Optimal. Leaf size=137 \[ -\frac{\log (x) \left (a+b x^2\right ) (b d-a e)}{a^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (a+b x^2\right ) (b d-a e) \log \left (a+b x^2\right )}{2 a^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d \left (a+b x^2\right )}{2 a x^2 \sqrt{a^2+2 a b x^2+b^2 x^4}} \]
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Rubi [A] time = 0.262672, antiderivative size = 137, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ -\frac{\log (x) \left (a+b x^2\right ) (b d-a e)}{a^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (a+b x^2\right ) (b d-a e) \log \left (a+b x^2\right )}{2 a^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d \left (a+b x^2\right )}{2 a x^2 \sqrt{a^2+2 a b x^2+b^2 x^4}} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x^2)/(x^3*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]),x]
[Out]
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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((e*x**2+d)/x**3/((b*x**2+a)**2)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0560914, size = 70, normalized size = 0.51 \[ \frac{\left (a+b x^2\right ) \left (2 x^2 \log (x) (a e-b d)+x^2 (b d-a e) \log \left (a+b x^2\right )-a d\right )}{2 a^2 x^2 \sqrt{\left (a+b x^2\right )^2}} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x^2)/(x^3*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]),x]
[Out]
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Maple [A] time = 0.017, size = 78, normalized size = 0.6 \[ -{\frac{ \left ( b{x}^{2}+a \right ) \left ( \ln \left ( b{x}^{2}+a \right ){x}^{2}ae-\ln \left ( b{x}^{2}+a \right ){x}^{2}bd-2\,\ln \left ( x \right ){x}^{2}ae+2\,\ln \left ( x \right ){x}^{2}bd+ad \right ) }{2\,{x}^{2}{a}^{2}}{\frac{1}{\sqrt{ \left ( b{x}^{2}+a \right ) ^{2}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x^2+d)/x^3/((b*x^2+a)^2)^(1/2),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)/(sqrt((b*x^2 + a)^2)*x^3),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.266636, size = 65, normalized size = 0.47 \[ \frac{{\left (b d - a e\right )} x^{2} \log \left (b x^{2} + a\right ) - 2 \,{\left (b d - a e\right )} x^{2} \log \left (x\right ) - a d}{2 \, a^{2} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x^2 + d)/(sqrt((b*x^2 + a)^2)*x^3),x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.9685, size = 41, normalized size = 0.3 \[ - \frac{d}{2 a x^{2}} + \frac{\left (a e - b d\right ) \log{\left (x \right )}}{a^{2}} - \frac{\left (a e - b d\right ) \log{\left (\frac{a}{b} + x^{2} \right )}}{2 a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x**2+d)/x**3/((b*x**2+a)**2)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.265249, size = 177, normalized size = 1.29 \[ -\frac{{\left (b d{\rm sign}\left (b x^{2} + a\right ) - a e{\rm sign}\left (b x^{2} + a\right )\right )}{\rm ln}\left (x^{2}\right )}{2 \, a^{2}} + \frac{{\left (b^{2} d{\rm sign}\left (b x^{2} + a\right ) - a b e{\rm sign}\left (b x^{2} + a\right )\right )}{\rm ln}\left ({\left | b x^{2} + a \right |}\right )}{2 \, a^{2} b} + \frac{b d x^{2}{\rm sign}\left (b x^{2} + a\right ) - a x^{2} e{\rm sign}\left (b x^{2} + a\right ) - a d{\rm sign}\left (b x^{2} + a\right )}{2 \, a^{2} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x^2 + d)/(sqrt((b*x^2 + a)^2)*x^3),x, algorithm="giac")
[Out]