Optimal. Leaf size=75 \[ \frac{\sqrt{a^2+2 a b x^2+b^2 x^4}}{2 b^2}-\frac{a \left (a+b x^2\right ) \log \left (a+b x^2\right )}{2 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.146885, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{\sqrt{a^2+2 a b x^2+b^2 x^4}}{2 b^2}-\frac{a \left (a+b x^2\right ) \log \left (a+b x^2\right )}{2 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}} \]
Antiderivative was successfully verified.
[In] Int[x^3/Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4],x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{3}}{\sqrt{\left (a + b x^{2}\right )^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**3/((b*x**2+a)**2)**(1/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.020955, size = 44, normalized size = 0.59 \[ \frac{\left (a+b x^2\right ) \left (b x^2-a \log \left (a+b x^2\right )\right )}{2 b^2 \sqrt{\left (a+b x^2\right )^2}} \]
Antiderivative was successfully verified.
[In] Integrate[x^3/Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4],x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.011, size = 41, normalized size = 0.6 \[ -{\frac{ \left ( b{x}^{2}+a \right ) \left ( -b{x}^{2}+a\ln \left ( b{x}^{2}+a \right ) \right ) }{2\,{b}^{2}}{\frac{1}{\sqrt{ \left ( b{x}^{2}+a \right ) ^{2}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^3/((b*x^2+a)^2)^(1/2),x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 0.704752, size = 63, normalized size = 0.84 \[ -\frac{a \sqrt{\frac{1}{b^{2}}} \log \left (x^{2} + \frac{a}{b}\right )}{2 \, b} + \frac{\sqrt{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}}{2 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^3/sqrt((b*x^2 + a)^2),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.257117, size = 30, normalized size = 0.4 \[ \frac{b x^{2} - a \log \left (b x^{2} + a\right )}{2 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^3/sqrt((b*x^2 + a)^2),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 1.16651, size = 20, normalized size = 0.27 \[ - \frac{a \log{\left (a + b x^{2} \right )}}{2 b^{2}} + \frac{x^{2}}{2 b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**3/((b*x**2+a)**2)**(1/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.270255, size = 45, normalized size = 0.6 \[ \frac{1}{2} \,{\left (\frac{x^{2}}{b} - \frac{a{\rm ln}\left ({\left | b x^{2} + a \right |}\right )}{b^{2}}\right )}{\rm sign}\left (b x^{2} + a\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^3/sqrt((b*x^2 + a)^2),x, algorithm="giac")
[Out]