Optimal. Leaf size=81 \[ \frac{3 c \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{b^{5/2}}-\frac{3 c \sqrt{x}}{b^2 \sqrt{b x+c x^2}}-\frac{1}{b \sqrt{x} \sqrt{b x+c x^2}} \]
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Rubi [A] time = 0.102254, antiderivative size = 81, 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 \[ \frac{3 c \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{b^{5/2}}-\frac{3 c \sqrt{x}}{b^2 \sqrt{b x+c x^2}}-\frac{1}{b \sqrt{x} \sqrt{b x+c x^2}} \]
Antiderivative was successfully verified.
[In] Int[1/(Sqrt[x]*(b*x + c*x^2)^(3/2)),x]
[Out]
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Rubi in Sympy [A] time = 11.9305, size = 75, normalized size = 0.93 \[ - \frac{1}{b \sqrt{x} \sqrt{b x + c x^{2}}} - \frac{3 c \sqrt{x}}{b^{2} \sqrt{b x + c x^{2}}} + \frac{3 c \operatorname{atanh}{\left (\frac{\sqrt{b x + c x^{2}}}{\sqrt{b} \sqrt{x}} \right )}}{b^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**(1/2)/(c*x**2+b*x)**(3/2),x)
[Out]
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Mathematica [A] time = 0.0555433, size = 65, normalized size = 0.8 \[ \frac{3 c x \sqrt{b+c x} \tanh ^{-1}\left (\frac{\sqrt{b+c x}}{\sqrt{b}}\right )-\sqrt{b} (b+3 c x)}{b^{5/2} \sqrt{x} \sqrt{x (b+c x)}} \]
Antiderivative was successfully verified.
[In] Integrate[1/(Sqrt[x]*(b*x + c*x^2)^(3/2)),x]
[Out]
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Maple [A] time = 0.014, size = 60, normalized size = 0.7 \[{\frac{1}{cx+b}\sqrt{x \left ( cx+b \right ) } \left ( 3\,{\it Artanh} \left ({\frac{\sqrt{cx+b}}{\sqrt{b}}} \right ) \sqrt{cx+b}xc-3\,cx\sqrt{b}-{b}^{{\frac{3}{2}}} \right ){x}^{-{\frac{3}{2}}}{b}^{-{\frac{5}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^(1/2)/(c*x^2+b*x)^(3/2),x)
[Out]
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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(1/((c*x^2 + b*x)^(3/2)*sqrt(x)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.231356, size = 1, normalized size = 0.01 \[ \left [-\frac{2 \, \sqrt{c x^{2} + b x}{\left (3 \, c x + b\right )} \sqrt{b} \sqrt{x} - 3 \,{\left (c^{2} x^{3} + b c x^{2}\right )} \log \left (-\frac{2 \, \sqrt{c x^{2} + b x} b \sqrt{x} +{\left (c x^{2} + 2 \, b x\right )} \sqrt{b}}{x^{2}}\right )}{2 \,{\left (b^{2} c x^{3} + b^{3} x^{2}\right )} \sqrt{b}}, -\frac{\sqrt{c x^{2} + b x}{\left (3 \, c x + b\right )} \sqrt{-b} \sqrt{x} - 3 \,{\left (c^{2} x^{3} + b c x^{2}\right )} \arctan \left (\frac{\sqrt{-b} \sqrt{x}}{\sqrt{c x^{2} + b x}}\right )}{{\left (b^{2} c x^{3} + b^{3} x^{2}\right )} \sqrt{-b}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((c*x^2 + b*x)^(3/2)*sqrt(x)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{x} \left (x \left (b + c x\right )\right )^{\frac{3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**(1/2)/(c*x**2+b*x)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.229448, size = 78, normalized size = 0.96 \[ -c{\left (\frac{3 \, \arctan \left (\frac{\sqrt{c x + b}}{\sqrt{-b}}\right )}{\sqrt{-b} b^{2}} + \frac{3 \, c x + b}{{\left ({\left (c x + b\right )}^{\frac{3}{2}} - \sqrt{c x + b} b\right )} b^{2}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((c*x^2 + b*x)^(3/2)*sqrt(x)),x, algorithm="giac")
[Out]