Optimal. Leaf size=136 \[ -\frac{\sqrt{e} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (3 c d^2-e (5 b d-7 a e)\right )}{2 d^{9/2}}-\frac{e x \left (a e^2-b d e+c d^2\right )}{2 d^4 \left (d+e x^2\right )}-\frac{c d^2-e (2 b d-3 a e)}{d^4 x}-\frac{b d-2 a e}{3 d^3 x^3}-\frac{a}{5 d^2 x^5} \]
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Rubi [A] time = 0.423074, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12 \[ -\frac{\sqrt{e} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (3 c d^2-e (5 b d-7 a e)\right )}{2 d^{9/2}}-\frac{e x \left (a e^2-b d e+c d^2\right )}{2 d^4 \left (d+e x^2\right )}-\frac{c d^2-e (2 b d-3 a e)}{d^4 x}-\frac{b d-2 a e}{3 d^3 x^3}-\frac{a}{5 d^2 x^5} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x^2 + c*x^4)/(x^6*(d + e*x^2)^2),x]
[Out]
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Rubi in Sympy [A] time = 66.2802, size = 126, normalized size = 0.93 \[ - \frac{a}{5 d^{2} x^{5}} + \frac{2 a e - b d}{3 d^{3} x^{3}} - \frac{e x \left (a e^{2} - b d e + c d^{2}\right )}{2 d^{4} \left (d + e x^{2}\right )} - \frac{3 a e^{2} - 2 b d e + c d^{2}}{d^{4} x} - \frac{\sqrt{e} \left (7 a e^{2} - 5 b d e + 3 c d^{2}\right ) \operatorname{atan}{\left (\frac{\sqrt{e} x}{\sqrt{d}} \right )}}{2 d^{\frac{9}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**4+b*x**2+a)/x**6/(e*x**2+d)**2,x)
[Out]
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Mathematica [A] time = 0.153693, size = 135, normalized size = 0.99 \[ -\frac{\sqrt{e} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (7 a e^2-5 b d e+3 c d^2\right )}{2 d^{9/2}}-\frac{e x \left (a e^2-b d e+c d^2\right )}{2 d^4 \left (d+e x^2\right )}+\frac{-3 a e^2+2 b d e-c d^2}{d^4 x}+\frac{2 a e-b d}{3 d^3 x^3}-\frac{a}{5 d^2 x^5} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x^2 + c*x^4)/(x^6*(d + e*x^2)^2),x]
[Out]
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Maple [A] time = 0.021, size = 183, normalized size = 1.4 \[ -{\frac{a}{5\,{d}^{2}{x}^{5}}}+{\frac{2\,ae}{3\,{d}^{3}{x}^{3}}}-{\frac{b}{3\,{d}^{2}{x}^{3}}}-3\,{\frac{a{e}^{2}}{{d}^{4}x}}+2\,{\frac{be}{{d}^{3}x}}-{\frac{c}{{d}^{2}x}}-{\frac{{e}^{3}xa}{2\,{d}^{4} \left ( e{x}^{2}+d \right ) }}+{\frac{{e}^{2}xb}{2\,{d}^{3} \left ( e{x}^{2}+d \right ) }}-{\frac{cex}{2\,{d}^{2} \left ( e{x}^{2}+d \right ) }}-{\frac{7\,{e}^{3}a}{2\,{d}^{4}}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}+{\frac{5\,b{e}^{2}}{2\,{d}^{3}}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}-{\frac{3\,ce}{2\,{d}^{2}}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^4+b*x^2+a)/x^6/(e*x^2+d)^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((c*x^4 + b*x^2 + a)/((e*x^2 + d)^2*x^6),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.274425, size = 1, normalized size = 0.01 \[ \left [-\frac{30 \,{\left (3 \, c d^{2} e - 5 \, b d e^{2} + 7 \, a e^{3}\right )} x^{6} + 20 \,{\left (3 \, c d^{3} - 5 \, b d^{2} e + 7 \, a d e^{2}\right )} x^{4} + 12 \, a d^{3} + 4 \,{\left (5 \, b d^{3} - 7 \, a d^{2} e\right )} x^{2} - 15 \,{\left ({\left (3 \, c d^{2} e - 5 \, b d e^{2} + 7 \, a e^{3}\right )} x^{7} +{\left (3 \, c d^{3} - 5 \, b d^{2} e + 7 \, a d e^{2}\right )} x^{5}\right )} \sqrt{-\frac{e}{d}} \log \left (\frac{e x^{2} - 2 \, d x \sqrt{-\frac{e}{d}} - d}{e x^{2} + d}\right )}{60 \,{\left (d^{4} e x^{7} + d^{5} x^{5}\right )}}, -\frac{15 \,{\left (3 \, c d^{2} e - 5 \, b d e^{2} + 7 \, a e^{3}\right )} x^{6} + 10 \,{\left (3 \, c d^{3} - 5 \, b d^{2} e + 7 \, a d e^{2}\right )} x^{4} + 6 \, a d^{3} + 2 \,{\left (5 \, b d^{3} - 7 \, a d^{2} e\right )} x^{2} + 15 \,{\left ({\left (3 \, c d^{2} e - 5 \, b d e^{2} + 7 \, a e^{3}\right )} x^{7} +{\left (3 \, c d^{3} - 5 \, b d^{2} e + 7 \, a d e^{2}\right )} x^{5}\right )} \sqrt{\frac{e}{d}} \arctan \left (\frac{e x}{d \sqrt{\frac{e}{d}}}\right )}{30 \,{\left (d^{4} e x^{7} + d^{5} x^{5}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^4 + b*x^2 + a)/((e*x^2 + d)^2*x^6),x, algorithm="fricas")
[Out]
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Sympy [A] time = 9.70136, size = 284, normalized size = 2.09 \[ \frac{\sqrt{- \frac{e}{d^{9}}} \left (7 a e^{2} - 5 b d e + 3 c d^{2}\right ) \log{\left (- \frac{d^{5} \sqrt{- \frac{e}{d^{9}}} \left (7 a e^{2} - 5 b d e + 3 c d^{2}\right )}{7 a e^{3} - 5 b d e^{2} + 3 c d^{2} e} + x \right )}}{4} - \frac{\sqrt{- \frac{e}{d^{9}}} \left (7 a e^{2} - 5 b d e + 3 c d^{2}\right ) \log{\left (\frac{d^{5} \sqrt{- \frac{e}{d^{9}}} \left (7 a e^{2} - 5 b d e + 3 c d^{2}\right )}{7 a e^{3} - 5 b d e^{2} + 3 c d^{2} e} + x \right )}}{4} - \frac{6 a d^{3} + x^{6} \left (105 a e^{3} - 75 b d e^{2} + 45 c d^{2} e\right ) + x^{4} \left (70 a d e^{2} - 50 b d^{2} e + 30 c d^{3}\right ) + x^{2} \left (- 14 a d^{2} e + 10 b d^{3}\right )}{30 d^{5} x^{5} + 30 d^{4} e x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**4+b*x**2+a)/x**6/(e*x**2+d)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.270252, size = 177, normalized size = 1.3 \[ -\frac{{\left (3 \, c d^{2} e - 5 \, b d e^{2} + 7 \, a e^{3}\right )} \arctan \left (\frac{x e^{\frac{1}{2}}}{\sqrt{d}}\right ) e^{\left (-\frac{1}{2}\right )}}{2 \, d^{\frac{9}{2}}} - \frac{c d^{2} x e - b d x e^{2} + a x e^{3}}{2 \,{\left (x^{2} e + d\right )} d^{4}} - \frac{15 \, c d^{2} x^{4} - 30 \, b d x^{4} e + 45 \, a x^{4} e^{2} + 5 \, b d^{2} x^{2} - 10 \, a d x^{2} e + 3 \, a d^{2}}{15 \, d^{4} x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^4 + b*x^2 + a)/((e*x^2 + d)^2*x^6),x, algorithm="giac")
[Out]