Optimal. Leaf size=143 \[ \frac{191 \sqrt{-3 x^2-5 x-2} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right ),-\frac{2}{3}\right )}{18 \sqrt{3} \sqrt{3 x^2+5 x+2}}+\frac{1}{45} \sqrt{2 x+3} \sqrt{3 x^2+5 x+2} (88-9 x)-\frac{761 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{90 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]
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Rubi [A] time = 0.0808324, antiderivative size = 143, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172, Rules used = {814, 843, 718, 424, 419} \[ \frac{1}{45} \sqrt{2 x+3} \sqrt{3 x^2+5 x+2} (88-9 x)+\frac{191 \sqrt{-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{18 \sqrt{3} \sqrt{3 x^2+5 x+2}}-\frac{761 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{90 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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Rule 814
Rule 843
Rule 718
Rule 424
Rule 419
Rubi steps
\begin{align*} \int \frac{(5-x) \sqrt{2+5 x+3 x^2}}{\sqrt{3+2 x}} \, dx &=\frac{1}{45} (88-9 x) \sqrt{3+2 x} \sqrt{2+5 x+3 x^2}-\frac{1}{90} \int \frac{664+761 x}{\sqrt{3+2 x} \sqrt{2+5 x+3 x^2}} \, dx\\ &=\frac{1}{45} (88-9 x) \sqrt{3+2 x} \sqrt{2+5 x+3 x^2}-\frac{761}{180} \int \frac{\sqrt{3+2 x}}{\sqrt{2+5 x+3 x^2}} \, dx+\frac{191}{36} \int \frac{1}{\sqrt{3+2 x} \sqrt{2+5 x+3 x^2}} \, dx\\ &=\frac{1}{45} (88-9 x) \sqrt{3+2 x} \sqrt{2+5 x+3 x^2}-\frac{\left (761 \sqrt{-2-5 x-3 x^2}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{2 x^2}{3}}}{\sqrt{1-x^2}} \, dx,x,\frac{\sqrt{6+6 x}}{\sqrt{2}}\right )}{90 \sqrt{3} \sqrt{2+5 x+3 x^2}}+\frac{\left (191 \sqrt{-2-5 x-3 x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2} \sqrt{1+\frac{2 x^2}{3}}} \, dx,x,\frac{\sqrt{6+6 x}}{\sqrt{2}}\right )}{18 \sqrt{3} \sqrt{2+5 x+3 x^2}}\\ &=\frac{1}{45} (88-9 x) \sqrt{3+2 x} \sqrt{2+5 x+3 x^2}-\frac{761 \sqrt{-2-5 x-3 x^2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{1+x}\right )|-\frac{2}{3}\right )}{90 \sqrt{3} \sqrt{2+5 x+3 x^2}}+\frac{191 \sqrt{-2-5 x-3 x^2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{1+x}\right )|-\frac{2}{3}\right )}{18 \sqrt{3} \sqrt{2+5 x+3 x^2}}\\ \end{align*}
Mathematica [A] time = 0.329322, size = 193, normalized size = 1.35 \[ -\frac{-188 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{\frac{3 x+2}{2 x+3}} (2 x+3)^2 \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right ),\frac{3}{5}\right )+2 \left (162 x^4-1071 x^3-2220 x^2-1049 x-62\right ) \sqrt{2 x+3}+761 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{\frac{3 x+2}{2 x+3}} (2 x+3)^2 E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right )|\frac{3}{5}\right )}{270 (2 x+3) \sqrt{3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.02, size = 141, normalized size = 1. \begin{align*}{\frac{1}{16200\,{x}^{3}+51300\,{x}^{2}+51300\,x+16200}\sqrt{3+2\,x}\sqrt{3\,{x}^{2}+5\,x+2} \left ( 194\,\sqrt{3+2\,x}\sqrt{15}\sqrt{-2-2\,x}\sqrt{-20-30\,x}{\it EllipticF} \left ( 1/5\,\sqrt{30\,x+45},1/3\,\sqrt{15} \right ) +761\,\sqrt{3+2\,x}\sqrt{15}\sqrt{-2-2\,x}\sqrt{-20-30\,x}{\it EllipticE} \left ( 1/5\,\sqrt{30\,x+45},1/3\,\sqrt{15} \right ) -3240\,{x}^{4}+21420\,{x}^{3}+90060\,{x}^{2}+97080\,x+31680 \right ) } \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{\sqrt{3 \, x^{2} + 5 \, x + 2}{\left (x - 5\right )}}{\sqrt{2 \, x + 3}}\,{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 (x - 5\right )}}{\sqrt{2 \, x + 3}}, 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{5 \sqrt{3 x^{2} + 5 x + 2}}{\sqrt{2 x + 3}}\, dx - \int \frac{x \sqrt{3 x^{2} + 5 x + 2}}{\sqrt{2 x + 3}}\, 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{\sqrt{3 \, x^{2} + 5 \, x + 2}{\left (x - 5\right )}}{\sqrt{2 \, x + 3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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