Optimal. Leaf size=213 \[ \frac{7 x \left (a+b x^2\right )^2}{48 a^2 \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}+\frac{x \left (a+b x^2\right )}{8 a \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}+\frac{35 \left (a+b x^2\right )^5 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{128 a^{9/2} \sqrt{b} \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}+\frac{35 x \left (a+b x^2\right )^4}{128 a^4 \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}+\frac{35 x \left (a+b x^2\right )^3}{192 a^3 \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.1777, antiderivative size = 213, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ \frac{7 x \left (a+b x^2\right )^2}{48 a^2 \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}+\frac{x \left (a+b x^2\right )}{8 a \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}+\frac{35 \left (a+b x^2\right )^5 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{128 a^{9/2} \sqrt{b} \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}+\frac{35 x \left (a+b x^2\right )^4}{128 a^4 \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}+\frac{35 x \left (a+b x^2\right )^3}{192 a^3 \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \]
Antiderivative was successfully verified.
[In] Int[(a^2 + 2*a*b*x^2 + b^2*x^4)^(-5/2),x]
[Out]
_______________________________________________________________________________________
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(1/(b**2*x**4+2*a*b*x**2+a**2)**(5/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0725017, size = 105, normalized size = 0.49 \[ \frac{\sqrt{a} \sqrt{b} x \left (279 a^3+511 a^2 b x^2+385 a b^2 x^4+105 b^3 x^6\right )+105 \left (a+b x^2\right )^4 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{384 a^{9/2} \sqrt{b} \left (a+b x^2\right )^3 \sqrt{\left (a+b x^2\right )^2}} \]
Antiderivative was successfully verified.
[In] Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^(-5/2),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.012, size = 169, normalized size = 0.8 \[{\frac{b{x}^{2}+a}{384\,{a}^{4}} \left ( 105\,\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ){x}^{8}{b}^{4}+105\,\sqrt{ab}{x}^{7}{b}^{3}+420\,\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ){x}^{6}a{b}^{3}+385\,\sqrt{ab}{x}^{5}a{b}^{2}+630\,\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ){x}^{4}{a}^{2}{b}^{2}+511\,\sqrt{ab}{x}^{3}{a}^{2}b+420\,\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ){x}^{2}{a}^{3}b+279\,\sqrt{ab}x{a}^{3}+105\,\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ){a}^{4} \right ){\frac{1}{\sqrt{ab}}} \left ( \left ( b{x}^{2}+a \right ) ^{2} \right ) ^{-{\frac{5}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x)
[Out]
_______________________________________________________________________________________
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((b^2*x^4 + 2*a*b*x^2 + a^2)^(-5/2),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.269975, size = 1, normalized size = 0. \[ \left [\frac{105 \,{\left (b^{4} x^{8} + 4 \, a b^{3} x^{6} + 6 \, a^{2} b^{2} x^{4} + 4 \, a^{3} b x^{2} + a^{4}\right )} \log \left (\frac{2 \, a b x +{\left (b x^{2} - a\right )} \sqrt{-a b}}{b x^{2} + a}\right ) + 2 \,{\left (105 \, b^{3} x^{7} + 385 \, a b^{2} x^{5} + 511 \, a^{2} b x^{3} + 279 \, a^{3} x\right )} \sqrt{-a b}}{768 \,{\left (a^{4} b^{4} x^{8} + 4 \, a^{5} b^{3} x^{6} + 6 \, a^{6} b^{2} x^{4} + 4 \, a^{7} b x^{2} + a^{8}\right )} \sqrt{-a b}}, \frac{105 \,{\left (b^{4} x^{8} + 4 \, a b^{3} x^{6} + 6 \, a^{2} b^{2} x^{4} + 4 \, a^{3} b x^{2} + a^{4}\right )} \arctan \left (\frac{\sqrt{a b} x}{a}\right ) +{\left (105 \, b^{3} x^{7} + 385 \, a b^{2} x^{5} + 511 \, a^{2} b x^{3} + 279 \, a^{3} x\right )} \sqrt{a b}}{384 \,{\left (a^{4} b^{4} x^{8} + 4 \, a^{5} b^{3} x^{6} + 6 \, a^{6} b^{2} x^{4} + 4 \, a^{7} b x^{2} + a^{8}\right )} \sqrt{a b}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^(-5/2),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(b**2*x**4+2*a*b*x**2+a**2)**(5/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.60368, size = 4, normalized size = 0.02 \[ \mathit{sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^(-5/2),x, algorithm="giac")
[Out]