Optimal. Leaf size=55 \[ \frac{2 a^2}{b^3 \sqrt{a+\frac{b}{x}}}+\frac{4 a \sqrt{a+\frac{b}{x}}}{b^3}-\frac{2 \left (a+\frac{b}{x}\right )^{3/2}}{3 b^3} \]
[Out]
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Rubi [A] time = 0.0781181, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{2 a^2}{b^3 \sqrt{a+\frac{b}{x}}}+\frac{4 a \sqrt{a+\frac{b}{x}}}{b^3}-\frac{2 \left (a+\frac{b}{x}\right )^{3/2}}{3 b^3} \]
Antiderivative was successfully verified.
[In] Int[1/((a + b/x)^(3/2)*x^4),x]
[Out]
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Rubi in Sympy [A] time = 9.74647, size = 46, normalized size = 0.84 \[ \frac{2 a^{2}}{b^{3} \sqrt{a + \frac{b}{x}}} + \frac{4 a \sqrt{a + \frac{b}{x}}}{b^{3}} - \frac{2 \left (a + \frac{b}{x}\right )^{\frac{3}{2}}}{3 b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(a+b/x)**(3/2)/x**4,x)
[Out]
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Mathematica [A] time = 0.0411181, size = 46, normalized size = 0.84 \[ \sqrt{\frac{a x+b}{x}} \left (\frac{16 a}{3 b^3}-\frac{2 a}{b^2 (a x+b)}-\frac{2}{3 b^2 x}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[1/((a + b/x)^(3/2)*x^4),x]
[Out]
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Maple [A] time = 0.009, size = 44, normalized size = 0.8 \[{\frac{ \left ( 2\,ax+2\,b \right ) \left ( 8\,{a}^{2}{x}^{2}+4\,abx-{b}^{2} \right ) }{3\,{b}^{3}{x}^{3}} \left ({\frac{ax+b}{x}} \right ) ^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(a+b/x)^(3/2)/x^4,x)
[Out]
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Maxima [A] time = 1.43559, size = 63, normalized size = 1.15 \[ -\frac{2 \,{\left (a + \frac{b}{x}\right )}^{\frac{3}{2}}}{3 \, b^{3}} + \frac{4 \, \sqrt{a + \frac{b}{x}} a}{b^{3}} + \frac{2 \, a^{2}}{\sqrt{a + \frac{b}{x}} b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((a + b/x)^(3/2)*x^4),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.236646, size = 51, normalized size = 0.93 \[ \frac{2 \,{\left (8 \, a^{2} x^{2} + 4 \, a b x - b^{2}\right )}}{3 \, b^{3} x^{2} \sqrt{\frac{a x + b}{x}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((a + b/x)^(3/2)*x^4),x, algorithm="fricas")
[Out]
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Sympy [A] time = 7.9295, size = 457, normalized size = 8.31 \[ \frac{16 a^{\frac{9}{2}} b^{\frac{7}{2}} x^{3} \sqrt{\frac{a x}{b} + 1}}{3 a^{\frac{7}{2}} b^{6} x^{\frac{7}{2}} + 6 a^{\frac{5}{2}} b^{7} x^{\frac{5}{2}} + 3 a^{\frac{3}{2}} b^{8} x^{\frac{3}{2}}} + \frac{24 a^{\frac{7}{2}} b^{\frac{9}{2}} x^{2} \sqrt{\frac{a x}{b} + 1}}{3 a^{\frac{7}{2}} b^{6} x^{\frac{7}{2}} + 6 a^{\frac{5}{2}} b^{7} x^{\frac{5}{2}} + 3 a^{\frac{3}{2}} b^{8} x^{\frac{3}{2}}} + \frac{6 a^{\frac{5}{2}} b^{\frac{11}{2}} x \sqrt{\frac{a x}{b} + 1}}{3 a^{\frac{7}{2}} b^{6} x^{\frac{7}{2}} + 6 a^{\frac{5}{2}} b^{7} x^{\frac{5}{2}} + 3 a^{\frac{3}{2}} b^{8} x^{\frac{3}{2}}} - \frac{2 a^{\frac{3}{2}} b^{\frac{13}{2}} \sqrt{\frac{a x}{b} + 1}}{3 a^{\frac{7}{2}} b^{6} x^{\frac{7}{2}} + 6 a^{\frac{5}{2}} b^{7} x^{\frac{5}{2}} + 3 a^{\frac{3}{2}} b^{8} x^{\frac{3}{2}}} - \frac{16 a^{5} b^{3} x^{\frac{7}{2}}}{3 a^{\frac{7}{2}} b^{6} x^{\frac{7}{2}} + 6 a^{\frac{5}{2}} b^{7} x^{\frac{5}{2}} + 3 a^{\frac{3}{2}} b^{8} x^{\frac{3}{2}}} - \frac{32 a^{4} b^{4} x^{\frac{5}{2}}}{3 a^{\frac{7}{2}} b^{6} x^{\frac{7}{2}} + 6 a^{\frac{5}{2}} b^{7} x^{\frac{5}{2}} + 3 a^{\frac{3}{2}} b^{8} x^{\frac{3}{2}}} - \frac{16 a^{3} b^{5} x^{\frac{3}{2}}}{3 a^{\frac{7}{2}} b^{6} x^{\frac{7}{2}} + 6 a^{\frac{5}{2}} b^{7} x^{\frac{5}{2}} + 3 a^{\frac{3}{2}} b^{8} x^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(a+b/x)**(3/2)/x**4,x)
[Out]
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GIAC/XCAS [A] time = 0.256802, size = 93, normalized size = 1.69 \[ \frac{2}{3} \, b{\left (\frac{3 \, a^{2}}{b^{4} \sqrt{\frac{a x + b}{x}}} + \frac{6 \, a b^{8} \sqrt{\frac{a x + b}{x}} - \frac{{\left (a x + b\right )} b^{8} \sqrt{\frac{a x + b}{x}}}{x}}{b^{12}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((a + b/x)^(3/2)*x^4),x, algorithm="giac")
[Out]