Optimal. Leaf size=60 \[ \frac{4 \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{a x+b \sqrt{x}}}\right )}{a^{3/2}}-\frac{4 \sqrt{x}}{a \sqrt{a x+b \sqrt{x}}} \]
[Out]
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Rubi [A] time = 0.140339, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19 \[ \frac{4 \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{a x+b \sqrt{x}}}\right )}{a^{3/2}}-\frac{4 \sqrt{x}}{a \sqrt{a x+b \sqrt{x}}} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[x]/(b*Sqrt[x] + a*x)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 12.2549, size = 53, normalized size = 0.88 \[ - \frac{4 \sqrt{x}}{a \sqrt{a x + b \sqrt{x}}} + \frac{4 \operatorname{atanh}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{a x + b \sqrt{x}}} \right )}}{a^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(1/2)/(b*x**(1/2)+a*x)**(3/2),x)
[Out]
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Mathematica [A] time = 0.0732259, size = 72, normalized size = 1.2 \[ \frac{2 \log \left (2 \sqrt{a} \sqrt{a x+b \sqrt{x}}+2 a \sqrt{x}+b\right )}{a^{3/2}}-\frac{4 \sqrt{a x+b \sqrt{x}}}{a \left (a \sqrt{x}+b\right )} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[x]/(b*Sqrt[x] + a*x)^(3/2),x]
[Out]
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Maple [B] time = 0.006, size = 238, normalized size = 4. \[ 2\,{\frac{\sqrt{b\sqrt{x}+ax}}{{a}^{3/2}\sqrt{\sqrt{x} \left ( b+\sqrt{x}a \right ) }b \left ( b+\sqrt{x}a \right ) ^{2}} \left ( x\ln \left ( 1/2\,{\frac{2\,\sqrt{\sqrt{x} \left ( b+\sqrt{x}a \right ) }\sqrt{a}+2\,\sqrt{x}a+b}{\sqrt{a}}} \right ){a}^{2}b-2\,\sqrt{\sqrt{x} \left ( b+\sqrt{x}a \right ) }x{a}^{5/2}+2\,\sqrt{x}\ln \left ( 1/2\,{\frac{2\,\sqrt{\sqrt{x} \left ( b+\sqrt{x}a \right ) }\sqrt{a}+2\,\sqrt{x}a+b}{\sqrt{a}}} \right ) a{b}^{2}+2\,{a}^{3/2} \left ( \sqrt{x} \left ( b+\sqrt{x}a \right ) \right ) ^{3/2}-4\,\sqrt{\sqrt{x} \left ( b+\sqrt{x}a \right ) }\sqrt{x}{a}^{3/2}b+\ln \left ( 1/2\,{\frac{2\,\sqrt{\sqrt{x} \left ( b+\sqrt{x}a \right ) }\sqrt{a}+2\,\sqrt{x}a+b}{\sqrt{a}}} \right ){b}^{3}-2\,\sqrt{a}{b}^{2}\sqrt{\sqrt{x} \left ( b+\sqrt{x}a \right ) } \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(1/2)/(b*x^(1/2)+a*x)^(3/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{x}}{{\left (a x + b \sqrt{x}\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(x)/(a*x + b*sqrt(x))^(3/2),x, algorithm="maxima")
[Out]
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(x)/(a*x + b*sqrt(x))^(3/2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{x}}{\left (a x + b \sqrt{x}\right )^{\frac{3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**(1/2)/(b*x**(1/2)+a*x)**(3/2),x)
[Out]
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GIAC/XCAS [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(x)/(a*x + b*sqrt(x))^(3/2),x, algorithm="giac")
[Out]