Optimal. Leaf size=573 \[ -\frac{5 b^2-18 a c}{2 a^2 \sqrt{x} \left (b^2-4 a c\right )}+\frac{\sqrt [4]{c} \left (-\left (5 b^2-18 a c\right ) \sqrt{b^2-4 a c}-28 a b c+5 b^3\right ) \tan ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{-\sqrt{b^2-4 a c}-b}}\right )}{4\ 2^{3/4} a^2 \left (b^2-4 a c\right )^{3/2} \sqrt [4]{-\sqrt{b^2-4 a c}-b}}-\frac{\sqrt [4]{c} \left (\left (5 b^2-18 a c\right ) \sqrt{b^2-4 a c}-28 a b c+5 b^3\right ) \tan ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{\sqrt{b^2-4 a c}-b}}\right )}{4\ 2^{3/4} a^2 \left (b^2-4 a c\right )^{3/2} \sqrt [4]{\sqrt{b^2-4 a c}-b}}-\frac{\sqrt [4]{c} \left (-\left (5 b^2-18 a c\right ) \sqrt{b^2-4 a c}-28 a b c+5 b^3\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{-\sqrt{b^2-4 a c}-b}}\right )}{4\ 2^{3/4} a^2 \left (b^2-4 a c\right )^{3/2} \sqrt [4]{-\sqrt{b^2-4 a c}-b}}+\frac{\sqrt [4]{c} \left (\left (5 b^2-18 a c\right ) \sqrt{b^2-4 a c}-28 a b c+5 b^3\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{\sqrt{b^2-4 a c}-b}}\right )}{4\ 2^{3/4} a^2 \left (b^2-4 a c\right )^{3/2} \sqrt [4]{\sqrt{b^2-4 a c}-b}}+\frac{-2 a c+b^2+b c x^2}{2 a \sqrt{x} \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )} \]
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Rubi [A] time = 4.34032, antiderivative size = 573, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35 \[ -\frac{5 b^2-18 a c}{2 a^2 \sqrt{x} \left (b^2-4 a c\right )}+\frac{\sqrt [4]{c} \left (-\left (5 b^2-18 a c\right ) \sqrt{b^2-4 a c}-28 a b c+5 b^3\right ) \tan ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{-\sqrt{b^2-4 a c}-b}}\right )}{4\ 2^{3/4} a^2 \left (b^2-4 a c\right )^{3/2} \sqrt [4]{-\sqrt{b^2-4 a c}-b}}-\frac{\sqrt [4]{c} \left (\left (5 b^2-18 a c\right ) \sqrt{b^2-4 a c}-28 a b c+5 b^3\right ) \tan ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{\sqrt{b^2-4 a c}-b}}\right )}{4\ 2^{3/4} a^2 \left (b^2-4 a c\right )^{3/2} \sqrt [4]{\sqrt{b^2-4 a c}-b}}-\frac{\sqrt [4]{c} \left (-\left (5 b^2-18 a c\right ) \sqrt{b^2-4 a c}-28 a b c+5 b^3\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{-\sqrt{b^2-4 a c}-b}}\right )}{4\ 2^{3/4} a^2 \left (b^2-4 a c\right )^{3/2} \sqrt [4]{-\sqrt{b^2-4 a c}-b}}+\frac{\sqrt [4]{c} \left (\left (5 b^2-18 a c\right ) \sqrt{b^2-4 a c}-28 a b c+5 b^3\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{\sqrt{b^2-4 a c}-b}}\right )}{4\ 2^{3/4} a^2 \left (b^2-4 a c\right )^{3/2} \sqrt [4]{\sqrt{b^2-4 a c}-b}}+\frac{-2 a c+b^2+b c x^2}{2 a \sqrt{x} \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )} \]
Antiderivative was successfully verified.
[In] Int[1/(x^(3/2)*(a + b*x^2 + c*x^4)^2),x]
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Rubi in Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**(3/2)/(c*x**4+b*x**2+a)**2,x)
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Mathematica [C] time = 0.409386, size = 190, normalized size = 0.33 \[ -\frac{\frac{\text{RootSum}\left [\text{$\#$1}^8 c+\text{$\#$1}^4 b+a\&,\frac{-18 \text{$\#$1}^4 a c^2 \log \left (\sqrt{x}-\text{$\#$1}\right )+5 \text{$\#$1}^4 b^2 c \log \left (\sqrt{x}-\text{$\#$1}\right )-23 a b c \log \left (\sqrt{x}-\text{$\#$1}\right )+5 b^3 \log \left (\sqrt{x}-\text{$\#$1}\right )}{2 \text{$\#$1}^5 c+\text{$\#$1} b}\&\right ]}{b^2-4 a c}+\frac{4 x^{3/2} \left (-3 a b c-2 a c^2 x^2+b^3+b^2 c x^2\right )}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{16}{\sqrt{x}}}{8 a^2} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^(3/2)*(a + b*x^2 + c*x^4)^2),x]
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Maple [C] time = 0.035, size = 245, normalized size = 0.4 \[ -{\frac{{c}^{2}}{a \left ( c{x}^{4}+b{x}^{2}+a \right ) \left ( 4\,ac-{b}^{2} \right ) }{x}^{{\frac{7}{2}}}}+{\frac{{b}^{2}c}{2\,{a}^{2} \left ( c{x}^{4}+b{x}^{2}+a \right ) \left ( 4\,ac-{b}^{2} \right ) }{x}^{{\frac{7}{2}}}}-{\frac{3\,bc}{2\,a \left ( c{x}^{4}+b{x}^{2}+a \right ) \left ( 4\,ac-{b}^{2} \right ) }{x}^{{\frac{3}{2}}}}+{\frac{{b}^{3}}{2\,{a}^{2} \left ( c{x}^{4}+b{x}^{2}+a \right ) \left ( 4\,ac-{b}^{2} \right ) }{x}^{{\frac{3}{2}}}}-{\frac{1}{8\,{a}^{2}}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{8}c+{{\it \_Z}}^{4}b+a \right ) }{\frac{c \left ( 18\,ac-5\,{b}^{2} \right ){{\it \_R}}^{6}+b \left ( 23\,ac-5\,{b}^{2} \right ){{\it \_R}}^{2}}{ \left ( 4\,ac-{b}^{2} \right ) \left ( 2\,{{\it \_R}}^{7}c+{{\it \_R}}^{3}b \right ) }\ln \left ( \sqrt{x}-{\it \_R} \right ) }}-2\,{\frac{1}{{a}^{2}\sqrt{x}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^(3/2)/(c*x^4+b*x^2+a)^2,x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ -\frac{{\left (5 \, b^{2} c - 18 \, a c^{2}\right )} x^{\frac{7}{2}} +{\left (5 \, b^{3} - 19 \, a b c\right )} x^{\frac{3}{2}} + \frac{4 \,{\left (a b^{2} - 4 \, a^{2} c\right )}}{\sqrt{x}}}{2 \,{\left (a^{3} b^{2} - 4 \, a^{4} c +{\left (a^{2} b^{2} c - 4 \, a^{3} c^{2}\right )} x^{4} +{\left (a^{2} b^{3} - 4 \, a^{3} b c\right )} x^{2}\right )}} - \int \frac{{\left (5 \, b^{2} c - 18 \, a c^{2}\right )} x^{\frac{5}{2}} +{\left (5 \, b^{3} - 23 \, a b c\right )} \sqrt{x}}{4 \,{\left (a^{3} b^{2} - 4 \, a^{4} c +{\left (a^{2} b^{2} c - 4 \, a^{3} c^{2}\right )} x^{4} +{\left (a^{2} b^{3} - 4 \, a^{3} b c\right )} x^{2}\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((c*x^4 + b*x^2 + a)^2*x^(3/2)),x, algorithm="maxima")
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((c*x^4 + b*x^2 + a)^2*x^(3/2)),x, algorithm="fricas")
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**(3/2)/(c*x**4+b*x**2+a)**2,x)
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (c x^{4} + b x^{2} + a\right )}^{2} x^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((c*x^4 + b*x^2 + a)^2*x^(3/2)),x, algorithm="giac")
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