Optimal. Leaf size=224 \[ -\frac{30734 \sqrt{-3 x^2-5 x-2} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right ),-\frac{2}{3}\right )}{625 \sqrt{3} \sqrt{3 x^2+5 x+2}}-\frac{6 (47 x+37)}{5 (2 x+3)^{5/2} \sqrt{3 x^2+5 x+2}}-\frac{426748 \sqrt{3 x^2+5 x+2}}{9375 \sqrt{2 x+3}}-\frac{61468 \sqrt{3 x^2+5 x+2}}{1875 (2 x+3)^{3/2}}-\frac{4124 \sqrt{3 x^2+5 x+2}}{125 (2 x+3)^{5/2}}+\frac{213374 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{3125 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]
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Rubi [A] time = 0.154737, antiderivative size = 224, normalized size of antiderivative = 1., number of steps used = 9, 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} \[ -\frac{6 (47 x+37)}{5 (2 x+3)^{5/2} \sqrt{3 x^2+5 x+2}}-\frac{426748 \sqrt{3 x^2+5 x+2}}{9375 \sqrt{2 x+3}}-\frac{61468 \sqrt{3 x^2+5 x+2}}{1875 (2 x+3)^{3/2}}-\frac{4124 \sqrt{3 x^2+5 x+2}}{125 (2 x+3)^{5/2}}-\frac{30734 \sqrt{-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{625 \sqrt{3} \sqrt{3 x^2+5 x+2}}+\frac{213374 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{3125 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]
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 )^{3/2}} \, dx &=-\frac{6 (37+47 x)}{5 (3+2 x)^{5/2} \sqrt{2+5 x+3 x^2}}-\frac{2}{5} \int \frac{542+705 x}{(3+2 x)^{7/2} \sqrt{2+5 x+3 x^2}} \, dx\\ &=-\frac{6 (37+47 x)}{5 (3+2 x)^{5/2} \sqrt{2+5 x+3 x^2}}-\frac{4124 \sqrt{2+5 x+3 x^2}}{125 (3+2 x)^{5/2}}+\frac{4}{125} \int \frac{-\frac{6235}{2}-\frac{9279 x}{2}}{(3+2 x)^{5/2} \sqrt{2+5 x+3 x^2}} \, dx\\ &=-\frac{6 (37+47 x)}{5 (3+2 x)^{5/2} \sqrt{2+5 x+3 x^2}}-\frac{4124 \sqrt{2+5 x+3 x^2}}{125 (3+2 x)^{5/2}}-\frac{61468 \sqrt{2+5 x+3 x^2}}{1875 (3+2 x)^{3/2}}-\frac{8 \int \frac{3952+\frac{46101 x}{4}}{(3+2 x)^{3/2} \sqrt{2+5 x+3 x^2}} \, dx}{1875}\\ &=-\frac{6 (37+47 x)}{5 (3+2 x)^{5/2} \sqrt{2+5 x+3 x^2}}-\frac{4124 \sqrt{2+5 x+3 x^2}}{125 (3+2 x)^{5/2}}-\frac{61468 \sqrt{2+5 x+3 x^2}}{1875 (3+2 x)^{3/2}}-\frac{426748 \sqrt{2+5 x+3 x^2}}{9375 \sqrt{3+2 x}}+\frac{16 \int \frac{\frac{364839}{8}+\frac{320061 x}{8}}{\sqrt{3+2 x} \sqrt{2+5 x+3 x^2}} \, dx}{9375}\\ &=-\frac{6 (37+47 x)}{5 (3+2 x)^{5/2} \sqrt{2+5 x+3 x^2}}-\frac{4124 \sqrt{2+5 x+3 x^2}}{125 (3+2 x)^{5/2}}-\frac{61468 \sqrt{2+5 x+3 x^2}}{1875 (3+2 x)^{3/2}}-\frac{426748 \sqrt{2+5 x+3 x^2}}{9375 \sqrt{3+2 x}}-\frac{15367}{625} \int \frac{1}{\sqrt{3+2 x} \sqrt{2+5 x+3 x^2}} \, dx+\frac{106687 \int \frac{\sqrt{3+2 x}}{\sqrt{2+5 x+3 x^2}} \, dx}{3125}\\ &=-\frac{6 (37+47 x)}{5 (3+2 x)^{5/2} \sqrt{2+5 x+3 x^2}}-\frac{4124 \sqrt{2+5 x+3 x^2}}{125 (3+2 x)^{5/2}}-\frac{61468 \sqrt{2+5 x+3 x^2}}{1875 (3+2 x)^{3/2}}-\frac{426748 \sqrt{2+5 x+3 x^2}}{9375 \sqrt{3+2 x}}-\frac{\left (30734 \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 )}{625 \sqrt{3} \sqrt{2+5 x+3 x^2}}+\frac{\left (213374 \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 )}{3125 \sqrt{3} \sqrt{2+5 x+3 x^2}}\\ &=-\frac{6 (37+47 x)}{5 (3+2 x)^{5/2} \sqrt{2+5 x+3 x^2}}-\frac{4124 \sqrt{2+5 x+3 x^2}}{125 (3+2 x)^{5/2}}-\frac{61468 \sqrt{2+5 x+3 x^2}}{1875 (3+2 x)^{3/2}}-\frac{426748 \sqrt{2+5 x+3 x^2}}{9375 \sqrt{3+2 x}}+\frac{213374 \sqrt{-2-5 x-3 x^2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{1+x}\right )|-\frac{2}{3}\right )}{3125 \sqrt{3} \sqrt{2+5 x+3 x^2}}-\frac{30734 \sqrt{-2-5 x-3 x^2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{1+x}\right )|-\frac{2}{3}\right )}{625 \sqrt{3} \sqrt{2+5 x+3 x^2}}\\ \end{align*}
Mathematica [A] time = 0.389061, size = 182, normalized size = 0.81 \[ -\frac{2 \left (60586 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{\frac{3 x+2}{2 x+3}} (2 x+3)^{7/2} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right ),\frac{3}{5}\right )+922020 x^3+3383680 x^2+3957355 x-106687 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{\frac{3 x+2}{2 x+3}} (2 x+3)^{7/2} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right )|\frac{3}{5}\right )+1439445\right )}{9375 (2 x+3)^{5/2} \sqrt{3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.028, size = 296, normalized size = 1.3 \begin{align*} -{\frac{1}{46875} \left ( 426748\,\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}-119408\,\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}+1280244\,\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}-358224\,\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}+960183\,\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 ) -268668\,\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 ) +25604880\,{x}^{4}+128709640\,{x}^{3}+236542100\,{x}^{2}+186801610\,x+52801770 \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{x - 5}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{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 )}}{144 \, x^{8} + 1344 \, x^{7} + 5416 \, x^{6} + 12296 \, x^{5} + 17185 \, x^{4} + 15126 \, x^{3} + 8181 \, x^{2} + 2484 \, x + 324}, 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{3}{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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