Optimal. Leaf size=133 \[ \frac{5 b^4 \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{x}}}{\sqrt{a}}\right )}{32 a^{7/2}}-\frac{5 b^3 \sqrt{a+b \sqrt{x}}}{32 a^3 \sqrt{x}}+\frac{5 b^2 \sqrt{a+b \sqrt{x}}}{48 a^2 x}-\frac{b \sqrt{a+b \sqrt{x}}}{12 a x^{3/2}}-\frac{\sqrt{a+b \sqrt{x}}}{2 x^2} \]
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Rubi [A] time = 0.172506, antiderivative size = 133, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294 \[ \frac{5 b^4 \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{x}}}{\sqrt{a}}\right )}{32 a^{7/2}}-\frac{5 b^3 \sqrt{a+b \sqrt{x}}}{32 a^3 \sqrt{x}}+\frac{5 b^2 \sqrt{a+b \sqrt{x}}}{48 a^2 x}-\frac{b \sqrt{a+b \sqrt{x}}}{12 a x^{3/2}}-\frac{\sqrt{a+b \sqrt{x}}}{2 x^2} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[a + b*Sqrt[x]]/x^3,x]
[Out]
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Rubi in Sympy [A] time = 17.9616, size = 119, normalized size = 0.89 \[ - \frac{\sqrt{a + b \sqrt{x}}}{2 x^{2}} - \frac{b \sqrt{a + b \sqrt{x}}}{12 a x^{\frac{3}{2}}} + \frac{5 b^{2} \sqrt{a + b \sqrt{x}}}{48 a^{2} x} - \frac{5 b^{3} \sqrt{a + b \sqrt{x}}}{32 a^{3} \sqrt{x}} + \frac{5 b^{4} \operatorname{atanh}{\left (\frac{\sqrt{a + b \sqrt{x}}}{\sqrt{a}} \right )}}{32 a^{\frac{7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a+b*x**(1/2))**(1/2)/x**3,x)
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Mathematica [A] time = 0.0882718, size = 90, normalized size = 0.68 \[ \frac{5 b^4 \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{x}}}{\sqrt{a}}\right )}{32 a^{7/2}}-\frac{\sqrt{a+b \sqrt{x}} \left (48 a^3+8 a^2 b \sqrt{x}-10 a b^2 x+15 b^3 x^{3/2}\right )}{96 a^3 x^2} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[a + b*Sqrt[x]]/x^3,x]
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Maple [A] time = 0.015, size = 87, normalized size = 0.7 \[ 4\,{b}^{4} \left ({\frac{1}{{x}^{2}{b}^{4}} \left ( -{\frac{5\, \left ( a+b\sqrt{x} \right ) ^{7/2}}{128\,{a}^{3}}}+{\frac{55\, \left ( a+b\sqrt{x} \right ) ^{5/2}}{384\,{a}^{2}}}-{\frac{73\, \left ( a+b\sqrt{x} \right ) ^{3/2}}{384\,a}}-{\frac{5\,\sqrt{a+b\sqrt{x}}}{128}} \right ) }+{\frac{5}{128\,{a}^{7/2}}{\it Artanh} \left ({\frac{\sqrt{a+b\sqrt{x}}}{\sqrt{a}}} \right ) } \right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a+b*x^(1/2))^(1/2)/x^3,x)
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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(sqrt(b*sqrt(x) + a)/x^3,x, algorithm="maxima")
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Fricas [A] time = 0.255883, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, b^{4} x^{2} \log \left (\frac{\sqrt{a} b \sqrt{x} + 2 \, \sqrt{b \sqrt{x} + a} a + 2 \, a^{\frac{3}{2}}}{\sqrt{x}}\right ) - 2 \,{\left ({\left (15 \, b^{3} x + 8 \, a^{2} b\right )} \sqrt{a} \sqrt{x} - 2 \,{\left (5 \, a b^{2} x - 24 \, a^{3}\right )} \sqrt{a}\right )} \sqrt{b \sqrt{x} + a}}{192 \, a^{\frac{7}{2}} x^{2}}, -\frac{15 \, b^{4} x^{2} \arctan \left (\frac{a}{\sqrt{b \sqrt{x} + a} \sqrt{-a}}\right ) +{\left ({\left (15 \, b^{3} x + 8 \, a^{2} b\right )} \sqrt{-a} \sqrt{x} - 2 \,{\left (5 \, a b^{2} x - 24 \, a^{3}\right )} \sqrt{-a}\right )} \sqrt{b \sqrt{x} + a}}{96 \, \sqrt{-a} a^{3} x^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(b*sqrt(x) + a)/x^3,x, algorithm="fricas")
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Sympy [A] time = 35.6483, size = 170, normalized size = 1.28 \[ - \frac{a}{2 \sqrt{b} x^{\frac{9}{4}} \sqrt{\frac{a}{b \sqrt{x}} + 1}} - \frac{7 \sqrt{b}}{12 x^{\frac{7}{4}} \sqrt{\frac{a}{b \sqrt{x}} + 1}} + \frac{b^{\frac{3}{2}}}{48 a x^{\frac{5}{4}} \sqrt{\frac{a}{b \sqrt{x}} + 1}} - \frac{5 b^{\frac{5}{2}}}{96 a^{2} x^{\frac{3}{4}} \sqrt{\frac{a}{b \sqrt{x}} + 1}} - \frac{5 b^{\frac{7}{2}}}{32 a^{3} \sqrt [4]{x} \sqrt{\frac{a}{b \sqrt{x}} + 1}} + \frac{5 b^{4} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt [4]{x}} \right )}}{32 a^{\frac{7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a+b*x**(1/2))**(1/2)/x**3,x)
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GIAC/XCAS [A] time = 0.228737, size = 127, normalized size = 0.95 \[ -\frac{1}{96} \, b^{4}{\left (\frac{15 \, \arctan \left (\frac{\sqrt{b \sqrt{x} + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{3}} + \frac{15 \,{\left (b \sqrt{x} + a\right )}^{\frac{7}{2}} - 55 \,{\left (b \sqrt{x} + a\right )}^{\frac{5}{2}} a + 73 \,{\left (b \sqrt{x} + a\right )}^{\frac{3}{2}} a^{2} + 15 \, \sqrt{b \sqrt{x} + a} a^{3}}{a^{3} b^{4} x^{2}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(b*sqrt(x) + a)/x^3,x, algorithm="giac")
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