Optimal. Leaf size=136 \[ -\frac{\sqrt{-3 x^2-5 x-2} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right ),-\frac{2}{3}\right )}{\sqrt{3} \sqrt{3 x^2+5 x+2}}-\frac{26 \sqrt{3 x^2+5 x+2}}{5 \sqrt{2 x+3}}+\frac{13 \sqrt{3} \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{5 \sqrt{3 x^2+5 x+2}} \]
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Rubi [A] time = 0.0811321, antiderivative size = 136, 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 = {834, 843, 718, 424, 419} \[ -\frac{26 \sqrt{3 x^2+5 x+2}}{5 \sqrt{2 x+3}}-\frac{\sqrt{-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{\sqrt{3} \sqrt{3 x^2+5 x+2}}+\frac{13 \sqrt{3} \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{5 \sqrt{3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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Rule 834
Rule 843
Rule 718
Rule 424
Rule 419
Rubi steps
\begin{align*} \int \frac{5-x}{(3+2 x)^{3/2} \sqrt{2+5 x+3 x^2}} \, dx &=-\frac{26 \sqrt{2+5 x+3 x^2}}{5 \sqrt{3+2 x}}-\frac{2}{5} \int \frac{-28-\frac{39 x}{2}}{\sqrt{3+2 x} \sqrt{2+5 x+3 x^2}} \, dx\\ &=-\frac{26 \sqrt{2+5 x+3 x^2}}{5 \sqrt{3+2 x}}-\frac{1}{2} \int \frac{1}{\sqrt{3+2 x} \sqrt{2+5 x+3 x^2}} \, dx+\frac{39}{10} \int \frac{\sqrt{3+2 x}}{\sqrt{2+5 x+3 x^2}} \, dx\\ &=-\frac{26 \sqrt{2+5 x+3 x^2}}{5 \sqrt{3+2 x}}-\frac{\sqrt{-2-5 x-3 x^2} \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 )}{\sqrt{3} \sqrt{2+5 x+3 x^2}}+\frac{\left (13 \sqrt{3} \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 )}{5 \sqrt{2+5 x+3 x^2}}\\ &=-\frac{26 \sqrt{2+5 x+3 x^2}}{5 \sqrt{3+2 x}}+\frac{13 \sqrt{3} \sqrt{-2-5 x-3 x^2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{1+x}\right )|-\frac{2}{3}\right )}{5 \sqrt{2+5 x+3 x^2}}-\frac{\sqrt{-2-5 x-3 x^2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{1+x}\right )|-\frac{2}{3}\right )}{\sqrt{3} \sqrt{2+5 x+3 x^2}}\\ \end{align*}
Mathematica [A] time = 0.17741, size = 104, normalized size = 0.76 \[ \frac{(x+1) \sqrt{\frac{3 x+2}{2 x+3}} \left (13 E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right )|\frac{3}{5}\right )-12 \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right ),\frac{3}{5}\right )\right )}{\sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.02, size = 131, normalized size = 1. \begin{align*}{\frac{1}{900\,{x}^{3}+2850\,{x}^{2}+2850\,x+900}\sqrt{3+2\,x}\sqrt{3\,{x}^{2}+5\,x+2} \left ( 34\,\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 ) -39\,\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 ) -2340\,{x}^{2}-3900\,x-1560 \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{x - 5}{\sqrt{3 \, x^{2} + 5 \, x + 2}{\left (2 \, x + 3\right )}^{\frac{3}{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} \sqrt{2 \, x + 3}{\left (x - 5\right )}}{12 \, x^{4} + 56 \, x^{3} + 95 \, x^{2} + 69 \, x + 18}, 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{x}{2 x \sqrt{2 x + 3} \sqrt{3 x^{2} + 5 x + 2} + 3 \sqrt{2 x + 3} \sqrt{3 x^{2} + 5 x + 2}}\, dx - \int - \frac{5}{2 x \sqrt{2 x + 3} \sqrt{3 x^{2} + 5 x + 2} + 3 \sqrt{2 x + 3} \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{x - 5}{\sqrt{3 \, x^{2} + 5 \, x + 2}{\left (2 \, x + 3\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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