Optimal. Leaf size=179 \[ \frac {6 a^2 b \sqrt {x} \sqrt {a^2+\frac {2 a b}{\sqrt {x}}+\frac {b^2}{x}}}{a+\frac {b}{\sqrt {x}}}+\frac {6 a b^2 \log \left (\sqrt {x}\right ) \sqrt {a^2+\frac {2 a b}{\sqrt {x}}+\frac {b^2}{x}}}{a+\frac {b}{\sqrt {x}}}-\frac {2 b^3 \sqrt {a^2+\frac {2 a b}{\sqrt {x}}+\frac {b^2}{x}}}{\sqrt {x} \left (a+\frac {b}{\sqrt {x}}\right )}+\frac {a^3 x \sqrt {a^2+\frac {2 a b}{\sqrt {x}}+\frac {b^2}{x}}}{a+\frac {b}{\sqrt {x}}} \]
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Rubi [A] time = 0.09, antiderivative size = 179, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {1341, 1355, 263, 43} \[ \frac {a^3 x \sqrt {a^2+\frac {2 a b}{\sqrt {x}}+\frac {b^2}{x}}}{a+\frac {b}{\sqrt {x}}}+\frac {6 a^2 b \sqrt {x} \sqrt {a^2+\frac {2 a b}{\sqrt {x}}+\frac {b^2}{x}}}{a+\frac {b}{\sqrt {x}}}-\frac {2 b^3 \sqrt {a^2+\frac {2 a b}{\sqrt {x}}+\frac {b^2}{x}}}{\sqrt {x} \left (a+\frac {b}{\sqrt {x}}\right )}+\frac {6 a b^2 \log \left (\sqrt {x}\right ) \sqrt {a^2+\frac {2 a b}{\sqrt {x}}+\frac {b^2}{x}}}{a+\frac {b}{\sqrt {x}}} \]
Antiderivative was successfully verified.
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Rule 43
Rule 263
Rule 1341
Rule 1355
Rubi steps
\begin {align*} \int \left (a^2+\frac {b^2}{x}+\frac {2 a b}{\sqrt {x}}\right )^{3/2} \, dx &=2 \operatorname {Subst}\left (\int \left (a^2+\frac {b^2}{x^2}+\frac {2 a b}{x}\right )^{3/2} x \, dx,x,\sqrt {x}\right )\\ &=\frac {\left (2 \sqrt {a^2+\frac {b^2}{x}+\frac {2 a b}{\sqrt {x}}}\right ) \operatorname {Subst}\left (\int \left (a b+\frac {b^2}{x}\right )^3 x \, dx,x,\sqrt {x}\right )}{b^2 \left (a b+\frac {b^2}{\sqrt {x}}\right )}\\ &=\frac {\left (2 \sqrt {a^2+\frac {b^2}{x}+\frac {2 a b}{\sqrt {x}}}\right ) \operatorname {Subst}\left (\int \frac {\left (b^2+a b x\right )^3}{x^2} \, dx,x,\sqrt {x}\right )}{b^2 \left (a b+\frac {b^2}{\sqrt {x}}\right )}\\ &=\frac {\left (2 \sqrt {a^2+\frac {b^2}{x}+\frac {2 a b}{\sqrt {x}}}\right ) \operatorname {Subst}\left (\int \left (3 a^2 b^4+\frac {b^6}{x^2}+\frac {3 a b^5}{x}+a^3 b^3 x\right ) \, dx,x,\sqrt {x}\right )}{b^2 \left (a b+\frac {b^2}{\sqrt {x}}\right )}\\ &=-\frac {2 b^4 \sqrt {a^2+\frac {b^2}{x}+\frac {2 a b}{\sqrt {x}}}}{\left (a b+\frac {b^2}{\sqrt {x}}\right ) \sqrt {x}}+\frac {6 a^2 b^2 \sqrt {a^2+\frac {b^2}{x}+\frac {2 a b}{\sqrt {x}}} \sqrt {x}}{a b+\frac {b^2}{\sqrt {x}}}+\frac {a^3 \sqrt {a^2+\frac {b^2}{x}+\frac {2 a b}{\sqrt {x}}} x}{a+\frac {b}{\sqrt {x}}}+\frac {3 a b^3 \sqrt {a^2+\frac {b^2}{x}+\frac {2 a b}{\sqrt {x}}} \log (x)}{a b+\frac {b^2}{\sqrt {x}}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 66, normalized size = 0.37 \[ \frac {\sqrt {\frac {\left (a \sqrt {x}+b\right )^2}{x}} \left (a^3 x^{3/2}+6 a^2 b x+3 a b^2 \sqrt {x} \log (x)-2 b^3\right )}{a \sqrt {x}+b} \]
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.39, size = 80, normalized size = 0.45 \[ a^{3} x \mathrm {sgn}\left (a x + b \sqrt {x}\right ) \mathrm {sgn}\relax (x) + 3 \, a b^{2} \log \left ({\left | x \right |}\right ) \mathrm {sgn}\left (a x + b \sqrt {x}\right ) \mathrm {sgn}\relax (x) + 6 \, a^{2} b \sqrt {x} \mathrm {sgn}\left (a x + b \sqrt {x}\right ) \mathrm {sgn}\relax (x) - \frac {2 \, b^{3} \mathrm {sgn}\left (a x + b \sqrt {x}\right ) \mathrm {sgn}\relax (x)}{\sqrt {x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 68, normalized size = 0.38 \[ \frac {\sqrt {\frac {a^{2} x^{\frac {3}{2}}+2 a b x +b^{2} \sqrt {x}}{x^{\frac {3}{2}}}}\, \left (a^{3} x^{\frac {3}{2}}+3 a \,b^{2} \sqrt {x}\, \ln \relax (x )+6 a^{2} b x -2 b^{3}\right )}{a \sqrt {x}+b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ a^{3} x + 3 \, a b^{2} \int \frac {1}{x}\,{d x} + 6 \, a^{2} b \sqrt {x} - \frac {2 \, b^{3}}{\sqrt {x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (a^2+\frac {b^2}{x}+\frac {2\,a\,b}{\sqrt {x}}\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a^{2} + \frac {2 a b}{\sqrt {x}} + \frac {b^{2}}{x}\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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