Optimal. Leaf size=387 \[ \frac{1463 \sqrt{d x}}{4096 a^5 d \left (a+b x^2\right )}+\frac{209 \sqrt{d x}}{1024 a^4 d \left (a+b x^2\right )^2}+\frac{19 \sqrt{d x}}{128 a^3 d \left (a+b x^2\right )^3}+\frac{19 \sqrt{d x}}{160 a^2 d \left (a+b x^2\right )^4}-\frac{4389 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{16384 \sqrt{2} a^{23/4} \sqrt [4]{b} \sqrt{d}}+\frac{4389 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{16384 \sqrt{2} a^{23/4} \sqrt [4]{b} \sqrt{d}}-\frac{4389 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{8192 \sqrt{2} a^{23/4} \sqrt [4]{b} \sqrt{d}}+\frac{4389 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{8192 \sqrt{2} a^{23/4} \sqrt [4]{b} \sqrt{d}}+\frac{\sqrt{d x}}{10 a d \left (a+b x^2\right )^5} \]
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Rubi [A] time = 0.495858, 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, 211, 1165, 628, 1162, 617, 204} \[ \frac{1463 \sqrt{d x}}{4096 a^5 d \left (a+b x^2\right )}+\frac{209 \sqrt{d x}}{1024 a^4 d \left (a+b x^2\right )^2}+\frac{19 \sqrt{d x}}{128 a^3 d \left (a+b x^2\right )^3}+\frac{19 \sqrt{d x}}{160 a^2 d \left (a+b x^2\right )^4}-\frac{4389 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{16384 \sqrt{2} a^{23/4} \sqrt [4]{b} \sqrt{d}}+\frac{4389 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{16384 \sqrt{2} a^{23/4} \sqrt [4]{b} \sqrt{d}}-\frac{4389 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{8192 \sqrt{2} a^{23/4} \sqrt [4]{b} \sqrt{d}}+\frac{4389 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{8192 \sqrt{2} a^{23/4} \sqrt [4]{b} \sqrt{d}}+\frac{\sqrt{d x}}{10 a d \left (a+b x^2\right )^5} \]
Antiderivative was successfully verified.
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Rule 28
Rule 290
Rule 329
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{d x} \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx &=b^6 \int \frac{1}{\sqrt{d x} \left (a b+b^2 x^2\right )^6} \, dx\\ &=\frac{\sqrt{d x}}{10 a d \left (a+b x^2\right )^5}+\frac{\left (19 b^5\right ) \int \frac{1}{\sqrt{d x} \left (a b+b^2 x^2\right )^5} \, dx}{20 a}\\ &=\frac{\sqrt{d x}}{10 a d \left (a+b x^2\right )^5}+\frac{19 \sqrt{d x}}{160 a^2 d \left (a+b x^2\right )^4}+\frac{\left (57 b^4\right ) \int \frac{1}{\sqrt{d x} \left (a b+b^2 x^2\right )^4} \, dx}{64 a^2}\\ &=\frac{\sqrt{d x}}{10 a d \left (a+b x^2\right )^5}+\frac{19 \sqrt{d x}}{160 a^2 d \left (a+b x^2\right )^4}+\frac{19 \sqrt{d x}}{128 a^3 d \left (a+b x^2\right )^3}+\frac{\left (209 b^3\right ) \int \frac{1}{\sqrt{d x} \left (a b+b^2 x^2\right )^3} \, dx}{256 a^3}\\ &=\frac{\sqrt{d x}}{10 a d \left (a+b x^2\right )^5}+\frac{19 \sqrt{d x}}{160 a^2 d \left (a+b x^2\right )^4}+\frac{19 \sqrt{d x}}{128 a^3 d \left (a+b x^2\right )^3}+\frac{209 \sqrt{d x}}{1024 a^4 d \left (a+b x^2\right )^2}+\frac{\left (1463 b^2\right ) \int \frac{1}{\sqrt{d x} \left (a b+b^2 x^2\right )^2} \, dx}{2048 a^4}\\ &=\frac{\sqrt{d x}}{10 a d \left (a+b x^2\right )^5}+\frac{19 \sqrt{d x}}{160 a^2 d \left (a+b x^2\right )^4}+\frac{19 \sqrt{d x}}{128 a^3 d \left (a+b x^2\right )^3}+\frac{209 \sqrt{d x}}{1024 a^4 d \left (a+b x^2\right )^2}+\frac{1463 \sqrt{d x}}{4096 a^5 d \left (a+b x^2\right )}+\frac{(4389 b) \int \frac{1}{\sqrt{d x} \left (a b+b^2 x^2\right )} \, dx}{8192 a^5}\\ &=\frac{\sqrt{d x}}{10 a d \left (a+b x^2\right )^5}+\frac{19 \sqrt{d x}}{160 a^2 d \left (a+b x^2\right )^4}+\frac{19 \sqrt{d x}}{128 a^3 d \left (a+b x^2\right )^3}+\frac{209 \sqrt{d x}}{1024 a^4 d \left (a+b x^2\right )^2}+\frac{1463 \sqrt{d x}}{4096 a^5 d \left (a+b x^2\right )}+\frac{(4389 b) \operatorname{Subst}\left (\int \frac{1}{a b+\frac{b^2 x^4}{d^2}} \, dx,x,\sqrt{d x}\right )}{4096 a^5 d}\\ &=\frac{\sqrt{d x}}{10 a d \left (a+b x^2\right )^5}+\frac{19 \sqrt{d x}}{160 a^2 d \left (a+b x^2\right )^4}+\frac{19 \sqrt{d x}}{128 a^3 d \left (a+b x^2\right )^3}+\frac{209 \sqrt{d x}}{1024 a^4 d \left (a+b x^2\right )^2}+\frac{1463 \sqrt{d x}}{4096 a^5 d \left (a+b x^2\right )}+\frac{(4389 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 )}{8192 a^{11/2} d^2}+\frac{(4389 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 )}{8192 a^{11/2} d^2}\\ &=\frac{\sqrt{d x}}{10 a d \left (a+b x^2\right )^5}+\frac{19 \sqrt{d x}}{160 a^2 d \left (a+b x^2\right )^4}+\frac{19 \sqrt{d x}}{128 a^3 d \left (a+b x^2\right )^3}+\frac{209 \sqrt{d x}}{1024 a^4 d \left (a+b x^2\right )^2}+\frac{1463 \sqrt{d x}}{4096 a^5 d \left (a+b x^2\right )}+\frac{4389 \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^{11/2} \sqrt{b}}+\frac{4389 \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^{11/2} \sqrt{b}}-\frac{4389 \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^{23/4} \sqrt [4]{b} \sqrt{d}}-\frac{4389 \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^{23/4} \sqrt [4]{b} \sqrt{d}}\\ &=\frac{\sqrt{d x}}{10 a d \left (a+b x^2\right )^5}+\frac{19 \sqrt{d x}}{160 a^2 d \left (a+b x^2\right )^4}+\frac{19 \sqrt{d x}}{128 a^3 d \left (a+b x^2\right )^3}+\frac{209 \sqrt{d x}}{1024 a^4 d \left (a+b x^2\right )^2}+\frac{1463 \sqrt{d x}}{4096 a^5 d \left (a+b x^2\right )}-\frac{4389 \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^{23/4} \sqrt [4]{b} \sqrt{d}}+\frac{4389 \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^{23/4} \sqrt [4]{b} \sqrt{d}}+\frac{4389 \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^{23/4} \sqrt [4]{b} \sqrt{d}}-\frac{4389 \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^{23/4} \sqrt [4]{b} \sqrt{d}}\\ &=\frac{\sqrt{d x}}{10 a d \left (a+b x^2\right )^5}+\frac{19 \sqrt{d x}}{160 a^2 d \left (a+b x^2\right )^4}+\frac{19 \sqrt{d x}}{128 a^3 d \left (a+b x^2\right )^3}+\frac{209 \sqrt{d x}}{1024 a^4 d \left (a+b x^2\right )^2}+\frac{1463 \sqrt{d x}}{4096 a^5 d \left (a+b x^2\right )}-\frac{4389 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{8192 \sqrt{2} a^{23/4} \sqrt [4]{b} \sqrt{d}}+\frac{4389 \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{8192 \sqrt{2} a^{23/4} \sqrt [4]{b} \sqrt{d}}-\frac{4389 \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^{23/4} \sqrt [4]{b} \sqrt{d}}+\frac{4389 \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^{23/4} \sqrt [4]{b} \sqrt{d}}\\ \end{align*}
Mathematica [A] time = 0.162617, size = 295, normalized size = 0.76 \[ \frac{\sqrt{x} \left (\frac{16384 a^{19/4} \sqrt{x}}{\left (a+b x^2\right )^5}+\frac{19456 a^{15/4} \sqrt{x}}{\left (a+b x^2\right )^4}+\frac{24320 a^{11/4} \sqrt{x}}{\left (a+b x^2\right )^3}+\frac{33440 a^{7/4} \sqrt{x}}{\left (a+b x^2\right )^2}+\frac{58520 a^{3/4} \sqrt{x}}{a+b x^2}-\frac{21945 \sqrt{2} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{\sqrt [4]{b}}+\frac{21945 \sqrt{2} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{\sqrt [4]{b}}-\frac{43890 \sqrt{2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt [4]{b}}+\frac{43890 \sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt [4]{b}}\right )}{163840 a^{23/4} \sqrt{d x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.069, size = 333, normalized size = 0.9 \begin{align*}{\frac{3803\,{d}^{9}}{4096\, \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{5}a}\sqrt{dx}}+{\frac{6289\,{d}^{7}b}{2560\, \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{5}{a}^{2}} \left ( dx \right ) ^{{\frac{5}{2}}}}+{\frac{5947\,{d}^{5}{b}^{2}}{2048\, \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{5}{a}^{3}} \left ( dx \right ) ^{{\frac{9}{2}}}}+{\frac{209\,{d}^{3}{b}^{3}}{128\, \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{5}{a}^{4}} \left ( dx \right ) ^{{\frac{13}{2}}}}+{\frac{1463\,{b}^{4}d}{4096\, \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{5}{a}^{5}} \left ( dx \right ) ^{{\frac{17}{2}}}}+{\frac{4389\,\sqrt{2}}{32768\,d{a}^{6}}\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{4389\,\sqrt{2}}{16384\,d{a}^{6}}\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{4389\,\sqrt{2}}{16384\,d{a}^{6}}\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 = 1.47251, size = 1104, normalized size = 2.85 \begin{align*} \frac{87780 \,{\left (a^{5} b^{5} d x^{10} + 5 \, a^{6} b^{4} d x^{8} + 10 \, a^{7} b^{3} d x^{6} + 10 \, a^{8} b^{2} d x^{4} + 5 \, a^{9} b d x^{2} + a^{10} d\right )} \left (-\frac{1}{a^{23} b d^{2}}\right )^{\frac{1}{4}} \arctan \left (\sqrt{a^{12} d^{2} \sqrt{-\frac{1}{a^{23} b d^{2}}} + d x} a^{17} b d \left (-\frac{1}{a^{23} b d^{2}}\right )^{\frac{3}{4}} - \sqrt{d x} a^{17} b d \left (-\frac{1}{a^{23} b d^{2}}\right )^{\frac{3}{4}}\right ) + 21945 \,{\left (a^{5} b^{5} d x^{10} + 5 \, a^{6} b^{4} d x^{8} + 10 \, a^{7} b^{3} d x^{6} + 10 \, a^{8} b^{2} d x^{4} + 5 \, a^{9} b d x^{2} + a^{10} d\right )} \left (-\frac{1}{a^{23} b d^{2}}\right )^{\frac{1}{4}} \log \left (a^{6} d \left (-\frac{1}{a^{23} b d^{2}}\right )^{\frac{1}{4}} + \sqrt{d x}\right ) - 21945 \,{\left (a^{5} b^{5} d x^{10} + 5 \, a^{6} b^{4} d x^{8} + 10 \, a^{7} b^{3} d x^{6} + 10 \, a^{8} b^{2} d x^{4} + 5 \, a^{9} b d x^{2} + a^{10} d\right )} \left (-\frac{1}{a^{23} b d^{2}}\right )^{\frac{1}{4}} \log \left (-a^{6} d \left (-\frac{1}{a^{23} b d^{2}}\right )^{\frac{1}{4}} + \sqrt{d x}\right ) + 4 \,{\left (7315 \, b^{4} x^{8} + 33440 \, a b^{3} x^{6} + 59470 \, a^{2} b^{2} x^{4} + 50312 \, a^{3} b x^{2} + 19015 \, a^{4}\right )} \sqrt{d x}}{81920 \,{\left (a^{5} b^{5} d x^{10} + 5 \, a^{6} b^{4} d x^{8} + 10 \, a^{7} b^{3} d x^{6} + 10 \, a^{8} b^{2} d x^{4} + 5 \, a^{9} b d x^{2} + a^{10} d\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 = 1.27412, size = 467, normalized size = 1.21 \begin{align*} \frac{4389 \, \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 )}{16384 \, a^{6} b d} + \frac{4389 \, \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 )}{16384 \, a^{6} b d} + \frac{4389 \, \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 )}{32768 \, a^{6} b d} - \frac{4389 \, \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 )}{32768 \, a^{6} b d} + \frac{7315 \, \sqrt{d x} b^{4} d^{9} x^{8} + 33440 \, \sqrt{d x} a b^{3} d^{9} x^{6} + 59470 \, \sqrt{d x} a^{2} b^{2} d^{9} x^{4} + 50312 \, \sqrt{d x} a^{3} b d^{9} x^{2} + 19015 \, \sqrt{d x} a^{4} d^{9}}{20480 \,{\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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