Optimal. Leaf size=209 \[ \frac{1}{4 a x^3 \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{7}{8 a^2 x^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{35 b^{3/2} \left (a+b x^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{9/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{35 b \left (a+b x^2\right )}{8 a^4 x \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{35 \left (a+b x^2\right )}{24 a^3 x^3 \sqrt{a^2+2 a b x^2+b^2 x^4}} \]
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Rubi [A] time = 0.222925, antiderivative size = 209, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{1}{4 a x^3 \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{7}{8 a^2 x^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{35 b^{3/2} \left (a+b x^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{9/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{35 b \left (a+b x^2\right )}{8 a^4 x \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{35 \left (a+b x^2\right )}{24 a^3 x^3 \sqrt{a^2+2 a b x^2+b^2 x^4}} \]
Antiderivative was successfully verified.
[In] Int[1/(x^4*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2)),x]
[Out]
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Rubi in Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: RecursionError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**4/(b**2*x**4+2*a*b*x**2+a**2)**(3/2),x)
[Out]
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Mathematica [A] time = 0.0705342, size = 105, normalized size = 0.5 \[ \frac{\sqrt{a} \left (-8 a^3+56 a^2 b x^2+175 a b^2 x^4+105 b^3 x^6\right )+105 b^{3/2} x^3 \left (a+b x^2\right )^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{24 a^{9/2} x^3 \left (a+b x^2\right ) \sqrt{\left (a+b x^2\right )^2}} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^4*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2)),x]
[Out]
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Maple [A] time = 0.027, size = 139, normalized size = 0.7 \[{\frac{b{x}^{2}+a}{24\,{a}^{4}{x}^{3}} \left ( 105\,\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ){x}^{7}{b}^{4}+105\,\sqrt{ab}{x}^{6}{b}^{3}+210\,\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ){x}^{5}a{b}^{3}+175\,\sqrt{ab}{x}^{4}a{b}^{2}+105\,\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ){x}^{3}{a}^{2}{b}^{2}+56\,\sqrt{ab}{x}^{2}{a}^{2}b-8\,\sqrt{ab}{a}^{3} \right ){\frac{1}{\sqrt{ab}}} \left ( \left ( b{x}^{2}+a \right ) ^{2} \right ) ^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^4/(b^2*x^4+2*a*b*x^2+a^2)^(3/2),x)
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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(1/((b^2*x^4 + 2*a*b*x^2 + a^2)^(3/2)*x^4),x, algorithm="maxima")
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Fricas [A] time = 0.274802, size = 1, normalized size = 0. \[ \left [\frac{210 \, b^{3} x^{6} + 350 \, a b^{2} x^{4} + 112 \, a^{2} b x^{2} - 16 \, a^{3} + 105 \,{\left (b^{3} x^{7} + 2 \, a b^{2} x^{5} + a^{2} b x^{3}\right )} \sqrt{-\frac{b}{a}} \log \left (\frac{b x^{2} + 2 \, a x \sqrt{-\frac{b}{a}} - a}{b x^{2} + a}\right )}{48 \,{\left (a^{4} b^{2} x^{7} + 2 \, a^{5} b x^{5} + a^{6} x^{3}\right )}}, \frac{105 \, b^{3} x^{6} + 175 \, a b^{2} x^{4} + 56 \, a^{2} b x^{2} - 8 \, a^{3} + 105 \,{\left (b^{3} x^{7} + 2 \, a b^{2} x^{5} + a^{2} b x^{3}\right )} \sqrt{\frac{b}{a}} \arctan \left (\frac{b x}{a \sqrt{\frac{b}{a}}}\right )}{24 \,{\left (a^{4} b^{2} x^{7} + 2 \, a^{5} b x^{5} + a^{6} x^{3}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b^2*x^4 + 2*a*b*x^2 + a^2)^(3/2)*x^4),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{4} \left (\left (a + b x^{2}\right )^{2}\right )^{\frac{3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**4/(b**2*x**4+2*a*b*x**2+a**2)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.619549, size = 4, normalized size = 0.02 \[ \mathit{sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b^2*x^4 + 2*a*b*x^2 + a^2)^(3/2)*x^4),x, algorithm="giac")
[Out]