Optimal. Leaf size=156 \[ \frac{x (a e+3 b d)}{8 a^2 b \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{x (b d-a e)}{4 a b \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (a+b x^2\right ) (a e+3 b d) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{5/2} b^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}} \]
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Rubi [A] time = 0.201955, antiderivative size = 156, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{x (a e+3 b d)}{8 a^2 b \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{x (b d-a e)}{4 a b \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (a+b x^2\right ) (a e+3 b d) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{5/2} b^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x^2)/(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((e*x**2+d)/(b**2*x**4+2*a*b*x**2+a**2)**(3/2),x)
[Out]
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Mathematica [A] time = 0.0971276, size = 108, normalized size = 0.69 \[ \frac{\sqrt{a} \sqrt{b} x \left (a^2 (-e)+a b \left (5 d+e x^2\right )+3 b^2 d x^2\right )+\left (a+b x^2\right )^2 (a e+3 b d) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{5/2} b^{3/2} \left (a+b x^2\right ) \sqrt{\left (a+b x^2\right )^2}} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x^2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2),x]
[Out]
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Maple [A] time = 0.02, size = 186, normalized size = 1.2 \[{\frac{b{x}^{2}+a}{8\,{a}^{2}b} \left ( \arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){x}^{4}a{b}^{2}e+3\,\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ){x}^{4}{b}^{3}d+\sqrt{ab}{x}^{3}abe+3\,\sqrt{ab}{x}^{3}{b}^{2}d+2\,\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ){x}^{2}{a}^{2}be+6\,\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ){x}^{2}a{b}^{2}d-\sqrt{ab}x{a}^{2}e+5\,\sqrt{ab}xabd+\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){a}^{3}e+3\,\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ){a}^{2}bd \right ){\frac{1}{\sqrt{ab}}} \left ( \left ( b{x}^{2}+a \right ) ^{2} \right ) ^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x^2+d)/(b^2*x^4+2*a*b*x^2+a^2)^(3/2),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((e*x^2 + d)/(b^2*x^4 + 2*a*b*x^2 + a^2)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.272678, size = 1, normalized size = 0.01 \[ \left [\frac{{\left ({\left (3 \, b^{3} d + a b^{2} e\right )} x^{4} + 3 \, a^{2} b d + a^{3} e + 2 \,{\left (3 \, a b^{2} d + a^{2} b e\right )} x^{2}\right )} \log \left (\frac{2 \, a b x +{\left (b x^{2} - a\right )} \sqrt{-a b}}{b x^{2} + a}\right ) + 2 \,{\left ({\left (3 \, b^{2} d + a b e\right )} x^{3} +{\left (5 \, a b d - a^{2} e\right )} x\right )} \sqrt{-a b}}{16 \,{\left (a^{2} b^{3} x^{4} + 2 \, a^{3} b^{2} x^{2} + a^{4} b\right )} \sqrt{-a b}}, \frac{{\left ({\left (3 \, b^{3} d + a b^{2} e\right )} x^{4} + 3 \, a^{2} b d + a^{3} e + 2 \,{\left (3 \, a b^{2} d + a^{2} b e\right )} x^{2}\right )} \arctan \left (\frac{\sqrt{a b} x}{a}\right ) +{\left ({\left (3 \, b^{2} d + a b e\right )} x^{3} +{\left (5 \, a b d - a^{2} e\right )} x\right )} \sqrt{a b}}{8 \,{\left (a^{2} b^{3} x^{4} + 2 \, a^{3} b^{2} x^{2} + a^{4} b\right )} \sqrt{a b}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x^2 + d)/(b^2*x^4 + 2*a*b*x^2 + a^2)^(3/2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{d + e x^{2}}{\left (\left (a + b x^{2}\right )^{2}\right )^{\frac{3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x**2+d)/(b**2*x**4+2*a*b*x**2+a**2)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.627621, size = 4, normalized size = 0.03 \[ \mathit{sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x^2 + d)/(b^2*x^4 + 2*a*b*x^2 + a^2)^(3/2),x, algorithm="giac")
[Out]