Optimal. Leaf size=170 \[ \frac{9 \sqrt{3} \sqrt{-3 x^2-5 x-2} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right ),-\frac{2}{3}\right )}{50 \sqrt{3 x^2+5 x+2}}+\frac{\sqrt{3 x^2+5 x+2} (43 x+32)}{25 (2 x+3)^{5/2}}+\frac{49 \sqrt{3 x^2+5 x+2}}{125 \sqrt{2 x+3}}-\frac{49 \sqrt{3} \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{250 \sqrt{3 x^2+5 x+2}} \]
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Rubi [A] time = 0.107984, antiderivative size = 170, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207, Rules used = {810, 834, 843, 718, 424, 419} \[ \frac{\sqrt{3 x^2+5 x+2} (43 x+32)}{25 (2 x+3)^{5/2}}+\frac{49 \sqrt{3 x^2+5 x+2}}{125 \sqrt{2 x+3}}+\frac{9 \sqrt{3} \sqrt{-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{50 \sqrt{3 x^2+5 x+2}}-\frac{49 \sqrt{3} \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{250 \sqrt{3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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Rule 810
Rule 834
Rule 843
Rule 718
Rule 424
Rule 419
Rubi steps
\begin{align*} \int \frac{(5-x) \sqrt{2+5 x+3 x^2}}{(3+2 x)^{7/2}} \, dx &=\frac{(32+43 x) \sqrt{2+5 x+3 x^2}}{25 (3+2 x)^{5/2}}-\frac{1}{150} \int \frac{-48-81 x}{(3+2 x)^{3/2} \sqrt{2+5 x+3 x^2}} \, dx\\ &=\frac{49 \sqrt{2+5 x+3 x^2}}{125 \sqrt{3+2 x}}+\frac{(32+43 x) \sqrt{2+5 x+3 x^2}}{25 (3+2 x)^{5/2}}+\frac{1}{375} \int \frac{-\frac{459}{2}-\frac{441 x}{2}}{\sqrt{3+2 x} \sqrt{2+5 x+3 x^2}} \, dx\\ &=\frac{49 \sqrt{2+5 x+3 x^2}}{125 \sqrt{3+2 x}}+\frac{(32+43 x) \sqrt{2+5 x+3 x^2}}{25 (3+2 x)^{5/2}}+\frac{27}{100} \int \frac{1}{\sqrt{3+2 x} \sqrt{2+5 x+3 x^2}} \, dx-\frac{147}{500} \int \frac{\sqrt{3+2 x}}{\sqrt{2+5 x+3 x^2}} \, dx\\ &=\frac{49 \sqrt{2+5 x+3 x^2}}{125 \sqrt{3+2 x}}+\frac{(32+43 x) \sqrt{2+5 x+3 x^2}}{25 (3+2 x)^{5/2}}+\frac{\left (9 \sqrt{3} \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 )}{50 \sqrt{2+5 x+3 x^2}}-\frac{\left (49 \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 )}{250 \sqrt{2+5 x+3 x^2}}\\ &=\frac{49 \sqrt{2+5 x+3 x^2}}{125 \sqrt{3+2 x}}+\frac{(32+43 x) \sqrt{2+5 x+3 x^2}}{25 (3+2 x)^{5/2}}-\frac{49 \sqrt{3} \sqrt{-2-5 x-3 x^2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{1+x}\right )|-\frac{2}{3}\right )}{250 \sqrt{2+5 x+3 x^2}}+\frac{9 \sqrt{3} \sqrt{-2-5 x-3 x^2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{1+x}\right )|-\frac{2}{3}\right )}{50 \sqrt{2+5 x+3 x^2}}\\ \end{align*}
Mathematica [A] time = 0.30659, size = 182, normalized size = 1.07 \[ \frac{22 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{\frac{3 x+2}{2 x+3}} (2 x+3)^{7/2} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right ),\frac{3}{5}\right )+1290 x^3+3110 x^2+2460 x-49 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{\frac{3 x+2}{2 x+3}} (2 x+3)^{7/2} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right )|\frac{3}{5}\right )+640}{250 (2 x+3)^{5/2} \sqrt{3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.036, size = 296, normalized size = 1.7 \begin{align*} -{\frac{1}{2500} \left ( 16\,\sqrt{15}{\it EllipticF} \left ( 1/5\,\sqrt{30\,x+45},1/3\,\sqrt{15} \right ){x}^{2}\sqrt{3+2\,x}\sqrt{-2-2\,x}\sqrt{-20-30\,x}-196\,\sqrt{15}{\it EllipticE} \left ( 1/5\,\sqrt{30\,x+45},1/3\,\sqrt{15} \right ){x}^{2}\sqrt{3+2\,x}\sqrt{-2-2\,x}\sqrt{-20-30\,x}+48\,\sqrt{15}{\it EllipticF} \left ( 1/5\,\sqrt{30\,x+45},1/3\,\sqrt{15} \right ) x\sqrt{3+2\,x}\sqrt{-2-2\,x}\sqrt{-20-30\,x}-588\,\sqrt{15}{\it EllipticE} \left ( 1/5\,\sqrt{30\,x+45},1/3\,\sqrt{15} \right ) x\sqrt{3+2\,x}\sqrt{-2-2\,x}\sqrt{-20-30\,x}+36\,\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 ) -441\,\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 ) -11760\,{x}^{4}-67780\,{x}^{3}-124200\,{x}^{2}-92220\,x-24040 \right ) \left ( 3+2\,x \right ) ^{-{\frac{5}{2}}}{\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{\sqrt{3 \, x^{2} + 5 \, x + 2}{\left (x - 5\right )}}{{\left (2 \, x + 3\right )}^{\frac{7}{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 )}}{16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81}, 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}}{8 x^{3} \sqrt{2 x + 3} + 36 x^{2} \sqrt{2 x + 3} + 54 x \sqrt{2 x + 3} + 27 \sqrt{2 x + 3}}\, dx - \int \frac{x \sqrt{3 x^{2} + 5 x + 2}}{8 x^{3} \sqrt{2 x + 3} + 36 x^{2} \sqrt{2 x + 3} + 54 x \sqrt{2 x + 3} + 27 \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 )}}{{\left (2 \, x + 3\right )}^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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