Optimal. Leaf size=335 \[ \frac{15 \sqrt{d} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{256 \sqrt{2} a^{13/4} b^{3/4}}-\frac{15 \sqrt{d} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{256 \sqrt{2} a^{13/4} b^{3/4}}-\frac{15 \sqrt{d} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{128 \sqrt{2} a^{13/4} b^{3/4}}+\frac{15 \sqrt{d} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{128 \sqrt{2} a^{13/4} b^{3/4}}+\frac{15 (d x)^{3/2}}{64 a^3 d \left (a+b x^2\right )}+\frac{3 (d x)^{3/2}}{16 a^2 d \left (a+b x^2\right )^2}+\frac{(d x)^{3/2}}{6 a d \left (a+b x^2\right )^3} \]
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Rubi [A] time = 0.729592, antiderivative size = 335, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 9, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.321 \[ \frac{15 \sqrt{d} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{256 \sqrt{2} a^{13/4} b^{3/4}}-\frac{15 \sqrt{d} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{256 \sqrt{2} a^{13/4} b^{3/4}}-\frac{15 \sqrt{d} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{128 \sqrt{2} a^{13/4} b^{3/4}}+\frac{15 \sqrt{d} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{128 \sqrt{2} a^{13/4} b^{3/4}}+\frac{15 (d x)^{3/2}}{64 a^3 d \left (a+b x^2\right )}+\frac{3 (d x)^{3/2}}{16 a^2 d \left (a+b x^2\right )^2}+\frac{(d x)^{3/2}}{6 a d \left (a+b x^2\right )^3} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[d*x]/(a^2 + 2*a*b*x^2 + b^2*x^4)^2,x]
[Out]
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Rubi in Sympy [A] time = 147.626, size = 309, normalized size = 0.92 \[ \frac{\left (d x\right )^{\frac{3}{2}}}{6 a d \left (a + b x^{2}\right )^{3}} + \frac{3 \left (d x\right )^{\frac{3}{2}}}{16 a^{2} d \left (a + b x^{2}\right )^{2}} + \frac{15 \left (d x\right )^{\frac{3}{2}}}{64 a^{3} d \left (a + b x^{2}\right )} + \frac{15 \sqrt{2} \sqrt{d} \log{\left (- \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d} \sqrt{d x} + \sqrt{a} d + \sqrt{b} d x \right )}}{512 a^{\frac{13}{4}} b^{\frac{3}{4}}} - \frac{15 \sqrt{2} \sqrt{d} \log{\left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d} \sqrt{d x} + \sqrt{a} d + \sqrt{b} d x \right )}}{512 a^{\frac{13}{4}} b^{\frac{3}{4}}} - \frac{15 \sqrt{2} \sqrt{d} \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}} \right )}}{256 a^{\frac{13}{4}} b^{\frac{3}{4}}} + \frac{15 \sqrt{2} \sqrt{d} \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}} \right )}}{256 a^{\frac{13}{4}} b^{\frac{3}{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x)**(1/2)/(b**2*x**4+2*a*b*x**2+a**2)**2,x)
[Out]
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Mathematica [A] time = 0.260261, size = 253, normalized size = 0.76 \[ \frac{\sqrt{d x} \left (\frac{256 a^{9/4} x^{3/2}}{\left (a+b x^2\right )^3}+\frac{288 a^{5/4} x^{3/2}}{\left (a+b x^2\right )^2}+\frac{45 \sqrt{2} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{b^{3/4}}-\frac{45 \sqrt{2} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{b^{3/4}}-\frac{90 \sqrt{2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{b^{3/4}}+\frac{90 \sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{b^{3/4}}+\frac{360 \sqrt [4]{a} x^{3/2}}{a+b x^2}\right )}{1536 a^{13/4} \sqrt{x}} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[d*x]/(a^2 + 2*a*b*x^2 + b^2*x^4)^2,x]
[Out]
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Maple [A] time = 0.026, size = 272, normalized size = 0.8 \[{\frac{15\,{b}^{2}d}{64\, \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{3}{a}^{3}} \left ( dx \right ) ^{{\frac{11}{2}}}}+{\frac{21\,{d}^{3}b}{32\, \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{3}{a}^{2}} \left ( dx \right ) ^{{\frac{7}{2}}}}+{\frac{113\,{d}^{5}}{192\, \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{3}a} \left ( dx \right ) ^{{\frac{3}{2}}}}+{\frac{15\,d\sqrt{2}}{512\,{a}^{3}b}\ln \left ({1 \left ( dx-\sqrt [4]{{\frac{a{d}^{2}}{b}}}\sqrt{dx}\sqrt{2}+\sqrt{{\frac{a{d}^{2}}{b}}} \right ) \left ( dx+\sqrt [4]{{\frac{a{d}^{2}}{b}}}\sqrt{dx}\sqrt{2}+\sqrt{{\frac{a{d}^{2}}{b}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}}+{\frac{15\,d\sqrt{2}}{256\,{a}^{3}b}\arctan \left ({\sqrt{2}\sqrt{dx}{\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}}+{\frac{15\,d\sqrt{2}}{256\,{a}^{3}b}\arctan \left ({\sqrt{2}\sqrt{dx}{\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x)^(1/2)/(b^2*x^4+2*a*b*x^2+a^2)^2,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(d*x)/(b^2*x^4 + 2*a*b*x^2 + a^2)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.288892, size = 460, normalized size = 1.37 \[ \frac{180 \,{\left (a^{3} b^{3} x^{6} + 3 \, a^{4} b^{2} x^{4} + 3 \, a^{5} b x^{2} + a^{6}\right )} \left (-\frac{d^{2}}{a^{13} b^{3}}\right )^{\frac{1}{4}} \arctan \left (\frac{3375 \, a^{10} b^{2} \left (-\frac{d^{2}}{a^{13} b^{3}}\right )^{\frac{3}{4}}}{3375 \, \sqrt{d x} d + \sqrt{-11390625 \, a^{7} b d^{2} \sqrt{-\frac{d^{2}}{a^{13} b^{3}}} + 11390625 \, d^{3} x}}\right ) + 45 \,{\left (a^{3} b^{3} x^{6} + 3 \, a^{4} b^{2} x^{4} + 3 \, a^{5} b x^{2} + a^{6}\right )} \left (-\frac{d^{2}}{a^{13} b^{3}}\right )^{\frac{1}{4}} \log \left (3375 \, a^{10} b^{2} \left (-\frac{d^{2}}{a^{13} b^{3}}\right )^{\frac{3}{4}} + 3375 \, \sqrt{d x} d\right ) - 45 \,{\left (a^{3} b^{3} x^{6} + 3 \, a^{4} b^{2} x^{4} + 3 \, a^{5} b x^{2} + a^{6}\right )} \left (-\frac{d^{2}}{a^{13} b^{3}}\right )^{\frac{1}{4}} \log \left (-3375 \, a^{10} b^{2} \left (-\frac{d^{2}}{a^{13} b^{3}}\right )^{\frac{3}{4}} + 3375 \, \sqrt{d x} d\right ) + 4 \,{\left (45 \, b^{2} x^{5} + 126 \, a b x^{3} + 113 \, a^{2} x\right )} \sqrt{d x}}{768 \,{\left (a^{3} b^{3} x^{6} + 3 \, a^{4} b^{2} x^{4} + 3 \, a^{5} b x^{2} + a^{6}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(d*x)/(b^2*x^4 + 2*a*b*x^2 + a^2)^2,x, algorithm="fricas")
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Sympy [A] time = 136.218, size = 252, normalized size = 0.75 \[ \frac{226 a^{2} d^{11} \left (d x\right )^{\frac{3}{2}}}{384 a^{6} d^{12} + 1152 a^{5} b d^{12} x^{2} + 1152 a^{4} b^{2} d^{12} x^{4} + 384 a^{3} b^{3} d^{12} x^{6}} + \frac{252 a b d^{9} \left (d x\right )^{\frac{7}{2}}}{384 a^{6} d^{12} + 1152 a^{5} b d^{12} x^{2} + 1152 a^{4} b^{2} d^{12} x^{4} + 384 a^{3} b^{3} d^{12} x^{6}} + \frac{90 b^{2} d^{7} \left (d x\right )^{\frac{11}{2}}}{384 a^{6} d^{12} + 1152 a^{5} b d^{12} x^{2} + 1152 a^{4} b^{2} d^{12} x^{4} + 384 a^{3} b^{3} d^{12} x^{6}} + 2 d^{7} \operatorname{RootSum}{\left (68719476736 t^{4} a^{13} b^{3} d^{26} + 50625, \left ( t \mapsto t \log{\left (\frac{134217728 t^{3} a^{10} b^{2} d^{20}}{3375} + \sqrt{d x} \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x)**(1/2)/(b**2*x**4+2*a*b*x**2+a**2)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.278439, size = 417, normalized size = 1.24 \[ \frac{45 \, \sqrt{d x} b^{2} d^{6} x^{5} + 126 \, \sqrt{d x} a b d^{6} x^{3} + 113 \, \sqrt{d x} a^{2} d^{6} x}{192 \,{\left (b d^{2} x^{2} + a d^{2}\right )}^{3} a^{3}} + \frac{15 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{3}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} + 2 \, \sqrt{d x}\right )}}{2 \, \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}}}\right )}{256 \, a^{4} b^{3} d} + \frac{15 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{3}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} - 2 \, \sqrt{d x}\right )}}{2 \, \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}}}\right )}{256 \, a^{4} b^{3} d} - \frac{15 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{3}{4}}{\rm ln}\left (d x + \sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} \sqrt{d x} + \sqrt{\frac{a d^{2}}{b}}\right )}{512 \, a^{4} b^{3} d} + \frac{15 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{3}{4}}{\rm ln}\left (d x - \sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} \sqrt{d x} + \sqrt{\frac{a d^{2}}{b}}\right )}{512 \, a^{4} b^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(d*x)/(b^2*x^4 + 2*a*b*x^2 + a^2)^2,x, algorithm="giac")
[Out]