Optimal. Leaf size=179 \[ -\frac{245 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} \text{EllipticF}\left (\tan ^{-1}\left (\sqrt{x}\right ),-\frac{1}{2}\right )}{\sqrt{3 x^2+5 x+2}}-\frac{198 \sqrt{x} (3 x+2)}{\sqrt{3 x^2+5 x+2}}+\frac{2 \sqrt{x} (297 x+250)}{\sqrt{3 x^2+5 x+2}}-\frac{2 \sqrt{x} (37 x+30)}{3 \left (3 x^2+5 x+2\right )^{3/2}}+\frac{198 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{\sqrt{3 x^2+5 x+2}} \]
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Rubi [A] time = 0.112693, antiderivative size = 179, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24, Rules used = {820, 822, 839, 1189, 1100, 1136} \[ -\frac{198 \sqrt{x} (3 x+2)}{\sqrt{3 x^2+5 x+2}}+\frac{2 \sqrt{x} (297 x+250)}{\sqrt{3 x^2+5 x+2}}-\frac{2 \sqrt{x} (37 x+30)}{3 \left (3 x^2+5 x+2\right )^{3/2}}-\frac{245 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{\sqrt{3 x^2+5 x+2}}+\frac{198 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{\sqrt{3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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Rule 820
Rule 822
Rule 839
Rule 1189
Rule 1100
Rule 1136
Rubi steps
\begin{align*} \int \frac{(2-5 x) \sqrt{x}}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx &=-\frac{2 \sqrt{x} (30+37 x)}{3 \left (2+5 x+3 x^2\right )^{3/2}}-\frac{2}{3} \int \frac{-15+\frac{111 x}{2}}{\sqrt{x} \left (2+5 x+3 x^2\right )^{3/2}} \, dx\\ &=-\frac{2 \sqrt{x} (30+37 x)}{3 \left (2+5 x+3 x^2\right )^{3/2}}+\frac{2 \sqrt{x} (250+297 x)}{\sqrt{2+5 x+3 x^2}}+\frac{2}{3} \int \frac{-\frac{735}{2}-\frac{891 x}{2}}{\sqrt{x} \sqrt{2+5 x+3 x^2}} \, dx\\ &=-\frac{2 \sqrt{x} (30+37 x)}{3 \left (2+5 x+3 x^2\right )^{3/2}}+\frac{2 \sqrt{x} (250+297 x)}{\sqrt{2+5 x+3 x^2}}+\frac{4}{3} \operatorname{Subst}\left (\int \frac{-\frac{735}{2}-\frac{891 x^2}{2}}{\sqrt{2+5 x^2+3 x^4}} \, dx,x,\sqrt{x}\right )\\ &=-\frac{2 \sqrt{x} (30+37 x)}{3 \left (2+5 x+3 x^2\right )^{3/2}}+\frac{2 \sqrt{x} (250+297 x)}{\sqrt{2+5 x+3 x^2}}-490 \operatorname{Subst}\left (\int \frac{1}{\sqrt{2+5 x^2+3 x^4}} \, dx,x,\sqrt{x}\right )-594 \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{2+5 x^2+3 x^4}} \, dx,x,\sqrt{x}\right )\\ &=-\frac{2 \sqrt{x} (30+37 x)}{3 \left (2+5 x+3 x^2\right )^{3/2}}-\frac{198 \sqrt{x} (2+3 x)}{\sqrt{2+5 x+3 x^2}}+\frac{2 \sqrt{x} (250+297 x)}{\sqrt{2+5 x+3 x^2}}+\frac{198 \sqrt{2} (1+x) \sqrt{\frac{2+3 x}{1+x}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{\sqrt{2+5 x+3 x^2}}-\frac{245 \sqrt{2} (1+x) \sqrt{\frac{2+3 x}{1+x}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{\sqrt{2+5 x+3 x^2}}\\ \end{align*}
Mathematica [C] time = 0.306502, size = 165, normalized size = 0.92 \[ -\frac{47 i \sqrt{\frac{2}{x}+2} \sqrt{\frac{2}{x}+3} x \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right ),\frac{3}{2}\right )}{\sqrt{3 x^2+5 x+2}}-\frac{2 \left (2205 x^3+5494 x^2+4470 x+1188\right )}{3 \sqrt{x} \left (3 x^2+5 x+2\right )^{3/2}}-\frac{198 i \sqrt{\frac{2}{x}+2} \sqrt{\frac{2}{x}+3} x E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right )|\frac{3}{2}\right )}{\sqrt{3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.022, size = 297, normalized size = 1.7 \begin{align*}{\frac{1}{3\, \left ( 1+x \right ) ^{2} \left ( 2+3\,x \right ) ^{2}} \left ( 156\,\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 ){x}^{2}-297\,\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 ){x}^{2}+260\,\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 ) x-495\,\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 ) x+104\,\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 ) -198\,\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 ) +5346\,{x}^{4}+13410\,{x}^{3}+10990\,{x}^{2}+2940\,x \right ) \sqrt{3\,{x}^{2}+5\,x+2}{\frac{1}{\sqrt{x}}}} \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 (5 \, x - 2\right )} \sqrt{x}}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}}}\,{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{\sqrt{3 \, x^{2} + 5 \, x + 2}{\left (5 \, x - 2\right )} \sqrt{x}}{27 \, x^{6} + 135 \, x^{5} + 279 \, x^{4} + 305 \, x^{3} + 186 \, x^{2} + 60 \, x + 8}, 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{2 \sqrt{x}}{9 x^{4} \sqrt{3 x^{2} + 5 x + 2} + 30 x^{3} \sqrt{3 x^{2} + 5 x + 2} + 37 x^{2} \sqrt{3 x^{2} + 5 x + 2} + 20 x \sqrt{3 x^{2} + 5 x + 2} + 4 \sqrt{3 x^{2} + 5 x + 2}}\, dx - \int \frac{5 x^{\frac{3}{2}}}{9 x^{4} \sqrt{3 x^{2} + 5 x + 2} + 30 x^{3} \sqrt{3 x^{2} + 5 x + 2} + 37 x^{2} \sqrt{3 x^{2} + 5 x + 2} + 20 x \sqrt{3 x^{2} + 5 x + 2} + 4 \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 (5 \, x - 2\right )} \sqrt{x}}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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