Optimal. Leaf size=205 \[ -\frac{10 a^2}{b^6 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{5 a (a+b x) \log (a+b x)}{b^6 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{x (a+b x)}{b^5 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{a^5}{4 b^6 (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{5 a^4}{3 b^6 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{5 a^3}{b^6 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.241021, antiderivative size = 205, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ -\frac{10 a^2}{b^6 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{5 a (a+b x) \log (a+b x)}{b^6 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{x (a+b x)}{b^5 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{a^5}{4 b^6 (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{5 a^4}{3 b^6 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{5 a^3}{b^6 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}} \]
Antiderivative was successfully verified.
[In] Int[x^5/(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 24.5336, size = 199, normalized size = 0.97 \[ - \frac{5 a \left (a + b x\right ) \log{\left (a + b x \right )}}{b^{6} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} - \frac{x^{5} \left (2 a + 2 b x\right )}{8 b \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}} - \frac{5 x^{4}}{12 b^{2} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}} - \frac{5 x^{3} \left (2 a + 2 b x\right )}{12 b^{3} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}} - \frac{5 x^{2}}{2 b^{4} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} + \frac{5 \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{b^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**5/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0595892, size = 93, normalized size = 0.45 \[ \frac{-77 a^5-248 a^4 b x-252 a^3 b^2 x^2-48 a^2 b^3 x^3+48 a b^4 x^4-60 a (a+b x)^4 \log (a+b x)+12 b^5 x^5}{12 b^6 (a+b x)^3 \sqrt{(a+b x)^2}} \]
Antiderivative was successfully verified.
[In] Integrate[x^5/(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.01, size = 145, normalized size = 0.7 \[ -{\frac{ \left ( 60\,\ln \left ( bx+a \right ){x}^{4}a{b}^{4}-12\,{b}^{5}{x}^{5}+240\,\ln \left ( bx+a \right ){x}^{3}{a}^{2}{b}^{3}-48\,a{b}^{4}{x}^{4}+360\,\ln \left ( bx+a \right ){x}^{2}{a}^{3}{b}^{2}+48\,{a}^{2}{b}^{3}{x}^{3}+240\,\ln \left ( bx+a \right ) x{a}^{4}b+252\,{a}^{3}{b}^{2}{x}^{2}+60\,\ln \left ( bx+a \right ){a}^{5}+248\,{a}^{4}bx+77\,{a}^{5} \right ) \left ( bx+a \right ) }{12\,{b}^{6}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{5}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^5/(b^2*x^2+2*a*b*x+a^2)^(5/2),x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 0.75791, size = 153, normalized size = 0.75 \[ \frac{12 \, b^{5} x^{5} + 48 \, a b^{4} x^{4} - 48 \, a^{2} b^{3} x^{3} - 252 \, a^{3} b^{2} x^{2} - 248 \, a^{4} b x - 77 \, a^{5}}{12 \,{\left (b^{10} x^{4} + 4 \, a b^{9} x^{3} + 6 \, a^{2} b^{8} x^{2} + 4 \, a^{3} b^{7} x + a^{4} b^{6}\right )}} - \frac{5 \, a \log \left (b x + a\right )}{b^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^5/(b^2*x^2 + 2*a*b*x + a^2)^(5/2),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.22903, size = 201, normalized size = 0.98 \[ \frac{12 \, b^{5} x^{5} + 48 \, a b^{4} x^{4} - 48 \, a^{2} b^{3} x^{3} - 252 \, a^{3} b^{2} x^{2} - 248 \, a^{4} b x - 77 \, a^{5} - 60 \,{\left (a b^{4} x^{4} + 4 \, a^{2} b^{3} x^{3} + 6 \, a^{3} b^{2} x^{2} + 4 \, a^{4} b x + a^{5}\right )} \log \left (b x + a\right )}{12 \,{\left (b^{10} x^{4} + 4 \, a b^{9} x^{3} + 6 \, a^{2} b^{8} x^{2} + 4 \, a^{3} b^{7} x + a^{4} b^{6}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^5/(b^2*x^2 + 2*a*b*x + a^2)^(5/2),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{5}}{\left (\left (a + b x\right )^{2}\right )^{\frac{5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**5/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.558379, size = 4, normalized size = 0.02 \[ \mathit{sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^5/(b^2*x^2 + 2*a*b*x + a^2)^(5/2),x, algorithm="giac")
[Out]