Optimal. Leaf size=146 \[ -\frac{35 b^3 \sqrt{a x+b \sqrt{x}}}{32 a^4}+\frac{35 b^2 \sqrt{x} \sqrt{a x+b \sqrt{x}}}{48 a^3}+\frac{35 b^4 \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{a x+b \sqrt{x}}}\right )}{32 a^{9/2}}-\frac{7 b x \sqrt{a x+b \sqrt{x}}}{12 a^2}+\frac{x^{3/2} \sqrt{a x+b \sqrt{x}}}{2 a} \]
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Rubi [A] time = 0.119912, antiderivative size = 146, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {2018, 670, 640, 620, 206} \[ -\frac{35 b^3 \sqrt{a x+b \sqrt{x}}}{32 a^4}+\frac{35 b^2 \sqrt{x} \sqrt{a x+b \sqrt{x}}}{48 a^3}+\frac{35 b^4 \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{a x+b \sqrt{x}}}\right )}{32 a^{9/2}}-\frac{7 b x \sqrt{a x+b \sqrt{x}}}{12 a^2}+\frac{x^{3/2} \sqrt{a x+b \sqrt{x}}}{2 a} \]
Antiderivative was successfully verified.
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Rule 2018
Rule 670
Rule 640
Rule 620
Rule 206
Rubi steps
\begin{align*} \int \frac{x^{3/2}}{\sqrt{b \sqrt{x}+a x}} \, dx &=2 \operatorname{Subst}\left (\int \frac{x^4}{\sqrt{b x+a x^2}} \, dx,x,\sqrt{x}\right )\\ &=\frac{x^{3/2} \sqrt{b \sqrt{x}+a x}}{2 a}-\frac{(7 b) \operatorname{Subst}\left (\int \frac{x^3}{\sqrt{b x+a x^2}} \, dx,x,\sqrt{x}\right )}{4 a}\\ &=-\frac{7 b x \sqrt{b \sqrt{x}+a x}}{12 a^2}+\frac{x^{3/2} \sqrt{b \sqrt{x}+a x}}{2 a}+\frac{\left (35 b^2\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{b x+a x^2}} \, dx,x,\sqrt{x}\right )}{24 a^2}\\ &=\frac{35 b^2 \sqrt{x} \sqrt{b \sqrt{x}+a x}}{48 a^3}-\frac{7 b x \sqrt{b \sqrt{x}+a x}}{12 a^2}+\frac{x^{3/2} \sqrt{b \sqrt{x}+a x}}{2 a}-\frac{\left (35 b^3\right ) \operatorname{Subst}\left (\int \frac{x}{\sqrt{b x+a x^2}} \, dx,x,\sqrt{x}\right )}{32 a^3}\\ &=-\frac{35 b^3 \sqrt{b \sqrt{x}+a x}}{32 a^4}+\frac{35 b^2 \sqrt{x} \sqrt{b \sqrt{x}+a x}}{48 a^3}-\frac{7 b x \sqrt{b \sqrt{x}+a x}}{12 a^2}+\frac{x^{3/2} \sqrt{b \sqrt{x}+a x}}{2 a}+\frac{\left (35 b^4\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{b x+a x^2}} \, dx,x,\sqrt{x}\right )}{64 a^4}\\ &=-\frac{35 b^3 \sqrt{b \sqrt{x}+a x}}{32 a^4}+\frac{35 b^2 \sqrt{x} \sqrt{b \sqrt{x}+a x}}{48 a^3}-\frac{7 b x \sqrt{b \sqrt{x}+a x}}{12 a^2}+\frac{x^{3/2} \sqrt{b \sqrt{x}+a x}}{2 a}+\frac{\left (35 b^4\right ) \operatorname{Subst}\left (\int \frac{1}{1-a x^2} \, dx,x,\frac{\sqrt{x}}{\sqrt{b \sqrt{x}+a x}}\right )}{32 a^4}\\ &=-\frac{35 b^3 \sqrt{b \sqrt{x}+a x}}{32 a^4}+\frac{35 b^2 \sqrt{x} \sqrt{b \sqrt{x}+a x}}{48 a^3}-\frac{7 b x \sqrt{b \sqrt{x}+a x}}{12 a^2}+\frac{x^{3/2} \sqrt{b \sqrt{x}+a x}}{2 a}+\frac{35 b^4 \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b \sqrt{x}+a x}}\right )}{32 a^{9/2}}\\ \end{align*}
Mathematica [A] time = 0.137471, size = 142, normalized size = 0.97 \[ -\frac{35 b^5 \left (\frac{a \sqrt{x}}{b}+1\right ) \left (-\frac{32 a^4 x^2}{35 b^4}+\frac{16 a^3 x^{3/2}}{15 b^3}-\frac{4 a^2 x}{3 b^2}+\frac{2 a \sqrt{x}}{b}-\frac{2 \sqrt{a} \sqrt [4]{x} \sinh ^{-1}\left (\frac{\sqrt{a} \sqrt [4]{x}}{\sqrt{b}}\right )}{\sqrt{b} \sqrt{\frac{a \sqrt{x}}{b}+1}}\right )}{64 a^5 \sqrt{\sqrt{x} \left (a \sqrt{x}+b\right )}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 203, normalized size = 1.4 \begin{align*}{\frac{1}{192}\sqrt{b\sqrt{x}+ax} \left ( 96\,\sqrt{x} \left ( b\sqrt{x}+ax \right ) ^{3/2}{a}^{7/2}-208\, \left ( b\sqrt{x}+ax \right ) ^{3/2}{a}^{5/2}b+348\,\sqrt{b\sqrt{x}+ax}{a}^{5/2}\sqrt{x}{b}^{2}+174\,\sqrt{b\sqrt{x}+ax}{a}^{3/2}{b}^{3}-384\,{a}^{3/2}\sqrt{\sqrt{x} \left ( b+a\sqrt{x} \right ) }{b}^{3}+192\,a\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 ){b}^{4}-87\,\ln \left ( 1/2\,{\frac{2\,a\sqrt{x}+2\,\sqrt{b\sqrt{x}+ax}\sqrt{a}+b}{\sqrt{a}}} \right ) a{b}^{4} \right ){\frac{1}{\sqrt{\sqrt{x} \left ( b+a\sqrt{x} \right ) }}}{a}^{-{\frac{11}{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{x^{\frac{3}{2}}}{\sqrt{a x + b \sqrt{x}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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{x^{\frac{3}{2}}}{\sqrt{a x + b \sqrt{x}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.31092, size = 131, normalized size = 0.9 \begin{align*} \frac{1}{96} \, \sqrt{a x + b \sqrt{x}}{\left (2 \,{\left (4 \, \sqrt{x}{\left (\frac{6 \, \sqrt{x}}{a} - \frac{7 \, b}{a^{2}}\right )} + \frac{35 \, b^{2}}{a^{3}}\right )} \sqrt{x} - \frac{105 \, b^{3}}{a^{4}}\right )} - \frac{35 \, b^{4} \log \left ({\left | -2 \, \sqrt{a}{\left (\sqrt{a} \sqrt{x} - \sqrt{a x + b \sqrt{x}}\right )} - b \right |}\right )}{64 \, a^{\frac{9}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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