Optimal. Leaf size=82 \[ -\frac{15 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{4 b^{7/2}}+\frac{5}{4 b^2 \sqrt{x} (a x+b)}+\frac{1}{2 b \sqrt{x} (a x+b)^2}-\frac{15}{4 b^3 \sqrt{x}} \]
[Out]
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Rubi [A] time = 0.0843754, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ -\frac{15 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{4 b^{7/2}}+\frac{5}{4 b^2 \sqrt{x} (a x+b)}+\frac{1}{2 b \sqrt{x} (a x+b)^2}-\frac{15}{4 b^3 \sqrt{x}} \]
Antiderivative was successfully verified.
[In] Int[1/((a + b/x)^3*x^(9/2)),x]
[Out]
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Rubi in Sympy [A] time = 14.3435, size = 75, normalized size = 0.91 \[ - \frac{15 \sqrt{a} \operatorname{atan}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}} \right )}}{4 b^{\frac{7}{2}}} + \frac{1}{2 b \sqrt{x} \left (a x + b\right )^{2}} + \frac{5}{4 b^{2} \sqrt{x} \left (a x + b\right )} - \frac{15}{4 b^{3} \sqrt{x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(a+b/x)**3/x**(9/2),x)
[Out]
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Mathematica [A] time = 0.0706817, size = 70, normalized size = 0.85 \[ -\frac{15 a^2 x^2+25 a b x+8 b^2}{4 b^3 \sqrt{x} (a x+b)^2}-\frac{15 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{4 b^{7/2}} \]
Antiderivative was successfully verified.
[In] Integrate[1/((a + b/x)^3*x^(9/2)),x]
[Out]
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Maple [A] time = 0.02, size = 66, normalized size = 0.8 \[ -2\,{\frac{1}{{b}^{3}\sqrt{x}}}-{\frac{7\,{a}^{2}}{4\,{b}^{3} \left ( ax+b \right ) ^{2}}{x}^{{\frac{3}{2}}}}-{\frac{9\,a}{4\,{b}^{2} \left ( ax+b \right ) ^{2}}\sqrt{x}}-{\frac{15\,a}{4\,{b}^{3}}\arctan \left ({a\sqrt{x}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(a+b/x)^3/x^(9/2),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/((a + b/x)^3*x^(9/2)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.246333, size = 1, normalized size = 0.01 \[ \left [-\frac{30 \, a^{2} x^{2} + 50 \, a b x - 15 \,{\left (a^{2} x^{2} + 2 \, a b x + b^{2}\right )} \sqrt{x} \sqrt{-\frac{a}{b}} \log \left (\frac{a x - 2 \, b \sqrt{x} \sqrt{-\frac{a}{b}} - b}{a x + b}\right ) + 16 \, b^{2}}{8 \,{\left (a^{2} b^{3} x^{2} + 2 \, a b^{4} x + b^{5}\right )} \sqrt{x}}, -\frac{15 \, a^{2} x^{2} + 25 \, a b x - 15 \,{\left (a^{2} x^{2} + 2 \, a b x + b^{2}\right )} \sqrt{x} \sqrt{\frac{a}{b}} \arctan \left (\frac{b \sqrt{\frac{a}{b}}}{a \sqrt{x}}\right ) + 8 \, b^{2}}{4 \,{\left (a^{2} b^{3} x^{2} + 2 \, a b^{4} x + b^{5}\right )} \sqrt{x}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((a + b/x)^3*x^(9/2)),x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(a+b/x)**3/x**(9/2),x)
[Out]
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GIAC/XCAS [A] time = 0.216672, size = 80, normalized size = 0.98 \[ -\frac{15 \, a \arctan \left (\frac{a \sqrt{x}}{\sqrt{a b}}\right )}{4 \, \sqrt{a b} b^{3}} - \frac{2}{b^{3} \sqrt{x}} - \frac{7 \, a^{2} x^{\frac{3}{2}} + 9 \, a b \sqrt{x}}{4 \,{\left (a x + b\right )}^{2} b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((a + b/x)^3*x^(9/2)),x, algorithm="giac")
[Out]