x^2\right )^3}+\frac{663 (d x)^{3/2}}{5120 a^4 d \left (a+b x^2\right )^2}+\frac{663 (d x)^{3/2}}{4096 a^5 d \left (a+b x^2\right )}-\frac{\left (663 \sqrt{b}\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 )}{8192 a^5 d}+\frac{\left (663 \sqrt{b}\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 )}{8192 a^5 d}\\ &=\frac{(d x)^{3/2}}{10 a d \left (a+b x^2\right )^5}+\frac{17 (d x)^{3/2}}{160 a^2 d \left (a+b x^2\right )^4}+\frac{221 (d x)^{3/2}}{1920 a^3 d \left (a+b x^2\right )^3}+\frac{663 (d x)^{3/2}}{5120 a^4 d \left (a+b x^2\right )^2}+\frac{663 (d x)^{3/2}}{4096 a^5 d \left (a+b x^2\right )}+\frac{\left (663 \sqrt{d}\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 )}{16384 \sqrt{2} a^{21/4} b^{3/4}}+\frac{\left (663 \sqrt{d}\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 )}{16384 \sqrt{2} a^{21/4} b^{3/4}}+\frac{(663 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 )}{16384 a^5 b}+\frac{(663 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 )}{16384 a^5 b}\\ &=\frac{(d x)^{3/2}}{10 a d \left (a+b x^2\right )^5}+\frac{17 (d x)^{3/2}}{160 a^2 d \left (a+b x^2\right )^4}+\frac{221 (d x)^{3/2}}{1920 a^3 d \left (a+b x^2\right )^3}+\frac{663 (d x)^{3/2}}{5120 a^4 d \left (a+b x^2\right )^2}+\frac{663 (d x)^{3/2}}{4096 a^5 d \left (a+b x^2\right )}+\frac{663 \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 )}{16384 \sqrt{2} a^{21/4} b^{3/4}}-\frac{663 \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 )}{16384 \sqrt{2} a^{21/4} b^{3/4}}+\frac{\left (663 \sqrt{d}\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 )}{8192 \sqrt{2} a^{21/4} b^{3/4}}-\frac{\left (663 \sqrt{d}\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 )}{8192 \sqrt{2} a^{21/4} b^{3/4}}\\ &=\frac{(d x)^{3/2}}{10 a d \left (a+b x^2\right )^5}+\frac{17 (d x)^{3/2}}{160 a^2 d \left (a+b x^2\right )^4}+\frac{221 (d x)^{3/2}}{1920 a^3 d \left (a+b x^2\right )^3}+\frac{663 (d x)^{3/2}}{5120 a^4 d \left (a+b x^2\right )^2}+\frac{663 (d x)^{3/2}}{4096 a^5 d \left (a+b x^2\right )}-\frac{663 \sqrt{d} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{8192 \sqrt{2} a^{21/4} b^{3/4}}+\frac{663 \sqrt{d} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{8192 \sqrt{2} a^{21/4} b^{3/4}}+\frac{663 \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 )}{16384 \sqrt{2} a^{21/4} b^{3/4}}-\frac{663 \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 )}{16384 \sqrt{2} a^{21/4} b^{3/4}}\\ \end{align*}
Mathematica [C] time = 0.0062739, size = 32, normalized size = 0.08 \[ \frac{2 x \sqrt{d x} \, _2F_1\left (\frac{3}{4},6;\frac{7}{4};-\frac{b x^2}{a}\right )}{3 a^6} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.067, size = 336, normalized size = 0.9 \begin{align*}{\frac{7529\,{d}^{9}}{12288\, \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{5}a} \left ( dx \right ) ^{{\frac{3}{2}}}}+{\frac{527\,{d}^{7}b}{384\, \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{5}{a}^{2}} \left ( dx \right ) ^{{\frac{7}{2}}}}+{\frac{9061\,{d}^{5}{b}^{2}}{6144\, \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{5}{a}^{3}} \left ( dx \right ) ^{{\frac{11}{2}}}}+{\frac{1989\,{d}^{3}{b}^{3}}{2560\, \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{5}{a}^{4}} \left ( dx \right ) ^{{\frac{15}{2}}}}+{\frac{663\,{b}^{4}d}{4096\, \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{5}{a}^{5}} \left ( dx \right ) ^{{\frac{19}{2}}}}+{\frac{663\,d\sqrt{2}}{32768\,{a}^{5}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{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}}+{\frac{663\,d\sqrt{2}}{16384\,{a}^{5}b}\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{663\,d\sqrt{2}}{16384\,{a}^{5}b}\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 = 1.47173, size = 1199, normalized size = 3.1 \begin{align*} -\frac{39780 \,{\left (a^{5} b^{5} x^{10} + 5 \, a^{6} b^{4} x^{8} + 10 \, a^{7} b^{3} x^{6} + 10 \, a^{8} b^{2} x^{4} + 5 \, a^{9} b x^{2} + a^{10}\right )} \left (-\frac{d^{2}}{a^{21} b^{3}}\right )^{\frac{1}{4}} \arctan \left (-\frac{291434247 \, \sqrt{d x} a^{5} b d \left (-\frac{d^{2}}{a^{21} b^{3}}\right )^{\frac{1}{4}} - \sqrt{-84933920324457009 \, a^{11} b d^{2} \sqrt{-\frac{d^{2}}{a^{21} b^{3}}} + 84933920324457009 \, d^{3} x} a^{5} b \left (-\frac{d^{2}}{a^{21} b^{3}}\right )^{\frac{1}{4}}}{291434247 \, d^{2}}\right ) - 9945 \,{\left (a^{5} b^{5} x^{10} + 5 \, a^{6} b^{4} x^{8} + 10 \, a^{7} b^{3} x^{6} + 10 \, a^{8} b^{2} x^{4} + 5 \, a^{9} b x^{2} + a^{10}\right )} \left (-\frac{d^{2}}{a^{21} b^{3}}\right )^{\frac{1}{4}} \log \left (291434247 \, a^{16} b^{2} \left (-\frac{d^{2}}{a^{21} b^{3}}\right )^{\frac{3}{4}} + 291434247 \, \sqrt{d x} d\right ) + 9945 \,{\left (a^{5} b^{5} x^{10} + 5 \, a^{6} b^{4} x^{8} + 10 \, a^{7} b^{3} x^{6} + 10 \, a^{8} b^{2} x^{4} + 5 \, a^{9} b x^{2} + a^{10}\right )} \left (-\frac{d^{2}}{a^{21} b^{3}}\right )^{\frac{1}{4}} \log \left (-291434247 \, a^{16} b^{2} \left (-\frac{d^{2}}{a^{21} b^{3}}\right )^{\frac{3}{4}} + 291434247 \, \sqrt{d x} d\right ) - 4 \,{\left (9945 \, b^{4} x^{9} + 47736 \, a b^{3} x^{7} + 90610 \, a^{2} b^{2} x^{5} + 84320 \, a^{3} b x^{3} + 37645 \, a^{4} x\right )} \sqrt{d x}}{245760 \,{\left (a^{5} b^{5} x^{10} + 5 \, a^{6} b^{4} x^{8} + 10 \, a^{7} b^{3} x^{6} + 10 \, a^{8} b^{2} x^{4} + 5 \, a^{9} b x^{2} + a^{10}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.2259, size = 468, normalized size = 1.21 \begin{align*} \frac{663 \, \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 )}{16384 \, a^{6} b^{3} d} + \frac{663 \, \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 )}{16384 \, a^{6} b^{3} d} - \frac{663 \, \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 )}{32768 \, a^{6} b^{3} d} + \frac{663 \, \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 )}{32768 \, a^{6} b^{3} d} + \frac{9945 \, \sqrt{d x} b^{4} d^{10} x^{9} + 47736 \, \sqrt{d x} a b^{3} d^{10} x^{7} + 90610 \, \sqrt{d x} a^{2} b^{2} d^{10} x^{5} + 84320 \, \sqrt{d x} a^{3} b d^{10} x^{3} + 37645 \, \sqrt{d x} a^{4} d^{10} x}{61440 \,{\left (b d^{2} x^{2} + a d^{2}\right )}^{5} a^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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