Optimal. Leaf size=204 \[ \frac{33 b^2 x^{3/2} \sqrt{a x+b \sqrt{x}}}{80 a^3}-\frac{231 b^5 \sqrt{a x+b \sqrt{x}}}{256 a^6}+\frac{77 b^4 \sqrt{x} \sqrt{a x+b \sqrt{x}}}{128 a^5}-\frac{77 b^3 x \sqrt{a x+b \sqrt{x}}}{160 a^4}+\frac{231 b^6 \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{a x+b \sqrt{x}}}\right )}{256 a^{13/2}}-\frac{11 b x^2 \sqrt{a x+b \sqrt{x}}}{30 a^2}+\frac{x^{5/2} \sqrt{a x+b \sqrt{x}}}{3 a} \]
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Rubi [A] time = 0.17155, antiderivative size = 204, normalized size of antiderivative = 1., number of steps used = 9, 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} \[ \frac{33 b^2 x^{3/2} \sqrt{a x+b \sqrt{x}}}{80 a^3}-\frac{231 b^5 \sqrt{a x+b \sqrt{x}}}{256 a^6}+\frac{77 b^4 \sqrt{x} \sqrt{a x+b \sqrt{x}}}{128 a^5}-\frac{77 b^3 x \sqrt{a x+b \sqrt{x}}}{160 a^4}+\frac{231 b^6 \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{a x+b \sqrt{x}}}\right )}{256 a^{13/2}}-\frac{11 b x^2 \sqrt{a x+b \sqrt{x}}}{30 a^2}+\frac{x^{5/2} \sqrt{a x+b \sqrt{x}}}{3 a} \]
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^{5/2}}{\sqrt{b \sqrt{x}+a x}} \, dx &=2 \operatorname{Subst}\left (\int \frac{x^6}{\sqrt{b x+a x^2}} \, dx,x,\sqrt{x}\right )\\ &=\frac{x^{5/2} \sqrt{b \sqrt{x}+a x}}{3 a}-\frac{(11 b) \operatorname{Subst}\left (\int \frac{x^5}{\sqrt{b x+a x^2}} \, dx,x,\sqrt{x}\right )}{6 a}\\ &=-\frac{11 b x^2 \sqrt{b \sqrt{x}+a x}}{30 a^2}+\frac{x^{5/2} \sqrt{b \sqrt{x}+a x}}{3 a}+\frac{\left (33 b^2\right ) \operatorname{Subst}\left (\int \frac{x^4}{\sqrt{b x+a x^2}} \, dx,x,\sqrt{x}\right )}{20 a^2}\\ &=\frac{33 b^2 x^{3/2} \sqrt{b \sqrt{x}+a x}}{80 a^3}-\frac{11 b x^2 \sqrt{b \sqrt{x}+a x}}{30 a^2}+\frac{x^{5/2} \sqrt{b \sqrt{x}+a x}}{3 a}-\frac{\left (231 b^3\right ) \operatorname{Subst}\left (\int \frac{x^3}{\sqrt{b x+a x^2}} \, dx,x,\sqrt{x}\right )}{160 a^3}\\ &=-\frac{77 b^3 x \sqrt{b \sqrt{x}+a x}}{160 a^4}+\frac{33 b^2 x^{3/2} \sqrt{b \sqrt{x}+a x}}{80 a^3}-\frac{11 b x^2 \sqrt{b \sqrt{x}+a x}}{30 a^2}+\frac{x^{5/2} \sqrt{b \sqrt{x}+a x}}{3 a}+\frac{\left (77 b^4\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{b x+a x^2}} \, dx,x,\sqrt{x}\right )}{64 a^4}\\ &=\frac{77 b^4 \sqrt{x} \sqrt{b \sqrt{x}+a x}}{128 a^5}-\frac{77 b^3 x \sqrt{b \sqrt{x}+a x}}{160 a^4}+\frac{33 b^2 x^{3/2} \sqrt{b \sqrt{x}+a x}}{80 a^3}-\frac{11 b x^2 \sqrt{b \sqrt{x}+a x}}{30 a^2}+\frac{x^{5/2} \sqrt{b \sqrt{x}+a x}}{3 a}-\frac{\left (231 b^5\right ) \operatorname{Subst}\left (\int \frac{x}{\sqrt{b x+a x^2}} \, dx,x,\sqrt{x}\right )}{256 a^5}\\ &=-\frac{231 b^5 \sqrt{b \sqrt{x}+a x}}{256 a^6}+\frac{77 b^4 \sqrt{x} \sqrt{b \sqrt{x}+a x}}{128 a^5}-\frac{77 b^3 x \sqrt{b \sqrt{x}+a x}}{160 a^4}+\frac{33 b^2 x^{3/2} \sqrt{b \sqrt{x}+a x}}{80 a^3}-\frac{11 b x^2 \sqrt{b \sqrt{x}+a x}}{30 a^2}+\frac{x^{5/2} \sqrt{b \sqrt{x}+a x}}{3 a}+\frac{\left (231 b^6\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{b x+a x^2}} \, dx,x,\sqrt{x}\right )}{512 a^6}\\ &=-\frac{231 b^5 \sqrt{b \sqrt{x}+a x}}{256 a^6}+\frac{77 b^4 \sqrt{x} \sqrt{b \sqrt{x}+a x}}{128 a^5}-\frac{77 b^3 x \sqrt{b \sqrt{x}+a x}}{160 a^4}+\frac{33 b^2 x^{3/2} \sqrt{b \sqrt{x}+a x}}{80 a^3}-\frac{11 b x^2 \sqrt{b \sqrt{x}+a x}}{30 a^2}+\frac{x^{5/2} \sqrt{b \sqrt{x}+a x}}{3 a}+\frac{\left (231 b^6\right ) \operatorname{Subst}\left (\int \frac{1}{1-a x^2} \, dx,x,\frac{\sqrt{x}}{\sqrt{b \sqrt{x}+a x}}\right )}{256 a^6}\\ &=-\frac{231 b^5 \sqrt{b \sqrt{x}+a x}}{256 a^6}+\frac{77 b^4 \sqrt{x} \sqrt{b \sqrt{x}+a x}}{128 a^5}-\frac{77 b^3 x \sqrt{b \sqrt{x}+a x}}{160 a^4}+\frac{33 b^2 x^{3/2} \sqrt{b \sqrt{x}+a x}}{80 a^3}-\frac{11 b x^2 \sqrt{b \sqrt{x}+a x}}{30 a^2}+\frac{x^{5/2} \sqrt{b \sqrt{x}+a x}}{3 a}+\frac{231 b^6 \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b \sqrt{x}+a x}}\right )}{256 a^{13/2}}\\ \end{align*}
Mathematica [A] time = 0.179312, size = 164, normalized size = 0.8 \[ \frac{\left (a \sqrt{x}+b\right ) \left (\sqrt{a} \sqrt{x} \sqrt{\frac{a \sqrt{x}}{b}+1} \left (1584 a^3 b^2 x^{3/2}-1848 a^2 b^3 x-1408 a^4 b x^2+1280 a^5 x^{5/2}+2310 a b^4 \sqrt{x}-3465 b^5\right )+3465 b^{11/2} \sqrt [4]{x} \sinh ^{-1}\left (\frac{\sqrt{a} \sqrt [4]{x}}{\sqrt{b}}\right )\right )}{3840 a^{13/2} \sqrt{\frac{a \sqrt{x}}{b}+1} \sqrt{a x+b \sqrt{x}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 245, normalized size = 1.2 \begin{align*}{\frac{1}{7680}\sqrt{b\sqrt{x}+ax} \left ( 2560\,{x}^{3/2} \left ( b\sqrt{x}+ax \right ) ^{3/2}{a}^{11/2}+8544\, \left ( b\sqrt{x}+ax \right ) ^{3/2}{a}^{7/2}\sqrt{x}{b}^{2}-5376\, \left ( b\sqrt{x}+ax \right ) ^{3/2}{a}^{9/2}xb-12240\, \left ( b\sqrt{x}+ax \right ) ^{3/2}{a}^{5/2}{b}^{3}+16860\,\sqrt{b\sqrt{x}+ax}{a}^{5/2}\sqrt{x}{b}^{4}+8430\,\sqrt{b\sqrt{x}+ax}{a}^{3/2}{b}^{5}-15360\,{a}^{3/2}\sqrt{\sqrt{x} \left ( b+a\sqrt{x} \right ) }{b}^{5}+7680\,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}^{6}-4215\,\ln \left ( 1/2\,{\frac{2\,a\sqrt{x}+2\,\sqrt{b\sqrt{x}+ax}\sqrt{a}+b}{\sqrt{a}}} \right ) a{b}^{6} \right ){\frac{1}{\sqrt{\sqrt{x} \left ( b+a\sqrt{x} \right ) }}}{a}^{-{\frac{15}{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{5}{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{5}{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 = 1.32523, size = 169, normalized size = 0.83 \begin{align*} \frac{1}{3840} \, \sqrt{a x + b \sqrt{x}}{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \, \sqrt{x}{\left (\frac{10 \, \sqrt{x}}{a} - \frac{11 \, b}{a^{2}}\right )} + \frac{99 \, b^{2}}{a^{3}}\right )} \sqrt{x} - \frac{231 \, b^{3}}{a^{4}}\right )} \sqrt{x} + \frac{1155 \, b^{4}}{a^{5}}\right )} \sqrt{x} - \frac{3465 \, b^{5}}{a^{6}}\right )} - \frac{231 \, b^{6} \log \left ({\left | -2 \, \sqrt{a}{\left (\sqrt{a} \sqrt{x} - \sqrt{a x + b \sqrt{x}}\right )} - b \right |}\right )}{512 \, a^{\frac{13}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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