Optimal. Leaf size=25 \[ \frac {4 \sqrt {x}}{b \sqrt {a x+b \sqrt {x}}} \]
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Rubi [A] time = 0.01, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2000} \[ \frac {4 \sqrt {x}}{b \sqrt {a x+b \sqrt {x}}} \]
Antiderivative was successfully verified.
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Rule 2000
Rubi steps
\begin {align*} \int \frac {1}{\left (b \sqrt {x}+a x\right )^{3/2}} \, dx &=\frac {4 \sqrt {x}}{b \sqrt {b \sqrt {x}+a x}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 25, normalized size = 1.00 \[ \frac {4 \sqrt {x}}{b \sqrt {a x+b \sqrt {x}}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.88, size = 36, normalized size = 1.44 \[ \frac {4 \, \sqrt {a x + b \sqrt {x}} {\left (a \sqrt {x} - b\right )}}{a^{2} b x - b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 34, normalized size = 1.36 \[ \frac {4}{{\left (\sqrt {a} {\left (\sqrt {a} \sqrt {x} - \sqrt {a x + b \sqrt {x}}\right )} + b\right )} \sqrt {a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.06, size = 404, normalized size = 16.16 \[ \frac {\sqrt {a x +b \sqrt {x}}\, \left (-a^{2} b x \ln \left (\frac {2 a \sqrt {x}+b +2 \sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, \sqrt {a}}{2 \sqrt {a}}\right )+a^{2} b x \ln \left (\frac {2 a \sqrt {x}+b +2 \sqrt {a x +b \sqrt {x}}\, \sqrt {a}}{2 \sqrt {a}}\right )-2 a \,b^{2} \sqrt {x}\, \ln \left (\frac {2 a \sqrt {x}+b +2 \sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, \sqrt {a}}{2 \sqrt {a}}\right )+2 a \,b^{2} \sqrt {x}\, \ln \left (\frac {2 a \sqrt {x}+b +2 \sqrt {a x +b \sqrt {x}}\, \sqrt {a}}{2 \sqrt {a}}\right )-b^{3} \ln \left (\frac {2 a \sqrt {x}+b +2 \sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, \sqrt {a}}{2 \sqrt {a}}\right )+b^{3} \ln \left (\frac {2 a \sqrt {x}+b +2 \sqrt {a x +b \sqrt {x}}\, \sqrt {a}}{2 \sqrt {a}}\right )+2 \sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, a^{\frac {5}{2}} x +2 \sqrt {a x +b \sqrt {x}}\, a^{\frac {5}{2}} x +4 \sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, a^{\frac {3}{2}} b \sqrt {x}+4 \sqrt {a x +b \sqrt {x}}\, a^{\frac {3}{2}} b \sqrt {x}+2 \sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, \sqrt {a}\, b^{2}+2 \sqrt {a x +b \sqrt {x}}\, \sqrt {a}\, b^{2}-4 \left (\left (a \sqrt {x}+b \right ) \sqrt {x}\right )^{\frac {3}{2}} a^{\frac {3}{2}}\right )}{\sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, \left (a \sqrt {x}+b \right )^{2} \sqrt {a}\, b^{2}} \]
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 \[ \int \frac {1}{{\left (a x + b \sqrt {x}\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.43, size = 40, normalized size = 1.60 \[ -\frac {4\,x\,\left (\frac {b}{a\,\sqrt {x}}+1\right )}{{\left (a\,x+b\,\sqrt {x}\right )}^{3/2}\,\left (\sqrt {\frac {b}{a\,\sqrt {x}}+1}+1\right )} \]
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 \[ \int \frac {1}{\left (a x + b \sqrt {x}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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