Optimal. Leaf size=79 \[ \frac{32 a \sqrt{a x+b \sqrt{x}}}{3 b^3 \sqrt{x}}-\frac{16 \sqrt{a x+b \sqrt{x}}}{3 b^2 x}+\frac{4}{b \sqrt{x} \sqrt{a x+b \sqrt{x}}} \]
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Rubi [A] time = 0.122479, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {2015, 2016, 2014} \[ \frac{32 a \sqrt{a x+b \sqrt{x}}}{3 b^3 \sqrt{x}}-\frac{16 \sqrt{a x+b \sqrt{x}}}{3 b^2 x}+\frac{4}{b \sqrt{x} \sqrt{a x+b \sqrt{x}}} \]
Antiderivative was successfully verified.
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Rule 2015
Rule 2016
Rule 2014
Rubi steps
\begin{align*} \int \frac{1}{x \left (b \sqrt{x}+a x\right )^{3/2}} \, dx &=\frac{4}{b \sqrt{x} \sqrt{b \sqrt{x}+a x}}+\frac{4 \int \frac{1}{x^{3/2} \sqrt{b \sqrt{x}+a x}} \, dx}{b}\\ &=\frac{4}{b \sqrt{x} \sqrt{b \sqrt{x}+a x}}-\frac{16 \sqrt{b \sqrt{x}+a x}}{3 b^2 x}-\frac{(8 a) \int \frac{1}{x \sqrt{b \sqrt{x}+a x}} \, dx}{3 b^2}\\ &=\frac{4}{b \sqrt{x} \sqrt{b \sqrt{x}+a x}}-\frac{16 \sqrt{b \sqrt{x}+a x}}{3 b^2 x}+\frac{32 a \sqrt{b \sqrt{x}+a x}}{3 b^3 \sqrt{x}}\\ \end{align*}
Mathematica [A] time = 0.0524482, size = 48, normalized size = 0.61 \[ \frac{4 \left (8 a^2 x+4 a b \sqrt{x}-b^2\right )}{3 b^3 \sqrt{x} \sqrt{a x+b \sqrt{x}}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.011, size = 524, normalized size = 6.6 \begin{align*}{\frac{1}{3\,{b}^{4}}\sqrt{b\sqrt{x}+ax} \left ( 24\, \left ( b\sqrt{x}+ax \right ) ^{3/2}{a}^{7/2}{x}^{5/2}-6\,\sqrt{b\sqrt{x}+ax}{a}^{9/2}{x}^{7/2}-3\,\ln \left ( 1/2\,{\frac{2\,a\sqrt{x}+2\,\sqrt{b\sqrt{x}+ax}\sqrt{a}+b}{\sqrt{a}}} \right ){x}^{7/2}{a}^{4}b-6\,{a}^{9/2}{x}^{7/2}\sqrt{\sqrt{x} \left ( b+a\sqrt{x} \right ) }+3\,\ln \left ( 1/2\,{\frac{2\,\sqrt{\sqrt{x} \left ( b+a\sqrt{x} \right ) }\sqrt{a}+2\,a\sqrt{x}+b}{\sqrt{a}}} \right ){x}^{7/2}{a}^{4}b+44\, \left ( b\sqrt{x}+ax \right ) ^{3/2}{a}^{5/2}{x}^{2}b-12\,\sqrt{b\sqrt{x}+ax}{a}^{7/2}{x}^{3}b-6\,\ln \left ( 1/2\,{\frac{2\,a\sqrt{x}+2\,\sqrt{b\sqrt{x}+ax}\sqrt{a}+b}{\sqrt{a}}} \right ){x}^{3}{a}^{3}{b}^{2}-12\,{a}^{7/2}{x}^{3}\sqrt{\sqrt{x} \left ( b+a\sqrt{x} \right ) }b-12\,{a}^{7/2}{x}^{5/2} \left ( \sqrt{x} \left ( b+a\sqrt{x} \right ) \right ) ^{3/2}+6\,\ln \left ( 1/2\,{\frac{2\,\sqrt{\sqrt{x} \left ( b+a\sqrt{x} \right ) }\sqrt{a}+2\,a\sqrt{x}+b}{\sqrt{a}}} \right ){x}^{3}{a}^{3}{b}^{2}+16\, \left ( b\sqrt{x}+ax \right ) ^{3/2}{a}^{3/2}{x}^{3/2}{b}^{2}-6\,\sqrt{b\sqrt{x}+ax}{a}^{5/2}{x}^{5/2}{b}^{2}-3\,\ln \left ( 1/2\,{\frac{2\,a\sqrt{x}+2\,\sqrt{b\sqrt{x}+ax}\sqrt{a}+b}{\sqrt{a}}} \right ){x}^{5/2}{a}^{2}{b}^{3}-6\,{a}^{5/2}{x}^{5/2}\sqrt{\sqrt{x} \left ( b+a\sqrt{x} \right ) }{b}^{2}+3\,\ln \left ( 1/2\,{\frac{2\,\sqrt{\sqrt{x} \left ( b+a\sqrt{x} \right ) }\sqrt{a}+2\,a\sqrt{x}+b}{\sqrt{a}}} \right ){x}^{5/2}{a}^{2}{b}^{3}-4\, \left ( b\sqrt{x}+ax \right ) ^{3/2}\sqrt{a}x{b}^{3} \right ){\frac{1}{\sqrt{\sqrt{x} \left ( b+a\sqrt{x} \right ) }}}{\frac{1}{\sqrt{a}}} \left ( b+a\sqrt{x} \right ) ^{-2}{x}^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (a x + b \sqrt{x}\right )}^{\frac{3}{2}} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.87056, size = 132, normalized size = 1.67 \begin{align*} -\frac{4 \,{\left (4 \, a^{2} b x - b^{3} -{\left (8 \, a^{3} x - 5 \, a b^{2}\right )} \sqrt{x}\right )} \sqrt{a x + b \sqrt{x}}}{3 \,{\left (a^{2} b^{3} x^{2} - b^{5} x\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x \left (a x + b \sqrt{x}\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (a x + b \sqrt{x}\right )}^{\frac{3}{2}} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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