Optimal. Leaf size=192 \[ -\frac{2525 \sqrt{-3 x^2-5 x-2} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right ),-\frac{2}{3}\right )}{189 \sqrt{3} \sqrt{3 x^2+5 x+2}}-\frac{2}{21} \sqrt{3 x^2+5 x+2} (2 x+3)^{5/2}+\frac{10}{7} \sqrt{3 x^2+5 x+2} (2 x+3)^{3/2}+\frac{1010}{189} \sqrt{3 x^2+5 x+2} \sqrt{2 x+3}+\frac{865 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{27 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]
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Rubi [A] time = 0.12962, antiderivative size = 192, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172, Rules used = {832, 843, 718, 424, 419} \[ -\frac{2}{21} \sqrt{3 x^2+5 x+2} (2 x+3)^{5/2}+\frac{10}{7} \sqrt{3 x^2+5 x+2} (2 x+3)^{3/2}+\frac{1010}{189} \sqrt{3 x^2+5 x+2} \sqrt{2 x+3}-\frac{2525 \sqrt{-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{189 \sqrt{3} \sqrt{3 x^2+5 x+2}}+\frac{865 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{27 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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Rule 832
Rule 843
Rule 718
Rule 424
Rule 419
Rubi steps
\begin{align*} \int \frac{(5-x) (3+2 x)^{5/2}}{\sqrt{2+5 x+3 x^2}} \, dx &=-\frac{2}{21} (3+2 x)^{5/2} \sqrt{2+5 x+3 x^2}+\frac{2}{21} \int \frac{(3+2 x)^{3/2} \left (175+\frac{225 x}{2}\right )}{\sqrt{2+5 x+3 x^2}} \, dx\\ &=\frac{10}{7} (3+2 x)^{3/2} \sqrt{2+5 x+3 x^2}-\frac{2}{21} (3+2 x)^{5/2} \sqrt{2+5 x+3 x^2}+\frac{4}{315} \int \frac{\sqrt{3+2 x} \left (\frac{9675}{4}+\frac{7575 x}{4}\right )}{\sqrt{2+5 x+3 x^2}} \, dx\\ &=\frac{1010}{189} \sqrt{3+2 x} \sqrt{2+5 x+3 x^2}+\frac{10}{7} (3+2 x)^{3/2} \sqrt{2+5 x+3 x^2}-\frac{2}{21} (3+2 x)^{5/2} \sqrt{2+5 x+3 x^2}+\frac{8 \int \frac{\frac{29325}{2}+\frac{90825 x}{8}}{\sqrt{3+2 x} \sqrt{2+5 x+3 x^2}} \, dx}{2835}\\ &=\frac{1010}{189} \sqrt{3+2 x} \sqrt{2+5 x+3 x^2}+\frac{10}{7} (3+2 x)^{3/2} \sqrt{2+5 x+3 x^2}-\frac{2}{21} (3+2 x)^{5/2} \sqrt{2+5 x+3 x^2}-\frac{2525}{378} \int \frac{1}{\sqrt{3+2 x} \sqrt{2+5 x+3 x^2}} \, dx+\frac{865}{54} \int \frac{\sqrt{3+2 x}}{\sqrt{2+5 x+3 x^2}} \, dx\\ &=\frac{1010}{189} \sqrt{3+2 x} \sqrt{2+5 x+3 x^2}+\frac{10}{7} (3+2 x)^{3/2} \sqrt{2+5 x+3 x^2}-\frac{2}{21} (3+2 x)^{5/2} \sqrt{2+5 x+3 x^2}-\frac{\left (2525 \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 )}{189 \sqrt{3} \sqrt{2+5 x+3 x^2}}+\frac{\left (865 \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 )}{27 \sqrt{3} \sqrt{2+5 x+3 x^2}}\\ &=\frac{1010}{189} \sqrt{3+2 x} \sqrt{2+5 x+3 x^2}+\frac{10}{7} (3+2 x)^{3/2} \sqrt{2+5 x+3 x^2}-\frac{2}{21} (3+2 x)^{5/2} \sqrt{2+5 x+3 x^2}+\frac{865 \sqrt{-2-5 x-3 x^2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{1+x}\right )|-\frac{2}{3}\right )}{27 \sqrt{3} \sqrt{2+5 x+3 x^2}}-\frac{2525 \sqrt{-2-5 x-3 x^2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{1+x}\right )|-\frac{2}{3}\right )}{189 \sqrt{3} \sqrt{2+5 x+3 x^2}}\\ \end{align*}
Mathematica [A] time = 0.329684, size = 198, normalized size = 1.03 \[ -\frac{4540 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{\frac{3 x+2}{2 x+3}} (2 x+3)^2 \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right ),\frac{3}{5}\right )+8 \left (162 x^5-216 x^4-5526 x^3-18501 x^2-20111 x-6758\right ) \sqrt{2 x+3}-6055 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{\frac{3 x+2}{2 x+3}} (2 x+3)^2 E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right )|\frac{3}{5}\right )}{567 (2 x+3) \sqrt{3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.029, size = 146, normalized size = 0.8 \begin{align*}{\frac{1}{6804\,{x}^{3}+21546\,{x}^{2}+21546\,x+6804}\sqrt{3+2\,x}\sqrt{3\,{x}^{2}+5\,x+2} \left ( -2592\,{x}^{5}+706\,\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 ) -1211\,\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 ) +3456\,{x}^{4}+88416\,{x}^{3}+223356\,{x}^{2}+200676\,x+59688 \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{{\left (2 \, x + 3\right )}^{\frac{5}{2}}{\left (x - 5\right )}}{\sqrt{3 \, x^{2} + 5 \, x + 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{{\left (4 \, x^{3} - 8 \, x^{2} - 51 \, x - 45\right )} \sqrt{2 \, x + 3}}{\sqrt{3 \, x^{2} + 5 \, x + 2}}, 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{45 \sqrt{2 x + 3}}{\sqrt{3 x^{2} + 5 x + 2}}\, dx - \int - \frac{51 x \sqrt{2 x + 3}}{\sqrt{3 x^{2} + 5 x + 2}}\, dx - \int - \frac{8 x^{2} \sqrt{2 x + 3}}{\sqrt{3 x^{2} + 5 x + 2}}\, dx - \int \frac{4 x^{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{{\left (2 \, x + 3\right )}^{\frac{5}{2}}{\left (x - 5\right )}}{\sqrt{3 \, x^{2} + 5 \, x + 2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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