Optimal. Leaf size=187 \[ \frac{356 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} \text{EllipticF}\left (\tan ^{-1}\left (\sqrt{x}\right ),-\frac{1}{2}\right )}{81 \sqrt{3 x^2+5 x+2}}-\frac{2}{9} \sqrt{x} (5 x+1) \left (3 x^2+5 x+2\right )^{3/2}+\frac{4}{81} \sqrt{x} (45 x+82) \sqrt{3 x^2+5 x+2}+\frac{860 \sqrt{x} (3 x+2)}{243 \sqrt{3 x^2+5 x+2}}-\frac{860 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{243 \sqrt{3 x^2+5 x+2}} \]
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Rubi [A] time = 0.120124, antiderivative size = 187, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {814, 839, 1189, 1100, 1136} \[ -\frac{2}{9} \sqrt{x} (5 x+1) \left (3 x^2+5 x+2\right )^{3/2}+\frac{4}{81} \sqrt{x} (45 x+82) \sqrt{3 x^2+5 x+2}+\frac{860 \sqrt{x} (3 x+2)}{243 \sqrt{3 x^2+5 x+2}}+\frac{356 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{81 \sqrt{3 x^2+5 x+2}}-\frac{860 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{243 \sqrt{3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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Rule 814
Rule 839
Rule 1189
Rule 1100
Rule 1136
Rubi steps
\begin{align*} \int \frac{(2-5 x) \left (2+5 x+3 x^2\right )^{3/2}}{\sqrt{x}} \, dx &=-\frac{2}{9} \sqrt{x} (1+5 x) \left (2+5 x+3 x^2\right )^{3/2}-\frac{2}{63} \int \frac{(-133-175 x) \sqrt{2+5 x+3 x^2}}{\sqrt{x}} \, dx\\ &=\frac{4}{81} \sqrt{x} (82+45 x) \sqrt{2+5 x+3 x^2}-\frac{2}{9} \sqrt{x} (1+5 x) \left (2+5 x+3 x^2\right )^{3/2}+\frac{4 \int \frac{3115+\frac{7525 x}{2}}{\sqrt{x} \sqrt{2+5 x+3 x^2}} \, dx}{2835}\\ &=\frac{4}{81} \sqrt{x} (82+45 x) \sqrt{2+5 x+3 x^2}-\frac{2}{9} \sqrt{x} (1+5 x) \left (2+5 x+3 x^2\right )^{3/2}+\frac{8 \operatorname{Subst}\left (\int \frac{3115+\frac{7525 x^2}{2}}{\sqrt{2+5 x^2+3 x^4}} \, dx,x,\sqrt{x}\right )}{2835}\\ &=\frac{4}{81} \sqrt{x} (82+45 x) \sqrt{2+5 x+3 x^2}-\frac{2}{9} \sqrt{x} (1+5 x) \left (2+5 x+3 x^2\right )^{3/2}+\frac{712}{81} \operatorname{Subst}\left (\int \frac{1}{\sqrt{2+5 x^2+3 x^4}} \, dx,x,\sqrt{x}\right )+\frac{860}{81} \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{2+5 x^2+3 x^4}} \, dx,x,\sqrt{x}\right )\\ &=\frac{860 \sqrt{x} (2+3 x)}{243 \sqrt{2+5 x+3 x^2}}+\frac{4}{81} \sqrt{x} (82+45 x) \sqrt{2+5 x+3 x^2}-\frac{2}{9} \sqrt{x} (1+5 x) \left (2+5 x+3 x^2\right )^{3/2}-\frac{860 \sqrt{2} (1+x) \sqrt{\frac{2+3 x}{1+x}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{243 \sqrt{2+5 x+3 x^2}}+\frac{356 \sqrt{2} (1+x) \sqrt{\frac{2+3 x}{1+x}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{81 \sqrt{2+5 x+3 x^2}}\\ \end{align*}
Mathematica [C] time = 0.170308, size = 165, normalized size = 0.88 \[ \frac{208 i \sqrt{2} \sqrt{\frac{1}{x}+1} \sqrt{\frac{2}{x}+3} x^{3/2} \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right ),\frac{3}{2}\right )-2430 x^6-8586 x^5-9990 x^4-1746 x^3+6420 x^2+860 i \sqrt{2} \sqrt{\frac{1}{x}+1} \sqrt{\frac{2}{x}+3} x^{3/2} E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right )|\frac{3}{2}\right )+6052 x+1720}{243 \sqrt{x} \sqrt{3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 127, normalized size = 0.7 \begin{align*}{\frac{2}{729} \left ( -3645\,{x}^{6}+215\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{6}\sqrt{-x}{\it EllipticE} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ) -111\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{6}\sqrt{-x}{\it EllipticF} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ) -12879\,{x}^{5}-14985\,{x}^{4}-2619\,{x}^{3}+5760\,{x}^{2}+2628\,x \right ){\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{3\,{x}^{2}+5\,x+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{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}{\left (5 \, x - 2\right )}}{\sqrt{x}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{{\left (15 \, x^{3} + 19 \, x^{2} - 4\right )} \sqrt{3 \, x^{2} + 5 \, x + 2}}{\sqrt{x}}, 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{4 \sqrt{3 x^{2} + 5 x + 2}}{\sqrt{x}}\, dx - \int 19 x^{\frac{3}{2}} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int 15 x^{\frac{5}{2}} \sqrt{3 x^{2} + 5 x + 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{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}{\left (5 \, x - 2\right )}}{\sqrt{x}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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