Optimal. Leaf size=86 \[ \frac{c^2 \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{4 b^{3/2}}-\frac{c \sqrt{b x+c x^2}}{4 b x^{3/2}}-\frac{\sqrt{b x+c x^2}}{2 x^{5/2}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.10688, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ \frac{c^2 \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{4 b^{3/2}}-\frac{c \sqrt{b x+c x^2}}{4 b x^{3/2}}-\frac{\sqrt{b x+c x^2}}{2 x^{5/2}} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[b*x + c*x^2]/x^(7/2),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 13.0746, size = 73, normalized size = 0.85 \[ - \frac{\sqrt{b x + c x^{2}}}{2 x^{\frac{5}{2}}} - \frac{c \sqrt{b x + c x^{2}}}{4 b x^{\frac{3}{2}}} + \frac{c^{2} \operatorname{atanh}{\left (\frac{\sqrt{b x + c x^{2}}}{\sqrt{b} \sqrt{x}} \right )}}{4 b^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**2+b*x)**(1/2)/x**(7/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0628863, size = 81, normalized size = 0.94 \[ \frac{\sqrt{x (b+c x)} \left (c^2 x^2 \tanh ^{-1}\left (\frac{\sqrt{b+c x}}{\sqrt{b}}\right )-\sqrt{b} \sqrt{b+c x} (2 b+c x)\right )}{4 b^{3/2} x^{5/2} \sqrt{b+c x}} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[b*x + c*x^2]/x^(7/2),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.013, size = 71, normalized size = 0.8 \[{\frac{1}{4}\sqrt{x \left ( cx+b \right ) } \left ({\it Artanh} \left ({1\sqrt{cx+b}{\frac{1}{\sqrt{b}}}} \right ){c}^{2}{x}^{2}-xc\sqrt{cx+b}\sqrt{b}-2\,{b}^{3/2}\sqrt{cx+b} \right ){b}^{-{\frac{3}{2}}}{x}^{-{\frac{5}{2}}}{\frac{1}{\sqrt{cx+b}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^2+b*x)^(1/2)/x^(7/2),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(sqrt(c*x^2 + b*x)/x^(7/2),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.230496, size = 1, normalized size = 0.01 \[ \left [\frac{c^{2} x^{3} \log \left (-\frac{2 \, \sqrt{c x^{2} + b x} b \sqrt{x} +{\left (c x^{2} + 2 \, b x\right )} \sqrt{b}}{x^{2}}\right ) - 2 \, \sqrt{c x^{2} + b x}{\left (c x + 2 \, b\right )} \sqrt{b} \sqrt{x}}{8 \, b^{\frac{3}{2}} x^{3}}, \frac{c^{2} x^{3} \arctan \left (\frac{\sqrt{-b} \sqrt{x}}{\sqrt{c x^{2} + b x}}\right ) - \sqrt{c x^{2} + b x}{\left (c x + 2 \, b\right )} \sqrt{-b} \sqrt{x}}{4 \, \sqrt{-b} b x^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^2 + b*x)/x^(7/2),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{x \left (b + c x\right )}}{x^{\frac{7}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**2+b*x)**(1/2)/x**(7/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.22872, size = 76, normalized size = 0.88 \[ -\frac{1}{4} \, c^{2}{\left (\frac{\arctan \left (\frac{\sqrt{c x + b}}{\sqrt{-b}}\right )}{\sqrt{-b} b} + \frac{{\left (c x + b\right )}^{\frac{3}{2}} + \sqrt{c x + b} b}{b c^{2} x^{2}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^2 + b*x)/x^(7/2),x, algorithm="giac")
[Out]