Optimal. Leaf size=81 \[ -\frac{3 b^2 \tanh ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{a x^2+b x^3}}\right )}{4 \sqrt{a}}-\frac{3 b \sqrt{a x^2+b x^3}}{4 x^2}-\frac{\left (a x^2+b x^3\right )^{3/2}}{2 x^5} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.160324, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ -\frac{3 b^2 \tanh ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{a x^2+b x^3}}\right )}{4 \sqrt{a}}-\frac{3 b \sqrt{a x^2+b x^3}}{4 x^2}-\frac{\left (a x^2+b x^3\right )^{3/2}}{2 x^5} \]
Antiderivative was successfully verified.
[In] Int[(a*x^2 + b*x^3)^(3/2)/x^6,x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 16.3087, size = 75, normalized size = 0.93 \[ - \frac{3 b \sqrt{a x^{2} + b x^{3}}}{4 x^{2}} - \frac{\left (a x^{2} + b x^{3}\right )^{\frac{3}{2}}}{2 x^{5}} - \frac{3 b^{2} \operatorname{atanh}{\left (\frac{\sqrt{a} x}{\sqrt{a x^{2} + b x^{3}}} \right )}}{4 \sqrt{a}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**3+a*x**2)**(3/2)/x**6,x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.069556, size = 82, normalized size = 1.01 \[ -\frac{\sqrt{x^2 (a+b x)} \left (3 b^2 x^2 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )+\sqrt{a} \sqrt{a+b x} (2 a+5 b x)\right )}{4 \sqrt{a} x^3 \sqrt{a+b x}} \]
Antiderivative was successfully verified.
[In] Integrate[(a*x^2 + b*x^3)^(3/2)/x^6,x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.017, size = 74, normalized size = 0.9 \[ -{\frac{1}{4\,{x}^{5}} \left ( b{x}^{3}+a{x}^{2} \right ) ^{{\frac{3}{2}}} \left ( 3\,{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ){x}^{2}{b}^{2}+5\, \left ( bx+a \right ) ^{3/2}\sqrt{a}-3\,\sqrt{bx+a}{a}^{3/2} \right ) \left ( bx+a \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{a}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^3+a*x^2)^(3/2)/x^6,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*x^3 + a*x^2)^(3/2)/x^6,x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.25484, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, \sqrt{a} b^{2} x^{3} \log \left (\frac{{\left (b x^{2} + 2 \, a x\right )} \sqrt{a} - 2 \, \sqrt{b x^{3} + a x^{2}} a}{x^{2}}\right ) - 2 \, \sqrt{b x^{3} + a x^{2}}{\left (5 \, a b x + 2 \, a^{2}\right )}}{8 \, a x^{3}}, -\frac{3 \, \sqrt{-a} b^{2} x^{3} \arctan \left (\frac{a x}{\sqrt{b x^{3} + a x^{2}} \sqrt{-a}}\right ) + \sqrt{b x^{3} + a x^{2}}{\left (5 \, a b x + 2 \, a^{2}\right )}}{4 \, a x^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^3 + a*x^2)^(3/2)/x^6,x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (x^{2} \left (a + b x\right )\right )^{\frac{3}{2}}}{x^{6}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x**3+a*x**2)**(3/2)/x**6,x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.248436, size = 95, normalized size = 1.17 \[ \frac{\frac{3 \, b^{3} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right ){\rm sign}\left (x\right )}{\sqrt{-a}} - \frac{5 \,{\left (b x + a\right )}^{\frac{3}{2}} b^{3}{\rm sign}\left (x\right ) - 3 \, \sqrt{b x + a} a b^{3}{\rm sign}\left (x\right )}{b^{2} x^{2}}}{4 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^3 + a*x^2)^(3/2)/x^6,x, algorithm="giac")
[Out]