Optimal. Leaf size=107 \[ -\frac{64 a^2 \sqrt{a x+b \sqrt{x}}}{5 b^4 \sqrt{x}}+\frac{32 a \sqrt{a x+b \sqrt{x}}}{5 b^3 x}-\frac{24 \sqrt{a x+b \sqrt{x}}}{5 b^2 x^{3/2}}+\frac{4}{b x \sqrt{a x+b \sqrt{x}}} \]
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Rubi [A] time = 0.267868, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ -\frac{64 a^2 \sqrt{a x+b \sqrt{x}}}{5 b^4 \sqrt{x}}+\frac{32 a \sqrt{a x+b \sqrt{x}}}{5 b^3 x}-\frac{24 \sqrt{a x+b \sqrt{x}}}{5 b^2 x^{3/2}}+\frac{4}{b x \sqrt{a x+b \sqrt{x}}} \]
Antiderivative was successfully verified.
[In] Int[1/(x^(3/2)*(b*Sqrt[x] + a*x)^(3/2)),x]
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Rubi in Sympy [A] time = 25.0075, size = 95, normalized size = 0.89 \[ - \frac{64 a^{2} \sqrt{a x + b \sqrt{x}}}{5 b^{4} \sqrt{x}} + \frac{32 a \sqrt{a x + b \sqrt{x}}}{5 b^{3} x} + \frac{4}{b x \sqrt{a x + b \sqrt{x}}} - \frac{24 \sqrt{a x + b \sqrt{x}}}{5 b^{2} x^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**(3/2)/(b*x**(1/2)+a*x)**(3/2),x)
[Out]
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Mathematica [A] time = 0.0505471, size = 70, normalized size = 0.65 \[ -\frac{4 \sqrt{a x+b \sqrt{x}} \left (16 a^3 x^{3/2}+8 a^2 b x-2 a b^2 \sqrt{x}+b^3\right )}{5 b^4 x^{3/2} \left (a \sqrt{x}+b\right )} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^(3/2)*(b*Sqrt[x] + a*x)^(3/2)),x]
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Maple [C] time = 0.017, size = 540, normalized size = 5.1 \[{\frac{2}{5\,{b}^{5}}\sqrt{b\sqrt{x}+ax} \left ( -5\,{x}^{9/2}\ln \left ( 1/2\,{\frac{2\,\sqrt{\sqrt{x} \left ( b+\sqrt{x}a \right ) }\sqrt{a}+2\,\sqrt{x}a+b}{\sqrt{a}}} \right ){a}^{9/2}b+5\,{x}^{9/2}\ln \left ( 1/2\,{\frac{2\,\sqrt{b\sqrt{x}+ax}\sqrt{a}+2\,\sqrt{x}a+b}{\sqrt{a}}} \right ){a}^{9/2}b-10\,{x}^{4}\ln \left ( 1/2\,{\frac{2\,\sqrt{\sqrt{x} \left ( b+\sqrt{x}a \right ) }\sqrt{a}+2\,\sqrt{x}a+b}{\sqrt{a}}} \right ){a}^{7/2}{b}^{2}+10\,{x}^{4}\ln \left ( 1/2\,{\frac{2\,\sqrt{b\sqrt{x}+ax}\sqrt{a}+2\,\sqrt{x}a+b}{\sqrt{a}}} \right ){a}^{7/2}{b}^{2}-5\,{x}^{7/2}\ln \left ( 1/2\,{\frac{2\,\sqrt{\sqrt{x} \left ( b+\sqrt{x}a \right ) }\sqrt{a}+2\,\sqrt{x}a+b}{\sqrt{a}}} \right ){a}^{5/2}{b}^{3}+5\,{x}^{7/2}\ln \left ( 1/2\,{\frac{2\,\sqrt{b\sqrt{x}+ax}\sqrt{a}+2\,\sqrt{x}a+b}{\sqrt{a}}} \right ){a}^{5/2}{b}^{3}+10\,\sqrt{\sqrt{x} \left ( b+\sqrt{x}a \right ) }{x}^{9/2}{a}^{5}-30\,{x}^{7/2} \left ( b\sqrt{x}+ax \right ) ^{3/2}{a}^{4}+10\,{x}^{9/2}\sqrt{b\sqrt{x}+ax}{a}^{5}+10\,{a}^{4} \left ( \sqrt{x} \left ( b+\sqrt{x}a \right ) \right ) ^{3/2}{x}^{7/2}+20\,\sqrt{\sqrt{x} \left ( b+\sqrt{x}a \right ) }{x}^{4}{a}^{4}b-16\,{x}^{5/2} \left ( b\sqrt{x}+ax \right ) ^{3/2}{a}^{2}{b}^{2}-52\,{x}^{3} \left ( b\sqrt{x}+ax \right ) ^{3/2}{a}^{3}b+20\,{x}^{4}\sqrt{b\sqrt{x}+ax}{a}^{4}b+10\,\sqrt{\sqrt{x} \left ( b+\sqrt{x}a \right ) }{x}^{7/2}{a}^{3}{b}^{2}+4\,{x}^{2} \left ( b\sqrt{x}+ax \right ) ^{3/2}a{b}^{3}+10\,{x}^{7/2}\sqrt{b\sqrt{x}+ax}{a}^{3}{b}^{2}-2\,{x}^{3/2} \left ( b\sqrt{x}+ax \right ) ^{3/2}{b}^{4} \right ){\frac{1}{\sqrt{\sqrt{x} \left ( b+\sqrt{x}a \right ) }}}{x}^{-{\frac{7}{2}}} \left ( b+\sqrt{x}a \right ) ^{-2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^(3/2)/(b*x^(1/2)+a*x)^(3/2),x)
[Out]
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Maxima [A] time = 1.51746, size = 61, normalized size = 0.57 \[ -\frac{4 \,{\left (16 \, a^{3} x^{\frac{3}{2}} + 8 \, a^{2} b x - 2 \, a b^{2} \sqrt{x} + b^{3}\right )}}{5 \, \sqrt{a \sqrt{x} + b} b^{4} x^{\frac{5}{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((a*x + b*sqrt(x))^(3/2)*x^(3/2)),x, algorithm="maxima")
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Fricas [A] time = 0.25508, size = 107, normalized size = 1. \[ \frac{4 \,{\left (8 \, a^{3} b x^{2} - 3 \, a b^{3} x -{\left (16 \, a^{4} x^{2} - 10 \, a^{2} b^{2} x - b^{4}\right )} \sqrt{x}\right )} \sqrt{a x + b \sqrt{x}}}{5 \,{\left (a^{2} b^{4} x^{3} - b^{6} x^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((a*x + b*sqrt(x))^(3/2)*x^(3/2)),x, algorithm="fricas")
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{\frac{3}{2}} \left (a x + b \sqrt{x}\right )^{\frac{3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**(3/2)/(b*x**(1/2)+a*x)**(3/2),x)
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (a x + b \sqrt{x}\right )}^{\frac{3}{2}} x^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((a*x + b*sqrt(x))^(3/2)*x^(3/2)),x, algorithm="giac")
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