Optimal. Leaf size=72 \[ -\frac{3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{4 a^{5/2} \sqrt{b}}-\frac{3 \sqrt{x}}{4 a^2 (a-b x)}-\frac{\sqrt{x}}{2 a (a-b x)^2} \]
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Rubi [A] time = 0.0543917, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{4 a^{5/2} \sqrt{b}}-\frac{3 \sqrt{x}}{4 a^2 (a-b x)}-\frac{\sqrt{x}}{2 a (a-b x)^2} \]
Antiderivative was successfully verified.
[In] Int[1/(Sqrt[x]*(-a + b*x)^3),x]
[Out]
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Rubi in Sympy [A] time = 10.3299, size = 63, normalized size = 0.88 \[ - \frac{\sqrt{x}}{2 a \left (a - b x\right )^{2}} - \frac{3 \sqrt{x}}{4 a^{2} \left (a - b x\right )} - \frac{3 \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{4 a^{\frac{5}{2}} \sqrt{b}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(b*x-a)**3/x**(1/2),x)
[Out]
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Mathematica [A] time = 0.0487644, size = 60, normalized size = 0.83 \[ \frac{\sqrt{x} (3 b x-5 a)}{4 a^2 (a-b x)^2}-\frac{3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{4 a^{5/2} \sqrt{b}} \]
Antiderivative was successfully verified.
[In] Integrate[1/(Sqrt[x]*(-a + b*x)^3),x]
[Out]
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Maple [A] time = 0.011, size = 63, normalized size = 0.9 \[ -{\frac{1}{2\,a \left ( bx-a \right ) ^{2}}\sqrt{x}}-{\frac{3}{2\,a} \left ( -{\frac{1}{2\,a \left ( bx-a \right ) }\sqrt{x}}+{\frac{1}{2\,a}{\it Artanh} \left ({b\sqrt{x}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(b*x-a)^3/x^(1/2),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/((b*x - a)^3*sqrt(x)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.219454, size = 1, normalized size = 0.01 \[ \left [\frac{2 \, \sqrt{a b}{\left (3 \, b x - 5 \, a\right )} \sqrt{x} + 3 \,{\left (b^{2} x^{2} - 2 \, a b x + a^{2}\right )} \log \left (-\frac{2 \, a b \sqrt{x} - \sqrt{a b}{\left (b x + a\right )}}{b x - a}\right )}{8 \,{\left (a^{2} b^{2} x^{2} - 2 \, a^{3} b x + a^{4}\right )} \sqrt{a b}}, \frac{\sqrt{-a b}{\left (3 \, b x - 5 \, a\right )} \sqrt{x} + 3 \,{\left (b^{2} x^{2} - 2 \, a b x + a^{2}\right )} \arctan \left (\frac{a}{\sqrt{-a b} \sqrt{x}}\right )}{4 \,{\left (a^{2} b^{2} x^{2} - 2 \, a^{3} b x + a^{4}\right )} \sqrt{-a b}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x - a)^3*sqrt(x)),x, algorithm="fricas")
[Out]
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Sympy [A] time = 10.597, size = 1501, normalized size = 20.85 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(b*x-a)**3/x**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.204877, size = 69, normalized size = 0.96 \[ \frac{3 \, \arctan \left (\frac{b \sqrt{x}}{\sqrt{-a b}}\right )}{4 \, \sqrt{-a b} a^{2}} + \frac{3 \, b x^{\frac{3}{2}} - 5 \, a \sqrt{x}}{4 \,{\left (b x - a\right )}^{2} a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x - a)^3*sqrt(x)),x, algorithm="giac")
[Out]