Optimal. Leaf size=256 \[ \frac{129268 \sqrt{3} \sqrt{-3 x^2-5 x-2} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right ),-\frac{2}{3}\right )}{3125 \sqrt{3 x^2+5 x+2}}-\frac{2 (47 x+37)}{5 (2 x+3)^{5/2} \left (3 x^2+5 x+2\right )^{3/2}}+\frac{215096 \sqrt{3 x^2+5 x+2}}{15625 \sqrt{2 x+3}}+\frac{258536 \sqrt{3 x^2+5 x+2}}{3125 (2 x+3)^{3/2}}+\frac{87144 \sqrt{3 x^2+5 x+2}}{625 (2 x+3)^{5/2}}+\frac{4 (2013 x+1858)}{25 (2 x+3)^{5/2} \sqrt{3 x^2+5 x+2}}-\frac{107548 \sqrt{3} \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{15625 \sqrt{3 x^2+5 x+2}} \]
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Rubi [A] time = 0.188904, antiderivative size = 256, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207, Rules used = {822, 834, 843, 718, 424, 419} \[ -\frac{2 (47 x+37)}{5 (2 x+3)^{5/2} \left (3 x^2+5 x+2\right )^{3/2}}+\frac{215096 \sqrt{3 x^2+5 x+2}}{15625 \sqrt{2 x+3}}+\frac{258536 \sqrt{3 x^2+5 x+2}}{3125 (2 x+3)^{3/2}}+\frac{87144 \sqrt{3 x^2+5 x+2}}{625 (2 x+3)^{5/2}}+\frac{4 (2013 x+1858)}{25 (2 x+3)^{5/2} \sqrt{3 x^2+5 x+2}}+\frac{129268 \sqrt{3} \sqrt{-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{3125 \sqrt{3 x^2+5 x+2}}-\frac{107548 \sqrt{3} \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{15625 \sqrt{3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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Rule 822
Rule 834
Rule 843
Rule 718
Rule 424
Rule 419
Rubi steps
\begin{align*} \int \frac{5-x}{(3+2 x)^{7/2} \left (2+5 x+3 x^2\right )^{5/2}} \, dx &=-\frac{2 (37+47 x)}{5 (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^{3/2}}-\frac{2}{15} \int \frac{1362+1269 x}{(3+2 x)^{7/2} \left (2+5 x+3 x^2\right )^{3/2}} \, dx\\ &=-\frac{2 (37+47 x)}{5 (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^{3/2}}+\frac{4 (1858+2013 x)}{25 (3+2 x)^{5/2} \sqrt{2+5 x+3 x^2}}+\frac{4}{75} \int \frac{28953+30195 x}{(3+2 x)^{7/2} \sqrt{2+5 x+3 x^2}} \, dx\\ &=-\frac{2 (37+47 x)}{5 (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^{3/2}}+\frac{4 (1858+2013 x)}{25 (3+2 x)^{5/2} \sqrt{2+5 x+3 x^2}}+\frac{87144 \sqrt{2+5 x+3 x^2}}{625 (3+2 x)^{5/2}}-\frac{8 \int \frac{-147870-\frac{294111 x}{2}}{(3+2 x)^{5/2} \sqrt{2+5 x+3 x^2}} \, dx}{1875}\\ &=-\frac{2 (37+47 x)}{5 (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^{3/2}}+\frac{4 (1858+2013 x)}{25 (3+2 x)^{5/2} \sqrt{2+5 x+3 x^2}}+\frac{87144 \sqrt{2+5 x+3 x^2}}{625 (3+2 x)^{5/2}}+\frac{258536 \sqrt{2+5 x+3 x^2}}{3125 (3+2 x)^{3/2}}+\frac{16 \int \frac{\frac{1187847}{4}+\frac{872559 x}{4}}{(3+2 x)^{3/2} \sqrt{2+5 x+3 x^2}} \, dx}{28125}\\ &=-\frac{2 (37+47 x)}{5 (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^{3/2}}+\frac{4 (1858+2013 x)}{25 (3+2 x)^{5/2} \sqrt{2+5 x+3 x^2}}+\frac{87144 \sqrt{2+5 x+3 x^2}}{625 (3+2 x)^{5/2}}+\frac{258536 \sqrt{2+5 x+3 x^2}}{3125 (3+2 x)^{3/2}}+\frac{215096 \sqrt{2+5 x+3 x^2}}{15625 \sqrt{3+2 x}}-\frac{32 \int \frac{-\frac{546237}{4}+\frac{725949 x}{8}}{\sqrt{3+2 x} \sqrt{2+5 x+3 x^2}} \, dx}{140625}\\ &=-\frac{2 (37+47 x)}{5 (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^{3/2}}+\frac{4 (1858+2013 x)}{25 (3+2 x)^{5/2} \sqrt{2+5 x+3 x^2}}+\frac{87144 \sqrt{2+5 x+3 x^2}}{625 (3+2 x)^{5/2}}+\frac{258536 \sqrt{2+5 x+3 x^2}}{3125 (3+2 x)^{3/2}}+\frac{215096 \sqrt{2+5 x+3 x^2}}{15625 \sqrt{3+2 x}}-\frac{161322 \int \frac{\sqrt{3+2 x}}{\sqrt{2+5 x+3 x^2}} \, dx}{15625}+\frac{193902 \int \frac{1}{\sqrt{3+2 x} \sqrt{2+5 x+3 x^2}} \, dx}{3125}\\ &=-\frac{2 (37+47 x)}{5 (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^{3/2}}+\frac{4 (1858+2013 x)}{25 (3+2 x)^{5/2} \sqrt{2+5 x+3 x^2}}+\frac{87144 \sqrt{2+5 x+3 x^2}}{625 (3+2 x)^{5/2}}+\frac{258536 \sqrt{2+5 x+3 x^2}}{3125 (3+2 x)^{3/2}}+\frac{215096 \sqrt{2+5 x+3 x^2}}{15625 \sqrt{3+2 x}}-\frac{\left (107548 \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 )}{15625 \sqrt{2+5 x+3 x^2}}+\frac{\left (129268 \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 )}{3125 \sqrt{2+5 x+3 x^2}}\\ &=-\frac{2 (37+47 x)}{5 (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^{3/2}}+\frac{4 (1858+2013 x)}{25 (3+2 x)^{5/2} \sqrt{2+5 x+3 x^2}}+\frac{87144 \sqrt{2+5 x+3 x^2}}{625 (3+2 x)^{5/2}}+\frac{258536 \sqrt{2+5 x+3 x^2}}{3125 (3+2 x)^{3/2}}+\frac{215096 \sqrt{2+5 x+3 x^2}}{15625 \sqrt{3+2 x}}-\frac{107548 \sqrt{3} \sqrt{-2-5 x-3 x^2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{1+x}\right )|-\frac{2}{3}\right )}{15625 \sqrt{2+5 x+3 x^2}}+\frac{129268 \sqrt{3} \sqrt{-2-5 x-3 x^2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{1+x}\right )|-\frac{2}{3}\right )}{3125 \sqrt{2+5 x+3 x^2}}\\ \end{align*}
Mathematica [A] time = 0.52517, size = 229, normalized size = 0.89 \[ \frac{2 \left (-2 (2 x+3)^2 \left (3 x^2+5 x+2\right ) \left (70064 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{\frac{3 x+2}{2 x+3}} (2 x+3)^{3/2} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right ),\frac{3}{5}\right )+53774 \left (3 x^2+5 x+2\right )+26887 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{\frac{3 x+2}{2 x+3}} (2 x+3)^{3/2} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right )|\frac{3}{5}\right )\right )+3871728 x^6+36155064 x^5+129381052 x^4+231620622 x^3+220795962 x^2+106756189 x+20514383\right )}{15625 (2 x+3)^{5/2} \left (3 x^2+5 x+2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.03, size = 494, normalized size = 1.9 \begin{align*}{\frac{2}{78125\, \left ( 2+3\,x \right ) ^{2} \left ( 1+x \right ) ^{2}}\sqrt{3\,{x}^{2}+5\,x+2} \left ( 1616376\,\sqrt{15}{\it EllipticF} \left ( 1/5\,\sqrt{30\,x+45},1/3\,\sqrt{15} \right ){x}^{4}\sqrt{3+2\,x}\sqrt{-2-2\,x}\sqrt{-20-30\,x}+322644\,\sqrt{15}{\it EllipticE} \left ( 1/5\,\sqrt{30\,x+45},1/3\,\sqrt{15} \right ){x}^{4}\sqrt{3+2\,x}\sqrt{-2-2\,x}\sqrt{-20-30\,x}+7543088\,\sqrt{15}{\it EllipticF} \left ( 1/5\,\sqrt{30\,x+45},1/3\,\sqrt{15} \right ){x}^{3}\sqrt{3+2\,x}\sqrt{-2-2\,x}\sqrt{-20-30\,x}+1505672\,\sqrt{15}{\it EllipticE} \left ( 1/5\,\sqrt{30\,x+45},1/3\,\sqrt{15} \right ){x}^{3}\sqrt{3+2\,x}\sqrt{-2-2\,x}\sqrt{-20-30\,x}+12796310\,\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}+2554265\,\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}+9294162\,\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}+1855203\,\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}+19358640\,{x}^{6}+2424564\,\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 ) +483966\,\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 ) +180775320\,{x}^{5}+646905260\,{x}^{4}+1158103110\,{x}^{3}+1103979810\,{x}^{2}+533780945\,x+102571915 \right ) \left ( 3+2\,x \right ) ^{-{\frac{5}{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{x - 5}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}}{\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 )}}{432 \, x^{10} + 4752 \, x^{9} + 23256 \, x^{8} + 66656 \, x^{7} + 123867 \, x^{6} + 155895 \, x^{5} + 134543 \, x^{4} + 78609 \, x^{3} + 29754 \, x^{2} + 6588 \, x + 648}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}}{\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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