Optimal. Leaf size=136 \[ -\frac{35 b^4 \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{x}}}{\sqrt{a}}\right )}{32 a^{9/2}}+\frac{35 b^3 \sqrt{a+b \sqrt{x}}}{32 a^4 \sqrt{x}}-\frac{35 b^2 \sqrt{a+b \sqrt{x}}}{48 a^3 x}+\frac{7 b \sqrt{a+b \sqrt{x}}}{12 a^2 x^{3/2}}-\frac{\sqrt{a+b \sqrt{x}}}{2 a x^2} \]
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Rubi [A] time = 0.174682, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235 \[ -\frac{35 b^4 \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{x}}}{\sqrt{a}}\right )}{32 a^{9/2}}+\frac{35 b^3 \sqrt{a+b \sqrt{x}}}{32 a^4 \sqrt{x}}-\frac{35 b^2 \sqrt{a+b \sqrt{x}}}{48 a^3 x}+\frac{7 b \sqrt{a+b \sqrt{x}}}{12 a^2 x^{3/2}}-\frac{\sqrt{a+b \sqrt{x}}}{2 a x^2} \]
Antiderivative was successfully verified.
[In] Int[1/(Sqrt[a + b*Sqrt[x]]*x^3),x]
[Out]
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Rubi in Sympy [A] time = 18.6339, size = 124, normalized size = 0.91 \[ - \frac{\sqrt{a + b \sqrt{x}}}{2 a x^{2}} + \frac{7 b \sqrt{a + b \sqrt{x}}}{12 a^{2} x^{\frac{3}{2}}} - \frac{35 b^{2} \sqrt{a + b \sqrt{x}}}{48 a^{3} x} + \frac{35 b^{3} \sqrt{a + b \sqrt{x}}}{32 a^{4} \sqrt{x}} - \frac{35 b^{4} \operatorname{atanh}{\left (\frac{\sqrt{a + b \sqrt{x}}}{\sqrt{a}} \right )}}{32 a^{\frac{9}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**3/(a+b*x**(1/2))**(1/2),x)
[Out]
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Mathematica [A] time = 0.0978351, size = 90, normalized size = 0.66 \[ \frac{\sqrt{a+b \sqrt{x}} \left (-48 a^3+56 a^2 b \sqrt{x}-70 a b^2 x+105 b^3 x^{3/2}\right )}{96 a^4 x^2}-\frac{35 b^4 \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{x}}}{\sqrt{a}}\right )}{32 a^{9/2}} \]
Antiderivative was successfully verified.
[In] Integrate[1/(Sqrt[a + b*Sqrt[x]]*x^3),x]
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Maple [A] time = 0.008, size = 124, normalized size = 0.9 \[ 4\,{b}^{4} \left ( -1/8\,{\frac{\sqrt{a+b\sqrt{x}}}{a{b}^{4}{x}^{2}}}-{\frac{7}{8\,a} \left ( -1/6\,{\frac{\sqrt{a+b\sqrt{x}}}{a{x}^{3/2}{b}^{3}}}-5/6\,{\frac{1}{a} \left ( -1/4\,{\frac{\sqrt{a+b\sqrt{x}}}{a{b}^{2}x}}-3/4\,{\frac{1}{a} \left ( -1/2\,{\frac{\sqrt{a+b\sqrt{x}}}{ab\sqrt{x}}}+1/2\,{\frac{1}{{a}^{3/2}}{\it Artanh} \left ({\frac{\sqrt{a+b\sqrt{x}}}{\sqrt{a}}} \right ) } \right ) } \right ) } \right ) } \right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^3/(a+b*x^(1/2))^(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/(sqrt(b*sqrt(x) + a)*x^3),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.255696, size = 1, normalized size = 0.01 \[ \left [\frac{105 \, b^{4} x^{2} \log \left (\frac{\sqrt{a} b \sqrt{x} - 2 \, \sqrt{b \sqrt{x} + a} a + 2 \, a^{\frac{3}{2}}}{\sqrt{x}}\right ) + 2 \,{\left (7 \,{\left (15 \, b^{3} x + 8 \, a^{2} b\right )} \sqrt{a} \sqrt{x} - 2 \,{\left (35 \, a b^{2} x + 24 \, a^{3}\right )} \sqrt{a}\right )} \sqrt{b \sqrt{x} + a}}{192 \, a^{\frac{9}{2}} x^{2}}, \frac{105 \, b^{4} x^{2} \arctan \left (\frac{a}{\sqrt{b \sqrt{x} + a} \sqrt{-a}}\right ) +{\left (7 \,{\left (15 \, b^{3} x + 8 \, a^{2} b\right )} \sqrt{-a} \sqrt{x} - 2 \,{\left (35 \, a b^{2} x + 24 \, a^{3}\right )} \sqrt{-a}\right )} \sqrt{b \sqrt{x} + a}}{96 \, \sqrt{-a} a^{4} x^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(b*sqrt(x) + a)*x^3),x, algorithm="fricas")
[Out]
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Sympy [A] time = 43.1338, size = 173, normalized size = 1.27 \[ - \frac{1}{2 \sqrt{b} x^{\frac{9}{4}} \sqrt{\frac{a}{b \sqrt{x}} + 1}} + \frac{\sqrt{b}}{12 a x^{\frac{7}{4}} \sqrt{\frac{a}{b \sqrt{x}} + 1}} - \frac{7 b^{\frac{3}{2}}}{48 a^{2} x^{\frac{5}{4}} \sqrt{\frac{a}{b \sqrt{x}} + 1}} + \frac{35 b^{\frac{5}{2}}}{96 a^{3} x^{\frac{3}{4}} \sqrt{\frac{a}{b \sqrt{x}} + 1}} + \frac{35 b^{\frac{7}{2}}}{32 a^{4} \sqrt [4]{x} \sqrt{\frac{a}{b \sqrt{x}} + 1}} - \frac{35 b^{4} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt [4]{x}} \right )}}{32 a^{\frac{9}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**3/(a+b*x**(1/2))**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.276574, size = 127, normalized size = 0.93 \[ \frac{1}{96} \, b^{4}{\left (\frac{105 \, \arctan \left (\frac{\sqrt{b \sqrt{x} + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{4}} + \frac{105 \,{\left (b \sqrt{x} + a\right )}^{\frac{7}{2}} - 385 \,{\left (b \sqrt{x} + a\right )}^{\frac{5}{2}} a + 511 \,{\left (b \sqrt{x} + a\right )}^{\frac{3}{2}} a^{2} - 279 \, \sqrt{b \sqrt{x} + a} a^{3}}{a^{4} b^{4} x^{2}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(b*sqrt(x) + a)*x^3),x, algorithm="giac")
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