Optimal. Leaf size=89 \[ -\frac{2 \log \left (a+b \sqrt{x}\right )}{a^5}+\frac{\log (x)}{a^5}+\frac{2}{a^4 \left (a+b \sqrt{x}\right )}+\frac{1}{a^3 \left (a+b \sqrt{x}\right )^2}+\frac{2}{3 a^2 \left (a+b \sqrt{x}\right )^3}+\frac{1}{2 a \left (a+b \sqrt{x}\right )^4} \]
[Out]
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Rubi [A] time = 0.120346, antiderivative size = 89, 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 \[ -\frac{2 \log \left (a+b \sqrt{x}\right )}{a^5}+\frac{\log (x)}{a^5}+\frac{2}{a^4 \left (a+b \sqrt{x}\right )}+\frac{1}{a^3 \left (a+b \sqrt{x}\right )^2}+\frac{2}{3 a^2 \left (a+b \sqrt{x}\right )^3}+\frac{1}{2 a \left (a+b \sqrt{x}\right )^4} \]
Antiderivative was successfully verified.
[In] Int[1/((a + b*Sqrt[x])^5*x),x]
[Out]
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Rubi in Sympy [A] time = 17.3848, size = 87, normalized size = 0.98 \[ \frac{1}{2 a \left (a + b \sqrt{x}\right )^{4}} + \frac{2}{3 a^{2} \left (a + b \sqrt{x}\right )^{3}} + \frac{1}{a^{3} \left (a + b \sqrt{x}\right )^{2}} + \frac{2}{a^{4} \left (a + b \sqrt{x}\right )} + \frac{2 \log{\left (\sqrt{x} \right )}}{a^{5}} - \frac{2 \log{\left (a + b \sqrt{x} \right )}}{a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x/(a+b*x**(1/2))**5,x)
[Out]
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Mathematica [A] time = 0.10164, size = 71, normalized size = 0.8 \[ \frac{\frac{a \left (25 a^3+52 a^2 b \sqrt{x}+42 a b^2 x+12 b^3 x^{3/2}\right )}{\left (a+b \sqrt{x}\right )^4}-12 \log \left (a+b \sqrt{x}\right )+6 \log (x)}{6 a^5} \]
Antiderivative was successfully verified.
[In] Integrate[1/((a + b*Sqrt[x])^5*x),x]
[Out]
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Maple [A] time = 0.014, size = 76, normalized size = 0.9 \[{\frac{\ln \left ( x \right ) }{{a}^{5}}}-2\,{\frac{\ln \left ( a+b\sqrt{x} \right ) }{{a}^{5}}}+{\frac{1}{2\,a} \left ( a+b\sqrt{x} \right ) ^{-4}}+{\frac{2}{3\,{a}^{2}} \left ( a+b\sqrt{x} \right ) ^{-3}}+{\frac{1}{{a}^{3}} \left ( a+b\sqrt{x} \right ) ^{-2}}+2\,{\frac{1}{{a}^{4} \left ( a+b\sqrt{x} \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x/(a+b*x^(1/2))^5,x)
[Out]
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Maxima [A] time = 1.44198, size = 131, normalized size = 1.47 \[ \frac{12 \, b^{3} x^{\frac{3}{2}} + 42 \, a b^{2} x + 52 \, a^{2} b \sqrt{x} + 25 \, a^{3}}{6 \,{\left (a^{4} b^{4} x^{2} + 4 \, a^{5} b^{3} x^{\frac{3}{2}} + 6 \, a^{6} b^{2} x + 4 \, a^{7} b \sqrt{x} + a^{8}\right )}} - \frac{2 \, \log \left (b \sqrt{x} + a\right )}{a^{5}} + \frac{\log \left (x\right )}{a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*sqrt(x) + a)^5*x),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.243501, size = 230, normalized size = 2.58 \[ \frac{42 \, a^{2} b^{2} x + 25 \, a^{4} - 12 \,{\left (b^{4} x^{2} + 6 \, a^{2} b^{2} x + a^{4} + 4 \,{\left (a b^{3} x + a^{3} b\right )} \sqrt{x}\right )} \log \left (b \sqrt{x} + a\right ) + 12 \,{\left (b^{4} x^{2} + 6 \, a^{2} b^{2} x + a^{4} + 4 \,{\left (a b^{3} x + a^{3} b\right )} \sqrt{x}\right )} \log \left (\sqrt{x}\right ) + 4 \,{\left (3 \, a b^{3} x + 13 \, a^{3} b\right )} \sqrt{x}}{6 \,{\left (a^{5} b^{4} x^{2} + 6 \, a^{7} b^{2} x + a^{9} + 4 \,{\left (a^{6} b^{3} x + a^{8} b\right )} \sqrt{x}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*sqrt(x) + a)^5*x),x, algorithm="fricas")
[Out]
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Sympy [A] time = 18.3514, size = 1049, normalized size = 11.79 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x/(a+b*x**(1/2))**5,x)
[Out]
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GIAC/XCAS [A] time = 0.277833, size = 93, normalized size = 1.04 \[ -\frac{2 \,{\rm ln}\left ({\left | b \sqrt{x} + a \right |}\right )}{a^{5}} + \frac{{\rm ln}\left ({\left | x \right |}\right )}{a^{5}} + \frac{12 \, a b^{3} x^{\frac{3}{2}} + 42 \, a^{2} b^{2} x + 52 \, a^{3} b \sqrt{x} + 25 \, a^{4}}{6 \,{\left (b \sqrt{x} + a\right )}^{4} a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*sqrt(x) + a)^5*x),x, algorithm="giac")
[Out]