Optimal. Leaf size=94 \[ -\frac{5 a \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a x+b x^2}}\right )}{b^{7/2}}+\frac{5 \sqrt{a x+b x^2}}{b^3}-\frac{10 x^2}{3 b^2 \sqrt{a x+b x^2}}-\frac{2 x^4}{3 b \left (a x+b x^2\right )^{3/2}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.130669, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235 \[ -\frac{5 a \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a x+b x^2}}\right )}{b^{7/2}}+\frac{5 \sqrt{a x+b x^2}}{b^3}-\frac{10 x^2}{3 b^2 \sqrt{a x+b x^2}}-\frac{2 x^4}{3 b \left (a x+b x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[x^5/(a*x + b*x^2)^(5/2),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 15.0416, size = 87, normalized size = 0.93 \[ - \frac{5 a \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a x + b x^{2}}} \right )}}{b^{\frac{7}{2}}} - \frac{2 x^{4}}{3 b \left (a x + b x^{2}\right )^{\frac{3}{2}}} - \frac{10 x^{2}}{3 b^{2} \sqrt{a x + b x^{2}}} + \frac{5 \sqrt{a x + b x^{2}}}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**5/(b*x**2+a*x)**(5/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0778944, size = 89, normalized size = 0.95 \[ \frac{x \left (\sqrt{b} x \left (15 a^2+20 a b x+3 b^2 x^2\right )-15 a \sqrt{x} (a+b x)^{3/2} \log \left (\sqrt{b} \sqrt{a+b x}+b \sqrt{x}\right )\right )}{3 b^{7/2} (x (a+b x))^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[x^5/(a*x + b*x^2)^(5/2),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.008, size = 149, normalized size = 1.6 \[{\frac{{x}^{4}}{b} \left ( b{x}^{2}+ax \right ) ^{-{\frac{3}{2}}}}+{\frac{5\,a{x}^{3}}{6\,{b}^{2}} \left ( b{x}^{2}+ax \right ) ^{-{\frac{3}{2}}}}-{\frac{5\,{a}^{2}{x}^{2}}{4\,{b}^{3}} \left ( b{x}^{2}+ax \right ) ^{-{\frac{3}{2}}}}-{\frac{5\,{a}^{3}x}{12\,{b}^{4}} \left ( b{x}^{2}+ax \right ) ^{-{\frac{3}{2}}}}+{\frac{35\,ax}{6\,{b}^{3}}{\frac{1}{\sqrt{b{x}^{2}+ax}}}}+{\frac{5\,{a}^{2}}{12\,{b}^{4}}{\frac{1}{\sqrt{b{x}^{2}+ax}}}}-{\frac{5\,a}{2}\ln \left ({1 \left ({\frac{a}{2}}+bx \right ){\frac{1}{\sqrt{b}}}}+\sqrt{b{x}^{2}+ax} \right ){b}^{-{\frac{7}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^5/(b*x^2+a*x)^(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(x^5/(b*x^2 + a*x)^(5/2),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.240203, size = 1, normalized size = 0.01 \[ \left [\frac{15 \,{\left (a b x + a^{2}\right )} \sqrt{b x^{2} + a x} \log \left ({\left (2 \, b x + a\right )} \sqrt{b} - 2 \, \sqrt{b x^{2} + a x} b\right ) + 2 \,{\left (3 \, b^{2} x^{3} + 20 \, a b x^{2} + 15 \, a^{2} x\right )} \sqrt{b}}{6 \,{\left (b^{4} x + a b^{3}\right )} \sqrt{b x^{2} + a x} \sqrt{b}}, -\frac{15 \,{\left (a b x + a^{2}\right )} \sqrt{b x^{2} + a x} \arctan \left (\frac{\sqrt{b x^{2} + a x} \sqrt{-b}}{b x}\right ) -{\left (3 \, b^{2} x^{3} + 20 \, a b x^{2} + 15 \, a^{2} x\right )} \sqrt{-b}}{3 \,{\left (b^{4} x + a b^{3}\right )} \sqrt{b x^{2} + a x} \sqrt{-b}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^5/(b*x^2 + a*x)^(5/2),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{5}}{\left (x \left (a + b x\right )\right )^{\frac{5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**5/(b*x**2+a*x)**(5/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^5/(b*x^2 + a*x)^(5/2),x, algorithm="giac")
[Out]