Optimal. Leaf size=172 \[ \frac{2 a^9}{7 b^{10} \left (a+b \sqrt{x}\right )^7}-\frac{3 a^8}{b^{10} \left (a+b \sqrt{x}\right )^6}+\frac{72 a^7}{5 b^{10} \left (a+b \sqrt{x}\right )^5}-\frac{42 a^6}{b^{10} \left (a+b \sqrt{x}\right )^4}+\frac{84 a^5}{b^{10} \left (a+b \sqrt{x}\right )^3}-\frac{126 a^4}{b^{10} \left (a+b \sqrt{x}\right )^2}+\frac{168 a^3}{b^{10} \left (a+b \sqrt{x}\right )}+\frac{72 a^2 \log \left (a+b \sqrt{x}\right )}{b^{10}}-\frac{16 a \sqrt{x}}{b^9}+\frac{x}{b^8} \]
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Rubi [A] time = 0.330569, antiderivative size = 172, 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 \[ \frac{2 a^9}{7 b^{10} \left (a+b \sqrt{x}\right )^7}-\frac{3 a^8}{b^{10} \left (a+b \sqrt{x}\right )^6}+\frac{72 a^7}{5 b^{10} \left (a+b \sqrt{x}\right )^5}-\frac{42 a^6}{b^{10} \left (a+b \sqrt{x}\right )^4}+\frac{84 a^5}{b^{10} \left (a+b \sqrt{x}\right )^3}-\frac{126 a^4}{b^{10} \left (a+b \sqrt{x}\right )^2}+\frac{168 a^3}{b^{10} \left (a+b \sqrt{x}\right )}+\frac{72 a^2 \log \left (a+b \sqrt{x}\right )}{b^{10}}-\frac{16 a \sqrt{x}}{b^9}+\frac{x}{b^8} \]
Antiderivative was successfully verified.
[In] Int[x^4/(a + b*Sqrt[x])^8,x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{2 a^{9}}{7 b^{10} \left (a + b \sqrt{x}\right )^{7}} - \frac{3 a^{8}}{b^{10} \left (a + b \sqrt{x}\right )^{6}} + \frac{72 a^{7}}{5 b^{10} \left (a + b \sqrt{x}\right )^{5}} - \frac{42 a^{6}}{b^{10} \left (a + b \sqrt{x}\right )^{4}} + \frac{84 a^{5}}{b^{10} \left (a + b \sqrt{x}\right )^{3}} - \frac{126 a^{4}}{b^{10} \left (a + b \sqrt{x}\right )^{2}} + \frac{168 a^{3}}{b^{10} \left (a + b \sqrt{x}\right )} + \frac{72 a^{2} \log{\left (a + b \sqrt{x} \right )}}{b^{10}} - \frac{16 a \sqrt{x}}{b^{9}} + \frac{2 \int ^{\sqrt{x}} x\, dx}{b^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**4/(a+b*x**(1/2))**8,x)
[Out]
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Mathematica [A] time = 0.0693656, size = 150, normalized size = 0.87 \[ \frac{3349 a^9+20923 a^8 b \sqrt{x}+53949 a^7 b^2 x+72275 a^6 b^3 x^{3/2}+50225 a^5 b^4 x^2+12495 a^4 b^5 x^{5/2}-4655 a^3 b^6 x^3-3185 a^2 b^7 x^{7/2}+2520 a^2 \left (a+b \sqrt{x}\right )^7 \log \left (a+b \sqrt{x}\right )-315 a b^8 x^4+35 b^9 x^{9/2}}{35 b^{10} \left (a+b \sqrt{x}\right )^7} \]
Antiderivative was successfully verified.
[In] Integrate[x^4/(a + b*Sqrt[x])^8,x]
[Out]
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Maple [A] time = 0.016, size = 151, normalized size = 0.9 \[{\frac{x}{{b}^{8}}}+72\,{\frac{{a}^{2}\ln \left ( a+b\sqrt{x} \right ) }{{b}^{10}}}-16\,{\frac{a\sqrt{x}}{{b}^{9}}}+{\frac{2\,{a}^{9}}{7\,{b}^{10}} \left ( a+b\sqrt{x} \right ) ^{-7}}-3\,{\frac{{a}^{8}}{{b}^{10} \left ( a+b\sqrt{x} \right ) ^{6}}}+{\frac{72\,{a}^{7}}{5\,{b}^{10}} \left ( a+b\sqrt{x} \right ) ^{-5}}-42\,{\frac{{a}^{6}}{{b}^{10} \left ( a+b\sqrt{x} \right ) ^{4}}}+84\,{\frac{{a}^{5}}{{b}^{10} \left ( a+b\sqrt{x} \right ) ^{3}}}-126\,{\frac{{a}^{4}}{{b}^{10} \left ( a+b\sqrt{x} \right ) ^{2}}}+168\,{\frac{{a}^{3}}{{b}^{10} \left ( a+b\sqrt{x} \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^4/(a+b*x^(1/2))^8,x)
[Out]
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Maxima [A] time = 1.44218, size = 219, normalized size = 1.27 \[ \frac{72 \, a^{2} \log \left (b \sqrt{x} + a\right )}{b^{10}} + \frac{{\left (b \sqrt{x} + a\right )}^{2}}{b^{10}} - \frac{18 \,{\left (b \sqrt{x} + a\right )} a}{b^{10}} + \frac{168 \, a^{3}}{{\left (b \sqrt{x} + a\right )} b^{10}} - \frac{126 \, a^{4}}{{\left (b \sqrt{x} + a\right )}^{2} b^{10}} + \frac{84 \, a^{5}}{{\left (b \sqrt{x} + a\right )}^{3} b^{10}} - \frac{42 \, a^{6}}{{\left (b \sqrt{x} + a\right )}^{4} b^{10}} + \frac{72 \, a^{7}}{5 \,{\left (b \sqrt{x} + a\right )}^{5} b^{10}} - \frac{3 \, a^{8}}{{\left (b \sqrt{x} + a\right )}^{6} b^{10}} + \frac{2 \, a^{9}}{7 \,{\left (b \sqrt{x} + a\right )}^{7} b^{10}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^4/(b*sqrt(x) + a)^8,x, algorithm="maxima")
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Fricas [A] time = 0.244276, size = 356, normalized size = 2.07 \[ -\frac{315 \, a b^{8} x^{4} + 4655 \, a^{3} b^{6} x^{3} - 50225 \, a^{5} b^{4} x^{2} - 53949 \, a^{7} b^{2} x - 3349 \, a^{9} - 2520 \,{\left (7 \, a^{3} b^{6} x^{3} + 35 \, a^{5} b^{4} x^{2} + 21 \, a^{7} b^{2} x + a^{9} +{\left (a^{2} b^{7} x^{3} + 21 \, a^{4} b^{5} x^{2} + 35 \, a^{6} b^{3} x + 7 \, a^{8} b\right )} \sqrt{x}\right )} \log \left (b \sqrt{x} + a\right ) - 7 \,{\left (5 \, b^{9} x^{4} - 455 \, a^{2} b^{7} x^{3} + 1785 \, a^{4} b^{5} x^{2} + 10325 \, a^{6} b^{3} x + 2989 \, a^{8} b\right )} \sqrt{x}}{35 \,{\left (7 \, a b^{16} x^{3} + 35 \, a^{3} b^{14} x^{2} + 21 \, a^{5} b^{12} x + a^{7} b^{10} +{\left (b^{17} x^{3} + 21 \, a^{2} b^{15} x^{2} + 35 \, a^{4} b^{13} x + 7 \, a^{6} b^{11}\right )} \sqrt{x}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^4/(b*sqrt(x) + a)^8,x, algorithm="fricas")
[Out]
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Sympy [A] time = 34.9145, size = 1945, normalized size = 11.31 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**4/(a+b*x**(1/2))**8,x)
[Out]
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GIAC/XCAS [A] time = 0.219014, size = 161, normalized size = 0.94 \[ \frac{72 \, a^{2}{\rm ln}\left ({\left | b \sqrt{x} + a \right |}\right )}{b^{10}} + \frac{b^{8} x - 16 \, a b^{7} \sqrt{x}}{b^{16}} + \frac{5880 \, a^{3} b^{6} x^{3} + 30870 \, a^{4} b^{5} x^{\frac{5}{2}} + 69090 \, a^{5} b^{4} x^{2} + 83790 \, a^{6} b^{3} x^{\frac{3}{2}} + 57834 \, a^{7} b^{2} x + 21483 \, a^{8} b \sqrt{x} + 3349 \, a^{9}}{35 \,{\left (b \sqrt{x} + a\right )}^{7} b^{10}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^4/(b*sqrt(x) + a)^8,x, algorithm="giac")
[Out]