Optimal. Leaf size=83 \[ \frac{\left (a+b x^2\right ) (b d-a e) \log \left (a+b x^2\right )}{2 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{e \sqrt{a^2+2 a b x^2+b^2 x^4}}{2 b^2} \]
[Out]
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Rubi [A] time = 0.191917, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.129 \[ \frac{\left (a+b x^2\right ) (b d-a e) \log \left (a+b x^2\right )}{2 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{e \sqrt{a^2+2 a b x^2+b^2 x^4}}{2 b^2} \]
Antiderivative was successfully verified.
[In] Int[(x*(d + e*x^2))/Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4],x]
[Out]
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Rubi in Sympy [A] time = 24.5963, size = 76, normalized size = 0.92 \[ \frac{e \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{2 b^{2}} - \frac{\left (a + b x^{2}\right ) \left (a e - b d\right ) \log{\left (a + b x^{2} \right )}}{2 b^{2} \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x*(e*x**2+d)/((b*x**2+a)**2)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0390962, size = 51, normalized size = 0.61 \[ \frac{\left (a+b x^2\right ) \left ((b d-a e) \log \left (a+b x^2\right )+b e x^2\right )}{2 b^2 \sqrt{\left (a+b x^2\right )^2}} \]
Antiderivative was successfully verified.
[In] Integrate[(x*(d + e*x^2))/Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4],x]
[Out]
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Maple [A] time = 0.01, size = 55, normalized size = 0.7 \[ -{\frac{ \left ( b{x}^{2}+a \right ) \left ( -{x}^{2}be+\ln \left ( b{x}^{2}+a \right ) ae-\ln \left ( b{x}^{2}+a \right ) bd \right ) }{2\,{b}^{2}}{\frac{1}{\sqrt{ \left ( b{x}^{2}+a \right ) ^{2}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x*(e*x^2+d)/((b*x^2+a)^2)^(1/2),x)
[Out]
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Maxima [A] time = 0.71101, size = 92, normalized size = 1.11 \[ \frac{1}{2} \, \sqrt{\frac{1}{b^{2}}} d \log \left (x^{2} + \frac{a}{b}\right ) - \frac{1}{2} \,{\left (\frac{a \sqrt{\frac{1}{b^{2}}} \log \left (x^{2} + \frac{a}{b}\right )}{b} - \frac{\sqrt{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}}{b^{2}}\right )} e \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x^2 + d)*x/sqrt((b*x^2 + a)^2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.28958, size = 39, normalized size = 0.47 \[ \frac{b e x^{2} +{\left (b d - a e\right )} \log \left (b x^{2} + a\right )}{2 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x^2 + d)*x/sqrt((b*x^2 + a)^2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 1.51513, size = 27, normalized size = 0.33 \[ \frac{e x^{2}}{2 b} - \frac{\left (a e - b d\right ) \log{\left (a + b x^{2} \right )}}{2 b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*(e*x**2+d)/((b*x**2+a)**2)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.263942, size = 57, normalized size = 0.69 \[ \frac{1}{2} \,{\left (\frac{x^{2} e}{b} + \frac{{\left (b d - a e\right )}{\rm ln}\left ({\left | b x^{2} + a \right |}\right )}{b^{2}}\right )}{\rm sign}\left (b x^{2} + a\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x^2 + d)*x/sqrt((b*x^2 + a)^2),x, algorithm="giac")
[Out]