Optimal. Leaf size=110 \[ -\frac{63 a^{5/2} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{4 b^{11/2}}-\frac{63 a^2}{4 b^5 \sqrt{x}}+\frac{21 a}{4 b^4 x^{3/2}}+\frac{9}{4 b^2 x^{5/2} (a x+b)}+\frac{1}{2 b x^{5/2} (a x+b)^2}-\frac{63}{20 b^3 x^{5/2}} \]
[Out]
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Rubi [A] time = 0.123342, antiderivative size = 110, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ -\frac{63 a^{5/2} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{4 b^{11/2}}-\frac{63 a^2}{4 b^5 \sqrt{x}}+\frac{21 a}{4 b^4 x^{3/2}}+\frac{9}{4 b^2 x^{5/2} (a x+b)}+\frac{1}{2 b x^{5/2} (a x+b)^2}-\frac{63}{20 b^3 x^{5/2}} \]
Antiderivative was successfully verified.
[In] Int[1/((a + b/x)^3*x^(13/2)),x]
[Out]
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Rubi in Sympy [A] time = 22.2454, size = 104, normalized size = 0.95 \[ - \frac{63 a^{\frac{5}{2}} \operatorname{atan}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}} \right )}}{4 b^{\frac{11}{2}}} - \frac{63 a^{2}}{4 b^{5} \sqrt{x}} + \frac{21 a}{4 b^{4} x^{\frac{3}{2}}} + \frac{1}{2 b x^{\frac{5}{2}} \left (a x + b\right )^{2}} + \frac{9}{4 b^{2} x^{\frac{5}{2}} \left (a x + b\right )} - \frac{63}{20 b^{3} x^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(a+b/x)**3/x**(13/2),x)
[Out]
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Mathematica [A] time = 0.0951834, size = 92, normalized size = 0.84 \[ -\frac{63 a^{5/2} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{4 b^{11/2}}-\frac{315 a^4 x^4+525 a^3 b x^3+168 a^2 b^2 x^2-24 a b^3 x+8 b^4}{20 b^5 x^{5/2} (a x+b)^2} \]
Antiderivative was successfully verified.
[In] Integrate[1/((a + b/x)^3*x^(13/2)),x]
[Out]
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Maple [A] time = 0.023, size = 90, normalized size = 0.8 \[ -{\frac{2}{5\,{b}^{3}}{x}^{-{\frac{5}{2}}}}-12\,{\frac{{a}^{2}}{{b}^{5}\sqrt{x}}}+2\,{\frac{a}{{b}^{4}{x}^{3/2}}}-{\frac{15\,{a}^{4}}{4\,{b}^{5} \left ( ax+b \right ) ^{2}}{x}^{{\frac{3}{2}}}}-{\frac{17\,{a}^{3}}{4\,{b}^{4} \left ( ax+b \right ) ^{2}}\sqrt{x}}-{\frac{63\,{a}^{3}}{4\,{b}^{5}}\arctan \left ({a\sqrt{x}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(a+b/x)^3/x^(13/2),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^(13/2)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.241508, size = 1, normalized size = 0.01 \[ \left [-\frac{630 \, a^{4} x^{4} + 1050 \, a^{3} b x^{3} + 336 \, a^{2} b^{2} x^{2} - 48 \, a b^{3} x + 16 \, b^{4} - 315 \,{\left (a^{4} x^{4} + 2 \, a^{3} b x^{3} + a^{2} b^{2} x^{2}\right )} \sqrt{x} \sqrt{-\frac{a}{b}} \log \left (\frac{a x - 2 \, b \sqrt{x} \sqrt{-\frac{a}{b}} - b}{a x + b}\right )}{40 \,{\left (a^{2} b^{5} x^{4} + 2 \, a b^{6} x^{3} + b^{7} x^{2}\right )} \sqrt{x}}, -\frac{315 \, a^{4} x^{4} + 525 \, a^{3} b x^{3} + 168 \, a^{2} b^{2} x^{2} - 24 \, a b^{3} x + 8 \, b^{4} - 315 \,{\left (a^{4} x^{4} + 2 \, a^{3} b x^{3} + a^{2} b^{2} x^{2}\right )} \sqrt{x} \sqrt{\frac{a}{b}} \arctan \left (\frac{b \sqrt{\frac{a}{b}}}{a \sqrt{x}}\right )}{20 \,{\left (a^{2} b^{5} x^{4} + 2 \, a b^{6} x^{3} + b^{7} x^{2}\right )} \sqrt{x}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((a + b/x)^3*x^(13/2)),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**(13/2),x)
[Out]
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GIAC/XCAS [A] time = 0.220981, size = 108, normalized size = 0.98 \[ -\frac{63 \, a^{3} \arctan \left (\frac{a \sqrt{x}}{\sqrt{a b}}\right )}{4 \, \sqrt{a b} b^{5}} - \frac{15 \, a^{4} x^{\frac{3}{2}} + 17 \, a^{3} b \sqrt{x}}{4 \,{\left (a x + b\right )}^{2} b^{5}} - \frac{2 \,{\left (30 \, a^{2} x^{2} - 5 \, a b x + b^{2}\right )}}{5 \, b^{5} x^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((a + b/x)^3*x^(13/2)),x, algorithm="giac")
[Out]