Optimal. Leaf size=161 \[ \frac{\left (b^2-4 a c\right ) \left (5 b^2-4 a c\right ) \tanh ^{-1}\left (\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right )}{256 a^{7/2}}-\frac{\left (5 b^2-4 a c\right ) \left (2 a+b x^2\right ) \sqrt{a+b x^2+c x^4}}{128 a^3 x^4}+\frac{5 b \left (a+b x^2+c x^4\right )^{3/2}}{48 a^2 x^6}-\frac{\left (a+b x^2+c x^4\right )^{3/2}}{8 a x^8} \]
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Rubi [A] time = 0.386265, antiderivative size = 161, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3 \[ \frac{\left (b^2-4 a c\right ) \left (5 b^2-4 a c\right ) \tanh ^{-1}\left (\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right )}{256 a^{7/2}}-\frac{\left (5 b^2-4 a c\right ) \left (2 a+b x^2\right ) \sqrt{a+b x^2+c x^4}}{128 a^3 x^4}+\frac{5 b \left (a+b x^2+c x^4\right )^{3/2}}{48 a^2 x^6}-\frac{\left (a+b x^2+c x^4\right )^{3/2}}{8 a x^8} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[a + b*x^2 + c*x^4]/x^9,x]
[Out]
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Rubi in Sympy [A] time = 30.3439, size = 148, normalized size = 0.92 \[ - \frac{\left (a + b x^{2} + c x^{4}\right )^{\frac{3}{2}}}{8 a x^{8}} + \frac{5 b \left (a + b x^{2} + c x^{4}\right )^{\frac{3}{2}}}{48 a^{2} x^{6}} - \frac{\left (2 a + b x^{2}\right ) \left (- 4 a c + 5 b^{2}\right ) \sqrt{a + b x^{2} + c x^{4}}}{128 a^{3} x^{4}} + \frac{\left (- 4 a c + b^{2}\right ) \left (- 4 a c + 5 b^{2}\right ) \operatorname{atanh}{\left (\frac{2 a + b x^{2}}{2 \sqrt{a} \sqrt{a + b x^{2} + c x^{4}}} \right )}}{256 a^{\frac{7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**4+b*x**2+a)**(1/2)/x**9,x)
[Out]
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Mathematica [A] time = 0.228904, size = 143, normalized size = 0.89 \[ \frac{-\frac{2 \sqrt{a} \sqrt{a+b x^2+c x^4} \left (48 a^3+8 a^2 x^2 \left (b+3 c x^2\right )-2 a b x^4 \left (5 b+26 c x^2\right )+15 b^3 x^6\right )}{x^8}-3 \left (b^2-4 a c\right ) \left (5 b^2-4 a c\right ) \left (\log \left (x^2\right )-\log \left (2 \sqrt{a} \sqrt{a+b x^2+c x^4}+2 a+b x^2\right )\right )}{768 a^{7/2}} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[a + b*x^2 + c*x^4]/x^9,x]
[Out]
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Maple [B] time = 0.023, size = 387, normalized size = 2.4 \[ -{\frac{1}{8\,a{x}^{8}} \left ( c{x}^{4}+b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{5\,b}{48\,{a}^{2}{x}^{6}} \left ( c{x}^{4}+b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{5\,{b}^{2}}{64\,{a}^{3}{x}^{4}} \left ( c{x}^{4}+b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{5\,{b}^{3}}{128\,{a}^{4}{x}^{2}} \left ( c{x}^{4}+b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{5\,{b}^{4}}{128\,{a}^{4}}\sqrt{c{x}^{4}+b{x}^{2}+a}}+{\frac{5\,{b}^{4}}{256}\ln \left ({\frac{1}{{x}^{2}} \left ( 2\,a+b{x}^{2}+2\,\sqrt{a}\sqrt{c{x}^{4}+b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{7}{2}}}}-{\frac{5\,{x}^{2}{b}^{3}c}{128\,{a}^{4}}\sqrt{c{x}^{4}+b{x}^{2}+a}}+{\frac{7\,{b}^{2}c}{64\,{a}^{3}}\sqrt{c{x}^{4}+b{x}^{2}+a}}-{\frac{3\,{b}^{2}c}{32}\ln \left ({\frac{1}{{x}^{2}} \left ( 2\,a+b{x}^{2}+2\,\sqrt{a}\sqrt{c{x}^{4}+b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{5}{2}}}}+{\frac{c}{16\,{a}^{2}{x}^{4}} \left ( c{x}^{4}+b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{bc}{32\,{a}^{3}{x}^{2}} \left ( c{x}^{4}+b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{b{c}^{2}{x}^{2}}{32\,{a}^{3}}\sqrt{c{x}^{4}+b{x}^{2}+a}}-{\frac{{c}^{2}}{16\,{a}^{2}}\sqrt{c{x}^{4}+b{x}^{2}+a}}+{\frac{{c}^{2}}{16}\ln \left ({\frac{1}{{x}^{2}} \left ( 2\,a+b{x}^{2}+2\,\sqrt{a}\sqrt{c{x}^{4}+b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^4+b*x^2+a)^(1/2)/x^9,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(c*x^4 + b*x^2 + a)/x^9,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.310012, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (5 \, b^{4} - 24 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} x^{8} \log \left (-\frac{4 \, \sqrt{c x^{4} + b x^{2} + a}{\left (a b x^{2} + 2 \, a^{2}\right )} +{\left ({\left (b^{2} + 4 \, a c\right )} x^{4} + 8 \, a b x^{2} + 8 \, a^{2}\right )} \sqrt{a}}{x^{4}}\right ) - 4 \,{\left ({\left (15 \, b^{3} - 52 \, a b c\right )} x^{6} + 8 \, a^{2} b x^{2} - 2 \,{\left (5 \, a b^{2} - 12 \, a^{2} c\right )} x^{4} + 48 \, a^{3}\right )} \sqrt{c x^{4} + b x^{2} + a} \sqrt{a}}{1536 \, a^{\frac{7}{2}} x^{8}}, \frac{3 \,{\left (5 \, b^{4} - 24 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} x^{8} \arctan \left (\frac{{\left (b x^{2} + 2 \, a\right )} \sqrt{-a}}{2 \, \sqrt{c x^{4} + b x^{2} + a} a}\right ) - 2 \,{\left ({\left (15 \, b^{3} - 52 \, a b c\right )} x^{6} + 8 \, a^{2} b x^{2} - 2 \,{\left (5 \, a b^{2} - 12 \, a^{2} c\right )} x^{4} + 48 \, a^{3}\right )} \sqrt{c x^{4} + b x^{2} + a} \sqrt{-a}}{768 \, \sqrt{-a} a^{3} x^{8}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^4 + b*x^2 + a)/x^9,x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{a + b x^{2} + c x^{4}}}{x^{9}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**4+b*x**2+a)**(1/2)/x**9,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{c x^{4} + b x^{2} + a}}{x^{9}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^4 + b*x^2 + a)/x^9,x, algorithm="giac")
[Out]